.:;^:^. Conttibuttons to Ebucation Eescfjeri College ^rief 'Wi 1^ fyxmW Winivmii^ J Ixhux^ BOUGHT WITH THE INCOME | FROM THE SAGE ENDOWMENT FUND THE GIFT OF Hetirg W. Si^ge 1891 f\.%^^W] ]2 ^11 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924030592970 THE REASONING ABILITY OF CHILDREN OF THE FOURTH, FIFTH, AND SIXTH SCHOOL GRADES BY FREDERICK G. BONSER, Ph. D. TEACHERS COLLEGE, COLUMBIA UNIVERSITY CONTRIBUTIONS TO EDUCATION, NO. 37 PUBLISHED BY ^sutifsta (EalltQe. (Htslnmbia Intorsttg NEW YORK CITY 1910 4'- Copyright, 1910, by Frederick G. Bonser ACKNOWLEDGMENTS For permission to use the time of the children in making the tests, I am indebted to Superintendent I. O. Woodley of the City Schools of Passaic, New Jersey. To him, and to the prin- cipals and teachers of the schools used in the tests, I am very grateful for the every possible courtesy and cooperative aid they could lend in making conditions satisfactory for the work. My obligation is cheerfully tendered to Dr. E. L. Thorndike, under whose supervision the study has been made, for suggestive help and criticism in the development of the tests and of the study throughout. For much helpful assistance in many phases of the study, it would be ungrateful not to mention the cooperation of my wife, Edna MacDonald Bonser. F. G. B. TABLE OF CONTENTS Page Acknowledgments iii INTRODUCTORY The Problem i Description of Tests i Mathematical Judgment — ^Tests I and II i Controlled Association — Tests, III and IV 3 Selective Judgment — ^Tests V and VI 5 Literary Interpretation — Test VII 8 Spelling 8 Children Tested 8 Procedure in Making Tests 5 What These Tests Really Measure 11 Meaning of the Term Reasoning 11 Factors in Reasoning Ability Here Measured 14 Mathematical Judgment 14 Controlled Association 14 Selective Judgment 15 Literary Interpretation 15 Scoring 16 Different Plans of Scoring Tested — Table 1 17 Distribution of Children by Grade, Age, and Sex — Tables II, III. ... 19 Statistical Technique Employed in Computations 21 Array of Original Scorings — Table IV 22 RESULTS AND THEIR ANALYSIS Mathematical Judgment — ^Tests I and II 24 Tabulation of Results by Grades — Tables V and VI 24 Tabulation of Results by Ages — Tables VII to X 26 Differences between Tests I and II — Table XI 30 Summary for Mathematical Judgment Tests 31 Grrade Differences 31 Age Differences 32 Sex Differences 33 Controlled Association — Tests III and IV 34 Test III 3S Tabulation of Results by Grades — Tables XII and XIII. . . 35 Tabulation of Results by Ages — Tables XIV to XVII 36 Differences between Tests III A and III B— Table XVIII . . 40 Test IV 42 Tabulation of Results by Grades — Tables XIX and XX ... 42 Tabulation of Results by Ages — Tables XXI to XXIV 44 VI Contents Page Summary for Controlled Association Tests 48 Grade DiflEerences 48 Age Differences 49 Sex Differences 5° Selective Judgment — Tests V and VI Si Test V 51 Tabulation of Results by Grades— Tables XXV and XXVI. 51 Tabulation of Results by Ages— Tables XXVII to XXX.. . 53 Test VI 57 Tabulation of Results by Grades — Tables XXXI and XXXII 57 Tabulation of Results by Ages— Tables XXXIII to XXXVI. 59 Summary for Selective Judgment Tests 63 Grade Differences 63 Age Differences 64 Sex Differences 64 Intellectual Interpretation of Poems — Test VII 65 Tabulation of Results by Grades — Tables XXXVII and XXXVIII 6s Tabulation of Results by Ages— Tables XXXIX to XLII 67 Summary for Intellectual Interpretation Test 70 Grade Differences 70 Age Differences 71 Sex Differences 71 Array of Combined Results for the Seven Tests 72 Tabulation of Results by Grades — Tables XLIII and XLIV. . . 74 Tabulation of Results by Ages — Tables XLV to XL VIII 75 Summary for the Combined Results 78 Grade Differences 78 Age Differences 80 Sex Differences 80 Relative Progress in the Grades— Table XLIX 82 Relative Progress with Age — ^Table L 84 Prominent Sex Differences among the Tests 84 Variability — Table LI 85 Resemblances and Differences in the Tests — Tables LII and LIU ... 88 Separation of Native Capacity from School Training 89 General Conclusions 90 CORRELATION OF INDIVIDUAL ABILITIES Example of Corrected Coefficients — Table LIV 92 Correlations by Grade, Age, and Sex — Tables LV to LVIII 94 Age Differences within the Grades — Table LIX 97 Comparison of Median and Extreme Abilities — Table LX 98 Summary for Individual Correlations 100 Grade Differences loi Age Differences 102 Differences between Grade and Age Distributions 102 Sex Differences 103 Contents vii Page Separation of Native Capacity from School Training 103 Summarized Conclusions for Individual Correlations loi; APPENDIX Results for the Spelling Test 106 Tabulation of Results by Grades' — ^Tables LXI and LXII 106 Tabulation of Results by Ages— Tables LXIII to LXVI 107 Summary for the Spelling Test iii Grade Differences in Age Differences iii Sex Differences 112 Comparison of Spelling Results with Those in Reasoning 112 Bibliography 113 Array of Original Scorings — Table IV 115 FIGURES Tests I and II — Figures i-io 29 Test III — Figures 1 1-20 39 Test IV — Figures 2 1-30 47 Test V — Figures 31-40 56 Test VI — Figures 41-50 61 Test VII — Figures 51-60 69 Totals — Figures 61-70 79 Ability and Variability by Grades — Figures 71-74 81 Ability and Variability by Ages — Figures 75-78 83 Sex Differences for All Tests — Figures 79-80 86 Spelling Test — Figures 81-90 no THE REASONING ABILITY OF CHILDREN OF THE FOURTH, FIFTH, AND SIXTH SCHOOL GRADES INTRODUCTORY The Problem This study is an attempt to secure a concrete basis for some insight into the capacity of children of the fourth, fifth, and sixth school grades for those forms of purposive thinking which we call reasoning. Its scope is comprehended by a consideration of the following questions : 1. The development of tests which may be relied upon as measures of these forms of intellectual activity. 2. Differences, and development of these forms of activity in these three grades of school life. 3. Whether sex differences exist during these years in these types of mental activity. 4. The relation of age to school grade as an index of these phases of capacity. 5. The kind and extent of correlation of these forms of ability one with another. Description of Tests The tests employed were made up of a series of problems and questions designed to exercise the most fundamental four pihases of reasoning activity, namely : The mathematical judgment ; con- trolled association; selective judgment; and that complex of analytic and synthetic thinking used in the intellectual interpre- tation of literature. Mathematical Judgment The problems for testing the mathematical judgment were of two kinds, two sets of five each, I, A and B, stated in the -form 2 The Reasoning Ability of Children usually followed in current text-books in arithmetic; and two sets of five each, II, A and B, of the same difficulty as the pre- ceding in processes involved but stated in a less conventional way. Each of the ten problems of the first type may be called a " two-step " problem— it requires a preliminary operation for securing the intermediate datum necessary before the final oper- ation can be accomplished. Tests I and II I. A. Get the answers to these problems as quickly as you can. I . If l of a gallon of oil costs 9 cents, what will 7 gallons cost ? i!. John sold 4 sheep for $5 each. He kept J of the money and with the other J he bought lambs at $2 each. How many did he buy? 3. A pint of water weighs a pound. What does a gallon weigh? 4. At 12J cents each, how much more will 6 tablets cost than 10 pens at 5 cents each? 5. At IS cents a yard, how much will 7 feet of cloth cost? B. 1. A man whose salary is $20 a week spends $14 a week. In how many weeks can he save $300? 2. How many pencils can you buy for 50 cents at the rate of 2 for 5 cents. 3. A man bought land for $100. He sold it for $120, gaining $5 an acre. How many acres were there? 4. A man spent f of his money and had $8 left. How much had he at first? 5. The uniforms for a baseball nine cost $2.50 each. The shoes cost $2 a pair. What was the total cost of uniforms and shoes for the nine? II. A. 1. 132 plus what number equals 36? 2. If John had 15 cents more than he spent today he would have 40 cents. How much did he spend today? 3. What number minus 7 equals 23 ? 4. If James had 4 times as much money as George, he would have $r6. How much money has George? 5. What number added to 16 gives a number 4 less than 27? B. .. What number subtracted 12 times from 30 will leave a remain- der of 6? 2. If a train travels haU a mile in a minute, what is its rate per hour? 3. What number minus 16 equals 20? 4. What number doubled equals 2 times 3 ? 5. If 7 multiplied by some number equals 63, what is the number? of the Fourth, Fifth, and Sixth School Grades 3 In the original blanks, immediately following each problem space was left for its solution. Controlled Association For controlled association, three types of tests were used. First, two sets of ten sentences each. III, A, a and b, were given with a significant word omitted from each to be filled in by the pupil. Second, two sets of ten sentences each. III, B, a and b, were given in each of which two significant words were placed, one above the other, one giving a correct meaning to the sen- tence, the other an erroneous meaning, the pupil to draw a line through the wrong word leaving the sentence so that it would read correctly. Third, three sets of twenty words each, IV, A, B. and C, were given to pupils, they to write beside each respec- tive word a word just its opposite in meaning — the familiar " opposites " test. Tests III and IV III. A. a. Complete the following sentences as quickly as you can by filling the blank spaces with appropriate words : 1 . always comes in the last week in December, 2. A is one who plays a musical instrument. 3. The city is in Russia. 4. are large, visible bodies of watery vapor floating about in the air. 5. used for building houses are made of clay. 6. The machine used on a railroad for drawing cars is an 7. is the most useful metal for blacksmiths. 8. live and swim about in the water. 9. Most light, summer clothing is made of goods. 10. is a holiday. III. A. b. I. The flesh of cattle used for food is called . 2. The months are June, July and August. 3. The makes it light during the day. 4. catch many mice and birds. 5. A is a large stream of water flowing through the land. 6. Men who live in the country and till the soil are called . 7. is a mineral which we burn. 8. The Ocean is east of the United States. 9. sell sugar, vegetables and other foods. to. There are hours in half a day. 4 The Reasoning Ability of Children in. B. a. As quickly as you can, make these sentences cor- rect by drawing a line through the wrong word where two words occur, one above the other : -r-, longer . j_, ■ ■ , 1. Days are , ° in summer than m winter. snorter 2. Water always flows , hill. down 3. Glass breaks , easily than tin. less ^ 4. The sun rises ^ in January than in July. , T • softer ,, 5. iron is , J than wood. harder , -r. . warmer . _, . , 0. it IS , in Florida than in Maine. 7. Anything that floats is ,. , ^ than water. lighter 8. Oranges grow . satisfactorily in California than in New Jersey. 9. Shadows are , in summer than in winter. longer 10. Plants grow . readily in warm sunshine than in the cool shade. III. B. b. 1. Men are usually , than women. weaker 2. A pound of iron is worth than a pound of copper. 3. Christmas comes , Thanksgiving day. 4. Cotton clothing is ^'^^^'^ than wool. cooler „ Less , . Mo"re '^ ^^ ^ ^" summer than in winter. 6. Bankers are P°°''^'" than cab drivers. More ^' Fewer ""'"^^^ ^'^^ mules are used for driving purposes. Q T^i, more ». ihere are ^^^^^ teachers than preachers. e-\ more 9. Oranges are ^^^^ sweet than lemons. More , , , '°' Less °'^^^^ ^^^ ca-ke is eaten in this city. of the Fourth, Fifths and Sixth School Grades IV. As quickly as you can write beside each of th a word that means. exactly its opposite: A. B. C. day- great bad asleep hot inside absent dirty slow brother heavy short best late little above first soft big left black backwards morning dark buy much sad come near true cheap north dislike broad open poor dead round well land sharp sorry country east thick tall known full son something peace here- stay few less push below mine nowhere enemy Selective Judgment Two types of tests were used for selective judgment. First, two sets, V, A and B, of two series each of ten reasons why some given fact is true, some of which reasons are correct, the others incorrect or irrelevant, were given. The pupil was to select, by checking, the correct reasons. Second, there were given similarly two sets, VI, A and B, of three series each, of five definitions for a given thing or term, some of which were correct, the others incorrect or irrelevant. Tests V and VI V. A. The following reasons have been given to show why New York has become a larger city than Boston. As quickly as you can, place a cross like this, +, before each reason you think a good one: 1 . New York is on an island. 2. More foreigners live in New York than in Boston. 3. New York is on a large river coming from a rich agricultural region. 4. Mr. Rockefeller has a fine home in New York. 6 The Reasoning Ability of Children 5. New York has more churches than Boston. 6. New York has better communication with the States lying to the west. 7. New York has elevated railroads. 8. New York is in the midst of a rich fruit and agricultural district. 9. New York is nine or ten years older than Boston. 10. New York has a republican governor. B. These reasons have been given to show that oak wood is better than pine for making furniture. Check the good reasons. I. Oak wood is harder than pine. ■i. Oak trees have acorns, pine trees do not. 3. Oak wood takes a finer polish than pine. 4. Oak trees have more beautiful leaves. 5. Oak trees make good homes for squirrels. 6. Pine wood will not last so long as oak. 7. Pine is more easily dented and defaced than oak. 8. When polished and varnished, oak is much more beautiful than pine. 9. Pine trees are sometimes used for Christmas trees. 10. Oak trees are easier to climb than pine trees. C. The following reasons have been given to show why oranges grow better in Florida than in New Jersey. Check the good reasons. 1. There are many negroes in Florida who work very cheaply. 2. Florida has warm summer weather almost the whole year. 3. There are no alligators in New Jersey. 4. Florida very rarely has hard frosts. 5. New Jersey is not so large as Florida. 6. Florida was settled earlier than New Jersey. 7. New Jersey grows many fine peaches. 8. Florida has a very moist, warm climate. 9. Florida is a word meaning the land of flowers. 10. Florida is a popular winter resort. D. Among these reasons why horses are better than cattle for driving and working animals, check those which you think are good reasons. 1. Horses are more intelligent than cattle. 2. Cattle are not so tall as horses. 3- Horses like corn, oats and hay. 4- Horses are much more active and walk faster than cattle. 5- Cattle are extensively used for food. 6. Horses are much more beautiful and graceful than cattle. 7- The skms of horses are sometimes made into gloves. 8. Horses are more easily trained and controlled than cattle. 9. President Roosevelt likes to ride on horseback. 10. Horses have more rapid and varied gaits than cattle. of the Fourth, Fifth, and Sixth School Grades 7 VI. A. In the following definitions, place a small cross, like this, +, before those which you think are good ones, doing it as quickly as you can. a. Definitions of a shoe. 1. A portion of clothing. 2 . Something black made of leather. 3. A protective covering for the feet, usually made of leather, having a firm bottom or sole and flexible upper portions, an opening for the foot being fastened by lacings, buttons or buckles. 4. Something to wear on the feet. J. A necessary article costing from one to five or six dollars. b. Definitions of an island. 1 . A piece of land out in the water. 2. A small body of land. 3. A body of land entirely surrounded by water. 4. Cuba is an island. 5. A portion of land rising above the surrounding level. 0. Definitions of to explode. 1. To burst suddenly with a loud noise. 2. To knock all to pieces. 3. To make a very loud noise. 4. To fill the air with a tumultuous roar. 5. To blow up. a. Definitions of a chair. I . A piece of household furniture. ^. A movable seat with a back intended for one person. 3. A piece of furniture on which to sit. 4. Rocking chairs are comfortable chairs. 5. A single seat having a back. b. Definitions of to write. 1. To make marks with a pen or pencil. 2 . To make characters which stand for ideas. 3. To use a pen or pencil. 4. To make marks on any kind of surface with any kind of an instrument which will express one's ideas so that another may understand them. 5. To write a letter. c. Definitions of a buggy. 1. A buggy is black. 2. A buggy is something to ride in. 3. A buggy is a light, four wheeled vehicle, with or without a top or covering, designed for carrying two or three persons. 4. A buggy is drawn by horses. 5. A buggy may have rubber tires. 8 The Reasoning Ability, of Children Literary Interpretation For literarjr interpretation, two stanzas of poetry, V'll, A and B, were used, the pupil to write the meaning of each in his own words. These poems are taken from a third reader and a second reader respectively, each from a different standard series published within a decade of the time of these tests. Test VII VII. A. Read carefully the following stanza, then write its meaning in your own words. "This little rill, that from the springs Of yonder grove its current brings, P'lays on the slope awhile, and then Groes prattling into groves again, Oft to its warbling waters drew My little feet, when life was new." B. Read carefully the following stanza, then write its mean- ing in your own words : "Under the greenwood tree. Who loves to lie with me. And tune his merry note Unto the sweet bird's throat. Come hither, come hither, come hither; Here shall he see No enemy But winter and rough weather." Spelling As an incidental problem for correlation, the opportunity offered for a test in spelling was taken. Two papers, B and C, from test V, the opposites test, were graded in spelling for each pupil. As the pupils did not know that the papers were to be graded in spelling, it had little of the disadvantages of the formal spelling test, yet the words were practically predetermined and uniform. Children Tested The children of the upper division of the fourth grade, and of the fifth and s ixth grades of public schools number 2, 3, 4, 6, and 9 of^assaiqf New Jersey, were given these tests. These Glj of the Fourth, Fifth, and Sixth School Grades 9 schools furnish types of children representative of about all grades of families usually found in American cities of from twenty-five thousand to fifty thousand population, typically cos- mopolitan communities. Children of parents in the various pro- fessions, in manufacturing industries, and in the general trades are all found within the group. Complete sets of tests were received from 757 children, 385 boys, and 372 girls. Papers from 834 pupils were received, but because of absence when some of the tests were given the results from 77 were incom- plete and were therefore discarded entirely. These were so dis- tributed through the different grades and the different schools that there is no disturbance of the representative character of any group by their omission. Procedure in Making Tests The tests were all given within the period of February 13 to 27, inclusive, 1906. They were divided into two groups as fol- lows : Set I : I, A ; II, A ; III, A, a ; III, B, a ; III, B, b ; IV, A ; V, A ; V, B ; VI, A ; VII, A. Set II : I, B ; II, B ; III, A, b ; IV, B ; IV, C; V, C; V, D; VI, B; VII, B. The parts of the first set were given consecutively on the same day ; after an interval of two or three days the parts of the second set were similarly given. All of the tests were given by the writer or under his direct supervision. Some assistance was given on two days, one each respectively by Dr. W. C. Ruediger, and Dr. A. M. Stowe, both graduate students in Columbia University at that time and both trained in observation and experimentation. Teachers were asked to eliminate themselves entirely from the situation and were very often absent from the room during the tests. If not absent, they busied themselves usually with routine work of their own. The greatest care was used to preserve the most strict uniformity in making tests and it is believed that a high degree of success was attained in this. Pupils were given the printed papers containing the questions, one test at a time, face downward, upon their desks. Space was provided upon the papers themselves for all answers. Pupils had been directed to get pencils ready for writing before papers were distributed. When all had received copies of the test, the lo The Reasoning Ability of Children children were told to turn the papers over and to write their names and ages at their last birthday at the top of the pages, but to make no other marks upon them until a signal to begin was given. The printed directions at the top of the papers were read aloud to the pupils and the signal to begin was at once given unless experience had indicated a need for some additional word of explanation which was given before the signal to begin. A series of preliminary tests had been given to the respective grades of the Speyer School of New York City for the purpose of testing the tests as to points of manipulation so that any diffi- culties might be anticipated. When the first pupil to finish had completed his work, in all of the tests but that of IV, the oppo- sites, all turned the papers over, face downward, and they were collected. For the opposites, two minutes were given for each test. There are certain objections to setting time limits to tests in thinking ability. As Dr. F. L. Wells has said, " Only where the association is so closely controlled as to render but one associa- tion possible can the time factor be taken into account."^ Of the foregoing, tests III and IV do fairly conform to this cri- terion. For the rest, it is probably true that they are fully " valid only when everyone has done his best regardless of time."^ However, in a group test some arbitrary limit is neces- sary. If the slowest pupil were given all the time he wished in which to do his best it would make such a series of tests as here given practically impossible with large groups of children. By the plan used, results show a large proportion of finished papers. Viewed from the standpoint of practical efficiency, the factor of time may be as important as many other factors in any field save that of " pure scholarship." As a check upon age, grade, and name, after all of the tests had been given each child was furnished a blank form in which he wrote his name, his age at his last birthday, the date of his birth, the name of his school, his school grade, and the name of his teacher. ' Wells, Linguistic Ability and Intellectual Efficiency, Journal of Philosophy, Psychology, and Scientific Methods, III, 682. ' Wells, Ibid. of the Fourth, Fifth, and Sixth School Grades ii What These Tests Really Measure Meaning of the Term Reasoning That the solution of any problem, which may be but one form of " response by analogy," is a valid case of reasoning seems to be supported by Spencer who holds that, " It is clear that ordinary syllogistic deductions differ from analogical ones simply in degree."^ " The foundation of all reasoning is the recognition of similitude; reasoning may be roughly defined as the transition from a known fact to a second unknown fact, by means of a resemblance."* " The association of images which is established by reasoning takes place through the medium of pre-existing associations which are called premises."^ " Percep- tion is evidently composed of the same parts as formal reason- ing."* " Reasoning is that act of the mind which recognizes those relations of any content of consciousness through which it has, or is, what it is."'' Implicit reasoning " simply goes from the likeness of one case to another without recognizing in what the likeness consists. . . . Explicit reasoning discovers the universal element, the relation of identity, which is at work in implicit reasoning."' In all of these expressions, there is a measure of unity in the view that reasoning is but a form of thinking involving a new situation, a kind of apperception, differ- ing from ordinary purposive thinking in degree only, not in kind. But Lloyd Morgan maintains that reasoning applies only where the why is involved, not the what alone. Reasoning involves the thinking of the therefore.^ To limit the term reasoning to those instances in thinking in which the justification of the conclusion becomes the subject of reflective thinking, to those in which the thinker raises to con- sciousness the why of Lloyd Morgan as against the what, would limit the term to the critical reflections of formal logic and to a relatively few cases in other critical thinking. " The process ' Spencer, Principles of Psychology, II, Part 4, 76. ■* Binet, The Psychology of Reasoning, tr. from 2& French Ed. by Adam Gowans Whyte, 86. "Ibid., 151. "Ibid., iss- 'Dewey, Psychology, 221, 222. ' Morgan, Introduction to Comparative Psychology, Chapters XV and XVI. 1 2 The Reasoning Ability of Children of thought which the syllogism seeks to describe, is not that by which the inference is reached, but that by which it is justified; and in its totality is not gone through at all, unless the needs for justification is suggested."* In his most acute reasonings, the scientist or philosopher no more becomes conscious of the process of his thinking than does the child of ten unless a need arises for its justification or validity. In neither does the question of " sufficient reason " come to consciousness save when the validity of a conclusion is challenged or must be verified. The challenge is met by both in the same way — they look back upon the whats to see whether the assumed resemblance is valid. If it is not, then a further search is made for a principle, or what, which is true to the purpose. If the identity is further estab- lished, the why, the therefore, is but the expression of faith in the uniformity of nature. The child naively believes in this uniformity and acts, implicitly or unconsciously, upon this belief, just as the scientist acts upon it explicitly and consciously. " This is the way to work problems like that," is a statement based upon an assumption as implicitly appreciated by the child as it is explicitly depended upon by the critical logician. When validity is challenged, both are satisfied in the same way, " everything is explained by being brought into connection with something else "^° taken as valid. "All inference is a transition from par- ticulars to particulars and not, as is usually supposed, the appli- cation of a general affirmation to the special affirmations included in it."^^ Recognition that a given particular is identical with another particular gives a pragmatic right, whether implicit or explicit, to treat the given as we treat the other. By thus ful- filling all of the conditions in the given situation, the pupil is led to say that his answer " has " to be right just as the astronomer had a right to say that Neptune " had " to be out there so far away and in such a direction at a given time. That highest phase of reasoning involved in discovery and invention, to quote Royce, " involves no essentially new mental tendency. It is a continuation of the consciousness which char- acterizes the whole thinking process, only of this consciousness ' Spencer, Ibid., 98. "Ladd, Psychology, Descriptive and Explanatory, 485. " Mill, John Stuart, quoted by Bain, The Senses and the Intellect, 520. of the Fourth, Fifth, and Sixth School Grades 1 3 on a higher level."^^ " Discovery " and " invention " are but the recognition of previously unnoticed resemblances. The " sagacity " of the original thinker of the highest order consists in his sharpness of attention and in his keenness in recognizing resemblances, common elements. The observation and abstrac- tion in a number of cases of a relationship previously unnoticed lead to an hypothesis, a scientific "guess," that the observed relationship is universal under similar conditions. Newton notices the law of squares, Darwin the survival of the fittest.^' Up to the point of generalization, the reasoning is inductive— the search for and the recognition of common elements. From the moment the hypothesis is reached, the reasoning is deductive — ^the application of the discovered resemblance to further indi- viduals by way of verification. The element wholly unique is simply the noticing of a previously unnoticed resemblance. That the mind itself does not " invent " the relationship seems clear. It but takes that which comes, recognizing in a degree measured by its " sagacity " any resemblances which may appear, but helpless to produce anything that does not spontaneously come. " Newton, Mozart, Richard Wagner, and others say that thoughts, melodies, and harmonies had poured in upon them and that they had simply retained the right ones."^* In kind, all of these higher activities in purposive thinking are the same as the lower, the difference being one of degree. The differences in degree are measured by differences in sharpness and persistence of attention, and in keenness in recognizing resemblances. Prob- ably individual differences are matters of both nature and nurture. Individuals are born with different degrees of " sagacity," while critical selective judgment is partly a matter of training and development. Summarizing these observations on the meaning of reasoning in relation to the tests under consideration, we believe that an analysis of the four types of thinking involved will reveal a basis for measures of the most essential elements in the reason- ing ability of children of these three grades. Practically every problem employs deductive thinking. So far as evidence is •2 Royce, Outlines of Psychology, 295. " Adapted from James, Principles of Psychology, II, 343. " Mach, Accident in Invention and Discovery, Motdst, VI, 174. 14 The Reasoning Ability of Children available, inductive reasoning is not tested at all. It may have functioned in answering the questions in selective judgment, the choosing of good reasons or definitions from the given lists. The pupil may have read all of the definitions for a given term and derived his own " standard " definition inductively from these. Results provide no possible evidence on this point. How- ever, practically all of the elements in inductive thinking are involved in deductive thinking. Fixed and sharpened attention, keenness in recognizing resemblances, critical selective judg- ment, — ^these are vital in both. Factors in Reasoning Ability Here Measured Mathematical Judgment Tests I and II, the problems in arithmetic, test the mathe- matical judgment, in general that form of deductive reasoning most closely resembling the syllogistic movement of formal logic. The steps here involved are three: First, the analysis of the situation by which the essential features of the problem are conceived and abstracted; second, the recall of an appropriate principle to be applied to the abstracted problem, a search among various principles which may suggest themselves for the right one ; and third, involved in the second, the inference, the recognition of identity between the known principle and the new situation. While this process goes on as implicit, explicitly there are made the concrete applications in the resolution of the problems. Clearly these are examples of deductive reasoning of the usual scientific type, involving data, principles, and infer- ences. The only element omitted is that of verification, which, by the nature of the tests, can not here be brought out. Controlled Association Tests III and IV involve the presentation of a definite cue in each case with a very definite field for correct association. In the sentences with the omitted words, one word, or one of a pos- sible very few, only, can be used to make the sentence correct in meaning. In the sentences with the word to be stricken out. the selection is absolutely limited to a choice between two given responses to the given cue. In the opposites test, the mind set given by instructions directs the association definitely from the of the Fourth, Fifth, and Sixth School Grades i s cue to a single term, or at most to one of a possible two or three. There is involved, therefore, in all three phases of this test, recognition and selection on the basis of similitude or fit- ness to purpose as the dominant factor. The activity tested, that of spontaneity and accuracy in recognizing resemblances between the known of experience and the unknown of new situations, is vitally significant in all reasoning. Selective Judgment The checking of the good reasons for a given condition or fact, or of good definitions for a given term, among a number of erroneous or poor reasons or definitions, tests V and VI, involves a form of selective judgment, and the movement is one of implicit, deductive reasoning. To judge whether a given reason for a certain condition is good implies general principles or notions of why the condition is as it is. Each given reason must be measured by the appropriate general principle or notion, although this principle or notion will usually be typified by some well-known example of its application. The thinking process here is in the search for the appropriate principles, the compari- son of the given reasons with these, the search for and recogni- tion of resemblances, and selection or rejection respectively on the basis of coincidence or irrelevance in meanings. In terms of deductive logic, the given reasons constitute the minor premises, the general notions the major premises, and the inference pro- duces the conclusions in the form of acceptance or rejection of the given reasons. A like process applies in the selection among the definitions. Literary Interpretation For the intellectual interpretation of the two poems, test VII, the pupil must abstract the essential thoughts and rethink them in terms of his own phrasing. The problematic character of these tests lies almost wholly in their complex and unfamiliar form of structure. The number of subordinate modifiers with their inverted order and metrical phrasing require, first, a type of grammatical analysis, implicit in children of this age, to be sure, but no less essential. The pupil must break up the total " given " into several parts, connect in mind principal elements remotely separated, then relate in their conventional order the principal 1 6 The Reasoning Ability of Children elements and subordinate elements. A second type of analysis lies in the pupil's abstracting the essential thought units in the reconstructed selection, eliminating any elements of expression used solely for the sake of poetic form. Following this is a synthesis of these thought units and an expression of them in familiar terms. In stanza A, the child must be able to pass over in thought from the first phrase in the first line, " This little rill," to line five from wrhich it takes, " Oft drew," then to line six from which it adds, " My little feet," then back to line five for, " to its warbling waters," to get the principal thought in its normal order of expression. The degree of success with which this, and the remaining portions of the analysis and the synthesis are accomplished is certainly a measure of the pupil's ability to break up a complex, and properly to evaluate and relate its parts. A new product has been constructed from given material. As viewed by Binet, " This construction of images according to a new plan is really reasoning."^^ In these tests then, these pupils have been subjected to various forms of each of four types of reaction which furnish a measure for four somewhat overlapping and related, though quite com- plex, forms of purposive thinking. It is believed that measures of these qualities properly related will provide fairly reliable measures of what we usually mean by reasoning ability. Scoring Tests I and II. For each problem in arithmetic, a grade of 2 was given for each correct solution. If a two-step problem, and one part was right, the other not, the grade given was i. No detraction was made for inaccuracies in operations. Test III. In the filling of blanks, and the choice of words, a grade of i was given for each correct answer, o for each wrong. Test IV. For the opposites, 2 was given for the correct word, i when it was partly right in meaning, and o for wrong and omitted words. Tests V and VI. For choice of reasons and definitions, the scale used was as follows, the grade in each case being the alge- braic sum : " Binet, The Psychology of Reasoning, 170. of the Fourth, Fifth, and Sixth School Grades 1 7 V. A. Numbers 3, 6, and 8, each 3 points ; i, 2, 5, 7, and 9, each -I ; 4 and 10, each -2. B. I, 3, 6, 7, and 8, each 2 ; 2, 4, 5, 9, and 10, each -2. C. 2, 4, and 8, each 3 ; i, 3, 7, 9, and 10, each -i ; 5 and 6 each -2. D. I, 4, 6, 8, and 10, each 2 ; 2, 3, 5, 7, and 9, each -2. VI. A. a. Number 3, 7 points; 4, 2; i, -2; 2, -3; 5, -4. b, I, 2 points ; 3, 5 ; 4, i ; 2 and 5, each -4. c. I, 6 points ; 5, 3 ; 2, 3, and 4, each -3. B. a. 2, 5 points; 3, i ; 5, 2; i, and 4, each -4. b. 2, 2 points ; 4, 5 ; 5, i ; i, and 3, each -4. c. 2, 2 points; 3, 7; i, 4, and 5, each -3. Test VII. From o to 10 on basis of estimated merit for each part. Test VIII. Spelling. Subtract i for each misspelled word from the arbitrary standard of 15 for each of the two sets of papers used. The aims in scoring were twofold, to use a scale giving grades small enough not to be cumbersome in manipulation, and to weight in such a manner as to make grades rational. Since the results were to be rated in terms of relative position only, the standards used could be entirely arbitrary. But, for a given mental quality, the rating of each particular problem should be such as to show its real relationship to the standard adopted in terms of steps of the scale. Hence, an attempt was made to give such weighting in the plan of scoring as would give due regard to rational relationships. Different Plans of Scoring Tested As a means of showing that any arbitrary plan of scoring brings results practically the same as any other plan, providing individual items are weighted proportionate to their relative value in terms of the unit used, a sample of different methods of scoring and their effects is here given. For test IV, A, the papers of the 25 pupils of School No. 2, grade 6 B, were scored by four different plans, as follows: I.. That actually used throughout -3 for Nos. 3, 6, and 8 ; -i for I, 2, 5, 7, and 9. II. I for each of 3, 6, and 8 ; -i for each of the others. 1 8 The Reasoning Ability of Children III. 2 for each of 3, 6, and 8 ; -i for each of the others. IV. 2 for each of 3, 6, and 8 ; ignore all of the others. Arranging results by plan II in comparison with plan I, by relative position, only 3 papers were displaced. Each of the 3 was moved up one step. The steps are smaller, the scale finer. Arranging results by plan III similarly, the order is ex- actly as by plan I. In the arrangement of results by plan IV, two papers are moved, each up two steps. But the grade of the whole set is moved upward. The scale is coarser and the highest grade is gotten by 17 of the 25 children. The grades upon the different bases are given in the table below. TABLE I Scores op 25 Children in Test IV, A, by Four :Plans of Scoring Child Plan I Plan II Plan III Plan IV I 6 I 4. 6 26146 33024 42—2 o 4 5 -2 -3 -I a 65036 7 7 I 46 87146 98256 10. 5 03 ft 11 I —I o 2 12 7 I 4 6 13 2 -3 Q 6 14 —2 —2 —2 o 15 3-126 16 6 o 3 6 17 5 -I 2 6 18 I —2 o 4 19 S -I 2 6 20 3 -2 I 6 21 -6 -4 -5 2 22 6 2 44 23 S o 3 6 24 2-3 o 6 25 9 3 6 6 Correlations by like and unlike signs of plam I with each of the others give coefficients of .88 with plan II ; .92 with plan III ; and .80 with plan IV. While plan III is evidently most like that used throughout, plan I, either plan II or III could be used with- out materially changing the total of results, in this test. Applying plan II to a sample grade for test IV, B, the rela- tive position is exactly the same, but the grades gotten by plan I of the Fourth, Fifth, and Sixth School Grades 19 are divided by 2. For plan III, the lower limit of the scale is raised from -10 to -5 and all of the grades are raised in gen- eral. Most of all, it gives a wholly wrong result when, in any paper, all of the items are checked, a grade of 5 whereas it should be o, no selective judgment at all. In test VI, A, a sample grade was scored by plan II. Al- though it gave a relation coefficient of ."jy, the scoring is quite unfair as it fails to weight in a test where the items are of mark- edly different value. For this test, plan III gives a correlation of .92. But it does not weight individual items and is therefore faulty where they really differ. From the standpoints of both rational weighting and manipulation, the scales adopted for scor- ing seem fairly satisfactory. Distribution of Children by Grade, Age, and Sex The problem comprehends an investigation of any differences which may exist in children on the basis of school grade, age, and sex. In all computations, therefore, the sexes are kept sepa- rate, and the same computations are made on distributions of results by both grades and ages. For grade studies, computa- tions are first made by the half schqol years, designated as A and B, then combinations of these are made for each year. For age distribution, the computations are first made upon the basis of half years excepting only the two extreme limits which are represented by numbers so small that results are meaningless save as individual records. Then follow combinations of results for years, and at the upper and lower ends of the distribution for still longer periods. The following tabulations indicate the distribution of pupils by age, grade, and sex, and are explained sufficiently by titles. 20 The Reasoning Ability of Children TABLE II Distribution of Children by Grade, Age, and Sex Boys Girls Age Gr. 4A 5B sA 6B 6A T. 4A 5B 5A 6B 6A T. G.T. 8 to 92 2211 46 9 " 9-6 7 3 I I 12 12 2 2 16 28 9-6 "10 s 5 I I I 13 II 7 5 23 36 10 " 10-6 12 8 7 4 I 32 12 16 5 2 I 36 68 10-6 " II 18 II 10 13 o 52 15 13 5 8 4 45 97 11 " II— 6 17 8 12 17 9 63 12 12 14 II 5 54 117 11-6 "12 5 8 4 15 6 38 8 7 9 II 4 39 77 12 " 12-6 II 10 15 14 9 59 7 12 7 18 II 55 114 12-6 "13 5 9 5 9 9 37 3 S 6 II 13 38 75 13 13-6 4 12 6 14 3 39 3 6 2 12 II 34 73 13-6 "14 4 3 I 4 2 14 2 3 o 5 2 12 26 14 " 14—6 2 I 2 6 415 I I I 2 I 621 14-6 "is ioi 3S 001238 15 " 16 I 34 II 5 7 II 93 79 65 98 50 385 88 86 58 81 Stmimary by Grades and Sexes S9 372 757 mA 5B 5A 6B 6A 4 S 6 T. l\ 79 65 98 50 93 144 86 58 81 59 88 144 148 38s 140 372 Boys Girls 181 165 123 179 109 181 288 288 757 TABLE III Median Age and Variability op Each Grade" Boys Girls Older M Q M Q Boys Girls Grade Yrs. Mo. Yrs. Mo. Yrs. Mo. Yrs. Mo. Mo. Mo. 4A II .5 I 5.0 10 7.8 10.16 4.7 sB II 8.8 I 1. 14 II 0.0 II. 17 8.8 SA II 9.2 I 5.19 II 4.0 9.37 s-i= 6B II 10. o 9-91 12 I.I 9.06 3.1 6A 12 4.0 I .5 12 7.5 8.23 3.S The Same with Grade Sections Combined 4A II .5 I 5.0 10 7.8 10.16 4.7 5 II 9.0 I 3.16 II 2.0 10.27 7.0 6 121. o 11.20 123.2 8. 64 2.2 For all II 8.3s I 2.45 II 6.0 9-69 2.35 The number of children, both in distribution by grades and by years, is sufficiently large and sufficiently representative to give results which may be rated as really typical and comparable, each with the others. " For explanation of technical terms, see the section on statistical technique following. oj the Fourth, Fifth, and Sixth School Grades 2 1 There are 288 children in the fifth grade and the same number in the sixth grade. In the A division of each grade, the number is considerably smaller than in the B division, probably indicat- ing the selection of the better pupils of the grade as a whole, and perhaps also indicating a kind of retardation and repetition. The sex distribution is also surprisingly nearly equal — exactly equal in the fifth grade with 144 pupils of each sex' — ^and almost equal in the sixth grade — 148 boys and 140 girls. The median ages of both boys and girls are interesting in showing the very small differences in passing from grade to grade. Table III. For four half school years, the median age of the boys lies between 11 years and 11 years, 10 months. The variability in age is seen to be much greater among the boys than among the girls, as shown by a comparison of the Q's. In the first three half years studied, the median ages of the boys are 4.7 mo., 8.8 mo., and 5.2 mo., higher, respectively, than those of the girls, while in the remaining two half years the median ages of the girls are respectively 3.1 mo., and 3.5 mo. higher than those of the boys. This dropping behind of girls is coincident with that acceleration of physical growth that usu- ally begins at about 12 in girls but which is not evident in boys until about a year later, and is indicative of that inverse corre- lation usually found between mental ability and rapid physical growth. Statistical Technique Employed in Computations No exposition can here be made of statistical methods. A definition of terms used will be given, however, which will en- able the reader not versed in statistical technique to interpret intelligently the tables, summaries, and conclusions. For a full treatment of statistical methodology, the reader is referred to " Mental and Social Measurements," by E. L. Thorndike. For central tendencies, the median, M, the measure above which and below which exactly half of the separate measures lie, is used rather than the average in most cases. It is readily found, conveys no erroneous impression, and gives less weight to extreme cases. For variability, the 25 per centile, quartile, or Q, is used, rather than the average deviation or standard deviation. The Q is 22 The Reasoning Ability of Children gotten by counting in from the low end of the distribution until 25 per cent, of the cases are covered ; and likewise from the high end of the distribution until the point marking 75 per cent, of the cases is reached. These two values give the limits within which exactly 50 per cent, of all the cases lie. Subtracting the lower value from the higher and dividing the result by 2 gives the variability in steps of the unit of measure used above or below the median within which any individual case will probably be found. Where the number of cases is large, as in this study, this measure is very reliable, and it is easily computed. The single term for variability showing the relationship between the median ability and the Q is the coefficient of variability which is gotten by dividing the Q by the median. For showing differences of two groups in the same test, or of two tests by the same group, the percentage of the ability in the one test or group reached or exceeded by 50 per cent, of those in the other test or group is frequently used. It is a value easily computed, and its meaning and implications are clear. For the same purpose, the percentage of those in one group who reach or exceed the ability reached or exceeded by the highest 25 per cent, of another group is occasionally used. This compares those above the 75 per centile, those in the highest quartile of ability, and is a good index of relative superiority. For relationships of abilities, the coefficient of correlation de- rived from the method of like and unlike signs is used. A high coefficient of correlation between two abilities means that a given degree in one implies a corresponding degree in the other ; a low coefficient implies little correspondence of ability ; and a negative coefficient implies that any, ability above the median or the aver- age in the one involves a corresponding position below the median or average in the other. Array of Original Scorings Table IV As it would be obtrusive to introduce such a volume of data in this place, Table IV is included in an appendix at the close of this study. All original scorings are given in full. All computa- tions made in the study may be verified, or the data may be used in further studies in the thinking abilities of children or for pur- oj the Fourth, Fifth, and Sixth School Grades 23 poses of correlation with other abilities measured. But a be- ginning has been made in the accurate measurement of the various elements of mental ability. Only by the aggregation of immense numbers of data can valid and adequate conclusions be derived. Such studies involve hundreds of hours of patient labor, and it is decidedly unfortunate that any of the data secured at such cost should be lost. The scorings for this study are therefore given in full with the hope that they may be of service in further investigations. For the distribution in age, an entirely new array was made bf the 757 children on the basis of differences of six months excepting for the groups from 8 years to 9, and from 15 years to 16, which were too small to be subdivided — results from them as group tests would be meaningless. RESULTS AND THEIR ANALYSIS Mathematical Judgment — Tests I and II The results of tests I and II were combined, a single quantity thus representing the summarized valuation of each child's mathe- matical judgment. However, since the two tests are slightly different, both in degrees of difficulty and in the form of activity measured, an analysis and comparison of the two taken sepa- rately follows the treatment of combined results. Belo\jr are the tables and summaries of results for grade, age, and sex dif- ferences. TABLE V Frequency OF Abilities BY Grades Grade 4A sB SA 6B 6A Ability B G B G B G B G B G 2 I I 2 3 I 3 I 4 4 6 2 2 I 5 1 3 2 I I 6 6 4 I 4 2 I I I 7 S I 2 I 8 8 4 I 3 I I 9 I 6 I I 10 7 5 I S 2 3 I I 2 II 4 3 3 2 I I I 12 7 II 4 9 2 4 3 I I I 13 2 3 I 2 I 2 I I 14 4 4 6 3 3 s 2 3 3 IS I 5 I 3 3 2 I 16 4 7 7 6 2 2 2 6 2 17 2 I I 3 2 2 I 2 18 8 2 I 2 2 3 I 3 2 19 3 5 I S I I 20 6 4 8 9 4 2 4 S 2 3 21 2 I 2 2 4 I 22 2 2 9 S S 3 6 8 I 5 23 6 I I 2 3 2 3 2 I 24 6 3 II 3 3 2 6 6 4 I 25 c 3 2 I 4 I 26 3 I 8 4 S 3 II 4 7 7 27 2 I I I I 3 28 I I I 6 3 II 6 6 29 I I 3 2 I 2 6 3° 4 I 2 2 4 8 2 I 31 I 2 I I 3 32 I 4 I 4 2 10 5 6 8 24 SA 6B 6A B G B G B G 2 2 I 3 I I 3 3 4 I I I 2 I 6 4 3 3 I I I I 2 2 3 I The Reasoning Ability of Children 25 Grade 4A 5B Ability B G B G B" 33 34 I 35 36 I 37 38 I 39 40 II Cases 93 88 79 86 65 58 98 81 50 59 TABLE VI Median Ability and Variability for Each Grade M's Higher Q's Co'ts. of Var. Grades Boys Girls Boys Girls Boys Girls Boys Girls 4A 14.50 11-36 3-14 5-39 4-21 -37 -37 SB 21.39 15.66 S.73 4.75 4.71 .22 .30 SA 22.83 19.00 3.83 5.58 6.46 .24 .34 6B 25.63 24.08 1.55 4.95 6.42 .19 .26 6A 28.00 25.92 2.08 3.96 5.95 .14 -22 Combining the half years of Grades 5 and 6 5 22.11 17.33 4.78 5.10 5.58 .25 .32 6 26.81 25.00 1. 81 4-45 6.18 .16 .24 For all 22.60 17.7s 4.85 6.07 6.96 .26 .38 Per Cent, of Each Grade Reaching the 75 Per Gentile of All Asking this question, If we wish to select from each grade those pupils who reach or exceed the ability reached by the highest 25 per cent, of all children of each sex taken together, what per cent, will be found in each grade, respectively, we get the following answer: 5B 5A 6B 6A 12.65 32.30 44.89 62.00 6.97 20.68 37.02 44.04 That is, 5.37 per cent, of the boys of grade 4 A reach or exceed the ability reached by the highest 25 per cent, of all of the boys taken together in this test; 12.65 P^i" cent, of the boys of grade 5B ; and so on. By a mere accident of coincidence, the number of boys in 25 per cent, of the whole number, 385, is 96, so that these per cents, also roughly represent the per cents, of the whole group above the 75 per centile which are from each respective grade. Thus, 5 per cent., more exactly 5.2 per cent., of- all of the highest quarter of boys are from grade 4A, and so on. As the number of girls is 472, the approximations are also true for them. Grade 4A Boys Girls 5-37 3-40 26 The Reasoning Ability of Children Sex Diflferences by Grades Other than the evident differences in the foregoing, the simplest summary of sex differences is in a statement of the per cents, of the boys who reach or exceed the ability reached by 50 per cent, of the girls, as given below : Grade 4A 5B sA 6B 6A Per cent. 61.8 71.3 70.7 60.8 65.1 TABLE VI] Frequen CY OF Abilities by Ages Age 8- 10 10- -II II- -12 12- -h '3" -14 14- -16 Abilit y B G B G B G B B G B G 2 I I I I 3 2 I I I 4 I 4 I I I I 4 I I S 3 I I I 2 6 2 3 3 3 2 I 2 I I I I 7 I 2 3 2 I 8 3 I 3 I I 2 3 3 I 9 3 2 I I I I 10 2 2 5 5 5 I 3 I I 2 II I I 2 2 3 2 I I 2 12 S 5 5 5 S 5 S 2 4 2 13 3 2 I 2 2 I I 14 4 4 3 3 6 3 3 5 2 15 I 4 3 4 I 2 I 16 3 5 7 6 8 2 4 2 I I 17 3 I 2 5 I 2 18 2 4 I 2 7 4 2 I I I 19 I 3 I 4 I I I I 2 I 20 2 I 8 6 4 2 7 9 3 5 I 21 I I 2 2 2 2 I I 22 5 S 6 6 10 8 I 3 I 23 I 2 4 6 2 2 I 3 24 2 2 7 3 8 3 6 3 2 3 5 I 25 2 2 I I I 3 I 26 2 4 7 3 7 3 8 7 4 2 4 27 I I 3 I I I I 28 I 5 3 3 3 II I 4 2 2 29 I 2 6 3 I I I I 30 3 I 2 7 2 I 5 2 31 I I I I 2 2 32 2 4 5 4 2 8 8 s 2 2 33 I 2 2 I I I I 34 2 I 2 3 I 2 2 I 35 E 2 I 36 2 I 2 I s I I 4 2 I 37 I I I I .38 I 2 2 2 I I I 39 40 — I I _ Cases 27 43 84 81 loi 93 96 93 53 46 24 16 of the Fourth, Fifth, and Sixth School Grades 2 7 TABLE VIII Median Ability and Variability for Each Age M 's Higher Q's Co'ts. Of Var Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 16.50 15-50 1 .00 7 .00 7-So .42 -48 9 " 9- -6 21 .00 10 .50 10 .50 10.70 7-25 •5° .69 9-6 " 10 23-50 13-75 9-75 10.81 5-25 .46 •38 10 " 10- -6 19 .00 15.40 3.60 7.72 5-50 .40 -35 10-6 " II 21-75 16.50 5-25 4.00 6-39 .18 -38 11 " II- -6 21.50 15-50 6.00 5-74 6.20 .26 .40 11-6 " 12 23-5° 20 .50 3.00 6.89 6-7S •29 -32 12 " 12- -6 23.10 21.58 1-52 5.08 8.37 .22 •38 12-6 " 13 22. so 19.50 3.00 6.06 6.04 -27 •30 13 " 13- -6 22.25 21 .00 1-25 7 .06 6.12 -31 -29 13-6 " 14 21 .50 19-50 •z .00 9.00 II .50 -41 -58 14 " 14- -6 23.87 12 .50 11-37 2.68 4.00 .11 -32 14-6 " IS 31-50 15-50 16 .00 3-75 6.25 .12 .40 IS " 16 24-50 II .50 13.00 14.50 3-75 -58 -32 Combining Half Years, and Smaller Groups : 8 to 10 22 .50 13-37 9-13 10 .02 6 .00 • 44 -44 10 " II 21 .40 15.64 5-76 5.20 5-78 .24 ■36 II " 12 22 .41 17-50 4.91 6.19 7-03 -27 .40 12 ;; 13 23.00 20.7s 2.25 5-40 7-83 -23 -37 13 14 22 .16 20 .00 2 .16 7.09 7-37 -32 -36 14 " 16 25 .00 13-50 II .50 3.00 3-75 .12 .27 Combining Further the Extreme Groups : 8 to 11 21.50 15 .20 6.30 6.05 6 .21 •23 -40 13 " 16 23-50 19 .00 4-50 6.98 7-33 .29 -38 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-1 1 11-12 12-13 13-16 Boys 27.02 26.13 34.37 29.87 Girls 17.74 27.95 35.48 29.01 This summary tells us that if we should select all the boys in these grades who are in the highest 25 per cent, of all of them taken together in this test, 27.02 per cent, of all of the boys whose age lies between 8 and 11 years would be in the group; 34.37 per cent, of the boys who are from 12 to 13 would be in the group ; and so on. Comparison of Youngest and Oldest 25 Per Cents, af the A Grades To compare youngest pupils and oldest pupils in several grades, groupings were made of the youngest 25 per cent, and the oldest 25 per cent, of the pupils, by sexes, in grades 4A, 5A, and 6A, respectively. The summary of results is shown in the following table: 28 The Reasoning Ability of Children TABLE IX Youngest and Oldest Pupils DP Several Grades Compared Boys Grade Median Age Higher Yrs. Mo. M's Younger Older Co'ts. Q's ofVar. 4A Younger 25% Older « Whole Grade 9 12 II 8.0 9-S •S 11.50 12.66 14.50 I 16 5-87 4.50 S-39 .51 •35 -37 SA Younger 25% Older Whole Grade 10 13 II 2-S •5 9.2 25-50 22.75 22.83 2-75 6.68 3-93 5-58 .26 •17 -24 6A Younger 25% Older Whole Grade II 14 12 5-S 4.0 28 .00 27 .50 28 .00 ■50 5 -SO 4.62 3-96 .19 .16 • 14 4A Younger 25% Older « Whole Grade 9 12 10 Girls 4.0 11.50 3.5 II -60 7.8 11.36 .10 4.18 5.00 4.21 -36 •43 •37 SA Younger 25% Older Whole Grade 9 12 II 10.5 6-5 4.0 21 .50 19-75 19 .00 1-75 4.69 12 .25 6.46 .21 .62 •34 6A Younger 25% Older Whole Grade 11 14 12 .0 6.5 7-S 31.00 16.00 25.92 15.00 2.50 5-75 5-95 .08 ■36 .22 From this table it will be observed that, for both boys and girls, the group of the youngest 25 per cent, in each grade reaches a higher median ability than an equal number highest in age for that grade excepting in grade 4A. The median ability of the whole respective grade is reached or exceeded by the older 25 per cent, of the girls in grade 4A, by both boys and girls in grade 5A excepting the group of older boys, and by both the younger boys and the younger girls of grade 6A. Six Diiferences by Age TABLE X Per Cent, op Boys Reaching Median Ability op Girls Combining Half Years and Smaller Groups Age 8 to 9 9 9-6 For Half Years 10 10-6 II 11-6 9-6 10 10—6 II 1 1-6 12 Per Cent. 50.00 75.00 61.53 65.62 80.76 73-01 65.78 Age to 12-6 12-6 13 13-6 14 14-6 IS 13 13-6 14 14-6 15 16 In every age group, the boys ability of the girls, in that group. Per Cent. 67-79 67.56 56.41 57-14 86.66 80 .00 75.00 reach or Age 8 to 10 10 II 12 13 14 II 12 13 14 16 Per Cent. 66.66 78.56 70.28 62.50 60.37 83-33 exceed the median of the Fourth, Fifth, and Sixth School Grades 29 1- t 13. S.' a.s 36 3-11 11-11. 1113 l3-/h 10 Figures i-io. Tests I-II I. Frequencies of Abilities, Boys. 3. Median Ability, by Grades. 5. Variability, by Grades. 7. Per Cent, of Boys Reaching the so Per Centile of Girls, by GTa,des.' 9. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Grades. ^^^^— - Boys. 2. Frequencies of Abilities, Girls. 4. Median Ability, by Age. 6. Variability, by Age. 8. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Age. 10. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Age. Girls. 30 The Reasoning Ability of Children Differences Between Tests I and II Test I involves two-step problems of the conventional type found in school practice, while in test II all of the problems but two involve a single operation each, but the statement of the problem is unusual in form. Processes involved are alike and of approximately equal difficulty. The following table will indi- cate differences in results for the two tests. TABLE : XI Median Ability and Variability of Tests I and II Grades Boys M's Q'l I II I 3 II Girls M's Q'; I II I s II 4A SB SA 6B 6A 6.12 7.53 9 .87 II .02 II .05 II .61 13.00 13.41 14.50 15.13 3-77 2.88 3.20 3-48 2-37 3-14 3 -OS 3.18 2.86 3-64 2.07 2 .96 4.80 5.87 5-5° 9-25 7.16 10.00 9.88 13.04 11.87 14-64 7.84 10.50 3-11 3-47 3-45 3-23 2-73 3-19 2.08 2-94 3 -OS 3.28 3-17 Werages 10 .90 II .74 2 .90 A comparison of medians for individual grades, and of aver- ages, indicates that for both boys and girls test I is clearly more diflficult than test II. The average variability is also greater for test I, as it is in the individual grades excepting 5B and 6B for boys, and 6B and 6A for girls. The difference may be brought out further by the following summary which shows the per cent, of those who in test II reach or exceed the ability reached by 50 per cent, of them in test I. Grades 4A sB sA 6B 6A Averages Boys 59.13 63.29 58.46 59.18 56.00 59.21 Girls 70.45 87.20 79.31 76.54 72.88 78.67 For both boys and girls, the difference in the difficulty of the two tests is greatest in grade 5B. For girls, there is progressive diminution in difference through the remaining grades. For boys, there is a break in the progress in grade 6B. Sex Dififerences between the Tests The foregoing indicates that throughout both tests are more difficult for the girls than for the boys, and that test I is pro- portionately harder throughout for the girls. The per cent, of of the Fourth, Fifth, and Sixth School Grades 31 boys who reach or exceed the ability reached by 50 per cent, of the girls expresses this relative superiority for the boys. Grades 4A 56 5A 6B 6A Averages Test I 58.2 84.4 74.1 64.4 71.3 70.4 Test II 73.8 60.4 60.0 56.4 54.7 6r.6 In test n, the sex difference progressively diminishes through these five groups from the relative position 73.8 in grade 4A to 54.7 in grade 6A. Summary for Mathematical Judgment — Tests T and II Grade Differences A study of the median abilities of the respective grades, Table VI, shows progress through these from grade to grade for both boys and girls. The boys make a gain of 93.10 per cent, on their median ability for grade 4A, the girls 128.16 per cent. The greatest gain in per cent, from one grade to the next for both is from 4A to 5B. The smallest gain is, for boys, from grade 5B to sA ; for girls, from 6B to 6A. An examination of the summary of those from each grade who reach the 75 per centile of all, page 25, indicates that we might make a new grouping of children having the common abil- ity indicated by the 75 per centile of these tests, taking a certain proportion of the children from each grade. Results here show that in these grades we would take 5.37 per cent, of the boys from grade 4A ; 12.65 P^'' cent, from 5B ; and so on, up to 62.00 per cent, for grade 6A. In other words, in the abilities measured by these tests, 5.37 per cent, of the boys in grade 4A belong to the same group with 62.00 per cent, of the boys of grade 6A. For girl's, a similar new grouping is in evidence, but the proportion from every grade would be smaller. Such a new grouping is valid in just the measure in which these tests are valid as measures of certain forms of ability which we call mathematical. In these tests, variability diminishes from grade 4A upward. However, it is a little higher for grade 5A than for grade 5B for both sexes. Reference to the Q's and coefficients of vari- ability in Table VI, and to Figure 5 will show these tendencies. 32 The Reasoning Ability of Children Age Differences From the array of median abilities by half years, Table VIII, the regularity of progression found on the basis of school grades is not at all in evidence. A rhythm in ability is fairly apparent for the boys with its first crest at about 9 yrs. 6 mo., the second at about 12 yrs., and the third at about 14 yrs. 6 mo., each crest a little higher than the preceding. The median for the period 13 yrs. 6 mo. to 14 yrs., is just the same as that for the period II yrs. to II yrs. 6 mo., each 21.50, the interval being the same as that measuring the crests of the rhythm, 2 yrs. 6 mo. For the girls, there is evident,. though not so clearly, a rhythm with its crests about coincident with the valleys of the rhythm for the boys, excepting at the period 11 yrs. 6 mo. to 12 yrs. 6 mo., where the crests become nearly parallel. In general, however, the more evident element in the array of median abilities of girls may be described as their general rise from median 10.50 at 9 yrs. to median 21.58 at 12 yrs., then their fall to median 12.50 at 14 yrs. For both the rhythm appearing in the array for boys and the general rise and fall in the array for girls, the age distribution for years instead of half years lends additional clearness. From the per cent, of each age group reaching the 75 per centile of all, page 27, the relation of the distribution of higher abilities to age is very evident. If we should group these pupils reaching the 75 per centile in ability, the smallest proportion of any age group would come from those from 11 to 12 years in age, the largest from the group from 12 to 13 years, for boys. For the boys, there is a larger percentage of the group of higher ability in the age period from 8 to 11 years than in that of II to 12 years. For girls, the largest percentage also comes from the year 12 to 13 while the smallest is from the year 8 to II. For the years 11 to 12, and 12 to 13, the percentage is larger in both cases than that of boys. For the period 13 to 16, they are almost the same— there is a difference of less than i per cent. In so far as these tests and these children are typical, then, we can predict that of any group of children of these grades a larger proportion of those pupils who are from 12 to 13 will be found to be in the highest group as to ability than of those who are from 13 to 16 for both boys and girls, and for of the Fourth, Fifth, and Sixth School Grades 33 boys, more who are from 8 to 11 than who are from 11 to 12. The rhythm of Table VIII is evident in the selected group. In the arrays of results by half years and years, in the com- parison of the youngest and oldest 25 per cent, of the pupils in the respective grades, and in the arrays of children reaching or exceeding the 75 per centile of all, either by grade or age, we have a contrast of the higher degrees of native ability of the youngest pupils and of lower degrees of native ability in the oldest pupils. Retardation seems evident in the pupils of each respective grade who are from two to four years older than the median age for that grade. There also seems evident another type of retardation in these special abilities, perhaps quite as important, in those pupils who are from two to three years younger than the median age for their respective grades, but who are in the group exceeding in ability the 75 per centile for all. In tests in which progress from grade to grade and year to year is so very evident, in large groups as here shown, these wide divergencies in ability of lowest and highest quarters of these respective groups indicate that native ability is measured by the tests quite as much as school training. Children with from 3 to 5 years of experience and training more than a corre- sponding number of other children frequently do not do so well as the group of fewer years and less experience but with greater ability by nature. With the same high coefficient of .44 in the age group 8 to 10 years, the variability of both sexes diminishes with age, excepting that from 11 to 12 it is slightly higher for both than from ID to II, and that it rises rather abruptly for boys at 13 to 14. The general fall, and the rhythm in variability for the boys, are evident in Figure 6. Sex Differences Other than the specific differences pointed out in the foregoing, the one marked sex difference is that of the superiority of the boys in these tests. By every distribution, in every one of its respective divisions, the boys are shown to be more able than the girls excepting in two cases of selected groups, one from grade 4A, the other from grade 6A. In the youngest 25 per cent, of the former grade, the girls just equal the boys in median abil- 34 The Reasoning Ability of Children ity, while for the corresponding group for the latter grade the girls slightly excel. The percentage of all of the boys reaching or exceeding the ability reached by 50 per cent, of all of the girls is 71.43. The median ability of all of the boys taken together is 22.60 with a coefficient of variability of .26; of the girls, is 17.75 with a coefficient of .39. While the median ability of the boys is 27.73 P^r cent, higher than that of the girls, the coefficient of variability is 33.33 per cent, lower. As shown in Table VI, the sex difference diminishes as we proceed up the grades from 4 to 6. Should we proceed far enough, we might reach the condition found by Fox and Thorndike in a study of 28 boys and 49 girls of high school age where " girls do about 5 per cent, better on the whole than boys."^^ Controlled Association — Tests III and IV .The three parts of the tests for controlled association are III A, the filling of blanks in sentences with appropriate words ; III B, the crossing out of one of two significant words in sen- tences leaving the word which makes the sentence read cor- rectly; and IV, three sets of easy opposites. The combined results of III A and III B are here used but the differences between the results of the two parts of the test are summarized after the tables and results for the combined values. " Fox and Thorndike, The Relations between the Different Abilities Involved in the Study of Arithmetic, Columbia University Contributions to Philosophy, Psychology, and Education, ix. No. 2, 38. o/ the Fourth, Fifth, and Sixth School Grades 35 Test III TABLE XII Frequency of Abilities by Grades Grade 4A sB 5A 6B 6A AbiUty B GBGBGBGBG 9 2 10 I II 12 I 13 2 14 IS I I 16 I ^l I 18 3 3 I I 19 I 2 4 I 20 I 3 2 2 I 21 5 3 I I I I 22 2 6 3 2 3 23 3 7 2 I 2 2 I 24 3 3 I 4 4 3 2 2 25 12 4 6 I 2 I I 26 3 5 4 4 I 2 2 2 I 27 6 4 2 6 4 2 I 4 28 7 4 7 II 2 2 7 5 3 3 29 II 9 6 9 4 S s 6 2 3 30 4 6 2 II 3 4 8 5 4 4 31 5 4 6 9 12 II 8 4 2 5 32 3 6 10 12 7 5 9 10 s 5 33 3 7 S S 2 S 13 12 8 8 34 7 6 3 3 9 7 10 3 9 6 3S 6 3 II 3 S I 13 10 6 4 36 2 I I 2 2 3 10 7 4 9 37 I I 2 2 5 6 4 6 38 2 I 4 3 3 Cases 93 88 79 86 65 58 98 81 50 59 TABLE XIII Median Ability and Variability for Each Grade 11. [■s m% ;her Q's Co'ts. of Var. Gh-ades Boys Girls Boys Girls Boys Girls Boys Girls 4A 27-3S 27.50 -IS 3-36 4-37 .12 -IS SB 29-7S 29.36 -39 3-73 2. 1 1 .12 .07 SA 3°-S4 30-36 .18 3.19 V!.62 .10 .08 6B 32.46 3I-7S -71 2.38 2.9s .07 .09 6A 33-11 32-81 -30 1.72 2.50 •OS .07 Combining the Half Years of Grades 5 and 6 : 5 30-14 29.86 .28 3.46 ^.36 .11 .08 6 32.78 32.28 .50 2.05 2.72 .06 .08 For all 30.54 29.61 .93 3.25 3.40 .16 .11 36 The Reasoning Ability of Children Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 5B sA 6B 6A Boys 17.20 22.77 27.68 42.85 52.00 Girls 19.31 17-43 29.30 46.89 60.98 The diminished ability of the girls in grade 5B, and the rela- tive superiority of the girls throughout excepting in this grade, are the most significant features of the array for this selected group. Sex Differences by Grades Per Cent, of the Boys Reaching the 50 Per Centile of the Girls Grade 4A 5B 5 A 6B 6A Per Cent. 48.9 50.9 53-3 58-3 55-2 TABLE XIV Frequency OF Abilities BY Ages Age 8- -10 10- -II II- -12 12 -13 13- -14 14- -16 \bilil ty B G B G B G B G B G B G 9 I I 10 I II 12 I 13 I I 14 I IS I 2 I I I 16 I I I 17 2 I 3 2 18 2 I 2 I 2 3 I 2 19 I 2 I 4 20 2 I 2 I I I I I 21 I I 3 I I I 2 I 22 4 I 2 I 2 I I I I I 23 3 2 2 2 3 I 2 I 2 24 I I 4 3 I 4 2 I 2 2 25 2 7 5 3 6 2 2 I 26 2 I 2 2 3 6 2 4 I I 'I I I 2 I S 7 3 4 2 I 2 28 2 2 S 4 6 8 7 6 4 I 2 I 29 S 4 5 9 8 8 4 S 6 3 2 3° 3 4 9 3 4 12 II 4 2 31 2 4 8 4 10 12 9 4 3 6 I I 32 2 7 ? 9 S S 12 10 6 2 2 33 3 I 6 7 7 9 9 12 2 4 3 34 I 4 10 8 13 4 6 4 4 2 4 I 35 2 i: II 5 IS 3 6 3 s 4 2 36 I I 3 4 4 6 S S 2 2 4 I 37 I 4 4 4 3 2 4 2 [ I 38 I — I 4 1 2 2 I I Cases 27 43 84 81 loi 93 96 93 S3 46 24 16 of the Fourth, Fifth, and Sixth School Grades 37 TABLE XV Median Ability and Variability for Each Age M's Higher Q's Co'ts.ofVar. Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 24.50 27.00 2.5° 1 .00 8.25 .04 .30 9 " 9-6 29.50 27.50 2.00 3.50 4.56 •!! .16 9-6 " 10 28.83 29.75 .92 2.31 3.34 .08 .11 10 " 10-6 31.00 31.00 4.30 2.83 .14 .09 10-6 " II 30.7s 29.64 I. II 3-07 2.64 .10 .09 11 " 11-6 31.12 29.50 1.62 3.60 3.10 .11 .10 11-6 " 12 32.00 28.90 3.10 2.86 2.76 .09 .09 12 " 12-6 30.37 30.16 .21 2.68 3.13 .08 .13 12-6 " 13 30.50 29.00 1.50 ;i .40 4.29 .07 .14 13 " 13-6 28.70 29.33 -63 4-52 4.56 -IS -IS 13-6 " 14 31.00 31.00 4.00 6.66 .13 .21 14 " 14-6 33-37 27.50 5.87 2.16 5.00 .06 .18 14-6 " 15 29.50 31.50 2.00 2.25 7.75 .07 .24 15 " 16 32.50 27.50 5.00 6.50 2.25 .20 .oS Combining Half Years and Smaller Groups ; 8 to 10 28.70 28.87 ■'^1 3-27 4-27 .11 .14 10 " II 30.87 27.12 3.7s 3-25 2.76 .10 .10 11 " 12 31-30 29.37 1.93 3.28 3.12 .10 .10 12 " 13 3°-44 29.59 .85 2.67 3.41 .08 .11 13 " 14 28.91 30.00 1.09 4.40 4.81 .15 .16 14 " 16 33.00 28.00 5.00 3-5° 4.00 .10 .1^ Combining Further the Extreme Groups : 8 to II 30-55 29.75 -80 3.35 3.42 .16 .11 13 " 16 3°-87 29.50 1.37 3.92 4.62 .12 .15 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-11 11-12 12-13 13-16 Boys 27.92 39.60 22.90 35.04 Girls 28.22 29.02 30.10 29.01 The rhythm for boys in this selected group is very apparent, as is also the general rise for the girls to the 12-13 y^^^ period with the subsequent fall. The maximum for the girls is one year later than for the boys. 38 The Reasoning Ability of Children Comparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XVI Youngest and Oldest Pupils of Grades A Compared Boys Grade 4A Younger 25% Older Whole Grade Median Age Yrs. Mo. 9 8.0 12 9-5 II -S M's 26 .00 27 .00 27-3S High Younger er Older 1 ,00 Q's 1 5 -SO 3-50 3-36 Co'ts. 3f Var. .21 •13 .12 SA Younger 25% Older " Whole Grade 10 13 u 2-5 ■5 9.2 30.16 30.16 30-54 1-93 3-35 3-19 .06 .11 .10 6A Younger 25% Older Whole Grade II 14 12 i-S 5-S 4.0 32.5° 33-50 33-11 I .00 1 .00 1.58 1.72 •03 -05 •05 Girls 4A Younger 25% Older Whole Grade 9 12 10 4.0 3-S 7.8 25-50 24.50 27.50 1 .00 4.21 S-25 4.37 .16 .21 •IS SA Younger 25% Older Whole Grade 9 12 II 10. 5 6.5 4.0 31-25 30.16 30.36 1 .09 2.28 3.16 2 .62 -07 .14 .08 6A Younger 25% Older Whole Grade II 14 12 .0 6.5 7-S 31-50 31-33 32.81 -17 8.50 3-25 2.50 .27 •13 .07 In this test, the pupils of the younger groups equal or exceed the older pupils in median grade in all but the 25 per cent, for boys in grades 4A and 6A. In these grades, the superiority of the older groups is very small, less than 4 per cent. None of these groups reaches the median ability of its whole respective grade excepting the younger girls in grade 5A and the older boys in grade 6A. of the Fourth, Fifth, and Sixth School Grades 39 VA 5S SA tB 6A 15 ?-/<» loii I in a- 13 in*- fhih 16 'TK~T% ZX^ f-IO 18 t-n ii-a u-n ii-tt. 20 Figures 11-20. Test III II. Frequencies of Abilities, Boys. 13. Median Ability, by Grades. IS. Variability, by Grades. 17. Per Cent, of Boys Reaching the so Per Gentile of Girls, by Grades. 19. Per Cent, of Pupils Reaching the 7s Per Centile of All, by Grades. — — — ^ Boys. 12. Frequencies of Abilities, Girls. 14. Median Ability, by Age. 16. Variability, by Age. 18. Per Cent, of Boys Reaching the so Per Centile of Girls, by Age. 20. Per Cent, of Pupils Reaching the 75 Per Centile of AU, by Age. Girls. 40 The Reasoning Ability of Children Sex Differences by Age TABLE XVII Per Cent, of the Boys Reaching the 50 Per Centile of the Girls Combining Half Years For Half Years and Sma ler Groups Age Per Cent. Age Per Cent. Age Per Cent. 8 to 9 00.00 12 to 12-6 52 .54 8 to 10 62 .96 9 " 9-6 75.00 12-6 " 13 75.67 10 " II 75 -oo 9-6 " 10 46.15 13 " 13-6 46.15 II " 12 64.36 10 « 10-6 62 .50 13-6 " 14 57.14 12 " 13 66.66 10-6 " II 63.46 14 " 14-6 86.66 13 " 14 49.06 II " 11-6 61 .go 14-6 " 15 40.00 14 " 16 83.33 11-6 " 12 78.94 15 " 16 75.00 In this test, more than 50 per cent, of the boys in each age group by half years reaches or exceeds the median ability of the respective groups of girls excepting in the three, 9-6 to 10, 13 to 13-6, and 14-6 to 15, and the small group, 8 to 9. In the larger grouping by years, the boys excel excepting for the year 13 to 14, but the difference here is very small, 49.06 per cent, of the boys reaching the girls' median ability. Differences Between Tests III A and III B In test III A, the filling of the blanks, the appropriate term must be recalled and written, while in III B, the correct term is given, the association requiring merely a sufficient recall of mean- ing to enable the pupil to select it from the two. Differences in difficulty are evident in the following table of results for the two tests. TABLE XVIII Median Ability and Variability of Tests IIIA and IIIB Boys Gir Is M's Q's M's Q's ! Grades IIIA IIIB IIIA IIIB IIIA IIIB IIIA IIIB 4A 11.72 15.25 2.85 1.44 11.25 iS-io 3.28 1. 41 SB 13.50 15.66 3.06 .99 13.21 15.68 1-74 1 .12 SA 14-31 15-73 2-08 .85 13.87 15.92 1-79 1.23 63 16.20 16.14 1.79 .78 14.9s 16.18 2 .06 1 .01 6A 16.60 16.13 1-50 .85 16.21 16.52 1. 91 .86 Averages 14.46 15.78 2.25 .98 13.89 15.88 2.15 1.12 of the Fourth, Fifth, and Sixth School Grades 41 Test III A is clearly harder throughout excepting in grade 6 A and grade 6 B for boys where it seems slightly easier. Both tests grow easier as we pass up the grades, but III A much more so proportionately than III B. By the time grade 6A is reached, the tests are practically equal in difficulty. Test III A is evi- dently one in which school training enters as a large factor. The percentage of those who in test III B reach or exceed the median ability of their respective groups in test III A is here given, although because of the coarseness of the scale its results are slightly misleading; they are too high. Grades 4A SB SA 6B 6A Averages Boys 88.17 89.87 89.92 55- 10 54.00 75.41 Girls 93.18 93.02 82.7s 87.65 66.00 84.52 Sex Differences between the Tests The foregoing indicates that test III B is proportionately easier for the girls than for the boys. Excepting in grade 4A, the girls are absolutely as well as relatively superior to the boys in test III B, while in test III A the boys are superior through- out, although the difference is very small. The following array of the percentage of boys who reach or exceed the ability reached by 50 per cent, of the girls impresses these differences: Grades 4A SB SA 6B 6A Averages Test III A 54-5 5I-I 54-8 63.8 57-8 56-4 Test III B 53-5 49-3 44.1 48.8 38.4 46.8 For both boys and girls, the variability is greater for test III A than for III B. Relatively, it is greater for boys in III A and for girls in III B. In both tests and for both boys and girls there is a diminution in variability as we pass from the lower to the higher grades. 42 The Reasoning Ability of Children Test IV TABLE XIX Frequency of Abilities by Grades Grade 4A sB sA 6B 6A Ability B G B G B G B G B G to 5 6 I 2 I S " 10 5 4 I 10 " IS I 3 I I I IS " 20 2 I I I 20 « 25 4 I I 's ;; 30 I I i 3 I I I 3° 35 2 2 3 35 40 4 I I 3 40 " 45 2 I I 2 I 45 :, 5° 3 I I 3 I I 5° " 55 I 2 I I 2 3 55 " 60 6 6 I 3 I 4 I 60 " 65 7 4 4 2 I 2 I I ^s ;: 70 5 4 6 S 6 3 S 2 70 " 75 S 4 4 7 I I 5 I 2 7S " 80 6 II 7 5 4 3 5 I I 80 " 85 3 8 9 9 7 4 6 4 I 5 8s « 90 8 8 8 9 6 6 7 7 2 3 9° 95 9 12 8 4 6 8 8 6 8 S 95 " 100 7 8 II 14 9 7 10 18 10 8 100 " 105 3 5 S 12 9 ? 18 IS II 14 105 " no 2 2 7 9 4 6 9 14 9 14 no " IIS I I 3 9 8 S 4 IIS " 120 I 3 I I Cases 93 88 79 86 65 58 98 81 SO 59 TABLE XX 12 .16 25.00 15.81 •37 •19 3-50 13.00 13-50 •IS ■IS s-'^s 14.83 12 .46 .16 ■13 4.58 12 .19 7-2S .12 •07 I-2S 5-94 7-45 •OS .07 Median Ability and Variability for Each Grade M's Higher Q's Co'ts. of Var. Grades Boys Girls Boys Girls Boys Girls Boys Girls. 4A 67.50 79.66 SB 85.50 89.00 SA 87.25 92.50 6B 95 -oo 99.58 6A 100 .00 loi .25 Combining the Half Years of Grades 5 and 6 : 5 86.37 90-75 4.38 13-91 12-98 -16 .14 6 97-50 100.41 2.91 9.06 7.3s -09 .07 For all 88.93 93-00 4-07 16.05 12.24 -18 .13 Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 5B 5A 6B 6A Boys 7.52 18.97 24.56 42.85 54.00 Girls 4.54 22.71 20.68 37.02 47.43 of the Fourth, Fifth, and Sixth School Grades 43 Here, in the opposites test, the girls diminish in ability in grade 5A and are strong, relatively, in 5B, while for test III they were weakest in grade 5B, in this selected group. Sex Differences by Grades Per Cent, of the Boys Reaching the 50 Per Centile of the Girls Grade 4A 5B 5A 6B 6A Per Cent. 35.8 43.0 43.0 41-0 44-5 This array shows plainly the superiority of the girls in median ability. But to tell the whole story, the preceding two arrays must be taken into account. Note that in every grade excepting 5B a larger percentage of the boys than of the girls reach the 75 per centile of all of each sex. Referring to the Q's in Table XX, and the array of results for the combined divisions of grades 5 and 6 immediately following, it is seen that the variability of the boys is, on the whole, much higher than that of the girls. Here the comparison of the sexes by the central tendencies for the large groups in which the girls are found to be superior to the boys conceals a sub-group as large as 25 per cent, in each large group in which the superiority is reversed in four cases of five. This enforces the necessity that, when used as bases of compari- son, median values, or other central tendencies, where variability is at all great, must be checked by careful interpretation of that variability, or the comparison may be misleading and quite erroneous beyond the limits of the probable error in each array. Stress upon central tendencies to the neglect of variability may conceal sub-groups within the larger group having quite oppo- site central tendencies. 44 The Reasoning Ability of Children TABLE XXI Frequency op Abilities by Ages Age 8- 10 10- II II— 12 12— 13 r 14 14- 16 Ability B G B G B G B G G B G o to 5 2 2 2 3 I 5 " 10 2 2 I I 2 I I lO « 15 I I 2 I I 2 IS " 20 2 I 2 20 « 25 I I 2 2 25 " 3° I 2 3 I 3 3° ;; 35 I 2 3 I 3S 40 2 3 2 I I 40 « 45 I 2 I I 2 45 " 50 3 I I I 4 I so to SS I 4 I I 2 55 " 60 2 I 4 4 3 2 3 2 2 I 60 " 65 I 3 3 2 3 6 3 I I 65 " 70 4 2 9 5 4 2 3 4 I 7° « P 3 3 3 3 I 3 4 3 5 I I " 80 2 4 5 5 8 6 3 5 S I 80 " 85 I 3 6 4 4 8 10 8 4 S I 2 85 ;; 90 5 8 5 10 II 7 7 5 5 I 9° 95 3 6 10 9 9 6 9 10 4 4 4 95 100 4 5 IS 13 9 12 12 12 3 12 4 E 100 " 105 2 5 10 II 12 13 19 17 I 6 2 2 los ;; no 3 4 8 II 9 9 2 IS 6 4 3 I no " 115 I I I 6 9 4 3 4 I I I I IIS " 120 I _ I I I I I Cases 27 43 84 81 loi 93 96 93 S3 46 24 16 TABLE XXII Median Ability and Variability for Each Age M' 's Higher Q'i 3 Co'ts. of Var. Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 12 .00 93.00 81 .00 4.50 47-50 .37 .51 9 « 9-6 99.00 85.00 14.00 15.00 14.00 .15 .16 9-6 " 10 84.50 91.50 7-50 13.75 9.7s .16 .10 10 " 10-6 93-5° 95.50 2 .00 19.50 12.50 .21 •13 10-6 " II 93-5° 95.50 ^ .00 10 .00 12.83 .10 .13 II " I 1-6 88.7s 88.00 •75 17.19 10 .62 .19 .12 11-6 " 12 89.50 89.50 16.25 15.68 .18 .17 12 " 12-6 89.50 94-75 5.25 17.25 11.43 .19 .12 12-6 " 13 87.50 98.50 II .00 10.83 10.50 .12 .10 13 ^ ;; 13-6 77-5° 92.50 15.00 19.00 16.00 .24 .17 13-6 14 76 .00 93.50 17.50 21 .50 19.00 .28 .20 14 " 14-6 89.50 91 .00 1.50 19.75 4.50 .22 .05 14-6 " IS 94.50 105.50 II .00 22.75 12.7s •33 .12 IS " 16 98.50 76.50 22 .00 3.50 28.75 .03 .37 of the Fourth, Fifth, and Sixth School Grades TABLE XXII— Continued Combining Half Years, and Smaller Groups : 45 8 to lo 90.50 89.7s •75 19.18 11-43 .21 .12 lO " II 93-5° 95-5° 2 .00 II .40 11.83 .12 .12 II « 12 89.25 88.83 .42 16.85 13-56 .18 -IS 12 ;; 13 88.50 97.25 8.75 16.50 10.81 .18 .11 13 14 77-75 90 -oo 12 .25 19.12 16.18 .24 •17 14 " i6 93.00 91.00 ^ .00 18.50 21 .50 .19 .23 Combining Further the Extreme Groups : 8 " 10 91.83 93.00 1. 17 12 .56 12 .50 .136 •134 13 " 16 81.50 91.00 9.50 20 .81 16.18 •25 .17 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-1 1 11-12 12-13 13-16 Boys 29.72 32.67 27.06 19.47 Girls 24.99 25.80 31.17 19.43 The rhythm for both boys and girls, the general diminution in ability for both, the general superiority of the girls over the boys, and the greater variability of the boys are all quite evident in the foregoing distributions. In the array for the percentage of the age groups reaching the 75 per centile of all, the maximum for this selected group is found in the age period 11 to 12 for boys, and one year later for girls. After the maximum, both boys and girls decline in ability rather markedly, the percentage showing superior ability being less in the age period 13 to 16 than at any other age, and almost exactly the same for both sexes. Comparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XXIII Youngest and Oldest Pupils of Grades A Compared Boys Jradf Median Age Yrs. Mo. Higher M's Younger Older Co'ts. Q's ofVar 4A Younger 25% Older Whole Grade 9 8.0 12 9.5 II .5 74.50 55 -oo 67.50 19.5° 39.00 .52 19-00 .34 25.00 .37 5A Younger 25% Older Whole Grade 10 i! .5 13 -5 11 g.2 93.50 76.50 87.25 17 .00 13.68 .14 22.94 .29 14-83 .16 6A Younger 25% Older Whole Grade 11 1-5 14 5-5 12 4 .0 99 .00 98.50 100 .00 •5° 5-75 .58 5.25 -53 5.94 .05 Girls Younger 2 5% Older Whole Grade 9 12 10 4.0 83.50 3.5 69.00 7.8 79.66 14.50 9 .00 25-50 15.81 -17 -37 .19 Younger 2 5% Older Whole Grade 9 12 II 10.5 100.50 6.5 87-50 4.0 92.50 13.00 10 .62 11 .19 12 .46 .10 •13 -13 Younger 25% Older Whole Grade II 12 0.0 103.50 6.5 95-50 7-5 101.25 8.00 5-25 9 .00 7-45 -05 .09 .07 46 The Reasoning Ability of Children TABLE XXIII— Continued 4A 5B 6A The superiority of the younger group over the older is marked in this test for every grade. But this superiority is greatest in grade 4A and progressively diminishes as we pass upward. Tliis is in harmony with what we should expect from the array for the percentage of each age group who reach the 75 per centile of all. The marked superiority of the younger groups in the lower grades in contrast with the very greatly diminished superi- ority in upper grades would indicate that this test is one which clearly reveals a certain type of native ability. This ability becomes concealed in upper grades where the validity of these particular opposites as a test becomes very small because of school training and experience. With these easy opposites, we reach a point where good and poor pupils make grades about equally good. Here, the girls are superior to the boys, also, in every group excepting that of the older pupils of grade 6A. The sex difference is greater for the group of older pupils except in grade 6A. In no case does the older group of pupils reach the median ability of its grade as a whole, while all of the groups of younger pupils except that of the boys in grade 6A reach or exceed the median ability of their respective grades. In general, variability is greater for the older groups, for the lower grades, and slightly so for the boys. of the Fourth, Fifth, and Sixth School Grades 47 3. to ao ^ 40 ffo too no lo %o v« 40 90 too izo ai 22 6A t-lo ■2* it-/i *A 5B 5A 6B ^A 29 ^^ 3S- f-// //-/A /iV3 /3-« 30 Figures 21-30. Tbst IV 21. Frequencies of Abilities, Boys. 23. Median Ability, by Grades. 25. Variability, by Grades. 27. Per Cent, of Boys Reaching tbe so Per Gentile of Girls, by Grades. 29. Per Cent, of Pupils Reaching the 7S Per Gentile of AU, by Grades. -^^-^— Boys. 22. Frequencies of Abilities, Girls. 24. Median Ability, by Age. 26. Variability, by Age. 28. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Age. 30. Per Cent, of Pupils Reaching the 7? Per Centile of All, by Age. Girls. 48 The Reasoning Ability of Children Sex Differences by Age TABLE XXIV Per Cent, of the Boys Reaching the 50 Per Centile of the Girls Combining Half Years For Half Years and Smaller Groups Age Per Cent. Age Per Cent. Age Per Cent. 8 to 9 00.00 12 to 12-6 42.37 Stoic 51 -85 9 " 9-6 66.66 12-6 " 13 24.32 10 " II 41-66 9-6 " 10 38.45 13 " 13-6 28.20 II " 12 51.48 10 " 10-6 46.87 13-6 " 14 21.43 12 " 13 31-25 10-6 " II 38.46 14 " 14-6 46.66 13 " 14 32.08 11 " 11-6 55.55 14-6 " 15 20.00 14 " 16 58.33 11—6 " 12 52.63 15 " 16 100.00 In but four of the fourteen respective age groups, 9 to 9-6, II to 11-6, 11-6 to 12, and 15 to 16, do 50 per cent, of the boys reach the median ability of the girls. These periods, it may be noted, coincide fairly closely with the crests of the rhjrthm observed in the several arrays of the boys' records for age dis- tribution. From 12-6 to 14 is a period of especially low relative success for the boys — the period of especial superiority for the girls in several tests. Summary for Controlled Association Tests Grade Differences In both tests III and IV, there is general progress from grade to grade in both sexes. The greatest relative progress in both tests is from grade 4A to grade 5B. In general, there is also diminishing variability in passing upward from grade 4A. Median ability and variability are shown for the respective tests in Tables XIII and XX. The gain in per cent, for test III is, from grade 4A to grade 6A, 21.13 ior boys, 19.30 for girls; for test IV, 48.14 for boys, 26.97 fo^" gi^s. This is a much smaller gain in median ability than was made in the mathematical tests in every case. In the selected groups from the grades of those reaching the 75 per centile of all, pages 36 and 42, test III is seen to be much easier relatively for a larger portion of the lower grade pupils than is test IV, especially so in grade 4A. In the upper grades, the percentages are very similar — exactly the same for boys in grade 6B. In both tests, the high percentage of grade 6A who reach the 75 per centile for all indicates that of the Fourth, Fifth, and Sixth School Grades 49 both would probably be too easy to be valid as tests much beyond this grade. Age Differences The general statement for age differences for tests I and II holds fairly applicable for test III. Progress with age is rather irregular. A rhythm is observed for boys with crests at about 9 yrs. 6 mo., 12 yrs., and 14 yrs., each crest a little higher than the preceding. For the girls a more distinct rhythm is observed than in tests I and II. Its crests are fairly clear at 10 yrs., 12 yrs., 13 yrs. 6 mo., and 14 yrs. 6 mo., almost coincident with the rhythm of the boys at 12 yrs., but not so in earlier or later years. In the condensed form of age distribution immediately following Table XV, the very small changes with age are apparent. The group of boys from 13 yrs. up is very little superior to that of from 8 to 11 yrs., while the oldest group of girls is slightly below the youngest group in median ability. Reference to Table XVI indicates further these differences of groups within the grades. These small differences appearing in the age distributions in contrast with the more definite increase from grade to grade suggest that school training rather than life experience or native difference in capacity is a large factor in determining success in the test. In general, a pupil who is 14 does almost no better than one who is 11, but one who has been to school six years does better than one who has been to school but four years. In test IV, there is a marked rhythm in the age distribution, but with a general diminution in ability for boys, taking results as a whole, and very little increase for girls. In the results for boys. Table XXII, crests of the rhythm appear at 9 yrs., 10 yrs. 6 mo., 12 yrs., and 14 yrs. 6 mo. But the second and third crests are lower than the first ; the third is lower than the second ; and the fourth is no higher than the first. In the closer distribu- tion into four groups, there is diminution from group to group, the lowest result for boys being for the year 13 to 14. For the girls, the rhythm shows crests at 9 yrs., 10 yrs. 6 mo., 12 yrs. 6 mo., 13 yrs. 6 mo., and 14 yrs. 6 mo. The closer distribution, combining half years and smaller groups, shows this rhythm more central in the groups at 10 yrs. 6 mo., 12 yrs. 6 mo., and 14 yrs. 6 mo., intervals of two years. The grade for the year so The Reasoning Ability of Children 12 to 13 is especially high, exceeded only by the small group from 14 yrs. to 14 yrs. 6 mo. The parallelism of the rhythm for girls in this test and test III is quite marked as it is also for boys from 12 years. But in test III, up to 14 years, the boys are best from II to 12, in test IV from 10 to 11. Girls are best in both from 12 to 13. In variability, the older children of both sexes are slightly higher in test III, the younger in test IV. Sex Differences By the grade distribution, the boys are superior to the girls in median ability in every grade except 4A, in test III, as is shown in Table XIII ; and for test IV, as seen in Table XX, the girls are superior in every grade and most so in grade 4A. By the age distribution, practically the same results are in evi- dence, Tables XV and XXII. The arrays showing the percent- age of boys who reach or exceed the median ability of the girls, pages 36 and 43, for the grade distribution, and Tables XVII and XXIV for the age distribution, indicate this more definitely. In variability, the Q's of the girls are higher than those of the boys in 9 of the 14 half year groups for test III, Table XV, while for test IV, Table XXII, the boys are higher in 11 of the 14 half year groups. In grade distribution, the girls have a higher variability than the boys in 3 of the 5 grades in test III, Table XIII, while in test IV, Table XX, the variability of the boys is higher in 4 of the 5 grades. This variability asserts itself here in an interesting way. While it is true that in median ability 4 of the 5 grades show the boys to be superior to the girls in test III, reference to the array giving the percentage from each grade who reach the 75 per centile of all, page 36, will show that in all of these grades but 5B a larger percentage of girls than of boys is found in this highest quartile of ability. Similarly for test IV, although in all of the 5 grades the median ability of the girls is higher, in all of them but 5B there is a larger percentage of boys than of girls who are in the highest quartile of ability. This but intensifies the unreliability of the central tendency of a large group as a valid index of the quality of the group measured unless it is checked by an adequate interpretation of the variability of the group. As social capital and as social problems, the most important individuals or sub- of the Fourth, Fifth, and Sixth School Grades SI groups of individuals in large groups may often lie beyond the limits of the probable error. The greater the variability the more important is its adequate interpretation. Selective Judgment — Tests V and VI The two tests, V and VI, in selective judgment are treated separately throughout. Though the tests in selecting good rea- sons from among those given to account for the stated conditions seem very much like those in selecting good definitions from among those given for certain terms, and though many resemblances are shown in the results, a number of differences are also evident in the arrays of summaries. Test V TABLE XXV Frequency OF Abilities by Grades Grade 4A 5B SA 6B 6A Ability B G B G B G B G B G -20 to -18 I -18 ' -16 -16 ' -14 -14 ' -12 I I I I -12 ' ' -10 r I I I 2 -10 ' -8 I 4 3 2 I I -8 ' -6 I 5 2 4 3 4 -6 ' -4 I 3 3 3 3 I I -4 ' —2 17 9 3 6 4 4 I 2 -2 u 9 II 7 7 4 4 6 I 2 ' 2 13 10 6 10 4 4 3 6 3 I 2 4 8 13 8 10 6 3 6 6 I 4 ' 6 6 9 9 4 4 I 2 I 2 4 6 ' 8 8 S 2 6 2 2 6 2 I I 8 ' 10 6 2 2 4 2 2 I 3 2 10 ' 12 3 3 2 6 4 3 2 6 I 3 12 ' ' 14 2 I 4 2 2 2 I 3 2 14 ' ' 16 3 2 I 4 I 2 7 3 16 ' ' 18 I I s 3 5 4 S S 18 ' ' 20 2 2 2 4 4 4 2 4 20 ' 2.2 I I 2 3 6 2 s 3 22 ' ' 24 3 4 3 2 3 I 8 4 3 6 24 ' 26 2 4 2 3 6 II 5 4 4 26 ' 28 2 7 2 I 3 6 8 5 7 28 ' ' 3° 2 I 6 4 4 2 7 9 8 8 30 ' 32 2 4 2 3 2 6 5 2 6 32 ' 34 I I I 2 6 2 2 3 34 ' ' 36 I I I 3 I 2 36 * ' 38 I Cases 93 88 79 86 65 58 98 81 50 S9 52 The Reasoning Ability of Children TABLE XXVI Median Ability and Variability for Each Grade M's Higher Q's Co'ts. of Var. Grade Boys Girls Boys Girls Boys Girls Boys Girls 4A 2.62 2. II .51 5.57 4.12 2.12 1. 95 5B 5.90 4.00 1.90 12.28 6.83 2.08 1.70 5A 9.7s 14.00 4-2S 9-41 11.25 -96 -So 6B 21.50 17.50 4.00 10.31 10.53 -47 -60 6A 23.00 24.16 1. 16 7.29 5.94 .31 .24 Combining the Half Years of Grades 5 and 6 : 5 7.82 9.00 J.. 18 10.84 9-04 1.38 1. 00 6 22.25 20.83 1.42 8.80 8.23 .396 .395 For all 10.93 8.14 2.79 11-79 n-Si i .07 1.41 Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 56 5A 6B 6A Boys 7.52 29.09 16.91 39.79 42.00 Girls 7.9s 13.94 17.58 39.48 55.90 In this test, the large proportion of boys of grade 5B in the selected group is noticeable. In the two B grades, the boys are superior in numbers for this highest quartile while the girls are highest in the three A grades. Sex Differences by Grades Per Cent, of Boys Reaching the 50 Per Centile of the Girls Grade 4A 56 5A 6B 6A Per Cent. 52.2 60.7 38.1 55.1 47.6 Here, as in the selected group above, the relative superiority of the boys is greatest in grade 5B. In grades 5A and 6A, the girls are superior. of the Fourth, Fifth, and Sixth School Grades S3 TABLE XXVII Frequency of Abilities by Ages Age 8- 10 10- II II- 12 12- 13 13- 14 14- 16 Ability B G B G B G B G B G B G -20 to -18 I -18 " -16 -16 " -14 -14 " -12 I I I I -12 " —10 I 2 I I I -10 " -8 2 I 4 I 2 I I -8 " -6 I 2 I 3 I 5 3 2 2 -6 « -4 2 I 4 3 I 2 I I -4 " -2 3 7 13 4 6 3 3 4 4 I -2 " u I 3 4 6 4 7 10 4 8 2 I " 2 3 4 4 9 9 7 6 7 2 4 4 I 2 " 4 I 3 6 S 6 7 7 II 8 5 I I 4 " 6 I 2 4 2 6 6 8 4 2 4 2 6 " 8 2 5 2 7 2 S S I 3 I 2 8 " 10 3 3 3 I 4 2 3 I I 10 " 12 3 3 5 3 4 2 3 4 S I I 12 " 14 3 I 4 2 3 I I I I 14 " 16 I 2 2 2 3 5 7 2 I I I 16 " 18 I 3 2 S 3 3 4 2 18 " 20 3 3 2 4 3 3 3 I 4 I 20 " 22 I 2 2 4 3 4 4 22 " 24 2 I 7 3 4 7 2 5 4 3 24 " 26 2 2 S 5 S 4 8 4 I 2 [ I 26 " 28 2 I 3 4 7 4 3 8 3 2 2 I 28 " 30 2 2 6 4 6 7 6 7 3 2 I 2 3° ;; 32 2 I 5 3 8 4 7 4 2 3 ^'^ 34 I I I I 4 3 2 2 2 I 34 " 36 3 I 2 2 3 36 " 38 I Cases 27 43 84 81 loi 93 96 93 S3 46 24 16 TABLE XXVIII Median Ability and Variability for Each Age M' s Higher c 's Co'ts. of Var Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 3-5° 10 .00 6.50 5-50 16.50 1-57 1. 6s 9 '• 9-6 9-5° i-So 8.00 12 .50 9 .16 ^■31 6 .10 9-6 10 17-50 S.oo 12 .00 13-25 8.68 -75 1-73 10 " 10-6 9 .00 6.50 2.50 15.00 II .00 1.66 1 .69 IO-6 " II 7-5° 9-25 1-75 9.66 12 .72 1.28 1-37 11 " 11-6 9-5° 12 .00 2.50 II .46 11.87 1 .20 .98 11-6 " 12 16.00 10-75 S-2S 11-75 12 .16 -73 1-13 12 " 12-6 I4-S° 16. SO ■z .00 II .70 12 .92 .80 -77 12-6 " 13 14.25 7 .00 7-25 10 .24 8.94 ■''I i.27 13 ^ ;; 13-6 7-5° 9 .00 1-50 11.85 8.54 1.58 • 94 13-6 " 14 II .00 10. so •50 10 .00 13-50 .90 1.28 14 " 14-6 7-So II .50 4.00 "-7S 4 .00 1.56 •34 14-6 " IS 15-5° 26.50 II .00 5-25 14.75 -34 ■55 IS " 16 6. so 22.50 16 .00 17 .00 7-93 i=.6i ■35 54 The Reasoning Ability of Children TABLE XXVIII— Continued Combining Half Years, and Smaller Groups : 8 ' 10 9.83 2.16 7.67 12.87 8.91 1.30 4.11 lO ' II 8.00 8.50 .50 12.66 11.89 i-S^ 1. 31 II ' 12 12.37 11-25 1. 12 12.25 II-94 -99 1 .06 12 ' 13 14.33 11-50 2.83 11.26 12.00 .78 X .04 13 ' 14 9.50 10.33 .83 10.73 8.25 1. 12 •79 14 ' 16 10.50 17.00 6.50 11.00 8.50 1.04 Combining Further the Extreme Groups : •50 8 ' II 9.30 5.00 4.30 12.58 11.90 1.35 2.38 13 ' 16 9.75 11.00 1.25 10.82 10.44 i.IO •95 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-1 1 11-12 12-13 13-16 Boys 28.82 31.68 28.10 20.76 Girls 17.74 27.95 32.25 30.62 The foregoing arrays indicate rhythms of ability for both boys and girls; the maximal ability for boys in the year 12 to 13; a high plateau for girls from 11 to 12-6, then a decline, then a maximal high plateau from 14 to 16; high variability for both boys and girls; for the higher quartile of ability, a maximum of boys from 11 to 12, of girls from 12 to 13 ; a larger percentage of boys in the higher quartile up to 12 years, of girls from 12 upward, and smaller percentages for both from 13 to 16 than from 12 to 13; and suggests, in connection with Table XXVI, that this test is a measure of both native ability and school training. Progress through the grades makes for better results ; having good ability, though young, makes for good results; but merely growing older does not make for better results. Cbmparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XXIX Youngest and Oldest Pupils of the A Grades Compared Boys Grade Median Age Yrs. Mo. M's Higher Younger Older Co'ts. Q's of Var. 4 A Younger 25% Older Whole Grade 9 8.0 12 9.5 II •S 2.50 2.50 :..62 7.12 2.84 4.00 1.60 5 .11 2.12 5 A Younger 25% Older Whole Grade 10 2.S 13 -S 11 9.2 7-7S 9-7S 9-75 2 .0 12.18 1.57 10.25 '^ •°i 9.41 .96 6 A Younger 25% Older Whole Grade 11 1. 5 14 5-5 12 4.0 28.50 18.00 23.00 10 ■50 6.33 .22 8.50 .47 7.29 .31 of the Fourth, Fifth, and Sixth School Grades s S TABLE XXIX— Continued 4A Younger 2 s% Older Whole Grade 9 12 10 4.0 3-5 7.8 Girls •5° 2 .00 ^ .11 7-54 I .SO 2 .s6 4.12 15-08 1.28 I -95 SA Younger 25% Older Whole Grade 9 12 II 10.5 6.5 4.0 16. so 13-75 14 .00 2-75 9-93 II-7S II -2S .60 -85 .80 6A Younger 25% Older Whole Grade II 14 12 .0 6.5 7-S 27 .66 25-5° 24.16 ;..l6 3-5° 4.5° 5-94 .12 •17 -24 The younger groups here exceed the older in grade 6A for boys, and grades 5A and 6A for girls; for grade 4A the two groups are equal for boys ; and in grade 5A for boys and 4A for girls the older groups are superior. In grade 4A not one of the four groups reaches the median of ability for the whole grade; in grade 5 A, the older boys and the younger girls reach the respective grade medians ; and in grade 6A the younger boys and both groups of girls reach higher medians than the grade as a whole. In grade 4A for girls, and 5A for boys, the younger groups are more variable ; for the others, the older groups vary most. Here it is true that in every case where the younger group is superior in median ability the older group is higher in variability. This array tends to isolate groups on the basis of native ability, and also tends to establish a generalization to the effect that the higher the ability in a given test for a group in these school grades the lower the variability of the group for that test. Sex Differences by Age TABLE XXX Per Cent. OP Boys Reaching the so Per Centile of the Girls Combining Half Years For Half Years and Smaller Groups Age Per Cent. Age Per Cent. Age Per Cent. 8 to 9 SO .00 12 to 12-6 45-76 8 to 10 66.66 9 " 9-6 91-75 12-6 " 13 59-45 10 " II 45-23 9-6 « 10 61-53 13 ;; 13-6 48.72 II " 12 52-47 10 " 10-6 S6-2S 13-6 14 57-14 12 " 13 51-04 10-6 " II 48.07 14 14-6 40 .00 13 " 14 49 .06 II " 11-6 46 .03 14-6 " IS 20 .00 14 " 16 37-50 I 1-6 " 12 65-78 IS " 16 25 .00 In 7 of the 14 half year groups, fewer than half of the boys reach the median ability of the girls. These groups so alter- 56 The Reasoning Ability of Children M 5B M 6B 6A 39 s-ii ii-ii. is.-n i3-/t 40 Figures 31-40. Test V 31. Frequencies of Abilities, Boys. 33. Median Ability, by Grades. 35. Variability, by Grades. 37, Per Cent, of Boys Reaching the Per Centile of Girls, by Grades. Per Cent, of Pupils Reaching the Per Centile of All, by Grades. ^^^^^^ Boys. 39. 32. Frequencies of Abilities, Girls. ■ 34. Median Ability, by Age. 36. Variability, by Age, 38. Per Cent, of Boys Reaching the so Per Centile of Girls, by Age. 40. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Age. Girls of the Fourth, Fifth, and Sixth School Grades 57 nate that a distinct rhythm is observed, showing the relative ability of the boys to diminish progressively with age, the second crest and the second valley both being lower than the first crest and valley, respectively. Test VI TABLE XXXI Frequence OF Abilities BY Grades Grade 4A SB SA 6B 6A Ability B G B G B G B G B G -24 to —22 I -22 ' -20 -20 ' ' -18 I I I -18 ' ' -16 I 3 -16 ' ' -14 I I -14 ' ' —12 I I I -12 ' ' —10 2 2 I -10 ' -8 4 I I I 2 I I -8 ' ' -6 2 6 2 I I 2 -6 ' -4 2 4 2 4 I 3 I I -4 ' ' -2 8 5 3 4 4 3 I 3 I -2 ' ' IS 12 7 16 3 4 3 3 ' ' 2 6 4 6 8 8 3 ,S I 3 3 2 ' ' 4 10 8 8 II 6 7 4 4 2 4 ' ' 6 8 7 3 4 4 6 7 3 4 I 6 ' ' 8 2 8 8 5 3 6 6 S 4 I 8 ' 10 6 7 6 6 6 S 4 S 2 I 10 ' ' 12 5 6 7 2 2 I s s S S 12 ' ' 14 2 2 S 3 4 I 4 6 2 8 14 ' ' 16 2 3 2 6 4 3 s 7 I S 16 ' ' 18 5 4 3 2 6 14 7 I S 18 ' 20 3 I I 2 3 2 s 5 I 6 20 ' 22 2 2 2 2 S 7 8 4 6 22 ' ' 24 I 4 2 I 2 4 I I 24 ' ' 26 2 4 I 2 7 3 3 I 26 ' 28 I 2 I I I 3 3 4 28 ' 30 I I 2 I S 4 2 I 30 ' 32 I I I S 2 I 3 32 34 I I I I 6 3 34 ' 36 I 2 2 2 I I 36 ' ! 38 I I I 38 ' 40 I I I I 40 ' 42 I I 42 ' ' 44 I 44 ' 46 I Cases 93 88 79 86 6s 58 98 8t so S9 S8 The Reasoning Ability of Children TABLE XXXII Median Ability and Variability for Each Grade M's Higher Q's Co'ts. of Var. Grade Boys Girls Boys Girls Boys Girls Boys Girls 4A 2.90 3.33 .43 6.77 5.80 2.33 1.74 5B 7.50 3.28 4.22 6.29 6.06 .83 1.84 5A 7-50 6.00 1.50 6.84 7.25 .91 i .20 6B 16.50 15.75 .75 8.33 6.23 .50 .38 6A 14.00 17.37 3-37 i°-37 S-oS -74 .29 Combining the Half Years of Grades 5 and 6 : 5 7-50 4.64 ^.86 6.56 6.65 .87 1.43 6 15.25 16.56 i.31 9.35 5.64 .61 .34 For all 8.90 9.00 .10 8. II 7.84 .91 .87 Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 5B sA 6B 6A Boys 15.05 22.77 24.56 54.08 48.00 Girls 10.22 17.66 24.13 53.06 57-59 Here it will be noted that the maximum percentage of boys in the highest quartile is in grade 6B, of girls in grade 6A. For both sexes there is progress made from grade to grade excepting from 6B to 6A for boys where there is a loss. Sex Differences by Grades Per Cent, of the Boys Reaching the 50 Per Centile of the Girls Grade 4A 5B 5A 6B 6A Per Cent. 47.6 65.5 53.8 54.5 46.0 In median ability, the boys are superior to the girls except in grades 4A and 6A. In the higher quartile this superiority extends to grade 4A, also. Excepting in grade 5A, the variabil- ity of the boys is also greater than that of the girls, as shown by Table XXXII. In median ability as well as in ability in the highest quartile, the boys reach their maximum in grade 6B, the girls in 6A. Both boys and girls make a very marked advance from grade 5A to 6B. Boys are at a stand-still from grade 5B to 5A in median ability, and the girls lose in median ability from grade 4A to 5B, though in the highest quartile both gain for these respective periods. of the Fourth, Fifth, and Sixth School Grades 59 TABLE XXXIII Frequency of Abilities by Ages Age 8-10 lo-ii 11-12 12-13 13-14 '4-16 Ability BGBGBGBGBGBG ■24 to -22 ■2 2 (1 —20 •20 u -18 -18 u -16 -16 a -14 -14 u —12 -12 » -10 -10 it -8 -8 u -6 -6 u -4 -4 u -2 -2 u u 2 2 tt 4 4 u 6 6 u 8 8 a 10 10 tt 12 12 tt 14 14 u 16 16 tt 18 18 " 20 20 tt 22 22 tt 24 24 K 26 26 tl 28 28 It 30 30 tt 32 32 tt 34 34 tt 36 36 tt 38 38 It 40 40 tt 42 42 tt 44 44 tt 46 III 112 I 2 11 I 131^ 2 3 2 3 2 I I 521 42421 2 4155333111 3 5 10 13 5 7 8 7 3 3 434575644 22 43 10 10 82374513 45467833241 4286510542 I I I 4 5 3 8 9 8 4 I 4 I 13653444034 I 4157472213 64472923 I 2243 II 495241 3243262421 1134654823 21113322 2 5 51512211 1224313 ^ I 42212212 I 13221 I I II 11152 I I I 2 I 2 I I 2 Cases 27 43 84 81 loi 93 96 93 53 46 24 16 I 6o The Reasoning Ability of Children TABLE XXXIV Median Ability and Variability for Each Age M 's High er Q' s Co'ts. ofVar Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 7-5° 6. so 1 .00 9-So S-oo 1 .26 .76 9 ;; 9- 6 7-5° I. SO 6 .00 13.70 S-So 1.82 ■36 9-6 " 10 z -so 4.2s 1-75 8.2s S-i8 3-30 1 .21 lO " lO- ■6 8.50 4-5° 4 .00 7-53 9-S6 .88 2 .12 10-6 " II 9-5° 7-37 2.13 8.16 7-31 .86 •99 II " II- ■6 10 .50 9-75 •75 8.6s 7 .00 .82 ■71 II-6 " 12 10.00 9-5° ■SO 7-So 9.68 •7S 1 .02 12 " 12- -6 9.87 12 .12 2 .2S 7-2S 10.3s ■73 •8S 12-6 " 13 7-S° II. SO 4.00 8.25 8.44 1 .10 •73 '3 , ;; ^3- -6 9 -5° 14.00 4-50 8.75 7-SS .92 ■54 13-6 " 14 6. so 10 .50 4.00 8.00 9 -SO 1.23 .90 14 " 14- -6 II .2S 2.50 8.7s 6.68 7 -SO ■S9 3.00 14-6 ;; IS 4.75 12 .50 7-7S 2.44 14.25 ■SI 1. 14 IS " 16 7-50 12. so S-oo 20 .00 6.25 Z.66 •SO Combining Half Years and Smaller Groups : 8 to 10 3 -SO 3-5° 10.18 6.81 2 .90 1.94 10 « II 8.66 6.7s 1. 91 7-83 7-31 .90 1.08 II " 12 10. 2S 9.70 •ss 8.38 8. S3 .81 .88 12 ;; 13 9 .00 11.83 -.83 7-50 9.17 •83 •77 13 u 7.83 12 .00 4-17 8.S6 7.71 1 .09 .64 14 " 16 10 .00 10 .00 7 .00 7-83 .70 .78 Combining Further the Extreme Groups 8 to II 8.12 S-2S ..87 8.12 6.77 1 .00 X .29 13 « 16 8.25 12 .00 3-75 8.31 7-83 1 .00 .65 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-1 1 11-12 12-13 13-16 Boys 25.22 35.64 29.14 22.06 Girls 21.77 29.02 32.25 35.46 In the highest quartile, the girls progress with age throughout and are superior to the boys from 12 years upward; the boys reach their maximum in this quartile at 12, then decline to a lower percentage from 13 to 16 than from 8 to 11. In median ability, the rhythm is marked and corresponds closely to that noted in the mathematics tests. For the girls there is a less dis- tinct rhythm and also a general rise to a maximum in median ability at 13 yrs. 6 mo., then a slight diminution. The variabiUty is slightly greater for the younger groups. of the Fourth, Fifth and Sixth School Grades 6i dl -/a -6 *2 9 ly/^ iiHt *A SB SA 6B M 49 Figures Test 41. Frequencies of Abilities, Boys, 43- Median Ability, by Grades. 45. Variability, by Grades. 47. Per Cent, of Boys Reaching the so Per Centile of Girls, by Grades. 49. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Grades. — — ^^-^ Boys. ^ ^-// //-/A /li3 n-f(t 60 41-50. IV 42. Frequencies of Abilities, Girls. 44. Median Ability, by Age. 46, Variability, by Age, 48. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Age. SO. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Age. Girls, 62 The Reasoning Ability of Children Comparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XXXV Youngest and Oldest Pupils of Grades A Compared Grade 4 A Younger 25% Older Whole Grade 5 A Younger 25% Older Whole Grade 6 A Younger 25% Older Whole Grade 4A Younger 25% Older Whole Grade SA Younger 25% Older Whole Grade 6 A Younger 25% Older Whole Grade Median Age Yrs. Mo. 9 12 II 10 13 II 14 12 Boys M's -■25 .66 2 .90 8.25 ^■75 7-50 Higher Younger Older .91 5-5° 5 14-50 S 11-5° o 14.00 Girls 9 4.0 4.50 12 3.5 -.66 10 7-8 3-33 9 i°-S 7-2S 12 6.S 3.83 11 4.0 6.00 11 0.0 14 .50 14 6.5 12.50 12 7-5 17-37 3.00 S-16 3-42 Q's 7-25 S.83 6.77 6.94 9.18 6.84 9-75 9 .00 10-37 Co'ts. of Var. 29 .00 8.83 2-33 .84 3-33 .91 .67 .78 ■74 5.62 1.24 4.12 .62 5.80 1.74 8.25 S-94 7-25 9-7S S-oS 1-13 i-SS 1 .20 .68 .78 .29 In every case excepting for the boys in grade 4A, the younger groups reach a higher median ability than the older. From the younger groups, the boys of grades 5A and 6A, and the girls of 5A, reach a higher median than that for their whole respective grades, while none of the older groups do this. The variability is greater for the older group in every case but that of the boys for grade 4A. of the Fourth, Fifth, and Sixth School Grades 63 Sex Differences by Age TABLE XXXVI Per Cent, of Boys Reaching the 50 Per Centile of the Girls Combining Half Years For Half Years and Smaller Groups Age Per Cent. Age. Per Cent Age Per Cent. 8 to 9 50 .00 12 to 12-6 42-37 8 to 10 55-55 9 " 9-6 91-75 12-6 " 13 40-54 10 " II 52-37 9-6 " 10 38-45 13 " 13-6 35-89 II " 12 53-46 10 " 10-6 56.25 13-6 " 14 42.85 12 " 13 43-75 10-6 " 11 53-84 14 " 14-6 80.00 13 ;; 14 43-40 II " 11-6 50-79 14-6 " 15 20 .00 14 " 16 54-16 11-6 " 12 57-89 15 " 16 25.00 In 7 of the 14 groups, half or more of the boys reach or exceed the median ability of the respective groups of girls. The groups in which the boys are superior are all below 12 years but one. In general, for the younger pupils the boys are superior, for the older the girls, the dividing line being, for these groups, at about 12 years. However, the relative superiority of the sexes so alter- nates in the groups as to emphasize the rhythm already men- tioned in this test for both sexes. Summary for Selective Judgment Tests Grade Differences Reference to Tables XXVI and XXXII shows very marked progress in median ability from grade 4A to 6A, in test V a gain of 777.86 per cent, for the boys, 1045.02 per cent, for the girls, and in test VI a gain of 382.75 per cent, for the boys, 421.62 per cent, for the girls, the girls making larger gains in both tests. In both tests, there is a striking advance in ability from grade 5A to 6B excepting in girls for test V where the corresponding advance is from 5B to 5A. In test VI there is not always progress from grade to grade. For boys, the median ability is the same in grades 5B and 5A, and less in 6A than in 6B, while for girls there is a slight loss from grade 4A to 5B. That test V is more difficult than test VI is evidenced by the per cent, of each grade who reach the 75 per centile for all in the respective tests; in every grade for both sexes, the percentage reaching the 75 per centile is higher for test VI excepting for the boys of grade 5B. The variability, on the whole, is slightly higher in test V. Both of these tests are measures of acquired information and training as well as of native ability as is evi- 64 The Reasoning Ability of Children denced by the marked progress from grade 4A to 6A in contrast with the much smaller advance from 9 to 14 in age. Age Differences The progress from younger pupils to older, though consider- able, is not so marked as from lower grades to higher. The maximum of median ability, save for very small groups at the higher age extreme, is at about 12 years for boys, 12 yrs. 6 mo. for girls, in test V, and at about 11 yrs. 6 mo. for boys, and 13 yrs. for girls, in test VI. In both tests and for both boys and girls, a rhythm is very evident in the age distributions. While only a larger parallelism is noted in the two arrays for boys, the arrays for girls are parallel in much greater detail and both cor- respond closely to the array for girls in the mathematics tests. In general, it may be said that there is increased ability with age up to about 12 years for boys and 12 yrs. 6 mo. for girls, then a slight decrease until 14 for both. In the arrays for those of each age group reaching the 75 per centile of all, pages 54 and 60, the highest percentage of boys is seen to be in the group from 11 to 12, the lowest from 13 to 16, in both tests; for girls, the highest percentage is in the group from 12 to 13 for test V and from 13 to 16 in test VI, while it is lowest from 8 to 11 in both tests. A comparison of these arrays with those for the per cents, of each grade reaching the 75 per centile for these tests, pages 52 and 58, will impress how clearly the tests isolate native ability or successful school training. Forty- two per cent, of the boys of grade 6A reach the 75 per centile for test V, but only 20.75 P^'^ cent, of the boys who are from 13 to 16 do so. These percentages are computed on the same number of boys — all of those who took the test. In the com- parison of the youngest and oldest 25 per cents, of the three A grades, in all of the grades for both tests, excepting 4A for test V, the younger group of girls exceeds the median ability of the older; in all but grade 4A and sA in test V, and 4A in test VI, the younger boys exceed the median ability of the older, and in grade 4A for test V they reach it. Sex Differences As shown by the arrays on pages 52 and 58, in both tests over 60 per cent, of the boys reach or exceed the median ability of the girls in grade sB and about 55 per cent, in grade 6B, while o/ the Fourth, Fifth, and Sixth School Grades 65 in grade 6A in both tests the percentage is but about 47. In grade 4A the boys are superior in test V, the girls in test VI, while in grade sA the girls are superior in test V, the boys in test VI. In all of the grades excepting 4A, 5A, and 6A in test V, and 6A in test VI, a larger percentage of boys than of girls reaches the 75 per centile for all. In these grades in which the girls reach a higher percentage the difference is very small except in grade 6A in which, for both tests, the superiority of the girls is very marked. In general, on an age basis, the boys up to 12 years are superior, the girls from 12 years upward, for both tests. The superiority of the boys is greatest for both tests in the age period from 8 to 10 years, of the girls from 14 to 16 years in test V, and from 13 to 14 in test VI. For those reaching the 75 per centile of all, the percentage is highest for the boys in both tests from 11 to 12 years, lowest from 13 to 16 years; for girls it is highest from 12 to 13 years for test V and from 13 to 16 years in test VI, while it is lowest in both tests from 8 to 11 years. In variability, the averages for the sexes are about the same. For the girls, there is progressive diminution with age, while for the boys, variability is greater in the lowest and highest age groups. Intellectual Interpretation of Poems — Test VII TABLE XXXVII Frequency of Abilities by Grades Grade 4A 5B sA 6B 6A Ability B G B G B ' G B G B G 23 9 16 10 IS 6 II 3 S I 24 28 17 8 13 16 8 4 5 4 2 16 23 15 19 16 6 16 13 5 7 3 9 10 9 12 3 13 16 13 3 9 4 6 4 6 12 7 4 10 7 4 7 S I 3 I 8 3 7 12 12 10 S 6 4 6 4 3 2 I 9 8 4 7 7 4 3 4 3 I I 7 S 4 3 8 2 I 3 1 3 3 3 I 9 I 2 I 4 2 2 3 4 2 7 10 I 3 I I 2 2 4 2 4 II I 2 3 4 12 I I I 13 2 I 14 I 15 I I I 16 17 I 18 I Cases 93 88 79 86 65 58 98 81 50 59 66 The Reasoning Ability of Children TABLE XXXVIII Median Ability and Variability for Each Grade M's Higher Q's Co'ts. of Var. Grade Boys Girls Boys Girls Boys Girls Boys Girls 4A SA 6B 6A •98 ^^3° 1.43 :j.SO 1.28 2 .07 2.87 4.04 4.30 4-5° •32 i.37 1.07 1.58 •79 ^•57 1. 17 1.86 • 20 2.43 X.07 i.62 ^•54 2 .06 2.88 1-39 1 .10 1 .22 .64 •56 .82 .64 •74 •SO .64 Combining the Half Years of Grades 5 and 6: S 6 rail 1.3s 2.28 3^58 4^27 1.83 2.46 •93 i^S7 .69 2.14 .63 1.97 1.58 2.47 1.92 i.i6 •59 1.07 .69 •57 •77 Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 56 5A 6B 6A Boys 16.12 20.24 16.91 37-75 56.00 Girls 15.90 31.55 22.41 50.59 54.20 In this selected group, the same rise from grade 4A to 5B and fall from 5B to 5 A as seen in the median ability for the grades is noted for both boys and girls. Both advance through grades 6B and 6A. Sex Differences by Grades Per Cent, of Boys Reaching the 50 Per Centile of the Girls Grade 4A 5B 5A 6B 6A Per Cent. 44-30 37^50 31-90 37.20 46-00 In every grade the girls are superior to the boys in median ability. The boys make their highest percentages in grades 6A and 4A, in order. In the percentages of each grade who reach the 75 per centile of all, the boys exceed the girls in grades 4A and 6A but in none of the others. The greatest sex difference in median ability is in grade 5A, the smallest in grade 6A. of the Fourth, Fifth, and Sixth School Grades 67 TABLE XXXIX Frequency op Abilities by Ages Age 8 -10 10 -II II- 12 12 -13 13-14 14- i6 Ability B G B G B G B G B G B G 8 10 14 7 19 9 12 5 13 I 3 2 I 6 8 13 18 12 16 21 8 II 5 3 2 2 2 7 19 15 16 15 19 18 II 13 3 I 3 2 3 6 12 14 16 10 19 5 5 4 I 4 I 3 10 6 13 II 5 7 3 5 I I S I 3 6 6 8 6 4 9 5 7 3 3 6 5 4 4 3 3 " 6 2 3 3 4 7 2 I 2 2 7 5 8 5 I I 8 I [ 2 2 3 2 2 I I I 9 2 I 4 I 5 2 6 2 2 I 2 10 2 I 4 I 3 I 7 I II I I I I 3 I 2 I 12 I I 13 2 I 14 I IS I I I 16 17 I 18 I Cases 27 43 84 81 lOI 93 96 93 53 46 24 16 TABLE XL Median Ability AND Variability FOR Each Age M's Higher Q's Co'ts. of Var Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 S-So 3 00 2 .50 3.00 4-50 •54 1.50 9 ;; 9- -6 3.00 2 00 1 .00 3-33 2.66 1 .11 1-33 9- -6 " 10 .16 I 5° 1-34 .60 2 .06 3-75 1-37 10 " 10- -6 1.50 I 50 2 .02 1.72 1^34 1. 14 lo- -6 " II I .92 2 45 •53 1-57 1.83 .81 •74 ll " II- -6 ^.o^ -^ ss .48 1.83 i.6i .88 •63 II- -6 " 12 2 .42 •£ 21 .21 1.24 I .98 •51 .89 12 " 12- -6 1.86 2 86 I .00 2 .22 1-97 1. 19 .68 12- -6 ;; 13 1.68 2 75 1.07 2 .28 2.25 1^36 .81 13 '3- -6 1.08 3 00 I .92 1. 91 1.76 1.77 .58 13- -6 " 14 1 .40 2 66 i.26 J. .12 1.50 .80 •56 14 " 14- -6 2 .50 4 00 1.50 2 .12 2-37 ■93 •59 14- -6 " IS I-7S S 25 3-5° 2.44 •43 1-39 .08 15 " 16 6.50 2 50 4 .00 2.50 4.68 •38 1.87 Combining Half Years, and Smaller Groups : 8 " 10 .91 I 50 •59 i! .64 2.16 2 .90 1.44 10 " II 1.79 2 04 •25 I. 81 1.86 I .01 .91 II " 12 2.2s 2 41 .16 I .84 1.78 .81 •73 12 " 13 1.79 2 . 81 1 .02 2.26 2.03 I .26 •72 13 " 14 J. .22 ^ . 80 1.58 I .46 1.68 X.19 .60 14 " 16 2-75 4 33 1^58 2 .16 ■i .00 .78 .46 Combining Further the Extreme Groups : 8 " II 1 .69 I 86 •17 2 .01 1^95 1. 19 1 .04 13 " 16 1 .60 3 16 1.56 1-93 ■^ .84 I .20 .89 68 The Reasoning Ability of Children Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-1 1 11-12 12-13 13-16 Boys 27.02 27.72 29.14 25.96 Girls 28.22 27.95 38-70 41.91 In all of the foregoing arrays, the general superiority of the girls is in evidence. That this superiority increases relatively with age is apparent in the array for combined half years and extreme groups. It is also noted in the percentages of each group reaching the 75 per centile in which the girls are superior in every group but most so in the group from 13 to 16. Except in the extreme groups, the boys reach their maximum of median ability in the period from 11 to 12 years, the girls from 12 to 13 years. After the maximum is reached, there is a decline in median ability for both boys and girls, the ability for boys from 13 to 14 being less than from 10 to 11. For the girls there is a rhythm in ability, but each valley in it is higher than the preceding. Comparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XLI Youngest and Oldest Pupils op the A Grades Compared Grade Boys Median Age High Yrs. Mo. M's Younger er Older Co'ts. Q's ofVar 4A Younger 25% Older Whole Grade 9 12 II 8 9 5 S ■83 .90 .98 .07 1-57 I •73 1-37 I 89 81 39 SA Younger 25% Older Whole Grade 10 13 II 9 5 5 2 i.i6 1.28 ■34 1.68 I 1-33 ^•57 >■ 44 88 22 6A Younger 25% Older Whole Grade II 14 12 I 5 4 S 5 3-5° 4-33 4-3° •83 3-25 2-75 2-43 92 63 S6 Girls 4A Younger 25% Older Whole Grade 9 12 10 4.0 1-75 I. 25 1.30 •5° ^•15 .76 1.07 65 60 82 SA Younger 25% Older Whole Grade 9 12 II 10.5 6.5 4.0 2 .12 2 .16 2.07 .04 1. 41 1.72 '■•54 66 79 74 6A Younger 25% Older Whole Grade II 14 12 6 7 5 5 ^.66 4.66 4.50 i .00 :..25 2 .50 2.88 84 53 64 lit of the Fourth, Fifth, and Sixth School Grades 69 SI *^ SB 60 ^A 6B iA • J-// II-I3L /1-I3 li-lf^ 60 Figures 51-60. Test VII SI. Frequencies of Abilities, Boys. S3. Median Ability, by Grades. 55. Variability, by Grades. 57. Per Cent.fof Boys Reaching the so Per Centile of Girls, by Grades. 59. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Grades. — — ^— Boys. 52. Frequencies of Abilities, Girls. 54. Median Ability, by Age. S6. Variability, by Age. S8. Per Cent, of Boys Reaching the so Per Centile of Girls, by Age. 60. Per Cent, of Pupils Reaching the 75 Per Centile of All, bv Age. Girls. 70 The Reasoning Ability of Children In every case except that of the girls in grade 4A, in this test, the older group reaches a higher median ability than the younger. In general, this is in contrast to results for the other tests. The relative superiority of the older groups also increases with advance through the grades. The median ability for the grade as a whole is reached by the older groups of boys in grades 5A and 6A, and of the girls by all but the older group of grade 4A and the younger group of grade 6A. Sex Diflferences by Age TABLE XLII Per Cent. OF Boys Reaching the 50 Per Centile op the Girls Combining Half Years For Half Years and Smaller Groups Age Per Cent. Age Per Cent. Age Per Cent. 8 to 9 100 .00 12 to 12-6 47 .45 8 to 10 48.14 9 " 9-6 58.33 12-6 " 13 43-24 10 " II 45-23 9-6 " 10 30.76 13 " 13-6 35.89 II " 12 53-46 10 " I 0-6 59-37 13-6 14 28.57 12 " 13 45-83 I 0-6 "11 48.07 14 " 14-6 40.00 13 " 14 35-85 II " 11-6 50-79 14-6 "15 40.00 14 "16 41 .66 11-6 " 12 57-89 15 "16 100.00 In 6 of the 14 half year groups, more than 50 per cent, of the boys reach or exceed the median ability of the respective groups of girls. However, two of these are in the lowest and highest age groups, respectively, where the numbers are very small. All of these six higher percentages for boys are below the 12 year period but one. In the array combining half years, the boys show superiority in the one year, 11 to 12, only. It is also in this age period that the percentage of boys in the highest quartile is greatest, although even here the girls show a larger percentage for superior ability, page 68. Summary for Test VII Grade Differences The gain in per cent, from grade 4A to 6A is 338.77 for boys and 246.15 for girls. Reference to Table XXXVIII shows that for both boys and girls, the general progress from grade to grade is broken by a slight loss from grade 5B to sA, but that it is regained by an increased advance to grade 6B. The array oj the Fourth, Fifth, and Sixth School Grades 71 for the per cent, of each grade reaching the 75 per centile of all shows coincidence from grade to grade with the changes in median ability. The variability also decreases through the grades with the same slight fluctuation in grade 5A. Age Differences Following Table XL, it was pointed out that, excepting in the small groups at the extremes of the array, the boys reach a maximum in median ability in the period from 11 to 12 years, the girls from 12 to 13 years. Beyond the maximum, there is a decline in ability for both boys and girls, the ability for boys being less from 13 to 14 than from 10 to 11. The decrease is much smaller for the girls. Both sexes advance again after the decrease, reaching their highest ability in the small groups from 14 to 16 years. The highest percentage reaching the 75 per centile for all is in the age period 12 to 13 for boys, and 13 to 16 for girls. For the girls a larger percentage of this superior ability is found in the age period from 8 to il than from II to 12. The rhythm in the array for median ability is as evident as in the other tests. The more marked progress made through the grades is not so apparent in the age distri- bution. In general, boys of 13 do almost no better than boys of ID, but boys of the sixth grade do do better than those of the fourth and fifth grades. School training seems to be a factor in the test. However, in a given grade, the pupils in the oldest quartile do better than those in the youngest, as seen in Table XLI. The variability for both sexes is a little higher up to 12 years than from 12 years forward, the younger children vary more than the older in this test. Sex Dififerences In median ability, the girls are superior to the boys in every grade. In the age distribution, 6 of the 14 groups of boys show more than 50 per cent, reaching the 50 per centile of the respective groups of girls, but in combining the half years these percentages all fall below 50 except for the period, 11 years to 12. In the half year array, all of the higher percentages for boys are for periods under 12 years but one, that from 15 to 16. The sex difference is greater above 12 years than below. A higher 7 2 The Reasoning Ability of Children percentage of girls than of boys is found in the highest quartile of ability in all of the grades excepting 4A and 5A, and in all of the age periods. The percentage of the whole group of 385 boys who reach the median ability of the whole group of girls is 46.75. The variability of the boys is greater than that of the girls in every age group, and in every grade excepting 6A where it is slightly higher for the girls. For all of the boys taken together, the coefficient of variabiUty is 1.07, for the girls, .78. In the arrays for smaller groupings, there are no significant differences apparent. Array of Combined Results of the Seven Tests As a type of gross summary, from which too much must not be inferred, the sums of the results for the several separate tests have been taken and arrayed by the same distributions as the results of each separate test. These results will form one basis for comparisons of grades, ages, and sexes in the several abilities all taken together. of the Fourth, Fifth, and Sixth School Grades 73 TABLE XLIII Frequency of Abilities by Grades Grade 4A 5B 5A 6B 6A Ability B G B G B G B G B G 20 to 25 I 25 30 2 30 ' 35 2 I 35 ' 40 3 I I 40 ' 45 I 2 45 50 2 I 50 ' 55 3 I I 55 ' 60 I I I 60 ' 65 3 I I 65 70 4 I I 70 ; i^ 3 I 2 I 75 ' 80 3 1 I I I 80 ; 85 3 I 85 90 4 3 I 90 ; 95 5 2 I 2 2 I I 95 100 3 S 2 I 100 ' los 1 7 2 loS ' no 5 2 3 I 3 no ' 115 2 2 3 2 I I I "5 ' 120 I 7 6 6 2 2 4 I 120 ' 125 4 3 4 3 I 5 I I 125 ' 130 4 I 3 4 I 2 I I 13° ' 135 3 5 3 8 5 2 I 13s ' 140 s 7 3 7 5 2 140 ' 145 4 9 2 4 2 S 2 I I 145 ' 150 4 3 I 4 I 2 4 I I 150 ' 155 2 3 5 6 6 I 4 I 2 I 15s ' 160 3 4 4 3 I 2 2 5 I I 160 ' 16s I 5 4 3 4 2 2 2 I i6s ' 170 I I 2 5 4 3 8 5 3 I 170 " 175 2 I 4 2 4 3 3 3 3 4 17s " 180 I 2 3 3 3 3 6 4 4 180 " 185 I 1 4 3 I 2 4 7 3 3 185 " 190 I I I 4 2 3 5 5 I 3 190 ; 195 I 3 3 2 2 4 3 2 19s 200 2 4 2 5 2 6 6 3 3 200 • 20s 3 4 4 3 5 205 ' 210 2 I 3 2 I 5 7 4 I 2 210 ' 215 3 2 2 2 5 5 3 5 215 ' 220 2 I I [ 2 4 2 3 220 ' 22s I I I I I 3 I 6 225 ' 230 I 3 2 4 2 230 23s I I I 2 3 4 I 235 240 I I I 3 2 7 240 ' 245 I 2 I 4 245 ' 250 4 I I 250 ' 2S5 I 2 2 2 255 ' 260 2 I I 260 ' 265 2 I Cases 93 88 79 86 65 58 98 81 50 59 Grade Boys s Higher Girls Boys Girls 4A SA 6B 117.50 155-62 160 .62 187 .00 130.00 12.50 146.25 9.37 165.00 4.38 188 .50 I .50 6A 205.00 204.50 .50 74 The Reasoning Ability of Children TABLE XLIV Median Ability and Variability for Each Grade Q's Co'ts. of Var. Boys Girls Boys Girls 34.01 22.08 .28 .16 29.84 23.31 .19 .15 32.81 32.50 .20 .19 24.62 22.25 -13 -11 27.50 21.14 .13 .10 Porall 165.83 159.25 6.58 37.03 34.08 .22 .21 Per Cent, of Each Grade Reaching the 75 Per Centile of All Grade 4A 5B 5A 6B 6A Boys 9.67 18.97 19-95 42.85 62.00 Girls 2.27 10.45 24.13 45.65 64.37 For both sexes, there is progress from grade to grade, both in median ability and in the percentages reaching the highest quartile for all. A larger percentage of the boys reach the high- est quartile for all in grades 4A and 56, while the percentage is larger for girls in the upper three grades. Sex Differences by Grades Per Cent, of Boys Reaching the 50 Per Centile of the Girls Grade 4A 5B 5A 6B 6A Per Cent. 45.1 58.2 44.6 50.0 50.0 In grade 5B only are the boys decidedly superior. In grades 6B and 6A, the two sexes are practically the same in ability, just 50 per cent, of the boys reaching the median ability of the girls in each grade. of the Fourth, Fifth, and Sixth School Grades 75 TABLE XLV Frequency of Abilities by Ages Age Ability : 8-10 lO-II II-I2 1 2-1 3 y~' 4 1 4-1 6 G B G B G B G B G B G B 20 to 25 I 25 " 3° I I 30 " 35 I I I 35 "■ 40 I 2 I I 40 " 45 I I c I 45 " 5° I I 5° " 55 I I 2 £ 55 " 60 I 2 60 " 65 I I I I I 65 " 70 I I I 2 I 70 " 75 I 2 I I I I c 75 " 80 I I 2 I I I I I 80 " 85 I I I 85 " 90 I I 2 I 2 2 90 " 95 2 I I I 3 I 3 2 I I I 95 " 100 I I 3 3 2 I 100 " 105 I I I S I I 105 " no I 3 2 I 3 I I I no " 115 I 3 I I 5 I I 115 " 120 7 3 I 4 4 I 2 2 I 2 I 120 " 125 I 2 2 I 2 4 2 2 2 I I 2 125 " 13° I I 3 2 3 2 2 I I I 130 " 135 I 4 4 7 I 6 6 2 I 135 " 140 2 4 6 3 4 3 I I 140 " 145 4 4 4 5 5 6 I I I 145 " 150 2 5 I I 4 7 I I I 150 " 155 2 5 5 3 5 2 3 2 2 I 155 " 160 I 4 I 3 4 4 2 3 2 I I 160 " 165 I 4 3 4 2 2 3 2 3 165 " 170 I 5 2 3 8 5 4 2 3 2 170 " 17s 2 4 2 6 I 3 6 I I I 175 " 180 I I 2 2 2 4 7 4 3 I I 180 " 185 I I 4 2 6 6 4 3 2 I I 185 " 190 I 3 4 I 2 3 3 2 2 2 190 " 195 I 2 4 2 2 4 3 I I 195 " 200 3 4 3 I 7 2 3 3 3 3 I 200 " 205 2 I 4 3 2 2 I 2 I I 205 " 210 I I 5 4 5 2 3 5 I 2 210 " 215 4 3 4 5 3 4 2 2 215 " 220 3 I I 3 3 I I 2 I 220 " 225 I I I 2 2 5 I 2 225 " 230 I I 3 2 I 2 2 I 230 " 235 I 4 4 I 2 I 235 " 240 I 2 2 2 2 4 I 240 " 245 I I I 3 2 245 " 250 2 I 2 I 250 " 255 3 I I I I 255 " 260 I I I I 260 " 265 I I _ _ _ 1 — — Cases 27 43 84 81 loi 93 96 93 53 46 24 16 76 The Reasoning Ability of Children TABLE XLVI Median Ability and Variability for Each Age M's Higher Q's Co'ts. of Var. Age Boys Girls Boys Girls Boys Girls Boys Girls 8 to 9 72.00 158.50 86.50 5.50 77.50 .07 .48 9 " 9-6 184.50 124.50 60.00 48.50 17.20 .26 .13 9-6 " 10 171.50 144.5° 27-0° 41.7s 33-44 .24 .23 10 " 10-6 168.50 155.00 13.50 48.00 29.20 .28 .18 10-6 " II 164.50 156.50 8.00 28.00 31.75 .17 .20 11 " 11-6 165.75 164.50 1.25 33.25 32.00 .20 .19 11-6 " 12 172.00 166.50 5.50 35-12 40.75 .20 .24 12 " 12-6 167.75 175.50 7.7s 33-25 32.83 .19 .18 12-6 " 13 167.75 167.00 .75 26.37 25.29 .15 .15 13 " 13-6 155.50 168.00 12.50 39.50 34.12 .25 .20 13-6 " 14 152.50 166.50 14.00 35.00 56.50 .23 .33 14 " 14-6 154.75 151.00 3.75 47.25 15.00 .30 .09 14-6 " 15 180.50 203.50 23.00 36.25 52.25 .20 .25 15 " 16 180.50 123.50 57.00 50.50 33.25 .28 .27 8 10 II 12 13 14 Combining Half Years and Smaller Groups : 10 169.50 138.50 31.00 50.18 26.67 .29 .12 11 165.00 154.7s 10.25 3S.20 30.67 .21 .19 12 169.50 165.75 3-75 33-89 35.S6 .19 -21 13 168.50 172.16 3.66 31.75 31.37 .18 .18 14 153-50 167.00 13-50 34-83 38-50 -22 .23 16 163.00 151.00 12.00 47.00 32.00 .28 .21 Combining Further the Extreme Groups : 8 " II 166.16 150.50 15.66 3S-52 32.50 .213 .215 13 " 16 155.50 165.50 10.00 41.17 37. SO .26 .22 Per Cent, of Each Age Group Reaching the 75 Per Centile of All Age 8-11 11-12 12-13 13-16 Boys 26.12 29.70 21.86 23.36 Girls 19.35 25.80 32.25 27.60 In the percentages reaching the highest quartile, the boys are superior up to 12 years, the girls afterward. The boys reach their maximum from 11 to 12 years, the girls a year later. Both are inferior, from 13 to 16 years, to their earlier years, the boys showing larger percentages at both 8 to 11 and 11 to 12, than at either 12 to 13 or 13 to 16. In the arrays for age periods, the rhythms mentioned for individual tests are very evident. of the Fourth, Fifth, and Sixth School Grades 77 Comparison of Youngest and Oldest 25 Per Cents, of the A Grades TABLE XLVII Youngest and Oldest Pupils op the A Grades Compared Grade 4 A Younger 25% Older Whole Grade Median Age Yrs. Mo. 9 12 II SA Younger 25% 10 Older " 13 Whole Grade 11 6 A Younger 25% 11 Older " 14 Whole Grade 12 4 A Younger 25% 9 Older " 12 Whole Grade 10 5 A Younger 25% 9 Older " 12 Whole Grade 11 6 A Younger 25% 11 Older " 14 Whole Grade 12 Boys M's Higher Younger Older 4.0 3-5 7.8 10. 5 6.S 4.0 o .0 7-5 114.00 97 .00 117.50 162 .50 132.50 160 .62 202 .50 205 .00 205 .00 Girls 123-50 III .00 130 .00 184.50 149-50 165 .00 17 .00 30 .00 12.50 35 -°o 216.50 184 .50 32 .00 204.50 Q's 58.25 30-25 34.01 35-25 41.18 32.81 Co'ts. of Var. -51 •31 .28 .21 •31 21 .50 2.50 18.75 27-50 20.50 36 .62 22 .08 30.75 24.7s 32.50 19.50 43-75 21 .14 .10 .09 •13 .16 •33 ,16 .16 .16 .19 .09 -23 .10 In every case except that of the boys for grade 6A, the young- est 25 per cent, of the pupils of these three grades reach a higher total in median ability than the oldest 25 per cent, in the respective grades. This is in harmony with the general tendency shown in all of the tests excepting VII. Of the 12 groups, 4 only reach or exceed the median of their whole respective grades, the younger boys of grade 5A, the younger girls of 5A and 6A, and the younger boys of 6A. 78 The Reasoning Ability of Children Sex Differences by Age TABLE XL VIII Per Cent. OP Boys Reaching the 50 Per Centile op the Girls Combining Half Years For Half Years and Smaller Groups Age Per Cent Age Per Cent. Age Per Cent. 8 to 9 00 .oo 12 to 12-6 42.37 8 to 10 59-25 9 " 9-6 83-33 12-6 " 13 54.05 10 " II 58-33 9-6 " lo 6I-S3 13 ;: 13-6 38.46 II " 12 52-47 lo " IO-6 S6-2S 13-6 " 14 42.85 12 " 13 47.91 10-6 " II 57-69 14 " 14-6 60 .00 13 " 14 39-62 II " I 1-6 53-96 14-6 " IS 20 .00 14 " 16 62 .50 II-6 " 12 52.63 15 "16 75.00 In 9 of the 14 half year groups, more than half of the boys reach or exceed the median ability of the girls in this array. Up to 12 years the boys are superior in every period excepting the first, that from 8 to 9 years. In the array for combined half years, the three groups up to 12 years, and the one from 14 to 16 years, show the boys as superior. From 10 years to 14, the girls gain on the boys each year, surpassing them from 12 years to 14, when the boys again forge ahead. Summary for the Combined Results Grade Differences For both boys and girls, there is progress from grade to grade, both in median ability and in the percentages reaching the highest quartile for all. The per cent, of gain is 74.46 for boys and 57.30 for girls. The distribution of these gains through the grades is very irregular, as may be seen in the following sum- mary of gains in per cent, from grade to grade : Grades Per Cents. Boys Girls Higher Boys Girls 4A to 5B sB " sA 5A " 6B 6B " 6A 32.44 12.50 3.21 12.82 16.43 14-24 9.62 8.48 19.94 9 .61 2.19 1. 14 The variability for both boys and girls is slightly lower in the upper grades than in the lower. of the Fourth, Fifth, and Sixth School Grades 79 /i-/3 If-lt to TO to 1 1 ifA fA iA 67 ^ VA SB SA iB 6A «9 //-/i /Z-/3 /J-/fc 70 Figures 61-70. Totals 61. Frequencies of Abilities, Boys. 63. Median Ability, by Grades. 65. Variability by Grades. 67. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Grades. 69. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Grades. — ^^— Boys. 62. Frequencies of Abilities, Girls. 64. Median Ability, by Age. 66. Variability, by Age. 68. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Age. 70. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Age. Girls. 8o The Reasoning Ability of Children Age Differences In the arrays for half years and the combination of these into years, the rhythmic variation in the individual tests is noticed. As a whole, there is seen to be a loss in ability with the progress of age for the boys, and a gain for the girls. The boys reach their maximum, save for the two small extreme groups from 14 to 16, at 12 years, the girls theirs one year later. In the com- parison of youngest and oldest 25 per cents, of each grade A, the youngest pupils are superior in every case excepting that of the boys in grade 6A. Variability is just slightly higher for the younger boys, and for the older girls. Sex Differences In Table XLIV it is seen that in two of the five grades, 5B and 6A, the boys reach a higher median ability than the girls. Taking all of the boys together, they reach a higher median, 165.83, than all of the girls taken together, 159.25. Half or more of the boys reach the median ability of the girls in grades 5B, 6B, and 6A. A larger percentage of boys is found in the upper quartile in grades 4A and 5B, but of girls in the upper three grades. In the age distribution, the median ability of the boys is higher in 9 of the 14 groups, in general the boys being superior in the earlier years, up to 12, the girls superior above 12 years. This is also true of the percentage for each age group found in the highest quartile, and the percentage of boys who reach the 50 per centile of the girls, Table XLVIII. In the com- parison of the youngest and oldest 25 per cents, of the three A grades, the girls reach a higher median ability than the boys in every one of the six groups except that of the older for grade 6A. Evidently the generalizations possible upon the basis of central tendencies alone would not apply at all rigidly to the more variable lowest and highest 25 per cents, of each grade. Beyond the limits of the probable error, the girls seem to be superior in every grade excepting in the upper quartile of grade 6A. In variability, the boys are greater in every grade, the difference being greatest in the grades at the ends of the distribution, 4A and 5B, and 6A. In the age distribution, there are 9 of the 14 groups in which the boys show greater variability, the 5 groups higher for girls being so interspersed as to produce a rythmic alternation, as seen in Table XLVI. of the Fourth, Fifth, and Sixth School Grades 8i ¥A SB 74. Figures 71-74. Summaries of All Tests, by Grades 71. Median Ability, Boys. 73. Variability. Boys. 72. Median Ability, Girls. 74. Variability, Girls. 82 The Reasoning Ability of Children Relative Progress in the Grades Table XLIV, and Figure 63, show that there is progress from grade to grade in the totals for the tests but in very different proportions. A tabulation of these gains in per cents, will in- dicate the differences. TABLE XLIX Per Cents, of Gain from Grade to Grade Grades Per Cents. Boys Girls Grades Per Cents. Boys Girls 4A to sB SB " sA 32.44 12.50 3.21 12.82 5A to 6B 6B " 6A 16.43 14-24 9.62 8.48 For boys, there is a very marked difference in passing from the A section of one grade to the B section of the other and but little difference in passing from the B section to the A section of the same grade. The boys of the two sections of one grade are much more alike than the boys of the upper section of that grade and the lower section of the following grade. Here, the pupils of the divisions of grade 5 are a much more homo- geneous group than those of sA and 6B taken together would be. At mid-year, grade 5B automatically becomes grade 5 A, while a group from grade 4A comes into grade 5 as the new 5B. Usually much more care is taken in the control of school ability in promoting from the one grade into the next than in permitting continuance in a grade of those of weak ability when they are once in that grade. Thus, by a more careful selection on the basis of school ability for the B section, and a greater tolerance for the A section, the condition here revealed is, at least partly, explained. For girls, the difference here noted does not appear until the transition from grade 6B to 6A is made. The greater tendency toward regularity in the work and school progress of the girls asserts itself here as the probable explanation. of the Fourth, Fifth, and Sixth School Grades 83 i-io 10-11 ii-ii n-13 ti-/M if-n, <-.io 10-11 iHt ix-n n-tt- it-i 75 76 l-li SUM. »-/« /*-// IHi. Ii.-I3 n'l¥ l*-lh i-IO lO-ll IMS. ti-13 l}-lt It'll, 77 78 Figures 75-78. Summaries op All Tests, by Age 75. Median Ability, Boys. 77. Variability, Boys. 76. Median Ability, Girls. 78. Variability, Girls. 84 The Reasoning Ability of Children Relative Progress with Age In the array of results for the totals of the tests with age, it is seen that there is just as often a decline as an advance : TABLE L Per Cents, of Gain or Loss FROM Year to Year Per Cents. Years Boys Girls 8-IO to 10— II -2.6 II. 6 lO-II " II-I2 2.7 7-1 II-I2 " 12-13 - -5 3-7 12-13 " 13-14 -8.9 -2.9 13-14 " 14-15 6.1 -9-S For the girls, there is a diminution in the gain until it becomes negative at 13. For the boys, there is the rise to 12, then the fall to 14, then the rise again. These general fluctuations are represented graphically in Figure 64. Prominent Sex Differences Among the Tests That the boys are superior to the girls in mathematical judg- ment, and in the form of selective judgment represented by test V, is clearly true, and that the girls are superior in the form of controlled association represented by the opposites test, and in the literary interpretation, is also evident. In the form of con- trolled association represented by test III, the boys are also slightly superior. In kind, this test is more like the mathematical test and that of selective judgment as found in test V than is test IV, the opposites. In test VI, in selective judgment among definitions, the girls are just slightly superior. Here again is a test having elements somewhat like those of the opposites test — it deals with words. It may be noted especially that in the three tests dealing with words in a rather special way, the opposites test, the test using definitions, and the test in intel- lectual interpretation of literature, the girls show superiority. There is certainly tested here an ability that is different in the sexes, or at least an attitude of mind that brings superior results from girls. This may be the element of attention to detail, or patience in analysis, or a keener sense for the meanings of words and phrases, or a more accurate form of expressing their mean- ing, or a combination of these all lending their influence. It of the Fourth, Fifth, and Sixth School Grades 85 has been generally found that in all of these elements girls seem to be slightly superior to boys. In looking over the papers of test VII, that of literary interpretation, it seems to me that the two elements, attention to detail, and patience in analysis, are the dominant factors in that test. The girls seem to work a little harder, attend more to details, and give a more patient analysis to the poems than do the boys. A negative factor on the part of the boys also probably must be taken into account in the poorer showing of the boys — in these grades boys do not gen- erally like poetry. To be rid of the matter as soon as possible would be the attitude of the boys much more than of the girls in this test, and much more so than in any of the other tests. That the median ability of the boys is slightly higher than that of the girls, all taken together, and that 53.50 per cent, of the boys reach or exceed the median ability of the girls, all tests taken together, would indicate a slight superiority for the boys. The ratio of the median ability of the boys to that of the girls, however, is but 1.041; and the percentage of boys found in the highest quartile, taking all results into account, is 25.4 while that of the girls is 25.5. In variability, the boys are slightly higher, the ratio being 1.047. However, these summaries by combining results of the numerous individual tests must not be taken at nearly so high a value, for validity, as are the details of the individual tests. Mental abilities are too complex to be rated very extensively in the mass. One very marked sex dif- ference shown in the totals is that of the relative fluctuations of ability between the sexes. This difference is graphically shown in Figures 67 and 68, where the very marked superiority of the boys in grade 5B is evident, and where the diminishing ability of boys, relatively as well as absolutely, from 12 to 14 is brought out. That, on the whole, the resemblances between the sexes are far greater than the differences, is very evident in all of the figures in the general parallelism in fluctuations in median ability, and the even closer parallelism in variability. Variability These tests all show that the variability of both boys and girls diminishes from grade 4A upward to grade 6A. In most of the tests there is a slight fluctuation upward from grade 5B to 5A, 86 The Reasoning Ability of Children d. S 0) W >;>. O -^^ M §1 ■ CO l-* tH aJ .>, < ■^^^^^^^y^'"'^ YA ^A fcA 83 .Ok y\ /IS J>, -^\ .at- ^^\ / .113 ^"^^^i::^ *A JB 6B 85 CO 30 iS M 87 5/\ (,^ I3-Ih It'll. 3t> IS f=n: ¥A Si SA 6B ss e,A f II 11-11^ ix-13 n-n- id IS SO Figures 81-90. Spelling Test 81. Frequencies of Abilities, Boys. 83. Median Ability, by Grades. 85. Variability, by Grades. 87. Per Cent, of Boys Reaching the so Per Centile of Girls, by Grades. 89. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Grades. — ^^^— Boys. 82. Frequencies of Abilities, Girls. 84. Median Ability, by Age. 86. Variability, by Age. 88. Per Cent, of Boys Reaching the 50 Per Centile of Girls, by Age. 90. Per Cent, of Pupils Reaching the 75 Per Centile of All, by Age. Girls. of the Fourth, Fifth, and Sixth School Grades 1 1 1 Sex Differences by Age TABLE LXVI Per Cent. OF Boys Reaching the 50 Per Cbntile of the Girls For Half Years Combining Half Years and Smaller Groups Age Per Cent Age Per Cent. Age Per Cent. 8 to 9 oo .oo 12 to 12-6 SS .91 8 to 10 44.44 9 " 9-6 75-00 12-6 '• 13 48.64 10 " II 60 .71 9-6 " lo 23.07 13 ;; 13-6 46.15 II " 12 68.32 10 " 10-6 53-12 13-6 " 14 21.43 12 " 13 53-24 10-6 " II 65-38 14 " 14-6 6.66 13 " 14 45-28 II " II-6 66.66 14-6 " IS 60.00 14 " 16 50.00 II-6 " 12 71 -°5 IS " 16 75.00 In the age distribution, the superiority of the girls is not so marked as in the grade array. From about ten to about twelve, the boys are superior. Summary for the Spelling Test Grade Differences There is progress from grade to grade in this test excepting for the slight drop for boys in grade 5B and for girls in grade 6A. The gain in per cent, from grade 4A to 6A is 4.05 for boys and 5.05 for girls. In the percentage of pupils from each grade who are found in the highest quartile for all, there are more from grade 4A than from 5B, for both boys and girls. After 6B, there is progress through the remaining grades. Variability is greater in the lower grades than in the higher. Age Differences The boys average slightly lower for the years above twelve than for the years below, while the older girls average a little higher than the younger. Excepting for the small groups from 14 to 16, boys make their best record from 11 years to 11 years 6 mo. Girls make their best record from 13 yrs. 6 mo. to 14 years. Of the three A grades, the youngest 25 per cent, stand higher than the oldest 25 per cent, in every case excepting for the boys of grade 4A. In the percentages from each group found in the highest quartile for all, that of the period from 11 to 12 is much the highest for boys. The percentage for the 112 The Reasoning Ability of Children period 8 to ii is higher than that for from 12 to 13, and this higher than that from 13 to 16. For girls, the proportionate percentages are parallel excepting that for the period from 13 to 16 the percentage is highest of all, and that for every period the percentage of girls is higher than that of the boys. A rhythm in the half year array is quite apparent for boys with crests at 9 yrs., 11 yrs. 6 mo., and 14 yrs. 6 mo. For girls, there are small fluctuations with fairly prominent crests at 8 to 9 years, 12 yrs. 6 mo., and 13 yrs. 6 mo. Sex Differences In every grade but 4A, the median ability of the girls is higher than that of the boys. The relative superiority dimin- ishes from grade 5B upward. In the age distribution the girls are also superior in every group excepting those from 11 yrs. to II yrs. 6 mo., and from 14 yrs. to 16 yrs. In every grade, the percentage of girls who reach the 75 per centile of the abil- ity of all is larger than that of the boys. The same is also true of every age group excepting that from 11 to 12 years. In variability, the boys are a little higher, on the whole. Comparison of Spelling Results with Those on Reasoning Comparison of tables and of the graphical representations of general tendencies shows that the results for spelling most resemble those of test III excepting for the fall in ability of boys in grade 5B. In the percentage of each group reaching the 75 per centile, the test is also most like III. In sex differences as indicated by the per cent, of the boys reaching the 50 per centile of the girls, it is most like test VII. However, the test is not nearly so much like any of the other tests as they are like each other. The test is not nearly so adequate as the others in that too many of the pupils make a perfect grade in it. It is not difficult enough. Its value in relation to the other tests lies more in its possibilities for individual correlations than for group correlations. of the Fourth, Fifth, and Sixth School Grades 113 Bibliography Books Bain, Alexander, The Senses and the Intellect, Bk. 11, Chap. 11. John W. Parker & Son, London, 1855. BiNET, Alfred, The Psychology of Reasoning. Based upon Experi- mental Researches in Hypnotism. Tr. from 2nd French Edition by Adam Gowans Whyte. Open Court Pub. Co., Chicago, 1899. Dewey, John, Psychology, Chap. viii. Harper & Bros., New York, 1890. James, William, Principles of Psychology, Vol. 11, Chap. xxii. Henry Holt & Co., New York, 1890. Ladd, G. T., Psychology, Descriptive and Explanatory, Chap. xx. Chas. Scribner's Sons, New York, 1894. Miller, Irving, The Psychology of Thinking, Chaps, xxi and xxii The Macmillan Co., New York, 1908. Morgan, C. Lloyd, Introduction to Comparative Psychology, Chaps. XV and xvi. Chas. Scribner's Sons, New York, 1902. RiBOT, Th., The Evolution of General Ideas. Tr. by Frances A. Welby; Open Court Publishing Co., Chicago, 1899. RoYCE, JosiAH, Outlines of Psychology, Chaps, xii and xiii The Mac- millan Co., New York, 1903. Spencer, Herbert, Principles of Psychology, Vol. 11, Part iv. D. Appleton & Co., New York, 1896. Thorndike, E; L., The Elements of Psychology, Chaps, xvi and xvii. A. G. Seller, New York, 1905. Theory of Mental and Social Measurements. The Science Press, New York, 1904; Teachers College, 1910. Educational Psychology. Teachers College, 1910. Articles from Periodicals AiKiNS, H. A., and Thorndike, E. L., Correlations in the Perceptive and Associative Processes, Psycholog. Rev., ix, 374-382. Baldwin, J. Mark, Selective Thinking, Psycholog. Rev., v, 1-25. BiNET, Alfred, Attention et Adaptation, L'Annee Psychologtqtte, vi, 249-404. Brown, H. W., Thoughts and Reasonings of Children, Ped. Sem., 11, 358-396. Burt, Cyril, Experimental Tests of General Intelligence, Brit. Jour. Psychol, III, Parts i and 11, 94-177. Carlile, William W., Some Points in the Theory of Inference, Phil- osoph. Rev., IV, 407-422. Decroly, M. O., et Degard, Mile. J., Tests de Binet et Simon pour La Mesure de L'Intelligence, Archives de Psychologie, vi, 27-130. 114 The Reasoning Ability of Children La Mesure de L'Intelligence chez des Enfants Normaux. D'Apres Les Tests de MM. Binet et Simon, Archives de Psychologie, ix, 81-108. Fox, W. A., and Thorndike, E. L., The Relations between the DiflEerent AbiHties Involved in the Study of Arithmetic. Sex Differences in Arithmetical Ability, Columbia University Contrib. to Philos., Psy. and Ed., Vol. ix, No. 2, 32-40. Gard, Willis L., A Preliminary Study in the Psychology of Reasoning, Am. Jour. Psy., xviii, 490-503. Grogs, Karl, Experimental BeitrSge zur Psychologie des Erkennens, Zeitschrift fur Psy. und Physiol, der Sinnesorgane, xxvi, 145-167, and XXIX, 358-371. Hancock, John A., Children's Ability to Reason, Ed. Rev., xii, 261-268. King, C. H., The Reasoning Power of Children, Paidologist, v, 157-162. LiNDLEY, E. H., A Study of Puzzles, Am. Jour. Psy., viii, 431-493. Mach, E., Accident in Invention and Discovery, Monist, vi, 161-175. Ogdbn, John, Conception as a Mental Act, Education, xvi, 227-229. Sharp, F. C, Teaching Reasoning as a Fine Art, Ed. Rev., vi, 493-497. Smith, C. F., The Mathematical Consciousness, Ed. Rev., viii, 277-288. Wells, F. L., Linguistic Ability and Intellectual Efficiency, Jour. Philos., Psy., and Sci. Meth., in, 680-687. of the Fourth, Fifth, and Sixth School Grades 115 ^2 MMHHHHMWMHMMMMHH HMHMHHHmHW HMMHMHHH pq OOHHhmoOQOhoOhmh «tOfOOO««H'OV) (NnwimoOhwj ^'lHH 'J-^O HHO«W«'tN lOlOWOOOl'lH P3 -^ N fO **3^ t^WOp'*'*OMY10NN« O fOCO OO00>Oiot^r0 WN«OM00Ot^ > OOcimOOOtOOOf^fOOOOOO fO t^vO o *0 !^ ^ O «*0 tOOO«0000 *H .O^^HO«0<500Y1H ^CO MOO OOOWO'O'^iOvo'O^ OOwCO^^OOooO I N « t» O ^ H o t^^ t^NMco Ort-OeoO'fNO m o H O o > n o > n m " Oq 5 <: a o« « H Bi " o .9 < ^ Ot Oi r« 0»a Qi Oi OtCO 00 Oi Q'OO CO O>00 0> Ot QlOO r^ 0> OioO Ot t^ O>o0 Oi Ot r^OO 00 Ov pj CO Oi »fl t^ ^.00 \0 t~-CO ^«CO H OiCO &00 00 O* Ov»0 ^00 00 \0 00 O 00 o Oi t^*0 OOO Oi H ^»Or-0M>O«'O Ov* 0»>0 000^0*0 0000 lO'O 'O aoO OOi 0000O>cqO ^00 w n) f^ rt ''lO >0\0 ro (O ■* O O O r*0 C5 t».00 Tf W T}- lO-O «^t*t- OQioOi'*fOt> **00 nq ■*« « 'O o *0 '^'O ««\OOOOONOO ^OO^W'^ ^00 «•*« 'tNN^ejoOO''*" 0>0 1^ t*00 Ov O M « to ^ jg "^ HMMHMWtH'«HMHHH««C1«««'^«««tOtOfOfOtO si I 1 •s a -s ii6 The Reasoning Ability of Children _g HHMMWI-IIH MMHHHHHHHHMMHHMHHMHHMHHMMHMM HHHm =3 > < iHHHi/}C4MH 000000000'-*0000»''0MH00eiHtHH00t-'0 hmmh OOi-tHHOO 0**30MWO0OHOOO0Ot0N0HOO0'-'O'^'^*^O mOmo Q o > << t^ -^ Oioo 1^ & Oi ©■'O 0000 ^ON t^totnoo t»^o ooo tooo O0t^00t^0i>000'*&0000 00 ^00 r^ m < rt OiOOOOOOQO^t^ 0000OlC^C0t^^- OiOOeo O »ot*Oi>')OiOit^ ^-00 t^OOt*OOOiOOt^ "Ot^OiOi n Oi^'OOOom'o 000000 lot^'o t*) rooo t^^n ^^oo^o x^oo >o »/i »o Oi t^ Oioo »')o^*^ ^»^oo»o oo^'*o^>^'+ o >^oo 00 00 Oi i>oo o \o o o 00 oioo m*o fc ^ h ■* »o "loo Ht^m^l- oownvi H Oo^r»«00CO^ ^CO^ M 00 O O ^*0 ^O «0'*00«00«>ort-«ne<0 '^^ « « d^O OMOOO^^O 0«^0 H <] O H N 00 r^^O N NOO^««^«'*'^N fOO >0 ■* ^ O fO WJ « ■* N O 'O ^^O ■^OO 00 O 'O 00 ^ d »0 ■^00 lO M lO O fOOO ■^OO^iH'*PO*0.0000»0&HOOOMOWHOO>00« «tHCOO SS iiiiiiV imiiiiimiiiiiiiii'TViiTmii mm ^ • «H«OWNOi MMMHOOiO'-'OOiOH'^OOOOHOW'-lOcOWO^OOOH t*3HMOl •^ d d _, d —I 17 lo'Or^OOOOHiyPtPO^m^Of^OOO'O'-'Wco^iO'O *^00 OO h « w^iovo t^OO Oit:^ O H « fO 6 11 1 ■8 -S -S ma) to of the Fourth, Fifth, and Sixth School Grades 117 COOOHOt^-HCOOHOHOOCOHOHlOO H«MC*)0*nWt^tONeOH HOWOOMOOHWMO^ CO^Hf^MOHMHOOOnHOC^OnH MHn^OH m'O ^WOCON ot*««iOnOMOO« 'OOT*'fOOO\Or*j«0*OOr*OOwOt*^ t*0«*^Oi'*mt»0«r-Oi 't0i00i0"p0««*30t0 M00^^t*OO*HHO"-'MOOCOHTtO »OCp >nof*OHWHlo*tMVlC| coow<**o»rtMOOnio I I I I I I I I I I I I 1 III H H Yi*0 O O 00 O w^^O wOOWOwOwir^ 't^MiO^HTfMCOHvjM r-HOOH^woOO *co III I III I I I I I III I oooooo«oo^mO'OP>OMOooa«o «o*oofoo*o«oo (o*o o>o>o>o 0*0 o 000 m « \0 t- « O * lO^O ^0«CO^He1c^o*0 "100 m (^ m ^00 fO«ui«Hior- 000000'*o*]tO«OO'*' «0«On«OCiNciO^OOOTj-000000 OOOOMOO 'J-OO « « 00 "O 00 O '*'0 O O ei O « « 00 I I H M M I H I M I I 0vH«w)«0«'l«fCWOiiO0000Oi'^0*0 ^OoOiWQiMOOOiH Oi^O Oi hOiM«'00.««cOOO o « o ^'O on-^Nw^^wMowooooo oooowoonooo ^ooo o 00 *« « O rt *^ ^ o H| III II M HH Ml MHHl W H ■^t»M(^M H Oifoi>cow it"to wi*0 H Tt"© o ooM(*)«(om«twi m^O *o i> h o> « v) 10*0 fo « (^ >/i « t«.Mt*)H^f»5t>.**iO(*)«00*^^*0t^0 t<5io«Tft^ O>00 oo»OOiOi »«.NHW5'-'000^0'-'«0*0« 01*0 'O * »*3 M ■^^ W) « O. «t ^\o OiOO H Oi >0 00 Oi lO-O »0 o O- H "t O" a; ui« *N ^wipot^fio ^"O ^ I'loo 'to wj«« '3 r-^foOioo r-'O \o o ^ o oo *o tooo 00 o fo o •-« w 6 00 00 00 00 00 aoo 00 00 00 t^oo 00 o> ^00 00 00 00 00 ov*o a 000 o a o>oo t^oo o. too. o>oo o>oo 00 o> t- a 0,vO 00 O O<00 OOOOOOOOOO r«0>0,^t^t»Ott>* OlOO 0> 0>W OvOO OiO Oi OOO O 0> 0>00 t»00 O>00 O>00 0> O"! O t^*0 Oi V)00>O t*00 «^-OloO»')0'0000 00OL0OO>^•OOl''l^-OO O00i0i>00t*0>0w0i ^tr)Vi -^O^O ■*«'lt^m*^^M0'^«00O00O >Or*iflOvfO ^00 00 (*5 O >0 * *» O »O00 «^0D « Oi r- t O ^^« no w O « nn « nq'O'ooO'O ^00 n vo-o-^oOWoo^nnw^ Ttoo O o ■* 'i' ^ O to o o « -^o ■^tWTj-oteiw ^o wtoooot oonooo^o>7 o r%oo o* o '-' w fo 1 100 to !>. I« bs t^ t^oo OOOOOOOOOOoOOOOOOOOiOiOiOi ^-* OiOiOiOiO.O.OOOOOO^OOOOMHHMMHH ^ M M W H H M^ HHMHMMHHHHM o o u u CO CO 1 1 8 The Reasoning Ability of Children ^gP, MHHHMH.MMHHH HHHMHMHHHHH^MHMHHMHHMHHHHHH Pi a. J— OOO'tOr-fOONHOH i-itoMOM>-'»OOOOOOOOHHO«H«OW>-'"00000 iH(ONT}-0'*Nr^ mt^t^i^i-i WOO « > Oj OWOOOtwro t*vO COO wiNO'000'*JiOOf^t».«> Q OH aO O^f^tO0»^Ht HM\OtO«HOiOlMO.>J^O'»OH t^CO Hl«li^H(*3«oOM100l>-' « > n O fO ^00 o »o Tj- rt H »o H § < ■I I o '^ J (ooo<'iMOt^'0"iMCO*oO v]r*i^**«iN>O' ooo oivo d oco moo oo ^ doo « t^oo o looo o* oioo oo t«. &oo c^oo ** o i«- o. o. rt coco CO 00 ^ a i> o^oo \o t>.co o. Oi a Oioo to »n oi\o Oioo vo 'O co oo oo («. t* OlOO oo t^ r. ooo o. OiOO oo \~^ M *"• ^ vo'OoaooO'Or^ooo oo^oa'ot^'oooi looo oico o oo co oo aoo oo o^oo vo »o r^^ *>■ o o> ^ cj lO >OVO ro Oi (^00 O r- (OO O »0 (O N ^00 NOO (ON »O^M «Xn^' ^OO W (oO'ittM f0(O^t^OiH m vOOOOO«OOOOONOiOO M ■^O «0\0 -^O -^lO ^00 N « GOO 'J-'O N « rfQ nO ^0*0« O M ^ TfO tJ-oOO OOO 't 'to OOO 00 ^ O ■^OOOO 0«Ot^0^t00000^0 0^Wi>^0 (00 00>o m o o fo o o 00 00 o 'fo ^ M 00 ^ « »o ■* too o oo « t ■* o ^o o «vO00 ^O tow^f^ **^ o niO«OIO»OlOIOln rg ^MHHMMMMHMHM W^ HMMMHMHHMMHHHHWMMHHMHHMMMHHHH 73 o CO m of the Fourth, Fifth, and Sixth School Grades 119 HNMCt^HOOHOHOnOH nOt^MCtHQO OOHMHOOOThMt^OHfOOO W0« H«ThHHHOOwNMO«HM MHlOCqCOHOM OOHTtHMWHOWHlHHHH OOO I 0OO0'^0'*'OO00'^OO»O O « H o '^'O OOi 'O«»O(*i0N*~-0H 00 CO t^ «t to m P CO NOD O>00 00 r^OO 00 Oi & t^OO nCO Qi C>00 r« 0< 0>C0 0> aCO 00 Oi ^00 00 O'OO 00 0> !-• Oi 0> N CO o>co 00 O>C0 0> 0*00 OiOO 00 O>d0 CO 00 ^0 00 o 0>C0 o>00 0> O O oO CO c a^ ^« N O>C0 00 N 0> M t^OO CO o> r> *^ r* O Oi 0> Oi Oi NOO O«0it^«*0 000 tN.00 VO r* Oi OiM 00 Oi lOO OsOiOiOOCOOO t^oo ^ e* o "100 O r» o OiOO o Oi O'OO ^"^ o 00 o^co 'O noo n o •" t^ n m-o ^ f-»0 »o^ ^ r- r~vo k « m ^ too \o * ^00 ooo« t^-^w n -^vo o Tj- fo ■* « o ^^ N 'toa 'Ow\oo«'*«'Ot owoo N 00 O if-OO n'OCOOOnOOnOOn >O0D'O00'*t^t OOONOOoOCOOOOOO'O'OrJ- \O«0Q MM HH HH HHHHH ■tJ-OO O M ^ O CO '*»0 tWoO"* r^'*WO>'^«'tt OOt fO'O ^tOO *0 r- ''0'-'«'«wOi tnWOHONOiH OONONH«OWOfOH>OOHt«- oOiOi . 00 Os O H CI to t '"^^ NOO O. O M n t^ (O ^ >'1'0 t-CO Oi 0*7 w « fo t who NOO O. O w « ro t vjty VO NOO I Mji'ic io«coo'0'0«'or».r*'«*- '^ t^f^rst^t-i^ t^oo (^ 00 00 00 00 oo 00 00 00 00 oi oi o* o a o*^ & Oi oi ■i -s tn « 120 The Reasoning Ability of Children _gra HMHHHHHHHHMMKHMHMH HHMHHHHHHHH HMMH^HMMH pa PQ M-^WHMOHOo^OwmooOOH 00'-''*0'*OMOwOWMMMOtO(*)OMO t-H ^J 0»n«OWHHOMOO«»nOOOHN OOHWdHOOOWNHMWMWM-lJ-OMH Q o W H III I Im m >- J Cm "^ M10M(^NMO\0«f^'OOiH l1t^N'^*'*0 ^ "^oo ^ r-l^H cSrorOnClN com **3M(^fOtON <*JW ^^N (»^mHM (*)W(*5«W(*)MHt^tf> WW) 5 ^ *f4 « moo t>. Oi t^ « lo'O 00 o to^o H H ■* moo *o (o m t^^ "ivo o« a "T^ oi ■*« oo a « lo »o*o m oo M ^ (OW)«MH rOM««HW}COt*5«H[*iM Wi«HCO*^M fOnCtMHnM ««0 MM < t-i .O ov Oi (^ O'Oo m Oi r^ a t^ tooo oo oo oiOO oo o 0.00 t«. o Oi t>. tooo 00 00 o a doo *»■» ^'OO »ooo a rt c* aoo 00 CO *o 00 00 o. *>■ aoo co o i*. ^ a t*- t^oo m 00 o> a t* a Oi t>oo 00 00 aoo 00 000 »>■ 0.00 "^ ^ f^ o *'>oo t^ »o Oi '^eooo ■*» o a a'^oo *^ o Tfo o a 000 00 o*>Oi« aaao o t-iooa . »ooo fo o» Tf-o r^ ^"o w ^ a'o so ■* VI ^ t«.{io 00 a « *^ a rn o 00 o to*o ^00 t'O >o « ■* o 00 m^o « 'O 00 tj- tj-co o t « 00 O 'O ^'O "^oo *o ^*o W a'O ^ f** M M MM i-t n o t^*o ^o to t^so m^oo Tt-n>ooocooo t»toM «'0\ooo m ro « 1-1 <1 0«^«Nt'*wa -^00 N0'Ori-'-''^« OO00Oi Oi OiCO 00 Oi OiOO C0l>0 \o0000t>iOiOi doO CO Oi r- 0> O. 0> 0.00 Oi t-00 Oi ■*<* W CO ooo d 00 dco t^ o o- a aoo om oi oico o^o r^oo 00 co o t>m o> ** 0,00 00 O O 00 0.00 f-OO oOOOtO'*'*'* O.O00oOO.O>OO>O.Ov OOOOor- acO 0> m O dO w o o o H MmMM HHH " WHM oi Oi tNoo r* ■*'0 « to (o o t^'o ^* "0 >o *o o o Oi doo oio>«- oooo>o»oo. doo 00 e*oo >o woo o. o ^M O COOW CI ^WOOO Oi^O 0000 O'OWOOOO'OOWOOVOO O0000O00*O^Om00'*«O000000 ^HH MM HHW Q 10 sj- O ■* Tf sJ-sO '<1"0 yDO*^0« OOOOOoOOO ^00 WOOD OO'OOOOOOOO'OOO ^"O « w o •* O « M H H "^ H H H 00 000 I^OW)*COO«0000 0«W° -OOr^W -too ■<*■ O >^00 CO 00 vo O M o o i^vo o « M •* »^* o> to 'Oo'OO'OTfTj-noooooooWO'o oo'*o\o'oooo- o « o « -^oo t-. otnooHxooMt*j«ootn-t«Hio tSooaoMWto'* mvo t^oo o. o w « to>5 ^ >oo t*oo a o h « t^ ^^ uro t^oo a o « « to J lo-o j^ M M n •s ■s CO £0 122 The Reasoning Ability of Children ^PQ HHHHHHHWMHH HMHMHMMHMMHHMMHMM MHMMMMMMMM p_ O 0^0>0<000 OiO O O t*! to O **)*0 O *0 fO O to^ r*i ro O Oi "^ t*)*© (OO'OO O O WtC «0'0 rooO fO « O CO < (Q 0«'*«0'OfONWoO fOWfO' Q 0C0«00^«>Cl'O'Oei0C0 0000'© o O O O^'O Oi « ^ rt Oi C* wioO 'O OOiw (OO^OifO OifiO Oi fO O' Oi Oi M O O>00 *O0000 t^fOM Oi > n , ^ ,-•« ^ CO ^o « ■*'0 oofOt^Oi ^Oinr-ioOt^-^r-T V li >m >H •"< w ►J < m < H ^ m 00 00 CO CO 00 aoo o>^o vo o> o>co c a^ oo o^ o>oo ooooooooooo,o>o> oo c>oo O'oo aw oo oo Oi rt Oioo 00 Oi Oi Oi Oi Oico 00 CO t^oo aoo oo e.>o 0.00 i> oi a o.co o<'>-'^ o>*o 00 00 aw a r^oo o o coaaoaoo aoo 00 Oi a a a\o 00 r*>o « a o t^ '^ oico q ©lOO o oco aoao ar^-o < n> t^oo o Oi r- a »o o t^oo 00 w "noo r*5 ^^^o *o t* 6i r* o r^oo »ooq 00 00 o O o Ovoo vo >o a t-. o ™ H M M H H H 03 «0'0*0\0rJ-0NO0*O 0000»OO«^ aoo OOOOOOWvOOO^ 00 ■^'O O 00 ^O ^O O CO O00'OtHOIO^»O W)00 M CO»Oc*3«te>1000*OOCOOO aoo COoOOO ^OOO'^OO'OOO'O^ -I 0*«W d ' < a*>no>^o aM <*) oi^o oao o oh ao*"! »5r H « po * »o*o t«.oo ao H>^ N (o^oo t-cc^ a o m « n ^ io*o *^oo •^ oocooooooooooococo a or^ aaaaaaaaoooooo ooc^ oihmhmwhmh m ,—1 N«««N«M««WN,-H««NW«N«NrOfOtO*0(0'*)fO(0 tOi— i Mmwarorop^f^fOtOwi 00 o 00 o ^ o o m CO m of the Fourth, Fifth, and Sixth School Grades 1 23 OOONMNOOONOOOO'OO'-'O OOOmh(1(«ooOmO «ohhOHHmhmo>-««mhho I I I I I I I ' m Tf t^ »>. to H ■*«o m-tfO rvt*or^N^t^:^'^7>f'0« itop M fO I I I I I I I I I I I I 111 00 00 00000 W) Oi « mO Oi "^ »0'*in« uiOiioO&H «Ooi« »fl'0 0> wOMoooooMio oo. »noo ■*oo ho omw^wa Tf« o »« to *«oo o w w h ^c^CtM »OC1CIWT*)cOfONOH«M (OWt^fOWtOfOWMfOWl Mi*l«f0«'O(O<*J'*)'*icO<*>tOtOfOHCO 00 00 « 00 00 r-00 r-oo o> o. a oioo 00 00 00 t^oo 00 00 aoo ooo 00 m t^ 00 0.00 o. a aoo 00 00 o o. o^oo 00 000 00 (^ or-oooo oi&»o o>oi o*-.eooo 0.O1 Oi f^oooooo a oooo t*oo o 000000000000 oioo o. o aoo 00 00 t^oooo e, f^oo cocoooooooeooioc^oto ooooii>ooo oioo 00 00 t^oo o a o ov 000 000 ooooooo o o> hmMihni m m h hhh hh hh mm m Oi Vi Vi<0 00 OiOO «^« Ot Oi t^ OiM ■* »0 Oi 00 00 Oi O>00 Oi Oi ^- O 00 V) r»*0 00 ^ r^OO O 00 o «^ »o o>oo r^ t^ W M M MM 00 ■* "t *>O00 *0 -^OoO OOOOO O^OOO O -^00 OOOOOOOOO^'O ■>tn00O00«*O«>O00 -^OO o ^0^0 ^'O M ■<}-Tf«vO"0CO OiO MOO'-O'O O ^00 O OOOOOOOOOOOOO •'to ^ OOWNOOOO'OOO'O'* OiO lovOVO^OO * M M H H M M M vQ \0 \e ^ t O ■*«> 00 <4 00 too ««0000 OOOOOOOOOt OvOO « wiOO 00 O fr-OO OOOOOnOOOO'O'O Oi*0 ro 0»W O * O H O O •*\0 WTtoOOO* of*'*OH(Ot^'*MO- HO\O0**000000000Tff0OvM»/l(*) v^ to to M to *0 »0 fO W) W) to (^ f*! PO (O (O (O (O J i I I eg I 124 The Reasoning Ability of Children ju, g. ^ *n vi \n in xn vi *r, V) V) in Vi \n ^ tn »oo»nm»om^ r*)W ^ vi «^»o wui m vi m o a mi I I I I I I II II 111 ^ ■^ O " O O (*)*o o- n M M 00 ^»o ^ n CI 00 ^o «»)m M00^*n'^otni^^^»4ci III I I II III J J, «)00 t«.0 nOOCO (*)0 »OvO ^ ift >0 QD n *0 ■too W)0 «CT^0'O0"*t*'«^w^ •-• o 0>C0 O>G0 («)>0 0> 0> O 0> « tOOO 00>Otntn «MMOOHHMMnov) ^__ OOOWCOO^ -^OO O O O CO <0 ■*« 0*0000 oO«*'*0'*^'*000 FQ MHW W HM MhI« II l|l»H 2 ^ 1 I I I I I I ^ mOw^i^-OO Oo Oi t^OO •OOO' r*«»»«OiH»n^ Mr-O0oorao*'^i-0 J >■_» ,2 «'*(*j««>0^ O-^O « M ^ « t^ ■* r-QO »CPO«r~t> "S ®««MO*^ t^QO «o O* t**0 Oi !-• « . (^ g J J i^oo t^ ViOO ^O H t»>o 00 ^ en "* to O Oi «^ ©"O *vOi 00O'*O«^'«t'O»^tO ^ •^ 0> QtOOOOCOO 0> O't^OO CMOOO <» & O>Q0 OOD 0> 0> 0> tQCO O>eO00 O>O00 Ot O>00 00 (^ 0> 0> 0> 0> O>0O 0> 0»0 t« OOO 1^ t^ O' 0> 0> O O>00 0> Ok O C^ t^ ^ O>00 r« t» V) oso 00 *"* -O OOOQOOOOOO OO r~ & OOO Oi 00000>0<0 ^ Vi*nOO t*"0 t >0 to 00 < d t>oo i^ *>. o 00 00 t^o r- o a o ^^oo owioo.OiOiOi mcjt^ ^oo to m m 1 1^ "o __ •ooocooooooopiooooooo CO too o o 00 t w t w to w t M o to CQ M MMHIH-H HI (—1 ^ >ooo C4 o too noo too OOOOOOi oooooooooo M«0 t too « 1 1 to n _. ^OOOOOO tOOO W«00OO0000 OOOOQOCOOOO MOOmOOOOOOWO < O 00 t>. t too 00 O 1 1 O 00 to 00 OOQOOOWOO 0«»fl'*«'-'OMOO» ^ Q O tn vi t>- tfio «»« OiO t^r»(«j r-wt^ooo*^ «Mf»3(*i *^o o »« •■ Si^ I V I I I I I I I 1 I I I I I I I I I I I I I I I I I i I I I t r I <(-(MlH«MNCI«WM^MmOOr»J OMOWM'-'OlOtO.MOOM CO MHMMMMMMIHI-IMHMI-IM mhMIIHIHW HHHMIHI-t H M >* do O. w 2 Q d ,-c _-i l^ t •''O t^OO Oi O w « fO t loO r-OO^ & O M « «j t *o w f^H \o>ooO'0*oi^t»r^t^t^t^r«.r» t-^^- r^-ooooooooooco o oj the Fourth, Fifth, and Sixth School Grades 125 n »n to wmw ViVitnvi^ tOO"'^HO'**»nrO'*[0»OHco inv)inio'*ioiJ^ioio»OTj-^iOTKo M H H M H H H H H H H (OO ^H H M rovtc^ [*)iO»/»"*0 M^ J^J^wmiow -^wiioto^^uT^t-wj MMHOM« t^H\OI-*« HMQ^OMOOOOOOHOHO OHOMO^OMQOOHtnHH I T r — It M M Th f^OO fO 0m00'OO> MWMloHWit^iOt moo MOitMO «C1^-HwNVlMMto(^ ^oo « * I I I < 11 > > I I I I CIOOOOO ^«000 OO«0?*O'jt0MN0N«^0W OOM'jtoONWtoOOOOW I I I I I I I I I I I I I t 00 0> r^ Oi*0 00 I". Oi O" OiOO O" t^ c* Oi ^^ r«00 00 >0 0> *^ Oi t^OO 00 »oOO Oi t»« o^ r*00 00 Oi Oi *^ Oi OiOO * 00 t» Oi*o 0.00 00 Oioo o. o >o \o r* oi *^oo m vj tJ-00 Oitct^OiO woooot^oi t^oo 000 »o 000 »00 00 w V) »n ooo Oi ooiOi»oOi Ov>o t^oo HOO»n«somOi»oOiwmo «^ ''loo j> o t^oo 00 r* oi Oi»o 00 * d H m>0 »rHO r- O lO'O ** >000 t*'*'^uiO*^« t*3>OW t>.\0 *0 h to ■* dOO OQ>OiOtOkOiO OiOiOiCOiOi ^OnnM^O^O 0&W«t n«««0Nc<«PIOt««« *t^ *O*«00 Th'^'^(NH«« NO^^ w« ■*« 0»0 ^ ro«'0* vOHHooMt^NiOO^^d-^WOOO 'O'tm^fw TfvO ^J-«OO^^Om O M N ■>*■ ■* »0 **« ^^ ^OO00«*^OOOW000 t^v> o o Q -^ t»aofoo r-ooooO'^ 'tw Ht-MTj-Qooooo ^.i^oHM mvo aoo mo 000 o o llVlVl IIIII lllTlllMIMIIII MMTillMll'lI MWOOOM OOiHHM O>nO>MMC4OiHH00^0OOmO> «OiOOOHHO.OOiOtoOOO 00 Oi o M n m^ * I'l'o t«-oo ly oi H ct m ^ >n*o t^oo Oi o h « m ^f^ lo^o t^oo a o m « m *t >o« i>oo oi O.OiOOOO^^OOOOO^QHMMMHMMWMHWNNWw'^WNNWwtommmmfocommm •s CO 126 The Reasoning Ability of Children ')rc>oioM« mpo^ ^-w HV)io(o»o mcowHi/iKjeiinMionrtio^ioirt^PO tioioiowioioioioiopo^iotoo^ «»nui«>nro^^ CQ OmO^i-iW^O'OcOmO'-iohm M>HnH-i00i- cq ■ ^■^ . . . r<.0 p^co T »0 ro o « c^ -* lOWl o o -r W I I I "^ ' ' (MM > < I I O t*) o ._J J WOOt-.wit^WOiM'-'H'Ot^HMrfHtO \00'+&»0 ■*«> * O ■* C1 wi ^vo h \0 *0 -» -» H < pq -» H n ■t rt Oi t*l OiOO 00 vj'O CO 00 OiCO GO 00 coo « 00 t^ O-OO ©"00 00 so r* ■* Ul lOOO »0 00 t^'O *>■ j3 N*o «w *^co m to oioo t>. « Oi »noo Oi n fj w ■^ to « w^o »o>o TtoO W O to O ^00 ^OMH«cOOt^ OO^'*^'*^ *"0 «'**00«0«** WOO^OOO^'iwwOHMioow viMMH^ctCirocO^OOn^OCiCt^ TJ■C*0^-OHtOMMT|■«H t^O «o vO0O««'Oei ^« C* ^0*0 t ^ ^*0 ^« goo « to fOO OiioowOt^OiWOO^O (OO^HO'O^OH*«.0»00'00000 H H H H Wl too. to fO 00 o*a 00 » Ov o.ao ^^ ^ O O Q ►z o M « (o * to'O t^oo oi o H N (o ■* >o^ o ^.oo o> o m « to * »oo t^oo o. o H « to ♦=? « mo »d '^ •^■^■*'*'»nlO»0 Ifl^^ lo WHO ui^O <0>0OO^0W0<0t^»^t<.CN. ^ At-t^ •g of the Fourth, Fifth, and Sixth School Grades 127 m f^^O N(*)*^fOO NM«OfONHH«MMOM OOOMMMMOHWmOOMOOOOMcnOfO I 1 I I I r I 11111 1 I I r lO^^wOwMOO NQOO'^OOuiOOOo'^ MOf^Hmi>r-HNOm(0*not»-^00«OOW O0*0f^0f^0t0 OoOrOOOCOtOmOotO «OmO(^0'OOHOOOf^Ot^OOOOOOW t>. ^ 000 CO 00 c^co c^oo oo ooo a^t^OiO> ocoooooo ooo oo i^oo Oioo CO a a j>co t^oo Ot o> o> r^ a Oi ooo o. doo oi CTooooco t- a t^ a foooo t^ a Oi t^ ooo a t**^ o t^cooooo r- a oi o o. Oi aoo o ^Otov)vitOir«« r^iNt^ too *o r* ^« ^ »o ■**o W r^-o ^n loO o vi t^ ■^ «o m t* t^-o m ^ t» wi wioo sow^N'oo'OM ^^^ ^00 « •* « >o Tfo woo oonnco^woo to « w o "+ Tf T^ 'to o t 0000t*»«0'* O'^M too « 00 O 00 O O too WCO « « «00 0000 « 000 CONOOO MOOO^-W M H «OC«Nt«t« WtO to OtOOOOMO^■ O^OOOONHO^lHNWO'HmOtOtOt tO rn tOOOtCfWOt^ tWOOOOOOOC^ttMO 0^0«Otw fOO 000 to o to r^« WOOOO ooHW)t»a«»rtw O'O to t o o !-• a r^oo ot>. *OH»o^.io«for* u^oo >oooo »oo^«ho*^oo IMMMI lllIfVlMMI IMIMIfltlllMVllllll wHoaMwa*^ aooao«o«woMMco oawwHo«oo«Nfoo«oNo«oofOM t»oo a o M « w) t*^ "J'O r»oo o o H « (*3 1 iflo 1^^ 00 a o M n to t •oo *«.oo a o "-i w « t "lo i^oo a ^% r» r-oo oooooooo^oooooooooo aaaaaaa a^^ aaooooooooooHHMMHHHHHH ^ ^ ^ ^ ^ i^ '^ ^_ ^ttttttttttt ti-H tt»oin»oin»oio>n>rt»o»oifl»no»o»o>n O O •s -g CO to 128 The Reasoning Ability of Children •*-(*)wOt*iH o Oh fO'O MMN MnoN^OMHt^toc4>OM«mHi-*>.(*)«/10HOWMMOt-'IHH(*JP'iOO'*0«M'0* u m c» pq 0 CO Oi Oi t^OO O>00 O CO O-'O 00 0> Oi t- »oW 00 00 1^00 00 t- t^ r* tOO Oi Oi t* f* Oi o> *• m flj 00 choe o. o a o> ©""o oioo a o o> t^ a i>oo oo o. r* ^ oioo tv aoo oo oo oo t^ a t^ ooo t>oo eo » o> t— I H Xi o f^^ v> r» »o r* o OiOO 00 Oi ^»oo *».'0 >o w aoo O'O o o t^ooo'O ^t^Oit^^ r^eo oo oo t^ o w Pj t»0 (*) (O (o^O »0 O «0 r«.*0 >0 (^ ^ t* t^lO'O'OOO » ^. Oi 0*0 00*0 »o «o »O00 00 « •O'O *o -^oo o> o> CQ 0««ThO'0*0«'*««>'*^« ■^ '*'0 ■* ^*0 VO -^OO 00 VO "O \0 00 -O 'fOOOO « t ■^Tf'O O t'O t— I >-i O00 « « « ooooo-* H H H H n O ^ m-O O 'O O O 00 00 O too t W«*t«rOOO'*««'*0'*N«00(*jW)«C«'0 (*j*0 O O O t-H o«oooo'*f^<^'*>o oowwei oonoHO^owo as z «OiHTj-&«*O«t».0'-'w«t ij^M(*iOOiO^O OH ■>i-o t^O'00'*'00000»'iHOOkO ,—1 ln»OV)IO»0>0»')»OtO»0»0>0 to VX-H >010»010»0»0>0»0>010>0»0»0»010>0»0>0>0*0»0>0>0»0»0»0 I of the Fourth, Fifth, and Sixth School Grades 129 M ^ >-< PO'O MM© «h«Ohohm^i-ih«h4WiOmw«i-> Mr-iOOHNO'-i'-'O^OOHOO I I I I I I I I III I OOiO'^OiOOo Of^ f^'O V0c*i00'0{00'00'00'0000' PO«OOOO^OOOf^fOOO*00 o o 00 "o ■*oo ooo cooooootsoooo^'ovo'f* ■^'O 00^ 00 -o ■*oo 00 o*oo 000 DON ^00 lO. O. OiOO Oi ■* 0>« O'^N wiPOO '-' roO^W ror/) t^O O O ^0 po O r^O t "1 r* * V "^ ^"^ "^"^ ^^ "^ ^ OOOOOOOw OOO^ONOONOtNOOnO'tM'O'OOO vO-'J-OO'^OOoO ■*oo o ««p '^ ]*" ^ \0 1^00 W so 00 m m t^O ff) ■* H r^ Wi "It t^^O M'-00 1^ OnM000000 t^OO 00 » »^ "H ■^ Ci m»o i-> O fOI^>or*»0« u-jM m o>(^o ^c^ d en »ftOO i-i\oO*OiM 000«'^« ■*'0 ■* « ro fr-'O t^^wO»~-»oO «0 t^OO P0>')'*000O'roM WO O O O t,.0»0>^0''^^*' 000. ■*00 OOOO^O r^CTCvCO. OOO -^t aoO 1^00 aO>OiTfOO>0'0-»/l Ooo 0> t^ -"t ''ICO Oi 000 Oi o>oooo o. oioooooooo 000 a oi t^ r». o> o Oi i-'o o-oo o 00 a d t^OOOO Oi 0.00 o o>oo m c t^oo o o 00 o* Oioo o Oi o. Oi t^ o >ooo o>ooat^ooo»t^oo>t^ 0000 aoo o^ t^ 0.00 oi o> r^ »o »o t-00 •O O 0> '^ Oi^ 0>vO 00 \o »/i r~ f^OO 00 m c i^ m t^O r^ « -o too m 10 »n rj- « »oco oo^o-o t^mrow wm 000'O*O't0N« 'OOO ■^00 ■*'0 O'OO^t'OoOTh^O^Ot WN »o>0 fOO "* Tl- W lO^O ■*'0 r*0 'Q OO'OOO^O'O^ OOOOOWO to OOONOOO»0'00"+000 00 't'O T^ uiCOOO '■trO't'O'O ooooooo f/)lO'*)'*« <*)" O 't'O 'O>On«rtUit^'tN'*t«0f^«TfO> N-^rONfOW^r^WNNoO^OOO"! OO'OW'+M't'tO 0000000'0'tOOO« 'too 0'*««'0000 NMvOwNNNOWO'^J-'tWNO'O i^iiiiii TiiiMiiiiTiiiiiiii iiMiiiiiiTiViii ^ n <-■ M m Ti- m^O r-*y 00 o. o •-* « ^o ■* *o*0 r^oo a o h « t* it wiO'^ t^oo o. o m « n ■* 100 t^oo a o h c« ^ vo vn\0'0 o>o^0 0'^'0'0 r^txr^i>.t^r~t^»^r* 1^.00 00 00 00 oo 00 00*^00 0000 oaoo-ovao>oo>Oio o o 130 The Reasoning Ability of Children »m l-H OWONWMOOHlnHi-i-^l-h-M Mf^HO'-'HHt^OO (NfJMONMroNOO «00 iHMWWrOfOOHNMOMMH't M«'^0»OW««« •«^0'-'WW^ 01>-O tJ-0>-IWO*0<^W^ "^ O oOOrOOOfOOOOOr^OOr^ roO'OPO(*:rOOO'0 <^^ -^•t'0'*l-OOf^"*00'*wo«M N^ 1^00 ■* M M NOO M 00 •^'O t- W -tt Tj-00 O f^- O '-' I « H I „ M I M *^ ■Ot^HOO'o-+H'?- ^co ^00 co^wto*-''^ woo V <5 '^ m > > J < NOO m 00 OiCO 0>COo>0>COoOt^OQOa O000f^00000it^0i0> 1^00 OioO & Oi OiOO & 00 OiCO •"• ^ \0 O "OCO O t>-O>o,O^00 « aco Oi r^ r-OO 00 O" O 000 O miOOooOOOOO-Ov OiOO < rt vioo t r^co r». 1 10 t-^ to ^^'0 -o 00 10 >o "o moo r* fn ^* O ^ t^ t^ OlOo t^oo "O «^ t^ o Oi rn M00-Tj-iHtttoN'0 t »oo o *ooo t f*)oo t ro po t^ n « t n m o >o oowotO'ooo -t to ttocsto ONt^wootooo noon ^ H M M d Viiiirriirii'iTiiiM ^ o o 00 Zro t "oo t^co Oi o I-* N (O t "oo I-- "Tf-ooo-OMenPot too^ r^oo o o h « f^ t >oipp \o f^oo OOOOOOOwwMMWMi-iM '^ MHNMWNNNW^WWNtorotor*)!^ r*)<^ r^ (^ m r3 __ •O'OOOOOO'OO'OOOO'OO ^H \00*0'OOOO^O^OT-100'00^00^0'0^^— ivOOO of the Fourth, Fifth, and Sixth School Grades 131 «Tl-MfOlO«OI-INM OMC4H^MlOMC>t^C4HC4iOO^O>^ *0« »O00 rf« ^M Mrfw OH Mt^W >-i O Oi ^^ OH^fH CTcOofO '^^ OiO t^ (^ ^'O O <^ fO O O '^ t^ ro'O o ^'O O t^ Oi^O Oi M t- M h ro c\0 °° ©■ H CT 10 Oi OiCO lo ^ O -^ o f^OO WOiOiMM oo^^o-o IS. Oi O Ov "^ O" ^00 O O>00 O>00 Oif^Oic'^OiO'Oi O'CO Oi OiOO 0> Oi t^ t^ O>00 'too 000 Oi 000 0> Oi Oi 00 00 Oi Ov Oi t^ Ov 0,00 00 00 OiOO O>00 Oi O" ^ Oi l^ Oi »^ OiOO Ct^O" OiOiOiCOif^r* OOO P* Oi Oi Oi & Oi o 00 O Oi OiOO o, &00 ©> Oi Oi O" 000 O'OO Oi ^ Oi r^OO r-OiOi(^t^Oi Ot^OOiOO 0>0 Oi O" O>00 00 t^ Oi-O rooo ^'i^ ^00 00 t^ oi H CO *o o t^oo 1-00 Oi c r»oo r^ O- t^oo r» o o^oo o o Oi -fl- m 't f^oo co "O « « co w irf- Tf- '^ n ■^lO'O \O00 00^^0*020 0,0 CO oO'OOOOOOON'* O't ^"O WOCO 000 « N « 'J-'*^''*- MM hHHH M •-< W O ■* O^ •^'Ooo O 'OOOODOOoO»0*Oo'^000 0'*0 ■*'0 00 ^NOOOO TfW 000 W^ ■^'OTf rtO W 't^ « ^ fO^ 0*0 000>00^*^000^00000 0000 ^^ 00 W "O o ^ « >tco ooon'o^ooo ^ ^vO ^WTj-Q^^O^ \0*000«f*^'*<^00«000«OOC>0« OOtJ- ^^ 00 to TfOO 00 O ^ ■* 't'O « O liOOO'^'^OONtO 00*0 r^O NO'0'*)'-cwo^OOiOWH O 'O'O HiorOt*OHCO>0>00 f) *-' h |1|[V|[|I lilllllMMriMMI TllTllllMMIlVl 0> O H M ro ^lOVO t-«y 00 O O M « m ^loO *^00 Oi O H « PO TfiO'T-'O l>-00 O O H N fO t >o^ f^CO O- O H pn^^rf'*^^'* ■^^ ■H-'S-iotov)ioioioV)ui»o io*o- vo*0>0\OvO'^'0\0«0 t^t*t^r^r^i>t^r*r» r*oo 00 sOO'0'00»0^0\0'<0.-i'0»0'0*0*0*0'OOOvOVO'0'0>0'00'0^0,-i'0»00»0'0^\0^0^0\0»0>00'0\0 132 The Reasoning Ability of Children -— O-r t--'--^r--', — , ctnciw^ — .O -* trci'r:^ — — oct^". — ^ -r*ro ^cc oc tnao * r I < ! > I II ! 1 II I _ d -2 O r^X Cweiii-T«---r(,c=C-" O "S"^ -^ao "-oc r^—OO >r— . CCt^«i/--r „ ^XnOXCCXXCCXCoc^iOX nOOCC'CXX'CXXX xoo«ocxxc ixt^r-f^-- —-=' X - « o >ro ^ -c=«:>Dr-=- DXX -» ?e^-"X=e-r- :? C X f^ C —.■X. -CXXX^ioO'?'^ -r*^ r-X X lo o c> o oc o f-x -rx X C X ' -r; >- 5 ' - - - . - _ - ._ < -, c. f-c r^xx t^D=-ci:-x='Mi/-.xx 3.r>='3>:>X5DO-CQ0 ^c-rococo r~X __ a:-^'^^'^"^*^ ''**' t^;>xc'3.cxx xddxcxsc t^oo c oor-cocr^^ _ Ot^L/:X=''2f^3CXC»Oot^OX XCCXOXCOXOO OOOOCQOCC < XZ-rCOCXXCCXCCxXOX wCXXXQCOO-^-S OQXCXXOO tCCXCCXOOX-^ -rX ::; O X X 00 r^X OCXC'-CXCO oco-=cxoo <^ — — *^ — — " — M-M — ^ cx^'C-rcx'S-xxxx *^oo coo ooxnCv: -^C^nC oxiotj-cgoo^ XCW'^CXC-CCCXnwoC'^C t^a>nOf^'20 r^X XX ooxxxcooc* ^ Oh.C-'=-X-*-~rX«0-='*nO u-»rX'*»0t^t^M002' inr*'«--OOt-t* - --- -- ^ -1 ri^. -^f^XDc-i-r-T-u-C t-X -?■ s^o-^v-^. -^^n-C t-X 3-4- c - ^ -t'rw, cr* - -f -r'f -*■:*:xxx^^:^;?3:^:^^>:? ^^ ^ooooooOOOO*^ — -•-- — ~ — — of the Fourth, Fifth, and Sixth School Grades 133 ioi^mio*oto *>ov Tt/iuitrttou-i i/^ioioinintoi' -, ro >0 »0 ■"*■-' T 10 »<^ « *t 0«o Ulr0>OO^iOlOVl^^uli/l^Tf^TfiOiO»f'J' u-, ^ioiot}-10^>0 ^«N'O»rtH00MM t»0 u^ n dO ■* M »OoO - r^ « rj- M 10 10 I e* M -<*■ h I CI « cy> w « "1 io« to Tj- m H •o « ■tf « loco fo •"'O "> >-< ►^ O «o iH o o CI •'J m n ^ t^ (vj-o 10 o wj'O ^'O (*)!>■ f- (*) ^o Q o "O r* o <^ I HW0"'*0O0 01*0 to -0*0 ^« * »0«0(000000*0o. (O>o M r-'O CO ^00 « n I lor-fOci tnioM r*^ ^oooowTfooooo "O-oooOoo ■*« cocooooOmoooooo'* 'too CO Y H -^00 ^OOO-O f^ « too r»to« r»»OM f*) OiOO O H CO M irt^O 00 OOKXSOmCOOOOoO 0**0 o>oo oit^CTOO ao>>-'OviOi« o^oooo coi>>Oiw lo^im^ t^oo » *o CO ■*0O -^ CI 00 o w m r* (oco tn •-< r« »o>o « ooo o'^'O^o ^ r^^o 'O ^ -OfO'HCiO<0« OO 00 00 00 OiOO A o> t^ 00 o>oo o> o o o> C^OO 00 Oi O O O'OO CO o o> o o OQCO O" O- o, c r,.oo a O. O* & O Oi Oi 0.00 r^ too &0O O, 0> CT O" o, o, o>00 Oi CT C* O'OO 'O CO Oi Oi O t^ ©> OOO Q,OiQ&0 r^r-O ^o toOr--00 Oir~r»-0 O>000 00 O>00 0> roOOOO 0> OOCOioOOOCO o> o o o c o> o>oo o> o vo t» r* '^'O o 00 00 *o so t~ « o^o oioo t-"© d o. p- o-oo r* 000 1/1 po m ^o o '^ w *oo "O co^o ■fo cooooO'Oto'Ooooo ■*«o "^oo to^o^owCT o pfocooOiot ■to 00 o 60 too O 'O ■to o o CO CO 00 t^oo o « o o w o « "joo o«'0««Moooo»ooo'0*ooo»o eoooot too O HM hHmH h m to to 00 coco o t e« "^O t\o w t>0«fle) ttM fO tO'OOO'tt oooooOmoooco H H M H H « t too "O t loo o t o * 1^0 00 00 M o CO t to wt^oooooNooor*- 000 too CTO too •or^o ooontoOiHio oOwOwo'^O'*'') &co o t^ t »^ c^ n to o O 00 ro n t t O O I PI CO t I C4 « n n - r^ i^ f^ t^ O tr^*^ 00 O* O * I W^ « M ro ' > r^,— I r~ r* r^ t* f» i^ t » t«. t^ «N. I^ t