AN EXPERIMENTAL STUDY OF THE EFFECT OF THE DIRECT SRFDY OF QUANTITATIVE VOCABULARY QM THE ARITHMETIC PROBLEM SOLVING ABILITY OF FIFTH GRADE PUFILS Thesis 5:» the. Deans of Ed. D. MICHIGAN SWEAT? UNIVERSITY Louis Frederick VanfierUndu I962 0-169 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIZIIIIIIIIIZIIIIIIIII 3 1293 10462 This is to certify that the thesis entitled AN EXPERIMENTAL STUDY OF THE EFFECT OF THE DIRECT STUDY OF QUANTITATIVE VOCABULARY ON THE ARITHMETIC PROBLEM SOLVING ABILITY OF FIFTH GRADE PUPILS presented by Louis Frederick VanderLinde has been accepted towards fulfillment of the requirements for Qoggor's degree in Education \ a . A -. Major profess r ' Datew._l9_62__ MSU LIBRARIES RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped\below. J, g. .931. 8 : .;. ‘ £91) ' .. .87 — o ,' AN EXPERIMENTAL STUDY OF THE EFFECT OF THE DIRECT STUDY OF QUANTITATIVE VOCABULARY ON THE ARITHMETIC PROBLEM SOLVING ABILITY OF FIFTH GRADE PUPIIS by Iouis Frederick Vanderldnde AN ABSTRACT OF A THESIS Submitted to Michigan State university in partial fulfillment of the requirements for the degree of DOCTOR OF EDUCATION College of Education 1962 ABSTRACT AN EXPERIMENTAL STUDY OF THE EFFECT OF THE DIRECT STUDY OF QUANTITATIVE VOCABULARY ON THE ARITHMETIC PROBLEM SOLVING ABILITY OF FIFTH GRADE PUPIIS by Iouis Frederick VanderLinde Statement of the Problem It was the purpose of this study to determine the effect of the direct study of quantitative vocabulary on the verbal problem solving ability of fifth grade pupils. Procedures Twelve of the 2A fifth grade classes in Bay City) Michigan were randomly assigned to the experimental group and 12 to the control group. A pre-test was administered to all pupils and the "t" test was used to analyze these data for the purpose of pairing classes. The final sample consisted of 211 pupils in 9 experimental classes and 183 pupils in 9 control classes. Four direct study techniques comprised the experimental method used in teaching the meanings of the quantitative terms selected for study; The length of the experimental period ranged from 20 to 2h weeks during which time experimental classes studied 8 quantitative terms each week. The total number of terms studied in a given classroom ranged, l 2 Louis Frederick-VanderLinde therefore, from 158 to 190. Control classes followed the regular instructional program in arithmetic. An alternate form of the pre-test was administered as the post-test. The "t" test and an analysis of variance technique were used to analyze the data and test hypotheses. Findings The findings of this study were interpreted in terms of the following hypotheses. H1 Pupils who have studied quantitative vocabulary by direct study techniques will attain higher mean achievement on a test of arithmetic problem solving than pupils who have not studied quantitative vocabulary by direct study techniques independent of the effects of sex and intelligence. H2 There will be no difference in mean gains over the eXperimental period on tests of arithmetic problem solving among experimental pupils who have above average intelligence, average intelligence, and below average intelligence. There will be no difference in mean gains over the experimental period on tests of arithmetic problem solving between male pupils in the experimental group and female pupils in the experimental group. With the exception of Hg, all hypotheses were supported when statistical tests were applied to the post-test data. Hypoth- esis Hé was partially rejected as a result of significant differ- ences in mean gains on tests of arithmetic problem solving between the high and low and the middle and low IQ categories. The difference in mean gains between the high and middle IQ categories was not significant and, to this degree, hypothesis 32 was supported. 3 Louis Frederick Vanderlinde Further analysis of the data revealed that the experimental group also attained significantly higher achievement on the post— test of arithmetic concepts. The differences in mean scores between methods on the post-tests of vocabulary and reading comprehension were not significant. Conclusions l. The experimental group and the control group were not significantly different at the beginning of the experi- ment in mean achievement on tests of vocabulary, reading comprehension, arithmetic concepts, and arithmetic pro- blem solving or in mean intelligence. 2. Pupils who have studied quantitative vocabulary using the direct study techniques described in this study achieve significantly higher on a test of arithmetic problem solving than pupils who have not devoted special attention to the study of quantitative vocabulary. 3. Pupils who have studied quantitative vocabulary using the direct study techniques described in this study achieve significantly higher on a test of arithmetic concepts than pupils who have not devoted special attention to the study of quantitative vocabulary. A. The direct study of quantitative vocabulary does not tend to result in improvement in general vocabulary or in reading comprehension. 5. The experimental method is not more effective with one sex than with the other. 6. The experimental method is more effective with pupils who have above average or average intelligence than with pupils who have below average intelligence. 7. Effective vocabulary study can be made a part of the regular arithmetic program without sacrificing pupil achievement in the subject matter of arithmetic. 8- The significant difference in mean scores between sexes on the post-test of arithmetic concepts was due to the significant difference between sexes on the pre-test of arithmetic concepts. AN EXPERIMENTAL.STUDY OF THE EFFECT OF THE DIRECT STUDY OF QUANTITATIVE VOCABULARY ON THE ARITHMETIC PROBLEM SOLVING ABILITY OF FIFTH GRADE PUPILS by Louis Frederick vanderldnde A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF EDUCATION College of Education 1962 ACKNOHLEDGEMENTS The writer would like to express his grateful appreciation to Dr. Calhoun C. Collier, his major professor, for his stimu- lation, encouragement and guidance throughout the conduct of the study. To the members of the doctoral guidance committee, Dr. Charles A. Blackman, Dr. Byron H. VanRoekel, and Dr. Orden C. Smucker for their helpful suggestions and criticism. To Dr. JOhn J. Paterson for his assistance with the treatment of the data. To the administrators and teachers in Bay City, Michigan for their graciousness and continued cooperation. And to his wife for her patience and inspiration. 11 TABLE OF CONTENTS A C KNOWMNTS O O I O O O O O O O O I O LET OF TABIES . O O C O O O O O O O O 0 Chapter I. II. 111. IV. ./ IN'I'RODIJC T ION O O O O O O O O O O 0 Need for the Study . . . . . . . Statement of the Problem . . . Statement of the Hypotheses . Definition of Terms. . . . . . . Nature and Design of Study . . . Limitations of the Study . . Plan of the Report . . . . . . . Summary . . . . . . . . . . REVIEW OF THE LITERATURE . . . . . Vocabulary as a Factor in Verbal Problem Solving Studies Related to the Present Investigation . . Summary . . . . . . . . . . . . METHOD OF THE STUDY . . . . . . . Selection of Terms to be Studied Selection of the Teaching Techniques . Nature of the Data . . . . . . . Length of the Experimental Period Selection of the Sample . . . . Orientation of the Teachers . Methods of Testing the Hypotheses ANALYSIS OF THE DATA . . . . Differences Between Methods Not Independent of Other Effects 0 O C O O O O 0 Differences Between Methods Independent of Other Effects. . . . . . . . Differences in Mean Gains on a Test of Arithmetic Problem Solving . iii Page ii 13 1b. 15 15 16 17 l9 19 2O 72 71+ 76 Chapter Differences Between Methods Independent of Other TABLE OF CONTENTS (continued) Among IQ Categories. . . . . . . . . . . . . Between Sexes . . . . . . . . . Effects on Post- Tests of Vocabulary, Reading Comprehension, and Arithmetic Concepts . . . Vocabulary . . . . . . . . . . . . . . . . Reading Comprehension . . . . . . . . . . . Arithmetic Concepts . . . . . . . . . . . . V. SWEY, CONCLUSIONS AND IMPLICATIONS . . . . Summary . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . Implications . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . APPENDIXA. APPENDIXB. Immune. APPENDIXD. iv Page . 76 78 . 78 - 79 . 79 . 83 87 . 87 . 9h . 98 . 101 . 112 . 119 . 133 . 11+8 Table LIST OF TABLES Number of Pupils, Means, Variances, and Results of Tests for the Significance Between Means for Nine Pairs of Experimental and Control Classes on the Vbcabulary Pre-Test . . . . Number of Pupils, Means, variances, and Results of Tests for the Significance Between Means for Nine Pairs of Experimental and Control Classes on the Pre-Test of Reading Comprehension . Number of Pupils, Means, Variances, and Results of Tests for the Significance Between Means for Nine Pairs of Experimental and Control Classes on the Pre-Test of Arithmetic Concepts . Number of Pupils, Means, variances, and Results of Tests for the Significance Between Means for Nine Pairs of Experimental and Control Classes on the Pre-Test of Arithmetic Problem SOlVing O I O O O O O O O O O O I O O 0 Number of Pupils, Means, variances, and Results of Tests for the Significance Between Means for Nine Pairs of Experimental and Control Classes on the Iorge-Thorndike Intelligence Test . . . . . . . . . . . . . . . . . . . . Means, Variances, and Results of Tests for the Significance Between Means from 211 Experi- mental Pupils and 183 Control Pupils on Four Pre-Tests and on the Iorge-Thorndike Intelli— gence Test, Level 3 . . . . . . . Number of Pupils and Mean Scores on Four Post- Tests for Pupils in the Experimental and Con- trol Groups by Sex and by IQ Level . . . 61 62 63 6h 65 66 69 LIST OF TABLES (continued) Table Page 8 Means, Variances, and Results of Tests for the Significance Between Means from 211 Experi- mental Pupils and 183 Control Pupils on Four Post- Tests . . . . . . . . . . . . . . . . . . . 73 9 Analysis of variance for Testing the Significance of the Effect of Methods on Mean Scores from the Post-Test of Arithmetic Problem Solving Independent of the Effects of Sex and Intelli- gence . . . . . . . . . . . . . . . . . . . . . . . 75 10 Number of Pupils, Variances, Means, and Results of Tests for the Significance of the Difference in Mean Gains for Experimental Pupils on Tests of Arithmetic Problem Solving Among IQ Cate- gories and Between Sexes . . . . . . . . . 77 11 Analysis of Variance for Testing the Significance of the Effects of Methods on Mean Scores from the Post-Test of vecabulary Independent of the Effects of Sex and Intelligence . . . . . . . . . . . 8O 12 Analysis of variance for Testing the Significance of the Effect of Methods on Mean Scores from the Post-Test of Reading Comprehension Independent of the Effects of Sex and Intelligence . . . . . . . . 81 13 Analysis of Variance for Testing the Significance of the Effect of Methods on Mean Scores from the Post-TESt of Arithmetic Concepts Independent of the Effects of Sex and Intelligence . . . . . . . . . 82 1h Number of Pupils, Sum of Raw Scores, and Means from the Post-Test of Arithmetic Concepts for Males and Females . . . . . . . . . . . . . . . . . . 8h 15 Analysis of variance for Testing the Significance of the Difference in Mean Scores Between Sexes from the Pre- Test of Arithmetic Concepts Inde- depdent of the Effects of Method and Intelli- gence . . . . . . . . . . . . . . . . . . . . . 86 vi CHAPTER I INTRODUCTION [A major responsibility of mathematics instruction beginning in the elementary grades is to assist pupils in deve10ping the ability to do quantitative thinking and to reason logically. 'With respect to this Grossnickle and Brueckner say: It is a primary function of the arithmetic program to arrange experiences that will deve10p in children the ability to "think through" problematic situations that they encounter and to deal with them intelligently and skillfully.l Problem solving is the highest level of quantitative thinking. Quantitative thinking involves the use of number and number relation- ships in dealing with those aspects of situations which are character— ized by quantity, magnitude, order, form, extent, relation, and position. It is the quantitative aspects of problematic situations that make it necessary for man to apply mathematical analysis and interpretative powers in attempting to solve problems arising in such situations. An arithmetic prOgram which expects to help pupils improve in quantitative thinking, critical thinking, and in logical reasoning is based largely on problem solving. Mbrton supports the opinion of many in stressing the importance of problem solving as a function of modern arithmetic programs. He says, "Inasmuch as teaching pupils to solve problems is the chief purpose of arithmetic instruction, the problem solv- ing program is perhaps the most important part of the entire arithmetic 1Fester E. Grossnickle and Ieo J. Brueckner, Discovering Meanings in Arithmetic (Philadelphia: John C. Winston Co., 1959), p. 308. l 2 curriculum." While it has been argued that genuine problems should take the place of textbook problems, it is unrealistic to assume that the school can consist entirely of real-life situations. Vicarious problem situ- ations in the form of verbal problems must be used if pupils are to become thoroughly familiar with the processes involved in problem solution. The deve10pment of skill in solving written verbal problems is not, however, an end in itself. Practice in solving many kinds of verbal problems assists pupils in developing a variety of problem solving skills and in making application of these skills in solving real problems arising in their every-day experiences. Other purposes of written verbal problems have been identified by Grossnickle and Brueckner: For many years one of the major objectives of arithmetic instruction has been to teach children to solve written verbal problems. The purposes of this work are: (l) to test the pupil's readiness for new work to be done, (2) to determine how efficiently he uses number facts and compu- tational skills, (3) to check and refine his concepts of mathematical relationships, (A) to challenge his ability to apply number processes and quantita- tive procedures, (5) to develop his ability to make estimations and approximations, and (6) to extend his arithmetical bagkground and sensitivity to the role of number in daily life. It is readily apparent that one of the major objectives in arithmetic in the elementary grades is the solving of written verbal problems. Yet one of the most persistent difficulties in arithmetic instruction is in the area of verbal or story problems. There is a 2Robert L. Marton, "The Place of Arithmetic in Various Types of Elementary-School Curriculums," Arithmetic 1919 (Supplementary Educa- tional Mbnographs, No. 70; Chicago: Uhiversity of Chicago Press, l9h9), p- 7- 3Grossnickle and Brueckner, loc. cit., p. 315. 3 considerable amount of opinion and data available which indicates that pupils do not achieve as well in arithmetic problem solving as they do in arithmetic computation. Approximately thirty years ago Newcomb was moved to say that it was, ". . . common knowledge of every teacher of arithmetic that the most difficult part of the subject is the securing h There is of satisfactory results in the solution of problems." little reason to believe that this condition has abated over the years. It was the Opinion of Spitzer and Flournoy in a 1956 article that, "The improvement of pupil achievement in verbal problem solving is an important objective of most upper grade arithmetic teachers. That this objective is not often reached with any degree of satisfaction is evident to all students of arithmetic teaching."5 MDst elementary school teachers today support the contention that verbal problems cause children difficulty and that pupils often become confused and frustrated when faced with the task of solving them. This frustration often leads to dislike for arithmetic as a school subject. Fundamental to the accurate and efficient solving of verbal arithmetic problems is the adequate comprehension of the statement of the problem situation. The first step in the solution of a verbal problem is to read the problem with understanding. If pupils cannot read with a relatively high degree of understanding they are unlikely to be able to interpret the meaning of the problem, analyze the relations between 1+R. S. newcomb, "Teaching Pupils Bbw to Solve Problems in Arithmetic," Elementary School Journal, XXIII (November, 1922), 183. 5Herbert F. Spitzer and Frances Flournoy, "Developing Facility in Solving Verbal Problems," Arithmetic Teacher, III (November, 1956), I77. h the elements, or engage in the level of quantitative thinking necessary to the selection of the appropriate computation or sequence of compu- tations. Greene6 undertook an analysis of the factors necessary to the solution of verbal problems and listed five fundamental steps. The first of these was comprehension, including understanding of the items, elements, and processes, either stated or implied. Other factors listed by Greene were rate of reading, vocabulary difficulties, and reading numerals. Stevenson reported that in a study made by a class of thirty- two elementary teachers one of the important causes of failure was the ". . . inability to read.which, of necessity, affects the ability to read arithmetic problems."7 Another cause of failure identified by this group was the lack of general and technical vocabulary which is closely associated with ability to read the written material found in arithmetic. It is the Opinion of a considerable number of others that adequate comprehension is necessary to the solution of verbal problems and that various factors related to reading ability are major causes of diffi- culty in comprehension. The comprehension of verbal problems is complicated by several of the characteristics of the language of mathematics. Many problems contain abbreviations, mathematical signs, and collocations, such as, 'total amount,’ 'scale drawing,‘ and 'on the average.' Abbreviations 6Herry A. Greene, "Directed Drill in The Comprehension of Verbal Problems in Arithmetic," Journal of Educational Research, x1 (January, 1925). 3%. 7P. R. Stevenson, "Difficulties in Problem Solving," Jburnal of Educational Research, XI (February, 1925), 95. 5 may cause difficulty because they are sometimes ambiguous, that is, the same abbreviation may have more than one referent. Collocations present a special difficulty because of the change in meaning of the terms when used in conjunction with other terms. Context is another significant factor contributing to the difficulty pupils have in comprehending the meaning of verbal problems. .An examdnation of verbal problems reveals that they are almost always stated briefly and concisely. A.mdnimum number of words is generally used to communicate intricate and often abstract relationships. Often only the bare essentials are given. Elaborate, descriptive statements are seldom used, therefore, context offers a minimum number of clues for unlocking the meanings of unfamdliar terms. Verbal problems usually follow no special pattern and the absenggiof continuity from problem to problem minimizes the possibility of context clues. .According to Young context may mitigate against comprehension because of the complex ideas expressed by otherwise easy words. He cited the following example of context interfering with comprehension in which all of the words are from the first twenty-five hundred in the Thorndike word list: "The square of the sum of two numbers is equal to the square of the first number added to twice the product of the first and second numbers, added to the square of the second number."8 A very significant source of difficulty in understanding verbal problems is caused by the unusual number of technical and semi- technical terms found in most problems. Cole suggests that: 8H. E. Young, "Language.Aspects of Arithmetic,“ School Science and thhemtics, 57 (March, 1957), 172. . 6 In about the third grade the child meets his first technical vocabulary. Here he begins to read problems in arithmetic which employ such special words as inch, yard, measure, loss, gain, rice, subtract, zero, times [underlines indicate italicized words 7g Many of these terms have a specialized meaning and are unlikely to be met in other subject areas. In the field of arithmetic pupils encounter many terms which are outside their experience. Some terms are known by pupils in one context, but when used in arithmetic have entirely unknown meanings. Other terms are unknown in form, but are used to express a meaning which is familiar to pupils. Still others are unfamiliar because they have not been seen or heard before and because pupils have not had experiences with those situations with which the terms may be appro- priately associated. Buswell and John conducted an extensive study of pupil knowledge of the vocabulary of arithmetic and it was their Opinion that: . . a considerable number of arithmetical terms are not likely to be encountered by the pupil in his work in reading and spelling. Since these are the two subjects in.which new words are generally presented, it is doubtful whether the pupil will learn many Of the technical terms in arithmetic unless they are taught in the arithmetic class or develOped.with greater fullness in the arithmetic text- book. The comprehension difficulties encountered by pupils in solving verbal problems are often due to the simple fact that they do not know what the worch mean. This fact is true Of the reading material found in all of the content areas. The understanding of the elementary concepts in the written material in any field of knowledge is essentially 9mella Cole, The Improvement of Reading (New York: Farrar and Rinehart, Inc., 1938), p. 126. loGuy Buswell and lenore John, The Vocabulary of Arithmetic (Supplementary Educational Lbnographs, No. 38; Chicago: ‘Ihe University of Chicago Press, 1931), p. 100. 7 the understanding of the meaning Of the terms used. Comprehension will necessarily be inadequate if pupils do not have a sufficient meaning vocabulary. Black,11 in a study involving college students, studied the difficulties faced in understanding what is read. He concluded that one Of the factors responsible for pupil difficulty in comprehension was that the pupils failed to grasp a complex, abstract, or unfamiliar idea represented by a word. Nolte stressed the importance of meaning vocabulary to comprehension in saying that, "VOcabulary has been considered by numerous investigators as the most important structural element affecting comprehension in reading."12 Knowledge of the vocabularyiof arithmetic is basic to an under- standing of its concepts, principles, and relationships. This knowledge is vital to a pupil's ability to think quantitatively and is recognized by some as one of the important aims Of arithmetic education. Brownell lists as an important aim of arithmetic instruction the acquisition Of ". . . a meaningful vocabulary of useful technical terms Of arithmetic which designate quantitative ideas and the relationships between them."13 Pressey and Elam stress the importance of the vocabulary of arithmetic, and ". . . are convinced that one outstanding source of error in arithmetic problems and of antagonism toward arithmetic lies 11E. L. Black, "The Difficulties of Training College Students in Uhderstanding What They Read," British Journal of Educational Psychology, XXIV (February, l95h), 30. 12K. F. Nolte, "Simplification of Vocabulary and Comprehension in Reading," ElementaryEnglish Review, XIV (April, 1937), 119. 13William A. Brownell, "The Evaluation of learning in Arithmetic," Arithmetic in General Education (Sixteenth Yearbook of the National . Council of Teachers Of.Mathematics; New York: Bureau of Publications, Teachers College, Columbia University, l9hl), p. 231. 8 in the fact that children do not know what the words mean."lh Many investigators are convinced that the predominant diffi- culties in comprehension are caused by technical and semi-technical vocabulary. The field of arithmetic employs many such terms and it cannot be left entirely up to the pupils to learn the meanings of these specialized terms. Thus, because of the unusually large number of technical terms and their importance to pupil comprehension Of the meaning of problems, vocabulary study becomes an important aspect of instruction in arithmetic. A.pupil's meaning vocabulary may deve10p incidentally as he reads, listens, and engages in conversation. These methods are un- likely to result in satisfactory improvement in the acquisition of the meanings Of important quantitative terms for the following reasons. It has already been noted that many terms are specific to arithmetic and are unlikely to be met in other content areas.15 For Similar reasons many arithmetic terms occur seldomly in out-Of-school experiences, therefore, such experiences contribute little toward the deve10pment of these concepts.16 The vocabulary load of unfamiliar words or new uses for familiar words is not always controlled in the reading matter found in arithmetic. Many terms do not appear a sufficient number of times to assist the reader in acquiring apprOpriate and enriched mean- ings. Many studies of the vocabulary content Of arithmetic texts th- C Pressey and M. K; Elam, “The Fundamental VOcabulary Of Elementary-School Arithmetic," Elementary School Journal, XXXIII (September, 1932): 50- 15Buswell and John, loc. cit., p. 100. l6Buswell and JOhn, loc. cit., p. 2. 9 support the findings of a study by O'Rourke and Mead.l7 These investi- gators examined five arithmetic textbooks and reported that many tech- nical terms appeared only once in the books in which they were used and fifty-five per cent appeared less than five times. They suggested that more repetition is needed if terms are to be meaningful. Johnson surveyed fifth grade texts in six subject areas and compiled a list of terms felt to be important to an understanding of the concepts in these subjects. She tested pupil knowledge of these terms and reported that, "The results of this study Show that a program of word enrichment is needed for the understanding of the textbooks used in the content subjects.1'8 It is not feasible to leave the learning of important terms to incidental methods. Under such circumstances there can be no guarantee that pupils will acquire meanings of important quantitative terms. The literature contains many suggestions for the improvement of meaning vocabularies, and there is a small body of experimental research which has attempted to determine the results Of conscious, direct efforts to increase the size Of pupilg"vocabulary. Eurich reported that prior to 1930 few studies were devoted to the problem.of word study.19 In the decades since 1930 more attention has been given to the problem, and the results of experimental studies indicate consistently l7Everett V. O'Rourke and Cyrus D. Mead, "Vocabulary Difficulties of Five Textbooks in.Third-Grade Arithmetic," Elementary School JOurnal, XLI (May, l9hl), 69o. laMary E Johnson, "The VOcabulary Difficulty of Content Subjects in Grade Five," Elementary English, XXIX (ray, 1952), 280. 19A. c. Eurich, "Enlarging the Vocabularies of College Freshnen," English Journal (college edition), 1x1 (February, 1932), 135. 10 superior gains in vocabulary achievement through the use of direct study techniques. A study by Johnson20 demonstrated the effectiveness of direct study techniques for increasing arithmetic vocabulary. It was concluded, however, that specific vocabulary exercises did not result in general improvement in arithmetic learnings. Drake21 con- ducted a study at the high school level to test the efficacy of certain direct study techniques for learning the vocabulary of Algebra and reported higher achievement by the experimental group as well as favorable attitudes on the part of teachers and pupils. Similar results were reported by Dresher22 and by Buckingham?3 in studies at the junior high school level. ,A considerable number of studies which indicate the value Of certain direct study techniques are reported in the literature. The conclusiveness Of the findings reported in a study by Gray and HOlmes indicates the efficacy of planned, deliberate efforts to increase meaning vocabulary. The findings of the study led Gray and Holmes to conclude that: . the evidence demonstrated clearly . . . the necessity for direct teaching of the meaning, recognition, and use of "basal" or fundamentally important words needed in study. It follows that teachers should supplement wide reading with carefully planned 2OHarry C. Johnson, "The Effect Of Instruction in Mathematical Vocabulary Upon Problem Solving in Arithmetic," Journal of Educational Research, XXXVIII (October, l9hh), p. 109. 21Richard M. Drake, "The Effect Of Teaching the Vocabulary of Algebra," Journal of Eflucational Research, XXXIII (April, l9h0), 608. 22Richard Dresher, "Training in Mathematics Vocabulary," Edu- cational Research Bulleth XIII (November 11+, 1931+), p. 201-1}. 23Guy E. Buckingham, "The Relationship Between VOcabulary and Ability in First Year Algebra," Mathematics Teacher, xxx (February, 1937): 76-79- ll instruction that promotes vocabulary deve10pment.2h Various analyses of research in the area of vocabulary growth indicate that direct study procedures are effective in increasing the size of pupils' meaning vocabularies. Relatively few studies, however, have attempted to determine if the direct study of vocabulary has effect on other aspects Of the curriculum such as subject matter gains in the content areas, increased reading ability in the content areas, or in the ability to achieve a particular task. In a summary Of research in this area Crosscup25 concluded that the accumulated evidence did not offer completely clear and convincing evidence that instruction in a specific vocabulary is beneficial in terms of attainment in the field from which that vocabulary is chosen. He states further that the evidence supports the conclusion that specific vocabulary instruction does not Significantly increase an individual's general vocabulary, as it is determined by the usual measures. A number of language factors have been identified as major sources of difficulty in the solution of verbal problems. The precise nature of the relationship between language factors and the successful solution of verbal problems, however, is not sufficiently understood. In a study reported by Hansen in 19hh superior achievers were compared with low achievers on a variety of reading factors, mental factors, and arithmetic factors. He reported that the factors most closely associated 2hWilliams. Gray and Eleanor Holmes, The DevelOpment of Meaning VOcabularies in Reading: An Experimental Study (Publications of the Laboratory Schools of the University of Chicago, No. 6; Chicago: Department of Education, university of Chicago), 1938. x’ “”f 25Richard B. Crosscup, "A.Survey and Analysis Of'thhods and Techniques for Fostering Growth of Meaning VOcabulary? (unpublished Master's thesis, Boston Uhiversity, l9h0), p. 160. 12 with superior achievers were arithmetic factors and mental factors and that there was a lack of relationship between certain reading skills and the ability to solve problems successfully. He felt that this: . may lead to the conclusion that skill in general reading and knowledge of general vocabulary are not essential for success in verbal problem solving in arithmetic, and that readin skills and vocabulary in arithmetic are specific in that field. The relationship of specific reading abilities and problem.solving ability is thought to be positive although the Specific reading Skills identified are not consistent from study to study, and there has been little research to determine the exact nature of this relationship. There is no evidence to date that relates problem solving success to a single Specific reading skill. Treacy conducted a study of the re- lationship of fifteen reading skills to verbal problem solving ability. He compared good and poor achievers in problem solving and found that four of the reading skills on which they differed significantly were associated in one way or another with vocabulary. These areas were, (1) quanti- tative relationships, (2) vocabulary in context, (3) vocabulary-isolated words, and (h) arithmetic vocabulary. He states in the implications of his study that, "This fact suggests the need for stressing the meaning of terms, general and mathematical, as an approach to improving pupil 2 ability in problem solving." 7 26Carl W Hansen, "Factors Associated.Hith Successful Achievement in Problem Solving in Sixth Grade Arithmetic," Journal of Educational Research, XXXVIII (October, l9hh), 115. 27John P. Treacy, "The Relationship of Reading Skills to The Ability to Solve Arithmetic Problems, "Journal of Educational Research, XXXVIII (October, l9hh), 93. 13 It has been shown in the preceding paragraphs that problem solving is a major objective of elementary school arithmetic instruction and that, on the whole, pupils do not achieve sufficiently well in this area. The adequate solution of verbal problems depends upon the pupil's ability to comprehend the problem statement, and basic to this ability is the understanding of the various concepts represented by the words used in the problem. It has been shown that the learning Of technical and unfamiliar vocabulary of arithmetic cannot be left to incidental methods but that conscious, direct methods must be utilized. It has been shown further that the relation between vocabulary facility and verbal problem solving ability is not clearly established. These factors suggest the need for the present study. Need for the Study Various analyses of research in arithmetic indicate that there is a need for improvement in the verbal problem solving ability of pupils. Many investigators stress the importance Of reading ability and vocabulary facility in solving verbal arithmetic problems. They point out that unless a pupil is able to comprehend the problem through an interpretation of the vocabulary used, he is unlikely to know what he is supposed to do. Analyses of research in the field invariably conclude that vocabulary is a vital, if not the most important, factor in comprehend- ing written problems. It is well known that the teaching of vocabulary is deemed important by those interested in improving the ability Of pupils to solve arithmetic problems. Research fails to demonstrate clearly, however, the relationship between increased vocabulary and ability it to solve verbal problems successfully. 'Wilson reviewed the research in arithmetic problem solving in the Engyclopedia Of Educational Research and summarized a portion of this research by saying: There is an accumulation of evidence which indicates that the reading Of verbal problems calls for some special reading Skills as well as for an acquaintance with the vocabulary and conventions employed in problem statements. . . . The question Of the nature of the reading instruction that should be given has received only limited attention8 and further research is needed before any conclusion can be stated.2 The literature is replete with references to vocabulary as a factor in verbal problem.solving in arithmetic, and to the need for research to determine the relationship Of vocabulary facility and other reading abilities to verbal problem Solving ability. Because of incon- clusive evidence in this important area Of learning, it has been deemed desirable to utilize the experimental method in seeking knowledge of the relationship of vocabulary facility to ability to solve verbal arith- metic problems. Statement of the Problem The experiment reported in this study has been purposely limited to a single aspect of the general problem of the difficulty encountered by pupils when solving verbal problems. The area of study has been narrowed to a consideration of the effect of the study Of quanti- tative vocabulary on the ability of pupils to solve verbal problems. The area of study was so limited because of the fundamental relation of 280ny Mg'Wilson, "Arithmetic," Encyclopedia Of Educational Research, ed, thter S. Mbnroe, (New York: The Macmillan.CO., 1950), P~ 5 - 15 vocabulary to comprehension and because few studies have attempted to determine the effect of the direct study of technical vocabulary on other abilities. It is the purpose Of this study to determine the effect of the direct study of quantitative vocabulary on the verbal problem solving ability of fifth grade pupils. Statement of Hypotheses The major hypotheses to be tested concern the effects of the experimental method. Two additional hypotheses, indirectly related to the study, are included. 1. It is a hypothesis Of this study that pupils who have studied quantitative vocabulary by direct study techniques will attain higher mean achievement on a test of arithmetic problem solving than pupils who have not studied quantitative vocabulary by direct study techniques independent Of the effects of sex and intelligence. 2. It is a hypothesis of this study that there will be no difference in mean gain over the experimental period on tests Of arithmetic problem solving among experimental pupils who have above average intelligence, average intelli- gence, and below average intelligence. It is a hypothesis Of this study that there will be no difference in mean gain over the experimental period on tests of arithmetic problem solving between.male pupils in the experimental group and female pupils in the experimental group. (.0 Definition of Terms 1. Mean achievement - arithmetic average Of test scores. 2. Mean gain - arithmetic average of the differences between pre-test and post-test scores. 3. That of Arithmetic Problem Solving - Test A-2: Arithmetic Problem Solving, IOwa TeSts of Basic Skills. is h. Pupils - boys and girls attending a public elementary school in the fifth grade. 5. Quantitative vocabulary - (1) terms which denote space, quantity, time, position, value, money, and degree, (2) mathematical signs, (3) technical terms such as, add, minuend, and partial product, which are used in Operations with numbers, (h) measurement terms, (5) Commercial terms, and (6) words identifying common spacial figures. 6. Direct study techniques - deliberate, systematically planned teaching procedures used in this study to teach the meanings Of the quantitative terms included for study. The word "direct" indicates that the methods of instruction used in this study involve conscious selection, introduction, and attention to the deve10pment of the meanings of the quanti- tative terms included for study. 7. Average intelligence - IQ score greater than 89 but less than 111. 8. Above average intelligence - IQ score greater than 110. 9. Below average intelligence - IQ score less than 90. Nature and Design of the Study This study was designed to determine the effect of the direct study of quantitative vocabulary on the ability of pupils to solve verbal arithmetic problems. The study is based on the following assumptions: 1. The adequate solution of verbal arithmetic problems is based on comprehension of the problem statement. 2. The comprehension of verbal problems is complicated by the presence of technical and semi-technical arithmetical terms, and depends, in part, on knowledge of the meanings of these terms. 3. The meanings Of important quantitative terms are best taught through direct study procedures rather than through incidental or indirect methods. The approach used to realize the Objectives of this study was the experimental method. The experimental design involved the selection of the population, random assignment Of the pOpulation to an experimental 17 group and a control group, administration Of a pre-test to Obtain data with which to determine differences in mean achievement, if any, between the experimental and control groups, the application Of a particular treatment to the experimental group, administration of a post-test, and the testing of hypotheses through an analysis of the data. The sample consisted of 39h fifth grade pupils attending the Bay City, Michigan public school district during the school year 1961-62. One-half Of the fifth grade classrooms were randomly assigned to the experimental group and one-half were assigned to the control group. Two hundred forty-two quantitative terms were selected for direct study by the experimental group. The eXperiment was designed to cover a period of thirty-one weeks. The Iowa Tests Of Basic Skills was used to secure pre-test and post-test data. Alternate forms of the IOwa Tests of Basic Skills were used as the pre-test and post-test. Limitations Of the Stigly An experimental study must necessarily be limited in certain respects in order that a sufficient number Of variables may be con- trolled. An experimental study is subject to errors from several sources. The limitations placed upon the study were an attempt to control all variables except the independent variables being studied. limitations afford the study a greater degree of control and precision. The study was limited to a pOpulation consisting Of all fifth grade pupils in attendance in one public school district. Limiting the study to a single school district and to a single grade level con- trolled the textbook factor. The same basal arithmetic textbook had been adOpted for use by all pupils in the experiment. Insofar as the 18 textbook was the course of study, the vocabulary content of the arithmetic program was more similar classroom to classroom than it would have been had the study included classrooms using different texts. The study was limited to a particular group of quantitative terms and by the procedures used in their selection. It was recognized that there were other terms present in the fifth grade text which were used in a quantitative sense, but since such terms did not meet the criteria for selection used in this study they were not included in the list Of terms to be taught. limitations were placed on the study in the choice Of the particular direct study techniques selected. Practical limitations of time available for attention to vocabulary during the regular arithmetic period made it impossible to include all of the techniques recognized as being effective in teaching vocabulary at this educational level. One of the criteria used in the selection of direct study techniques was that they would require only a small portion of the total time available for arithmetic instruction. It was felt that the four direct study techniques selected met this criterion. The amount of class time available for instruction was a limit- ing factor in this study. Notwithstanding the fact that the study was conducted over a school year, it must be recognized that a relatively small portion of the total time available for arithmetic instruction was used to meet the Objectives of this study. It was felt, however, that this was one of the strengths of the study in terms of discovering efficient as well as effective teaching methods designed to increase pupil ability to solve verbal problems. 19 Plan Of the Réport Chapter One provides an introduction to the study by orienting the reader to the problem.snd acquainting him with the general design. Chapter Two is a review of the literature related to the problem. Chapter Three presents the method Of the study and describes in detail the procedures used. Chapter Fbur is devoted to an analysis of the data. Chapter Five presents a general summary of the study, conclusions, and recommendations based on the results Of the study. Summagy It has been the purpose of this chapter to orient the reader to the study. A.presentation was made of the factors demonstrating the need for a study of this nature. The problem and the hypotheses were stated, terms were defined, and the nature and design of the study were described. The limitations Of the study were briefly discussed, and the plan of the report was presented. CHAPTER II REVIEW OF THE LITERATURE The purpose of this chapter is to review (1) selected liter- ature pertaining to vocabulary as a factor in verbal problem solving, and (2) other studies related to the present investigation. In addition to the literature reviewed in this chapter, certain references from.the literature pertaining to specific aspects of the study will be found in Chapter III. Vocabulary as a Factor in verbal Problem.Solving Over the past four decades investigators in the field have recognized the importance of quantitative vocabulary in the understand- ing Of written material in arithmetic and in the successful solution of verbal problems. Early studies of arithmetic vocabulary were devoted to frequency counts Of the words appearing in textbooks and other written materials. Hunt1 analyzed the words used in six third grade arithmetic texts and in ten third grade readers and compared each Of these with the words in Thorndike's The Teachers Wbrd Book. Heightshoe2 IAva Ferwell Hunt, TA Comparison of the Vocabularies of Third- Grade Textbooks in Arithmetic and Reading" (unpublished Master's thesis, University Of Chicago, 1926). 2 . Agnes Ethel Heightshoe, "A Comparison of the VOcabularies of Arithmetics and Readers of the Second and Third Grades" (unpublished Master‘s thesis, University Of Chicago, 1928). 20 21 conducted a Similar analysis of second and third grade textbooks. She compared the vocabularies of four texts in reading and four arithmetic texts. The results of the two studies are similar in showing that the technical vocabulary common to all arithmetics is relatively small and that these words appear seldomly in the reading texts. These facts suggest the necessity for teaching the meanings of some words in the arithmetic period. Brooks carried out an elaborate survey of the technical vocabulary found in five series Of arithmetic textbooks for grades three through eight. He found #29 different arithmetical terms in five third grade arithmetics and only 110 of these appeared in all five books. One hundred seventeen terms appeared in only one book. It seems evident from these facts that textbooks used at that time presented unnecessary difficulties for children by using unfamiliar technical terms infrequently. AS a result Of this extensive exam- ination Of arithmetic vocabulary Brooks was moved to say: Vocabulary difficulties . . . are real and are always present. They hinder the pupils in their understanding of the full import of the problems to be solved. The words and phrases used are often strictly mathematical and bear a sense peculiar to the subject. When pupils fail to interpret them correctly, they become a source of difficulty. Haldorsen compared the arithmetic vocabulary found in five word lists. The lists used in the comparison were the Ayres list, the Commonwealth List, Horn's A Basic writing Vocabulary, Pressey's list of the technical vocabulary found in elementary school arithmetic, and Thorndike's 3Samuel S. Brooks, "A Study of the Teohnical and Semi-Technical Vocabulary of Arithmetic" (unpublished Master's thesis, The Ohio State University, 1926), p. 3.. 22 The Teachers WOrd Book. The purpose of the study was to compare lists, formulate a comprehension list, and to classify according to certain criteria. A total of 2h30 different arithmetic terms were identified, and it is interesting to note that results similar to those found in textbook analyses were Obtained. Over 37 per cent Of the terms were found in one list and slightly less than four per cent were found in all five lists. As a result of the study Haldorsen stated that: It is an established fact that one of the major difficulties in solving arithmetic written problems is a lack Of arithmetic vocabulary. Pupils who cannot grasp the meaning of the words in a given problem, or have no idea of the concept represented by certain keygwords, cannot comprehend the problem, and hence cannot solve it. The question arises as to whether analyses Of more recent arith- metic textbooks would obtain similar results. In 19hl O'Rourke and Mead5 reported a study of vocabulary difficulties in five third grade arithmetic texts. The purpose of their analysis was to determine the difficulties presented by the vocabulary in the problems found in third grade arithmetic texts. The results Of their study were very similar to the studies conducted by Hunt, Heightshoe, and Brooks and an implication of the study was that textbooks used in the third grade seemed to increase rather than decrease the vocabulary load. Repp,6 in 1960, reported an analysis of five third grade arithmetic texts and the resudts were similar to those Obtained by previous investigators. It seems safe to hO.‘W'. Haldorsen, "Arithmetic Vocabulary in Standard‘WOrd lasts" (unpublished Master's thesis, University of Minnesota, 1935), p. 3. 5Everett V. O'Rourke and Cyrus D. Mead, "Vocabulary Difficulties Of Five Textbooks in Third-Grade Arithmetic," Elementary School Journal, XLI (May, 19M), 683-91. 6Florence C. Repp, "The Vocabularies of Five Recent Third Grade Arithmetic Textbooks," Arithmetic Teacher, VII (March, 1960), 128-32. 23 conclude that the technical and semi-technical vocabulary found in arithmetic texts at any elementary school grade level is a source of difficulty in comprehending the meaning of verbal problems. Some investigators have studied the extent and nature of vocabulary difficulties in arithmetic by testing pupil knowledge of Selected technical terms. In addition, two of the studies were concerned with tracing the development of concepts of arithmetic words. This was accomplished by studying rate and patterns of growth in word knowledge. These studies Show that correct concepts develop gradually, and they indicate the nature of partially corrected concepts and incorrect concepts. In 1930 Buswell and John conducted an extensive and thorough study Of arithmetic vocabulary for the purpose of determining the nature and development of pupils' concepts of arithmetical terms. This was accomplished by dividing the study into three major sections which made it possible for the investigators to present data with respect to (1) the general understanding of arithmetical terms by pupils in grades 1+, 5, and 6, (2) the development of concepts of words in the first six grades, and (3) the explanation Of new technical terms in textbooks. From an original list of 500 terms, 100 were selected for study. To realize the first objective group tests of the terms selected for study were constructed and administered to 1500 pupils in grades h, 5, and 6. Additional data were secured for a selected list of twenty- five terms through the construction of three types of tests. Buswell and JOhn summarize the results Of the group tests as follows: 1. The pupils in a given grade differ widely in the size of their arithmetical vocabularies. 2h 2. The difficulty of the terms studied as indicated by the percentages of pupils responding correctly Shows great variation. 3. The difficulty of the classes Of terms into which the list is divided indicates that, in general, the technical terms are the most difficult and that the terms relating to time, space, or quantity are the least difficult. h. The growth in the understanding Of terms . . . Shows great variation for different terms . 5. . . . for a given term the percentages of correct responses and the percentages Of omissions vary widely . 6. Analaysis Of the incorrect responses indicates that incorrect meanings are frequently associated with terms . 7. The lack of agreement between the results of Tests I, II, III, and IV suggests that ability to respond to a word correctly in one situation does not necessarily indicate that understanding is complete. Further experience with the word may be needed for complete understanding.7 Pressey and Moore conducted an extensive study of pupil knowledge of mathematical vocabulary from grade three through high school. It was the purpose of the investigation to discover what words were known in what grades under the teaching techniques then in use. They selected 2h3 terms to measure pupil knowledge of word meaning. The results of the study showed: (1) that some terms are learned early and are remembered through high school, (2) others are acquired slowly but steadily, (3) a few are learned in the grade in which they are taught and then forgotten, (h) some show gradual acquisition and never a high percentage Of mastery, (5) some show actual loss in the upper grades, and (6) some are never understood by more than a very few pupils. In 7Guy T. Buswell and Lahore John, The Vocabulary of Arithmetic (Supplementary Educational Mbnographs, NO. 38; Chicago: Uhiversity of Chicago, 1931), pp. hl-h2. 25 reporting the implications of these results the investigators offered the suggestion that: . inadequate mastery of fundamental terminology (as shown by these results) is one of the most important reasons for the difficulty encountered by so many persons of all ages and social strata in dealing with anything of a mathematical nature. They have never mastered these relatively simple, fundamental meanings, and their efforts to build on a shaky foundation have been so futile that they have come to regard mathematics as a subject which can be mastered only by those with a "genius" for figures.8 Unsatisfactory knowledge Of technical arithmetic vocabulary in grades seven and eight was reported by Geisler in 1953. Geisler Selected 168 terms common to eight textbooks used in these grades. A group test of these terms was administered to h00 subjects who were using the specified texts consulted in the compilation of the list of terms. The results of the test prompted the investigator to conclude that: The evidence indicates that as a group, the #00 children tested failed to show satisfactory understanding of the technical terms which presumably they had studied. These terms are all used in the eight textbooks analyzed, but the explanations found in these texts are Often too meagre and stated in language too formal tO be easily grasped by the student. It is clear that the mastery which is presupposed and the mastery which is actually shown by these results are quite divergent. A drive should be made by everyone concerned upon this matter of acquisition by the pupils of those meanings Of terms with which they must work. A number of investigations of arithmetic problem solving have been made for the purpose of gaining insight into the relation Of vo- cabulary to problem solving ability and the extent to which vocabulary '81“ C. Pressey and W. S. MOore, "The Growth of Mathematical Vocabulary from the Third Grade Through High School," School Review, XL (June, 1932), A53. 9Sister Mary Damian Geisler, RSM, "A Technical Vocabulary of Arithmetic in Grades Seven and Eight" (unpublished Master's thesis, Catholic Uhiversity Of America, 1953), p. #6. 26 may be a cause of pupil difficulty in solving verbal problems. The results of tests and the testimony of teachers in a study by Chase in 1917 indicate the difficulties pupils have with the vocabulary used in arithmetic problems found in texts of the day. In summarizing her study the writer said: In addition, some of the problems, even though they may be about things or situations that are familiar to children, are yet stated in words that children do not use and consequently may not under- stand. . . . The investigation here recorded has shown, after a careful study of numerous text-books, that many problems involve conditions that are quite untrue to life; that many of the words used were unfamiliar or even quite unknown to the one hundred children tested; and finally that forty-five experienced teachers from various school systems have found the subject matter and the vocabularies of the various texts which they have used quite un- suited to the capacities of their pupils.10 Roling, Blume, and Morehartll listed inability to read as one of six specific causes Of failure in problem solving. Hydle and Clappl2 studied several of the elements contributing to the difficulty of problem solving in grades four to eight and found that the use Of unfamiliar terms in problems made their solution slightly more 1h difficult. Chase13 and Lutes are in agreement on major reasons for 10Sara E. Chase, "waste in Arithmetic," Teachers College Record, XVIII (September, 1917), 369. 11‘Pearl Roling, Clara Blume, and Mary Morehart, "SpecificCauses Of Failure in Arithmetic Problems," Educational Research Bulletin, III (October 15, 192%), 271-72. 12 L L Hydle and F I. Clapp, Elements of Difficulty in the Interpretation of Concrete Problems in Arithmetic (University of Wisconsin Bureau of Educational Research Bulletin, NO. 9; Madison: university of Wisconsin, 1927). 13V. E. Chase, "The Diagnosis and Treatment of Some Common Difficulties in Solving Arithmetic Problems," Journal of Educational Research, XX (December, 1929), 335-h2. 11‘O. S. Lutes, "Where Pupils Fail in Verbal Problems," Journal of Educational Research, XIII (January, 1926), 71-72. 27 children not succeeding in solving problems, one Of which is inadequate understanding of vocabulary used in problems. Stevenson reported a study made by elementary school teachers who were concerned with identifying important causes of pupil failure in solving verbal problems. It was his Opinion that pupils often fail to solve arithmetic problems because one or more of the words are unfamiliar. He stated that: Children do not understand as much about the meaning Of words as their teachers give them credit for knowing. Not only are pupils deficient in general reading vocabulary but they aig also unfamiliar with the many technica1.words used in arithmetics. Georgesl6 studied the nature of difficulties in reading mathematics and found that predominant reading difficulties were caused by technical vocabulary and that a significant proportion of all difficulties was caused by technical vocabulary. Two of nine causes of difficulty in solving verbal problems listed by Bruecknerl7 are, (1) failure to comprehend the problem in whole or in part, due to inferior reading ability, inability tO visualize the situation, lack Of practice in solving problems, and Similar con- ditions, and (2) ignorance of quantitative relations due to the lack of vocabulary or Of understanding of principles. John 18 classified various errors made by pupils when solving verbal problems and suggested that pupil difficulties was due to: 15P. R. Stevenson, "Difficulties in Problem Solving," Journal of Educational Research, XI(February, 1925), 98. 6 l J. S. Georges, "The Nature of Difficulties Encountered in Reading Mathematics," School Review, XXXVII (March, 1929), 217-26. 1 17Leo J. Brueckner, "Improving Pupils' Ability to Solve Problems,‘ Journal Of the National Education Association, XXI (June, 1932), 176. 18Lenore John, "Difficulties in Solving Problems in Arithmetic:" E1ementarx.sehood_dnurnal, XXXI (November, 1930), 206. 28 (1) errors in reasoning, (2) errors in fundamentals, (3) errors in reading, and (h) miscellaneous errors. Stevens examined various aspects of the relationships between verbal problem solving ability and other abilities and stated that "the failure of a pupil in solving reason- ing problems in arithmetic are [fSic_7 due, in large measure, to deficiencies in reading abilities," and that a pupil who is ". . . unable to understand the situation described in a problem will find himself unable to solve it."19 FOran studied the nature of reading problems in arithmetic and reported on several phases of problem solving. He found that technical terms and unfamiliar terms at a given grade level inter- fere greatly with problem solving performance at that level. He arrived at the following generalization concerning the importance Of vocabulary to successful solution of verbal problems, "Should the unfamiliar words be important words in the sentence, there is little possibility that the meaning of the problem will be sufficiently clear to guide the selection of the methods leading to a solution. . . Y20 He went on to say that, "There is just as much necessity Of teaching the language of arithmetic as there is of teaching the arithmetical processes, for the latter cannot be understood without the former."21 Johnson examined the relationship of several faCtors to success in problem solving and believed vocabulary to be more important to 193. A, Stevens, "Problem Solving in.Arithmetic," Journal of Educational Research, XXV.(April-1~ay, 1932), 253. 20T- G. Foran, "The Reading of Problems in Arithmetic," Catholic Educational Review, XXXI (December, 1933), 602. . 2lI‘Did., p. 609. 29 successful problem solving than reasoning ability.22 In 19h5 Rogers administered the Woody-McCall Mixed Fundamentals Test to approximately 10,000 pupils in Grade 6A in Chicago public elementary schools for the purpose of comparing the ability of these pupils with the ability of pupils who had taken the same test in 1925. In discussing the results of a comparison Of the test scores with nine elementary school district superintendents, reasons for pupil difficulty in problem solving were examined. Rogers reported that it was the belief of some of the district superintendents that the “. . . key to arithmetic improvement would be an improvement Of the reading vocabulary of our children, with particular emphasis on words dealing with quantity."23 Two studies of the relation of mathematics vocabulary to aChieve- ment in mathematics at the high school and junior high school levels indicate a strong relation between knowledge of mathematical vocabulary and performance in.the subject area. Buckinghamgh conducted a correlation study between vocabulary and ability in first year Algebra which revealed that total scores in vocabulary and total scores in Algebra are closely related. Eagleas studied the relationship of several reading abilities tO success in junior high school mathematics and found that mathematics 22J. T. Johnson, "On the Nature of Problem-Solving in Arithmetic," Journal of Educational Researcp, XLIII (October, 1919), 110-15. . 23Don C. Rogers, "COOperative Inservice Studies," Arithmetic 19h9 (Supplementary Educational MonOgraphs, No. 70; Chicago: .Uhiversity of Chicago Press, 19h9), p. 75. 2hGuy E. BuCkingham, "The Relationship Between VOcabulary and Ability in First Year Algebra," Mathemtics Teacher, xxx (February, 1937), 79. 25Edwin Eagle, "The Relationship of Certain Reading Abilities to Success in Mathematics,” Mathematics Teacher, XLI (April, 19%), 179. 30 vocabulary is closely related to mathematical achievement. ‘While these studies do not indicate a direct relationship between knowledge of arith- metic vocabulary and achievement in arithmetic, many studies in other subject matter areas have resulted in findings which suggest a relation- ship between knowledge of vocabulary and achievement in that subject area. One may reason, therefore, that knowledge of vocabulary is closely related to achievement in the area in which that vocabulary is found. Morton discussed six language difficulties encountered in reading the written material found in arithmetic. While Morton did not directly relate these difficulties to pupils' problem solving ability, it seems safe to assume that similar language difficulties are to be found in problem statements. He identified the following diffi- culties: 1. Introduction and use of technical terms. 2. unknown terms or phrases without definition or explanation. 3. Explanatory statements . . . expressed in language which is beyond the pupils level of comprehension. h. Explanatory statements that are vague, inadequate, or incomplete. 5. Statements which involve familiar words but which use these words in unfamiliar ways. 6. Statements that are misleading or ingorrect or that may lead the pupil to wrong conclusion.2 A variety of approaches to problem solving has been suggested in the literature. The concern of investigators for determining the most 26Et 1" Morton, "Language and meaning in Arithmetic," Educational Research Bulletin, xmv.(November, 1955), l97-20h. 31 effective methods for attacking verbal problems has resulted in dis- cussions of the various factors contributing to successful problem solution, lists of competencies and abilities needed by pupils, and lists of steps to be followed in solving problems. In nearly every instance reading abilities and knowledge of technical terms are included. In 1922 Newcomb attempted to determine the best method of solving verbal arithmetic problems. His research design utilized a control group and an experimental group. Newcomb's recognition of the importance of adequate reading ability and knowledge of word meanings is evident in the series of eight steps which he outlined for the experi- mental group to follow. He suggested that: . . . in order to reason properly about problems and to solve them with accuracy and facility the pupil must be able to: Understand each word in the problem Read the problem intelligently [Add, subtract, multiply, and divide with speed and accuracy Determine what is given in the problem Determine the part required Select the different processes to be used in the solution and the order in which these processes are to be used Plan the solution wisely and systematically Check readi O O O O O Cb-q ovuuer turbid Greene undertook an analysis of factors in the solution of verbal problems and noted that: The first step in the solution of a verbal problem involves a thorough understanding of the items, elements, and processes which are stated or implied in it. It is a matter of comprehension [ritalichZ . . . This includes many factors, such as the rate of reading, difficulties of vocabulary, the reading of numerals, 27R. s. Newcomb, "Teaching Pupils How to Solve Problems in Arithmetic,‘ Elementary School Journal, XXIII (November, 1922), 185. . 32 and the organization of the problem, as well as its complexigg_ in regard to the number and order of the processes involved. ‘ ~ In an article devoted to a discussion of the modern conception of arithmetic, Schaaf discussed various aspects of problem solving. He suggests that, "The power to understand the problem situations which are presented in books and to resolve them effectively depends principally on (1) general reasoning ability, (2) reading comprehension, (3) vocabulary facility, and (h) rational use of number relationships and processes."29 Among the nine aspects of problem solving power listed by Schaaf were the following: 1. understanding the essence of a problem situation. 2. understanding the vocabulary used in the problem.30 .McSwain and Cooke were concerned with identifying essential mathematical meanings in arithmetic and listed as one of the abilities used in problem solving the ". . . ability to read and interpret and n3], define the problem situation. Corle, in an experimental study, investigated the problem solving behavior of sixth grade pupils. He identified five aspects related to pupil ability to arrive at successful solutions to verbal problems. These are: (l) concept formation, (2) reasoning, (3) con- fidence, (h) understanding of vocabulary, and (5) word recognition in oral reading. The results of the study indicate that ". . . as many 28Harry A. Greene, "Directed Drill in the Comprehension of Verbal Problems in.Arithmetic," Jburnal of Educational Research, XI (January, 1925), 3k. 29Hilliam L. Schaaf, "A Realistic Approach to Problem-Solving in Arithmetic," Elementary School Journal, XLVI (may, 19h6), #95. 30Ibid., p. #96. 31E. T. McSwain and Ralph J. Cooke, "Essential Mathematical Meanings in Arithmetic," Arithmetic Teacher, v (October, 1958), 192. 33 as one out of four problems was correctly solved without a clear concept 32 of the actual meaning of the problem." Conclusions reached by Corle were that the ability to interpret vocabulary correctly is related to problem solving efficiency, and poor interpretation of vocabulary may result in incorrect solutions. The nature of the specific reading abilities needed in compre- hending verbal problems has been examined by a number of investigators whose primary interest is in the teaching of reading. Tinker has identified several factors which make the reading of written material in arithmetic difficult. Among the complicating factors identified by him are technical vocabulary, common words with special meanings, and reading and interpreting pictures and diagrams. It is Tinker's Opinion that, "Reading arithmetic problems is one of the most difficult reading tasks encountered in the elementary school. Systematic guidance is needed, infused with a full realization of the reading difficulties which the child is up against."33 With respect to the adequate solution of verbal problems Tinker says, "A prerequisite to the solution of an arithmetic problem is that the pupil have an accurate command of all symbols it uses whether verbal Or in the form of condensed signs."3h Bond and Wagner, in TEaching Children to Read, describe the nature of reading in the content areas and suggest the following 32Clyde G. Corle, "Thought Processes in Grade Six Problems," Arithmetic Teacher, v (October, 1958), 198. 33Miles A. Tinker, Tsachinnglementary Reading (New York: Appleton-Century-Crofts, Inc., 1952 , p. 259. 31‘Ibid. 31+ difficulties in reading arithmetic material: 1. Difficulties of VOcabulary . . . 2. Difficulties of Abbreviation and Symbolization . 3. Difficulties Due to lack of Continuity . . . h. Difficulties Due to Inability to Form Sensory Impressions . 5. Difficulties Resulting from Failure to Reject Irrelevant Facts . . . 6. Difficulgée Due to the Omission Of Necessary Steps in a Problem. Strang, McCullough, and Traxler have directed their attention to an analysis of the problems involved in the improvement of reading in the content areas. It is their opinion that a major problem in the reading of mathematics is caused by difficulties with technical terms. They state that: The reading of mathematics requires an understanding Of quite an assortment of technical terms which present three major difficulties. For one thing, the same term is not always used to mean a particular Operation. . . . In addition to this, there are several technical terms in mathematics which have their uses in general conversation with quite a different meaning. . . . The third difficulty is that mathematics requires the understanding of many terms which remind the studeng of absolutely nothing and must be learned the long, hard'way.3 A review of the literature relative to vocabulary as a factor in verbal problem solving indicates that knowledge of the meaning of the terms used is essential to the adequate comprehension and solution of such problems. It is obvious that a pupil is unlikely to solve verbal problems correctly if he cannot comprehend the meaning of the problem statements. At the root of much of the difficulty experienced by pupils in reading verbal problems is lack of knowledge of the meaning of 350ny Bond and EVa Bond‘wagner, Teaching The Child to Read (revised; New York: The mcmillan Co., 1950), p. 312. 36Ruth Strang, C. M. McCullough, and A. E. Traxler, Problems in the anrovement of Reading (New York: McGraw-Hill Book Co., Inc., 1955): P- 172- 35 important words and mathematical symbols. Technical terms are a major cause of difficulty in comprehending the meaning of problems. The presence of abbreviations, signs, and collocations adds to the difficulty Of the reading task. The evidence suggests that arithmetic texts use many technical terms which are a source of difficulty in comprehending verbal problems, and that the texts do not adequately teach the meanings of many of these terms. It is readily apparent that training in the vocabulary of arithmetic must be given as part of the arithmetic class if pupils are to acquire satisfactory knowledge of the meanings of important terms. Studies Related to the Present Investigation .An examination of the reports of experimental studies was made to determine the extent and nature of research efforts to study the effects of special vocabulary instruction. In an early study Elenman37 paired high school SOphomores for general achievement and amount of training in foreign languages and drilled specific vocabulary items, such as, prefixes, suffixes, and roots for a period of twelve weeks. VOcabulary drills were based on a word study syllabus prepared by the teacher and covered all English periods during the experiment. unfortunately the exercises were not described in the report. At the end of the experimental period the pupils were tested for increase in vocabulary, ability to read with understanding, ability to choose words correctly, and ability to give accurate meanings. The results showed a statistically significant gain for the experimental grOup with the greatest gains on increase 37V. A. C. Henman, “An Experimental Study of the value of word Study," Journal of Educational Psycholo , XII (February, 1921), 98-102. 36 in vocabulary and ability to give word meanings. Symonds and Penney38 investigated the possibility of increasing the vocabulary used in English classes because they felt that the needs of individuals demand larger vocabularies as well as increased ability to use more appropriate words in conveying precise nuances of meaning. A total of thirty ninth-grade English students were used with fifteen each in the experimental and control groups. ‘Word knowledge of the subjects was tested by using an incomplete sentence arrangement and credit was given for any one of several words. The vocabulary training consisted of motivational techniques to arouse interest, making word lists to keep students at work studying vocabulary, and class discussion of a few words per day. The time devoted to vocabulary exercises was approximately fifteen minutes per day at the beginning of the experiment, but by the end of the four month experimental period the time spent per day was nearer to five minutes. While both groups showed improvement in vocabulary, the experimental group had the greater gains and the gains and differences were found to be significant. The investigators concluded that vocabulary gains were due largely to the interest aroused and subsequent word study in all areas. 38Percival Symonds, and.Edith M5 Penney, “The Increasing of English Vocabulary in the English Class," Journal of Educational Research, xv (February, 1927), 93-lO3. 37 Curoe,39 and Curoe and Wixted,1+0 carried on two experiments with college seniors enrolled in English courses. The investigators set aside three minutes of each class period for direct treatment of "Today's WOrds." The words were written on the chalkboard and discussed as to meaning in particular contexts, syllabication, spelling, and pro- nunciation. Eighteen students in the experimental group studied fifty- nine words over one semester. At the end of the semester a twenty item objective test was administered to the experimental group and the results were compared to those obtained from the twenty students in the control group who had not had three minute drill sessions as part of the course work. They found this method effective on words used in the experiment but not effective in increasing general vocabulary or other linguistic skills. A comparison of the upper and lower quartiles Of the experimental group revealed no significant differences between the gains made by these initially different groups. The median of the control group was lower than the lowest score of the experimental group. The highest score of the control group was below the median of the experimental group. Another investigation carried out in English classes was conducted by Eurich who worked with college freshman. Each week 39Philip R. V. Curoe, "An Experiment in Enriching the Active Vocabularies of College Seniors," School and Society, XLIX (April 22, 1939), 522-2h. . to Philip R. V. Curoe and William G. Wixted, “A Continuing Experi- ment in Enriching the Active Vocabularies of College Seniors," School and Society, III (October 19, l9uo), 372-76. . 38 the subjects were presented.with a list of 100 words selected from essays listed as required reading for the course. An examination of word knowledge was given at the end of each week. During each class period a ten minute discussion of the words in the list was held and general problems of vocabulary improvement were discussed. In addition English conferences were held regularly and included discussion of vocabulary. Eurich reports impressive gains made with the experimental group showing a decidedly significant gain as against a slight loss for the control group. The experimental group was significantly higher on a general vocabulary test where the words did not overlap the words taught. It was Eurich's Opinion that the gains made by the experimental group were relatively permanent and were in specific vocabulary. He states that, "The advantage of the experimental group appears to be unequivocal, for without any corresponding losses the students in this group have added approximately 12h words to their vocabularies in addition to the normal increment that might have taken place under ordinary conditions.”1 Dawha studied the effect Of direct teaching on vocabulary growth in an exPeriment in the junior high school. The experimental (group used the dictionary and class discussion in addition to help received from context. Words selected from the units being studied kl , A. C. Eurich, "Enlarging the Vocabularies of College Freshman,‘ English Journal, College Edition, xx: (February, 1932), l35-h1. “QSeward Emerson Daw, "The Effect of Direct Teaching Upon Vocabulary Growth in Junior High School" (unpublished.Master's thesis, The university of Chicago, 1933); as cited by: Richard B. Crosscup, "A Survey and Analysis of Methods and Techniques for Fostering Growth of Meaning VOcabulary" (unpublished Master's thesis, Boston University, l9h0), p. 1&6. 39 were written on the chalkboard and familiar words were eliminated by means of a cursory examination. Meanings of unfamiliar words were looked up in a dictionary. The meaning of a word best adapted to its use in the context being studied was agreed upon in class discussion. AS the word appeared in varying contexts, new uses were matched with the agreed-on definition. Some attention was given to analysis and derivation. Non-technical words were illustrated from every day usage. The words were left on the board during the study of the reading from which they had been selected. The initial period of word study was fifty minutes which was later increased approximately ten per cent. Improved results followed the presentation of words in smaller numbers and lengthened time of exposure to the learning materials. The results of the final vocabulary test caused Dew to conclude that direct teaching had a decided advantage over incidental learning. Iawsheh3 studied the relationship of certain factors to the rates of learning and retention of word meanings taught in isolation. The word teaching phase of the experiment was carried out in one day and tests of the knowledge of the words studied were given a number of times at different intervals. The method of teaching was: (1) dictation of words by the teacher; (2) individual use of dictionary; and (3) group discussion. The teacherdirected the discussion to the particular meaning that had been used as the correct response on the test. Students were required to keep a written record of the study in the form.of a list of h3Charles Iawshe, “A Study of the Relationship of Certain Factors to the Rates at Which Children learn and Forget WOrd Meanings as Indi- cated by ApprOpriate Vocabulary Tests," (unpublished Master's thesis, University of.Michigan, 1935). ho words discussed with one or more synonyms. At the end of the experi- mental period the score for girls was about thirty—five percent lower. The score for boys was forty percent lower. The percentage of loss of newly acquired words in the day of teaching is considerably greater. Approximately eighty-five percent of the words acquired were lost the same day and only fifteen percent retained. These results suggest that most of the meanings of words taught in isolation are lost when no provision is made for retention, review, or use. The results of a study carried out by Traxlerhh seem at variance with the results of Lawshe's study. Traxler had a group of junior high school pupils study twenty-six words during a single class period. Each word was presented with a definition and three sentences illus- trating uses. The word being studied was underlined in each instance of its use. For example: Aspire - have an ambition for something, desire earnestly, seek, rise high. - They aspire to rule. He aspires to be president. The artist aspires to create works of beauty. A final test of word knowledge given two months later resulted in significant improvement in knowledge of the words studied. From the results of the test Traxler concluded that vocabulary drill of the kind described can result in permanent knowledge Of many of the words studied in a single class period. thrthur E. Traxler, "Improvement in VOcabulary Through Drill," English Journal, High School Edition, XXVII (June, 1938), h9l-9u. hl hS conducted a thorough and extensive study to Gray and Holmes compare the relative merits of direct and incidental instruction in vocabulary using pupils in fourth grade history classes. All pupils were given a specific vocabulary test composed of words selected from two units taught during the experimental period. Pupils were assigned to two experimental groups and a control group on the basis of the scores made on the test of history vocabulary. The control group included pupils whose scores were distributed rather evenly from lowest to highest in Specific vocabulary mastery. Experimental group A was composed of pupils having higher vocabulary achievement while pupils of lower achievement were assigned to experimental group B. In the control classes the teacher provided no vocabulary guidance except as children asked for help. In the experimental classes the teacher pro- vided specific vocabulary help before and during the reading periods. The following procedures were used in the experimental groups: (1) use of specific words in meaningful situations; (2) illustrations and pictures; (3) new word forms written on the board and.pointed to only when pronounced; (h) pupils pronounce the words softly; (5) words used informally in class discussion; (6) correction of inaccuracies; (7) sentences written on board to assist pupils in deriving meaning from context; and (8) children encouraged to use new words in writing answers to guide questions. h 5William S. Gray and Eleanor Holmes, The Development of Meaning Vocabularies in Reading: An Experimental Study (Publication of the Laboratory Schools of the University of Chicago, No. 6; Chicago: University of Chicago Press, 1938). #2 Mean gain and percent of gain as measured by a retest of the specific vocabulary indicate that greater gains were made by pupils in the two experimental groups. The higher mean gain made by experimental group B indicates that direct teaching procedures are of relatively more value to pupils of limited initial achievement and of limited mental ability. Pupils of superior ability who received Specific help on new words made greater progress than pupils of similar ability in the control group who received no Specific help. Gray and Hblmes concluded that many pupils are in urgent need of help in acquiring word meanings and that the direct method of vocabulary development is far more effective in enriching word meanings than indirect methods. Drakeh6 used the results of diagnostic vocabulary tests to determine vocabulary difficulties of pupils in ninth grade Algebra classes. Special remedial exercises were used by the experimental group while the classes in the control group were taught under the customary procedure. WOrd lists were put on the board and discussed. Pupils made a copy of the word list with examples, and took frequent short vocabulary tests. Teachers were instructed to use expressions using the words when possible and to encourage pupils in using correct vocabulary. The direct study techniques to improve vocabulary were successful in that the results of achievement tests showed significant differences in favor of the experimental classes. In contrast to the h 6Richard M5 Drake, "The Effect of Teaching the VOcabulary of glgebra," Journal of Educational Research., XXXIII (April, 19ho), 01-10. 1+3 findings of some studies, the group initially better had greater gains from the vocabulary training. Relatively few studies have included a planned follow-up as part of the experimental design. Miles)+7 gave the Inglis Vocabulary Test to fifty high school students who discussed the results and selected for study forty words they thought they should know. The students were given one week of vocabulary drill for thirty minutes per day. After the first week approximately ten minutes of each class period were devoted to drill until the end of the semester. Retests on alternate forms of the test were given at the end of the semester and again after two and one-half years. The results of the study at the end of the semester showed that the experimental group scored higher than the control group with an increase from.forty-one words to 73.5 words. The planned follow-up test showed a loss of less than four words over two and one-half years. Miles concluded that direct study techniques have great potential over a longer period of time. Inconsistent findings are evident from a comparison of the results of studies which have attempted to determine if the direct study of vocabulary results in improvement of general vocabulary and subject matter achievement as well as an increase in knowledge of vocabulary studied. N'ewburnh8 studied the effects of two methods of h 7Isadora‘W. Miles, "An Experiment in Vocabulary Building in a High School," School and Society, LXI (April 28, l9h5), 285-86. hBHarry K5 Newburn, "The Relative Effect of Two Methods of Vocabulary Drill on Achievement in American History" (unpublished Doctor's thesis, University of Iowa, 1933). hh drill on words used in American History. Purposes of the study were to determine the relative value of two methods of presenting words to be studied, and the value of vocabulary drill in terms of increase in vocabulary and improvement in subject matter achievement. Two experimental groups and a control group were used. Experimental group A studied five words per day taken from the day's reading for a total of twenty-five days. Experimental group B also studied five words per day for twenty-five days, but the words were not necessarily taken from the daily reading in the text. The words for this group were selected on the basis of some unifying factor. Vocabulary instruction, otherwise, was identical for the two experimental groups. The control group received no special training. The procedures used in the experimental groups included the presentation of sentences with the term to be studied underlined, a short explanation, illustrations of use, and series of exercises. Newburn found that drill on vocabulary significantly improved vocabulary. Both experimental groups made significantly greater gains in vocabulary than did the control group. He did not find significant differences between the two experimental groups. Drill on vocabulary did not affect the achievement of pupils sufficiently to counteract the loss of time from subject matter study, that is, the control group made greater gains in achievement in American History. Greater gains were made by students of lower mentality than by students of higher mentality. Phippsug had sixth grade pupils study history vocabulary and, unlike the results of the Newburn study, found that these pupils did th. R. Phipps, "An Experimental Study in Developing History Read- ing Ability with Sixth Grade Pupils Through the DevelOpment of an Active History Vocabulary," Journal of Experimental Education, VII (September, 1938). 19-23. 1+5 increase their ability to read history material as a result of vocabulary training. He equated the experimental and control groups for history reading and history composition ability and for intelligence. unfortunately Phipps did not report the method of selection of words or the teaching technique used. Phipps concluded that giving training in the meaningful use of the vocabulary of history facilitates the reading of history material. Dresher50 studied the effects of extensive and specific vocabulary training at the junior high school level using mathematical terms found in study. Special attention was given to words appearing in the textbook and a list of technica1.words and their definitions was duplicated and given to each pupil. A series of tests was given to motivate pupils to seek meanings for unknown terms. Both the experimental and control groups showed gains though the greater gains made by the experimental classes were not shown to be statistically significant. It was Dresher's opinion that vocabulary training not only results in increased.word knowledge but also helps pupils to understand and work concrete problems. Dresher did not, however, Show experimental evidence that such was the caSe. A unique approach to vocabulary building was used by Petersen51 who established a separate vocabulary course at the high school level in 5ORichard Dresher, "Training in Mathematics VOcabulary," Edu- cational Research Bulletin, .2111 (November 1b., 1931+), 201-1». 51Olga C. Petersen, "A Vocabulary Course: An Experiment in Teaching VOcabulary as a Separate Course in High School" (unpublished Master's thesis, university of North Dakota, 1937). #6 an attempt to determine if such an approach would result in increased knowledge of word meanings. 'WOrds were selected for study from texts used in history, geography, literature, and present day problems. Additional sources of words were word lists taken from English textbooks, vocabulary tests and college entrance examinations, words incidentally brought in through association, classified lists on the basis of Origin, and words contributed by pupils. A.variety of techniques was used. Wbrds were defined, illustrated, and used in sentences, compositions, and conversation. The dictionary was consulted to Secure definitions and etymology. Attention was directed to the importance of context, to the selection of effective words to express specific meaning, to errors, and to excellent vocabularies of radio speakers. Various kinds of word study were used including roots, suffixes, prefixes, and compound words. Matching and grouping exercises were used. Petersen reported that all differences in relation to the measure used at the end of the experimental period were in favor of the experimental group, and further, that the results indicated that the word study techniques used led to higher achievement in studies. Other studies Show practically no gain in school achievement as a result of vocabulary study. KOmisarse carried on one of the few studies failing to report success with direct study techniques for improving vocabulary. The experimental and control subjects were college freshnen enrolled in Social Science Classes. The experimental subjects engaged in daily 52David Daniel Kbmisar, "The Effects of The Teaching of Social Science Vocabulary to College Freshman on Some Aspects of Their Academic Performance" (unpublished Doctor's dissertation, Columbia university, 1953)- 1+7 study of terms taken from current assigned readings. The instructors met periodically and compared lists of the terms taught. While more than seventy terms were selected for study, each instructor covered a common vocabulary of seventy terms by the end of one semester. From three to ten minutes at the beginning of each class period.were used by the instructor to clarify meanings of words. Explanations of words brought up in class were given. Short vocabulary tests were given period- ically as a motivational device. The hypotheses tested.were that special study would result in (1) the learning of more words studied, (2) the learning of more words in general, (3) an improvement in reading compre- hension, and (h) the learning of the content of the course. Komisar found no statistically significant differences in favor of either group for any of the hypotheses tested. He suggests that more time devoted to the study of a greater number of terms would result in significant gains for the experimental group. Variations in word study techniques have been used by some investigators. Blair53 used 101 college juniors and seniors as subjects in an experiment carried on primarily outside the classroom. JMembers of the class agreed to secure a dictionary, and look up new words. At the beginning of each class meeting they agreed to hand in a list of the new words encountered and the meaning, source, and a sentence in ‘which each was used. The students agreed to keep a vocabulary notebook and hand it in at the end of the semester. The results of the study indicate that greater gains were made by the experimental group, especially those of initial inferior ability. ‘53G1enn M. Blair, "An Experiment in Vocabulary Building," Journal of Higher Education, XII (February, 19h1), 99-101. #8 Another experiment in vocabulary building that was carried on largely outside the classroom was reported by Bernard.5h He combined pupil conferences with word lists handed in by students. For six weeks students were encouraged to read widely and to keep a vocabulary notebook. Definitions of words was required on weekly examinations. Five students were selected to work as a small group with a special instructor three times per week. During the conferences words were defined orally, root words were eXplained and written sentences were checked. The average gain of the experimental group was 1h.2 words compared to six words for the control group. Bernard concluded that vocabulary increases regardless of amount of specific attention, but that specific attention can result in gains twice as rapidly. A distinctive technique was used by Haefner in an experiment to determine the value of 'casual' techniques for the acquisition of word meanings. The experiment involved ninety-three pairs of adults and covered a period of thirty-nine weeks. Haefner felt that ". . . the learning situation must be presented as 'casual', and not as a type of experience which the individual believes he is under some obligation to put forth effort to acquire."55 Each day one word was written on the Chalkboard.with a definition and illustrative sentence. The word was underlined in both the definition and sentence. This material was left on the board during the five minute interval while the class assembled. The purpose for having this material on the chalkboard was never eXplained h I! 5 Harold‘W. Bernard, "An Experiment in VOcabulary Building, School and Society, LIII (June 7, 191+1), 7u2-h3. 55Ralph Haefner, "Casual Learning of Word Meanings," Journal of Educational Research, XXV.(April-May, 1932), 268. 1+9 and at no time were the words discussed. At the signal for the beginning of class the work was erased. .A comparison of the means of the two groups showed a significant difference in favor of the experimental group when intelligence was taken into account. SachsS6 reported the only study in which wide reading was used as a means to develop vocabulary. He selected twenty-five key words from Thorndike's list of 20,000 on the basis of frequency and counted the occurrence of these in texts read by high school students. Tests were administered at the end of the experimental.period to see if more knowledge resulted because of an increased number of contacts with words as recorded by the percentage of correct responses on the achievement test. Sachs reported that the results show ". . . that these students possess an astonishing immunity from learning words in contexts--that the majority of them, in fact, make no effort at all to learn words from "57 contexts. The experimental study reported by Sachs supports the view that vocabulary study should be direct rather than indirect or inci- dental. I Only one study reported in the literature is directly related to 58 in a study involving approximately the present investigation. Johnson, 600 seventh grade pupils, attempted to determine whether improvement in specific arithmetic vocabulary would lead to an improvement in the 56H, J. Sachs, "The Reading Method of Acquiring Vocabulary," Journal of Educational Research, XXXVI (February, l9h3), h57-6h. 57mm, p. 1.62. 58Harry G. JOhnson, "The Effect of Instruction in mathematical Vocabulary Upon Problem Solving in Arithmetic," Journal of Educational Research, xxxvnI (October, 19%), 97-110. 50 solution of verbal problems in which those words were used. The experi- ment was divided into three periods and ran fourteen weeks. A total of sixty terms was selected for study. The practice exercises included daily oral word drills, use of a dictionary, individual notebooks, and mimeOgraphed materials to supplement the text. The results of the study indicate that the experimenta1.group achieved significantly greater gains than did the control group in knowledge of vocabulary for all three periods. The experimental group achieved only slightly significant gains in problem solving during the first two periods. The difference was not significant for the third period and Johnson suggests that the reduced amount of time devoted to study is a probable cause. The superiority of the experimental group maintained for'pupils of practically all levels of mental ability and initial status in the area studied. Summary In this chapter the literature was reviewed for the purpose of determining (l) the importance of vocabulary knowledge to verbal problem solving ability, and (2) the nature of investigations which have reported the use of special efforts to increase vocabulary. A number of the causes of pupil difficulty in solving verbal problems have been identified and among the most significant are deficient and immature reading abilities and lack of vocabulary. It is rec0gnized that.a knowledge of the meanings of key words is essential to the comprehension and correct solution of verbal problems, and that special effort must be directed toward the acquisition of these meanings as part of the regular arithmetic period. 51 Experimental studies have demonstrated that specific words can be added to a pupil's vocabulary as a result of special efforts to increase vocabulary. A wide variety of direct and incidental techniques has been the subject of experimentation in many subject areas and at all educational levels. The weight of evidence clearly indicates that "direct" methods are more effective than "incidental" or "indirect" methods for improving vocabulary. Nearly every study reports differences in mean gain in favor of the experimental group and in most cases these differences are statistically significant. The experimental procedures vary from study to study with respect to the specific teaching techniques used, the number of subjects in the experimental and control groups, the number of words selected for study, the methods of selecting words, the length of the experimental period, the amount of time devoted to instruction per class period and for the total experiment, the testing procedures used and the treatment of data. while it is clearly evident that specific words can be added to a pupil's vocabulary by direct study techniques, the studies report conflicting results as to improvement in general vocabulary, effect on achievement in course content, improvement in ability to read materials used in the course, improvement in written composition, and the value of direct study techniques for pupils initially inferior or superior in intelligence or knowledge of vocabulary. 52 The recognition of the importance of vocabulary knowledge to the comprehension of written material and the inconclusiveness of available evidence of the relationship of vocabulary facility to verbal problem solving ability suggest, in the opinion of the writer, that further investigation of the problem is warranted. CHAPTER III METHOD OF THE STUDY The general design of the study presented in chapter one orientated the reader to the procedures used in conducting the study. This chapter will present a detailed description of (l) the selection of the terms to be studied, (2) the selection of the teaching techniques, (3) the nature of the data, (A) the length of the experimental period, (5) the selection of the sample, (6) the orientation of the teachers, and (7) methods of testing the hypotheses. Selection of Terms to be Studied The word 'terms' has been used in this study to denote mathe- matical signs and collocations as well as words. All of the terms to be studied were selected from the fifth grade arithmetic textbook in use in all fifth grade classrooms in the Bay City, Michigan public school district. This textbook is one of a series adopted by the Bay City school system for use as basic textbooks in arithmetic and consti- tutes the basic course of instruction in arithmetic at this level. An alphabetical list of the words and mathematical signs appearing in the textbook exclusive of those in the table of contents, index, and on the cover and title page was prepared. All terms in this list having, in the opinion of the writer, at least one quantitative meaning were listed with the exception of number words, such as one, five, first, and fifth, and abbreviations with the exception of AM. and P.M. 53 5h A total of uu7 terms exclusive of collocations was thus secured. Only those collocations appearing in the vocabulary exercises at the end of the chapters were included for study. It was apparent that all Rh? terms could not be taught during the experimental period without seriously reducing the amount of time to be devoted to other arithmetic objectives. To reduce the number of terms to manageable prOportions and to substantiate the selection of these terms, only those were included for study which had been selected by the investigator and which appeared in one or more of the following: (1) the Mathematics Dictionary}, (2) The Iaidlaw Glossary of Arithmetical- Mathematical Tarmse, or (3) the vocabulary study exercises found at the end of the chapters in the fifth grade arithmetic textbook. This pro- cedure resulted in the selection of 208 words and mathematical signs and thirty-four collocations. A list of these terms and the chapters in the fifth grade arithmetic textbook in which they appear is included in Appendix A. To facilitate study the 2h2 terms were divided into groups of eight terms each. One group of terms was studied each week with the exception of the eighth week when six mathematical signs were studied. The groups of terms were duplicated so that each pupil in the eXperi- Vmental group could have his own c0py (see Appendix B). Each term lGlenn James and Robert C. James, Mathematics Dictionary (Princeton, New Jersey: D. Vaanstrand Company, Inc., l959):7h7hp. 2Bernard H. Gundlach, The Laidlaw Glossary of Arithmetical- Mathematical Terms, (River FDrest, Illinois: Iaidlaw Brothers, 1961), pp. viii + 120. 55 was accompanied by a sentence illustrating a quantitative meaning. To help focus attention on the terms each was underlined in the sentence in which it appeared. Selection of the Toachinngechniques Many techniques for improving meaning vocabulary have been tried. It is especially significant to note that (1) some of these techniques are used in the initial presentation of terms, (2) some techniques are used to deve10p meanings, and (3) some are used to aid retention of meanings. It is important to point out, however, that while techniques which may be utilized in the initial presentation of terms to be learned are important to vocabulary deve10pment, they may not teach the meanings of those terms. While techniques used to aid retention are important, they may not aid i2_teaching meaning. Techniques used in the initial presentation of terms may contribute toward understanding if the terms are presented in context rather than in isolated lists. Tachniques used to aid retention make it possible for pupils to use newly acquired words in reports and discussions and in conversation. For some children the experience of listening to others use these terms in a variety of contexts will, in effect, be a means of acquiring meaning. Criteria for the selection of the teaching techniques to be used in this study were that they: (1) can be carried on as a regular part of the arithmetic period, (2) do not requirematerials and equipment not found in the average classroom, (3) do not require an undue amount of class time, (h) do not require an undue amount of preparation outside 56 the regular class period, and (5) have been utilized in experiments resulting in significant vocabulary gains. While there are many techniques and combinations thereof that might have been employed, it was felt that a few well selected techniques would enable the teacher to present the term, teach its meaning, and provide for its use to aid retention. The teaching techniques that were used in this study are (1) initial presentation, (2) class discussion, (3) teacher explanation, and (h) using the dictionary. A detailed description of the teaching techniques used in this study was duplicated and distributed to the teachers of the experimental cLasses. A copy of this material is included in Appendix C. Nature of the Data Before beginning the experiment in September, 1961, four sub- tests of the Iowa Tests of Basic Skills for grades 3-9 were administered to all fifth grade pupils. The four sub-tests were: Test V: VOcabulary, Test R: Reading Comprehension, Test Arl: Arithmetic Concepts, and Test A-2: Arithmetic Problem Solving. During the previous school year the Iorge-Thorndike Intelligence Tast, level 3, Grades h, 5, and 6 was administered to these pupils which yielded verbal and non-verbal raw scores, IQ scores, grade and age equivalents and grade percentiles. The verbal IQ scores were used in preference to the non-verbal scores. The four sub-test scores secured from the Iowa Taste of Basic Skills and the verbal IQ scores comprised the pre-test data used to pair experimental and control classes. Post-test data consisted of scores on the same four sub-tests of an alternate form of the Iowa Tbsts of Basic Skills 57 administered to all fifth grade pupils in March and April of 1962. The administration of the post-tests marked the termination of the experi- mental study. Included for final analysis were the data of only those pupils for whom IQ scores and all pre-test and post-test scores were available. The data of pupils who transferred to other fifth grade classrooms were not used. Lengph of the Experimental Period The teachers of experimental classes began the experiment upon completion of the pre-testing. The administration of the pre-tests and the administration of the post-tests followed staggered schedules and in both cases covered periods of approximately three weeks. Since it was not feasible in all cases to observe the same schedule in administering the pre-tests and post-tests, the length of the experimental period ranged from.twenty weeks in some classes to twenty-four weeks in others. .(The number of terms studied by experimental classes ranged, therefore, from 158 to 190). Selection of the Sample The original design of the study was based on the premise that all fifth grade pupils in attendance in the Bay City public school district during the school year 1961-62 would participate. Since it was not feasible to randomly assign pupils, 12 of the 2h fifth grade classes were randomly assigned to the experimental group and twelve to the control group. To demonstrate that the pupils in the experimental group and the pupils in the control group were not significantly different 58 in mean achievement on any of the four pre-tests or in mean IQ, the pre-test data were used to pair as many experimental and control classes as possible. For each of the twenty-four classes the following were determined: (1) the number of pupils, (2) the sum of the scores for each sub-test and for IQ, (3) the mean score for each sub-test and for IQ, (h) the sum of the squares of the scores for each sub-test and for DQ, and (5) the variance for each sub-test and for IQ. Twelve pairs of experimental and control classes were selected by comparing mean IQ scores and mean scores on the pre-test of arithmetic problem solving. Differences in mean IQ and in mean achievement on all four pre-tests for each pair of classes were tested for significance through the use of the "t" test. The results of the "t" test indicated that in no case was therea significant difference in mean IQ or in mean achievement on any of the four pre-tests for nine of the twelve pairs of classes. The number of pupils, means, variances, and "t" values for each of the nine pairs of classes for each of the pre-tests and for IQ are presented in Tables 1-5. The question arose as to whether a significant difference in mean IQ or in mean achievement on the four pre-tests would result when the nine experimental classes were taken as a whole and the nine control classes were taken as a whole. The "t” test was used to test these differences for significance and in no case was any of the differ- ences found to be significant. The means, variances, and "t" values for the nine experimental classes and for the nine control classes on the four pre-tests and for IQ are shown in Table 6. It was concluded that the nine experimental classes and the nine control classes were 59 not significantly different in mean IQ or in mean achievement on any of the four pre-tests at the beginning of the experiment. The final sample consisted of these nine pairs of classes with 211 pupils in the experimental group and 183 pupils in the control group. Orientation of the Teachers The orientation of teachers in the control group was limited to one meeting of all fifth grade teachers in May, 1961. At that time it was announced that an experiment would take place during the next school year, and that one-half of the teachers would be in the eXperi- mental group and one-half in the control group. The method of determin- ing which teachers would bein the experimental and control groups was described. The subject of the experimental study was announced to all fifth grade teachers but the purpose and plan of the study and the teaching techniques were described only to the experimental teachers. During the experimental period the control classes pursued the regular instructional program in arithmetic. During the course of the experiment a series of monthly meet- ings was held with the teachers of the experimental classes. The pur- pose of these meetings was to achieve a reasonable degree of con— sistency in their understanding of the experimental method, to coordinate the use of the teaching techniques, and to provide for communication of ideas and experiences so as to direct the efforts of each teacher toward a common objective. The first of these meetings was held during the pre-school conference period in September, 1961 at which time these teachers 60 were presented.with a detailed description of the purpose and design of the study. A rationale for the selection of the teaching techniques was presented and these techniques were thoroughly explained and demonstrated. Lists of the terms selected for study (see Appendix A) were distributed, reviewed, and discussed. Each teacher was provided with duplicated material (see Appendix C) describing the purpose and plan of the study and the teaching techniques which constituted the lesson. plan for developing the meanings of the terms selected for study. To assist the teachers in presenting the terms for study and in clarifying meanings, each of the 2h2 terms was defined and two or more sentences illustrating various meanings were given (see Appendix D). The terms with their definitions and illustrative sentences were arranged by chapter and duplicated for distribution to the teachers. The list of terms for a given chapter included only those terms appearing in the text for the first time. It was emphasized that as nearly as possible both the experi- mental and control classes should devote the same amount of time to the study of arithmetic. The experimental teachers were instructed to include vocabulary Study within the regular arithmetic period. It was suggested that approximately thirty minutes per week be devoted to vocabulary study. (The teachers reported an average of forty-five minutes devoted to vocabulary study each week). In subsequent group meetings the teachers discussed the progress of the experiment and the relative success of particular techniques with their group. Further explanation and demonstration 61 oaa.m n ao.p nous pmonwwn one human Honpsoo u some amonwfin use mmmao Honouaanoaxm c msm.a u no.9 mma Ham noossz Hosea Hoe. - mma.ss meo.ms mmo.am mmm.am mm mm m mam. . www.0s mmm.mm Hmm.am mmm.mm ma Hm m mom. - mam.mm www.mw 0mm.am wsm.am om mm a mmo. . mma.am >m>.os moo.om owo.om mm mm m moo.a - smo.mm mom.mma www.mm ooa.om mm mm m oao. . aaa.aoa mma.mm aam.mm 0mm.mm ma am a mom. - maa.aaa smm.oaa mmp.am Hem.ma mm Hm m was. . mmm.ma Hmm.mm maa.ea mam.ma ea ma m osw.a . msm.>m mam.mm oao.ma ooo.ma mm em H Hdvaufi Hdubvfi kubve Hoapsoo nanomxm Hospsoo nauogxm Hoapsoo tauomxm an: oosmfiad> one: mafimsm mo nooasz afinm BmMBnmmm Nm HEB zo mmmmfiuo nomazoo nz<_u¢fizm2HmmmNm ho mmHmm.oa mam.oa mm mm m sow. . oaa.mma 0mm.mma mum.mm mw>.am ma Hm m mmm.a - asm.msa mom.mam oos.mm oam.mm om mm s Ham. . 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Duplicated weekly lists of terms to be studied over a four week period were distributed and the meanings of unfamiliar terms or unknown meanings of familiar terms were clarified. Pupil and teacher attitudes toward the experimental study were dis- cussed. (The majority of the teachers indicated that both they and their pupils thought that the vocabulary study was of great value and that it contributed to the ability of pupils to solve verbal problems). Methods of Testing the Hypotheses The problem of primary concern in this study was to compare the effects of the experimental and control methods as measured by mean scores on a test of arithmetic problem solving. Additional problems investigated in this study were to determine whether there were sex and intelligence differences attributable to the eXperimental method. The following null hypotheses were tested: 1. There will be no difference in mean scores on a test of arithmetic problem solving between pupils who have studied quantitative vocabulary by direct study techniques and pupils who have not studied quantitative vocabulary by direct study techniques independent of the effects of sex and intelligence. 2. There will be no difference between mean gains over the experimental period on tests of arithmetic problem solving among experimental pupils who have above average scores, average scores, and below average scores on an intelligence test. 3. There will be no difference between mean gains over the eXperimental period on tests of arithmetic problem solving between male pupils in the experimental group and female pupils in the experimental group. To facilitate the analysis of the post-test data all pupils in the experimental and control groups were divided into sub-classes on the basis of sex--male and female, and IQ category--above llO, 68 between 90 and 110, and below 90. The number of pupils in the eXperi- mental and control groups by sex and by IQ is presented in Table 7. The first problem investigated required the use of the "t" test to determine the significance of differences in mean achieves ment between the experimental and control groups on the post-tests of vocabulary, reading comprehension, arithmetic concepts, and arithmetic problem solving. The "t" test provided evidence of the level of significance of differences in test results, but did not supply data for an analysis of the effect of method per se. Because the effect of method was of primary concern, the second problem investigated.was to test the differences between the means of the experimental and control groups on the post—test of arithmetic problem solving to determine if they were significantly different independent of the effects of sex and intelligence”. The analysis of variance was used to test the difference between means. Two additional problems concerning gains made by pupils in the experimental group were investigated. The "t" test was used to test the differences in mean gain over the experimental period on tests of arithmetic problem solving between experimental males and experimental females, and among experimental pupils who were above average, average, and below average on the Iorge-Thorndike Intelligence Test. It was decided, further, to report the results of the analysis of variance test of differences in mean achievement between the experi- mental and control groups on the post-tests of vocabulary, reading comprehension, and arithmetic concepts independent of the effects of sex and intelligence. Hypotheses regarding possible significant 69 oom.aa aaa.m Hoo.ma som.ma osm.sa mmm.sa ooa.HH mmm.aa eme.ma mmm.oa smo.mm mom.am some -omom msa>aom soaooam capes unpfih< so mono: oom.oa omm.ma oom.am amm.am smo.sm soa.am oom.aa mmm.ma mom.mm oms.om emo.wm mms.am omma-pmom mpmoosoo capes unpau< so memo: ooa.mm omm.om Haa.mm ooo.wm Hom.mm www.mm omm.om mmm.mm oao.aa Pom.as ows.am Hma.mm amoe-pmom scamsooonm uaoo mofiomom so mono: 00:.wa mam.ma 00m.am me.mm mm:.mm www.mm owm.:a mwm.wa mm>.Hm maw.mm www.mm m:m.am pmma upmom mama: nomoo> so memo: oa em on ma mm ma mm am mm as mm as maaaan mo noossz mamemm oan2_ vameoh ofimzn oamaoh mam: OHmEmh vaz mHmth mHmz vamaoh mam: oaa-oml. oHH-om‘ om soaom oH soosoom oH oaa osoom oH om abate oH soozoom 0H odd bacon 0H mocha aoapsoo macho Hopsosfiuomxm Nmoomado 0H Nm Qz<.Nmm Mm mMDomo Homazoo Qz¢_A¢Bzmszmmxm Ema ZH mAHmDm mom mBmMHuBmom mDOh zo mmmoom z¢mznnz< mHHmDm ho mmmzsz P mumea 7O differences in mean achievement between the experimental and control groups on these post-tests were not formulated. An investigation of these problems was thought to be warranted because of the lack of conclusive evidence as to the effects of the direct study of a special or technical vocabulary on other aspects of academic perform- ance, such as learning of course content, reading comprehension, general vocabulary knowledge and written composition. CHAPTER IV ANAIXSIS OF THE DATA The final analysis of results is based on the test data from 211 fifth grade pupils in nine experimental classes and 183 fifth grade pupils in nine control classes. All of these pupils had taken the Lorge-Thorndike Intelligence Test, level 3, and had completed the four sub-tests of the Iowa Tests of Basic Skills, for grades 3-9 administered before and after the experimental period. The analysis of the data is divided into four sections. In the first section of the analysis, results of tests_for_the significance ’1’“ of differences in mean scores between the experimental and control_groups "a!” if‘ \_-..___,,_._ A on the post-tests of vogapglagy, reading comprehension, arithmetic concepts and apithmetic problem solving are reported. These tests did 293’sta- tistically control the effects of sex and intelligence. In the second section of the analysis, the results of a test for the significance of the difference in mean scores between the experimental and control groups on the post-test of arithmetic problem solving independent of the effects of sex and intelligence are reported. In the third section of the analysis, the results of tests for the significance of differences in mean gains of experimental pupils are reported. Mean gains over the experimental period on 71 72 tests of arithmetic problem solving among three IQ categories and between sexes are examined. In the final section of the analysis, the results of tests for the significance of differences in mean scores between the experi- mental and control groups on the post-tests of vocabulary, reading comprehension, and arithmetic concepts independent of the effects of sex and intelligence are reported. Differences Between Methods Not Independent of Other Effects The first problem investigated did not involve the testing of any hypotheses. The question was whether or not there was a significant difference in mean scores between the experimental and control groups on the post-tests of vocabulary,_reading comprehension, arithmetic concepts, and arithmetic problem solving. The "t" test was used to test differences in mean scores on each of the post-tests without controlling the effects of sex and intelligence. The means, variances, and results of tests for the significance of differences in mean scores on each of the post-tests are reported in Table 8. For each of the post-tests the experimental group mean exceeded the control group mean. The differences between means on the post—tests were as follows: vocabulary - .591, reading comprehension - 1.221, arithmetic concepts - 1.711, and arithmetic problem solving - 2.955. The differences between mean scores on the post-tests of arithmetic concepts and arithmetic problem solving were significant at the .01 level. While the differences in mean scores on the post-tests of vocabulary and reading compre- hension favored the experimental group, neither was significant at the .05 level. 73 omm.m n Ho.» mao.a u,mo.o some smegma: one mooam Hoapsoo u some panama: one macaw Hopsoefinomxm 4 aam.m . 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The results of the test for the significance of the difference in mean scores on the vocabulary post-test indicated that the difference among IQ categories was significant at the .01 level. Differences in mean scores between methods and between sexes on the vocabulary post-test were not significant at the .05 level. The interaction was not significant. The results of the test for the significance of the difference in mean scores on the post-test of reading comprehension indicated that the difference among IQ categories was significant at the .01 level. Differences in mean scores between methods and between sexes on the post-test of reading comprehension were not significant. The inter- action was not significant. The results of the test for significance of the difference in mean scores on the post-test of arithmetic concepts indicated that the differences among IQ categories, between methods, and between sexes were significant at the .01 level. The interaction was not significant. The significant difference between sexes on the post- test of arithmetic concepts favored the male pupils. This unexpected difference was accounted for by testing for the significance of the difference between sexes on the pre-test of arithmetic concepts. The difference in mean scores between sexes on the pre-test of arithmetic concepts was in favor of the male pupils and was significant at the .01 level. 9h The following null hypotheses were tested through the analysis of the post-test data: 1. There will be no difference in mean scores on a test of arithmetic problem solving between pupils who have studied quantitative vocabulary by direct study techniques and pupils who have not studied quantitative vocabulary by direct study techniques independent of the effects of sex and intelligence. 2. There will be no difference between mean gains over the experimental period on tests of arithmetic problem solving among experimental pupils who have above average scores, average scores, and below average scores on an intelligence test. 3. There will be no difference between mean gains over the experimental period on tests of arithmetic problem solving between male pupils in the experimental group and female pupils in the experimental group. The findings of the analysis of the post-test data resulted in the rejection of null hypothesis 1. Null hypothesis 3 was accepted. Null hypothesis 2 was partially rejected as a result of the significant differences in mean gains between the high and low IQ categories and between the middle and low IQ categories at the .01 level. Conclusions Before stating any conclusions based on the findings from the statistical analysis of the post-test data, it seems necessary to discuss several relevant points. In studies of this nature it is often difficult to control all of the factors which may contribute to the experimental results. While the significant differences in mean achievement between the experimental and control groups on the post- tests of arithmetic problem solving and arithmetic concepts appear to be the result of the experimental method, it must be pointed out that it 95 is also possible that these differences are due, in part, to what is referred to as the 'Hawthorne effect'. It has been found in some experimental studies that one of the factors contributing to the experi- mental results was the increased effort on the part of those involved in the study as a result of their awareness of the fact that they were participating in an experimental study. While it is possible that the 'Hawthorne effect' Operated in this study, it may not be assumed that it was limited to teachers and pupils in the experimental group. It is entirely possible that individuals in the control group put forth greater effort as a result of their knowledge that they were partici- pating in an experimental study. Another factor which may affect the results of investigations of this nature is 'leakage' from the experimental group to the control group. The results of experimental studies have been known to be affected because individuals in the control group became acquainted with the experimental method and utilized the method in whole or in part. The result of 'leakage' may be to reduce or minimize the differences between the experimental and control methods. It is important to note that had this factor Operated in this investigation, the results are all the more significant since differences would have been larger had the control group not used the experimental method. It is possible that the monthly group meetings held.with the teachers of the experimental classes fUnctioned as a form of in- service education resulting in increased teaching competence on the part of‘these teachers. It is also possible that these meetings stimmlated «certain teachers to greater effort thereby contributing to the 96 'Hawthorne effect'. It seemed to this investigator that the advantages of these meetings (stated above) far outweighed any deleterious effects which they might have had. If the achievement of pupils can be improved as a result of such in—service education, this investigator is in favor of increasing the in-service education opportunities available to teachers. A wide variety of direct study techniques might have been employed in order to realize the objectives of this study. While the four direct study techniques, which comprised the experimental method used in this study, appear to have been effective, it may not be assumed that any and all direct study techniques would be equally effective with fifth grade pupils. Some techniques would be more effective with primary grade pupils while others would be more effective at the adult level. Any interpretation of the findings of this study should be made in terms of the direct study techniques used and not generalized to include direct study techniques not a part of the experimental method. It was demonstrated in this investigation that the direct study of quantitative vocabulary was accompanied by significantly higher achievement on the post-tests of arithmetic problem solving and arithmetic concepts in favor of the experimental group. From these findings, however, one may not assume that a causal relation- ship exists between the experimental method and the significantly higher achievement of the experimental group on these two post-tests. An interpretation of the findings of this investigation should be 97 based upon a recognition of the fact that a cause-effect relation- ship was not definitely established. On the basis of the statistical analysis of the data obtained in this investigation, the following conclusions appear to be warranted. l. The experimental group and the control group were not significantly different at the beginning of the experi- ment in mean achievement on tests of vocabulary, reading comprehension, arithmetic concepts, and arithmetic problem solving or in mean intelligence. Pupils who have studied quantitative vocabulary using the direct study techniques described in this study achieve significantly higher on a test of arithmetic problem solving than pupils who have not devoted special attention to the study of quantitative vocabulary. Pupils who have studied quantitative vocabulary using the direct study techniques described in this study achieve significantly higher on a test of arithmetic concepts than pupils who have not devoted special attention to the study of quantitative vocabulary. The direct study of quantitative vocabulary does not tend to result in improvement in general vocabulary or in reading comprehension. The experimental method is not more effective with one sex than with the other. The experimental method is more effective with pupils who have above average or average intelligence than with pupils who have below average intelligence. Effective vocabulary study can be made a part of the regular arithmetic program without sacrificing pupil achievement in the subject matter of arithmetic. The significant difference in mean scores between sexes on the post-test of arithmetic concepts is attributed to the significant difference between sexes on the pre- test of arithmetic concepts. 98 Implications Throughout the conduct Of this investigation certain edu- cational and research implications became apparent. 'With respect to curriculum development, the following recommendations are made. \1. \2. 33. That vocabulary study be made an integral part of the instructional program in arithmetic. That the study of quantitative vocabulary begin in the primary grades. That direct study techniques be used to teach the tech- nical and semi-technical vocabulary in each of the content subjects. That pupils be provided with more Opportunities to use quantitative terms in both written and oral communication. That more emphasis be placed on the need for, and importance Of, vocabulary study as it relates to academic achieve- ment and individual growth. That pupils be provided with rich and varied experiences which will.furnish a background for the new terms to be encountered. That elementary school classrooms be equippedeith a variety Of arithmetic teaching aids to assist in the clarification of the meanings Of quantitative terms. That textbook publishers give less attention to vocabulary control and more attention to vocabulary building by thorough explanation of new terms as they are introduced, and by including a variety of vocabulary exercises in their arithmetic textbook series. That in-service education opportunities, such as workshOps, be made available to teachers so that they might become acquainted with the technical vocabulary used in arithmetic, clarify the meanings Of unfamiliar terms, and become familiar with a variety of vocabulary building techniques suitable for use at various grade levels. i 10. 99 That instructors in arithmetic methodology courses in teacher training institutions devote more attention to helping pre-service teachers learn the meanings Of important arithmetic terms and become acquainted.with effective methods Of teaching the meanings Of terms tO elementary school pupils. A number of recommendations for future research are made. 1. That research be devoted to the study Of the relative effects of specific direct study techniques for learning vocabulary. That research be devoted to developing a basic arith- metic vocabulary for grades K-8. That research be devoted to studies similar to this one at other grade levels. That replications of this study be made with samples from different populations as additional tests of the hypotheses. That research be devoted to determining the reason for the significantly higher mean achievement of male pupils than female pupils on the arithmetic concepts sub- test of the Iowa Tbsts of Basic Skills for grades 3-9. That research be devoted to a determination of which direct study techniques are most effective with pupils who have below average intelligence. That research be devoted to the deve10pment Of a standardized» test Of quantitative vocabulary for use in the elementary grades. That research be devoted to similar investigations of the effects of incidental or indirect study Of quantitative vocabulary on verbal problem solving ability Of elementary school pupils. That research be devoted to determining which quantitative terms are learned primarily as a result of school eXperiences and which are learned primarily as a result Of out-Of-school experiences. 10. 11. 100 That research be devoted to determining the Optimum amount Of time that may profitably be given to the study of quantitative vocabulary at each grade level. That research be devoted to determining the relationship Of other language skills to arithmetic problem solving ability. BIBLIOGRAPHY lOl BIBLIOGRAPHY BOOKS Banks, J. Houston. learning and TBaching Arithmetic. Boston: Allyn and Bacon, Inc., 1959. Bond, Guy I", and Tinker, Miles At Readipngifficulties. New YOrk: .Appleton-Century-Crofts, Inc., 1957. Bond, Guy, and Wagner, Eva Bond. Teaching The Child to Read. New York: The Macmillan Company, 1950. Boyer, Ike 3., Brumfiel, Charles, and Higgins, William. "Defi- nitions in Arithmetic," Instruction in Arithmetic. 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"Arithmetic in Elementary and Junior High Schools," Arithmetic I951. Edited by Guy T. Buswell. ("Supplementary Educational.MOnographs," NO. 63.) Chicago: The university of Chicago Press, October, l9h7. pp. l-9. Brueckner, Ieo J}, and Grossnickle, Foster E. HOw to Make Arithmetic Meaningful. Philadelphia: John C. Winston CO., l9fl7. 102 103 Brune, Irvin H. "Language in Mathematics," The learning of Mathe- matics: Its Thquy and Practice. Twenty-first Yearbook of the National Council of Teachers Of Mathematics. Washington, D.C.: The National Council Of Teachers of mathematics, 1953. pp. 156-2oh. Buswell, Guy T., and John, Ienore. The VOcabulary of Arithmetic. ("Supplementary Educational Mbnographs;fiNO. 38.) Chicago: The Uhiversity of Chicago Press, 1931. Buswell, Guy '13., and Judd, Charles H. Summagyr Of Educational Investigations Relating to Arithmetic. ( Supplementary Educa- tional Monographs," No. 27.) Chicago: The University Of Chicago Press, 1925. Buswell, Guy T. "Arithmetic," Encyclopedia Of Educational Research. Edited by Chester W. Harris. New York: The Macmillan.Co., 1960. pp. 63-77. Diagnostic Studies in Arithmetic. ("Supplementary Educational Monographs:Tr NO. 30.) Chicago: The University of Chicago Press, 1926. "Methods of Studying Pupils' Thinking in Arithmetic," Arithmetic l9h9. Edited by Guy T. Buswell and Maurice L. Hartung. ("Supplementary Educational Monographs," NO. 70.) Chicago: The University or Chicago Press, l9u9. pp. 55-63. Cochran, William G., and Cox, Gertrude M5 Experimental Designs. .New York: JOhn'Wiley and Sons, Inc., 1950. Cole, Luella. The Elementary School Subjects. New York: Rinehart and CO., 1946. . e . The Igprovement of Readipg. New York: Farrar and Rinehart, Inc., 1938. The Teacher's Handbook Of Technical VOcabulagy. Bloomington, Illinois: Public School Publishing CO., 19h0. Cronbach, Lee J. "The Meanings of Problems," Arithmetic 19h8. Edited by Guy T. Buswell. ("Supplementary Educational Mano- graphs,” NO. 66.) Chicago: The university Of Chicago Press, October, l9h8. pp. 32-h3. Dale, Edgar, and Reichert, Donald. Bibliography of Vocabulary Studies. revised edition, A.Payne Fund Communication Project, Columbus, Ohio: Bureau of Educational Research, The Ohio State University, 1957. Dixon, Wilfrid J ., and Massey, Frank J. Jr. Introdmtion to Statistical Analysis. New York: McGraw-Hill Book CO., Inc., 1957. 10h Edwards, Allen I" Statistical Methods for the Behavioral Sciences. New York: Hblt, Rinehart and Winston, 1961. Fouch, Robert S., and Nichols, Eugene D. "Language and Symbolism in Mathematics," The Growth of Mathematical Ideas: Grades Krl2. Twenty-fourth Yearbook Of The National Council Of Teachers of Mathematics. Washington, D.C.: The National Council Of Teachers Of Mathematics, 1959. pp. 327-69. Gray, William S., and Holmes, Eleanor. The Development Of’Meaning Vocabularies in Reading;_ An Experimental Study. Publications Of the laboratory Schools Of the University Of Chicago, NO. 6. Chicago: Department of Education, University Of Chicago, 1938. Grossnickle, Foster E., and Brueckner, Leo J. Discoverinngeanings in Arithmetic. Philadelphia: The JOhn C. Winston CO., 1959. Gundlach, Bernard H. The Laidlaw Glossary Of Arithmetical-Mathe- matical Terms. River Forrest, Illinois: Iaidlaw Brothers, 1%]. o Hartung, Maurice L. "Advances in the Teaching of Problem.SOlving," Arithmetic l9h8. Edited by Guy T. Buswell. ("Supplementary Educational MOnographs," No. 66.) Chicago: The university of Chicago Press, l9h8. pp. hh-53. . "Major Instructional Problems in Arithmetic in the Middle Grades," Arithmetic l9h9. Edited by Guy T. Buswell. ("Supplementary Educational Monographs," NO. 70.) Chicago: The University Of Chicago Press, 19h9. pp. 80-86. Henderson, Kenneth B., and Pingry, Robert E. "Problem-Solving in Mathematics," The Learning of Mathematics: Its Theorypand Practice. Twenty-first Yearbook of the National Council Of Teachers Of Mathematics. Washington, D.C.: The National Council of Teachers of Mathematics, 1953. pp. 228-70. Hydle, I" I”, and Clapp, F. L. .Elements Of Difficulty in.the Inter- pretation of Concrete Problems in Arithmetic. university Of Wisconsin Bureau Of Educational Research Bulletin, NO. 9. Madison: University of Wisconsin, 1927. James, Glenn, and James, Robert C. Mathematics Dictionagy. Princeton, New Jersey: D. Van Nestrand Company, Inc., 1959. John, Lenore. "Clarifying and Enriching Meaning VOcabularies in Mathematics," Improving Reading in the Content Fields. Edited by William S. Gray. 8("Supp1ementary Educational Monographs," NO. 62.) Chicago: The University Of Chicago Press, 19h7. PP- 99-103- lOS Marks, John L., Purdy, C. Richard, and Kinney, Lucien B. Teaching Arithmetic for Understanding. New York: MCGraw-Hill Book 00., Inc., 1958. Morton, Robert L. Teaching Children Arithmetic. New York: Silver Burdett CO., 1953. "The Place Of Arithmetic in various Types of Elementary- School Curriculums," Arithmetic i949. Edited by Guy T. Buswell and Maurice L. Hartung. ("Supplementary Educational Monographs, NO. 70.) Chicago: The University of Chicago Press, l9h9. pp. 1-20. Rogers, Don C. "Co-Operative In-Service Studies Of Arithmetic," Arithmetic 1959. Edited by Guy T. Buswell and Maurice L. Hartung. (“Supplementary Educational MOnographs," NO. 70.) Chicago: The university Of Chicago Press, 19h9. pp. 75-79. Russell, David H. "Higher Mental Processes," Encyclgpedia Of Educational Research. Edited by Chester W. Harris. New York: The Macmillan Company, 1960. pp. 6h5-6l. Spencer, Peter L., and Russell, David H. "Reading in Arithmetic," Instruction In Arithmetic. Twenty-fifth Yearbook Of the National Council Of Teachers Of Mathematics. ‘Washington, D.C.: The National Council Of Teachers Of Mathematics, 1960. pp. 202-23. Spitzer, Herbert F. "learning and Teaching Arithmetic," 392.2 Teaching of Arithmetic. Edited by Nelson B. Henry. Fiftieth Yearbook of the National Society for the Study of Education, Part II. Chicago: The university Of Chicago Press, 1951. pp. 120-h2. . The Teaching of Arithmetic. second edition. Boston: Houghton Mifflin CO., 195h. Strang, Ruth, MCCullough, C. M., and Traxler, Arthur E. Problems in the Improvement of Reading. New York: MCGraw-Hill Book Company, Inc., 1955- Thiele, C. I" "Arithmetic in the Middle Grades," The Teaching Of Arithmetic. Edited by Nelson B. Henry. Fiftieth.YearbOOk of the National Society for the Study Of Education, Part II. Chicago: The University Of Chicago Press, 1951. pp. 76-102. Tinker, Miles At Teaching Elementary Reading. New York: Appleton- Century-Crofts, Inc., 1952. 106 Traxler, Arthur E., and Jungeblut, Ann. Research in Reading During Another Four Years. Educational Records Bulletin, NO. 75. New York: Educational Records Bulletin, 1960. Traxler, Arthur E., and Townsend, Agatha. Eight Mere Years of Research in Reading. Educational Records Bulletin, NO. OH. New York: Educational Records Bulletin, 1955. Van Engen, Henry. "The Formation Of Concepts," The Learning of Mathematics: Its Theory and Practice. Twenty-first Yearbook Of the National COuncil of Teachers Of Mathematics. Washington, D.C.: The National Council of Teachers of Mathematics, 1953. PP. 69-98- Walker, Helen M3, and.Iev, JOseph. Statistical Inference. New York: Henry Belt and Company, 1953. Wilson, Guy M5 "Arithmetic," Enoyglopgdia of Educational Research. Edited by Walter S. Mbnroe. New York: The Macmillan CO., l9ul. pp. u2-58. . "Arithmetic," Encyclopedia Of Educational Research. Edited by Walter 8. Monroe. New York: The Macmillan CO., 1950 0 Pp o M“ 58' ARTICLES Alexander, Burton F. "Language DevelOpment in Mathematics Through Vocabularies," Mathematics Teacher, XL (December, 19h7), 389-90. Bernard, Harold w; "An Experiment in Vocabulary Building," school and Society, LIII (June 7, l9hl), 782-h3- Black, E. L. "The Difficulties of Training College Students in Understanding What They Read," British JOurnal Of Educational ngchology, XXIV (February, l95h), 17-31. Blair, Glenan., “An Experiment in VOcabulary Building," JOurnal of Higher Education, x11 (February, l9hl), 99—101. Brueckner, IeO J. "Improving Pupils' Ability to Solve Problems," Journal of the National Education Association, xxx (June, 1932), 175-76. ’ Buckingham, Guy E. "The Relationship Between Vocabulary and Ability in First Year Algebra," Mathematics Teacher, xxx (February. 1937). 76-79. 107 Chase, Sara E. "Waste in Arithmetic," Teachers College Record, XVIII (September, 1917), 360-70. Chase, V. E. "The Diagnosis and Treatment of Some Common Diffi- culties in Solving Arithmetic Problems," Journal Of Educational Research, XX (December, 1929), 335-h2. Corle, Clyde G. "Thought Processes in Grade Six Problems," Arithmetic Teacher, v (October, 1958), 193-203. Curoe, Philip R. V. "An Experiment in Enriching the Active Vocabu- laries Of College Seniors," School and Society, XLIv (April 22, 1939), 522-2h. Curoe, Philip R. V., and.Wixted, William G. "A Continuing Experi- ment in Enriching the Active VOcabularies Of College Seniors," School and Society, LII (October 19, 19h0), 372-76. Drake, Richard M "The Effect of Teaching the Vocabulary of Algebra," Journal Of Educational Research, XXXIII (April, l9h0), 601-10. Dresher, Richard. "Training in Mathematics Vocabulary," Educational Research Bulletin, XIII (November 1h, 193A), sol-oh. Eagle, Edwin. "The Relationship of Certain Reading Abilities to Success in.Mhthematics," Mathematics Teacher,.XLI (April, 19h87. 175-79- Eurich, A. C. "Enlarging the Vocabularies of College Freshman," English Journal, college edition, XXI (February, 1932), 135-h1. Foran, Thomas G. "The Reading Of Problems in Arithmetic," The Catholic Educational Review, XXI (December, 1933), 601-12. Georges, J. S. "The Nature Of Difficulties Encountered in Reading Mathematics," School Review, XXXVII (March, 1929), 217-26. Greene, Harry A. "Directed Drill in the Comprehension Of Verbal Problems in Arithmetic," Journal of Educational Research, XI (January, 1925): 33-h0. Haefner, Ralph. "Casual Learning of Word Meanings," Journal Of Educational Research, XXV (April-May, 1932), 267-77. Hansen, Carl W} "Factors Associated.with Successful Achievement ' in Problem Solving in Sixth Grade Arithmetic," Journal Of Educational Research, XXXVIII (October, l9uh), 111-18. 108 Henman, V. A. C. "An Experimental Study of the value Of WOrd Study," Journal of Educational Payphology, XII (February, 1921), 98-102. John, Lenore. "Difficulties in Solving Problems in Arithmetic," Elementary School Journal, XXXI (November, 1930), 202-15. Johnson, Harry C. "The Effect of Instruction in Mathematical Vocabulary Upon Problem Solving in Arithmetic," Journal Of Educational Research, XXXVIII (October, l9uu), 97-110. Johnson, J. T. "On the Nature Of Problem-Solving in Arithmetic," Journal Of Educational Research, XLIII (October, l9h9), 110-15. Johnson, Nhry E. "The VOcabulary Difficulty of Content Subjects in Grade Five," Elementary Englioh, XXIX (May, 1952), 277-80. Lutes, O. 3. "Where Pupils Fail in Verbal Problems," Journal of Educational Research, XIII (January, 1926), 71-72. McSwain, E. T-, and Cooke, Ralph J. "Essential Mathematical Meanings in Arithmetic,” Arithmetic Teacher, V (October, 1958), 185-92. Miles, Isadora W3 "An Experiment in Vocabulary Building in a High School," School and Sociegy, LXI (April 28, 19h5), 285-86. Morton, Robert I” "language and Meaning in Arithmetic," Educational Research Bulletin, XXXIV (November, 1955), 197-20h. Newcomb, R. S. "Teaching Pupils How to Solve Problems in Arith- metic," Elementary School Journal, XXIII (November, 1922), 183-89. Nolte, Karl F. "Simplification of Vocabulary and Comprehension in Reading," Elementapy English Review, XIV (April, 1937), 119-2h. O'Rourke, Everett, and Mead, Cyrus D. "Vocabulary Difficulties Of Five Textbooks in Third-Grade Arithmetic," Elementary School Phipps, W. R. "An Experimental Study in Developing History Reading Ability with Sixth Grade Pupils Through the DevelOpment Of an Active History Vocabulary," Journal Of Experimental Education, VII (September, 1938), 19-23. Pressey, luella C., and Elam, M. K. "The Fundamental Vocabulary Of Elementary-School Arithmetic," Elementary School Journal, XXXIII (September, 1932), h6-5o. 109 Pressey, luella C., and Moore, W. S. "The Growth of Mathematical Vocabulary from the Third Grade Through High School," School Review, XL (June, 1932), hu9-5h. Repp, Florence C. "The Vocabularies Of Five Recent Third Grade Arithmetic Textbooks," Arithmetic Teacher, VII (March, 1960), 128-32. Roling, Pearl, Blume, Clara L., and Morehart, Mary 8. "Specific Causes Of Failure in Arithmetic Problems," Educational Research Bulletin, III (October 15, 192%), 271-72. Sachs, H. J. "The Reading Method Of Acquiring Vocabulary," Journal Of Educational Research, XXXVI (February, 19h3), h57:68. Schaaf, William L. "A Realistic Approach to Problem-Solving in Arithmetic," Elementary School Journal, XLVI (May, 1912.6), h9h-97. Spitzer, Herbert F., and Flournoy, Frances. "Developing Facility in Solving Verbal Problems," Arithmetic Teacher, III (November, 1956). 177-82. Stevens, B. A. "Problem Solving in Arithmetic," Journal Of Edu- cational Research, XXV (April-May, 1932), 253-60. Stevenson, P. R. "Difficulties in Problem Solving," Journal of Educational Research, XI (February, 1925), 95-103. Symonds, Percival, and Penney, Edith M. "The Increasing of English Vocabulary in the English Class," Journal Of Educational ‘ Research, XV (February, 1927), 93-103. Traxler, Arthur E. "Improvement in Vocabulary Through Drill," English Journal, High School Edition, XXVII (June, 1938), h91-9h. Treacy, John P. "The Relationship of Reading Skills to the Ability to Solve Arithmetic Problems," Journal of Educational Research, XXXVIII (October, 19th), 86-96. Young, William E. "language Aspects Of Arithmetic," School Science and Mathematics, LVII (March, 1957), 171-7h. _________ "Teaching Quantitative language," Education Digest, XXII (January, 1957), h7-9. llO UNPUBLISHED MATERIALS Breen, Margaret M., et al. "An Evaluation Of Exercises for the Development of Word Recognition and Word Meaning in Grade Five." Unpublished Master's thesis, Boston university, 1953. Brooks, Samuel S. "A Study Of the Technical and Semi-Technical Vocabulary of Arithmetic." Unpublished Master's thesis, The Ohio State University, 1926. Crosscup, Richard B. "A Survey and Analysis of Methods and Techniques for Fostering Growth of Meaning VOcabulary." unpublished Master's thesis, Boston university, l9h0. Drake, Richard M. "The Effect of Instruction in the VOcabulary of Algebra Upon Achievement in Ninth Grade Mathematics." Un- published Doctor's dissertation, university of Minnesota, 1938. Eicher, James Eugene. "An Experiment in Teaching the Vocabulary of General Science." Unpublished Master's thesis, University of Pittsburgh, 1932. Fahey, Nb:ion.Joan. "An Item-Analysis Vocabulary Study in Arith- metic at the Fifth Grade level." Unpublished Master's thesis, Boston College, 1953. Geisler, Sister.Mary Damian, RSM. "A Technical Vocabulary Of Arithmetic in Grades Seven and Eight." unpublished Master's thesis, Catholic university of America, 1953. Haldorsen, 0. W. "Arithmetic Vocabulary in Standard Word Lists," Unpublished Master's thesis, University Of Minnesota, 1935. Harrison, Irene G. "Survey of Meanings Of WOrds and Signs in Two Arithmetic Textbook Series." Unpublished Doctor's dissertation, Columbia university, 1953. Heightshoe, Agnes Ethel. "A Comparison Of the Vocabularies of Arithmetics and Readers Of the Second and Third Grades." Unpublished Master's thesis, university of Chicago, 1928. Hunt, Ava Farwell. "A Comparison Of the Vocabularies of Third- Grade Textbooks in Arithmetic and Reading." Unpublished Master's thesis, university Of Chicago, 1926. Komisar, David Daniel. "The Effects Of the Teaching Of Social Science Vocabulary to College Freshman on Some Aspects of Their Academic Performance." unpublished Doctor's disser- tation, Columbia University, 1953. 111 Lawshe, Charles. "A Study Of the Relationship Of Certain Factors to the Rates at Which Children Learn and Forget Word Meanings as Indicated by ApprOpriate VOcabulary Tests." Unpublished Master's thesis, university Of Michigan, 1935. Newburn, Harry K. "The Relative Effect Of Two Methods at Vocabulary Drill on Achievement in American History." Unpublished Doctor's dissertation, University Of Iowa, 1933. Petersen, Olga C. "A Vocabulary Course: An Experiment in Teaching Vocabulary as a Separate Course in High School." Unpublished Master's thesis, University of North Dakota, 1937. Reichart, Robert R. "A Suggested Method for Teaching Vocabulary Building." Unpublished Doctor's dissertation, University Of Oregon, 19h1. Tyler, Bertha M. "Arithmetical Vocabulary as a Factor in Problem Solving." unpublished Master's thesis, State University Of Iowa, July, 1927. APPENDIX A List of 2A2 Quantitative Terms and the Chapters in the Fifth Grade Arithmetic Text in Which They Appear 112 Arithmetic Vocabulary EXperiment Bay City, Michigan 2A2 Quantitative Terms and The Chapters in Which They Appear accurately acre add addition age almost A.M. amount annual approximate area arithmetic arrange average balance base bill borrowed carrying cash cent change charge check circle coin collect collecting collection columns combined (nommon (Rampare commess complete correct coumn; counting decimmd. deeyree denominator diameter Choppers 123E56789 x x (x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X x x x x x x x x x x x x x x x x x x x x x 113 111+ diameter difference digit dimensions directions distance divide dividend division divisor double dozen edge encircle end equal error estimate even exact exercises eXpenSes face fact figure foot form fraction fractional grade graph great greater greatest gross group half half-fare height higher highest hour inch incomplete increase into large larger Chapters 1 2 3 1+ 5 6 7 8 9 x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 115 least left length less limit line loan lower many matches measure measurement meet members meters mile minus minute multiplication multiplier multiply normal number numerals numerator Odd order outside pair partial parts payment perimeter period place plan plane plus P.M. point pound price principal problem process product profit prove Chapters 1 2 3 A 5 6 7 8 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 116 purchases quantity quarter quotient rate ream rectangle reduce reduction reference regular reminder rent repeat result right rise round row ruler scale score second section selling serial set shape sheet short side sign single small space speed square standard straight subtract subtraction sum surface system table tax terms time Chapters 1 2 3 h 5 6’ 7» 8 9 x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X x x x x x X x x x x x x x x x x x x x x x x x X x x x 117 times ton total trials triangle unit upper value weight whole width yard year zero a. balance due bar graph decimal fraction decimal point down payment draw to scale exact number imprOper fraction like fractions liquid measure lowest terms mixed decimal mixed number monthly payment on the average place holder place value price list . record Of expenses round numbers Chapters 123E567§9 x x x x x x x x x x x X x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x x x x x x x x x X x x x x x x x x x x x x x x x x x x x x x. x x x x x x x x x x x x x x x x x x X x x X x x x x x x x x 118 round trip scale drawing square foot square inch square measure square mile square yard tens' place three-place number time line total amount unit of measure who 1e number zero as a place holder Chapters 11+5 78 x x x x x x x x x x x x x x x x x x x x x x x x x APPENDIX B Weekly Lists of llerms and Illustrative Sentences 119 First Week base dividend figure limit Odd plus repeat system Second week compass foot greater lower P.M. point ruler single 120 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan We have a §§§g_ten number system. The dividend is the number to be divided. Sue wrote the figure A on her paper. The speet 11215 is 25 miles per hour. Three is an odd_number. The plgg sign tells us to add. In writing the number 111 we repeat the numeral one. Our system of numbers is based on ten. The needle Of the compass is a magnet. John caught a fish that was a foo§_long. Mary has a greater amount than Sue. He will buy a coat at the sale when prices are 12133, John has an appointment at 2 2111' You‘ score one poipp_for each right answer. Harry used his £2125 to draw a straight line. The children.walked in single file. Third Week collecting decimal larger order purchases table whole decimal point Fourth Week bill carrying counting height inch prove speed zero 121 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan The children have been collecting stamps. We have a decimal number system. Seven is a larger number than six. The names were put in alphabetical 2£§E£° They went shopping and made Six purchases. A table of weights and measures will tell how many pounds are in a ton. Four is a whole number. In writing $2.25 the decimal point separates dollars from cents. Our grocery pill was high last month. Adding 37 to 29 involves cargying. John is counting the new books for the library. Your height is measured in inches. John grew an gggh in one month. Bob was asked to p321: his answer. The boys ran at full gaggi. The zero in the number 20 means no ones. Fifth Week expenses members period quantity ton unit unit Of measure liquid measure Sixth WEek cent higher increase mile partial scale width 0 (degrees) 122 Arithmetic Vocabulary EXperiment Grade 5 Bay City, Michigan At the end Of the trip Mary totaled their egpenses. All the members of the group were there. In the number h96,27l, which pgglg§_contains a 9? The farmer sold a Quantity Of hay. I Dad bought a pop Of coal last week. A foot is a unit of length. The inch is a smaller unit Of measure than the foot. The pint, quart, and gallon are liqpid measures. Mary bought four 3 ggpp stamps. He climbed hlghgg than he had ever climbed before. Mike asked his dad to increase his allowance. It is a 9212 from here to the lake. Dad made a partial payment on the car. The map is drawn on a ggglg Of one inch to the mile. Our room is 25 feet in.!;gpp, Yesterday the temperature reached 6h0. Seventh Week collection error half length profit regular side year Eighth Week 4 123 Arithmetic Vocabulary EXperiment Grade 5 Bay City, Michigan Bob has a fine stamp collection. Mary found the 5532; she made in adding. Dick earned 2312.3 dollar. The lengph Of’a room is the longest way it can be measured. John can earn 10¢ profit on every box Of candy he sells. John learned the regplar procedure to use in multiplying. He sits on the other gigg_of the room. There are twelve months in one year. The plus sign tells us to add. The minus sign tells us we should subtract. The times sign means that we should multiply. The divide sign is read as "divided by". The frame in division separates the dividend and divisor. The equal sign tells us that what is written to the left is equivalent to or the same as what is written tO the right. Ninth Week age borrowed columns combined double loan multiplier result Tenth Week amount equal grade highest pair plane set place-value 12h Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan Mary is 10 years of 282° He borrowed a ten and exchanged it for 10 ones. Sue added four columns Of numbers. Two small groups were combined to form one large group. When you dOuble a number you have twice as much. Mr. Jones went to the bank to get a 1232' In 3 x A, three is the multiplier. What result did you get for exercise four? We need only a small amount Of paint. Two nickels are 333§l_to one dime. Farmers 53293 their eggs according to size. The highest temperature we had was 98 degrees. Dad bought a new p§i£_of shoes. The desk top is a plgpg surface. We have a complete pep Of those books. We used a place-value chart in arithmetic. Eleventh Week coin collect even plan sign upper weight total amount Twelfth Week arithmetic charge count difference group quarter yard place holder 125 Arithmetic VOcabulary Experiment Grade 5 Bay City, Michigan A dime is a coin. The teacher asked Mary to collect the milk money. Four is an SEER number. He drew a plan_of the house. The plus sign means that we should add. In subtraction the upper number is larger. The weight of the dOg is fifty pounds. John found the total amount by adding all the numbers together. Bob said that arithmetic is working with numbers. The charge for renting the boat is fifty cents. Mary can count to one hundred. The difference between seven and ten is three. A group of boys were playing marbles. Sam ate a Quarter of the pie for supper. A yard is equal to 36 inches. Zero is the numeral used most often as a place holder. 126 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan Thirteenth'Week accurately arrange edge face incomplete numerals reduction selling Mary worked the examples accurately. Mary will arrange the numbers in order from largest to smallest. He sat on the gdgg_of his chair. The £222.0f the mountain.was covered with ice. Three of the problems were incomplete. Roman numerals are sometimes used on clocks. The reduction of 2/h to g is easy. The selling price is marked on the price tag. Fourteenth Week Ankh denominator digit fact Ininus numerator straight School begins at 8:30 9:3. In 2/3, three is the denominator. John wrote a six digi§_number on the board. Sue had difficulty with a multiplication £393. Ten minus seven leaves three. In 3/8, 3 is the numerator. Can you draw a straight line without a ruler? Fifteenth Week end fraction measure reference second square ten's place three place number Sixteenth Week encircle greatest meters reduce rent like fractions mixed number whole number 127 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan He stood at the end of the line. 5 is a fraction. We can measure the room.with a yardstick. There is a reference table on page 369 of our arith- metic book. A Jet plane has flown 15 miles a second. John drew a sguare on his paper. Any of the numerals may be used in ten‘s place. The number 100 is the smallest threegplace number. They put up a fence to encircle the playground. Mary lives the gzeatest distance from school. A man 2 meters tall would be a little over six feet tall. We do not always have to 332233 a fraction. Mr. Smith pays his rent once a month. 1/6, h/6, and 2/6 are like fractions. 3 2/3 is a mixed number. A number like 7 is a whole number. Seventeenth Week balance cash ream serial half-fare price list round trip time line Eighteenth Week payment place rate short balance due down payment monthly payment record of expenses 128 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan After the down payment we owed a balance of $e0.00. The storekeeper took in ten dollars in gg§g_today. You can buy that paper only by the 3239. Bob knew the Eggigl number on his bicycle. Children under 12 can ride for half-fare. Bob looked at a price list to find out how much a light would cost for his bicycle. Mrs. anes bought a round trip ticket to New York. The children drew a time line on the chalkboard of important events in arithmetic. Mr. Jones made a $10.00 paypent on his bill. The figure 3 goes in ten's BEESE: The Jet flew at a great ggpg_of speed. It is only a ghggp distance to school. The amount left to be paid after the down payment is the balance due. John's dad made a $100.00 down_payment on a car. The monthly payment on Bob's bicycle is $5.00. On the trip Sue kept a record of expenses for her father. Nineteenth Week average change exact measurement outside price terms zero as a place holder Twentieth Week add addit ion remainder subtract subtraction sum total on the average 129 Arithmetic VOcabulary Experiment Grade 5 Bay City, Michigan The average weight of the boys is 92 pounds. Mary received ten cents in change. Some answers in arithmetic do not have to be exact. Degrees give us a measurement of temperature. They live Just outside of town. The price of the shoes is $5.00. The terms of a fraction are the numerator and the denominator. In the number 206, zero is used as a place holder. We can gdthhe numbers to find out how much we have. John had two addition problems to do. John subtracted four from six and had a remainder of two. We subtract to find out how many are left. Nancy likes to wonk subtraction examples. The sum of 6 and 5 is eleven. We can add the numbers to find the total. John sold 35 newspapers a day on the average. 130 Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan Twenty-First Week divide Mary said she would divide to find out how many twos are in ten. division Division is the Opposite of multiplication. divisor In 14%", 1+ is the divisor. multiplication She had two multiplication exercises to do. multiply Sally can multiply 27 times 16. product The product of 3 x h is 12. quotient The answer in a division problem is called the guotient. times Five timgg four is twenty. TWenty-Second.Week check The teacher said that we should ghggk_each answer. compare We can cogpare the numbers to see which is the largest. distance He lives a short distance from school. large One million is a laggg number. less The table is lggg than six feet long. number The figure 3 is called a EEEEEE! row Mary likes to sit in the front 52!, small A penny is a small amount of money, Twenty-Third Week correct hour left minute parts problem right time TwentyeFourth Week circle dozen least matches pound triangle value improper fraction l3l Arithmetic Vocabulary Experiment Grade 5 Bay City, Michigan All the children in the class had the problem correct. The movie will start in one £933. Bob spent some money and has ten cents lggp. A minute is equal to sixty seconds. One of the pgppg of the radio was missing. We must add in the first problem. t answer. John said he had the ri The time is now ll o'clock. He drew a circle that was four inches across. The grocer sells oranges by the QEEEEf The lg§§p_number of boys are in our room. Bob exclaimed, "My bicycle matches yours.'" Mother went to the store to buy a pggpg_of coffee. The sides of a triangle do not have to be the same length. The value of a dollar is less than it used to be. An imprgper fraction is equal to one or more than one. Twenty-Fiftpreek almost complete degree form fractional sheet exact number round number TWenty:Sixth Week annual area diameter normal rise space draw to scale scale drawing 132 Arithmetic Vocabulary EXperiment Grade 5 Bay City, Michigan Peter was almost late for school. Complete each example before you start the next one. The temperature has gone up 10 degrees. We learned a new fgpp for working division examples. A foot is a fractional part of a yard. JOhn wanted a E2223 of paper without lines. A number found by counting is called an exact number. Mr. Jones reported the attendance at the county fair in round numbers. The gpppgl school picnic is held in May. The E£E§.°f the room is 750 square feet. The diameter of the three is 2% inches. The normal temperature of the body is 98.6 degrees. The temperature will 33§3_when the sun comes up. Write your name in the §p232_at the t0p of the paper. The boys learned how to draw to scale in shOp class. Harry made a scale drawing_of the playground. APPENDIX C Description of Experimental Design and Direct Study Techniques 133 Fifth Grade Experimental Study Bay City, Michigan September 1961 During the 1961-62 school year the fifth grade classrooms in Bay City, Michigan will participate in a study to determine if direct study techniques directed toward arithmetic vocabulary deve10pment will affect the ability of pupils to solve arithmetic story problems. The enclosed materials are provided to assist the teachers of the experimental class- rooms. The first section is a rationale for the study briefly explaining the importance of vocabulary knowledge and its relationship to verbal problems and the ability of pupils to solve them. The second section contains a description of the techniques to be used in teaching the arithmetic vocabulary in- cluded in this study. 13h The Purpose and Plan for Developing Arithmetic Vocabulary PURPOSE OF THE STUDY By the time pupils have entered grade four the nature of the instructional program demands that each pupil develop proficiency in interpreting the written material found in each of the content areas. While it may have at first appeared that the ability to read well was the result of a generalized reading ability it is now known that the ability to comprehend material in the content areas is dependent upon a complex of interrelated skills. Some of the skills needed to read with understanding are common to all content areas. However, there are certain characteristics of each of the content subjects that call for specialized skills not needed in the other subjects. Certain skills are needed to comprehend the materials found- in arithmetic that are not needed to understand materials in science or in social studies. Each of the content fields has its unique technical vocabulary, its own set of concepts and relationships, its own organization, and its own purposes and aims. These unique features demand unique reading skills. A considerable number of studies have been made in.the various content areas for the purpose of analyzing the comprehension skills needed and in identifying the difficulties pupils have in understanding written material found in a particular subject. Most reviews of research in the content areas include an understanding of the technical vocabulary as an essential for the understanding of important concepts in the field. The vocabulary load of unfamiliar 135 136 words or new uses for familiar words is not always controlled in the reading matter found in the content subjects. These words do not always appear a sufficient number of times to assist the reader in acquiring appropriate meanings. It is not feasible to leave the learning of important words to incidental methods. Under such circumstances there can be no guarantee that pupils will acquire meanings of technical and semi-technical terms. Vivid and accurate meanings are essential to adequate comprehension, interpretation, and critical analysis. One of the most persistent problems in elementary school arithmetic instruction is in the area of story or verbal problems. There is a considerable amount of data available which indicates that pupils do not achieve as well in arithmetic problem solving as they do in arithmetic computation. This has been true in the past as well. Most elementary school teachers support the contention that verbal problems cause children difficulty and that pupils often become confused and frustrated when faced.with the task of solving them. This frustration often leads to dislike for arithmetic as a school subject. Many studies have been concerned with the role of specific comprehension abilities in pupils' success in solving verbal problems. Several such studies have indicated a strong relationship between pupil knowledge of mathematical vocabulary and problem solving ability. Lack of knowledge and familiarity with technical vocabulary have been shown to interfere greatly with problem solving performance. The conclusion of one study was that knowledge of technical vocabulary 137 is the main factor in problem solving, ranking in importance ahead of reasoning ability. While many factors have been the subject of experimentation, knowledge of vocabulary remains a most significant one in problem solving ability. Understanding,Story_Problems Fundamental to the accurate and efficient solving of story problems is the adequate comprehension of the statement of the problem situation. An examination of story problems reveals that they are almost always stated briefly and concisely. A minimum number of words is generally used to communicate intricate and often abstract re- lationships. Often only the bare essentials are given. Elaborate, descriptive statements are seldom used, therefore, context offers a minimum number of clues for unlocking the meanings of unfamiliar words. It is true that context always determines meaning but context does not always readily reveal meaning. Verbal problems follow no special pattern and the absence of continuity from problem to problem minimizes the possibility of context clues. Another source of difficulty in understanding verbal problems is caused by the unusual number of technical and semi-technical terms found in most problems. .Many of these terms have a specialized meaning and are unlikely to be met in other subject areas. Abbreviations and mathematical signs and symbols are used in many problems and are a source of difficulty. Each problem must be thought through carefully and it becomes evident then, that a knowledge of the terms in the statement of a problem is essential to its adequate comprehension. 138 In the content areas pupils encounter many terms which are outside their experience. Many terms are known by pupils in one context but when used in arithmetic have entirely unknown meanings. ~Other terms are unknown in form but are used to express a meaning which is familiar to pupils. Still others are unfamiliar because they have not been seen or heard before and because pupils have not had experiences with those situations in which the terms may be apprOpri- ately used. Some research in this area has shown that pupils experi- ence the most difficulty when they encounter known words used in unfamiliar contexts. The field of arithmetic employs many such terms and it cannot be left entirely up to the pupils to learn these specialized meanings for familiar words. Thus, because of the unusually large number of technical terms and the relative lack of context from'which to infer word meanings, vocabulary study becomes an important aspect of instruction in arithmetic. A considerable number of studies have been concerned with increasing the size of pupils' vocabularies at all educational levels. Relatively few studies, however, have attempted to determine if an increased vocabulary has effect on other aspects of the curriculum such as achievement in content areas or in the ability to achieve a particular task. References are continually made in the literature to quantitative vocabulary as a factor in verbal problem solving in arithmetic, and to the need for research to determine the relation- ship of vocabulary facility to verbal problem solving ability. 139 Value of Direct Study Techniques A considerable number of studies in many content fields and at many grade levels have been conducted to discover the effects of direct study techniques for improving meaning vocabulary. It has been found that significant gains can be made using such techniques. A person's meaning vocabulary may deve10p incidentally as he reads, listens, and engages in conversation. However, it is well known that conscious, direct efforts to build vocabulary result in greater achievement than if vocabulary deve10pment is left to incidental methods. A functional vocabulary may be develOped through a wide variety of direct methods. The direct study techniques to be used in this study to teach quantitative terms will be described in detail below. Quantitative terms as used in this study are: 1. space, time, position, value, money, quantity, and degree, 2. signs, 3. technical terms in arithmetic such as, add, minuend, and partial product, which are used in operations with numbers, A. measurement terms, 5. commercial terms, and 6. words identifying common spacial figures. It is the purpose of this study to utilize direct study tech- niques in the teaching of quantitative vocabulary and to determine the effects of these techniques on verbal problem solving ability in arithmetic. PLAN OF THE STUDY The study involves all fifth grade pupils in Bay City. Approximately one-half of the pupils will be assigned to the experimental IMO group and one-half to the control group. The pupils will be tested at the opening of school in the fall in several areas including verbal problem solving ability. The experimental group will utilize direct study techniques for learning quantitative vocabulary as a part of the regular program while the control group will pursue the regular instructional program in arithmetic. At the close of the school year the pupils in both the control group and the experimental group will again be tested in verbal problem solving ability and in vocabulary. A statistical analysis of the post-test scores will be made. Lesson plans have been constructed so as to become an integral part of the instructional program in the experimental group in the Bay City Schools.' The techniques described below have been developed for the fifth grade teachers in Bay City who are teaching the experimental classes. These lesson plans are to be used within the framework of the existing prOgram and are not to be considered an extension or addition to the regular arithmetic period. Fbr the results of the experiment to be significant both the control group and the experimental group must devote as nearly as possible the same amount of time to the study of arithmetic. This means that the direct study techniques utilized to teach the quantitative terms must use only a small part of the period devoted to arithmetic instruction. Criteria for the selection of techniques to be used are: 1. they can be carried on as a regular part of the arithmetic period, 2. they do not require materials and equipment not found in the average classroom, 3. they do not require a large amount of class time, lhl h. they do not require an undue amount of preparation outside the regular class period, and 5. they have been utilized in experiments resulting in significant vocabulary gains. The word terms rather than words has been used throughout the discussion because of the inclusion for study of some technical signs, such as, +, 'y x, f, and phrases, such as, liquid measure and mixed number. All of the terms to be studied are found in the fifth grade arithmetic text. .An alphabetical list of all terms will be provided for each teacher. For the convenience of the teacher the terms have been listed alphabetically. An examination of this list will indicate which terms are repeated in succeeding chapters. While it will not be necessary to devote undue time to terms previously studied, repetition is often helpful particularly for difficult terms. In another list the terms in each chapter have been defined and two or more illus- trations of their uses are given. This will assist the teacher in presenting terms for study and in clarifying meanings. It will be left to the teacher to determine how many terms should be presented at a given time. There are approximately 250 terms to be taught during the school year. Over a period of 30 weeks this would average approximately eight terms per week. There would probably be an advantage in varying this procedure. Some groups of terms may lend themselves readily to presentation as a group. At other times it may be best to introduce only a few at a time. Re- gardless of the number of terms presented at any one time it is 1&2 requested that teachers in the experimental classes devote some attention to study every day. Setting_the Stage At the beginning of the study and periodically throughout the year it will be helpful to the students to have their teacher discuss with them in a general way the importance of vocabulary development and the fact that they will be concerned throughout the year with discovering the meanings of important terms used in arithmetic. An important aspect of the discussion will be to help the pupils recognize that many terms used frequently by them may be used in a quantitative sense and for this reason may be associated with the field of arithmetic. The association of terms with a specialized area in which they may be found helps to develop the meanings of these terms. Techniques to be USed There are several important factors to be considered in any program of vocabulary development. The primary concern is, of course, to teach the meanings of unfamiliar terms and to extend the child's knowledge of words already familiar to him. Many techniques and methods have been utilized.with the intention of improving meaning vocabulary. However, it is important to distinguish between: 1. those techniques and methods used in initial presentation of terms, 2. those used to develop meanings, and 3. those used to aid retention of meanings. While these three areas are not discrete it 11.3 is important to point out that methods which may be utilized in the initial presentation of terms to be learned are important to vocabulary development, but may not teach the meanings of those terms. Methods used to aid retention are important, but may not aid in teaching meaning. Methods used in initial presentation of terms may contribute toward understanding if the terms are presented in context rather than in isolated lists. Terms have no unique meaning unless they appear in a context. Methods used to aid retention make it possible for pupils to use newly acquired words in reports and discussions and in conversation with others. Fer some children the experience of listen- ing to others use these terms in a variety of contexts will in effect, be a means of acquiring meaning. While there are many techniques and combinations thereof that might be employed, it is felt that a few well selected techni- ques will enable the teacher to present the term, teach its meaning, and provide for its use so as to aid retention. The techniques that we will use in this study are (1) initial presentation, (2) class discussion, (3) teacher explanation, and (h) using the dictionary. I. Initial Presentation These techniques are designed to focus the attention of the pupils upon the terms to be studied so as to enable each child to establish a three-way association between the written symbol, the sound of the term, and at least one of its meanings. For the convenience of the teachers each term is accompanied by an apprOpriate definition and sentences illustrating some of its meanings. For example: it}. add - to find the total of several numbers 1. We can add the numbers to find out how much we have. 2. Father must add a long column of numbers. 3. We can add more sugar to make it sweeter. h. Bob will add that butterfly to his collection. 5. The speaker said, "I have only this to add." a. Sentences using each of the terms in an arithmetic context may be written on the chalkboard, or duplicated and dis- tributed to each pupil. If the sentences are duplicated each should be numbered and the term being studied under- lined. b. Each term should be pronounced carefully by the teacher as the pupils direct their attention to its written form. If the chalkboard is used the attention of pupils may be directed by framing the term with the hands as it is pro- nounced or by underlining the term. If each pupil has a duplicated copy of the sentences, his attention may be drawn to the underlined word in sentence number one, for example. c. Have the pupils pronounce each term after having heard it pronounced. This may be done by individuals, or, in unison by small groups or the total group. Errors in pronunciation should be corrected. d. Attention may be called to unusual features of a term such as spelling, configuration, or pronunciation. II. Class Discussion The meanings of many terms are learned through new experiences. Encountering new terms in a variety of contexts is an effective means of developing breadth and depth of meaning and in refining shades of meaning. Encouraging pupils to exchange experiences orally is an efficient way to accomplish this. While it is not always possible to participate in or observe certain activities and actions it is worth- while to listen to oral descriptions of them given by others. A th variety of contacts with terms enable pupils to learn, not a single narrow meaning, but a generalized of "class" meaning of those terms. Involving pupils in class discussion makes it possible for self- correction to take place. Faulty concepts and mdsconceptions can be corrected through pupil interaction. Methods to be used are to ask pupils to: a. share their knowledge of the terms with their classmates. The question technique may be employed. 1. Where have you heard or seen this term and how was it used? 2. What did it mean in that situation? 3. Does it have the same meaning when used in arithmetic? give definitions in their own words. Encourage them to refer to objects in the room and to use chalkboard drawings, and so on, to make their definitions clear. restate in their own words the sentences using the terms. Substitute synonyms or equivalent phrases. Substitute antonyms to emphasize approximately Opposite meanings. compare and contrast various meanings for the same term. Distinguish between mathematical and non-mathe— matical meanings. Rec0gnize other terms with which the terms being studied might be confused. For example: add - ad cent - scent some " sum Make up sentences using the term and tell.whether or not it has an arithmetical meaning. III. Teacher Explanation Many times it is helpful for the teacher to explain the meaning of difficult terms. The teacher is often the only one present 116 in the classroom who has the capacity to sense instances where terms may be used incidentally, and can, therefore, capitalize on certain experiences which might otherwise go unnoticed. Often un- expected situations arise which make it possible for the terms being studied to be put into use naturally. It is within the teacher's capacity to clarify meanings by referring to a use of a term within the eXperience of the pupils. The teacher's explanation of terms often acts in a summarizing capacity and assists pupils in associ- ating a term with a generalized rather than specific meaning. Specific techniques to use are: a. oral definition b. verbal illustrations and analogies c. chalkboard drawings and sketches d. diagrams, charts, graphs, still pictures e. to associate terms with the materials and equipment in the classroom. f. to request pupils to watch for the teacher's use of the terms being studied. IV. USing the Dictionary Mbst fifth grade pupils are able to use guide words and entry words with sufficient skill so that they are able to locate a term in their dictionary. However, they may need to refine other 'dictionary skills which help to develop vocabulary. An important skill in acquiring the meanings of terms from a dictionary is that of distinguishing which of several meanings listed is the one appro- priate to the context in which the term was encountered. The 1&7 dictionary may assist in clarifying meanings of unfamiliar terms or to discover other meanings for the same term. A majority of the terms are to be found in the classroom dictionary. Techniques to use with these terms are: 8. b. locate terms in the dictionary notice that several uses for a term may be given and that phrases or sentences are used to place the term in context. distinguish which of the meanings given is apprOpriate and compare this dictionary definition with the way the term is used in arithmetical contexts. total class activity, or, have an individual or small group look up the term and read out loud the various definitions and uses given. The various uses may be discussed and the most appropriate one decided upon and written on the chalkboard. The definition may then be compared with meanings known by the class. APPENDIX D Chapter Iists of Terms, Definitions, and Illustrative Sentences 11+8 add addition amount average base bill lh9 Chapter 1 to find the total of several numbers; to combine two or more numbers into one sum \J'I-(f’UUl'Dl-J We can 3 g the numbers to find out how much we have. Father must Egg a long column of figures. We can E99 more sugar to make it sweeter. Bob will Edd_that butterfly to his collection. The speaker said, "I have only this to a_d_c_1_." the process of combining or Joining two or more numbers to obtain a number called the sum rWMH @014?me John had two addition problems to do. The workmen built an addition on the school. The singer was a pleasant addition to the program. In addition, there is Just this to say. total of two or more sums or quantities; a quantity We need only a small amount of paint. It takes a large amount of money to buy a house. No amount of help will be enough. He is sure to amount to something. The book I read did not amount to much. sum of the numbers divided by the number of numbers; usual thing; the most frequently encountered thing The average weight of the boys is 92 pounds. His baseball average is .3lh. My dad works an Ezerage of h5 hours a week. She is an average person. They thought the moving picture was about average. The man wears an average size shoe. a given number of units of one order make one unit of the next higher order; the bottom part of anything which supports the rest of it; foundation (TAD-FLUID!“ We have a base ten number system. The base of the triangle is four inches long. The batter ran to first base. The base of the statue was made of stone. My uncle is at an army base. It is five miles to the base of the mountain. a list of things bought with their cost; a statement of money due; a piece of paper money \DKUJIUH Our grocery bill was high last month. Sue found a dBIIar bill. The duck's bill is yellow. The doctor sent us a bill. He pulled on the bill of his cap. carrying cent change check coin collect collecting lSO transferring a number from one place to another place, as, from one's place to ten's place 1. Adding 37 to 29 involves carrying. 2. The next tOpic in arithmetic that we will study is cargying. 3. The little boy was carpyipg a heavy box. a coin or piece of money 1. Mary bought four 3 cent stamps. 2. Penny candy costs one cent. 3. The tax on a one dollar purchase Just went up one cent. money that is given back when you give an amount larger than the price of what you buy; smaller pieces of money given in exchange for a large piece of money. Mary received ten cents in change. Sue asked the man to change a dollar bill. Children often put their chapge in a bank. Wait for me while I chapge my clothes. When winter comes the weather will chapge. \fl-F’WNl-J a written order directing a bank to pay an amount of money; to do something to make sure you are right The teacher said that we should check each answer. Dad cashed the check at the bank. Mother bought a check table cloth. The man at the bus station gave me a baggage check. . The farmer went out to the barn to check on the cows. U'I-P'UURDH a piece of metal money; to change metal into money 1. A dime is a goin. 2. The government must coin new money every day. 3. John has a Japanese 33in. h. I heard him say he would coin a phrase. to gather things into one body or place; assemble; to obtain payment of money owed to you The teacher asked Mary to collect the milk money. John likes to collect stamp§T__'_ The paper boy goes out to collect once each week. The man came to collect shapes—era. . The farmer's wife went out to collect the eggs. \fl-F’UOI'DH the act of gathering things into one place; the act of gathering money 1. The children have been collecting stamps. 2. The teacher is collecting milk money. 3. The soldiers were sent to a collecting station. A. They are collecting money to build a new church. collection compare compass correct count 151 the act of collecting; a gathering of objects; a gathering of money 1. Bob has a fine stamp collection. 2. A collection was taken up to buy books for the library. 3. That is one of the finest paintings in the collection. to examine things to tell how they are alike and how they differ; to represent as similar 1. We can compare the numbers to see which is the largest. 2. He will compare the bicycles before buying one. 3. How does your new car compare to the old one? an instrument for showing directions on the earths surface by means of a magnetic needle; an instrument for drawing circles l. The needle of a compass is a magnet. 2. He used a compass to draw the circle. 3 A compass can be made with a flat piece of cork and a sewing needle. to remove errors; to make right; free from mistakes Synonyms: right, regular, prOper, true, faultless, perfect, strict, particular All the children in the class had the problem correct. We can correct our errors. Mether will correct them if they misbehave. The teacher had some papers to correct. He was correct when he said the answer was ten. \nerH to tell or name things or numerals one by one or by groups using the regular order of number names; to determine the whole or total number of things; to say the number names one by one in order; the whole or total number of things Mary can count to one hundred. You can count on me. Did you get a correct count? Jim likes to count the cars in a train as it passes. Taking a head count means to count the number of people present. \fl-IS’UJTOH 152 counting the act of telling or naming things or numerals one by one or by groups using the regular order of number names; the act of determining the whole or total number of things 1. John is counting the new books for the library. 2. The children made a counting frame for arithmetic chess. . 3. We are counting on you to be there. h. Bill said that counting by twos is faster. decimal numbered or proceeding by tens, each unit being ten times the next smaller unit; a number written as tenths, hundredths, thousandths, etc. 1. We have a decimal number system. 2. The number 17.06 has two decimal places. difference answer in a subtraction problem; the amount by which one quantity or number is larger or smaller than another l. The difference between seven and ten is three. 2. It does not make any difference. 3. What is the difference between the two cars? A. There is a slight difference in price. distance the space between two things; the length of a line; condition or quality of not being near He lives a short distance from school. . It is a great distance between Michigan and New York. The distance between first base and home plate is ninety feet. The distance is greater if you travel by way of Centerville. 43' bowl-J divide to find out how many times one number is contained in another; to separate into parts; to deal out something in portions or equal shares 1. Mary said she would divide to find out how many twos are in ten. 2. Bob will divide the candy among his friends. dividend the number or quantity to be divided; a sum of money to be distributed; the share of a sum that falls to each individual The dividend is the number to be divided. In 15 :7, 11; is the dividend. He received a baseball as a dividend for selling the most magazines. . The company did not pay a dividend last year. The number above the line in a fraction is the dividend. m4:- DUMP-4 division dozen equal even figure O\U'| to.) 153 act or process of finding out how many times one number or quantity is contained in another; inverse of multiplication; act or process of dividing anything into parts 1. Division is the opposite of multiplication. 2. 'E55_H§§_completed the division. 3. An army division contains many men. h. In which—HIVisian of the company does he work? a collection of 12 objects; to dozen, or, to make up into lots of 12 each The grocer sells oranges by the dozen. . There are twelve things in a dozen. Have you ever heard of a "bakers dozen"? . Bob would rather have a dozen pennies than one dime. "‘ 47'me having the same size, amount, or value, etc.; to match; not more or less 1. Two nickels are equal to one dime. 2. It is difficult to divide a candy bar into equal parts. 3. Bob said he could egual John's score. h. The teams scored an equal number of points. 5. He is the equal of any man. leaving no remainder when divided by 2; equal in size, number, or quantity; level; free from inequality Four is an even number. Twelve eggs make an even dozen. The water is even.with_the t0p of the pail. He has an evefiftemper. Even Bob can jump that far. \n-P'UJI'DH a written or printed character representing a number; a digit; a numeral ; to estimate; the form.or shape of anything; shape, outline; the representation of any form (as by drawing, painting, etc; value expressed in numbers; price; amount; sum; a pattern or design in cloth or paper, or in nature 1. Sue wrote the figgpe h on her paper. 2. The man sold his car at a high figure. . The grocer hast) figgre the cost. He drew the figure of a man. . Mary learned how to figpre skate. The man could figure things out. foot grade greater half height 15h a measure of length equal to twelve inches; one third of a yard John caught a fish that was a foot long. He grew a foot taller since I saw him last. He hurt his foot. Mother sat on the foot of the bed. They live at the foot of the hill. . He gets to work on foot. O\\J'1-¥='UJR)|—' a position in any scale of rank, quality or order; relative position or standing; to arrange in order according to size, quality or rank Farmers grade their eggs according to size. . He buys the Best grade of gasoline for his car. Bob is in the fifth grade. He drove his car up a steep grade. The teacher will grade the papers. . He received a grade of B in spelling. Owns—wand larger in size, quantity, or amount; longer in time; more important; bigger; larger in size or spatial dimensions; of much size; larger in quantity; larger in number3° more numerous; longer in duration or interval a great whilg7 Mary has a greater amount than Sue. l. 2. Some think he was a greater man than his father. 3. The distance to New York is greater than the distance to Chicago. h. A greater amount of time is needed to complete the building. ' 5. The post office serves gpeater Bay City. one of two equal parts into which anything is or may be divided; in an equal part or degree; a part of anything equal to the remainder 1. Dick earned half a dollar. 2. I will see you in half an hour. 3. The man sawed the board in half. h. The potatoes are only half done. 5. His glass is half full of milk. 6 She was only half asleep. how high a thing is; altitude; measurement from t0p to bottom 1. Your height is measured in inches. 2. They had never climbed to that height before. 3. He was very ill at the height of the fever. h The airplane flew to a great height. hour inch into large larger least 155 sixty minutes; a unit of measure of time; the time of day as expressed in hours and minutes indicated by a timepiece; l. The movie will start in one hour. 2. Our lunch hour begins at 12 F’ciock. . 3. The grandeEEEr clock strikes each hour. hm We live an hour from downtown. 5. This is the hour for decision. a measure of length equal to the twelfth part of a foot John grew an inch in one month. An inch of rain fell last night. Show me the inch mark on your rule. The man bought an 18 inch ruler. . We watched the worm inch along. \ne-oumI-a to the inside of a place or thing . He walked into the room. The magician changed a ball into a rabbit. The car turned into the driveway. . We drove into the city . The man will look into the matter. \n-t’UoIUH or great size, amount or number; ample in quantity or extent ' 1 One million is a large number. 2. My dad works for a large company. 3. A large number of birds were in our yard last night. h John found a large sum of money. greater in quantity, size or extent; greater in number or amount 1. Seven is a larger number than six. 2. A quart is larger than a pint. 3. Bay City is larger than Midland. smallest in size, amount, quantity, price, etc.; that which is of slightest or lowest possible value; lowest in value; shortest The least number of boys are in our room. The least amount you can pay is $2.00. Four is the least amount you can buy at one time. He liked that part least of all. That is the least he could do. via—wrote left less limit lower many multiplication 156 the side or direction to the west when you face north; direction; remaining Bob spent some money and has ten cents left. . John is left handed. We made a left turn at the corner. The peOple left the room. #‘UOI'DH smaller; not so much; shorter; fewer; a smaller portion, quantity or degree The table is less than six feet long. He earned less money this week than last week. The program is on for less than an hour. It takes less time to go there by bus. It is of less importance than you think. \n-p'UOI'UH a boundary or boundary line; a point not to be passed; a region, thing, or time defined by bounds The speed limit is 25 miles an hour. The fisherman caught his limit. The city limits end here. I heard his mother say, "He is the limit!" He had to limit his time to five mm. \h-fi’UUFOP relatively low in position, value, amount, etc.; to reduce in value, quantity, or amount, etc.; less in price, quality, strength, etc. 1. He will buy a coat at the sale when prices are lower. The temperature is lower today. Put the book on a lower shelf. You can lower the box with a rope. . Please lower the window. \nrwm consisting of a great number; numerous; not few; a large or considerable number 1. There were mapy children at the picnic. 2. Many is the Opposite of few. 3. They lived mapy years ago. act or process of multiplying; the process of repeating or adding any given number or quantity a certain number of times; the Opposite of division 1. She had two multiplication exercises to do. 2. Multiplication is a quick way to combine equal groups. 3. Multiplication is the opposite of division. multiply number odd order pair 157 to make or become greater in number, degree, or extent; to find the product of by multiplication; to take a number or quantity 3 given number of times 1. Sally can multiply 27 times 16. 2. If you must add the same number several times it is easier to multiply. 3. Some peOple say sdrrows multiply as one grows old. a word or symbol used in counting; a total amount of units or things; a numeral with which a person or thing is marked; to count or enumerate. The figure 3 is called a number. Number your paper from one to ten. A large number of peOple were there. John bought a number of things at the store. Our telephone'nfimher is not in the phone book. \fi-l-‘UUFDH not divisible by 2 without leaving a remainder; not paired with another; left over; Opposite of even Three is an odd number. There are five-odd numbers between one and ten. Bob lost one shSE‘and then threw the Egg one away. It was odd that he did not know where to go. He was leaking for an Egg job. \fl-t’me the way one thing follows another; to arrange in a series; a rank or class The names were put in alphabetical order. The room was left in order. In which order do the books go on the shelf? The captain gave an order. John bought a money order at the post office. MOther telephoned the store to order groceries. C\\fi-F'LJOTUI—’ two things of a kind, similar in form; a single thing composed of 2 corresponding pieces; a set of two, usually of the same kind; couple; to unite or arrange in a set of two Dad bought a new pair of shoes. John has a pair of skates. The teacher asked the chidren to pair off to play the game. He bought a pair of tickets for the ball game. They make a nice pair. \J‘I-t" UUI'UH parts place plan plus P.M. 158 two or more of the portions, equal or unequal into which anything is divided; several quantities into which a whole may be divided One of the parts of the radio was missing. The children learned their parts well. The watch has moving parts. The garage man has a parts list. We looked for him in all parts of the building. \n-P‘UUI'DH the position of a figure in a number; numerals or figures occuring after a decimal point; to put or set in a particular rank, position, office; a particular point or part of space The figure 3 goes in ten's place. There is only one place after the decimal point. The books are in place on the shelf. John won first place. The people place confidence in the President. \J'I-C'UJNI-J a drawing or a diagram showing the arrangement of parts; to form a method for doing something; a representation drawn on a plane surface such as a map 1. He drew a plan of the house. 2. We will plan for the picnic at 2 O'clock. 3. They plan to have a large crowd. indicating addition or requiring to be added; Opposite of minus; an added quantity; something additional or extra . The plus sign tells us to add. Five plus five equals ten. His job required intelligence plus experience. One side of a dry cell is marked with a plus. The gift cost one dollar plus tax. \J‘I-F'UORDH afternoon; the time from noon to midnight 1. John has an appointment at 2 Egg. 2. Mary usually goes to bed before 10 P.M. 3. Mather said we could play ball in the §;M° h. P.M. is an abbreviation for post meridian. 159 point a unit of counting in scoring a game; a particular or definite position, place, or time; a degree, step or stage; the tapering end of anything; anything having a tapering end; a unit, as, a unit of counting in the scoring of a game; a place having definite position but no extent in space; a spot; a geographical place or situation; a locality as a good point from which to start; (a position or condition attained; a degree; step; stage; as, a point of elevation, boiling point, freezing point); a dot used in writing or printing; the exact time of occurring, as, the point of death. You score one point for each right answer. The freezing point of water is 32 degrees. We asked the man to point the way. The speaker made his point. The point of the pencil is dull. \J‘l-P'UURDH pound a unit of weight equal to 16 ounces; unit of money in .England worth about $2.80. Mother went to the store to buy a pound of coffee. The package weighed less than one ound. An English pound is worth about $2.80. Please do not pound on the table. When you run fast you can feel your heart pound. \n-F‘UONI-J price an amount for which something is sold; the cost to the buyer; value; worth; to set a value on; to compare the value The price of the shoes is $5.00. The grocer marked the price on the cans. That is too high a price to pay. The price tag tells how much it costs. We went to three stores to price rugs. \J‘I-F'UOI'DH ;problem a question proposed for solution; something to be worked out; a question, doubt, or matter of difficulty 1. We must add in the first problem. 2. John thought that the second problem was easy. 3. It is a problem getting to work on time. gproduct answer in multiplication; a number or quantity resulting from multiplying; amount, quantity or total produced The product of 3 x h is 12. The answer in a multiplication is called the product. The company makes a good product. Wheat is his largest selling product. 43"me prove purchases quarter quotient rate regular 160 to put to a test; to show to be true; to test the correctness of an Operation or result Bob was asked to prove his answer. Can you prove where—ybfi.were last night? The hunter went out to prove his new rifle. Did your new coat prove to be warm? ~F’UUIDH things obtained by paying a price, usually money 1. They went shOpping and made six purchases. 2. The purchases totaled more than ten dollars. one of four equal parts into which anything is divided; the fourth part of a measure of weight, length, time, value, etc.; twenty—five cents; one fourth of a dollar; to divide into h equal or nearly equal parts Sam ate a quarter of the pie for supper. He will arrive in a quarter of an hour. The book cost a Quarter. Mother will quarter the orange. Dad bought a quarter of beef. The soldiers will quarter there for the winter. Chm-F'UOI'DI-J the answer in a division problem; the number resulting from the division of one number by another 1. The answer in a division problem is called the 3222.129!- 2. The Quotient of 6 divided by 3 is 2. 3. We must learn where to place the first quotient figure. a quantity;amount, or degree of a thing measured in prOportion to something else; price; a ratio; proportion The Jet flew at a great rate of speed. Oranges sold at the rate of 69 cents a dozen. He must pay the regular rate. The rate of production was increased. His rate of pay is two dollars an hour. She is a first rate musician. (hm-POOR)!“ normal; standard; correct; formed, built, or arranged according to some established rule or custom; steady or even in size, spacing or speed 1. John learned the regular procedure to use in multiply- ing. . He keeps regular hours. Dad bought-fegfilar gasoline. The sergeant—Ts—lh the regular army. Mary is a regular bookworm. Ul-IT'WI'D remainder repeat right I‘OV ruler set 161 the part left over or remaining; the answer to a subtraction example; the number carried over in a division example which cannot be divided evenly 1. John subtracted four from six and had a remainder of two. ' 2. Seven divided by three leaves a remainder of one. 3. Some went by car, and the remainder went by buy. A. He put the remainder of the books on the shelf. to do again; to perform again; to cause to happen or appear again; a repetition; to say or utter again 1. In division we repeat the process to find each quotient figure. 2. In writing the number 111 we repeat the numeral one. 3. Please repeat what you said. h. The act5?§_gave a repeat performance. correct; suitable; prOper; a direction Opposite of left; straight, direct, not crooked John said he had the right answer. Turn to the right at the next corner. He came right away. . They had to right the boat after it capsized. . He set the box on the floor right side up. 0 \n-F’wIUH a straight line; a line of plants or houses; to arrange or place in a row or line; a series of things in a line Mary likes to sit in the front row. He planted the seeds in a row.'__- Bob likes to row the boat.—_— . We have 3 523—5? trees along our street. 43‘me a straight piece of wood, metal, etc., with a smooth edge used in drawing lines; measuring instrument usually marked to show inches 1. Harry used his ruler to draw a straight line. 2. We have an 18 iHEh—Tuler. 3. My dad has a ruler that is divided into centimeters instead of inches. h. The king was a kind ruler. a group of things or peOple belonging tOgether; things of the same kind; a collection of objects which comple- ment each other and usually go together; . We have a complete ESE of those books. Mother bought a £33 of dishes. My little brother has a §§p_of blocks. Mary will Egg the table for supper. . John broke his leg and went to the doctor to have it set. \J‘l-F’LDNH sign single speed subtract subtraction sum 162 a mark or symbol used to tell what Operation to perform with numbers; a mark or symbol used to mean or represent an idea, as a word, letter, or mark. 1. The plus sign means we should add. 2. The times sign looks like one of the letters of the alphabet. 3. The road sign was painted yellow. h. Sign your name on the bottom line. only one; one and no more, individual; separate 1. The children walked in single file. 2. The button hung by a single thread. 3. The batter hit a single to right field. h. He sleeps on a single bed. 5. A bachelor is a single man. act or state of moving rapidly; swiftness; rate of motion; to make haste; to go fast The boys ran at full speed. . The speed limit is 25 miles per hour. The car traveled at high speed. . The doctors were able to speed his recovery. . The boy told his dad to speed up. \J'IJT'UUI'DH to take away one number or quantity from another; to withdraw or take away; to remove 1. We subtract to find out how many are left. 2. If you subtract h from 10 you have 6 left. 3. You can subtract the cost from'what I owe you. act or process of subtracting one number or quantity from another; the Opposite of addition; finding the difference between two numbers or quantities 1. Nancy likes to work subtraction examples. 2. Subtraction is the Opposite of addition. 3. We have five subtraction examples to do. the answer in an addition problem; the whole amount; the total of two or more numbers added tOgether; to bring to a total The ppm of 6 and 5 is eleven. He paid a large ppp_of money for the car. Judy will ppm up the main parts of her report. The answer in an addition problem is called the ppm. 4:"me system table time times total 163 a plan or method for getting things done; a group of objects or parts combined to form a whole 1. Our system of numbers is based on ten. 2. He wETEEE'Out a system for doing the job better. 3. His digestive system is in good order. h. We believe in BET-system of government. a smooth flat surface or a thin plate, tablet, or board, on which an inscription, drawing, or the like, may be produced; any collection and arrangement in a condensed form, for ready reference, of such things as weights, measures, time, money, etc. 1. A table of weights and measures will tell how many pounds are in a ton. 2. Multiplication tables are listed at the end of the book. 3. Look in the table of contents. h. The conversation officer said the water table was low . the period during which an action, process, or condition continues; the interval between leaving and returning; the moment or hour for something to happen; an exact moment in a day, week, month, etc.; to determine the interval during which an action continues. 1. The time is now 11 O'clock. . We set our clocks according to Eastern Standard time. He will time the runner with a stop watch. It is time to eat. We had a good time at the party. . We can keep time with the music. [U O\\J‘l to.) a number of occasions or repeated actions; a multiplying of one number or thing by another number or thing Five times four is twenty. We have gone there many times. I heard my dad say that times are hard. He ran around the house seven times. When we multiply three by four we take three four times. \IIJTLUMH the whole sum or amount of anything; complete; a whole or sum of all the parts; to add, as, to total the amounts We can add the number to find the total. He will total the column of figures. The total number to attend the fair was 15000. It was a total failure. There was a total eclipse of the sun. 0 \n-VUJNH upper value weight whole yard year 16h being farther up; higher in place, position, rank, etc.; superior; directed upward 1. In subtraction the upper number is larger. 2. The man cut his uppE?_lip. 3. Bob slept in an upper berth. h. They gained the upper hand. 5. The top of a shoe is called an upper. a fair return in money for something exchanged; that which is thought to be equivalent in worth; to rate highly; to estimate the worth of . The value of a dollar is less than it used to be. . The watch is of great value. . I value our friendship. . Milk has food value. . The store advertised the value of its merchandise. \n-fi'WNl-J how heavy a thing is; something heavy; a quantity or thing weighing a fixed, usually specified amount; to load or make heavy l. The weigpt of the dog is fifty pounds. 2. He put a five pound weight on the scales. 3. The teacher has a paper weight on her desk. h. His argument carried no weight. 5. The weight of the snow caused the roof to break. the entire thing; all of something; total; the sum of its parts; not divided into smaller parts Four is a whole number. John ate the whole pie. Dad worked the whole day. . Their whole attention was on their work. . He finished the whole thing in one hour. \fl-F'UUI'DH a measure of length equal to 36 inches or 3 feet; a unit of measure; a measuring stick A yard is equal to 36 inches. The table is just one yard wide. His dad works at the railroad yard. . The children are playing in the school yard. . The sailor crawled out the yard to fix the sail. \n-F’WNH a period of time equal to 365%-days or 12 months; the period of time between one birthday and the next; the period of time in which the earth completes a revolution around the sun There are 12 months in one year. The earth revolves around the sun once each year. 3. The weather is nice for this time of year. A school year is about ten months long. NH zero decimal point 165 a number meaning not any, or nothing; the starting point on a scale, such as a thermometer; the lowest point . The zero in the number 20 means no ones. . The temperature was 10 degrees below zero. . The game ended with a zero score. . The rocket took off at zero hour. . We will have zero weather this winter. \n-fi'UOIUI-J the plus sign; a sign used in arithmetic which means we should add; a sign used to mean positive, or the Opposite of negative. Sometimes we can substitute the word “and" for the plus sign in arithmetic. the minus sign; a sign used in arithmetic which tells us to subtract; a sign used to mean negative, or the Opposite of positive. Sometimes we can substitute the words "take away" for the minus sign in arithmetic. the times sign; a sign used in arithmetic which means we should multiply. a frame used in division to separate the dividend from the divisor and the quotient. In 2 , the dividend is written inside the frame, the divisor to the left of the frame, and the quotient above the frame. the equal sign; a sign used in arithmetic to tell that the amount written to the left is equivalent to or the same as the amount written to the right of the equal sign. 1. h 1. Examples: 2 ¥ 2 5 - 1 h a dot, or period, placed at the left of a decimal fraction; a dot, or period, placed between one's place and tenth's place to show the dividing point between the places in the whole number and the decimal places; a dot or period used to separate dollars from cents in money numbers. 1. In.writing $2.25 the decimal point separates dollars from cents. 2. The decimal point is placed to the right of one's place. place-holder place value ten's place three-place number total amount ETOUP 166 a digit or numeral used in a place to keep other digits or numerals in their proper place; zero is used as a place holder in 203 to mean that there are no tens and so that the 2 can be written in hundred's place to mean two hundreds. 1. Zero is the numeral used most Often as a place holder. 2. A zero in a multiplier is a place holder. 3. The zeros in #009 are place holders. the position that a numeral occupies in a number determines the value of that numeral. Example: A numeral in ten's place has ten times the value of the same numeral in one's place. 1. We used a place value chart in arithmetic. 2. Bob used bundles of ten sticks each to show that he understood place value. the place to the left of one's place and to the right of hundred's place; the second place to the left of the decimal point Example: In the number #6, the h is written in ten's place. 1. Any of the ten numerals may be used in ten's place. 2. In the number 32, the 3 is written in tenrstlace. a number having 3 digits or numerals, one each in one's place, ten's place and hundred's place. A number having a numeral in hundred's place, a numeral in ten's place, and a numeral in one's place. 1. The number 100 is the smallest three-place number. 2. The number 999 is the largest three-place number. 3. 65% is the largest three-place number that can be written using the numeralsifi, 5, and 6. all of anything; the sum of all the quantitites, amounts, or numbers 1. John found the total amount by adding all the numbers together. 2. Bob saved one half the total amount of money he earned last summer. to arrange or combine things together; a number of persons or things brought together 1. A group of boys were playing marbles. 2. We can group the children according to age. a. All of those children belong in the same group. . John was group leader in the air force. Arithmetic charge divisor error expenses fraction 167 Chapter 2 the art or science of adding, subtracting, multiplying and dividing with figures; a book containing the principles of this science 1. Bob said that arithmetic is working with numbers. 2. He has a new arithmetic book. 3. We have arithmetic class in the morning. the price demanded for a thing or service; to put down as a debt to be paid; an exPense . The charge for renting the boat is fifty cents. The repair charge on the watch was low. Mbther opened a charge account. Bob's mother left him in charg . The hunter put a charge in the gun. The captain ordered his men to charge. O\\J'I-¥='LJONI-' a number by which the dividend is divided; a number that divides another number without a remainder; a number or quantity by which another is divided 2 1. In thI h is the divisor. 2. In 6 g 3 = 2, 3 is the divisor. 3. The denominator of a fraction is a divisor. a mistake; something that is wrong; condition of being wrong or mistaken; an act involving a departure from truth or accuracy; a deviation from a standard ‘Mary found the error she made in adding. Bob made only one error in spelling. The first baseman made an error in the third inning. The carpenter made an error in sawing a board. 4:”me acts of expending; disbursements; outlays, costs or money paid out At the end of the trip Mary totaled their expenses. . The salesman turned in a list of expenses. . The expenses for repairs were high. . The eXpenses in time and energy were great. «F‘UUI'DH one or more of the equal part of a whole; a part or piece broken off; the indicated quotient of one number divided by another number l. g is a fraction. 2. He spent only a fraction of his time reading. 3. Only a fraction of the peOple were there. h. The fraction 273 is less than one. graph higher increase length 168 a line or a diagram showing relationships between quantities and things; any line or lines representing a series of relations; to indicate by means of a graph 1. A graph.was used to show the rise and fall of the temperature. 2. We can graph the height and weight of the boys and girls. 3. A graph in the newspaper showed how tax money is spent. h. John drew a graph to show how many newspapers he sold each week last year. taller; greater; reaching farther above the ground or floor, a place more elevated; a superior region; of relatively greater degree, size, amount . He climbed hi er than he had ever climbed before. Mary can sign a higher note than Sue. . Oranges are higher in price this week. . Put the dish up higher so the baby cannot get it. . A general has higher rank than a captain. \n-F-‘wNH to make or become greater in size, quantity, number, degree, value, intensity, power; to grow in number; advance; growth; act of increasing, as addition or enlargement in size, extent, quantity, number, value . Mike asked his dad to increase his allowance. . The deer herds are on the increase. . Cars will increase in price next year. . You can increase your skill by practicing. . Production will increase with the new machine. . The flowers will increase every year. O\\fl4F’U)I\)|-‘ the longest or longer dimension of an object; the distance from end to end; duration; extent in time, number, or quantity, as, an hour's length; quality or state of being long; extent, or duration, as a sermon's length; a distance equivalent to the extent of a thing; a portion of space or time, as a length of years 1. The lengph of a room is the longest way it can be measured. The piece of material is 3 yards in lengph. He bought a lengph of pipe. He spoke at lepgph about the weather. The lengph of the storm spoiled our vacation. \J'I-P‘UOI'D measure members mile minute partial 169 to find the size or amount of anything; an instrument for measuring; a unit of measurement; a due or given extent, degree, or quantity; to find the extent, degree, quantity, dimensions or capacity of, by comparing with a standard . We can measure the room with a yardstick. . Dad bought a tape measure. . Mother has a quart measure. . What is your wait measure? . Measure off three feet on the board. \nF‘UOIUH parts of a whole; elements of which a group or whole is composed; individuals who belong to a group; parts of an animal, as legs or arms 1. All the members of the group were there. 2. The church has over 500 members. 3. Our club has six new members. h. The arms and legs are sometimes called members of the body. "“"" a measure of distance; 1760 yards; 5280 feet; 320 rods 1. It is a mile from here to the lake. 2. It takes about twenty minutes to walk a mile. 3. There are 63,360 inches in a mile. h. An automobile can travel over a mile a minute. 5. A rocket can travel over a mile a second. sixty seconds; a measure of time; the sixtieth part of a unit especially of an hour or a degree; the distance one can travel in a minute, as, he was five minutes from school; a point or short space of time; a moment, as he arrived that very minute 1. A minute is equal to sixty seconds. 2. It takes only a minute to drive to school. . I will be there in a minute. . He arrived that very minute. . Bob yelled, “Just a minute!" . A minute is l/60 of a'degree. O\\fl #1» a part only; not total or entire; not complete; forming a part 1. Dad made a pgrtial payment on the car. 2. We add the partial product to find the total product. 3. Sometimes he seems partial to girls. h. Sam is partial to ice cream. 170 period one of several sets of figures usually marked off by commas placed at regular intervals; a portion of time marked off by events; length of existence; duration; a point, portion, or division of time; an end; conclusion; termination . In the number 196,271, which period contains a 9? He was in the army for a period of 10 years. . He visited us for a short period. . The Civil War was a period—ET—strife. . Put a period at the end of a sentence. \J'I-h‘UOMI-J plane a surface; a flat or level surface; without elevation or depressions; even; level; a level or stage; to make smooth or even 1. The desk top is a plane surface. 2. A carpenter uses a plane to smooth wood. 3. The plane flew at KOO—miles an hour. h. His work was on a high plane. profit the gain usually in money after all the expenses are subtracted from the total amount received; gain; advantage; excess of returns over expenditure; excess of income over expenditure; to take advantage; to make good use; to gain 1. John can earn 10¢ profit on every boy of candy he sells. 2 His business showed a large profit. 3. We can profit from our mistakes. h. 'We do not profit from worrying. quantity an amount or portion; a Specified or indefinite number of persons or things; great or considerable amount 1. The farmer sold a quantity of hay. 2. John won a large quantity of marbles. 3. The library received a Quantity of new books. h. We have a large quantity of drawing paper in the cupboard. scale a balance; an instrument for weighing things; a series of steps or degrees; a measuring instrument marked with lines at regular intervals; the size of a plan or map compared with what it represents The map is drawn on a scale of one inch to the mile. A balance scale is used to weigh chemicals. The scale of wages is from ten to twenty dollars a day. Did you ever scale a fish? She practiced a scale on the piano. He can scale the wall quickly. O\U1-F-"UO|\)l-‘ second side small ton unit 171 sixtieth part of a minute; a moment; an instant; immediately following the first; next to the first in order of place or time; occuring again; another; other; of the same kind as another; next to the first in value, power, dignity or degree, hence, subordinate; inferior; next to the best 1. A jet plane has flown 15 miles a second. 2. He came in second in the race. 3. I second the—matibn. h. Just a second and I will be ready. 5. The Jones family bought a second hand car. 6. My watch has a second hand. a border or edge of an object; the right or left part of the body; a part of an object that is not an end, tOp, or bottom; a part located in a particular direction from center, as, one side of a room; the position of a person regarded as Opposed to another person, as, choose up sides for soft ball . He sits on the other side of the room. John fell down and hurt his side. . Bob played ball on the winning side. . He rowed the boat to the other side of the lake. . Please use the side door of the house. . Write on just one €299 of your paper. OUT-F‘UOIDH having little size compared.with other things of the same kind; not large; little quantity, amount, value, duration etc.; short, brief, low; limited or slight in degree, intensity or sc0pe; of little consequence; trivial; l. A penny is a small amount of money. 2. The fish was too small to keep. 3. This is only a small matter. A. 5 John was hit in the small of the back. . PeOple often engage in small talk. a measure of weight; 2000 pounds in the United States and Canada; 22h0 pounds in England. 1. Dad bought a ton of coal last week. 2. Our car weighs nearly a ton and a half. 3. An English ton weighs more than an American ton. the smallest whole number; one; a single thing regarded as an undivided whole; a single thing or person; any group considered as one; a quantity adopted as a standard of measurement. width liquid measure unit of measure 172 A foot is a unit of length. The figure one is sometimes called a unit. In a unit fraction the numerator is one. The soldier must return to his unit. There was only one unit left at the motel when we arrived. . The heating unit on the stove is burned out. 0\ \n-lr'me the dimension of an object measured across from side to side, or in a direction at right angles to the length; distance across; how wide a thing is 1. Our room is 25 feet in width. 2. The width is shorter than the length. 3. He swam the width of the pool. h. You can buy that material in a 36 inch width. a small circle placed at the upper right of a number to indicate degrees of temperature or the degrees in a circle ' 1. Yesterday the temperature reached 6h0. 2. Water boils at 212° and freezes at 32°. 3. The ship captain ordered that the ship alter its course 20 south. a unit or system of units for measuring liquids; a container of specified or standard size used in measuring liquids l. The pint, quart, and gallon are liquid measures. 2. A liquid measure is not used to determine quantities of coal. 3. John used a liquid measure to find out how many quarts are in a gallon. a specified or standard amount or quantity of weight, time, value, distance, degree, volume, or liquid against which other amounts or quantities are compared or measured. 1. The inch is a smaller unit of measure than the foot. 2. A unit of measure used in measuring time is the hour. 3. The inch, foot, and yard are units of measure used in measuring distance. h. We need both large and small units of measure. age borrowed columns combined complete 173 Chapter 3 a time of life, length of life; one of the stages of life; a period of time; to become old; to grow older . Mary is 10 years of age. We live in the space—age. We can tell the age Of—a tree by counting the rings. Uncle Harry lived—to an old age. She has aged greatly since I‘last saw her. \fl-F’LAJMF" having taken one from the next higher place in a number in order to exchange it for ten units of the next lower and add those units to the next lower; gotten from another with the understanding that it must be returned 1. He borrowed a ten and exchanged it for 10 ones. 2. John's work showed that he had borrowed correctly. 3. He borrowed a book from the library. upright lines of figures or other symbols; upright bodies or masses, as, columns of air; a kind of supporting pillar Sue added four columns of numbers. Several columns of soldiers marched in the parade. The newspaper has 8 columns on a page. Columns of smoke rose lazily into the air. . We have four columns on our porch. \Jl-F’LAJIUH joined together; united or joined; linked closely together; united; formed by combination 1. Two small groups were combined to form one large group. ' 2. The ingredients for a cake must be combined in the right order. . Their combined efforts were needed to do a good job. Mrs. Harris teachers a combined first and second grade. All the children in the school combined would not fill the new auditorium. \J'l-F‘w filled up; with all the parts; entire; perfect; bring to an end, or to a final or intended condition; to bring to a state of entirety; to fulfill; finish Complete each example before you start the next one. We bought a complete set of dishes. It came as a complete surprise. The workmen will complete the work on time. #‘UOMH double and form greatest highest 17h twofold; multiplied by 2; made twice as large, as great, as much, as many; a number or amount twice as large; to increase by adding an equal number, quantity, length, value; to multiply by 2; to make twice as great; a turn or circuit in running to escape When you double a number you have twice as much. He walked thru the double doors. He will double back to escape his pursuers. He received double pay for working overtime. Many words have a double meaning. \n-lr'me a limit or boundary of any area; the part where a thing begins or stOps; the final place, point, or position; that which is left; to complete; to bring to a con- clusion He stood at the gpg_of the line. Our vacation will epd_on Labor Day. Please gpd_the argument. Measure from the_§pd of your ruler. Their house is at the_gpd of the road. xii-rooms: an image or likeness; shape or structure of anything; a manner or method; conduct regulated by custom; manner of performing or accomplishing something; to take or give shape; to construct; make; fashion; produce We learned a new form for working division examples. They watched the building take fgpm. Please form a circle on the playground. Ice is a form of water. His swimming form is very good. He filled out the form in ink. O\\fl-¥=‘WNI-' biggest; largest in size, importance, number; largest in spatial dimensions; of much size; numerous; highest in rank 1. Mary lives the greatest distance from school. 2. He is the greatest singer I have ever heard. 3. The greatest number of peOple were there on Saturday. tallest; with greatest altitude; farthest above the ground; with units of measurement having elevation or altitude, as, the highest note in music; chief; main, principal, as, to be known in highest circles The highest temperature we had was 98 degrees. . We climbed the biggest mountain. The price of strawberries is highest in winter. He was elected to the highest Office. . He put the globe on the_h1gh§§§ shelf. U1 43‘me loan multiplier result square on the average 175 to lend; the act of lending; to let another person have or use for a time with the understanding that it will be returned; anything that is lent, especially money 1. Mr. Jones went to the bank to get a loan 2. Bob said he would loan his bicycle to Jim. the number by which another number is to be multiplied; a thing that multiplies 1. In 3 x A, three is the multiplier. 2. The multiplier may be any number. 3. John bought a plastic multiplier at the dime store. an answer to a problem; to proceed or follow as a consequence; effect or conclusion; to terminate; to end; something achieved, obtained, brought about by calculation or investigation What result did you get for exercise four? The scientist reported the result of his experiment. The result of his fall was a broken arm. His accident was the result of ' careless driving. The farmer lost all his crop as a result of the storm. \I‘IFLAJMH a figure with h equal sides and h right angles; multiply a number by itself; to bring approximately to a right angle, as, to square ones' shoulders; even; leaving no balance, as, to square one's account John drew a square on his paper. Nine is the square of three. He ate three square meals every day. They often sit in the village square. He was told to square his shoulders. . Mom and dad.went to a square dance. OUT-POOR)?“ the divide sign; a sign used in arithmetic which tells us to divide. Sometimes we can substitute the words "divided by” for the divide sign. Example: 6:2 : 3 may be read six divided by two equals three. occurring or happening in an amount or rate equal to the average 1. John sold 35 newspapers a day on the average. 2. The airplane flew 600 miles per hour on the avepgge. 3. Mr. Stone earned two dollars an hour on the averagg. accurately ADM. arrange circle denominator 176 Chapter h free from failure or error; exactly; correctly; in exact or careful conformity to truth 1. Mary worked the examples accurately. 2. Dad measured the board accurately. 3. We can measure more accurately if our unit of measure is smaller. before noon; the time from midnight to noon. 1. School begins at 8:30 a. . 2. It is very dark at 3 a. . 3. Dad gets up at 5 a.m. so he can be to work at 7. to put in proper order; to adjust or settle, or, to arrange a program; to come to an agreement, understanding or settlement, as, arrange transportation or arrange an appointment; plan 1. Mary will arrgpge the numbers in order from largest to smallest. ‘We can arrapge the room the way we want it. . I will arrapge an appointment for you. His job is to arrange music for TV programs. Can you arrange to be there at noon? \n-SI‘UJN a curved line that is closed, every point of which is equally distant from a point within called the center; a flat figure that is round; a ring; something having a circular form; to encompass by a circle; to surround; to enclose . He drew a circle that was four inches across. . The children sat in a circle. . A jet plane can circle the earth in a short time. . They have a small circle of friends. ' . Dad will circle the shrubs with small fences. \n-t’wml-J The number below the line in a fraction; the number in a fraction that tells the size of the parts; the divisor, or that part of any expression under a fractional form which is situated below the horizontal line signifying division; in simple fractions it denominates or names into how many equal parts the unit is to be divided 1. In 2/3, three is the denominator. 2. The denominator tells the size of the parts into which something is divided. 3. The denominator is the number below the line in a fraction. digit edge face fact fractional 177 any number under ten; any of the 10 figures or symbols 0,132,391}, 5,6,798)9 1. John wrote a six digit number on the board. 2. In #68, the middle digit is 6. the part that is farthest from the middle; the border or part adjacent to the line of division; to furnish with a border He sat on the edge of his chair. They walked to the edge of the cliff. The scientist is on the edge of a discovery. That man has been on edge all day. Bob will edge the yard after school. The knife has a sharp edge. O\\J"|4=’UOI\)|-‘ the surface or most important surface of anything; a front surface; a working surface; the side or edge presented to view; the front part; the right siie The face of the mountain was covered with ice. There was only one like it on the face of the earth. John had candy on his face. You must face the east to watch the sun rise. The face of the clock had no numerals on it. \n-fi‘me a thing known to be true or to have happened; an expression in numerals or number symbols recognized as being true; a true statement . 'Sue had difficulty with a multiplication fact. The children learned the division facts. It is a fact that the sun rises in the east. The man said, "As a matter of fact, I saw him last night." -¥="L.\)l'\)l--l of or pertaining to fractions or a fraction; a small part of a whole thing; relatively small; inconsiderable; insignificant 1. A foot is a fractional part of a yard. 2. The box of arithmetic materials contains fractional pieces. 3. How many fractional pieces size 1/3 are needed to make one whole? incomplete line minus numerals numerator 178 not finished; not having all the parts; not complete; undone; not filled up; imperfect; defective . Three of the problems were incomplete. The shipment from the factory was incomplete. . The new house will be incomplete until it is painted. . The quarterback threw an incomplete pass. . John's report was incomplete. \n-p'UUNI-J a long narrow mark; anything that is like a long narrow mark; an edge or boundary; a cord for measuring Bob drew a straight line with his ruler. The boys formed in a line along the windows. We crossed the boundary line between the United States and Canada. He bought a new fish line. Our clothes line broke. . He will line the box with cloth. 0 Own-3r me indicating subtraction; negative; not positive; less; lacking; without; wanting; having lost 1. Ten minus seven leaves three. 2. Paul did not know what a minus sign looked like. 3. He is minus his coat today. h. The lowest temperature last winter was a minus ten degrees. 5. Mary wanted to know what the minus sign meant on a dry cell. expressing, denoting, or representing number, as numeral letters, words, or characters; figures or characters used to express a number; words or figures expressing a number Roman numerals are sometimes used on clocks. Number words, such as ten, are numerals. Our numbers are written with Aratic numerals. Both Roman and Arabic numerals are used in outlining. #‘UOI’DH The number above the line in a fraction; the term in a fraction'which indicates the number of fractional units taken; one who or that which numbers 1. In 3/8, 3 is the numerator. 2. The numerator is the numher above the line in a fraction. 3. The numerator tells how many parts of a particular size are taken. h. A person who counts things is sometimes called a numerator. reduce reduction reference selling 179 to change the form of numbers or quantities without changing the value, as, to reduce fraction; to change the denomination without changing the value, as, to reduce dollars to cents, hours to minutes; decrease; make smaller; to diminish especially in bulk, amount or extent, as, to reduce expenses; diminish; lower . We do not always have to reduce a fraction. If you reduce 1 hour to seconds, you have 60 seconds. . Our salesmen must reduce expenses. . She is trying to reduce her weight. . The store is having a sale to reduce the stock. . Cruel words will reduce her to tears. (hm-twink! change of form, as the reduction of fractions; the act of reducing; the amount or quantity by which a thing is reduced The reduction of 2/h to 5 is easy. In the reduction of fractions we change halves to fourths as well as fourths to halves. 3. There was a reduction in taxes last year. 4. A reduction was made in the price. NH act of referring, or state of being referred, as, a reference was made to a chart; statement referred to; something used for information or help; that which refers Or alludes to something; one who or that which is referred to or consulted; a written recommendation 1. There is a reference table on page 369 of our arithmetic book. . He sent me to the reference desk in the library. The man said he would.write a reference for me. Tom gave his principal as a reference. An atlas is a reference book. . He made reference to the talk he had heard. C\\fl-F"UUI\) act of one who sells; sale; engaged in selling; finding, offering, or making, a sale at a specified figure The selling price is marked on the price tag. The selling price includes the cost and the profit. The man said he was in the selling business. Apples are selling three for a dime. The sellipg price was $5.98. 0 \fl-F’LJUNH short straight terms like fractions mixed number 180 not long from end to end; of brief length; not tall; not a great distance; not extended in time; not coming up to measure, as, is short supply; in a short manner, abruptly, curtly It is only a short distance to school. A short man walked by the door. . Mbther read us a short story. . He stOpped short in his tracks. . The man said he was short of money. \n-F'UUI'DH not crooked or bent; direct; uninterrupted; unbroken; having an invariable direction; lying evenly through out its extent; not leaning, bending, inclining, or the like Can you draw a straight line without a ruler? . Mother told them {0 come straight home. . He sat in a straigpt back chair. . Come straight to the point! kWMH the numerator and denominator of a fraction; the parts of a fraction; that which limits the extent of any- thing; limit; bound; a definite extent of time; mutual relationship; words or groups of words used in connection with a special subject 1. The terms of a fraction are the numerator and the denominator. 2. In 3/h, and 3 and the h are called the terms of that fraction. 3. The President served two terms. h. We are on good terms with our neighbor. 5. What are the terms of the agreement? fractions having the same denominator indicating equal sized parts and making it possible for them to be added and subtracted 1. 1/6, h/6, and 2/6 are like fractions. 2. Like fractions have the same denominator. 3. like fractions may be added or subtracted while unlike fractions cannot. a whole number and a fraction together, usually written 1. A mixed number is a whole number and a fraction written together. 2. 3 2/3 is a mixed number. 3. Bill learned how to subtract a mixed number from a whole number. 181 whole number a number representing all of a thing or things with no fractional part remaining; an integer; not a fraction or mixed number 1. A number like 7 is a whole number. 2. It is easier to divide with whole numbers than with mixed numbers. balance cash exact encircle measurement 182 Chapter 5 remainder; an instrument for weighing; a weight used as a counter balance; a combination of factors or elements, as, in a diet; the excess on either side, or the differ- ence between two sides; to equal in number, weight, force 1. After the down payment we owed a balance of $20.00. 2. A balance is a kind of scales used for weighing things. 3. The coach put John on the other team to balance the sides. . h. The doctor told him to balance his diet. 5. The man's life hung in the balance. a sum of money; money in coins; to exchange for money, as, to cash a check 1. The storekeeper took in ten dollars in cash today. 2. You may cash your check at the bank. . 3. He always pays for things with money, so we call him a cash customer. h. The storekeeper put the money in a cash register. accurate; without any error; undeviating; strict; complete, not merely approximate; to require; to compel to yield or furnish 1. Some answers in arithmetic do not have to be exact. . He had the exact amount of change. . A scientist must be exact in his work. . He did not remember the exact date. . Measurements are never exact, only approximate. . The king‘will exact obedience from his subjects. O\\Jl ~F’UUI'D to enclose within a circle or ring; to surround; to form a circle about; to pass completely around 1. They put up a fence to encircle the playground. 2. The life preserver was designed to encircle the body. 3. The teacher told us to encircle the carry number. k. The new satellite will encircle the earth in one hour. act or result of measuring; the extent, size, capacity, amount or quantity determined by measuring . Degrees give us a measurement of temperature. The carpenter made an incorrect measurement. His waist measurement is 27 inches. The measgggment of distance in outer space is done with light years. #‘UOIUH meters outside payment ream rent 183 measures of length a little longer than a yard stick; measures of length equal to 39.37 inches or about 39% inches; things or machines that measure 1. A man 2 meters tall would be over six feet tall. 2. The man ran 100 meters in 10.5 seconds. 3. The gas company installed seven meters today. h. What are the dimensions of your school room in meters? - the surface or part that is out; the external part; space that is beyond or not inside; beyond the limits; as, outside the sc0pe of his knowledge; the most, or the furthest limit, as, it may last a week at the outside They live Just outside of town. Someday we may travel outside the earth's atmosphere. Dad will paint the outside of the house next summer. . The storm may last three—days at the outside. 4?me an amount paid; pay; that which is given to discharge a debt 1. Mr. Jones made a $10 payment on his hill. 2. The first payment is due next month. 3. The man accepted a TV set as payment for the work he had done. h. His gratitude was payment enough. a quantity of paper, usually 20 quires or #80 sheets (usually about 500 sheets); to widen the opening of a hole, as to ream a piece of pipe 1. You can buy that paper only by the ream. 2. A package of paper containing 500 sheets is called a ream. 3. The plumber must ream.the end of the pipe to make it smooth. a tear in cloth; to rend or tear; a schism; a rupture; a split; to let or hire out for pay usually money; an amount of money paid for the use of something; pay; reward; share . Mr. Smith pays his rent once a month. The sign in the window _said 'apartment for rent! Dad wants to rent a boat and go fishing. The man found a rent in his coat. 4?me serial' sheet balance due down payment half-fare monthly payment price list 18h arranged in a series or row; occurring in regular succession, appearing in successive parts 1. Bob knew the serial number on his bicycle. 2. The stock was—arranged on the shelf in serial order. 3. Mother was reading a serial in a magazine. h. A serial is being shown at the movie theater. single piece of paper; a broad stretch or surface . John wanted a sheet of paper without lines. He tore a hole in the bed sheet. Sailors refer to a sail as a sheet. . A sheet of ice covered the side walk. . My—dad’works for a sheet metal company. via-cum»- payments, usually money, remaining owed after a partial payment has been made 1. The amount left to be paid after the down payment is the balance due. 2. After each payment the balance due is less. 3. The larger the down payment the smaller the balance due. an amount paid which is less than the purchase price leaving an unpaid balance or balance due 1. John's dad made a $100.00 down payment. 2. The size of a down payment often depends on the price of what is bought. 3. Bob made a down payment on a new bicycle. one half the amount of money one usually pays to ride on a bus, train, airplane, etc. 1. Children under 12 can ride for half-fare. 2. Nary only paid half—fare to fly to Detroit. a payment made once each month on the unpaid balance, usually equal in amount from month to month 1. The monthlypayment on Bob's bicycle is $5.00. 2. The department store started a monthly payment plan. a list showing the prices of things to buy 1. Bob looked at a price list to find out how much a light would cost for his bicycle. 2. The cost of repairs are given in a price list. 3. The storekeeper sent a price list to his customers. record of expenses round-trip time line zero as a place holder 185 a written account of amounts of money spent; a list of the amount of money spent for things, such as on a vacation 1. 0n the trip Sue kept a record of expenses for her father. 2. Mr. Smith looked at his record of expenses to see how much money he had spent for gas. a trip to a place or places and back again; a trip to a destination and return to the starting point 1, Hus. anes bought a round-trip ticket to New York. 2. The round-trip_fare for children is 25 cents. 3. we made a round-trip to Mackinac Island on a ferry boat. a graduated line along which the dates or occurrences of important events or happenings are written in sequence and prOportionately according to the graduations l. The children drew a time line on the chalkboard of important events in arithmetic. 2. A time line in history helps us to know when people lived. 3. A time line will show the deve10pment of important inventions. the number symbol zero may be used to hold a place in a number Open to show that there are no groups of that size 1. In the number 206, zero is used as a place holder. 2. Zero is used as a place holder, as a starting point, and in scoring a game. ‘ matches rectangle Shape space . triangle 186 Chapter 6 equals; things equal to or similar to others; exact counter parts; those which are exactly like others; to make equal, prOportionate or suitable . Bob exclaimed, "my bicycle matches yours!" The color of her dress matches the color of her eyes. The man used safety matches to light the fire. . John likes to watch Wing matches. ¢?UJn)F4 a figure whose Opposite sides are equal and.which has «1 square corners ‘ 1. Mary drew a rectangle that was 6 inches long and p h inches wide. ¥ 2. John said, "A square is a rectangle but a rectangle may not be a square." i 3. The doorway is a rectangle. . form; mold; model; to form or create; to cut out; to bring about; appearance; condition 1. The shape of a triangle is different from that of a quEFET A new moon is in the shape of a crescent. The artist's picture Began to take shape. His old hat had lost its shape. The man said the used car was in good shap . via-win unlimited room or place extending in all directions; an empty place; quantity of time, duration; a short time; to place at intervals; to arrange with spaces between; a part marked off or bounded in some way; distance, area, or volume [.4 Write your name in the space at the top of the paper. .A carpenter needs plenty of space in which to work. The farmer will space the fence posts evenly. The rocket will travel in outer space. He laid 100 bricks in the space of an hour. \fl-F'UUR) a figure formed with three lines; a set of three (people); an object or thing with the shape of a triangle 1. The sides of a triaagle do not have to be the same length. 2. The playground is in the shape of a triangle. 3. Jim plays the triaagle in the school orchestra. 187 imprOper fraction a fraction in which the numerator is equal to, or larger than, the denominator; a fraction which is equivalent to one, or is more than one. Example: _2_ is an improper fraction. _5_ is an imprOper fraction. 3 1. An imprOper fraction is equal to one or more than one. 2. _3_ is an improper fraction. [0 I'D almost annual approximate area degree diameter 188 Chapter 7 nearly; in large part; all but; a little short of 1. Peter was almost late for school. 2. we have saved almost enough money to buy the present. 3. It was almost time for the game to start. occurring once each year; yearly; lasting only one year or one growing season; performed or accomplished in a year 1. The annual school picnic is held in May. 2. Mother planted an annual in her flower garden. 3. An annual report is one that is written once a year. h. The county fair is an annual event. nearly correct or exact; not perfectly accurate; to come near to; to approach; approaching 1. An approximate answer is one that is not quite correct. 2. What is the approximate number of books you will need? 3. The Jet approximated the speed of sound. amount of surface; an Open space in a building; extent; range; a particular extent on the earth's surface . The area of the room is 750 square feet. The workmen filled in the area with dirt. . Mr. Smith parked his car in area G. . The children play ball in the fenced in area. The desert is in the western area of the state. \n-P'UUMH a unit for measuring temperature; the opening of an angle, or the arc of a circle; amount; extent; a step or station in any series; a rank or grade . The temperature has gone up 10 degrees. . There are 60 minutes in one degree of a circle. . I agree with him to a large degree. . He had first degree burns on his hand. . My brother has a college degpee. \n-fi'UJIUH the length of a straight line thru the center of a circle or object; thickness, as, the diameter of a tree 1. The diameter of the tree is 2h inches. 2. The diameter of a dime is smaller than that of a nickel. dimensions estimate normal perimeter 189 3. The teacher asked the children to stand in a circle whose diameter was ten feet. DO you know how to measure the diameter Of a tire? measurements; measure in length and width, or, in length, width, and thickness or depth; size or extent 1. The dimensions Of the room are 2h x 30. 2. The dimensions of an Object are its width, length, and’HEpth. 3. What are the dimensions Of a ball? h. The length Of the hall is one of its dimensions. 5. The president discussed some Of the dimensions. aJudgment or Opinion as to how much, how many, etc.; to form a Judgment or Opinion; to fix the worth, value, size, or extent of; to calculate approximately; tO Judge; a statement Of the amount for which certain work will be done by one who undertakes to do it 1. The teacher taught the students how to estimate answers in arithmetic. 2. Mr. Brown said he would estimate the cost of the trip to be about $20.00.‘””“" 3. His estimate of the length of the train was a half mile. h. The carpenter gave us an estimate of the cost Of a garage. the ordinary or usual condition, degree, quantity or the like; of the usual standard; regular; not abnormal; natural; the usual state or level; Occurring naturally The normal temperature Of the body is 98.6 degrees. . His weight is two pounds above normal. . He is back to normal after his illness. . He followed the normal procedure in doing his Job. 43‘me the distance around the outer edge Of something; the whole Outer boundary of a surface or figure, or the measure of the same 1. The perimeter Of the playground is 1000 feet. 2. The perimeter of a rectangle is equal to two times the length plus two times the width. 3. The perimeter Of a circle is the circumference. h. Mr. Smith had perimeter heating installed. 5. The lieutenant formed his men in a perimeter defense. principal rise round section 190 a sum of money on which interest is paid; highest in rank, authority, or importance; chief; leading; out- standing; important 1. A sum Of money on which interest is paid is called the principal. 2. Mr. Holmes is the principal at our school. 3. Detroit is the principal city Of Nfichigan. moving upward; go up; come up; tO move from a lower place to a higher place; to increase, as in volume, price, degree, intensity; a going up; an increase; to become Of higher value; an upward lepe The temperature will rise when the sun comes up. There has been a rise in prices since the war. The mountains rise in the distance. They built their house on a rise. . The sun will rise at 6:00 a.m. \fl-F'UUMH shaped like a ball; circular; to make circular; having a curved outline or form; full; complete, as, a round trip; not fractional; approximately in even units, as, round numbers; the distance around 1. The class made a round trip to Detroit in one day. 2. Today we learned How tO round Off numbers. 3. They walked round the town window shOpping. h» We learned a round in music class called "Three Blind Mice." '5 5. The boxing match was in the fourth round. to cut or separate into parts; a part cut Off; a part separated; a division; portion; slice; 3 distinct part of a city, country or peOple; a portion Of land equal to a square mile or 6h0 acres 1. Mother cut the pie into 6 equal sections. 2. The farmer said that a section Of land was equal to 640 acres. 3. The drawing was a cross section Of a tree. h. We were asked to read section four. 5. Our town has a fine business section. a small line or mark placed at the upper right Of a number to indicate a measure Of length in feet. Examples: h'; 23'; 1000’; these are read as: four feet; twenty-three feet; one thousand feet. 1. Our room measures 36' long and 25' wide. 2. The playground measures 200' x 300'. bar graph draw to scale exact number round numbers scale drawing 191 two small lines or marks placed side by side at the upper right of a number to indicate a measure Of length in inches. Examples: 6"; #2". These are read as: six inches; forty-two inches. 1. The drawing paper measures 9" by 12". 2. Paul is 59" tall. a graph constructed Of bars which compares various amounts, quantities, values, rates, prices, etc. The bars are usually drawn in the form Of squares or rectangles. l. The children made a bar graph to compare the amounts Of money collected by each classroom. 2. John saw a bar graph in the newspaper comparing amounts of rainfall over the past 5 years. 3. The teacher made a bar graph to show how the attendance changes from day to day. to make a map, plan, or representation on which a smaller unit of measure is used to represent a larger unit Of measure. Example: draw to a scale Of 1 inch equal to 1 mile. 1. The boys learned how to draw to scale in shOp class. 2. The architect drew the plans on a scale Of 3 inch to the foot. 3. The map is drawn to a scale Of 1 inch for 1 mile. a number found by counting; not approximate l. A number found by counting is called an exact number. "““ 2. Numbers found by measuring are not exact numbers. numbers that are not exact, but which are close enough to be used in estimating 1. Mr. Jones reported the attendance at the county fair in round numbers. 2. Harry gave his answer in round numbers. 3. Round numbers may be used to estimate the cost of a vacation. a plan, map, or representation that is drawn to scale; a drawing that is made by using a smaller unit Of measure to represent a larger unit Of measure, as, one inch on the drawing equal to 100 feet. 1. Harry made a scale drawing Of the playground. 2. The maps in our book are scale drawings. square foot square inch square mile square yard 192 a square which measures one foot on each side; a unit of measure used in determining amounts or quantities Of area or surface 1. A square foot measures 12 inches on each side. 2. The size Of houses is Often given in square feet. 3. How many square inches are there in a_§quare foot? a square which measures one inch on each side; a unit of measure used in determining amounts or quantities Of area or surface 1. A postage stamp is about one square inch in size. 2. A square inpp measures one inch on each side. 3. Can you figure the size of your desk top in square inches? a square which measures one mile on each side; a section of land; 6&0 acres; a unit Of measure used in determining amounts or quantities Of area or surface 1. The area Of the United States is Often given in square miles. 2. A square mile measures one mile on each side. 3. There are 6&0 acres in a aquaregpgle. a square which measures one yard or three feet on each side; a unit of measure used in determining amounts or quantities of area or surface 1. Floor covering may be purchased by the square_yard. 2. A squaregyard measures 36 inches on each side. 3. There are nine sguare feet in a squareqyard. acre common lowest terms square measure 193 Chapter 8 a field; a measure Of land; 160 square rods; h8h0 square yards; h3,560 square feet 1. Dad bought an acre Of land for our new house. 2. The playground at school is about an acre in size. 3. The farmer planted one acre Of strawberries. h. A field #35 feet long and 100 feet wide is about an acre in size. belonging equalLy'tO two or more quantities; shared equally by two or more individuals or by all the members of a group; of frequent or ordinary occurrence or appearance; familiar; of the usual type; that which is general or usual 1. The children learned how to find the common denominator. 2 1/2, 2/3, and 3/h are common fractions. 3 What do those three boys have in common? h. long winters are common in this part of the country. 5. The president was elected by common consent. numerator and denominator can be divided evenly only by the number one; the result of reducing a fraction to its simplest form 1. The class learned to reduce fractions to lowest terms. 2. Dividing both terms in 3/6 by 3 will reduce the fraction to lowest terms. 3. The fraction—571s in lowest terms. a unit Of area having four equal sides and square corners used in measuring things having length and‘width or bredth, such as areas, fields, surfaces, and the like 1. The square yard used in measuring rugs is an example Of a aquare measure. 2. Square measure may be used tO determine the size Of a football field. 3. Square measure is not used in determining the length of something. Ii 'fflu H directions exercises great gross meet 19h Chapter 9 instructions or information telling how to do, where to go, etc.; ways in which one may face or point; guiding; managing; controls; orders; commands 1. North and south are in opposite directions. 2. John read the directions before building the model plane. 3. The directions on the package were not very clear. that which gives practice; trials; tests; that which is done for the sake of practicing, training, or promoting skill, health; act Of exercising; to put into practice or carry out in action, as, to exercise care or patience l. The children did ten practice exercises in sub- traction. 2. The teacher asked the children to do the exercises on page 10. 3. The flag salute is part of the Opening exercises. A. They learned some new exercises in gym class. 5. The president exercises the duties of his Office. big; very large; huge; vast; immense; more numerous than the average; extending thru a long time; constierable degree, extreme; important; excellent 1. A great number of people watched the football game. 2. A great storm swept over the land. 3. My‘EEEEt grandfather is 90 years old. h. Mr. Smith is a great musician. 5. Mary thought that was a great idea. number of twelve dozen; twelve times twelve; whole, entire, total, as, the gross sum or amount; of relatively great size; thick; bulky; massive; as a gross pillar; big; burly; fat, as, a gross man 1. A gross is equal to twelve dozen or lhh. 2. The storekeeper ordered a gross of paint brushes. 3. The gross income is all the money taken in. h. John makes gposs errors in spelling. 5. Mr. Smith is a gposs man. to Join or intersect; as, where the roads meet; to cOme fact to face with a person or thing coming from.the other direction; to come upon or across; to find; an assembling together; to come into contact or connection with; to come close to process score standard surface 195 Draw the lines so they meet at point A. The two highways meet at the bridge. Mary said, "I would like to have you meet my sister," . my brother ran in a track meet. . He will meet you at 2 O'clock. \J‘I-F" bowl-4 a set of Operations performed with numbers, usually according to an order; a set of actions or changes in a special order; prepare by some special method 1. The children learned the process of long division. 2. Bob did not know which process to use to solve the problem. 3. This material is made by a new process. h. What is the process used to develop film? the number Of points made by a team in a game or contest; a mark or a line made for the purpose of keeping account; a unit or group of 20; an amount owed l. The final score was 7 to 3. 2. A score of peOple were at the party. 3. I have a score to settle with you. h. Jim kept score by making a mark on the board for each right answer. 5. The music teacher told them to study the score. anything used as a basis Of comparison; that which is established as a model, example, or rule; used as a standard; according to rule; a structure built as a base or support for something 1. His work was below the standard. 2. Our standard of living is high. _ 3. I set my watch on Eastern standard time. h. Place the flag in the standard. 5. A.flag is sometimes called a standard. the exterior or outside of anything; the outtermost or uppermost boundary; the face or faces of a 3-dimentional thing; put a surface on; come to the surface 1. The surface of the table is smooth. 2. Most of the surface of the earth is water. 3. The surface of the lake is calm. . The submarine will surface at noon. . They will surface the parking lot with black tOp. . He is very nice below the surface. O\\J'I-F" tax trials 196 money paid by people for the support of the government; to put a tax on; to lay any burden or demand upon; strain 1. .ng money is used to build schools. 2. He paid two cents paa on his purchase. 3. Father pays an income pap every year. h. Hard work will EEE his strength. processes of trying or testing; tests; examinations; participation in contests or competitions; troubles; burdens 1. Each contestant was given three trials. 2. Many trials were needed to deve10p the rocket. 3. The soap box derby trials will be held on Saturday. A. The explorers faced many trials in crossing the country. 5. There were four trials in court last Monday. decimal fraction mixed decimal a number less than one that is written in the decimal places which are on the right hand side of the decimal point; a proper fraction whose denominator, not written, is a power of 10, the power being determined by the number of digits in the decimal places to the right of a period or dot called the decimal point. 1. .05 is an example of a decimal fraction. 2. Yhat is the decimal fraction that is equivalent to 27 3. A decimal fraction is often easier to work with than a common fraction. h. Engineers use decimal fractions in their work. a whole number and a decimal fraction together, usually written 1. A mixed decimal is a whole number and a decimal fraction written together. 2. 3h.5 is a mixed decimal. 3. Money numbers are examples of pixed‘gecimals. f‘"“’§ !‘{:C ”a?! ‘ U'ls“b VIE-Lu ..s [h '2." v 'v ' a. .. “8‘5 i985 hm 7 a) 3...}, ‘. \I «ll. "I7'Elllllllllllll'TS