A STUIW OF SOME FACTORS THAT CAUSE M AND DfSLlKE OF MATHEMATICS Thai: for flu Dam of Ed. D. MICHIGAN STATE UNIVERSSTY Leon A. M: Dermot? 1956' -..~_ _._,...,.._ . -‘~u:.‘.»-M.-.—~~" m—tzi‘s u I! um; {”211wa M} in Mm It; I 1:)! mg! n This is to certify that the thesis entitled " A Study of Some Factors that Cause Fear and Dislike .of Mathematics" presented by Leon Ans on McDemott has been accepted towards fulfillment of the requirements for Ed.D degree in Education //1 V _' Major professor Datem 0-169 I /— OVERDUE FINES: 25¢ per day per item RETUMIMS LIBRARY MATERIALS: Place in book netum to remove charge from circulation records ‘ “MP520242005 A STUDY OF SOME FACTORS THAT CAUSE FEAR AND DISLIKE OF KATHEHATICS BY Leon AKINcDermott A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of INMEROFEMEMHGJ College of Education 1956 AC KN OT. ILSDGICENI‘S The writer wishes to express his appreciation to the many people who helped him throughout the de- ve10pment of this study. The encouragement and guidance came primarily from Dr. Milosh Muntyan, Dr. Chester Lawson, Dru Walker Hill, and Dr. Walter Johnson. Their oOoper- ation, understanding attitude, and helpful sugges- tions are greatly appreciated. In addition, acknowledgment is given to those of the faculty of Central Michigan College in the counseling and mathematics departments for their co- operation, and eSpecially to Dr. Judson Foust, Vice- President, who made space available for interviews, and gave much help and encouragement. Likewise, credit is due the department of audio-visual aids for their cooperation and technical assistance. Furthermore, the writer wishes to eXpress his gratitude to the forty-one students of Central Mich- igan College who gave their time and c00peration for the interviews used in this study. Invaluable helpful criticism and encourage- ment was given the writer by his wife, Frances, whose time, efforts, and patience have carried the writer through his more discouraging times to the completion of this study. VITA Leon Anson HoDermott candidate for the degree of Doctor of Education Final Examination: July 30, 1956 Thesis: A Study of Some Factors that Cause Fear and Dislike of Kathematics Outline of Studies: Kajor subject -- Higher Education Cognate Field -- Physical Science Biographical Items: Born: Harch 15, 1906, Petoskey, Richigan Undergraduate Studies: Eastern Kichigan College, 1924-28 Graduate Studies: University of Chicago, Summer, 1929 Experience: Nember of:. Listed in: Master of Science, University of Michigan, 1937 ' Michigan State University, 1951-56 Science Teacher, Howell High School, Howell, Michigan, 1928-42 Analytical Research Chemist, The Dow Chemical Company, Hidland, Richigan, 1942-47 Assistant Professor of Physics and Chemistry, Central Michigan College, Rount Pleasant, hichigan, 1947-54 Associate Professor of Physics and Chemistry, Central Michigan College, Mount Pleasant, Eichigan, 1954- Kappa Delta Pi; American Chemical Society; Division of Chemical Education of the Amer— ioan Chemical Society; National Society for the Study of Education; National Association of Teachers of Science American Men of Science: The Physical Sciences; Who‘s Who in American Education A STUDY OF SOME FACTORS THAT CAUSE FEAR AND DISLIKE OF MATHENATICS By Leon A. HoDermott AN ABSTRACT Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF EDUCATION School of Education 1956 A fl/V '0 -/ 1! "‘ *7" .Approved_ -{fi¢a.- (éggyywyaigi5{_k t w v I Leon A. HcDermott Thesis Abstract This study was designed to determine what effect the following have had to make students fear or dislike mathematics: (1) first difficulties and attempted reme- dies; (2) the degree of ability in performing mathematical Operations; (3) the degree of understanding the function of mathematics; (4) the preconceived idea that success in mathematics cannot be attained; (5) any break in sequence in the study of mathematics; (6) dependency on others to solve mathematical problems; (7) the emotional relation- ship to studies in general and mathematics in particular; (8) conflicts with others; (9) non-mathematical reasoning ability; and (10) recreational patterns. The case study method was used to gather the data. A group of students at Central Michigan College was se- lected as subjects. These people who had developed fears and dislike of mathematics were referred to the investi- gator by remedial mathematics instructors and personnel counselors. In addition, some students who were profi— cient in mathematics were also interviewed. All students studied entered into the project on a voluntary basis. In all, 41 cases were studied, 34 of whom had developed a fear of mathematics and seven of whom were proficient in the subject. 2 Leon A. McDermott (Thesis Abstract Continued) Students participating in the study had education- al backgrounds ranging from one-room rural schools to large city school systems. All interviews were tape recorded. Further infor- mation pertaining to the student was obtained from tests and other records from the student personnel office of the college. Each case study was analyzed individually; all cases taken together were then studied as a group. The data indicated: (1) Host students having fear and dislike of mathematics met with frustration in the elementary grades; the remainder met with difficulty when they attempted the use of symbols in algebra and higher mathematics. (2) Students met difficulties by resorting to rote, by giving up entirely, by becoming hostile to the subject, by using inefficient methods, and by resorting to dishonest means to pass courses. (3) The students who developed fear think of mathematics as con- sisting of the four fundamental skills useful for commer- cial transactions; those proficient were only vaguely aware of its larger place in our society. (A) Most stu- dents who have deve10ped a dread of mathematics have con- vinced themselves that they cannot succeed in this area. (5) There is some evidence that fear has been developed 3 Leon A. McDermott (Thesis Abstract Concluded) because students have missed some part of the subject by loss of school time or by failing to graSp some area of mathematics. (6) Students who have a fear of mathematics have a tendency to rely on others for help. (7) Those who have deveIOped a fear of mathematics appear to prefer English, the social studies, and the arts, both fine and practical; they dislike the definiteness of mathematics. Those proficient in this subject frequently seem to be dissatisfied with what they take to be vagueness in the humanities, and are critical of those majoring in this area. (8) Both those who have develOped fear of mathe- matics and those proficient in it have been influenced by others--parents, siblings, and peers. (9)‘There seems to be no conclusive evidence that lack of ability in rea- soning is the sole cause of fear of mathematics. (10) There seems to be little difference in the recreational pattern of those who have develOped fear of mathematics and those proficient in it. TABLE OF CONTENTS CHAPTER PAGE I. II‘ITRODUCTION . . . . . . . . . . . . . . . . . l A. The Problem Statement of the problem. . . . . . . . . 3 Importance of the study . . . . . . . . . 4 3. Limitations. . . . . . . . . . . . . . . . 6 C. Definition of Terms Number combinations . . . . . . . . . . 7 The four fundamental Skills . . . . . . . 7 D. SumnarYo o o o o o o o o o o o o o o o o o 7 II. REVIEW OF THE LITERATURE. . . . . . . . . . . . 9 A. Inheritance and Special Ability in Mathematics Inheritance . . . . . . . . . . . . . . lO Iature of abilities in mathematics. . . . 12 B. Prediction of Mathematical Ability. . . . 15 C. Educational Psychology of Mathematics Nature of learning in mathematics . . . . 19 Methods of teaching . . . . . . . . . . . 22 D. Errors Encountered in Kathematics Fundamentals. . . . . . . . . . . . . . . 26 Other errors. . . . . . . . . . . . . . . 27 L11 . Causes of Failure in Mathematics Common causes of failure. . . . . . . . . 28 Psychological causes of failure . . . . . Bl GEE-.88 StlldieSo o o o e o o o o o o o o o 0 324 *a . Summary. . . . . . . . . . . . . . . . . . 37 CRAP III. IV. TEE METHOD OF CONDUCTING THE IEVESTIGATICN . A. Selection of Kethod. . . . . . . . . B. Selection of Subjects. . . . . . . . C. Collection of the Data . . . . . . . D. Treatment of the Data. . . . . . . . E. Summary. . . . . . . . . . . . . . . CASE STUDIES . . . . . . . . . . . . . . A. Studies of Those Having Difficulties Case number one . . . . . . . . . . Case number two . . . . . . . . . . Case number three . . . . . . . . . Case number four. . . . . . . . . . Case number six . . . . . . . . . . Case number seven . . . . . . . . . Case number eight . . . . . . . . . Case number nine. . . . . . . . . . Case number ten . . . . . . . . . . Case number eleven. . . . . . . . . Case number twelve. . . . . . . . . Case number thirteen. . . . . . . . Case number fourteen. . . . . . . . Case number sixteen . . . . . . . . Case number seventeen . . . . . . . Case number eighteen. . . . . . . . Case number nineteen. . . . . . . . Case number twenty. . . . . . . . . Case number twenty-one. . . . . . . Case number twenty-two. . . . . . . Cas number twenty-three. . . . . . Case number twenty—four . . . . . . Case number twenty-five . . . . . . Case number tw nty-six. . . . . . . Case number twenty-seven. . . . . . Case number twenty-eight. . . . . . Case number thirty. . . . . . . . . Case number thirty-one. . . . . . . ase number thirty-two. . . . . . . 122 125 CHAPTER B. Case number thirty-four. . . . . . . . Case number thirty-five. . . . . . . . Case number thirty-six . . . . . . . . Case number thirty-seven . . . . . . . Case number thirty—eight . . . . . . . Studies of Those Proficient in Nathematics Case number five . . . . . . . . . . . Case number fifteen. . . . . . . . . . Case number twenty-nine. . . . . . . . Case number thirty-three . . . . . . . Case number thirty—nine. . . . . . . . Case number forty. . . . . . . . . . . Case number forty-one. . . . . . . . . V. jXITIIXLYSIS OF T:::: Dli‘XTAo o o o c o o o . o o o o A. ’1.) Where Difficulties Have Started and How They Have Been Net Time of first difficulty . . . . . . . Reaction to difficulties . . . . . . . De gree of Understanding of Iia.thematics Those having difficulties. . . . . . . Those proficient in mathematics. . . . The Preconceived Idea that Success Cannot Be Attained Those convinced they cannot master mathemabiCS o o o o o o o o o o o o inhose convinced they are capable of mastering mathematics . . . . . . . . Loss of Sequence Loss by omission . . . . . . . . . . . Losses not due to omission . . . . . . Degree of Dependency . . . . . . . . . . Reaction to Studies in General Those havin q difficulties. . . . . . . r"hose proficient in mathematics. . . . 139 142 1L5 147 1M9 152 154 156 160 160 161 164 164 165 167 167 168 168 169 172 CRAP ‘ ER PAGE H] G. Influence of Other Individuals Those having difficulties . . . . . . . . 173 Those proficient in mathematics . . . . . 176 H. Differences in Eathematical and Non—mathematical Reasoning Those having difficulties . . . . . . . . 178 Those proficient in mathematics . . . . . 179 I. Recreation Patterns Those having difficulties . . . . . . . . 179 Those proficient in mathematics . . . . . 180 J. Summary . . . . . . . . . . . . . . . . . . 180 VI . C 02:7 C LUSI OTIS A231) REC OI‘II IEIJDAT I COTS . . . . . . . . . l 85 A. Principal Findings Time of frustration and reactions . . . . 185 Understanding the subject . . . . . . . . 186 The preconceived idea that success cannot be attained . . . . . . . . . . . 186 Reactions to studies in general . . . . . 187 Influence of others . . . . . . . . . . . 187 Differences in mathematical and non- mathematical reasoning . . . . . . . . . 189 Recreation patterns . . . . . . . . . . . 189 B. Correlation with Other Studies. . . . . . . 189 C. Educational Implications In elementary school. . . . . . . . . . . 191 In secondary school . . . . . . . . . . . 194 Education at all levels . . . . . . . . . 195 Classroom environment . . . . . . . . . . 195 demedial mathematics. . . . . . . . . . . 197 Social sciences . . . . . . . . . . . . 198 D. Suggestions for Further Study . . . . . . . 199 E. Summary 0 O O O O O O O O O O O O O O 0 O O 200 BIBLIOGliAPHY o o o o o o o o o o o o o o o o o o o 0 201+ APIDEI:DI;IC O O O I O O O O O O O O O O O O O O O O O O 216 CHAPTER I INTRODUCTION Many students come to college with fear or dislike of mathematics. Since they have such attitudes toward the subject, they tend to shun this discipline and all related subject matter. If they do elect or are required to take work in this area, they frequently fail. This problem is not a new one. Townsendl in 1902 was alarmed about the failures in college mathematics. At that time, as now, high schools were being blamed for the poor preparation in this field. In 1906 Foering2 seemed to think that the trouble stemmed from poor teaching in the elementary schools. For evidence he, as well as Townsend, was stating a personal Opinion, although Foering did send a questionnaire to other college professors of mathematics, who confirmed his viewpoint. Later Handcock3 made a bitter attack on colleges 1 E. J. Townsend, "Analysis of the Failure in Freshman Mathematics,” The School Review, 10:675-86, November, 1902. 2 H. A. Foering, "Some Causes Contributing to the Failure of Students in College Mathematics," Education, 27:143-49, November, 1906. 3 Harris Handcock, "The Defective Scholarship of Our Public Schools. The Pernicious Influence of the' Colleges for'Teachers,“ School and Society, 9:552-56, May 10, 1919. . 2 for teachers, in which he stated that the cause of indif- ference and failure in mathematics was due to overempha- sis on professional education courses and a dearth of solid subject matter in mathematics for those who would teach the subject. Handcock relied on his judgment and selected readings from the literature that suited his thesis. Mor- ris“ later pointed out that Handcock's authorities were not giving the whole answer. Buswell5 in his discussion of the problem feels much more research is needed today to learn the causes of fears and frustrations in mathematics. As Glaubiger6 points out, failure is not alone due to those characteristics so often mentioned, namely: (1) pupil carelessness, (2) poor habits of study, (3) poor teaching, and (4) lack of attention to individual differ- ences by the teacher. 'All these may contribute to failure in every subject. Yet, he says that the number of fail- ures in mathematics is higher than failures in other areas in New York schools. He feels this may be caused by some 4 J. V. L. Morris, "Why Teachers' Colleges?" School and Societ , 10:522-29, November 1, 1919. 5 Guy T. Buswell, "Needed Research on Arithmetic," {The Teaching 92 Arithmetic, Fiftieth Yearbook of the Na- ‘tional Society for the Study of Education, Part II. Chi- cago: University of Chicago Press, 1951. pp. 282-97. 6 Isadore Glaubiger, "Causes and Remedies for Pupil Failure in High School Mathematics," High Points, 23:26-30, September, 19 l. 3 factors peculiar to mathematics which are not so pronounced in other subjects. This is true not only of students in New York, but also is and has been a persistent problem everywhere. Since failure may lead to dislike and fears, those things which lead to failure in mathematics and consequent frus- trations should be investigated. A. The Prdblem Statement 23 the problem. This research is de- signed to study some factors that have caused a selected group of students of Central Michigan College to dislike and fear mathematics. The investigation will attempt to find answers to the following questions: 1. Where has difficulty in mathematics started and why at that particular point? 2. What degree of understanding do these people have of mathematics? Is it rote? Is it very superficial? .Are the methods of handling mathematics inefficient? 3. What do these peOple understand about the pur- jpose and function of mathematics in our society? h. To what degree have these peOple developed the .idea that they cannot succeed in mathematics, and.where have they learned this? 5. Have these people suffered a loss in sequence due to absence, changing schools, or failure to under- stand some units? 6. To what extent have these students been de- pendent on someone else in mathematics? 7. What is the emotional relationship of these students to studies in general and to mathematics in particular? 8. To what degree has conflict with teachers, parents, or others caused a dislike or dread of mathe- matics? 9. Is there any difference between their mathe- matical and non-mathematical reasoning? 10. What is the pattern of the active and non- active recreational interests of this group? gmportance 9§_the study. Although there has been a steady growth in college enrollment, eSpecially during recent years, there has at the same time been a decrease in enrollment in mathematics and the sciences. This is eSpecially true in the program for the training of science teachers. According to Maul,7 of those who do prepare to teach science, only 38 per cent appear in class rooms. 7 Ray C. Maul, "The Science Teacher Supply--Another Look,“ The Science Teacher, 21:172-76, September, l95#. Since there are so few training to be science and mathe- matics teachers, and since there is a great demand in these fields, Central Michigan College, with more than half its student body training to be teachers, is interested in. learning why this scarcity has deve10ped. Mathematics, being fundamental to all sciences, is the most logical place to start a study. In addition, this institution demands a certain competency in mathematics before the teaching certificate can be granted.8 Not only are teacher training institutions inter- ested because of a shortage of teachers in this area, but also all institutions are interested in the larger problem of finding and training people in the sciences (which must include mathematics), to meet the demands of our rapidly growing technological society. According to a report of the Manpower Committee9 it is difficult, if not impossible, to keep up with the expanding economy as far as the train- ing of scientists is concerned. If we know how to avoid these fears and frustrations which lead students to shun mathematics, we may be able to alleviate the shortages in our technical personnel. 8 Bulletin, 1254-55 SessionS. Central Michi an Col- ‘lege 9§,Education. Mount Pleasant, Michigan, 195 , p, 38. '9 "Science Manpower Short," Science Newsletter, 62:198, September 27, 1953. In addition, we need an informed public to make intelligent decisions about our society and, since it is so technical, some knowledge and purposes of science are necessary. Many who have a fear of mathematics fail to get this knowledge, since they will not study anything which they feel will deal with mathematics. Such a study as this, it is hOped, will serve as a help to teachers in this field. Perhaps it will help people to recognize the cause of the trouble which has in the past caused dislike of mathematics, and to elimin— ate this at an earlier time. The questions which this study attempts to answer have been selected as the result of consultations with interested persons, readings in the literature, and per- sonal eXperience. These questions seemed to be possible areas of investigation and, while many have been suggested in the literature, very little beyond speculation has been done. B. Limitations The sample of students studied is limited, since they are students of Central Michigan College who were adnuttedly afraid of mathematics and who were willing to «zoOperate in the study, together with a selected group of students adept in mathematics. Before analogies can be drawn with other groups of students, one must determine the extent to which these students are comparable to those at Central Michigan College. There are factors of nervousness, personal health, or problems that may have had undue influence on a student at the time of his interview. These were not explored. This study has been predicated on the assumption that students are not born with dislikes and fears of mathematics, but that they are learned from their envi- ronment. C. Definition 2: Terms 1. Number combinations. In this study when num- ber combination is used, it means the combinations of numbers in addition, such as seven plus six, or the com- binations in subtraction, such as thirteen minus six. 2. iggiiggg fundamental skills. In this study, the four fundamental skills refers to addition, subtrac- tion, multiplication, and division of numbers. D. Summary The problem in this study may be regarded as an attempt to determine what effect, if any, the following have had to make students fear or dislike mathematics: (1) first difficulties and attempted remedies; (2) the degree of ability in performing mathematical operations; (3) the degree of understanding of the function of mathe- matics; (4) the conceived idea that success in mathematics cannot be attained; (5) any break in sequence in the study of mathematics; (6) dependency on others to work mathemat- ioal problems; (7) the emotional relationship to studies in general and mathematics in particular; (8) conflicts with others; (9) non-mathematical reasoning ability; and (10) recreational patterns. This study should be of concern to institutions training mathematics and science teachers and also to colleges and high schools training any persons to meet the technological needs of our society. It may also have implications in both adult general education and elemen- tary education. CHAPTER II REVIEW OF THE LITERATURE In order to explore the literature pertinent to the present study, it was necessary not only to read published and unpublished research in the field of mathematics,l'5 but also to examine psychological, sociological and educa- tional writings.6'1° 1 H. E. Benz, "A Summary of Some Scientific Investi- gations of the Teaching of High School Mathematics," The Teaching of Mathematics in the Seconda_y School, The Eighth Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Colum- bia University, 1933. pp. 14-54. 2 Leo J. Brueckner, "Significant Trends in Research in Diagnosis in Arithmetic," Journal of Educational Research, 33: 460- 62, February, 1940. 3 Earl Boy Douglass, "Special Methods on High School Level: Mathematics,” Review 9; Educational Research, 2:?- 20, February, 1932. 4 , "Psychology and Methods in High School and College Mathematics," Review 9; Educational Research, 8:51-57, February, 1938. 5 Clifford Woody, "Arithmetic,” Review of Educa- tional Research, 5: 15-16, February, 1935. 6 Alice F. Moench and others, editors. The Inter- national Index to Periodicals Devoted Chiefly,tg the Human- ities and Sciences. New York: H. w. Wilson Company, Vols. I—XII, 1913-1953. 7 waiter S. Monroe, editor. Encyclopedia 9: Educa- ‘tional Research, Revised Edition. New York: The Macmillan (30mpany, 1950. 1520 pp. 10 The literature relating to this study will be re- viewed under the following categories: (1) inheritance and Special abilities in mathematics, (2) predictions of mathematical ability. (3) educational psychology of math- ematics, (4) errors in mathematics, and (5) causes of failure in mathematics. A. Inheritance and Special Ability ;n_Mathematics l. Inheritance. Carter11 made a study of 108 fam- ilies to determine family resemblances in numerical abili- ties. This study was based on a group of subjects very superior to the general level of the pOpulation, yet there was great variation present. He concluded that there was 8 Ralph P. Rosenberg, "Bibliographies of Theses in America," Bulletin 2§_Bibliography, 18:181-82, September- December, 1955. 9 Isabella Towner and Ross Carpenter, editors. The Education Index. A Cumulative Author and Subject Indexfl to a Selected List Lf Educational Periodicals, Books, and Pamphlets. New York: H. J Jilson Company, Vols. I-VIII, January, 1929-June, 1953. Also Education Index Monthly Check-List, July, 1953-June, 1955. 10 Sarita Robinson, Bertha Joel, and Zada Limerick, editOrs. Beaders' Guide 39 Periodical Literature. New 'York: The H. W. Wilson Company. Vols. I-XVIII, January, l900—March, 1953. Also Beaders' Guide §g_Period;gal Lgter- ziture Monthly Check List, April, l953-June, 1955. 11 Harold Dean Carter, Family Besemblenggg Lg Verbal ang,NUmericg;.Abilities. Genetic Psychology Monographs, 'Vol. XII, No. 1. Worchester, Mass.: Clark University Press, 1932. 10# pp. 11 resemblance between the siblings and between the children and their parents in numerical ability. He thinks this may be due to inheritance, but admits in his conclusion thaiznurtureimay'be an important answer to this question; he suggested that study of parental supervision, schooling, peer relations, and effects of changes of environment need to be undertaken. b12 made a study in which Along this same vein, Cob she found the child's ability in the four simple arithmet- ical skills and Speed of COpying figures tended to resemble that of one or the other of his parents. His skills ap- peared to be in direct prOportion to those of one of his parents. A child of either sex may bear this resemblance to either parent, but to one only. However, the child shows a greater resemblance to the like parent. She con- cluded that this is due to heredity. However, Starch13 in his study of siblings concluded that there was no evidence that Special abilities ran in families. He found that children from any family, on the average, were equally poor or equally good in all studies. 12 Margaret V. Cobb, "A Preliminary Study of the Inheritance of Arithmetical Ability," The Journal 9: Edu- cational Psychology, 8:1-20, January, 1917. 13 Daniel Starch, "The Inheritance of Abilities in School Studies," School and Society, 21608-10, October 23, 1915. 12 He felt that general ability was inherited like physical features rather than ability in any Special area of learn- ing. 2. Nature 23 abilities in mathematics. IS there a "born mathematician"? This question of Special factors involved in mathematical ability is closely related to the 1“ made problem of inheritance of such abilities. Cairns a careful study of this problem in which he attempted to determine if there was a Special faculty underlying mathe- matical ability, distinct from and with no close correla- tion to other forms of mental ability. He selected eighteen mental factors and tested his subjects for general factors and specific factors. He concluded that there was some evi- dence that the ability to do plane geometry was Specific. However, he also found that there was no Justification for the view "that mathematical ability is a separate capacity running throughout all branches of the subject and uncon- nected with other mental abilities." He says further that mathematical ability is a misnomer; the ability should be designated as arithmetical, algebraic, or geometrical. 1“ George J. Cairns, Ag Analytical Study 9: Mathe- matical Abilities. The Catholic UniverSity of America Education Research Monographs. Vol. VI, No. 3. Washing- ton, D. C.: The Catholic University Press, 1931. 104 pp. 13 A similar study was made by Oldhamls in which she sought to find some factor common or specific to the dif- ferent branches (arithmetic, algebra, and geometry) of mathematics. She found: (1) no evidence of any large group factor in the three branches either in pairs or run- ning throughout the branches, (2) overlapping factors, one of which was teaching method, which she found pronounced, (3) a low correlation of the three subjects with intelli— gence was found, (A) antipathy on the one hand and enthusiasm on the other did not always indicate differences in ability, (5) differences in early teaching and early attitudes con- tributed to fictitious differences in mathematical ability during several, if not the whole, of the school years. Somewhat in disagreement with Oldham's study, Steinl6 attributed ability in geometry to general intelligence; how- ever, Englehart,l7 investigating the relationship of general 15 Hilda 0.01dham, "A Psychological Study of Mathe- matical Ability with Special Reference to School Mathematics, " Part I, British Journal of Educational Psychology, 7: 269-86, November, 1937. Also Part II, British Journal of Education- al Ps cholo , 8:16-28, January, 1938. 16 Harry L. Stein, "Characteristic Differences in Mathematical Traits of Gobd, Average, and Poor Achievers in.Demonstrative Geometry,” The Mathematics Tgacher, 36: 16u-68, April, 1943. 17 Max D. Englehart, "The Relative Contributions of (Bertain Factors to Individual Differences in Arithmetical lProblem Solving Ability," Journal of Experimental Educa- 'tion, 1:14-27, September, 1932. 11+ intelligence, computational and reading ability with skill in solving arithmetic problems found that variance in abil- ity due to intelligence was present only in.about 26 per cent of his cases (568 fifth graders). He found that 3n per cent of the causes for individual differences in arith- metical ability were unknown and unmeasured factors in his eXperiment. Dexter18 tried to find some specialized factor that was peculiar to mathematics. She found that all cases in the study that appeared to have a Specific factor proved to be caused by erratic evaluation of the subject's ability; hence she also concluded there was no specialized ability needed for mathematics. Two studies were made, one by Washburn19 and the 20 concerning the relationship between rea- other by Davis, soning and mathematical ability. There seems to be some disagreement because Washburn found that mathematical and reasoning ability were related while Davis found a low cor- relation between computation and reasoning. However, Davis 18 Emily S. Dexter, "Does Mathematics Require Spe- cialized Endowment?" School and Societ , 44: 220-24, August 15. 1936 '19 Margaret F. Nashburn, "Mathematical Ability, Bea- soning, and Academic Standing," The American Journal of Psychology, 50: 484-88, November, 1937. 20 G. R. Davis, "Elements of Arithmetical Ability," The Journal of Educational Psychology, 5: 131— 40, March, 1914. 15 did not indicate his method of finding reasoning ability while Washburn gave her subjects a non-mathematical reason- ing test. In eight cases investigated by Bronner,21 defects in number work were found to be related to delinquency, but she was unable to determine which caused which. B. Prediction 2; Mathematical Ability Several studies have been made to determine a method of predicting the success or failure of students in mathe- matics when they enter college. Since this may have a bearing on the subject of this study, the investigations were reviewed. Most of the methods used were, understand- ably, based upon tests given incoming freshmen and the re- sults were correlated with their achievement in mathemat- ics later in their college careers. Bromley and Carter22 tried this scheme at the Galesburg division of the Univer- sity of Illinois. They gave a battery of tests including: (1) a psychological, (2) general achievement, (3) profi- ciency in social science, natural science, and mathematics, (M) COOperative engineering test, (5) silent reading, (6) 21 Augusta Fox Bronner, The Psychology 2: Special .Abilities and Disabilitigg, Boston: Little, Brown, and Company, 1923. pp. l-7h. 22 Ann Bromley and Gerald C. Carter, "Predictability of Success in Mathematics," Journal 9; Educational Research, 44:148-50, October, 1950. l6 mechanical comprehension, and (7) a two dimensional Spa- cial test. Their results were discouraging, as both in- dividual and multiple correlations were low. Bromley and Carter concluded that prediction of success involved more than was revealed in their studies. However, Kent and Schreurs23 in a study of the re- lationship of mental alertness, high School class quartile, number of units in mathematics earned in high School, and general achievement, in relation to success in college math- ematics, thought that mental alertness was of distinct value in predicting grades in mathematics. They further stated that quartile rank in high school and units presented for college entrance had no predictive value. A general schol- astic achievement test at the end of the college freshman year had value in prediction only in the cases of those who were low. Since, for prediction, they used high school grades presented on college entrance, they concluded that teachers' marks are not reliable. Contrary to this, Douglass and Michaelsonzu found 23 R. A. Kent and Esther Schreurs, "A Predictive Value of Four specialized Factors for Freshman English and Mathematics," School and Societ , 27:242-46, February 25, 1928. 2# Harl Roy Douglass and Jessie H. Michaelson, "The Relation of High School Mathematics to College Marks and of Other Factors to College Marks in Mathematics," Th; School Review, 4h:6l5-l9, October, 1936. 17 that ability in high school is materially, though not close- ly, associated with ability to do work in any field in col- lege. The average high school mark proved to be more close- ly related to success in college mathematics than did marks in high school mathematics. They also found that success in college mathematics cannot be predicted from the amount of high school mathematics taken. Crawford25 agrees with this study and found that class placement in high school had predictive value for col- lege achievement; however, he pointed out that the sustained record of good achievement meant more than an average. Perry26 attempted to relate standardized tests in mathematics, chemistry, and English, together with intelli- gence and personality ratings, with success in college math- ematics. From working with 1600 freshmen at Purdue Univer- sity, he found that intelligence played a very small part in predicting grades in mathematics when used in combination with other standardized tests of a mathematical nature. How- ever, for average scholarship the intelligence test ranked thigh in predictive value. He learned that personality was a strong predictor of marks in mathematics, and concluded 25 Albert Beecher Crawford, "Forecasting Freshman AChievement," School and Society, 30:125-32, January 25, 1930. 6 2 Rdbert D. Perry, Prediction Equations for Success ,gg College Mathematids. Nashville, Tenn.: George Peabody College for‘Teachers, 193“. 58 pp. 18 that this area deserved a place of equal or greater impor- tance than some of the standardized tests for prediction. He also indicated a need for more investigation. Seigle27 and Fredrickson28 used aptitude tests in mathematics to predict the success of entering freshmen in that area. Seigle found the entrance test in mathematics at Washburn University to be a satisfactory predicting agent and even better when combined with high school grades. Here again the amount of high school mathematics proved of little value as a predictor. Frederickson, working at Princeton, obtained the same results. He found that a survey mathemat- ics test together with high school grades made the best pre- dictor. In a study of the prediction of success in high school mathematics, Douglass29 also found that a prognostic test together with an intelligence test was a good predictor. Here Douglass hints that "character trait ratings" may have some value. 27 William F. Seigle, ”Prediction of Success in Col- lege Mathematics at Washburn University," Journal of Edu- cational Research, 47: 577-88, April, 1954. 28 Norman Fredrickson, "Predicting Mathematics Grades of Veteran and Non-Veteran Students," EduCational and Psy- chological Measurements, " 9: 77- -88, Spring, 1949. 29 Harl Roy Douglass, “The Prediction of Pupil Sucé cess in High School Mathematics,” The Mathematics Teacher, 28:489-504, December, 1935. 19 The prdblem of the relationship of traits to ability in mathematics was studied by Cattell30 with 123 persons in the army specialist training programs. He controlled 35 personality trait clusters and measured the mathematical ability by means of the Graduate Record Examinations. He found that he had slight correlation between mathematical ability and "surgency" (extroversion) and dominance. C. Educational Psychology 9; Mathematics 1. Nature 9; learning lg mathematics. In a discus- sion of the number system and symbolic thinking, Judd31 arrived at the following conclusions: (1) symbolic think- ing is economical because it is a substitute for concrete experience; (2) symbolic thinking requires conformity to certain rules which depend on systems of experience rather than individual items of experience; (3) mental processes involve external impressions to a diminishing degree as these processes reach a higher level; (4) one may acquire some of the rules of intellectual procedure without having 30Raymond B. Cattell, "Personality Traits Associa- ted with Abilities. II: With Verbal and Mathematical Abil- ities," The Journa;,Q§ Educational Psychology, 36:475-86, November, 1945. 1 . 3 Charles Hubbard Judd, Education.as Cultivation of the Higher Mental Process. New York: The Macmillan (30mpany, 193 . pp. 38-105. Also Educational PS cholo . INew York: Houghton Mifflin Company, 1939. pp. 260-332. 20 any true understanding of the system from which rules are derived; (5) one may acquire a certain limited understand- ing of a system of thinking without mastering the whole system; and (6) there is a wide variation in the form in which higher mental processes take place in the experiences of different individuals. All these conclusions seem to apply to the learning and using of mathematics. Brownell32 has made a study of practices in teaching mathematics and the learning process. He says that our "connectionism" in teaching mathematics does not lead to sound learning in this subject. In the past the student had te be made to identify the stimuli to which he was to react, and also to determine what reaction he was to make. He was rewarded for the prOper reaction, and received pun- ishment for failure. He then repeated the reSponse; in this way he learned to make the preper connection. Brownell felt that this led one away from the process of learning and made one more concerned about the product of learning. This led to too rapid instruction, and failure to give aids that would forestall difficulties. The evaluation of error and our treatment of errors are superficial. Instructors Should learn what the learner does, which is equally as important as ' 32 William A. Brownell, "The Progressive Nature of learning in Mathematics," The Mathematics Teacher, 37:147- 57, April, 1944. 21 getting results. Under the connective method the old adage "practice makes perfect" is not always true. There is dan- ger of repetitive practice being introduced too soon. Brown- ell does not mean to imply by this that drill has no place. He points out that remedial teaching may fail if only errors are shown while continued practice is wrong. If a student is forced to perform at a higher level than he has attained and is given no guidance in processes to reach this level he has three courses Open, namely: (1) he refuses to learn, (2) he blindly follows rules and appears fer a time to be successful (Spurious evidence of learning), (3) he fools the teacher by performing at his level (inefficient) and not at the eXpected level and may not be discovered until later when the work gets more complex. Hence education Should stress the notion of progressive reorganization in mathematics teaching. . In his study of children, Piaget33 states that there are three steps in learning number concepts. First the children lay out equal rows of chips regardless of numbers involved; secondly, they place chips Opposite those laid out; and thirdly, they lay down.a given number of chips re- gardless of their geometric pattern. Children must graSp 33Jean Piaget, "How Children Form Mathematical Concepts," Scientific American, 189:74-79, November, 1953. 22 the principle of conservation of quantity before they can develop the concept of number. In geometry the child first learns the difference between Open and closed figures. If shown a square or rectangle and asked to draw one, the child draws a circle; much later he can draw a triangle, and finally a square and a rectangle. Children do not appre- ciate the prinCiple of conservation of length or surface until they are approximately seven years old. They measure by using their hands, then measure on themselves before they use a rod of exact length. 34 point out that learning is a Kinney and Freeman reorganization of behavior and learning is by wholes, also that learning is colored by feelings and emotions. The student is not only learning mathematics but he is also learning to like or dislike the Subject; to like or dislike the school; to work as a COOperative member of a group, or to work by himself. 2. Methods pf teaching. In a study of class Size, Remmers and others35 compared three small classes (19-27) Lg, 3 Lucien.B. Kinney and Frank N. Freeman, "The Man- .mer in which Pupils Learn Mathematics," California Journal .2: Secondayy Education, 20:381-87, November, 1945. 35 Hermann H. Remmers, Laurence Hadley, and J. K. ILong, Learning, Effort, and Attitudes gg Affected by Class Size ;Q,Beginnigg College Engineering Mathematigg. Purdue University Studies in Higher Education, Number 19. Lafay- ette, Ind.: Purdue University, 1932. 31 pp. 23 with large classes (43-54) in mathematics at Purdue Uni- versity. They matched student to student as far as they were able to determine ability, and used the same instruct- or. They found that class Size was not a significant var- iable in the achievement of students. The students in the small classes unanimously favored the small class while only a minority in the larger classes favored the larger groups. The investigators wondered if the popularity of small classes was a social habit. The instructor involved in the experiment thought that class size had no effect on student attitudes toward mathematics. The average time needed for preparation.by the students was not Significantly different in the classes. In a discussion concerning the types of problems to be used in teaching mathematics, Everett36 said ...the educator who attempts to predicate his course . of study for any normal individual at any time of life upon the needs of the individual at the time must turn his back upon many of the responsibilities as well as the opportunities with which we are endowed by the fact that we have a social heritage. He further points out that when students can do equations until they come to a verbal problem, there is a problem of transfer. Teachers should attempt to 36 John P. Everett, "The Compatibility of Mind and Mathematics," Proceedings 2; the Ohio State Educational (Zonference, Eleventh Annual Session.’ Columbus, Ohio: The Ohio State University, September, 1931. pp. 355-65. 24 secure understanding of processes and transfer will take place. In this he is supported by Orato37 who states that the student learns his mathematics by analyzing the parts and putting them together again in different patterns. In other words, reconstruction is necessary for transfer. Hence there is a choice of being routine in the process with no reconstruction and merely making an application with no learning, or learning through means of reconstruc- tion. Orato also places emphasis on process rather than arriving at the answer. In a similar vein Mitchell38 found that being able to solve a Specific problem will not lead to a generaliza- tion, and that prOblems with definite numerical quantities seem to be more readily understood and_solved than those involving general principles. However, he did.not suggest ways to remedy the situation. A study in pupil reactions by McWilliams39 noted that 37 Pedro T. Orato, "Transfer of Training and Recon- struction cf EXperience," The Mathematics Teacher, 30:94- 109, March, 1937, and "Transfer of‘Training and Educational Pseudo-Science,"'Educational Administration and Supervision, 21:241-64, April, 1934. 38 Claude Mitchell, "Specific Type of Problem in Arithmetic versus the General Type of Problem,” The Ele- mentary School Journal, 29:594-96, April, 1929. 39 Luke E. McNilliams, ”A Study Of Pupil Reactions," The Mathematics Teacher, 22:284-92, May, 1929. 25 a great deal of class time was consumed in asking Simple questions and not much time was devoted to thought ques- tions. He believed that this was due to a lack of thought- provoking material presented in mathematics classes. In another study of secondary schools, made by Schu- nert,”o it was found that (l) mathematics classes in schools of 100 to 500 exceeded in achievement that of classes in larger and Smaller schools and (2) classes in algebra of 20 to 30 students exceeded the achievement of larger and smaller classes. In the latter he differs from the find- ings of Remmers.u1 He also found that teachers of more than eight years' experience were best. In teaching factors Schunert found the following: (1) differentiated assignments were better than general as- signments, (2) algebra classes with life applications were best, (3) reviews more than once a month were most effective, (4) algebra classes where teachers failed no more than two per cent were better than those in which ten per cent or more failed, (5) classes with 20 to 30 minutes of supervised study were better than those with no supervised study. 40 Jim Schunert, "The Association of Mathematical' Achievement with Certain FactOrs Resident in the Teacher, in the Teaching, in the Pupil, and in the School,“ Journal 9; Experimental Education, 19:219-38, March, 1951. “1 Remmers, Hadley and Long, 2p. cit. 26 D. Errors Encountered gp Mathematics 1. Fundamentals. Several studies in the common er- rors made in mathematics would lead to the conclusion that the difficulty may be in the fundamental operationS--addi- tion, subtraction, multiplication, and division. Coit42 made a study of four high schools with an average enrollment of 230 pupils. He found.that the mathematical difficulty consisted of weakness in the four fundamental skills and this persisted through all the higher levels of mathematics. Arthur,43 in a study of tests devised for minimum army needs applied to high school pupils, substantiated the same results twenty-one years later. MacRae and Uhluu found the four fundamental processes the common error in algebra as well as in arithmetic. In a study of number combinations, Hashburne and Voge145 found that there were some combinations more “2 Wilber Allen Coit, "A Preliminary Study of Mathe- matical Difficulties," The School Review, 36:504-09, Sep- tember, 1928. “3 Lee E. Arthur, "Diagnosis of Disabilities in Arithmetic Essentials," The Mathematics Teacher, 43:197- 202, May, 1950. 44 Margaret MacRae and Willis L. Uhl, ”Types of Er- rors and Remedial Work in the Fundamental Processes of {Algebra," Journal 9: Educationg; Research, 26:12-21, Sep- tember, 1932. 45 Carleton Washburne and Mabel Vogel, "Are Any ihnnber Combinations Inherently Difficult?" Journal 9: Edu- cational Research, 17:235-55, April, 1928. 27 difficult than others. Placement of combinations did not seem to be as great a factor as some of these inherent dif- ficulties. In general, they found that the mere largeness of the addend was the principal cause of difficulty in ad- dition. Eights or nines in subtraction seemed to be diffi- cult, while multiplication by zero or a division with a like number gave trouble. 2. Other errors. In a study of arithmetic, Benz46 found that the most errors occurred in the multiplication and division of fractions and decimals. Washburne and Morphettu7 found that problems deal- ing with unfamiliar situations caused difficulties in arith- metic. If the same type of problem was used with situations familiar to the student, results improved. They felt that this gave the student a reason for solving the problem. This was substantiated by a later study made by Lydan8 wherein he used students of all abilities and found that “6 H. E. Benz, "Diagnosis in Arithmetic," Journal pi Educationgl Research, 15:140-41, February, 1927. “7 Carleton Washburne and Mabel Vogel Morphett, "Un- familiar Situations as a Difficulty in Solving Arithmetic Problems," Journal 2: Educational Research, 18:220-24, October, 1928. “8 Wesley J. Lyda, "Direct, Practical Experiences in.Mathematics and Success in Solving Realistic Verbal iReasonin ' Problems in Arithmetic," Egg Mathematics Teach- e_r;, 40:16 -67, April, 1947. 28 realistic and practical situations proved more successful in eliminating errors, eSpecially for students below aver- age and of average intelligence. The relation of reading difficulties to mathematical errors was investigated by Georges“9 and later by Boyd50 who came to the conclusion that reading difficulties both in com- mon and mathematical terms contributed to errors in mathe- matics. E. Causes pi Failures ;Q_Mathematics Failure in mathematics is probably closely linked to the problem in this study, since no one likes to at- tempt work at which he fails. Several studies have been made on the cause of failing, most of which are Specula- tion and some of which are based on more concrete evidence. 1. Common causes 9; failure. A committee of math- ematics chairmen of New York City, under the leadership of L, 9 J. S. Georges, "The Nature of Difficulties En- countered in Reading Mathematics," The School Review. 37: 217-26, March, 1929. 50 Elizabeth M. Boyd, "A Diagnostic Study of Stu- dents' DiffiCulties in General Mathematics in First Year College Work," Teachegg College Record, 42:344-45, Janu- ary, 1941. 29 Eisner,51 made a study of the causes of failure in mathe- matics. They investigated records of the students, and, coupled with this, reports of the teachers and of the students. The teachers listed the causes as (1) truancy, (2) poor quality written work, (3) lack of class reSponse, (4) poor classroom cooperation, (5) insufficient time in home study, (6) too much time spent on non-school subjects, (7) after school employment, and (8) failure to request help. One should note that these reasons could apply as well to any other Subject. The students gave as reasons for failure: (1) lack of study and attention in and out of class, (2) poor work habits, (3) lack of mathematical ability, and (4) lack of interest. Many of these would also apply equally as well in any other area. However, the committee goes on to note that there are inherent dif- ficulties in mathematics such as: (1) its analytic and synthetic nature, (2) its generalizations, (3) its logic, (4) its sequential nature, and (5) its conciseness. 52 A Similar study by Hamza was made in Great Britain. 51 Harry Eisner, Marie Shapiro, Harry Sitomer, and Harry C. Wolfson, "A Study of Failure in Mathematics," Re- port of a Committee of the Association of Mathematics ' Chzirmen of New York City," High Points, 27:18-32, April, 19 5. 2 . 5 Mukhtar Hamza, "Retardation in Mathematics Amongst Grammar School Pupils," British Journal pf Psychol- ogy, 22:189-95, November, 1952. 30 His results paralleled those of the New York Committees3 in such areas as poor study habits, carelessness, inatten- ticn, and disregard for details, but he added such reasons as inefficient methods, frequent changes of Schools, and discontinuity of mathematical SUbject matter. However, he included teaching factors in his study in which he cited the following as reasons for failures: (1) no provision for individual differences, (2) overemphasis of the mechan- ical aSpectS of the subject, (3) faulty assumption of trans- fer of training, (4) poor textbooks, and (5) artificial division of the subject matter into branches. Here again we find reasons for failure that can be applied to any field of learning. Rudman5u came to the same conclusions when he stated that the causes of failures in mathematics were lack of ap- plication, inability to give sustained attention, poor study habits, and lack of preparation. He goes on further to state the causes of failures are known, a statement that certainly can be challenged. Lyda55 goes somewhat further by studying the process 53 Eisner and others, 9p. cip. 5“ Barnet Rudman, "Causes for Failure in Senior High School Mathematics and Sug ested Remedial Treatment," The Mathematics Teacher, 27:40 -11, December, 1934. 55 Wesley J. Lyda, "Arithmetic in the Secondary ischeol Curriculum," The Mathematics Teacher, 40:387-88, December, 1947. 31 by which students arrive at their conclusions. He found the trouble was due to inability to analyze a problem for these three factors: question asked, information given, and the use of this information. In addition students fail to outline a method of problem attack. They have a tendency to manipulate figures without understanding. Furthermore, many fail to note the reasonableness of an answer. In a study on work habits, Krathwohl56 made a sta- tistical comparison between industriousness and achieve- ment. He found the prediction of achievement could not be made from work habits. Individuals with low aptitude had high indexes of industriousness. This would Show that those with low aptitude have to and do work harder. 2. Psychological causes 9; failure. Judd57 claims that failures in mathematics are due to students getting "off the track" and finding it hard to get back again. Also, the subject covers so many rules in a short time that it overtaxes the Span of attention. This all causes the 56 William C. Krathwohl, "Relative Contributions of .Aptitude and Work Habits to Achievement in College Mathe- matics,” Th§_Journal 2; Educational PS cholo , 44:140-48, March, 1953. 57 Charles H. Judd, "A Psychological Explanation of Failures in High School Mathematics," The Mathematics Teacher, 25:185-92, April, 1932. 32 emotion of fear to enter into their approach to mathematics. creenr1e1d58 who was aided by Karlan59 studied the problems of failures in mathematics in the New York schools. The largest group of failures were those who reacted to problems with timidity, backwardness, and shyness. They gave up when they came to an obstacle. A second group iden- tified were those students who were impatient, restless, and would flare up and'become short tempered. A third group encountered were the emotionally immature, the "tough guy," and "indulged kids." A fourth group proved to be those who were pampered until they thought the world owed them a grade. Further study revealed that emotional problems ac- counted for failure among the high I.Q'S. Careful guid- ance helped most of them to adjust and succeed to some de- gree in mathematics. Another cause of failure in mathematics that has been advanced is the desire to take the path of least re- sistance, as has been pointed out by Zavitz.6O He further 58 Samuel C. Greenfield, "Failure in Mathematics: A Problem in Mental Hygiene," High Points, 17:16-22, De- cember, 1935. 59 Samuel C. Karlan, "Failure in Secondary School as a Mental Hygiene Problem: A Study of Thirty-One Cases," Mental H iene, 18:611-20, October, 1934. '60 A. S. Zavitz, "Reasons Why Pupils Fail in Mathe- matics," The School. Secondary Edition, 27:33-35, Septem- ber, 1938. 33 states that failure may be due to attempting to learn math- ematics by rote rather than by reasoning. On the subject of psychological causes of student failures and successes in mathematics, Allen61 has made some points that may be pertinent. He feels that some students are inhibited against success by a belief in an inherent lack of mathematical talent. The vieWpoint that one is natively strong in one field and weak in others fosters this belief. Allen thinks that parents may hinder their children by saying that poor ability in mathematics runs in the family. This may be further fostered by the teacher who implies that he is "dumb" in mathematics. Al- len further states that more people fail because of emo- tional attitude than from lack of ability. Another kind of psychological conditioning against mathematics is caused by superiority on the part of adults. When teachers or parents do the student's work for him, he is imprOperly con- ditioned; he must try his wings and not be shown "scorn- fully." Cases of lack of self-reliance and confidence (often mistaken for laziness or lack of talent) have a his- tory of over-activity by parents and teachers. Furthermore, students and teachers need to separate mathematical princi- ples from mechanical processes. He concludes by saying: 61 J. Eli Allen, "Some Psychological Phases of Student Success in High SChool Mathematics," The Mathe- matics Teacher, 30:322-25, November, 1937. 34 It may be that reasoning ability cannot be devel- Oped. But one can develop the habit of careful read- ing, the habit of looking up the meaning of unfamiliar words, the habit of supplying missing data or of dis- carding the irrelevant, the habit of approximating in advance reasonable results so as to establish faith in his own work. On the subject of frustration Buswell62 warns that there is a weakness in present-day education because there is a tendency to deemphasize mathematics due to its sup- posed damage to personality. Some schools have gone so far as to teach it as only incidental material. It is now thought that this personality warping is pure illusion Since personality arises from ability to adjust success- fully to frustrations. The frustrations in mathematics do not arise from content but rather from the formal applica- tions through which the subject has been presented. Bus- well goes on to say another cause of difficulty is that one should start with concrete examples and carry through to the abstract. Too often the work stops with the con- crete. A third cause of difficulty arises from narrow application. It may be desirable to Start from the stu- dent's personal problems or those of his immediate envi- ronment. One Should expand on these types of problems. 3. Case studies. Some individuals have reported 62 Guy T. Buswell, "Weakness in Present Day Arith- metic Programs," Schog; Science and Mathematics, 43:201-12, March, 1943. 35 their own experiences with mathematics in the literature. One person63 reports that he hated arithmetic and dreaded mathematics. Although he had a patient teacher in algebra who told him he could do it if he would, yet he failed. When he took geometry he began to understand the subject. In his teaching experience he had a class in physics but could not do the problems. He decided to learn how to do them and came to enjoy them; later he asked to teach arith- metic and algebra. He thought his trouble was due to im- maturity when he took mathematics in school, since he liked it later in life. Ann Terre11,6u a student at Peabody College for Tea- chers, stated that her reason for not being able to under- stand mathematics was that She had convinced herself She could not grasp it. Many others whom she knew felt as she did about the subject and used this as an excuse to avoid it. In a study of three cases Gough65 found one case of failure in mathematics due to ridicule of the student before 63 John R. F. French, "A Layman Looks at Mathematics," The Mathematics Teacher, 22:348-60, October, 1929. 64 Ann Terrell, "I Cannot Learn Mathematics," Peabody .Iournal 93 Education, 31:335-36, May, 1954. 6 . 5 Sr. Mary Fides Cough, "Mathmaphobia: Causes and Treatment," The Clearing House, 28:290-94, January, 1954. 36 strangers, one case where interruption of sequence, never made up, was the cause of difficulty, and a third case where the student feared the examinations in the subject.’ In another case study of 34 students in arithmetic, 66 Schmitt by means of interview found the following reasons for failure: eight had periods of ill health, seven had general ill health, fourteen lacked interest and failed to apply themselves, and five of the cases failed to reveal a reason. Graham67 reports the case of a New York University student who felt he had been cheated in his mathematical experience. The student hoped he had finished mathematics when he completed intermediate algebra. In reminiscing, three of his high school teachers had remarked that they were "dumb as donkeys" in mathematics and one English tea- cher was proud to report that she had "flunked" all her college courses in mathematics. He said, "She took on prestige in our eyes." (students to whom she was talking) Early he came to the conclusion that there was no stigma attached to being poor in mathematics. Algebra was only a subject to be passed. Later in college he was required to 66 Clara Schmitt, "Extreme Retardation in Arith-' metic," The Elementary_School Journal, 21:529-47, March, 1921. 67 P. H. Graham, ed., "A Student Who 'Found Him- self' in Mathematics," School and Society, 54:249-50, September 27, 1941. 37 take a mathematics survey course where he came to the con- clusion that the beauty in structure of pure mathematics is as esthetic as is any form of art. He felt that he and others had been "betrayed" because they had never received any knowledge of the true nature of mathematics. F. Summary Several studies have been made to determine the nature of mathematical ability. There seems to be no clear-cut evidence that mathematical ability is inherited, as such. All attempts to prove that this ability demands some special capacity common to all branches of the subject matter and distinct from other mental abilities, have failed thus far. It is probably safe to conjecture that none is likely to be found. The attempts to predict the success of a student before he enrolls in college mathematics give rise to dis- appointing and conflicting results. However, there seems to be general agreement that success in high school math- ematics to some extent indicates possible success in col- lege mathematics. The number of units in this subject presented at entrance to college plays little part in such success. There appears to be some feeling that "personality" and "character traits" need further study in this area. Psychologists report that in the learning of mathe- matics there is grave danger of overemphasis on the product 38 at the eXpense of the process. The process of learning is a process of reorganization of behavior and is colored by feelings and emotion. There is concomitant learning with mathematics-~the student learns to like or dislike the sub- Ject, to like or dislike school, to c00perate or work as an individual. Children learn to group before they are able ~‘to number; only later do they have an idea of surface and length. In teaching techniques there seems to be disagree- ment concerning size of classes and its effect on the learning of mathematics. Most educational psychologists agree that problems in mathematics should be realistic and based on eXperience of the students, as far as possible, but not all problems should be based on their narrow out- look. Also, mathematics should be taught to produce trans- fer of training rather than assuming such transfer will take place. Several investigators seem to feel that weakness in the four fundamental skills causes the greatest difficulty in mathematics. Also, that certain number combinations appear to be more difficult to learn than others. Other investigations lead to the conclusion that problems deal- ing with unfamiliar environment lead to needless errors. In considering the causes of failures, many of the conclusions made are also applicable to any subject. This 39 is true where causes are truancy, lack of class reSponse, insufficient time Spent on work, poor teaching, and others. However, some work has been done on psychological causes of failure, which include such characteristics as timidity, impatience, emotional immaturity, and pampered groups. Some workers have suggested that students have been condi- tioned against mathematics by the attitude of others. Only a few case studies have been attempted to de- termine the causes of failure in mathematics. These point to the possibility of immaturity, peer relationships, ill health, interrupted sequence, ridicule, and failure to un- derstand the nature of mathematics. CHAPTER III METHOD OF CONDUCTING THE INVESTIGATION In this chapter the account of the method of con- ducting the study will be divided into four parts as fol- lows: (1) selection of the method, (2) selection of the subjects to participate in the investigation, (3) col- lection of the data, and (4) treatment of the data. A. Selection 2: Method After reviewing the literature pertaining to meth- ods of attacking problems related to this subject, it was thought best to use the case study method of approach. This method was used to only a slight extent in earlier investi- gations, accounting for only two or three per cent of the' studies made prior to 1939, according to Olson.1 However, as he points out, this method is gaining greater dignity in education.‘ In discussing methods of approach in psycholog- ical study, Allport2 says Of the case study, "This method is...the most comprehensive of all, and lies closest to the initial starting point of common sense. . . . PrOperly used l Willard C. Olson, "The Case Study," Review 93 Ed- ucational Research, 9:483-90, December, 1939. szordon Willard Allport, Personali_y: [A Psycholog- ical_Interpretation. New York: Henry Holt and Company, 1937. p. 390. 41 it is the most revealing method of all." In addition he says, ". . . it [the case study] has the full value of both a work of science and a work of art."3 The case study method has a place in science because it is useful in providing certain information that cannot be obtained in any other way--in medicine and in other fields as well. In an attempt to diagnose difficulties in solving arithmetic problems, Chase“ found the case study to be the only reliable method in determining the nature of the er- rors in arithmetic. Brownell and Watson5 made a study of two methods of diagnosing errors in arithmetic. They tried the personal interview method and the analysis of written records of pupil performances. Although they felt their interview method was used under unfavorable circumstances, was sub- jective, and could not be standardized, it proved to be the better method except in superficial types of diagnosis. 3 ______1d~. p. 395. Vernon Emory Chase, "The Diagnosis and Treatment of SOme Common Difficulties in Solving Arithmetic Prob- Zlems," Journal 2: Educational Research 20:335-42, Decem- ber, 1929. 5 William A. Brownell and Brantly Watson, "The Com- Iwarative Worth of Two Diagnostic Techniques," Journal 9: IEducational Research, 29: 6A-76, May, 1936. 42 They found that the analysis of written work had only the advantage Of convenience. In his discussion of types of research in education, Good6 says, "There are real possibilities in wider use of the case method . . . in its application to education." In their very unique study of problem solving, Bloom andBroder7 found that they needed to resort to case stud- ies. They discovered that too much of the investigative work depended upon various types of tests. This gave em- phasis on the product rather than on the process. There are certain limitations and objections to the case study method. A number of these have been given by Young and Schmid,8 such as (l) the subject may tend to give the reSponse he feels the investigator wants, (2) the subject may try to be self-justificatory rather than fact- ual, (3) the investigator may tend to see only that for 6 Carter V. Good, "Fields and Types Of Research in Education 1918-1931," Journal 9; Educational Research, 24: 33-43, June, 1931. 7 Benjamin Samuel Bloom and Lois J. Broder, Problem- Solving Processes 21 College Studentg: fig Exploratory lg: vestigation, Supplementary Educational Monographs. Pub- lished.in conjunction With.The School Review and Thglglg- mentary Schog; Journgl, Number 73. Chicago: The Univer- sity of Chicago Press. July, 1950. 109 pp. 8 Pauline V. Young and Calvin F. Schmid, Scientific Social Surveys and Research, Second Edition. New York: LPrentice Hall, Inc., 1§h9. pp. 273-76. 43 which he is looking, (4) the investigator may help the subject, (5) since the subject is naive, the logical con- cepts and scientific classification have to be read into the statements by the investigator. In Spite of the lim- itations of case data, this method is capable of reveal- ing the interests, motives, and inner lives of persons "and the meaning the social world assumes in their outlook on, and reaction to life."9 Since the present study is concerned with the causes of fears and dislike of mathematics, rather than the re- sults of these fears, it was thought that the case study method would yield the best results. B. Selection 9: subjects The subjects selected for this study were all stu- -dents of Central Michigan College. Since all students must demonstrate a certain proficiency in mathematics be- fore a teaching certificate can be granted,10 those who had fears of the subject were more inclined to express this fear when they were confronted with the requirement. In this college all elementary teachers are required to 9 Ibid., p, 284. 10 gullgtin, l2§3:jj Sessions. Central Michigan Col- lege gflgducation. loc. cit. MM take a course in general mathematics and all others are required to pass an examination in which they must reach a given standard. For those who fail to reach such a stan- dard, a class in remedial mathematics is offered. The teachers of both groups are aware of students who are hav- ing difficulties and those who admit they have fears and dislike of mathematics. The investigator was allowed to ask for volunteers for this study. Most of these students were willing to act as subjects for the investigation. In addition to this source, the student counselors who were made aware of this investigation furnished names of persons who would be possible candidates for the study. These persons were approached to determine their willing- ness to participate. A third source of subjects consisted of volunteers who learned of the study and wished to take part. These had the hope that what they might discover about their fears would help them overcome some of their frustrations. In no case were students requested by teachers, counselors, or the investigator to participate unless they wanted to do so. Furthermore, several students who were exceptionally good in mathematics also volunteered to participate in the study. These were all selected from the Kappa Mu Epsilon fraternity, an honorary organization in mathematics. #5 Selections, limited in this manner, included a total of 41 persons. That more women than men were involved may be due to the fact that the majority of students used in the study were in the field of elementary teacher training. These students represented all classes from freshmen to Seniors. However, the largest group were freshmen and sophomores since they were the ones usually involved in meeting the requirements in mathematics. Transfer stu- dents and those who were previously unable to meet the mathematical requirements accounted for most of the others. This group included students with varying educa- tional backgrounds. Some had received elementary training in one-room rural schools and in high schools where the total enrollment, grades nine through twelve, did not ex- ceed seventy-five pupils. Others had been educated in large city systems, while the remainder had been trained in average-sized schools} C. Collection pf the Data Arrangements were made to meet each student indi- vidually for a personal interview. In order to decrease any tendency of the students to give answers which they might feel were eXpected of them, each person interviewed was asked to cOOperate by not revealing the nature of the interview to others. 46 Although there is some difference of Opinion con- cerning this technique,11 it was thought best to record the interview on tape. The data was collected on a Revere Iagnetic Tape Recorder, Studio Model T-70167, at 7:5 r.p.m. on 1200 foot rolls of 0.25 inch tape. In this manner the interview could be studied more carefully and the danger of the investigator using his own interpretation and point of view were minimized. Through the OOOperation of the administration of Central Michigan College, a counseling room was reserved, for the interviews. This room was furnished with a desk, two chairs, and a table on which the tape recorder was placed. This room insured privacy for both the student and the investigator. Both were seated at the desk be- fore the micrOphone, and paper and pencil were furnished for the student if he wished to use them. In order to get the student adjusted to the tape recorder, simple questions were asked first. In a few cases, the student seemed to feel anxious when the re- corder Operated while he tried to recall a fact or decide on a question. To relieve tension, the recorder was stop- ped and then turned on again when he was ready to answer. The interview was divided into two parts. The first part was a structured interview and all students 11 Young and Schmid, 93;. cit., p. 263. 1+7 were asked the same questions.12 This group of questions served as a background and the interviewer used additional questions to provoke a clearer answer when necessary. The second part of the interview consisted of a num- ber of mathematical exercises which the student was asked to do "out loud."13 The purpose of this was to discover the process used. These exercises were selected from the 1” and from the competency tests required by the college, Foust-Schorling Test of Functional Thinking in Mathemat- ics.15 The latter test showed a high reliability (.88) for college freshmen and supplemented the more routine ex- ercises asked on the competency test. These types of ques- tions, as the name implies, gave more insight into the stu- dent's ability to think in the language of mathematics. When difficulties were uncovered the investigator probed into the background to find the cause of the trouble. Since a number of examples used in the competency test were similar, i.e., covered the same area of mathe- matics, not all were used for each person interviewed. In the case of those students proficient in mathematics, the 12 Appendix A. 13 Appendix B. 1" Permission granted by the college authorities. 15 Foust-Schorling Test of Functional Thinking in Mathematics. Permission obtained from Dr. Judson Foust. 48 arithmetical competency questions were omitted and only the general questions and those involving functional thinking were asked. Other information concerning the student was obtained from the student's Guide Book.16 This latter gives data on the A.C.E. Psychological Test, the Guilfordeimmerman Apti- tude Test, Purdue Placement Test in English, California Progressive Reading Test, California Arithmetic Test, Co- Operative General Culture Test, and the Cooperative Con- temporary Affairs Test. In addition, the high school and college scholastic record, as well as the co-curricular activities record, is available from the Guide Book. D. Treatment 9f the Data Each individual case was reviewed from the tape recording and an analysis of the finding was made. These are reported in Chapter IV, where general descriptions and Specific illustrations are cited. As Allport17 states, "One without the other is incomplete." Finally, the case studies are compared to discover 16 Guide Book for Educational DevelOpment. Cen- tral Michigan College, Division of Student Personnel. Mount Pleasant, Michigan. PermiSsion for use granted by Dr. Judson Foust, Vice-President, Central Michigan Col- age. 17 Gordon Allport, 2p. cit., p. 393. 49 trends, if any, and their relationship to the questions raised in Chapter I. E. Summary After examination Of the literature and consulta- tion with others familiar with educational research, it was decided that the case study method would be the best approach to the problem. It was then decided that each subject in the in— vestigation would be interviewed and the interview record- ed on tape. All subjects were referred to the investiga- tor and were asked to cooperate in the study on a voluntary basis. These people were referred to the investigator by remedial mathematics teachers and student personnel coun- selors. A few students volunteered of their own accord. After experimentation, a structured interview was established in which each student answered the same gen- eral questions and any others needed to clarify answers. In addition, each student was asked to perform some math- ematical exercises aloud to determine his methods of pro- cedure. These methods were then further probed by the investigator. Each case study was analyzed and an analysis of the group in light of the questions raised by the inves- tigation was made. CHAPTER IV CASE STUDIES In this chapter case studies will be discussed indi- vidually. These studies are divided into two groups, those of students who are having difficulties in mathematics, and those of students who are proficient in the subject. A. Studies Q; Those Having Difficulty Case number one. This student ranks low in reason- ing ability, as it is measured by the California Mental Maturity Test. Her score on this test put her at the twenty-fifth percentile of her college group of 625. She has difficulty with the fundamental arithmetic processes. Her methods are inefficient. In addition she adds a few numbers in a column, writes the sum, and adds a few more; finally, she adds the sub-totals to obtain the answer. Her conception of mathematics is rather naive.1 Tc lier the subject is a study of numbers, useful for practi- tcal purposes, such as counting calories in food or calcu- ljrting the cost of purchases. However, she feels that she 1 Investigator: "What is the purpose of mathe- matics?" Student: "I don't know, but I guess it's useful tc>