AN ANALYSIS OF PUPILS’ MISTAKES 
 IN GEOMETRY 
 
 By 
 
 GRACE ERMINIE MADDEN 
 
 A. B. University of Illinois, 1917 
 
 THESIS 
 
 SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS 
 FOR THE DEGREE OF MASTER OF ARTS IN EDUCATION 
 IN THE GRADUATE SCHOOL OF THE UNIVERSITY 
 OF ILLINOIS, 1922 
 
 URBANA, ILLINOIS 
 
Digitized by the Internet Archive 
 in 2015 
 
 https://archive.org/details/anaiysisofpupilsOOmadd 
 

 
 UNIVERSITY OF ILLINOIS 
 
 THE GRADUATE SCHOOL 
 
 June _ 2 
 
 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY 
 
 SUPERVISION BY_^ GRACE ERUHIIS I-IADDEH - 
 
 ENl'ITLED^ _ M MIALYSIS OF PUPIL S* lIISTAiaS DT GEOMSTHY 
 
 BE ACCEPTED AS FULFILLING 'PHIS PART OF THE REQUIREMENTS FOR 
 THE DEGREE OF MSTER OF ARTS III 3DUCATI0II 
 
 In Charge of Thesis 
 
 Head /of Department 
 
 Recommendation concurred iiU 
 
 Committee 
 
 on 
 
 Final Examination'* 
 
 Required for doctor’s degree but not for master's 
 
 /lopcN n 
 
 .J' . o’ •->' ' / 
 
TABLE OP CQNTENIS 
 
 Page 
 
 Introduction 1 
 
 Chapter I. Similar Studies 2 
 
 Chapter II. An Analysis of the Errors 
 
 Tables. 
 
 The Nature of the Errors 
 
 Chapter III. Causes of Mistalces and Hemedial Suggestions 50 
 
 Appendix, The Examination Questions ....57 
 
 Bibliography. 70 
 

1 
 
 INTBODUCTION 
 
 The purpose of this Investigation is to analyze the difficulties of 
 high school pupils in Plane Geometry, by noting and classifying their errors, 
 and to suggest methods of teaching which may be useful in meeting these 
 difficulties. Examination papers written January 1922 were secured from 
 Decatur, Clinton, Jacksonville, Champaign, Urbana, and University of Illinois 
 hl^ schools. Some semester examination papers from January and June 1921 were 
 available at Champaign High School and were included. Quiz papers similar to 
 the semester examinations , except that they were shorter, from Decatur and 
 Champaign were used. Four hundred eighty-eight semester papers in response to 
 eleven sets of questions and 117 quiz papers based on three sets of questions 
 were examined. 
 
 The errors were analyzed and those of like nature listed together u.dei 
 a descriptive name. The papers had been previously marked by the teacher of eac]i 
 section. His corrections or suggestions to the pupil, noted on the paper, w’ere 
 used in analyzing the exact nature of the error. TJhere no comment was offered 
 the nature of the error was determined after a study of the question asked and 
 of satisfactory responses by other pupils for which full credit was given. 
 
 The errors were then classified according to the mental processes in- 
 volved and the per cents determined. Since the nature and comparative number | 
 of mistakes varied with the type of question asked, a large number of papers 
 in response to a variety of quest lonswere used and the per cent of error of 
 each type compared with the per cent of questions calling for responses of that 
 type. The semester grades of the pupils were secured also. These data and the 
 method of selection, all papers available in each school were used. Indicate a 
 normal selection of errors. 
 

 
 
 
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2 
 
 Chapter I 
 
 SIMILAR STUDIES 
 
 1. C.W. Odell, of the University of Illinois,^ (in ’’School and Society” 
 
 December 31,1921) reports ”A Study of One Thousand Errors in Latin Prose 
 Composition.” His data were obtained from quizzes, written, class and board 
 work, of pupils in a small middle western town, Pour teachers cooperated. Two 
 different beginning text books and one prose composition book plus 50 per cent 
 supplementary material, were used. Errors for each of the four years are 
 classified under the following general headings: Declension, Conjugation, Order, 
 Comparison, Analysis, Vocabulary, Omissions, and Insertions. The first two were 
 subdivided into Parts of Speech and under each the factors with which form 
 varies, for exanople. Case, Declension, Number, Stem, Tense, Voice, Person, etc. 
 
 A second table gives the same data in per cents, A third table classifies all 
 errors into three groups: (I) errors made through wrong analysis and imperfect 
 knowledge of syntax, (II) errors caused by ignorance of forms, words, and rules, 
 which were chiefly matters of mechanical memory, (HI) errors of pure careless- 
 ness, A consistent one-third of all errors was caused by lack of mechanical 
 memory. Less than one-fourth were due to so-called "lack of reasoning” and about 
 one-half to carelessness. No teaching suggestions are offered although the 
 author states that certain changes were made in his own teaching as a result 
 of study of these data. 
 
 2. Thorndike gives two reports of studies in reading. "The Understand- 
 2 
 
 ing of Eentences” is based on the answers of 500 pupils of elementary school 
 
 1. P.643-6. 
 
 2. Elementary Educ.J. , Vol. 18, p. 98-114 
 

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 elements were familiar, but whose sentence structure was more elaborate than 
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 ability of the pupil to understand totals, few of whose elements are untaiown, | 
 but whose internal relations are somewhat intricate and subtle." The pupil is i 
 aslced to study the paragraph and answer the questions, reading as often as he 
 wishes* Answers typical of the mistakes in the order of their frequency are | 
 given. A table of sixteen overpotent elements and the responses which they call 
 forth is included in the data. 
 
 As a result of these investigations, eight causes of error in reading 
 are suggested. 
 
 a. Failure to set the mind toward reading and writing the answers - 
 pupil more interested in something else and fails to try. Such a situation 
 arises only rarely, 
 
 b. Some other set of association, productive of writing, prepotent 
 over the "read and ansv«er" set. For example, a sixth grade pupil copies all the 
 questions v^lle another copies the last two words of each question. 
 
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 being made to the paragraph. Such prepotency may effect the total response or 
 it may, and more frequently is, only occasional. 
 
 d. The fourth is overpotency of some element. The wrong paragraph, | 
 or a wrong element of a paragraph may be used, or a wrong element may be added | 
 to the correct one. When a pupil does not understand, the known element becomes 
 overpotent, 
 
 e. Underpotency, the obvious complement of overpotency, occurs. 
 Frequently the underpotent element is a modifying phrase or clause, omission of 
 vdiich completely changes the meaning of the sentence. 
 
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 f. It Is also the case that elements in the question tend to become 
 overpotent and to determine response more or less irrespective of the rest of 
 the question. 
 
 g. Since three to four times as many gave vrcng answers as failed to 
 answer, Thorndike concludes that the pupil has no criterion for understanding or 
 accepting the correctness of an answer. He accepts a series of words as an 
 answer because of a superficial appropriateness. No examination is felt neces- 
 sary. The penalty for omission should be lighter than for incorrect answers. 
 Healthy criticism, but not such as to frighten the child, is needed. He should 
 be encouraged to add to his method a careful examination to see if the question 
 fits. 
 
 i. Correct elements may be transposed into wrong relations. Under- 
 standing may be described as "thinking things together". "The contributory 
 tendencies of each word and word group have to be right not only in nature, but 
 also in amount of potency or influence of force, each in comparison with othera,” 
 
 The commonest cause of error is underpotency or overpotency of elements 
 of the question or paragraph. 
 
 3. "Reading as Reasoning; A Study of Mistakes in Paragraph Reading," 
 similar in method and conclusions by the same author, is based on all responses 
 of 200 pupils of grade six to four questions about one paragraph. The data show 
 frequency and per cent of frequency for each response and a table of ten over- 
 potent words with the answers they influenced. 
 
 He shows that reading may be wrong or Inadequate from three different 
 
 causes: 
 
 a. Wrong connection with words singly. 
 
 b. Overpotency or underpotency of elements. 
 
 3. J. of Educ. Psych. ,Vol.8,p.323-332, June, 1917. 
 
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 c. Pallure to treat Ideas produced by reading as provisional, 
 and 80 to inspect and welcome or reject them as they appear. 
 
 The relation of these three elements of reading to the following three 
 elements of solving mathematical problems is pointed out: 
 
 a. Selecting right elements. 
 
 b. Putting them together in right relations. 
 
 c. hight amount of weight of influence or force of each. 
 
 Thorndike concludes that reading involves the same sort of organiza- 
 tion and analytic action of ideas as occurs in thinking of supposedly higher 
 sorts. "It appears likely, therefore, that many children fall in features 
 
 of other subjects not because they have understood and remembered the facts and 
 principles but have been unable to organize and use them; or because they have 
 understood them but have been unable to remember themj but because they never 
 
 4 
 
 understood them," 
 
 4. In connection with an analysis of mental processes Huger’ s 
 "Psychology of Efficiency, k Study of Mechanical Puzzles" is suggestive. His 
 problem was to analyze human methods of meeting relatively novel situations 
 and of reducing their control to acta of skill. It deals with the part dif- 
 ferent sorts of thought processes actually play in meeting novel situations, 
 and, so far as possible, with conditions favoring the development of variation. 
 Mechanical puzzles which Involve actual manipulation of materials were used. 
 
 All were possible of solution. For the most part they were made of wire and 
 Involved removing of some part of the apparatus, such as a ring, star, or | 
 
 heart, from the remaining portion. Most of them were analytical and tridimen- 
 sional, The movements required were rather complex. 
 
 The subject examined each puzzle separately for an average of fifty 
 trials of one to one and a half hours in the presence of the examiner, A 
 4. J. of Educ.Psych, ,Yol,8,p. 331. 
 
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triple account was taken of each trial. One account was written or dictated 
 by the subject at the close of each trial. Running accounts of several subjects 
 were record®d by the examiner In the first two or three trials with a new puzzle. 
 In each case the examiner wrote a description of all the movements as they were 
 made by the subject. 
 
 The function tested, analysis In tridimensional space, was relatively 
 poorly developed. The subject could not mentally construct in any completeness 
 the spatial transformations required. Discriminations were difficult since no 
 complete system of terms was available to stand for the discriminations when 
 once made. The ideas with which a subject sought to reason out the problem were 
 not closely enough related to the case In hand to be of much value. Some were 
 of negative value, eliminating one method as Impossible, others a positive hin- 
 drance starting a subject on a futile method. The lack: of Ideas closely related 
 to puzzles and capacity of constructing transformations in three dimensions was 
 reflected In the attitude of the pupils. Round about methods, no decrease In 
 time for repeated successes, som=tluies an increase, success the result of random 
 movements, all showing lack of analysis were characteristic of the behavior of 
 the subjects. 
 
 Types of analysis of the puzzles varied in explicitness from a vague 
 feeling of familiarity at chance recurrence to ability to use analysis In a 
 novel situation and give a general formula for Its use under varying conditions. 
 Analysis varied In extent from a partial, such as a locus analysis - simply re- 
 cognition of the part of the puzzle involved in an accidental solution - to a 
 total unified analysis, either factual Image type or general formula, recognizing 
 the process as a single structure. The relation of time of analysis to motor 
 
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 variation varied. Motor variation may be successful but unnoticed, or analysis 
 
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 may come first and motor response follow much later. Analysis and motor varia- 
 tion may be simultaneous yet distinguishable, or analysis may occur at some 
 point of the motor variation and the course of the movement be continued purpose- 
 ly. Both sensory or perceptual and image or ideational analysis occurred. The 
 former often came with a rush or flash. The latter was advantageous in lopping | 
 off irrelevant data. Verbal image has advantage over factual in still further j 
 foreshortening the process and in greater control in recall. | 
 
 Consciously adopted variation, with explicit analysis of variation was 
 the most efficient raethod. Anticipatory analysis was difficult due to the in- 
 ability of picturing transformations and the strong impulse to manipulate. 
 Greometric knowledge was of little value. Ideas of general scientific method 
 of procedure seemed to be of more importance in attacking new problems than ideas 
 of geometry. The factors entering into the problem attitude, confidence and 
 high intellectual activity and attention, freedom toward variation, and critical 
 evaluation of suggestions, were noted to be connected with efficient forms of 
 response. 
 
 5. Belated to Huger' s "Study of Mechanical Puzzles" is earth's 
 "Psychology of Biddle Solution; An Experiment in Purposive Thinking." The rid- 
 dles required a mental solution rather than mechanical manipulation. The pur- 
 pose of the study was to compare the methods of purposive thinking used in solv- 
 ing riddles with the methods used in learning by trial and error. The subjects 
 of this experiment were three hundred and thirty-one college and normal school 
 students. The records of other individuals who knew one or more of the riddles 
 were discarded. Ten riddles were selected because of their seeming fairness as 
 mental problems, Biddles capable of only one answer were used as far as possible. 
 The subject was allowed three minutes for each riddle. He recorded all guesses 
 both right and wrong. A sample record sheet is included in the data. One table 
 5, J.B, Garth, J, of Educ. Psych., Jan. 1920, Vol.II, p, 16-33. 
 
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arranges the riddles in order of difficulty as based on frequency of solution. 
 
 The following Is a typical easy riddle. '*If your aunt’s sister Is not your aunt , 
 then ^at kin Is she to you?" "What Is It that asks no questions yet requires 
 to be answered many times?" Illustrates a hard riddle. The Introspections of 
 five subjects on riddle one and four subjects on riddle two show the associa- 
 tions which suggest each guess. The guesses are tabulated to show frequency forj 
 
 I 
 
 each riddle. Additional tables show the relation of difficulty, frequency, and i 
 rare guesses. 
 
 Garth sets forth some rather definite conclusions from the experiment. 
 
 (l.) One must believe In the trial and error character of the method em- 
 ployed In riddle solution. 
 
 (2.) Speedy guessing tends, as this objectively determined, to militate 
 against successful guessing. 
 
 (3.^ Rare associations characterize speedy guessing. 
 
 (4.^ Usualness or homogeneity of guessing marks the average guesser’s 
 association and is not characteristic of the speedy guessers. 
 
 (5.'i Slow guessing Is not marked by successful solution or homogeneity of 
 associations. However, the unusualness as determined in this experiment Is not 
 so great as In the case of speedy guessers. 
 
 This experiment with riddles gives evidence for Thorndike’s statement,^ 
 "This process of attentive consideration and selection or rejection is clearly 
 shown In the search of a proper word to express a meaning. In attempts to solve | 
 problems of all sorts from the simplest riddle or puzzle to the most abstruse 
 question In mathematics or science and in summoning evidence to support argument!' 
 
 n 
 
 The results also bear out Dewey In his conclusion that "Too few sug- 
 gestions, Indicate a dry and meager mental habit On the other hand, sug- 
 
 gestions may be numerous and too varied for the beat interests of the mental 
 E. B.T .Thorndike, "Elements of Psychology," p.265. 
 
 7» John Dewey, "How We Think." P«36. 
 

 
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 habit, so many suggestions may rise that the person Is at a loss to select a- 
 mong them The best mental habit involves a balance between paucity and re- 
 
 dundancy of suggestions.” 
 
 6. Thorndike's ” Psychology of Arithmetic” contains much suggestive 
 material. The chapters on "The Constitution of Arithmetical Abilities”® 
 analyzes certain important or neglected abilities of arithmetic as samples. 
 
 As a first example the meaning of a fraction is considered. Seventeen minor 
 abilities are listed. Scientific teaching builds up the total ability to manip- 
 ulate with fractions as a fusion or organization of these lesser abilities. 
 
 At the same time that these minor abilities are being developed the pupil gets 
 his knowledge of the meaning of a fraction at zero cost. In the case of the 
 meaning of a fraction, the ability and so the learning, is much more elaborate 
 than common practice has assumed; ”ln the case of subtraction and division 
 tables the learning is ouch less so.” Except perhaps in the case of the dullest 
 twentieth of pupils, the bonds formed in the subtraction and division tables j 
 
 I 
 
 are somewhat facilitated by the already learned addition and multiplication. 
 Instead of memorizing these tables independently the pupil may by a properly 
 arranged set of exercises derive them fronl^the corresponding addition and multi- 
 plication. I 
 
 A thorough analysis of the mental function involved in arithmetical \ 
 
 t 
 
 t 
 
 learning turns into the question, ”7/hat are the elementary bonds that constitute' 
 
 ij 
 
 these functions?” The problem of teaching arithmetic is psychologically a j| 
 
 ! ! 
 
 problem in the development of a hierarchy of intellectual habits, which becomes 
 
 ii 
 
 in large measure a problem of the choice of bonds to be formed and of the dis- 
 
 t 
 
 covery of the best order into which to form them and the best means of forming 
 each in that order. 
 
 M 
 
 Pupils learn the method of manipulation of numbers not by deduction 
 8. Psychology of Arithmetic, p.59. 
 
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 from their taaowledge of decimal notation but rather by seeing the method employ- 
 ed and acquiring them as associative habits. They add the 3 of 23 to the 3 of 
 53 and the 2 of 23 to the 5 of 53 at the start, in nine cases out of ten be- 
 cause they see the teacher do so. All but the very dullest tv^entieth or so of 
 children come In the end to something more than rote knowledge,- to understand, 
 to know that the procedure is right. V/e who have already formed and long used 
 the right habits can hardly realize the force of mere association. The pupil 
 writes 61 for 16 for the same reason that he writes, 63, 64, 65, 26, 36, 56, and | 
 so on correctly. He has learned to write the 6 in the same order in which he 
 speaks It. If the pupil has been drilled In writing 28 after the sum of 8,6, 
 
 9, and 5, he will be sure to do so in two column addition, even though a second 
 column la added also, unless some counter force influences him. Such cases 
 illustrate the fact that the learner rarely can, and almost never does, survey 
 and analyze an arithmetical situation and justify what he is going to do by ar- 
 ticulate deductions from principles. He usually feels the situation more or lessi 
 vaguely and responds to It as he has responded to it or some situation like it 
 in the last. 
 
 Psychologists wish the pupil to reason not less than he has in the 
 9 
 
 first but more. ’’They find, however, that you do not secure reasoning in a 
 pupil by demanding it, and that his learning of a general truth without the pro- 
 per development of organized habits back of it is likely to be, not a rational 
 learning of that general truth, but only a mechanical memorizing of a verbal 
 statement cfjit. The newer pedagogy is careful to help him build up these con- 
 
 I 
 
 nections or bonds ahead of and along with the general truth or principle, so thali 
 he can better understand it. It secures more reasoning in reality by not pre- 
 tending to secure so much. The newer pedagogy of arithmetic, then, scrutinizes 
 every element of knowledge, every connection made in the mind of the learner, 
 
 9. Thorndike, "Psychology of Arithmetic," p.73. 
 

 
 
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 so as to choose those which provide the most instructive experiences, those 
 which will grow together into an orderly, rational system of thinking about 
 ntirabers and quantitative facta. It is not enough for a problem to be a case 
 of some rule; it must help review and consolidate habits already acquired or 
 lead up to and facilitate habits to be acquired. Every detail of the pupil’s 
 
 ( 
 
 work must do maximum service in arithmetical learning," | 
 
 10 
 
 Thorndike next lists desirable bonds now often neglected. 
 
 (1) , "In the case of all save the very gifted children, the additions with 
 higher decades - that is, the bonds, 16 plus 7 equals 23, 26 plus 7 eiguals 33, 
 
 14 plus 8 equals 22, 24 plus 8 equals 32, and the like need to be speci- 
 
 fically practiced until the tendency becomes generalized. The quotients with 
 remainders for the divisions of every number to 19 by 2, every number to 29 by 
 3, every number to 39 by 4, and so on should be taught as well as the even 
 divisions, 
 
 (2) . "The equation form is the simplest uniform way yet devised to state 
 a quantitative issue. Consequently this form should be employed widely in ac- 
 counting and the treatment of commercial problems, since it is a leading con- 
 tribution of algebra to business and industrial life. Arithmetic, hov/ever, 
 can present it nearly as well," 
 
 (3) , In multiplying and dividing with fractions, special bonds should be 
 formed to counteract the nov/ harmful influence of the "multiply = get a larger 
 number", and "divide = get a smaller number" bonds which all work with integers 
 has been reinforcing. The following rules for multiplying and dividing by a 
 fraction will counteract the old habit. 
 
 Mult iolication. 
 
 When you multiply a number by anything more than 1 the result is 
 larger than the number, 
 
 10, Thorndike, "Psychology of Arithmetic", p. 75. 
 
 11. Ibid, , p.79. 
 

 
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When you multiply a number by 1 the result Is the same as the number. 
 
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 ■ 
 
 than the number. | 
 
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 than the number, 
 
 (4), Bonds should early be formed between manipulations of numbers and 
 means of checking, A discussion of wasteful and harmful bonds follows. We in- 
 clude one example. The multiplications of 2 to 12 by 11 and 12 as single con- 
 nections should be left for the pupil to acquire by himself as he needs them. 
 These connections Interfere with the process of learning two place multiplica- 
 tion. 
 
 Be concludes, "When we have cured all of our faults in respect to 
 bonds now neglected that should be formed and useless or harmful bonds formed 
 for no valid reason, and found all the possibilities for wiser selection of 
 bonds we shall have enormously improved the teaching of arithmetic. The ideal 
 is such choice of bonds as will improve the function in question at the least 
 cost of time and effort. The guiding principles may be kept in mind in the form 
 of seven simple but golden rules:- 
 
 (1) , Consider the situation the pupil faces, 
 
 (2) . Consider the response you wish to connect with it. 
 
 (3) . Form the bond: do not expect it to come by a miracle, 
 
 (4) . Other things being equal, form no bond that will have to be broken, 
 
 12, Thorndike , "Psychology of Arithmetic", p. 101. 
 
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 (5) . Other things being equal, do not form tw5 or three bonds when one will 
 serve. 
 
 (6) . Other things being equal, form bohds in a way that they are required 
 to act. 
 
 (7) . flavor, therefore, the situation T^ich life Itself will offer, and 
 the response which life itself will demand.” 
 
 In the chapters on psychology of drill in arithmetic, Thorndike sug- 
 gests certain general facts. (1). The constituent bonds involved in the funda- 
 mental operation with numbers need to be much stronger than they are now. 
 
 These bonds should be strong enough to abolish errors in competition, excdpt 
 those due to temporary lapses. (2). Certain bonds are of service for only a 
 limited time and so need to be formed only to a limited degree of strength. The 
 general deductive theory of arithmetic should not be learned only to be forgot- 
 ten. What is learned should be learned much later than now and should be among 
 the most permanent of a pupil’s stock of knowledge. The formation of bonds to 
 a limited strength because they are to be lost in their first form, in order 
 to be worked over in different ways in other bonds to which they contribute is 
 the most important case of low permanence of bonds. (3). Bonds and abilities 
 should rarely be formed each by itself alone and never kept so. Every oond 
 formed should be formed with due consideration of every other bond that has been 
 or will be formed: every ability should be practiced in the most effective 
 possible relation with other abilities. 
 
 I 
 
 The chapter entitled ’’The Psychology of Thinking*' presents certain | 
 notions of abstractions, generalization, analysis, and reasoning in a manner 
 suggestive for our problem (errors in geometry). 
 
 Abstraction and generalization depend upon analysis and upon bonds 
 formed with more or less subtle elements rather than with gross total concrete 
 situations. Three means are employed to facilitate analysis. The first of 
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14 
 
 these Is having the learner respond to the total situation containing the 
 element In question with the attitude of piece-meal examination, and with alert 
 ness to one elenaent after another. The second means Is having the learner re^- 
 spond to many situations each containing the element in question but with vary- 
 ing concomitance. The third is having the learner respond to situations which, 
 pair by pair, present the element in a certain context and present that same 
 context with the opposite of the element in question, or something very unlike 
 the element, A child is taught to respond to "one fifth of a cake" and in con- 
 trast "five cakes". These means utilize the laws of use, disuse, satisfaction, 
 and discomfort, to disengage a response element from gross total responses and 
 to attach it to some situation element. "What happens in such cases is that 
 the response, by being connected with many situations, alike in the presence 
 of the element in question and different in other respects, is bound firmly 
 to that element and loosely to each of its concomitants. Conversely, any 
 element is bound firmly to any one response that is made to all situations con- 
 taining it and very, very loosely to each of these responses that are made to 
 only a few of the situations containing it," A situation then acquires bonds 
 not only with some response to it as a gross total, but also with responses to 
 any of its elements that have appeared in any other gross total. 
 
 Learning by analysis does not often proceed in the carefully organ- 
 ized way representated by the most ingenious marshalling of comparing and con- 
 trasting activities. The associations may come in a haphazard manner over a 
 long interval of time. The process of analysis is the same in such casual, un- 
 systematized formation of connections with elements as in the deliberately 
 managed piecemeal inspection, comparison, and contrast mentioned above. The 
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15 
 
 time and done more or less Incidentally in other connections. The systerastic 
 method is more formal and artificial. The opportxinist ic method takes advantage 
 of pupils* interests and genuine life situations. It is harder to manage but 
 more desirable i»dien it can be properly administered and results tested. 
 
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 j 
 
 general ideas in arithmetic from the application of three additional principles,! 
 They are: (1), provide enough actual experiences before asking the pupil to I 
 
 I 
 
 understand and use an abstract or general idea, (2), develop such ideas gradu- 
 ally, not attempting to give complete and perfect ideas all at once, (3), 
 develop such ideas so far as possible from experiences vdiich will be valuable 
 to the pupil in and of themselves, quite apart from their merits as aids in 
 developing the abstraction or general notion. 
 
 Working with qualities and relations that are only partly understood 
 does under certain conditions give control over them. The general process of 
 analytic learning in life is to respond as well as one can; to get a clearer 
 idea thereby; to respond better next time; and so on. What begins as a blind 
 habit of manipulation started by imitation may grow into the power of correct 
 response to the essential element, A pupil should not first master a principle 
 and then merely apply it. On the contrary, the applications should help to 
 establish, extend, and refine the principle - the work a pupil does with number^ 
 should be a main means of increasing his understanding of the principles of 
 arithmetic as a science, | 
 
 In proportion as thinking is purposive, with selection from the ideas 
 that come up, and in proportion as it deals with novel problems for which no 
 ready made habitual response is available, and in proportion as many bonds act 
 together in an organized way to produce response, we call it reasoning. In 
 both inductive and deductive reasoning the process involves the analysis of 
 
 facts into their elements, the selection of the elements that are deemed sig- 
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16 
 
 nlf leant for the question In hand, the attachment of a certain amount of im- 
 portance or 'weight to each of them, and their use in the right relations. 
 
 Thought may fail because it has not suitable facts or does not select from them 
 the right ones, or does not attach the right amount of weight to each, or does j 
 not put them together properly. f 
 
 If we laclc any of the necessary facts, the first task of reasoning is 
 to acquire these facts. Other things being equal, problems where some fact abou: 
 common measures must be brought to bear, or some table or prices or discoimts 
 must be consulted, or some business custom must be remembered or looked up are 
 somewhat better training in thinking than problems where all the data are given 
 in the ‘problem itself. At least it is unwise to have so many problems of the 
 latter sort that the pupil may come to think of a problem in applied arithmetic 
 as a problem where everything is given and he has only to manipulate the data. 
 Life does not present its problems so. 
 
 Arithmetical problems should not be so stated as to rule out all 
 quantitative elements except those vhich should be considered. If they are the 
 pupils are tempted to think that in every problem they must use all the quanti- 
 ties given. The elements selected must not only be right but also in the right 
 relations to one another. 
 
 Reasoning or selective, inferential thinking, is not at all opposed to, 
 or independent of, the laws of habit, but really is their necessary result under 
 the conditions imposed by man's nature and training, "It is true that man's 
 behavior In meeting novel problems goes beyond, or even against, the habits 
 represented by bonds leading from gross total situations and customarily ab- 
 stracted elements thereof. One reason... is simply that the finer, subtle, pre- 
 ferential bonds, with subtler and less often abstracted elements go beyond, and 
 at times against , the grosser and more usual bonds..,. The other reason is that 
 

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17 
 
 in meeting novel problems the mental set or attitude is likely to be one ^diich 
 rejects one after another response as its unfitness to satisfy a certain de- 
 siration appears."^® Successful responses to novel data, associations by sim- 
 ilarity and purposive behavior are in only apparent opposition to the fundamental 
 laws of associative learning. Association by similarity is simply the tendency 
 of an element to provoke the responses which have been bound to it. Purposive 
 behavior is the most important case of the influence of the attitude or set or 
 
 I 
 
 adjustment of an organism in determining (1) which bonds shall act and (2) vdiichj 
 
 results will satisfy, Reasoning is not a radically different sort of force | 
 
 operating against habit but the organization and cooperation of many habits , ! 
 
 thinking facts together* The pupil’s own total repertory of bonds relevant 
 
 to the problem is what selects and rejects. Almost everything in arithmetic 
 
 should be taught as a habit that has connections wdth habits already acquired 
 
 and will work in an organization with other habits to come. The use of this 
 
 organized hierarchy of habits to solve novel problems is reasoning. 
 
 14 
 
 7. The purpose of Percival M. Symonds in his ’’Psychology of Errors 
 in Algebra” is to show the psychology of recurring errors, and from this to 
 point out methods of eradicating them or better, of preventing the formation of 
 the wrong habits. Seven characteristics of behavior are enumerated. The first 
 is multiple response to the same external situation. The problem of teaching | 
 algebra is how to secure the desired response. A second characteristic of be- ! 
 havior is that the pupil has a set or determination to get the answer. There 
 are three common satisfactions that determine a pupil’s choice of his answer. 
 
 One is social approval evidenced by the teacher or member of the class. F.eli- 
 ance on approval is not wholly bad but for success we must have some other means 
 by which the pupil may determine the answer independently. A second satisfac- 
 tion is an answer with which he may compare his - the answer book or key. An 
 
 13. Thorndike, ”Psychology of Arithmetic”, p,191. 
 
 14. The Mathematics Teacher, Feb. 1922, Vol, 15, p.93ff. 
 
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 assignment such as '’Practice these until you can get 12 right in six minutes" 
 will remove all motive for misuse of the answer book. A third and more intrin- 
 
 I 
 
 sic satisfaction is that of checking the answer. 
 
 A third characteristic of behavior is a practical response to a 
 situation. a^+lOa +24 = (a+8)(a+2) is a case where the prepotency of getting 
 8a+2a=10a has overshadowed getting a product of 24. The law of partial response 
 works especially with signs; -4(3x-4)= -12 x - 16. The effect of the minus 
 sign of the first 4 or the minus sign of the second 4 is ignored. 
 
 A fourth characteristic of behavior is what Thorndilie calls "response 
 by analogy**. In any new situation one responds as he would respond to the 
 situation most nearly resembling it in his past experience. Nearly all cases 
 of unfinished examples can be traced back to a previous process where v&iat is nov 
 a step in our unfinished example was then the final result. 
 
 x2 - 81 = 0 
 ( x+9 ) { x-9 ) = 0 
 
 is a case of response by analogy - the usual response to x^-81. Sometimes 
 these errors show how well previous processes have been learned. They surely 
 show how v.« have failed tn emphasizing those elements of a problem that require 
 a new or different response. 
 
 A fifth characteristic of behavior is associative shifting, the learn- 
 ing process itself. Of the responses suggested that one is selected which gives 
 satisfaction. 3y drill and repetition, these correct responses become definite- j 
 
 ! 
 
 ly established habits. | 
 
 Perseveration - a tendency to repeat an act time after time, when once | 
 it has been aroused by some appropriate stimulus - is a sixth characteristic of | 
 behavior. Perseveration is apt to occur in algebra when a pupil is not working 
 at top notch enthusiasm or v;hen he is disturbed by other train of thoughts. 
 Occasionally it may result from a too great concentration upon a particular 
 

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19 
 
 feature of a problem. Errors in vTriting, such as 
 (2x-3)^*= 4y^ - 12 x+9 
 
 may be a perseveration from some previous example. When one does all the 
 work mentally some element is quite apt to be perseverated. In 
 y^+y®6 
 
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 it may have been the minus sign before the 6 which he has to imagine when the 
 
 6 is transposed or it may be the minus sign of the 3 -which he has just written, j 
 
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 Anticipation may be a seventh characteristic of behavior. By this we \ 
 mean focusing attention on some element ahead of the writing which causes the 
 reaction to occur before it should. These errors are always accidental. Anti- 
 cipation is slw’ays a large factor in causing careless errors in examinations. 
 
 This psychological analysis of errors enables us to improve our 
 teaching. First, it sho-ws us the need for a psychological analysis of algebraic 
 processes into the constituent connection or bond involved. Especially those 
 bonds that require a different response than in previous processes need analysis 
 in order to receive emphasis in future teaching and drill. 
 
 In the second place, our analysis shov;s the need for drill or practice 
 in various processes. Drill or practice is the remedy for attention to certain 
 elements of an example to the exclusion of certain other elements. There must 
 be differential drill, with more emphasis on those connections that are hindered 
 by past responses, and relatively less emphasis on those connections that are 
 helped by past responses. Since only correct responses help build up correct j 
 
 I 
 
 I 
 
 habits many easy examples should be given rather than a few difficult ones. 
 
 In the third place, since accidental errors are largely due to mind 
 wandering, our psychological analysis shows need for reorganization of our class 
 room procedure by timing all formal activities and rapid shifts in type of menta! 
 processes involved. 
 

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 We see mistalcea to be Inconsistent because we see the meanings and uses 
 of symbols and see their interrelations. The pupil does not see them to be errors 
 because he has not learned the meaning of the symbols and has not comprehended 
 
 I 
 
 their significance. Psychology teaches us that pupils first learn to carry out | 
 the processes and manipulations, and thereby acquire the meaning and significance! 
 of what they do. 
 
 8. Section II of Chapter Vi of Rugg and Clark’s ”Scientific Method in 
 the Reconstruction of Ninth Grade Mathematics” is a detailed analysis of pupils’ 
 errors as revealed by the test given. Table III in the appendix gives a com- 
 plete list of "recurring errors” for each test, together with frequency and per- 
 centage of occurrence. 
 
 I 
 
 Test 1: Collecting terms. A common error, mistake in signs in combin- 
 ing similar terms occurs frequently in this test. The error is due to lack 
 of elasticity or flexibility of attention which will permit the pupil to hold the 
 result of one operation while he is adjusting to a new one. 
 
 Test 2:Bvaluation or substitution. Relatively few errors are made in 
 this operation. Nearly one third are due to squaring the product of literal 
 factors Instead of the one factor designated by the exponent. 
 
 Test 3: Simple equations. The greatest difficulty in solving simple 
 equations is illustrated by c = 6 as a solution of 4c=6c+12; this is caused by 
 failure to hold the sign of the -2c in mind. 
 
 Test 5: Parenthesis. A rather high degree of frequency has been secured t 
 in the teaching of this operation. About one half of the mistakes made were in 
 the use of signs. 
 
 Test 6; Special products. The chief difficulty - one-fourth of all 
 recurring errors-ls in obtaining the sum of the cross products. This difficulty 
 may be explained by Improper emphasis in teaching and by a temporary lapse of at- 
 tention between the first and last terms of the result obtained. 
 
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21 
 
 Test 7J Exponents. Addition of exponents in involution involves the 
 greatest difficulty. Following this is first, (1) the error of squiring the 
 exponent instead of multiplying and (2) failure to raise all the factors of the 
 product to the required power. Experimentation will tend to increase marlcedly 
 the teacher’s effectiveness. 
 
 Test 8: Factoring, Thirty-nine percent of all the recurring errors 
 Indicate positive inability in particular types of factoring. Failure to get 
 the correct sum of the cross products and to recognize the highest common 
 monomial are the most frequent. Students are outstandingly weak in continuing 
 txie process until tue piime factOxs nave ueen founa. The reason is simply they 
 have not had sufficient practice in examples requiring the successive use of 
 several factoring processes. 
 
 Test 9 and 10: Fractions. Errors in use of signs predominate, indicat- 
 ing that this very simple process of collecting signs and numbers has not been 
 properly habituated earlier in the course. 
 
 Test 111 Formulas. The error most in evidence in this operation is 
 that involved in selecting the coefficient of the unknown in the solution of 
 fractional equations. The recurrence indicates a lack of clearcut practise in 
 naoituation (»o tnis type of work. 
 
 Teat 12T Quadratic equations. The sum of the cross products diffi- 
 culty occurs here again. Failure to find both roots gives considerable 
 difficulty also. 
 
 Test 13^ Simultaneous equations. Arithmetical errors and mistakes in 
 signs in addition and subtraction are the most frequent. 
 
 Test 14t fiadicals. Positive inability and incomplete extraction of 
 
 root { V = b^ are most frequent here. The evident lack of mastery 
 
 of formal skill indicates the need of devices to supplement the text book. The 
 teaching emphasis of the text book is thoroughly iincritical. The printed prac- 
 
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22 
 
 tlce exercise is a supplementary device for conducting drill economically. The 
 fundamental aim of instruction is the solution of problems primarily of the 
 reasoning or interpretive type. We, therefore, insist on habituation only in so 
 far as it contributes to efficiency in the solution of such problems, 
 
 9, The purpose of the study of Roger’s "Tests of Mathematical Ability 
 and their Prognostic Value" is to make an analysis of the abilities Involved in 
 high school mathematics, to determine their efficiency and status, their inter- 
 relation, and also their connection with certain other forma of mental capacity. 
 Primarily it is directed toward the discovery of dynamic and quantitative rela- 
 tion between mathematical abilities, rather than toward showing how we think in 
 mathematics from the standpoint of analytic or structural psychology. 
 
 Thirteen tests were selected to touch as many forms of mathematical a- 
 chlevements as possible. They can be divided into three classes, six tests of 
 algebraic ability, five tests of geometrical ability, and tests of language 
 ability. 
 
 The subjects were fifty-three girls attending Wadleigh High School and 
 sixty-one pupils in Horace Mann School for girls. The geometrical tests were as 
 follows;- A geometry test of six exercises requiring numerical solution or proof, 
 tests of superposition, symmetry, matching solids and surfaces, and geometrical 
 definitions. 
 
 The results show that algebra and geometry demand activities of dif- 
 ferent kinds, although algebraic and geometrical abilities are positively related 
 as is usual in the case of desirable traits. The experimental evidence obtained 
 suggests that a marked degree of the power to analyze a complex and abstract 
 situation and to seize upon its implications is the most indispensable element 
 in mathematical proficiency. The closeness of relationship between mathematical 
 ability and ability with words points out that in later mathematics as in primary 
 arithmetic, the problem of teaching children to reason is largely a matter of 
 
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23 
 
 teaching them language and how to use it. Reasoning in school problems has far 
 more to do with language involved in a problem than with number or combination 
 of numbers, 
 
 Miss Rogers concludes that these tests are valuable for general diag- 
 nosis and prognosis of individual ability in mathematics, 
 
 10, Minnick’s study, "An Investigation of Certain Abilities Pundamenta] 
 to the Study of Geometry", is limited to an investigation of certain fundamental 
 abilities Involved in the demonstration of theorems. Corresponding to the four 
 steps in a demonstration are the four mental abilities with which this study is 
 concerned; namely, 
 
 fl.) The ability to draw a figure for a theorem, 
 
 f2.) The ability to state the hypothesis and conclusion accurately in 
 terms of the figure. 
 
 The ability to recall additional Imown facts concerning the figure, 
 
 and 
 
 ('4.^ The ability to select the necessary facts and to arrange them so as 
 to produce a proof. 
 
 The purpose of this investigation is threefold: 
 
 (l,^ To determine the relation of each of these four abilities to 
 teachers* marks, 
 
 (2,) To determine the extent to vdilch these abilities are developed in 
 our high schools. 
 
 (3.) To develop tests which may be used for the purpose of diagnosis, of 
 determining whether or not the weaknesses of a class are due to lack of develop- 
 ment of one or more of these abilities, 
 
 Some of her conclusions are of Interest. Since we do not know that 
 the test measures the respective abilities in the same way, we can not compare 
 the results of the different tests. We can compare the results of giving the 
 

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24 
 
 same test in different schools. In the case of each test, the marks of some 
 schools are quite satisfactory while those of others are extremely low. Thus, 
 although it is possible to develop the abilities v;ith which their study is con- 
 cerned, some schools fail to do so. k diagnosis such as these tests provides 
 will enable the teachers to check the development of each ability and give atten- 
 tion to particular phases of the subject as needed. 
 
 11. Judd devotes one chapter of ’’The Psychology of High School Subjects 
 to "The Psychological Analysis of Geometry." The material for this analysis is 
 procured from three sources: first, a text bock; second, class recitation by 
 the pupils; and third, books on the teaching of mathematics. The text book is 
 considered first. In the presentation of definitions three processes are noted. 
 They are analysis, or distinguishihg the parts, abstraction or ability to go be- 
 yond the real experience, and use of symbols. One tries by the use of an ab- I 
 street idea to get rid of as much sensory content as he can in order to leave be- 
 hind the pure special elements of the experience. 
 
 Jbllowing the definitions in the text comes single experimentation and 
 experimentation through which the properties of figures and geometrical elements 
 are discovered. The pupil gains very little from these definitions and simple- 
 analyses until he has used them repeatedly in later proof. 
 
 An angle is used to illustrate the development of ideas. The angle is 
 first explained by considering one single definite specimen as "the opening be- 
 tween two straight lines." Later he presents the angle in its general special 
 relations as formed by rotating a line about a point. 
 
 Superposition and symmetry are introduced as methods of comparison in- 
 dispensable for further mental development and purposes of application. Symmetry 
 is not so direct as superposition. The former offers more possibility for 
 error. Geometry does not consider it as reliable as superposition. 
 

 
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25 
 
 Geometry Is more than the art of seeing space. The student coaipares a 
 few simple figures and then generalizes his findings in the form of a principle. 
 
 He must have enough space perception to furnish the basis for his geometrical 
 analyses and generalizations. The emphasis on axioms as opposed to postulates 
 illustrates the frequent neglect of space and the emphasis of logic by geometri- 
 cians. 
 
 A demonstration differs psychologically and pedagogically from a defi- 
 nition. The latter is a close compact statement while a demonstration is a de- 
 tailed and explicit unravelling of the situation, Harly theorems train the 
 student m the use of a form of analysis not demanded in definitions. Combining 
 the results of successive steps of analysts in the later theorems calls for a 
 "type and reach of attention which is higher than that required in learning a 
 definitioru'*^® 
 
 In an original exercise a student must first analyze the situation 
 
 1 
 
 which is propounded, and then he must be able to draw out of his fund of accumulaj 
 ted experiences the principles viiich will help him in solving his complex present 
 experience. The only way to teach a student to solve originals is to teach him 
 how to analyze a new problem and how to seek among his store of experiences. 
 
 Prom a presentation of more than one solution of a problem and a discussion of 
 the merits and demerits of each, the pupil will gain an insight into the methods 
 employed by the author in solving the problem. 
 
 In a class recitation the reactions noted were infinite in their 
 variety. Several types of difficulty arose. A lack of appreciation of the de- 
 mands of logic prevented the class from arranging a demonstration when all the 
 facts were known. In another case a student was not able to pass from a logical 
 demonstration to the spacial fact; still another had his mind fixed on a rule 
 of procedure somewhat removed from the fundamental fact. He was committed to 
 15. Judd, "Psychology of High School Subjects", p. 61. 
 

 
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26 
 
 the practical rule and could not get a point of departure suitable to the proof. 
 
 We have noted three spheres of experience, each of -rhich is so different 
 from the other that transition out of one into another is difficult. They are, 
 first, space perception; second, an abstract system of logical steps constituting 
 proof; and third, rule of practical procedure. The student aiust be trained to 
 pass from one to the other. There should be a conscious effort to view the matter 
 from each of the possible points of view and to pass from one to another. The 
 student can see that the fact which is essential for demonstration is often very 
 different from the point of highest attention to the practical operator or to the 
 observer of space. 
 
 Students prefer to go back to simple fundamental theorems rather than 
 to use a recently proved theorem. They have more confidence in simple theorems. 
 
 The part of the figure on which attention was last concentrated is likely 
 to be the starting point of the next step in observation and reasoning. Therefore 
 preparation is of great importance. 
 
 All through the lesson pupils were called on to use their memory and to 
 select from the propositions and facts stored up in memory. The criticisms of 
 memory are directed not against memory but against bad forms of remembering. To 
 a well ordered body of knowledge especially if it is well ordered in many directicni 
 there can be no objection. Such a question as ,'*What do you know about equilateral 
 triangles?" helps students to classify material. Ideas must be put into the 
 mind with their possible relationships clearly recognized if these relationships 
 are, at some later day, to be used productively in calling out the ideas. 
 
 Reasoning involves memory and classification of experiences and the com- 
 bining of the experiences which belong together in leading to a definite conclusion, 
 Some general maxims help the teacher to induce this process in students. "First 
 try to foresee in a general way what end you want to reach. Next marshal all the 
 

 
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27 
 
 facts you can find which are related to this end* Then arrange these facts in a 
 
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 progressive series." 
 
 Judd closes his discussion of geometry with a comparison of two hooks 
 on methods of teaching geometry, one making no attempt to deal with the psycho- 
 •w logy of the situation, the other colored by psychological Interest and discusslom , 
 
 16. Judd, "Psychology of High School Subjects" p.73 
 

 
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28 
 
 Chapter II. 
 
 AIT AIIALY5IS OP PUPILS' EHRORS. 
 
 Table I classifies the errors noted according to the type of responses 
 required, giving both frequency and per cent. The total per cent for each 
 tjrpe is computed on the grand total of errors. For example, 564 or 14.1 per 
 cent of the total 3989 errors occurred in response to questions requiring con- 
 structions. The miscellaneous group includes errors in arithmetic, algebra, 
 grammar, spelling, form, or error due to carelessness having little relation 
 to the type of response required. 
 
 Table II classifies the data of table I according to the mental pro- 
 cesses involved. As in Table I, the total per cent for each subdivision is 
 based on the grand total of errors. For exami)le , 886 or 22,2 per cent of the 
 3989 errors involved memory, 7hen an analysis of the error failed to give any 
 cue to its cause, it v/as included in the miscellaneous list. For example, a 
 fact wrongly assumed by construction may have been the result of incorrect 
 analysis of cons tiuct ion, of a failure to reco^mize the logical consequences of 
 the construction, or of oarelessness . The cause could not be determined from the 
 pupil's record on the paper. So also, assuming a theorem not yet proved may 
 result fron failure to recall the order of proofs given in the text book, or 
 
 » 
 
 from a non-critical acceptance of an idea which is suggested by the question. 
 
 The pupil's response failed to indicate v/hich. 
 
 In Table III the questions asted are classified according to 
 the nature of response required. Frequency and per cents are given for each 
 type of response. V/here a question was composed of two or more parts, each part 
 was counted as a separate question since it called for a distinct response. The 
 
I 
 
 
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29 
 
 total frequency is therefore not identical with the sum of the questions of the 
 examinations. Column 3 giveg,for comparison, the per cent of errors from Table 
 
 1 for each type of response. The second entry in column 3 is the total for 
 all proofs both theorems and exercises, and should be compared with the sum of 
 
 2 and 3 in column 2. In the same v/ay, the last entry in column 3 should be com- 
 pared with the sura of 5 and 6 in column 2. 
 
 Table 17 gives the semester grades of 477 pupils whose papers were used 
 in this investigation. The grades of the pupils of one section were not re- 
 ceived. "D” represents lov; passing" .,orh in most of the schools. represents 
 
 grades from 70 to 74 which is pass ing in one of the schools and conditional in 
 another. In the remaining schools, represents all failures, 74 and belov/. 
 Grades from 70 to 74 are not differentiated. Several of the schools exempt from 
 final examination pupils whose average reaches a certain standard. The best 
 pupils from those schools are excluded from this study because of the source of 
 data. Only one school out of this group has regular raid-year promotions, there- 
 fore, the majority of the pupils enrolled in Geometry II during the first semestea: 
 from which the majority of these data were secured, are irregulars, probably 
 having flunked at least one semester of mathematics. This fact is an important 
 factor in explaining the higher percentage of low grades in Geometry II. 
 
 Table III raa]©s clear the fact that the per cent of errors in proofs of 
 exercises and theorems is out of proportion to the percent of questions requiring 
 proof. It v;as impossible to detemine in some cases which statements requiring 
 proof had been presented to the pupil as theorem, which he had studied (proofs 
 of which he would be required to recall or work out again), and 7/hich were 
 exercises that the pupil may or may not hare seen before. In classifying the 
 question any statement which is given in any well kaov/n text as a theorem or 
 corollary was so counted. Others were termed exercises requiring; proof, "hen 
 
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30 
 
 an exercise required both numerical solution and proof it ^s classed with the 
 former. 
 
 The percents of error in construction, numerical exercises, and quo- 
 tation are somewhat or at least slightly below the corresponding percents of 
 questions. The comparatively low percents of errors in constructions and num- 
 erical exercises indicate that the pupil is more successful in answering those 
 types of questions. Several factors may tend toward their success. Construc- 
 tions and numerical exercises are concrete, and more closely related to exper- 
 ience outside of school, and therefore of more Interest to the average pupil. 
 Numerical exercises (many of them very similar to mensuration problems of gram- 
 mar school arithmetic or to those algebraic problems requiring application of a 
 principle or fornaulae) Involve elements in which the pupil had received previous 
 training. The outstanding difficulty in such exercises is failure to analyse, 
 that is, to plclc out the various factors presented, to reorganize them if neces- 
 sary so as to make clear the connection between the required result and the I 
 principle or formula which the pupil has learned. The close correspondence of 
 error to requirement in quotation indicates that memory for a fact called for 
 directly is a factor but not a dominating factor in influencing failure. 
 
t) 
 
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31 
 
 Table I 
 
 Classification of Errors According to Type of Response Required 
 
 in Order of Frequency. 
 
 Proofs Frequency 
 
 Percents 
 
 1. Wrong authority 
 
 457 
 
 20.0 
 
 2. Absolute inability 
 
 353 
 
 15.5 
 
 3. Omission of authority 
 
 296 
 
 13.0 
 
 4. Omission of step 
 
 296 
 
 13.0 
 
 5. Wrong step 
 
 271 
 
 12.3 
 
 6. Wrong recall of plan of proof 
 
 108 
 
 4.7 
 
 7. Wrong conclusicn 
 
 97 
 
 4.2 
 
 8. Failure to recall plan of proof 
 
 82 
 
 3.6 
 
 9. Omission of proof of construction 
 
 68 
 
 3.0 
 
 10. Misinterpret hypothesis 
 
 54 
 
 2.4 
 
 11. Pact wrongly assumed by hypothesis 
 
 47 
 
 2.1 
 
 12. Omission of part of construction 
 
 47 
 
 2.1 
 
 13. Wrong hypothesis 
 
 45 
 
 2.0 
 
 14. Use of theorem to prove itself 
 
 25 
 
 1.1 
 
 15. Unnecessary step 
 
 12 
 
 .5 
 
 16. Special case 
 
 8 
 
 .3 
 
 17. Step out of order 
 
 6 
 
 .2 
 
 18. Assume theorem not yet proven 
 
 19. Substitution of quantity for one not 
 
 5 
 
 .2 
 
 equal 
 
 3 
 
 .1 
 
 20. Example instead of proof 
 
 2 
 
 .1 
 
 21. Wrong assumption in indirect proof 
 
 1 
 
 .0 
 
 Totals 
 
 2284 
 
 57.2 
 
 Construction 
 
 1. Failure to analyze construction correctly 
 
 
 
 148 
 
 26.2 
 
 2. Omission of explanation of construction 121 
 
 21.4 
 
 3. Omission of part of construction 
 
 56 
 
 9.9 
 
 4. Wrong construction lines 
 
 5. Failure to recall necessary construe- 
 
 52 
 
 9.2 
 
 tion lines 
 
 49 
 
 8.7 
 
 6. Fact wrongly assumed by construction 
 
 41 
 
 7.2 
 
 7. Failure to recall correct method 
 
 26 
 
 4.6 1 
 
 8. Vague explanation of construction 
 
 20 
 
 3.6 
 
 9. Inaccurate freehand drawing 
 
 18 
 
 3.3 
 
 10. Special case 
 
 15 
 
 2.7 
 
 11. Complete inability to construct 
 
 11 
 
 2.0 
 
 12. Omission of statement of construct lor 
 
 6 
 
 1.0 
 
 13. Unnecessary construction 
 
 1 
 
 .2 
 
 Totals 
 
 564 
 
 14.1 
 
M .1 
 
32 
 
 Exercise 8 
 
 I. 
 
 Flallure to analyze correctly 
 
 289 
 
 61.4 
 
 2. 
 
 Wrong interpretation of ’’required'* 
 
 61 
 
 12.7 
 
 3. 
 
 Failure to recall formula correctly 
 
 38 
 
 8.1 
 
 4. 
 
 Misinterpretation of problem 
 
 32 
 
 6.8 
 
 5. 
 
 Wrong Interpretation of ’’given" 
 
 23 
 
 4.9 
 
 6. 
 
 Omission of proof of exercise 
 
 14 
 
 3.0 
 
 7. 
 
 Omission of part of exercise 
 
 6 
 
 1.3 
 
 8. 
 
 Complete inability 
 
 8 
 
 1.7 
 
 
 Totals 
 
 471 
 
 12.0 
 
 
 Miscellaneous 
 
 140 
 
 3.5 
 
 
 Grand Total 
 
 3989 
 
 100.0 
 
iiun mw 
 
 
 
 
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 j 
 
 •T 
 
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 p.'^ 
 
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33 
 
 Table II 
 
 Classification of Errors According to Mental Processes 
 Error 
 
 Memory Frequency Percent 
 
 1. Failure to quote theorem correctly 
 
 325 
 
 36.7 
 
 2. Failure to recall theorem 
 
 150 
 
 16.9 
 
 3. Failure to recall plan of proof correctly 108 
 
 12.4 
 
 4. Failure to recall plan of proof 
 
 82 
 
 9.2 
 
 5. Failure to recall cons’t lines correctly 
 
 52 
 
 5.9 
 
 6. Failure to recall necessary construction 
 
 
 
 lines 
 
 49 
 
 5.' 5 
 
 7. Failure to recall definition correctly 
 
 42 
 
 4.8 
 
 6. Failure to recall formula correctly 
 
 38 
 
 4.4 
 
 9. Failure to recall correct method of 
 
 
 
 construct ion 
 
 26 
 
 2.9 
 
 10. Failure to recall definition 
 
 13 
 
 1.5 
 
 11. Unnecessary construction 
 
 1 
 
 .1 
 
 Totals 
 
 886 
 
 22.2 
 
 inalysis 
 
 
 
 1. Failure to analyze exercise correctly 
 
 289 
 
 38.6 
 
 2. Failure to analyze construction 
 
 148 
 
 19.8 
 
 3. Conclusion wrong 
 
 97 
 
 13.0 
 
 4. Wrong interpretation of "required” in 
 
 
 
 exercise 
 
 61 
 
 8.1 
 
 5. Misinterpret hypothesis 
 
 54 
 
 7.2 
 
 6. Hypothesis wrong 
 
 45 
 
 6.0 
 
 7. Misinterpret problem 
 
 32 
 
 4.3 
 
 8. Wrong interpretation of "given" in 
 
 
 
 exercise 
 
 23 
 
 3.1 
 
 Totals 
 
 749 
 
 18.8 
 
 Illogical 
 
 
 
 1. Use of theorem to prove itself 
 
 25 
 
 58.1 
 
 2. Unnecessary step 
 
 12 
 
 27.9 
 
 3. Step out of order 
 
 6 
 
 14.0 
 
 Totals 
 
 43 
 
 1.1 
 
•, f 
 
 si,.' . I 
 
 6 
 
34 
 
 
 
 Omleelon 
 
 
 
 1. Omission of step 
 
 296 
 
 31.8 
 
 2. Omission of authority 
 
 296 
 
 31.8 
 
 3. Omission of explanation of construction 
 
 121 
 
 13.0 
 
 4, Omission of proof of construction 
 
 68 
 
 7.3 
 
 5. Omission of part of construction 
 
 56 
 
 6.0 
 
 6. Omission of part of proof 
 
 47 
 
 5.1 
 
 7. Vague explanation of construction 
 
 20 
 
 2.2 
 
 8. Omission of proof of exercise 
 
 14 
 
 1.5 
 
 9. Omission of part of exercise 
 
 6 
 
 .6 
 
 10* Omission of statement of construction 
 
 6 
 
 .6 
 
 Totals 
 
 930 
 
 23.3 
 
 Ibaolute Inability 
 
 
 
 1, Complete inability to prove 
 
 353 
 
 95.0 
 
 2. Complete inability to construct 
 
 11 
 
 2.9 
 
 3. Complete Inability to work exercise 
 
 8 
 
 2.1 
 
 Totals 
 
 372 
 
 9.3 
 
 Miscellaneous 
 
 
 
 !• Wrong authority 
 
 457 
 
 45.3 
 
 2. Wrong step 
 
 271 
 
 26.9 
 
 3. Carelessness 
 
 60 
 
 5.9 
 
 4. Pact wrongly assumed by hypothesis 
 
 47 
 
 4.6 
 
 5. Arithmetical error 
 
 44 
 
 4.4 
 
 6. Pact wrongly asstimed by construction 
 
 41 
 
 4.1 
 
 7. Algebraic error 
 
 22 
 
 2.2 
 
 8. Preehand drawing 
 
 18 
 
 1.8 
 
 9* Special case, construction problem 
 
 15 
 
 1.5 
 
 19* Special case, theorem 
 
 8 
 
 .8 
 
 11. Spelling 
 
 6 
 
 .6 
 
 12. Vague meaning 
 
 6 
 
 .6 
 
 13* Assumed theorem not yet proved 
 
 5 
 
 .5 
 
 14. Quantity substituted for one not equal 
 
 3 
 
 .3 
 
 15* Exaoqple instead of proof 
 
 2 
 
 .2 
 
 16. Wrong description of consti*uction 
 
 1 
 
 .1 
 
 17. Wrong assumption in Indirect proof 
 
 1 
 
 .1 
 
 18. Work in bad form 
 
 1 
 
 .1 
 
 19. Error in grammar 
 
 1 
 
 .1 
 
 Totals 
 
 1009 
 
 25.3 
 
 Grand total 
 
 3989 
 
 100.0 
 
 
 
 
I 
 
35 
 
 Table III 
 
 Classification of Questions 
 
 
 
 Frequency 
 
 Per cent 
 
 .Errors 
 
 Construction 
 
 
 33 
 
 24.3 
 
 14.1 
 
 Proof of theorem 
 
 32 
 
 23.5 
 
 
 
 
 
 
 57.2 
 
 Proof of original exercise 
 
 18 
 
 13.2 
 
 
 Numerical exercise 
 
 35 
 
 25.7 
 
 12.0 
 
 Quotation of 
 
 theorem 
 
 16 
 
 11.8 
 
 13.2 
 
 Definition of 
 
 terms 
 
 2 
 
 1.5 
 
 
 Totals 
 
 
 136 
 
 100.0 
 
 
 
 
 Table IV 
 
 
 
 
 Semester 
 
 Grades of Pupils 
 
 
 Orade 
 
 Geometry One 
 
 Geometry Tvo 
 
 
 Frequency 
 
 Per cent 
 
 Frequency 
 
 Percent 
 
 A 
 
 27 
 
 8.3 
 
 6 
 
 5.2 
 
 B 
 
 59 
 
 18.0 
 
 22 
 
 14.3 
 
 C 
 
 72 
 
 22.2 
 
 23 
 
 15.0 
 
 D 
 
 69 
 
 21.3 
 
 50 
 
 32.5 
 
 S 
 
 26 
 
 8.3 
 
 13 
 
 8.4 
 
 F 
 
 71 
 
 22.0 
 
 38 
 
 24.6 
 
 Total 
 
 323 
 
 100.0 
 
 154 
 
 100.0 
 
V 
 
 
 
 .. 1 
 
 
 i‘S 
 
 ■■Xi 
 
 
 ; ■ not^efouO 
 
 I 
 
 
 o 
 
 
 0 
 
 i 
 
 m 
 
36 
 
 The remainder of this chapter will he devoted to a consideration of 
 the natxire of each error as listed In Table I. Errors in construction are dis- 
 cussed first, In the order of their frequency. 
 
 S 
 
 The nature of the error in construction. 
 
 1. Failure to analyze construction correctly. Given side b, an^le B, 
 
 and altitude to side c. Required to construct triangle ABC. Two errors In 
 
 analysis occur here. The pupil sometimes does not know or falls to note the 
 
 significance of ’’given altitude to side c" and draws the altitude to side a or 
 b. Or he may sketch the triangle correctly but fail to realize that he has 
 
 given the hypotenuse and leg of a right triangle, under which conditions he has 
 
 previously learned how to construct a right triangle. Continued use of the tri- 
 angle notation will Increase its meaning for the pupil. Repeated experience 
 with varied construction dependent upon "a known right triangle" will develop a 
 habit of analyzing the situation to discover "a known right triangle." 
 
 2. Omission of explanation of construction. Several factors are 
 responsible for the high frequency of such omissions. The question did not 
 always ask for an explanation of a construction when the teacher felt it neces- 
 sary. In such cases the practice varied widely in different sections. Some 
 classes were accustomed to give complete explanations, otheis apparently felt 
 nothing except construction was necessary. A careful statement of response re- 
 quired would probably have greatly reduced the frequency of these omissions. 
 
 For such errors the teacher Is more to blame than the pupil. 
 
 3. Omission of part of construction. This is typically a case where 
 the pupil’s attention Is concentrated on one factor of the total response. 
 
 That factor may be prepotent because of Its position in the solution - first 
 step - or its familiarity to the pupil. His attention does not include the 
 

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37 
 
 other requirements of the problem probably because he has not had practice In 
 using these elements together. The pupil who is given the perimeter and asked 
 to construct an equilateral triangle divides the perimeter into three equal 
 parts and steps as he did when learning to divide a segment into a given number 
 of equal parts. For the pupil who has had experience in constructing a rectanglt 
 equal to two-thirds or three-fourths of a given square, the habitual response 
 may be prepotent. He falls to attend to the two-thirds. Developing abilities 
 in the forms and relations in which we wish to use them will avoid this error. 
 
 Note. We will discuss number 5 next, and then 4. 
 
 5. Failure to recall necessary construction line. This error refers 
 to the construction of the figure required for the proof of a theorem. A 
 typical case la in the proof of ”The sum of the angles of a triangle is equal 
 to two right angles.** Two constructions are possible, (1) extend one side to 
 form an exterior angle, or, (2) construct through one vertex a parallel to 
 the opposite side. Another frequent failure occurs in the proof of **If two 
 straight lines are parallel the alternate interior angles are equal.” The line 
 forming alternate Interior angles that are equal, if the given angles are as- 
 sumed unequal is omitted. Such lack of recall is accompanied by failure to 
 recall the plan of proof and results in either complete inability to prove or 
 a wrong plan. Special attention is necewsary to strengthen the bonds between 
 theorem, figure, and method. Rapid fire review at the blackboard may be used 
 to exercise the desired bonds. The pupil may sketch the figure with all 
 aul illary lines for each theorem stated. Such exercise will probably show a need 
 for a detailed analysis of the construction necessary for the proof of certain 
 theorems (especially for contrast and comparison of similar figures), followed 
 by additional drill. 
 
 4. Wrong construction lines. The figure for a theorem and the one for 
 
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38 
 
 Its converse are sometimes confused. The construction required for the indirect 
 proof that "Two lines are parallel if the alternate interior angles are equal," 
 is sometimes given for its converse and vice versa. This is another case where 
 care is necessary to connect the correct response to the situation and keep 
 the bonds strong by use. 
 
 6. Pact wrongly assumed by construction. Sometimes a line is con- 
 structed to satisfy several different conditions, one or two of which it may 
 satisfy, but all of which we are not justified in assuming that it can satisfy 
 at one time. A common example is "To construct a line through a given point 
 perpendicular to and bisecting a given line." Two of these conditions a line 
 may satisfy but all three can be true of one line only in very special cases. 
 
 One such error of high frequency occurred in proving that a circle may be con- 
 structed through three points which are not in a straight line. The text used 
 by several of these classes suggests that the point used as the center of the 
 required circle must be proved equally distant from the three given points, but 
 leaves the reason to the pupil. The circle at once suggests that all radii are 
 equal. If the book suggested or the pupil considered that he is to prove that 
 the circle passes through all three points, he would realize he had used as 
 given the very construction he is to prove true. Special preparation for prov- 
 ing construction is necessary in order to overcome the pupil's previous set,- 
 assumlng all constructions true by construction. 
 
 7. Failure to recall correct method. "Required to construct a mean 
 proportional to m and n." The pupil remembers the first step -to construct a 
 semi-circle with m plus n as diameter. He falls to recall a second step - 
 construct a perpendicular to the diameter at intersection of m and n. The 
 attempted proof of the construction shows that the pupil has an inadequate 
 notion of proportion and an Incorrect conception of the meaning of mean 
 
M- 
 
 
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39 
 
 proportional. A careful explanation of the nature of the mean proportional and 
 additional practice in its construction will avoid such error, '’Inscribe a 
 circle in a triangle,’* The perpendicular bisectors of the sides of a triangle 
 are constructed, to locate the center of the circle, instead of the bisectors of 
 the angles. The given situation has been connected with the wrong response, 
 
 A comparison of the methods of construction for inscribed and circumscribed 
 circles and practice in constructing both will aid in forming the correct bonds, 
 
 8, Vague explanation of construction. Such an error may occur in 
 the explanation of a construction line required for the proof of a theorem or 
 in a construction problem. For example, diameter that bisects the chord is 
 perpendicular to the chord and bisects the arc subtended by it,” The plan of 
 proof of this theorem is dependent on a clear understanding of the facts given. 
 The explanation, "draw a diameter” may lead to the false assiLmption that this 
 diameter is perpendicular to the chord, ’’Draw a diameter through the mid point 
 of the chord,” aids in setting forth correctly the given facts in contrast to 
 those requiring proof. Clear, definite explanation facilitates proof in a 
 construction problem. The bonds between a construction and concise, exact 
 explanations should be formed and maintained by use. Dissatisfaction with vague 
 explanation will tend to weaken the uiidesirable bonds, 
 
 9. Inaccurate, Under this head are classed freehand drawings or 
 approxiTiate sketches of figures whose construction is required. For example, in 
 response to '’From an outside point construct a tangent to a circle,” a pupil 
 using a straight edge, by trial and error process, draws a line which passes 
 through the point and appears to touch the circle at only one point. The wrong 
 response has been joined to the stimulus ’’construct”. This undesirable associa- 
 tion is due to two factors. First, an approximate sketch in response to the 
 stimulus ’’construct” has been accepted as satisfactory. Perhaps no definite 
 
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40 
 
 distinction has been made between "sketch” and "construct". And second, the 
 familiar element of the situation - tangent - and the frequent response - "A 
 line that touches the circle at only one point"- are prepotent. Special pre- 
 paration is necessary to overcome this association and to establish the con- 
 nection between the situation "construct a tangent" and the response "a perpen- 
 dicular to a radius at its outer extremity is tangent to the circle." 
 
 10. Special case. A method of construction that gives a satisfactory 
 result under special conditions, but which is not generally applicable to the 
 situation, is used. For example, a pupil constructs "on a given base a triangle 
 equal to a given parallelogram" by using the base of the parallelogram for the 
 base of the triangle and doubling the altitude. Doubling the altitude would 
 fall to give the required triangle if any other base was used. Sometimes this 
 is undoubtedly a bluff. The pupil does not know how to construct the figure 
 under any other conditions but hopes to "get by" by satisfying the letter rather 
 than the spirit of the requirement. In other cases the pupil is not aware that 
 he is limiting the conditions of the problem. In either case the response 
 should be discouraged. Calling attention to the respect in vdiich the response 
 is unsatisfactory and giving no credit for it will encourage the pupil to watch 
 for and avoid special cases. 
 
 11. Complete inability to construct. The pupil considers the ques- 
 he 
 
 tlon. After^ carefully numbers and copies the question, he looks at it for some 
 time, hoping for an inspiration. He finds nothing in his past experience sug- 
 gestive of a solution. Be records no attempt at a solution. He gives up be- 
 cause he does not know how to begin. Either the question is not suitable or his 
 past experience is at fault, A situation requiring the use of new elements or 
 new combinations of known elements is not suited to the ability of any except the 
 upper third of pupils. Past experience may be unsatisfactory from two points 
 
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of view. The bonds involved In needed past erperlence may be too weak to 
 facilitate recall. These bonds may have been formed in some way other than 
 that in vhich they are now required to act. Additional experience in genuine 
 situations will Increase the pupil's ability. 
 
 12. Omission of statement of construction. A definite statement that 
 
 ! 
 
 "IB is a line through a given point. A, tangent to the given circle at point | 
 B," makes the work clearer to another who inspects it. The pupil's comparison | 
 of such a statement with a requirement of the problem tells him that he has or i 
 has not fulfilled the requirements. The statement of the construction also 
 sets forth definitely the fact to be proved. It should be required for every 
 construction. 
 
 13. Unnecessary construction. A small penalty should be attached to 
 adding an unnecessary line to a figure. It tends to confuse a pupil. 
 
 The nature of the error in proof . 
 
 1. Wrong authority. For example, two sides of a triangle are 
 equal because "A bisector divides into two equal parts." The pupil evidently 
 feels that the bisector divides the triangle into two equal parts. The line 
 is the bisector of the vertex angle of the triangle. It divides the triangle 
 into two congruent triangles only in the case of an Isoscles triangle. It is 
 possible that the pupil is thinking of that case but falls to realize the limi- 
 tation of its application. It is more probable that the difficulty is due to a 
 confusion resulting from the similarity of the words "triangle" and "angle." 
 Considerable practice with illustrative material is sometimes needed before a 
 pupil gets clearly in mind the two meanings. Sven after he understands the die-! 
 tlnctlon a tendency to use the wrong name may persist. The Indiscriminate use 
 of the words results in a confusion in the meaning of the early theorem of angle 
 and triangle. His abbreviated statement of the theorem favors the persistence 
 of the difficulty. 
 
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42 
 
 The same authority Is used to prove two alternate interior angles 
 equal. Here the fact used has little meaning to the pupil. It Is one of j 
 
 several facts which may be used to prove two angles equal. It probably seems I 
 
 to the pupil as good a guess as any. This case is typical of a large group of j 
 
 errors made by a class of pupils found in every school - a class very slow to j 
 
 grasp the meaning of and ability to use a logical proof. Intuitional or con- S 
 
 I 
 
 struct ive geometry naay be interesting to them but offers no aid. Unless | 
 
 special supplementary experience with simple demonstrations and a slow rate of 
 advance can be arranged, these pupils can succeed in a class of varied abllitleE 
 only by repeating the course. 
 
 2. Absolute inability. Absolute inability is frequent among the grouj 
 of pupils mentioned above. Careful preparation for the bonds to be formed, and 
 a large amount of practice with simple situations is needed. This will necessi- 
 tate slow progress at the start. Either the time devoted to demonstrative 
 geometry must be increased or the number of theorems decreased. The mental 
 ability of the pupil and his probable future vocation must be considered in 
 determining which is preferable in each case. Absolute inability to prove an 
 exercise Involves the problems suggested for inability to construct. 
 
 3. Omission of authority. Authority for the last step of a proof or | 
 for a step following directly from the previous step is often omitted. The | 
 practice of requiring no authority for a step following directly from the pre- 
 vious one is confusing to the pupil. When is a reason necessary and when not? 
 
 A parallel column arrangement of steps and authorities facilitates the develop- 
 ment of the habit of giving a reason for each step. A gap in either column 
 suggests a search for a suitable fact. I^hilure to find the right fact is then 
 the only explanation for an omission. Failure may result because the stock of 
 facts is too limited or because they are not stored in usable form. 
 
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43 
 
 4. Omission of steps. What has been said of omission of authority ap- 
 
 plies to omission of steps also. 1 step is often omitted, however, when the 
 pupil is not aware of the omission. i?or example, a pupil proves two triangles 
 congruent by two angles and an Included side but shows only one pair of equal 
 angles, A group of steps constituting a distinct section of the proof may be 
 omitted. For example, ''In the same or equal circles the greater of two unequal 
 arcs is subtended by the greater chord,” Given; arc CD is less than arc AB, To 
 prove; chord CD less than chord AB, The pupil gives an Indirect proof assuming 
 that if CD is not less than AB it is equal to AB. He fails to consider the , 
 
 f 
 
 i 
 
 possibility that CD may be greater than AB, In both examples the attention is > 
 
 i 
 
 centered on one element of the proof to the exclusion of the other. A step ■ 
 necessary for a logical proof is sometimes omitted because it appears so obvious , 
 to the pupil. If his attention is called to his ovm difficulties in explaining | 
 a step considered obvious by the author of his text, the pupil readily compre- I 
 hends the additional clearness secured of writing down each step of the proof. 
 
 ”It is obvious” is sometimes used by a pupil to cover up vagueness in his 
 own mind. Such vagueness can be discovered and cleared up at once by completing 
 the proof, 
 
 1 
 
 5. Wrong step. A pupil says, ’’angle 1 equals angle 2 because homologous 
 
 ! 
 
 angles of congruent triangles are equal,” Neither angle 1 nor angle 2 is in i 
 either one of the pair of congruent triangles. Another makes ”CQ is identical 
 to CQ** one step preparatory to proving two triangles congruent, when C^ is a lin^ 
 part of which is in one triangle and part in the other. In each case the pupil 
 fails to examine critically the step used. The inconsistency is very apparent 
 to him when his attention is called to it, 
 
 6. Wrong recall of plan of proof. Two types of error are grouped to- 
 gether here, (1) a plan suitable to one theorem is used for the proof of another 
 to which it is not suited. A pupil tries by the indirect method to prove the 
 

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 base angles of an isoscles triangle are equal. The bond between theorem and 
 plan is not strong enough to function. The bond between some element of the 
 situation and a response may be prepotent. The connections between the sltuatioa 
 ”a theorem to prove” and the response ”what plan shall I follow?” leads the 
 
 f 
 
 pupil to try some known method. (2) A plan may be recalled up to a certain \ 
 
 j 
 
 point correctly. The pupil Is unable to determine the proper plan for the re- 
 mainder of the proof. For example, median of a trapezoid Is parallel and 
 equal to one-half the sum of the bases.” A pupil makes the correct construction. 
 He proves the constructed figure a parallelogram, and uses the auxiliary triangle 
 to prove the median parallel to the bases, but falls to recall how to prove it 
 equal to one-half their sum, 
 
 7 and 13. Wrong conclusion and hypothesis. The most frequent error 
 la confusion of the hypothesis and conclusion. For example, "If the diagonals 
 of a quadrilateral bisect each other, the figure Is a parallelogram.” Several 
 pupils gave as their conclusion, the diagonals bisect each other, assuming by 
 hypothesis that the figure was a parallelogram, A critical comparison of the 
 selected "given" and "to prove” with the theorem as stated will correct these 
 errors. 
 
 8. Failure to recall plan of proof. This error Is similar to failure 
 to recall method In a construction problem, 
 
 9, Omission of proof of construction. Prepotency of the construction I 
 or inability to prove may be the reason for these omissions, 
 
 10. Misinterpret hypothesis. In number 13 the wrong part of the 
 statement is chosen as the hypothesis. Here the correct part of the theorem Is 
 considered, but an incorrect meaning attached. 
 
 11, Fhct wrongly assumed by hypothesis. "If the diagonals of a quad- 
 rllateral bisect each other, the figure Is a parallelogram,” The hypothesis 
 
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45 
 
 and conclusion are stated correctly, In symbols, at the beginning of the proof. 
 Step four says, **iB equals CD, By definition of a parallelogram." The figure 
 la not a parallelogram according to the hypothesis. Critical examination of the 
 hypothesis enables the pupil to find and avoid the error, 
 
 12. Omission of part of construction. These errors are similar to | 
 those Involving Incomplete recall of plan of proof In number 6, Construction 
 lines and plan of proof both In complete form need to be strongly corjiected wlthj 
 the theorem. 
 
 13. See 7 above. 
 
 14. Use of theorem to prove Itself, Such an error usually occurs 
 In the last stop of a theorem. For example, to prove "the angles opposite the 
 equal sides of an isoscles triangle are equal." The last step Is angle A equals 
 angle B because "the angles opposite the equal sides of an Isoscles triangle are 
 equal." Also in proving the "median of a traoezold is parallel to the bases 
 and equal to one-half of their sum." The laat step Is GP equals l/2(AD plus BC) 
 beoauso"tho median of a trapezoid Is equal to one-half of the sum of the bases." 
 The symbolism used In the steps of proof raalres possible euch an error. No one 
 would say "the base angles of an isoscles triangle are equal" because "The 
 
 base angles of an Isoscles triangle are equal." A translation of these symbols 
 baclc to words will show the pupil the nature of the error. The error Is per- 
 fectly natural. The theorem Is fresh In his mind. The pupil realizes that he 
 has a situation which la Identical with that described In the hypothesis. 
 Therefore his theorem aoplies. The following principle will help to eliminate 
 such an error. Only facta which require no proof (axioms) and facts which have 
 been proved may be used as authorities. 
 
 15. Unnecessary step. A pupil proves two pairs of sides and two 
 pairs of angles equal In order to prove two triangles congruent. He does not 
 
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46 
 
 consider that three equal parts are sufficient to prove two triangles congruent, 
 
 16. Special case. This error is of the same nature as a special case 
 In construction. The proof applies under special conditions only. For example, 
 to prove **the common Internal tangent of two equal circles bisects their line 
 of centers," he uses two tangent circles. The proof based on this figure is 
 not generally applicable to two equal sides. 
 
 17. Step out of order. A common mistake Is, to say that two angles 
 are equal because they are homonogous angles of congruent triangle, and then 
 use the equal angles to show that the triangles are congruent. The pupil has 
 not yet acquired a satisfactory knowledge of the nature of a logical proof. 
 
 18. Assume a theorem not yet proved. A pupil assumes "the area of . 
 a rectangle equals the product of Its base by Its altitude," In order to prove 
 "the areas of two rectangles have the same ratio as the products of their 
 bases and altitudes." He does not realize that the former depends upon the 
 latter for Its proof. 
 
 19. Substitution of quantity for one not equal. A half of one lihe 
 Is substituted for a half of another line regardless of the fact that the two 
 lines are unequal. A critical examination of the situation would reveal the 
 inappropriateness of such a step, 
 
 20. Example Instead of proof. In one case a pupil drew a circle and 
 assigned niunerical values to the central angles and arcs in order to Illustrate 
 that "In the same or equal circles, equal central angles cut out equal arcs," 
 
 For him the meaning of the theorem was prepotent over the requirement of the 
 question. 
 
 21. Wrong assumption in Indirect proof. The negative of the hypothe- 
 sis Is assumed Instead of the negative of the conclusion, A pupil begins the 
 proof of "the alternate interior angles formed by two parallel lines are equal" 
 
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47 
 
 by assuming that the lines are not parallel. He has connected the wrong response 
 to the situation, "How shall I begin an indirect proof?" 
 
 The nature of the errors in exercises. 
 
 1. failure to analyze correctly. One may read an exercise so that he 
 con 5 >rehends the special facts in their proper relation but fails to connect the 
 given data with a known theorem upon which the solution of the exercise depends. 
 One case will illustrate. The pupil is asked to find the area of the square 
 inscribed in or circumscribed about a circle of radius 4. He draws the figure 
 and marks the length of the radius correctly. He falls to connect the Pythagor- 
 ean theorem (which he has proved in a previous question) with the Isoscles right 
 triangle vdiose hypotenuse is the required side of the circumscribed square. His 
 experience has been lacking in applications of this theorem to situations com- 
 plicated by other lines. 
 
 2. Wrong interpretation of "required". This error differs from (1) 
 above in that here the pupil does not get the correct meaning of the requirement 
 of the exercise. Errors of this kind are often due to failure to attend to what 
 is required. One pupil drew the correct figure in the exercise cited above and, 
 ignoring the square entirely, found the area of the circle. Another pupil sub- 
 tracts the circumferences of the two concentric circles and marks his answer 
 "difference in area." He stated the requirement of IPe problem correctly but 
 consciously computed the circumferences. (He marked it circumference.) 
 
 Other misinterpretations are due to wrong meanings of words Involved. I 
 "A school house is to be located forty feet from each of two intersecting 
 roads." Distance from a point to a line does not mean for the pupil the 
 perpendicular distance from the point to the line so he locates the school 
 house forty feet, measured along an oblique line, from each road. 
 
 3. failure to recall formulae correctly. The needed formulae may be 
 entirely forgotten, the wrong one of two possible formulae may come to mind, or 
 
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48 
 
 the needed formulae may be stated incorrectly, For example, ”How many sides 
 has a regular polygon if one of the interior angles contains 108 degrees?” re- 
 mains unanswered for lack of the formulae, "The sum of the interior angles of 
 a polygon of n sides is (n-2} straight angles”, or ”The sides of the quadri- 
 lateral ABCD are 10, 17, 13, and 20 respectively. The diagonal AC is 21, Find 
 the area,” The pupil attempts to determine the area of each triangle by the 
 
 formula A = a* b , He fails to consider A •V'st s-a) ( s-b) ( s-c) , Several 
 
 2 
 
 pupils state the area of a circle = Ellr, Additional experience with these 
 formulae in the situation in which they will be used is necessary for their 
 effective recall. If the formulae are not sufficiently important to be re- 
 called, such exercises should be omitted or the required formulae given as part 
 of the data, 
 
 4. Misinterpretation of the problem, ”A tree is to be planted 10 
 feet from the front wall and 15 feet from the corner of a house. How many 
 locations are possible? Show by diagram,” One pupil located a line of trees 
 10 feet in front of the house, and another group in a circle 15 feet from a 
 corner, A second puts tirees 15 feet from a rear corner of the house. A third 
 measures 10 feet from a front wall or fence bounding the lawn. A fourth tries 
 to locate a tree which will be 15 feet from each of the two front corners of 
 the house and also 10 feet from the front wall. 
 
 5. Wrong interpretation of "given”. The pupil states the given 
 facts correctly but fails to use them with due consideration for their meaning. 
 
 6. Omission of proof of exercise. What was said of "omission of 
 proof of a construction” above applies here also. 
 
 7. Omission of part of an exercise. A pupil often fails to hold 
 two parts of an exercise in his attention. He finds the ratio of the area 
 of two similar polygons whose sides are 3 and 9, but fails to consider the 
 ratio of their perimeter. Perhaps in his eagerness to find the answer he 
 

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49 
 
 fails to read the second part of the question. A critical examination of the 
 question and his response would reveal to him the omission. 
 
 8. Complete inability, A scanty supply of facts from which to choose 
 or an xmorganized stock results in frequent inability. 
 
 The nature of errors in quotation . 
 
 1. Failure to recall theorem correctly. This group includes theorems 
 misquoted in a proof and also theorems called for directly. A slight change 
 
 in wording ai&y change the meaning entirely. Fhr example, "If two triangles 
 have three angles (sides is correct) of one equal to three angles of the other 
 the triangles are congruent." Or, "The areas of two similar polygons have the 
 same ratio as (the squares of) their homologous sides." Frequent experience 
 with the correct meaning will strengthen the desired bonds. The bonds involved 
 in erroneous statements will weaken if continually greeted by dissatisfaction. 
 
 2. Failure to recall theorem. These differ from the errors in 
 (1) above in that no attempt was made to state the theorem required. 
 
 3. Failure to recall definition correctly. As in (1) above, this 
 group Includes definitions stated Incorrectly as authority in a proof as well 
 as those required directly. A typical example is "If one pair of opposite 
 sides are parallel the quadrilateral is a parallelogram," 
 
 4. Failure to recall definition. In these cases no attempt was made 
 
 to define 
 

 
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50 
 
 Chapter III 
 
 CAUSES OF MISTAKES AKU REMEDIAL SUG3ES7I0R3 
 
 The classification of errors in Table II according to mental processes 
 suggests a plan for the consideration of causes of pupils* mistakes and methods 
 of correcting or avoiding them. 
 
 Among the classified errors, omissions are most frequent. They are of 
 two kinds, conscious and unconscious. The former indicate inability to respond 
 to some part of a situation. The pupil goes through elaborate preparations for 
 a required analysis or proof of an exercise whose solution he has already intui- 
 tively discovered, setting down what is given and what is required, sometimes 
 drawing a figure which is a work of art, but proceeds no further. His careful 
 beginnings are a vain attempt to find, by going as far as he can, some suggestion 
 for doing the part he does not know how to do. 
 
 He consciously omits the proof of a construction or exercise because 
 he cannot shift his point of view from the facts on which the construction or 
 solution were based to a very different set of principles upon which the proof 
 depends. He omits the authority for a step in a proof because he fails to dis- 
 cover in his stock of stored up experiences any fact orprinciple which applies 
 to this new situation. 
 
 As previously suggested, this failure may be due to a scanty supply 
 or disorderly arrangement of experiences. In any case conscious omission is 
 partial inability and differs from complete inability in extent only. Addition- 
 al experiences will help him by increasing the number or strength of the bonds 
 between the situation and the response. Careful and elastic organization of past 
 experiences in as many ways as possible will make his stock more available. 
 

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51 
 
 Association by similarity will function freely If each situation Is viewed re- 
 peatedly from every possible angle before It Is stored away. For example, the 
 pupil who has classified, "In the same or equal circles equal central angles 
 Intercept equal arcs" under the three headings, equal circles, equal arcs, and 
 equal central angles, will be in a position to recall it in a situation involving 
 any one of the three. 
 
 In contrast to these omissions of which the pupil Is fully aware are 
 the cases where he utterly oblivious of the incompleteness of his response. When 
 his attention 1s called to the omission he may or may not readily recognize it as 
 a necessary part. He decides instantly, upon an examination of his response, that 
 he was very dull or very absent minded, to comoare the perimeters of two inscribed 
 squares but to fall to compare their areas, or that it was very careless of him 
 to omit the definition of a word whose meaning he knows. The explanation of such 
 omissions Is concentration on some other element of the situation, often a more 
 difficult one. Critical examination of the response to see if it meets the re- 
 quirements of the situation will eliminate such omissions. 
 
 Other unconscious omissions are not readily recognized as omissions of 
 vital elements. The pupil thinks it foolish to write down as a separate step 
 that "two radii are equal", a fact essential to the progressive arrangement of 
 the proof, but perfectly obvious to him; or he sees no sense in stating what he 
 has constructed, after telling what v;as required and how he did it. 
 
 Two factors contribute to this attitude. First, the pupil has a one- 
 sided view of the situation. Because the Incomplete form conveys the desired 
 meaning to him, he can see no reason for completeness. He does not realize that 
 a formal demonstration should present In full all the facts and relations neces- 
 sary to the conclusion. He fails to consider that the readers may not be able to 
 Insert between the steps of the proof those facts and relations which are^erfectlj 
 obvious to the vTlter. He expects the teacher to credit him with a complete 
 
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52 
 
 proof although he gives no evidence either that he recognizes its abbreviated 
 form or that he taiows what steps are implied. Personal experience with trying 
 to fill in the steps slurred over by the ’’hence it is obvious” phrase, common 
 in the older text books, is very effective in changing his point of view. 
 
 The outlines of proofs given in many of the modern texts, merely sug- 
 gesting the general plan of procedure and leaving the details to be filled in 
 by the pupil, suggest to him such an abridged form for presenting a demonstration 
 Special instruction is required to offset the force of this suggestion. The mean 
 ings ’’outline of proof” and ’’demonstration” must be clearly distinguished. 
 
 Outlines of proof often are very valuable. An outline of the geometric 
 proof of the Pythagorean theorem facilitates recall of the plan of proof and 
 enables the pupil at any time to work out the details for a complete demonstra- 
 tion. All the early theorems should be proved completely. Exercises involving 
 the use of congruent triangles offer a good opportunity to develop, in definite 
 and comparatively simple situations, the nature of a demonstration. Outline of 
 proof may be introduced as an aid to recall when the pupil comes in contact with 
 more complex proofs. The emphasis in review work should be put on plan or out- 
 line of proof rather than on complete demonstration. 
 
 Closely related to and to a large extent responsible for this one-sided 
 view of the requirements of demonstration is the somewhat vague and lax practice 
 commonly accepted as proof. If an indefinite statement of one or more of the 
 facts fundamental to a proof is accepted as a proof, we cannot expect the pupil 
 to acquire the correct idea of proof or to give a complete response to a situa- 
 tion requiring proof. Common practice must constantly reenforce the distinction 
 between outline and demonstration. Requirement should at ell times be very 
 specific for either the one or the other. Neither should be accepted as a 
 satisfactory response when the other is called for. 
 
 Such directions as, ”State two theorems required in the proof of this 
 

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 exercise," "Give the authority for the last step in the proof of this theorem", 
 "Give complete proof of the first case and outline the proof for cases two and 
 three of the following theorem", "Construct and prove", or "iinalyze the con- 
 struction and give proof", are specific as to requirement and conducive to 
 satisfactory responses. 
 
 Next after omission of frequency comes error involving memory. Some 
 of these mistakes are due to failure to recall, others to faulty recall. The 
 former is a case of bonds too weak to function. In the latter the wrong re- 
 sponse has been connected with a situation. 
 
 The function of memory in geometry and the degree to which it should 
 be developed has been a much discussed subject. Some writers on method of teach- 
 ing have decried memory and lauded reasoning. No one has listed improvement of 
 recall among the chief objectives of the study of geometry. Rote memory, ac- 
 companied by a minimum of meaning, has little or no place. No progress, hov/- 
 ever, is possible xonless the bonds that connect a stimulus and response are 
 developed to a strength sufficient to insure a prcmot response whenever the 
 stimulus occurs, ouch habitual response to a total situation or an element of 
 a situation is recall or memory. The more the pupil understands the full mean- 
 ing of the situation and of its elements, the more chance he has of recognizing 
 a' like situation in new surroundings. The study of geometry requires repetition 
 of a response in the same identical situation in some cases, A pupil should 
 habituate the connection between the theorem "If two lines are cut by a trans- 
 versal so that the alternate interior angles are equal the lines are parallel", 
 and the given proof because it is a useful model of an indirect proof. The 
 connection betw'een a theorem, the requisite construction, and plan of proof 
 oug^t to become habitual. 
 
 Novel geometrical situations demand the recognition of similar elements 
 joined to different concomitants and the selection of one of the responses with 
 
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54 
 
 which that element has been connected. In either case no response is possible 
 unless the bonds connecting stimulus and response are ready to act. Reasoning 
 or purposive thinking is impossible unless the pupil has a definite response at- 
 tached to those factors of a new situation which he has met in past experience. 
 Habitual response or recall, then, is the foundation on which reasoning is built. 
 There is no antagonism between the two, 
 
 A study of our data suggests that additional practice is desirable 
 especially to further (1) correct quotation of theorems, (2) ready recall of 
 theorems, and (3) connection between a theorem and correct plan of proof and 
 construction lines. In each case memorization of the idea is preferable to memor- 
 ization of bare words. We must not forget, however, that meanings develop with 
 use and full meaning can come only as a result of a maximum of varied experience. 
 
 The third group of mistakes are those related to analysis. Thorndike 
 suggests^ that ''All learning is analytic. (1) The bond formed never leads from 
 absolutely the entire situation or state of affairs at the moment. (2) Within 
 any bond formed there are always minor bonds from parts of the situation to parts 
 of the response, each of which has a certain degree of independence, so that if 
 that part of the situation occurs in a new context, that part of the response has 
 a certain tendency to appear without its old accompaniments." 
 
 When such a fact appears in a new context it tends to provoke the total 
 was 
 
 response that bound to it, or tends especially to provoke the minor features of 
 that total response which was especially bound to it. All behavior is selective, 
 but certain features of it are emphatically so, "In meeting novel problems the 
 mental set is likely to be one which rejects one after another response as their 
 unfitness to satisfy a certain desideratum appears." Separation of a subtle 
 element from the total situation in which it inheres and the acquisition of some 
 constant element of response to it is typical of analysis. 
 
 1, Educational Psychology, Briefer Course, p.l53. 
 
 2. Thorndike, "Educational Psychology", p. 158. 
 
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55 
 
 We have noted above, ("Psychology of irithmetlc" see page 14 ) the three 
 means which Thorndike suggests to facilitate analysis: (1) piecemeal examination 
 of situation, (2) response to many situations each containing the common element, 
 and (3) contrast of the element to an opposite or imlike element. We have sot 
 a situation before a pupil and asked him to analyze it. Our results will be more 
 satisfactory if we arrange the pupil’s experience formally or informally to pre- 
 pare him to abstract the desired element and attack a constant response to it. 
 
 We attempt this consciously now for a few elements by repeated application of 
 certain principles to varied situations. For example, "To prove four line seg- 
 ments proportional, try to prove them homologous sides of similar triangles." 
 
 Absolute inability ranks next in order of frequency. It may be due 
 to failure to recall or failure to analyze and is like partial inability except 
 more extensive. The emotional excitement of an examination may produce a mental 
 set of hopelessness or of discouragement. Under this influence a pupil may 
 give up a question with which under other conditions he may be successful at 
 least in part. 
 
 Illogical errors form a very small percent of the total errors. The 
 pupil more often does not know what to put in a proof than how to put it together. 
 
 Throughout our discussion of pupils’ mistakes certain difficulties 
 constantly reappear. (1). Bonds have been developed to insufficient strength. 
 
 (2). The pupil does not pick out the known elements of a situation. (3) The 
 allkeness of two situations is not recognized. (4). He has had too little train- 
 ing in abstraction. (5). Too little preparation is made for meeting novel 
 situations. 
 
 Several factors contribute to this situation: 
 
 1. The teacher does not have clearly in mind the objectives in teaching 
 geometry. (They are in dispute and constantly changing). 
 
 2. He is not aware of the different processes Involved in the study of 
 
 geometry. 
 
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56 
 
 3. His Icnowledge of the psychology of learning in general and the psych- 
 ology of geometry in particular is insufficient to the situation. As a result he 
 tells the pupil what to do but fails to prepare him so that he is able to do it, 
 
 4, He divides his effort over too large a field to be effective. 
 
 The following guiding principles are offered as remedial suggestions; 
 
 1, Present each fact, idea, and principle with as full meaning as 
 
 possible. 
 
 2, Develop further meanings through use in varied situations, 
 
 3, For bonds which will be required later, give sufficient practice 
 to insure readiness to act (especially bonds betvreen theorem, construction, and 
 plan of proof), 
 
 4, Progress slowly at the beginning. Provide supplementary help for 
 weak students, 
 
 5, To facilitate recall organize facts and principles in as many ways 
 as possible, 
 
 6, Provide specific practice in transition from the field of space per- 
 ception to logical demonstration and vice versa, 
 
 7, Follow Thorndike’ s suggestions, noted above, for improving analysis, 
 
 8, State requirements definitely. Hold pupil strictly to requirement, 
 thus use dissatisfaction to weaken the connection between wrong response and 
 situation, 
 
 9, Use parallel column arrangement of steps and authorities in formal 
 demonstration. It encourages definite and complete proofs, 
 
 10, Encourage critical examination of all results. 
 
 11, If a pupil fails to solve an original problem, give him the necessarji 
 help. Avoid connecting the response ”I can’t” with a novel situation. 
 

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57 
 
 APPENDIX 
 
 Examination Questions 
 Geometry I. 
 
 I. Construct 
 
 1. A triangle having given two angles and the included side. 
 
 2. Through a point outside, construct a line parallel to a given 
 line, 
 
 3. Bisect a given arc. 
 
 4. Inscribe a circle in a given triangle. 
 
 II. Any tv;o. 
 
 1, The sum of all the angles formed at a point in a line, and on 
 the sane side of a straight line, is equal to two right angles. 
 
 2, From any point D on the base of the isoscles triangle ABC,DS and 
 DP are drawn parallel to the equal sides BC and AD respectively. Prove that the 
 perimeter DEBP is equal to AB plus BG. 
 
 3. The exterior angle of a triangle is equal to the sum of two 
 opposite interior angles. 
 
 4. A diagonal of a parallelogram divides it into two triangles 
 equal in all respects. 
 
 III. Prove: If two strai^t lines are cut by a' transversal maxing the 
 
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4:. 
 
 5. . 6. V 58 
 
 o 
 
 Any three questions 
 
 A X 5 = 17.0® 
 (B ^ ? 
 
 l&= 80 " 
 A1^= 7 
 ^C=7 
 
 V. 1. state ten cases when two angles are equal, 
 
 2, State five cases when lines are parallel, 
 
 3. Define: x^arallelogram, rectangle, t ngent, circle, radius, 
 
 YI, Prove: Parallel lines intercept equal arcs on a circumference. (one 
 case. ) 
 
 YII. Prove; A diameter that "bisects the chord is perpendicular to the 
 chord and "bisects the arc su"btended by it. 
 
 YIII. Finish statement and prove: The sura of the interior angles of a 
 triangle is equal to 
 
 quadr i la t e ral ? 
 
 4, In the same circle or equal circles, equal arcs on a circumference 
 
 are intercepted "by..... 
 
 X. At a given point in a line only one perpendicular can "be erected 
 
 IX, 1. State three cases when two triangles are equal. 
 
 2, State three cases v/hen two right angles are equal. 
 
 3. How many right angles in the sum of the interior angles of a 
 
 throuch the line 
 
I'.i • 4- 
 
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59 
 
 Geometry I. 
 
 I. Construct; 
 
 1* Through three points not in the same strai^t line construct 
 a circumference. 
 
 2. Inscribe a circle in a given triangle. 
 
 3. Consti*uct a tangent to a circle from a given point outside. 
 
 4. Construct a ri^t angle triangle having given two sides about 
 the right angle. 
 
 II. Any two. 
 
 1. ABC is an isoscles triangle, B3 is parallel to AG. Prove that 
 BS bisects the exterior angle CBD. 
 
 2. Prove that the exterior ang^e of a triangle equals the sum of 
 two opposite interior angles. 
 
 3. If two angles at the extremities of one base of a trapezoid 
 are equal, the non-parallel sides are equal. 
 
 4. The sxim of all the angles formed about a point is four right' 
 
 angles . 
 
 III. Prove: and construct the figure. The line joining the middle points 
 
 of tv/o sides of a triangle is parallel to the thiri side, and equal to one-half 
 of it. 
 
 IT. 
 
I 
 
Any three. 
 
 7. 1. State eight cases when tv;o lines are parallel, 
 
 2. Give the nuinher of right angles in the sum of the interior angles 
 of a hexagon. 
 
 3, State four cases v/hen right angle triangles are equal. 
 
 VI. State all the cases when two triangles are equal and prove one. 
 
 7II, Construct the figure and prove; A diameter perpendicular to 
 a chord "bisects the chord and also the arc subtended by it. 
 
 7III. Prove: The diagonals of a parallelogram bisect each other, 
 
 IX. Prove: If two circumferences intersect the line joining their 
 centers bisects at right angles their common chord, 
 
 X, 1. Angles inscribed in a semicircle are 
 
 2. Squal arcs on a circumference are intersected by 
 
 3. In the same circle or equal circles, chords equally distant from 
 the center are equal, 
 
 4. Circumscribe a square about a circle. 
 
61 
 
 Any ton. 
 
 Geometry I. 
 
 I, Name five fundamental construction problems in Book I. 
 
 II, State the converse. If tv;o triangles have two angles of one equal 
 
 respectively to two angles of the other and the included side unequal 
 
 III, Prove: If two straight lines bisect each other and their extremities 
 are joined, how many pairs of equal triangles are formed? 
 
 17, Construct the figure and state the hypothesis and conclusion. 
 
 In the given triangle ABC the perpendiculars from the extremities of the base to 
 the sides of the triangle are equal, prove the angles which they make v;ith the 
 base are eqxml and the triangle is isoscles, 
 
 7. Prove: The diagonals of a rectangle are equal, 
 
 71, In triangle ABC, angle A equals 60 degrees, angle B equals 70 
 degrees, angle C equals 50 degrees, '.Thich is the longest side of the triangle? 
 
 The shortest? Give proof, 
 
 7II. Is this a correct statement? "When two lines are cut by a trans- 
 versal the alternate interior angles are equal," 
 
 7III, If the bisector of the exterior angle of a triangle is parallel 
 to the base, prove the triangle is isoscles, 
 
 IX, If AB is xjarallel to CD and 
 SP is parallel to GH, prove angle 1 equals 
 angle 3 and angle 3 plus angle 16 equal 
 two ri^t angles, 
 
 f 
 
 X, State five ways to prove that a quadrilateral is a parallelogram, 
 
 XI, Given the perimeter, to construct an equilateral triangle, 
 
 XII, Given one side, to construct a square. 
 

 
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 J; 
 
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62 
 
 Geometry I. 
 
 I. Define; Parallel linos, triangle, polygon, trapezoid, parallelogram, 
 chord, tangent, perpendicular, "bisect, congraent. 
 
 II. a. If two straight lines are cut by a transversal making the con- 
 secutive exterior angles supplementary, the lines are parallel. State the con- 
 verse. 
 
 "b. If two parallel lines are cut by a transversal, the alternate 
 exterior angles are eqiial. State the converse. 
 
 III. Prove; The common external tangent of two equal circles bisects 
 the line of centers. 
 
 17. Two triangles are congruent if three sides of one are equal to three 
 sides of the other. 
 
 7. How many degrees are there in the interior angles of a regular 
 polygon of seven sides? 
 
 71. Prove; If thie consecutive angles of a quadrilateral are supplement- 
 ary, the figure is a parallelogram. 
 
 7II. Prove; In the same circle or eqtial circles, chords equally distant 
 from the center are equal. 
 
 7III. A line throu^ the center of circle perpendicular to the chord, 
 bisects the chord and its subtended arc. 
 
 IX. Prove; The sum of the angles of a triangle equals 180 degrees. 
 
63 
 
 Geometry I. 
 
 I. An inscribed angle is measured by l/s of its intercepted arc. 
 
 II. AB and CD are two equal and intersecting ciiords. Prove triangles 
 ABC and BCD congruent . 
 
 III. If two circles are tangent to each other externally at point A, | 
 the comnon tangent which passes through A bisects the other two common tangents, i 
 
 IV. Construct a circle throu^ three given points, not on the same 
 strai^t line. 
 
 V. Divide a given segment into 5 equal parts. 
 
 VI. Hoy; many sides has a polygon tlie sum of Y/hose interior angles 
 exceeds the sura of the exterior angles by 720 degrees. 
 
 VII. If the diagonals of a quadrilateral bisect each other the figure 
 is a parallelogram. 
 
 VIII. If a line joins the raid-points of two sides of a triangle it is 
 parallel to the thiid side and equal to l/2 of it. 
 
 IX. If two triangles have two sides of one equal respectively to two 
 sides of the other, but the included angle of the first greater than the in- 
 cluded angle of the second, then the third side of the first is greater than 
 the third side of the second. 
 
 X. The sum of the angles of a triangle is one straight angle. 
 
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64 
 
 Geom. I . 
 
 I 
 
 If DC - BC 
 
 angle 3 - angle 4 
 
 Prove triangle ADC oongruent- triangle ABC 
 
 II 
 
 Given iJD-DC =CBsrBA 
 
 angle B • angle D 
 Prove; angle ln«angle 2 
 
 III 
 
 Hov; many sides has a polygonthe siun of whose interior angles is equal to 
 
 five times the sum of the exterior angles? 
 
 17 
 
 The sum of tlie three angles of any triangle is equal to one strai^t angle. 
 Give proof. 
 
 7. 
 
 Prove; If the diagonals of a quadrilateral hisect each other, the figure is a 
 parallelogram. 
 
 71 
 
 Prove; The tangents to a circle from an outside point are equal. 
 
 71 1 
 
 (a) Construct a circle which v/ill pass thiu three not in a line .Ihq)lain construc- 
 tion. 
 
 71 1 
 
 (h) Construct a triangle congruent toa given scalene triangle. 
 
 71 1 1 
 
 In the same circle or in equal circles, if two minor arcs are unequal, then 
 their chords are unequal, the greater arc being subtended by the greater chord.. 
 
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65 
 
 Geometry II 
 
 I. Sound travels 1100 feet per second. If a cannon is fired from a certain 
 
 point, v;hat is the locus oj^all persons who hear the report at the end of 
 3 seconds. ‘ 
 
 II. Given side b, angle B, and altitude to the side c, required to construct 
 triangle ABC. 
 
 III. Prove: Two triangles are similar if their horaologus sides are proportional. 
 
 17. Given AB a diameter of the circle 0; AB produced to 0 and PC perpendicular 
 to AG; AB a line intersecting the circle in Q. Prove triangle APG con- 
 gruent to triangle AQ3. 
 
 7. If the perimeter of a given field is 210 rods and a side of this field is 
 to a corresponding side of a similar field as 3;2, find the perimeter 
 of the second field. 
 
 71. Gonstract a square equal to a given parallelogram. Explain and prove. 
 
 7II. The sides of a triangle are 8, 26, and 30. Pind the radius of the inscribed 
 circle . 
 
 7III. If a secant and a tangent are drawn to a circle from the same point out- 
 side the circle, the square of the tangent is equal to the product of 
 the whole secant and its external se^nent. 
 
 IX. The area of a regular hexagon inscribed in a circle is 54 \ 3. V/liat is the 
 area of the circle. 
 
 X. Prove: The area of 2 similar triangles are to each other as the squares of 
 any 2 homologus sides. 
 
66 
 
 &eoraetry II 
 
 (Choose ei^t out of ten)* 
 
 1. Locate a school house 40 feet from each of two intersect roads. 
 
 2. Construct the triangle ABC having given, h,c,h^- 
 (Give analysis, construction and discussion.) 
 
 3. State six conditions which inahe triangles similar. 
 
 4. Prove that "If a secant and a tan^^nt are drawn to a circle from the same 
 point outside the circle, the square of the tangent is equal to the product 
 of the whole secant and its external segment." 
 
 5. Prove that "The area of a triangle equals one-half the product of its base 
 and altitude." 
 
 6. The sides AB, BC, CD, and DA of quadrilateral ABCD are 10, 17, 15 and 20 
 respectively, and the diagonal AC is 21. Find the area of the 
 quadrilateral . 
 
 7. Give the geometric proof of the Pythagorean Theorem. 
 
 Construct a rectangle having given base m and equal to two-thirds a given 
 square. 
 
 9. Prove that "The area of a regular polygon is equal to one-half the product 
 of the apothera and its perimeter." 
 
 10. A circular grass plot, 100 feet in diameter, is surrounded by a v;allc 4 
 feet wide. Find the area of the v/alk. 
 
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67 
 
 Geometry II 
 
 Choose eight questions out of nine. j 
 
 1 (a) The poles of a telephone line are to he each equidistant from two houses. 
 What is the locus of tv/o poles? Prove it. 
 
 (b) A tree is planted 10 ft from the front wall of a rectangular house and 
 15 ft. from the comer of this house. How many solutions are there? 
 
 Show by diagram. 
 
 2, Prove the theorem: If tv;o chords are drawn throu^ a fixed point within a 
 
 circle the prod\;ct of the segments of one of the chords is equal to 
 the product of the segments of the other. 
 
 3, Construct the mean proportional between tv/o unequal segnents. Prove it. 
 
 4, Prove: The square upon the hypotenuse of a rt. triangle is equal to the sum 
 
 of the squares ux/on the other two legs, 
 
 5, The area of Polygon I is 147 sq. in, and its shortest side is 3 in. V/hat is 
 
 the area of Polygon II whose shortest side is 9 inches? What is 
 the ratio of their perimeters? 
 
 6, Construct a triangle equal to a given parallelogram with its base equal to 
 
 the base of the given parallelogram. 
 
 7, Find the areas of the circumscribed and inscribed squares of a circle whose 
 
 radius is 4 inches. 
 
 8, Pind the areas of: 
 
 (a) a triangle of sides 4,6, and 8 inches, 
 
 (b) a tra-pezoid whose lower base is 16 inches and v/hose upper base is one- 
 half the lov/er base, and v/hose altitude is one-fourth of the s’wUQ 
 
 of its two bases. 
 
 9, State five ways in vdiich triangles may be proved similar and prove one of 
 
 them, 
 
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68 
 
 Geometry II 
 
 . I. A circus ring is fifty feet in diameter. How raa^iy times v/ill a 
 
 horse have to circle it close to the outer edge to ran one mile? 
 
 II. Hov; can you place a doily in the center of a square table? 
 
 III. Required to divide a line segment into five equal parts, using 
 a sheet of ruled paper. 
 
 IV. If a post six feet high casts a three foot shadow, how tall is a 
 tree wMch casts a thirty foot shadow? 
 
 V. V/liat is the area of a triangular piece of ground when one side 
 is forty feet and the hypotenuse is fifty-two feet? 
 
 VI. How much longer will it tahe a two inch pipe to empty a tank 
 than an ei^t inch one? 
 
 VII. Ho’w many square feet of cement in an equilateral triangle six 
 feet on a side? 
 
 VIII. How could you make a six sided bird house and cut the sides 
 so they would fit and not leave cracks? 
 
 IX. Hov/ can I find the center of a locket in order to put a stone there? 
 
 X. A man wished to place eig^t shrubs around his circular lily pond. 
 
 How did he find the places to set them? 
 

 
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69 
 
 Geometry II 
 
 I. Construct the perpendicular "bisectors of the sides of an o"btuse 
 triangle, "here do they meet? 
 
 II. Divide a given line segment into five equal parts. 
 
 III. On a given "base construct a triangle equal to a given parallelo- 
 
 i 
 
 I 
 
 f 
 
 gram. 
 
 IV. If two tangents are drawn throu^^ a circle from an oxitside point, 
 the line joining the point to the center of the circle bisects the arc "be- 
 tween the tangents. 
 
 V. The legs of a ri^t triangle are 21 and 18. Find the altitude to 
 the hypotenuse if the segments of the hypotenuse are 27 and 12. 
 
 VI. A baseball diamond is a square whose side is 90. Hov/ far will 
 the man on first base have to throw the ball to the man on third? 
 
 VII. Find the area of a trapezoid if the altitude is seven and the 
 bases 15 l/3 and 12 l/2, 
 
 VIII. ’That is the ratio of the perimeter of tvvo squares inscribed 
 in circles whose radii are two and eight? Find the ratio of the areas of 
 the squares. 
 
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70 
 
 BIBLIOGSAPHY 
 
 1. Garth, T.H., “The Psychology of Riddle Solution: An Experiment in Purposive 
 Thinking,** Journal of Educational Psychology, January 1920, Vol. ll,p. 16-S3. 
 
 2. Judd, C#H,, “Psychology of High School Subjects'*, Giim and Co,, N.Y.,1915. 
 
 3. Minnlch, J,H., “An Investigation of Certain Abilities Fundamental to the 
 Study of Geometry." University of Pennsylvania, 1918. 
 
 4. Odell, C.W,,“A Study of 1,000 Errors in Latin Prose Composition”, School and 
 Society, Dec. 31, 1921, p. 643-646. 
 
 5. Rogers, A.L,, “Teats of Mathematical Ability and their Prognostic 7alue,“ 
 Teachers College, Columbia University, N.Y., 1918. 
 
 6. Ruger, Henry A., “Psychology of Efficiency, Study of Mechanical Puzzles." y 
 Archives of Psychology, No. 15, Vol.2, Series 2, June 1910. 
 
 7. Rugg, H.O., and Clark, J.R., “Scientific Method in the Reconstruction of 
 Ninth Grade Mathematics," University of Chicago Press, Chicago, 1918, 
 
 8. Symonds, P.M., “The Psychology of Errors in Algebra," The Mathematics Teacher, 
 Feb. 1922, Vol.15, p.93. 
 
 9. Thorndike, E.L. , “The Psychology of Arithmetic." Macmillan, N.Y., 1922, 
 
 10. Thorndike, E.L., “Reading as Reasoning: A Study of Mistakes in Paragraph 
 Reading." Journal of Educational Psychology, June 1917, Vol.8,p.323. 
 
 11. Thorndike, S.L., “Understanding of Sentences: A Study of Errors in Reading." 
 
 Elementary School Journal. October 1917, Vol. 18, p.98.