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 MAR 8 2006 
 
Bulletin No. 58 
 
 BUREAU OF EDUCATIONAL RESEARCH 
 COLLEGE OF EDUCATION 
 
 A Critical Summary of Research 
 
 Relating to the Teaching of 
 
 Arithmetic 
 
 By 
 Walter S. Monroe 
 
 Director, Bureau of Educational Research 
 and 
 
 Max D. Engelhart 
 
 Assistant, Bureau of Educational Research 
 
 Published by the University of Illinois, Urbana 
 1931 
 
UNIVERSITY 
 OF ILLINOIS 
 :: PRESS :i 
 
PREFACE 
 
 In view of the fact that several summaries of research in the field 
 of arithmetic are available, the preparation and publication of 
 another one requires justification. As the title indicates, the present 
 summary is restricted to research relating to methods of learning and 
 teaching arithmetic. A more significant characteristic is the attempt 
 to effect a systematic and critical evaluation of the researches sum- 
 marized. There have been several assertions that a considerable 
 portion of the research reported during recent years is faulty, and a 
 few studies have been criticized by writers in educational periodicals. 
 In the preparation of summaries of research, however, there has been 
 very little evaluation of the studies included. Although the authors 
 of this bulletin have recognized certain specified criteria in their 
 evaluation, the judgments are largely subjective, and, consequently, 
 the conclusions relative to the dependable findings concerning the 
 teaching of arithmetic may not be entirely valid. It is hoped, how- 
 ever, that the publication of this bulletin will contribute to a more 
 adequate understanding of what a critical summary involves. 
 
 Controlled experimentation has been hailed as a means of securing 
 dependable evaluations of all factors of the teaching process. Careful 
 study, however, indicates certain significant difficulties, and it is 
 hoped that the discussion in the final chapter of this bulletin will con- 
 tribute to a saner understanding of experimentation. The expendi- 
 tures required for certain types of studies do not appear to represent 
 wise investments, and those who are interested in educational research 
 should give careful attention to the probable dependability of the 
 outcomes of the studies they undertake or sponsor. 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/criticalsummaryo58monr 
 

 TABLE OF CONTENTS 
 
 PAGE 
 
 Chapter I. Introduction 7 
 
 Chapter II. Methods of Teaching and Learning the 
 
 Fundamentals 14 
 
 Chapter III. Drill in the Fundamentals 30 
 
 Chapter IV. Methods of Teaching Pupils to Solve 
 
 Verbal Problems 50 
 
 Chapter V. Methods of Diagnosis and Remedial 
 
 Treatment 68 
 
 Chapter VI. Methods of Teaching Reading of Arithmet- 
 ical Subject-Matter 76 
 
 Chapter VII. Motivation of Learning in Arithmetic. . 81 
 
 Chapter VIII. General Summary and Conclusion ... 89 
 
 Chapter IX. Bibliography of Investigations Pertaining 
 
 to Methods of Teaching Arithmetic 108 
 
 
CHAPTER I 
 INTRODUCTION 
 
 General purpose of this bulletin. The general purpose of this bul- 
 letin is to present a summary and an evaluation of the research 
 relating to instructional methods employed in teaching arithmetic in 
 Grades I to VIII. For each group of investigations the discussion 
 appears under three heads: (1) summary of reported conclusions, 
 (2) evaluation of experiments, (3) justified conclusions. 
 
 Sources of references to investigations. The sources of practically 
 all of the references were the ''Summary of Educational Investiga- 
 tions Relating to Arithmetic" of Buswell and Judd 1 and the annual 
 supplements prepared by Buswell. 2 An investigation of Brownell on 
 the techniques employed in research on arithmetic was of service in 
 locating in the above summaries investigations of the types desired. 3 
 The writers were able to include in the present summary some refer- 
 ences not given in the sources cited above. 
 
 General types of research included. Most of the investigations 
 included in this summary may be characterized as experiments. 
 Many of these experiments are of the single-group type, and as such 
 may be labeled "uncontrolled" experiments. In investigations of 
 this kind, the experimenter subjects a single group of pupils to the 
 method or procedure which he wishes to try out, and estimates by 
 observation or by administering tests the improvement in achieve- 
 ment assumed to be due to the new method or procedure. Where the 
 gains in achievement are large, the new method may, with some jus- 
 tification, be claimed effective, but it is evident that usually an 
 unknown amount of the gain is due to the operation of other factors. 
 Investigations of this kind are termed "experiments," even though 
 uncontrolled factors are operative, because they possess one impor- 
 tant characteristic of all experimentation — that of trying something 
 out to see what happens. 
 
 A number of the experiments referred to in this summary are of 
 the controlled type. In place of a single group, two or more equiv- 
 alent groups of pupils are used. In the typical controlled experiment, 
 
 buswell, G. T. and Judd, C H. "Summary of Educational Investigations Relating to Arith- 
 metic," Supplementary Educational Monographs Xo. 27. Chicago: University of Chicago Press, 1925. 
 212 p. 
 
 2 These supplements are published in the Elementary School Journal, as for example: 
 
 Buswell, G. T. "Summary of Arithmetic Investigations, 1928," Elementary School Journal, 
 29:691-8, 737-47; May, June, 1929. 
 
 3 Brownell, W. A. "The Techniques of Research Employed in Arithmetic," Twenty-Ninth Year- 
 book of the National Societv for the Study of Education. Bloomington, Illinois: Public School Publishing 
 Company, 1930, p. 415-443. 
 
8 Bulletin No. 58 
 
 the two groups of pupils are equated with respect to intelligence or 
 achievement test scores, or both; hence, they are considered poten- 
 tially equivalent with respect to the planned instruction. These 
 groups are subjected to instruction differing only with respect to the 
 experimental factor. For example, one of the groups is taught to add 
 in the upward direction, while the other group is taught to add in the 
 downward direction. After a period of such instruction, in which 
 attempts are made to prevent irrelevant factors from operating un- 
 equally on the two groups, the final achievement test is given. The 
 difference in final-test scores, or in mean gains in achievement from 
 initial to final tests, is then computed, and interpretations are made 
 with respect to the relative superiority of the one method or of 
 the other. 
 
 Several laboratory experiments have been included in this sum- 
 mary. In these investigations, laboratory apparatus, such as that 
 used in recording eye-movements, is used to secure an understanding 
 of the characteristics of arithmetical learning activity. Some of the 
 investigations are of the type in which data are collected by means of 
 a single administration of a test. In a few places in this summary, 
 relevant "case studies" are cited. Previous summaries of research in 
 the field of arithmetic have occasionally been used to supplement the 
 judgments of the writers with respect to the original research. 
 
 It should be mentioned that investigations of the nature of pupil 
 responses, as for example, the researches on the relative difficulty of 
 the number combinations, have been excluded from this study. The 
 same is true of analyses of arithmetic texts and practice materials. 
 Research of this kind, however important, is, in the judgment of the 
 writers, more relevant to the problems of the arithmetical curriculum 
 than to problems of methods of teaching arithmetic. 
 
 Criteria recognized in the evaluation of investigations. Evalu- 
 ation of experiments is largely a subjective matter, but the utilization 
 of specified criteria will tend to make it more dependable. A critical 
 reader may apply these same criteria to the experiments evaluated in 
 this summary and determine, to his own satisfaction at least, whether 
 or not the evaluations of the present writers are justified. 
 
 1. Definition and restriction of the experimental factor. In ex- 
 perimental investigations of methods of teaching, the ideal procedure 
 is to vary one of the factors that affect pupil achievement while all 
 others are kept constant. The factor that is varied is designated as 
 "experimental," and, obviously, it must be defined in specific terms. 
 Otherwise the basis of the experimentation cannot be definitely 
 known. For example, if the method of instruction is the experimental 
 
Summary of Research Relating to the Teaching of Arithmetic 9 
 
 factor and is designated merely as "the project method versus the 
 traditional method," the precise nature of the variation is not clear. 
 Usually the experimental factor must be restricted to a single phase or 
 detail of method. If it is complex, the experimenter cannot know 
 which element of the method produced the observed effect in the 
 pupil achievement. Hence, the factor that is being made the basis of 
 experimentation must be defined and restricted in such a way that 
 the results may be interpreted in definite terms. 
 
 2. Control of pupil factors. Variation in the experimental factor 
 is secured by employing two or more groups of pupils and maintaining 
 a specified status of this factor for each of the groups. For example, 
 if the type of drill exercises on addition of integers is the experimental 
 factor, one type is used with Group A, a second type, with Group B, 
 a third type, with Group C, and so on. Since achievement is influ- 
 enced by the capacity of the pupils to learn, by their previous school 
 experience, by their interest in the field of learning, and the like, it is 
 obviously necessary that all significant pupil factors be controlled. 
 This control is usually secured by forming groups that are equivalent 
 with respect to all significant pupil factors. Hence, unless some other 
 means of control is effected, the degree of equivalence of the groups is 
 a criterion of the dependability of the results of the experiment. 
 
 3. Control of important non-experimental factors. The achieve- 
 ment of pupils is affected by several factors. The more important 
 ones appear to be the following: 
 
 1. Instructional techniques 
 
 2. Skill of the teacher in using the instructional techniques 
 
 3. Zeal of the teacher 
 
 4. Personality traits of the teacher 
 
 5. Instructional materials 
 
 6. Time spent in learning activity 
 
 The significance of these factors varies with the character of the 
 achievement, but usually none of them should be neglected. The 
 skill and the zeal of the teacher appear to be more significant than is 
 commonly realized. Control of these factors may be attained by 
 securing equivalence or by determining the effect of variation and by 
 making appropriate allowance for this effect in interpreting the 
 results. 
 
 4. Accuracy and validity of measures of differences in achieve- 
 ment. An index of the relative effectiveness of two methods of in- 
 struction or of two types of instructional materials is obtained by 
 computing the difference between the means of the scores on the test 
 administered at the close of the experiment, or, preferably, between 
 
 
10 Bulletin No. 58 
 
 the mean gains in achievement, obtained by subtracting the initial- 
 test means from the final-test means. The obtained difference is 
 affected by the variable and systematic errors of measurement. It is 
 possible, if the coefficients of reliability of the tests used are known, to 
 make appropriate allowances for variable errors of measurement. If 
 the test is administered to both groups under approximately the same 
 conditions, the possibly existing systematic errors of measurement, 
 while they may raise or lower the means similarly, will not influence 
 to a significant extent the difference of the means. It should be noted, 
 also, that fluctuations of testing conditions tending to create system- 
 atic errors in certain groups of scores will tend to produce variable 
 errors when several groups are combined. Hence, when the number 
 of pupils is large, the systematic errors are likely to be less significant 
 than when the group of pupils is small. It should be emphasized in 
 this connection, however, that, when the groups of pupils and the 
 obtained differences in achievement are relatively small, the system- 
 atic and variable errors of measurement are not likely to be of negli- 
 gible significance. It is, therefore, essential that adequate recognition 
 be given to their possible or probable influence. The probable effect 
 of systematic errors cannot be calculated by any formula, and for 
 this reason they are the more difficult to deal with. 
 
 The problem of an experiment usually specifies or implies the 
 nature of the achievement on which the evaluation of the experimen- 
 tal factor is to be based. Hence, it is necessary to consider the extent 
 to which the instruments used actually measure the specified or 
 implied pupil achievements. This may not be the same as the usual 
 validity of the test, because in this case one is concerned only with 
 the extent to which the test measures the achievement designated in 
 its specified or implied function. It is possible that a test may be more 
 valid with respect to the instructional methods or materials of one 
 group than of the other. For example, a test consisting of addition 
 and subtraction examples in a mixed order would be more valid for a 
 group that had had addition and subtraction taught together than it 
 would be for a group that had had these processes taught separately. 
 A test may also be valid with respect to the measurement of the more 
 specific abilities engendered in arithmetic and yet be quite invalid 
 with respect to such general outcomes as attitudes, ideals, and inter- 
 ests. If the achievement of one of the groups includes such outcomes, 
 the differences in achievement obtained will contain errors of validity. 
 The effect of invalidity is to introduce additional variable errors, and, 
 as in the case of the variable errors of measurement, the effect tends 
 to become negligible when the groups are large. However, the valid- 
 
Summary of Research Relating to the Teaching of Arithmetic 11 
 
 ity of the test used should not be neglected when interpreting smaller 
 differences in gains. 
 
 5. Justification of generalization. If the preceding criteria have 
 been satisfied, conclusion reported may be accepted as dependable 
 with respect to the pupils participating in the experiment. If, how- 
 ever, the investigator wishes to generalize, his data must satisfy an 
 additional criterion. They must be representative of the larger pop- 
 ulation to which the generalization is to be applied. If the sample of 
 pupils used in the experiment was random, the investigator is justified 
 in using the standard, or probable, error of sampling as an index of 
 the representativeness of his groups. If, on the other hand, the sam- 
 ple was not random, the investigator must use other means to show 
 the extent to which his sample is representative. While no specific 
 rules may be stated, the investigator should consider all of the avail- 
 able evidence relative to the traits of the groups concerned. For 
 example, if he has scores of his pupils on intelligence and standardized 
 achievement tests, he may compare the means and standard devia- 
 tions of these scores with the corresponding measures of the larger 
 population. If this comparison indicates that his sample is typical of 
 the larger population, generalizations may be accepted with a reason- 
 able degree of confidence. If the data do not satisfy this criterion of 
 representativeness, the investigator should refrain from generalizing, 
 or limit his generalizations appropriately. 
 
 The application of these criteria. In the evaluation of the studies 
 reviewed in this bulletin, the second and third criteria are most 
 prominent. The reader, however, should not infer that the other 
 criteria are not important. Usually the definition and restriction of 
 the experimental factor are obvious, and the instructional techniques 
 applicable in the teaching of arithmetic tend to be relatively specific 
 rather than general. Hence, a large proportion of the experiments in 
 the field being considered satisfy this criterion. 
 
 In the judgment of the dependability of the differences in achieve- 
 ment reported in the experiments summarized in this bulletin, some 
 attention has been given to their "statistical" significance. The com- 
 bined allowance to be made for variable errors of measurement and of 
 sampling may be determined through the use of appropriate for- 
 mulae. 4 The employment of this procedure yields either the probable, 
 or standard, error of the difference, and it is customary to recognize a 
 difference as "statistically" significant when it is equal to, or greater 
 than, 2.78 times its standard error or approximately 4.4 times its 
 
 *See pages 101 to 106. 
 
12 Bulletin No. 58 
 
 probable error. 5 When an obtained difference is 2.78 times its stand- 
 ard error, the chances are not less than 369 to 1 (interpreting the 
 standard error as a limit) that the difference would have the same 
 sign, or be in the same direction, as they would have been if variable 
 errors of measurement and of sampling were eliminated. The "sta- 
 tistical" significance of a difference is, therefore, not very meaningful, 
 since a difference may be "statistically" significant and yet be unde- 
 pendable because of other limitations of the data, such as lack of 
 equivalence, failure to control non-experimental factors, variable 
 errors of validity, and systematic errors of measurement, validity, 
 and sampling. It is a safe assumption that any difference not "sta- 
 tistically" significant in the customary usage would not be of accept- 
 able dependability if consideration is given to all of the probable 
 faults of the data. On the other hand, if an obtained difference is 
 "statistically" significant, its dependability is more certain because of 
 this, but it is by no means guaranteed. In the estimation of the 
 dependability of differences reported in the experiments reviewed in 
 this summary, "statistical" significance has been recognized, there- 
 fore, as but one aspect of the matter. 
 
 The magnitude of possible systematic errors due to lack of equiv- 
 alence, to failure to control non-experimental factors, to failure to 
 secure comparable testing conditions in experimental and control 
 groups, and to failure to measure the same outcomes in both groups 
 is difficult to determine from the report of an experiment, unless the 
 investigator explicitly refers to the matter. 
 
 Unless some unusual achievement is specified or implied, most 
 tests designed to measure calculation skills are probably of rather 
 high validity. They, of course, measure the current ability of pupils 
 rather than the permanent residue of achievement. It is likely that 
 the latter type of achievement should be considered, but few, if any, 
 investigators have attempted to base their conclusions on it. Conse- 
 quently, the present writers have not applied this more severe test 
 in their evaluations. When the achievement to be measured includes 
 abilities other than calculation skills, the validity of the measures is 
 an important matter, but it is very difficult to determine the degree 
 of validity. 
 
 The organization of the summary. This summary of research 
 relating to instructional methods in arithmetic has been divided into 
 six major divisions represented by the following rubrics: (1) methods 
 
 5 Monroe, W. S. and Engelhart, M. D. "Experimental Research in Education," University of 
 Illinois Bulletin, Vol. 27, No. 32, Bureau of Educational Research Bulletin No. 48. Urbana: University 
 of Illinois, 1930, p. 59-76. See also: 
 
 McCall, W. A. How to Measure in Education. New York: Macmillan Company, 1922, p. 404-5. 
 
Summary of Research Relating to the Teaching of Arithmetic 13 
 
 of learning and teaching the fundamentals, (2) methods of drill in the 
 fundamentals, (3) methods of teaching pupils to solve verbal prob- 
 lems, (4) methods of providing diagnosis and remedial treatment, 
 
 (5) methods of teaching the reading of arithmetical subject-matter, 
 
 (6) methods of motivating learning activity in arithmetic. A chapter 
 is devoted to each of these divisions. 
 
 
CHAPTER II 
 
 METHODS OF TEACHING AND LEARNING 
 THE FUNDAMENTALS 
 
 The general nature of the experimental factor. The experimental 
 factors of the studies summarized in this chapter are essentially 
 methods of learning or performing the fundamental operations of 
 arithmetic. Requesting pupils to add upward or to add downward, 
 as the case may be, may be thought of as a method of teaching, but 
 the essential element is the activity of the pupil. In the same way, 
 requesting pupils to use the subtractive method of subtraction in 
 which borrowing or decomposition is used, and directing pupils in the 
 use of this method, may be regarded as a method of learning. 
 
 The research summarized in this chapter has been classified under 
 
 the following heads: (1) addition; (2) subtraction; (3) division; 
 
 (4) fractions, decimals, percentage, proportion, and denominate 
 
 numbers. 1 
 
 ADDITION 
 
 1. Summary of conclusions as reported. The relative efficiency of 
 upward and downward addition has been studied in one experiment 
 and in two investigations of other types. In the experiment reported 
 by Buckingham 2 the group that was taught to add downward attained 
 greater, but not significantly greater, achievement in addition. From 
 an analysis of test results, Cole 3 reported that individuals add more 
 accurately downward but less rapidly. Buckingham 4 has also re- 
 ported the findings of a questionnaire study in which it was discovered 
 that while more people prefer to add upward when the column is long, 
 they add downward when the column is short. On the basis of the 
 logical advantages that he claims for downward addition, and be- 
 cause of this variation, Buckingham recommends that downward 
 addition be taught. 
 
 Procedures for adding a column of figures have been studied in 
 four experiments and in one investigation where an observation 
 technique was used. Overman 5 investigated the relative effectiveness 
 
 J The absence of multiplication from this classification should be noted. The present writers have 
 been unable to discover any experimental investigations of methods of teaching or learning multi- 
 plication. 
 
 Buckingham, B. R. "Upward versus Downward Addition," Journal of Educational Research, 
 16:315-22, December, 1927. (18) 
 
 3 Cole, L. W. "Adding Upward and Downward," Journal of Educational Psychology, 3:83-94, 
 February, 1912. (29) 
 
 Buckingham, B. R. "Adding Up or Down: A Discussion," Journal of Educational Research, 
 12:251-61, November, 1925. (15) 
 
 5 Overman, J. R. "An Experimental Study of the Effect of the Method of Instruction on Transfer 
 of Training in Arithmetic," Elementary School Journal, 31 : 183-90, November, 1930. (97) 
 
 14 
 
Summary of Research Relating to the Teaching of Arithmetic 15 
 
 of the following methods of teaching addition (and subtraction) of 
 two- and three-place numbers in terms of transfer to untaught types, 
 such as addition of four two-place numbers, two three-place numbers, 
 one three-place number, and one one-place number. 
 
 (1) In Method A the pupils were shown how to perform the process, and 
 there was no generalization or consideration of underlying principles .... 
 (2) In Method B (generalization) the pupils were helped to formulate general 
 methods of procedure from the specific types taught, and these generalizations 
 were constantly emphasized throughout the teaching .... (3) In Method C 
 (rationalization) the reasons and principles underlying the specific types taught 
 were discussed with the pupils. The formulation of general rules of procedure 
 was avoided as much as possible .... (4) In Method D (generalization and 
 rationalization) general methods of procedure were formulated, and the under- 
 lying principles were discussed. 
 
 Method B was reported as the most effective, Method D was 
 found to be almost as effective as Method A, and Method C, only 
 slightly more effective than Method A, the least effective of all. 
 In connection with his experiment on transfer of learning in addition 
 and subtraction Olander 6 investigated the effectiveness of instruction 
 in generalizing groups of combinations. "For example, these children 
 were led to recognize the law common to zero combinations. They 
 noted that combinations appeared in reverse form such as 6 + 7 and 
 7 + 6, and they observed that a combination such as 10 — 6 was 
 intimately related to 6 + 4." In his conclusions the investigator 
 states that short daily instruction of this character had no significant 
 effect on the arithmetic scores of the pupils taught by the method. 
 
 Conard and Arps 7 discovered that strikingly superior results were 
 secured by teaching children to "think results only." 8 Ballenger 9 
 concluded that it is effective to teach children, who have been having 
 difficulty with addition, to break long columns into two parts and to 
 add each part separately. Arnett 10 reported that the most rapid and 
 accurate individuals add the digits in regular serial order. Excessive 
 combination, or rearrangement, of digits is detrimental to rate and 
 accuracy, but a moderate amount proves beneficial to some individ- 
 uals. Finally, Clark and Vincent 11 found that teaching the pupils to 
 check their answers results in greater, but not significantly greater, 
 accuracy. 
 
 6 01ander, H. T. "Transfer of Learning in Simple Addition and Subtraction," Elementary School 
 Journal, 31:358-69, 427-37; January, February, 1931. (94) 
 
 T Conard, H. E. and Arps, G. F. "An Experimental Study of Economical Learning," American 
 Journal of Psychology, 27:507-29, October, 1916. (32) 
 
 8 In this method the individual in the process of adding 3, 4, 9, and 6 thinks 7, 16, and 22 rather 
 than 3 and 4 are 7, 7 and 9 are 16, and 16 and 6 are 22. 
 
 9 Ballenger, H. L. "Overcoming Some Addition Difficulties," Journal of Educational Research, 
 13:111-17, February, 1926. (6) 
 
 10 Arnett, L. D. "Counting and Adding," American Journal of Psychology, 16:327-36, July, 1905. (4) 
 
 n Clark, J. R. and Vincent, E. L. "A Study of the Effect of Checking Upon Accuracy in Addition," 
 Mathematics Teacher, 19:65-71, February, 1926. (27) 
 
16 Bulletin No. 58 
 
 2. Evaluation of the experiments. In the only experiment on 
 upward versus downward addition, Buckingham (18) used seven 
 pairs of groups of second- and third-grade pupils, varying in size from 
 eleven to twenty-eight pupils. The paired groups were equated with 
 respect to scores made on an initial test in addition. Each of the 
 teachers participating in this experiment taught a pair of groups, 
 rotated at the end of each week the time of day during which addition 
 was taught, assigned no home work in arithmetic, and introduced no 
 new arithmetic topics. The teacher administered the final test as 
 soon as her pair of groups had attained reasonable proficiency in 
 adding short columns of one-place numbers. The differences in mean 
 gains for the six pairs of groups favored the method of downward 
 addition, but in only one case was the difference "statistically" 12 
 significant when compared with its probable error. 
 
 In this investigation, the experimental factor, the direction of 
 adding a column of figures, is specific and appears to have been satis- 
 factorily isolated. The control of the pupil factors by grouping the 
 pupils on the basis of the scores made on an initial addition test prob- 
 ably was not entirely satisfactory. The general intelligence and the 
 addition habits of the pupils were not directly considered. The con- 
 trol of the teacher factors was attempted by having each teacher 
 instruct a pair of groups, one in adding upward and the other in 
 adding downward. This procedure, however, does not insure control, 
 because there may have been variations in zeal and skill. The 
 validity of the test was not explicitly considered ; it depends upon the 
 ability that is specified as the criterion of merit of the direction of 
 adding. If validity is defined as "ability to add throughout the 
 pupils' school experience" or "ability to add when he becomes an 
 adult," it must be admitted that the degree of validity is unknown. 
 In view of the relatively small differences in gains, it seems reasonable 
 to say that the findings, which are interpreted as favoring downward 
 addition, are not dependable. When one considers the information 
 vielded by Buckingham's questionnaire study (15) and by Cole's 
 experiment with adults (29), it still appears that the relative merit of 
 the two directions of adding has not been determined. Neither does 
 one have dependable evidence to support the common-sense view 
 that the direction makes little or no difference. 13 
 
 Cole (29) had thirty persons add the same problems both upward 
 and downward. The fact that the subjects were accustomed to use 
 
 12 See pages 11 and 12. 
 
 13 It may be somewhat immaterial whether children are taught to add upward or downward, si nee the 
 carefully controlled experiment of Beito and Brueckner (9) would seem to indicate that there is a large 
 amount of transfer of training from learning to add in one direction to learning to add in the other, 
 or reverse, direction. It is stated in their conclusions that "When pupils of any mental level are taught 
 
Summary of Research Relating to the Teaching of Arithmetic 17 
 
 the upward method causes one to question the results obtained in 
 this investigation. It is possible that they added downward more 
 accurately because they added more slowly and took greater pains 
 with an unfamiliar method. 
 
 Overman (97) used four groups of 112 second-grade pupils which 
 were equivalent with respect to sex, mental age, teacher's estimate of 
 general ability, and score on a preliminary test. The experimental 
 factors 14 appear to have been adequately defined, but there is some 
 uncertainty in regard to the control of the non-experimental factors, 
 especially teacher skill and zeal. Each of the groups were given 
 twenty minutes of practice a day for fifteen days, eight days being 
 used for testing, and seven, for instruction and practice. Tests were 
 given at the beginning and at the end of the experiment, and twice 
 during the experiment. The differences in achievement, as measured 
 by these tests, were "statistically" significant for Methods B and D 
 compared with Method A, but not for Method C. The conclusions 
 of the experiment seem reasonably dependable. They also, for the 
 most part, seem to be the conclusions one should logically expect. 
 That pupils should be stimulated to generalize is sufficiently well 
 established that an experimental comparison of a method with 
 generalization and a method without generalization seems somewhat 
 futile. One wonders in the case of this experiment why generalization 
 plus rationalization should have proved inferior to generalization 
 alone. Common sense would lead to the inference that a combi- 
 nation of both would be most effective. It would seem justifiable to 
 ascribe the apparent inferiority not to the method which combines 
 generalization with rationalization but to the limitations of the 
 experiment. 
 
 In evaluating the effectiveness of instruction in generalizing 
 groups of combinations, Olander (94) used three hundred pairs of 
 second-grade pupils equivalent with respect to growth in arithmetic 
 ability over a period of five weeks. The reason given for using this 
 technique is the following: 
 
 If two groups exhibit similar learning curves under similar instruction until a 
 certain point is reached, it can be assumed that the groups are equal in the 
 function in question. 
 
 The experiment was conducted for twelve more weeks, during 
 
 only the direct form of an addition combination, such as 7, as nearly as can be, the reverse form, 4, 
 
 is learned concomitantly at least as completely as the direct form." See: 
 
 Beito, E. A. and Brueckner, L. J. "A Measurement of Transfer in the Learning ot Manner 
 Combinations," Twenty-Ninth Yearbook of the National Society for the Study of Education. Bloomington, 
 Illinois: Public School Publishing Company, 1930, p. 569-87. (9) 
 
 u See page IS for a description of these factors. 
 
 
18 Bulletin No. 58 
 
 which time the pupils of the experimental group were given instruc- 
 tion in generalizing for three minutes of the daily twenty-minute 
 period. Achievement was tested at the start of the experiment, at 
 the end of five weeks, at the end of eleven weeks, and at the close of 
 the experiment— at the end of seventeen weeks. The tests included 
 the one hundred addition and the one hundred subtraction combi- 
 nations and were administered by the flash-card method. The pupils 
 in the group not given the three minutes of daily generalizing instruc- 
 tion were able to generalize practically as well as the pupils who were 
 given this instruction. It appears logical that the experimental 
 factor, the generalization instruction, was not applied long or inten- 
 sively enough to add materially to the generalizing abilities acquired 
 by the pupils on their own account. The interpretation that general- 
 ization instruction is not worth while, on the basis of dander's data, 
 does not seem to be justified. 15 
 
 In studying the effect of teaching pupils to "think results only," 
 Conard and Arps (32) used two groups of thirty-two grade-school 
 children whose approximate equivalence was shown by comparison of 
 scores on the Courtis test. After eight work periods of seven exam- 
 ples in each of the four fundamentals, the final test was administered. 
 This experiment does not appear to justify a very high rating with 
 reference to any of the criteria. The experimental factor was not 
 adequately defined, and the control of the non-experimental factors 
 probably was not sufficient to justify acceptance of the obtained 
 results as demonstrating the superiority of "thinking results only." 
 There is evidence that experimental conditions were in some respects 
 abnormal and that the experimental pupils sometimes forgot to 
 "think results only." It may be argued that these faults, for the 
 most part, were such as would tend to reduce, rather than to increase, 
 the difference in favor of the experimental method and, consequently, 
 that the findings should be accepted as dependable evidence. In view 
 of the limitations, however, this argument is not convincing, and the 
 reported conclusion probably should not be accepted as dependable. 
 
 Ballenger (6) used a single group of 130 fourth-, fifth-, and sixth- 
 grade children. These children were taught to divide long columns 
 of figures and to add each part separately. While they improved 
 significantly in accuracy, the results of this uncontrolled experiment 
 cannot be regarded as other than merely suggestive. Such a pro- 
 cedure might be effective for backward children; it probably should 
 not be recommended as a standard method of teaching addition. 
 Children should be taught to add columns of increasing length. 
 
 15 The other conclusions stated in this experiment appear to be reasonably dependable. 
 
 

 Summary of Research Relating to the Teaching of Arithmetic 19 
 
 Splitting columns, as advocated by Ballenger, would seem to be a 
 method of forming undesirable habits which would need to be 
 unlearned later. 
 
 Arnett (4) used chronoscopic apparatus in determining the meth- 
 ods of counting and of adding used by several adults in a psychological 
 laboratory. His results are suggestive, but they should be verified by 
 observation and by controlled experimentation with school children. 
 Clark and Vincent (27), in their study of the effect of checking on 
 accuracy, used two groups of fifth- and sixth-grade children which 
 were equated on the basis of M. A., I. Q., and initial addition test 
 scores. The size of these groups is not reported. After twenty days 
 of practice, the final test was administered. The principal limitations 
 of this experiment are to be found in the lack of control of non- 
 experimental factors, in the lack of control of special teacher zeal, 
 and in the unknown validity of the tests. The difference in final-test 
 means was in favor of the method of checking, but not significantly so. 
 This might be interpreted to mean that teaching pupils to check 
 additions may be expected to increase the accuracy of their work only 
 very slightly. This conclusion, however, probably is not justified. 
 3. Justified conclusions. It is evident that none of the experi- 
 ments satisfy completely the criteria stated in Chapter I. Those of 
 Buckingham (18), Overman (97), and Olander (94) come nearest to 
 doing so, but the limitations of these experiments render the conclu- 
 sions of somewhat doubtful dependability. More experiments must 
 be reported before justified conclusions can be expressed with respect 
 to such problems as adding upward versus adding downward, the 
 effect of checking on accuracy, and the like. The merits of instruction 
 involving generalization and rationalization should be tested in exper- 
 iments where failure to control important non-experimental factors 
 does not obscure the effectiveness of such instruction. 
 
 SUBTRACTION 
 
 The relative merits of the four principal methods of subtraction 
 
 have been studied in a number of investigations. These methods may 
 
 be described briefly by noting the steps in subtracting 25 from 43. 
 
 In using the subtractive, or take-away, method in which borrowing 
 
 or decomposition is employed, the steps are: 
 
 5 from 13 = 8 
 
 2 from 3 = 1 
 
 In the subtractive, or take-away, method in which carrying or equal 
 addition is used, the steps are: 
 
 5 from 13 = 8 
 
 3 from 4 = 1 
 
20 Bulletin No. 58 
 
 The additive method in which borrowing or decomposition is used 
 requires the following steps: 
 
 5 and what are 13, write 8 
 
 2 and what are 3, write 1 
 
 The additive method in which carrying or equal addition is used is 
 illustrated as follows: 
 
 5 and what are 13, write 8 
 
 3 and what are 4, write 1 
 Decomposition, usually when used as illustrated in the first of 
 
 these examples, has been called the "first Italian method," and equal 
 addition, when used as in the second example, has been called the 
 "second Italian method." No name is given to the third method, but 
 the fourth is well known as the "Austrian method." Irmina 16 has 
 described a "complementary method" in which either decomposition 
 or equal addition may be used. However, since no experimental 
 evidence has been presented with respect to its merits, this method is 
 not considered here. 
 
 1. Summary of reported conclusions. The conclusions of Buck- 
 ingham, 17 Mead and Sears, 18 and Taylor 19 favor the subtractive 
 methods in comparison with the additive methods. 20 The only con- 
 clusion favorable to the additive methods is that of Beatty 21 who 
 found that greater accuracy but less speed resulted from their use. 
 Ballard, 22 McClelland, 23 and Winch, 24 studied the relative merits of 
 decomposition, or borrowing, versus equal addition, or carrying, in 
 connection with the subtractive procedure. 25 In each case the results 
 favored the equal addition, or carrying, process. 26 Johnston's 27 
 pupils used both the subtractive and additive general methods. For 
 
 "Irmina, Sister M. "The Relative Merits of the Methods of Subtraction," Catholic University of 
 lS'p 4 5 " ^arch Bulletins, Vol. Ill, No. 9. Washington: Catholic Education Press, 
 
 F „ ( ^Buckingham B. R. ''The Additive versus the Take-Away Method of Teaching the Subtraction 
 hacts ;.,f durational Research Bulletin (Ohio State University), 6:265-69, September ?8 1927 (16^ 
 
 isMead C. D. and Sears, Isabel. "Additive Subtraction and Multiplicative Division Tested " 
 Journal of Educational Psychology, 7:261-70, May, 1916. (72) ' 
 
 "Taylor, J. S "Subtraction by the Addition Process," Elementary School Journal, 20:203-7, 
 INovember, 1919. (114) 
 
 rather°than'the kS?**' ^ conclusions favor the first two Procedures illustrated on pages 19 and 20 
 
 method^ following stud y- not accessible to the writers, also favored the subtractive equal additions 
 
 "Methods of Subtraction," St. Louis Public School Messenger, 26:28-32, September 1. 1928 (128) 
 ci„ ; ? eatt -y; ^/, W - '' Tl '? Add [<- iv e versus the Borrowing Method of Subtraction," Elementary 
 School Journal, 21:198-200, November, 1920, (8) 
 
 22 Ballard, P. B "Norms of Performance in the Fundamental Processes of Arithmetic, with 
 '; u n g f n st I° ( ns for Their Improvement," Journal of Experimental Pedagogy, 2:396-405, December 5, 1914- 
 3:9-2U, March 5, 1915. (5) 
 
 "McClelland, W. W "An Experimental Study of the Different Methods of Subtraction," 
 Journal of Experimental Pedagogy, 4:293-99, December 5, 1918. (69) 
 
 • 2n y inc £' W - **• "'Equal Additions' versus 'Decomposition' in Teaching Subtraction: An Ex- 
 192™ er (125) C '' U °f Ex P erimental Pedagogy, 5:207-20, 261-70; June 5, December 6, 
 
 26 The first two procedures illustrated on page 19. 
 
 26 The second of the procedures illustrated on page 19. 
 
 "Johnston, J. T. "The Merits of Different Methods of Subtraction," Journal of Educational 
 Research, 10:279-90, November, 1924. (52) 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 21 
 
 both of these groups equal addition, or carrying, was found to be 
 superior. 
 
 2. Evaluation of experiments. McClelland (69), Mead and Sears 
 (72), Winch (125), and Buckingham (16) experimented with school 
 children. In all cases the experimental factor was defined and suffi- 
 ciently restricted. The other criteria, however, were not fully satisfied. 
 McClelland (69) employed two groups of children between twelve and 
 one-half to thirteen and one-half years of age in an English school. 
 One group of thirty-four had been accustomed to use the method of 
 equal addition, and the other group of thirty-two, the method of 
 decomposition. After an initial program of testing, which revealed 
 that the equal-addition group was significantly superior, the groups 
 were practiced in their respective methods for a period of twenty 
 weeks. The equal-addition group achieved the greater per cent 
 increase in speed and accuracy. It is evident that McClelland is to be 
 criticized for failure to secure equivalence at the beginning of his 
 experiment. It is possible that the group using the equal-addition 
 method consisted of more intelligent children and, in consequence, 
 made the greater gain in achievement. Furthermore, the degree of 
 control of non-experimental factors is not known. 
 
 Winch (125) conducted two experiments with girls in English 
 schools. In the first, two groups of nineteen eleven-year-old girls 
 were equated on the basis of scores on a series of initial subtraction 
 tests. All of the children had previously used the decomposition 
 method. In the experiment, one group was practiced in this method, 
 while the other learned the equal-addition method. After eight 
 lessons of fifteen to twenty minutes each the achievement of the group 
 learning the equal-addition method slightly surpassed that of the 
 other, as shown by the scores on a series of final subtraction tests. 
 The second experiment was conducted with two groups of twenty- 
 three eight and one-half year old girls who had been accustomed to 
 the equal-addition method. After equivalence had been secured with 
 respect to ability to subtract, one group was practiced in the equal- 
 addition method, while the other group learned the method of decom- 
 position. After eight lessons of thirty minutes each, four final tests 
 were given. The difference between the final-test means in favor of 
 the equal-addition method is approximately seven times its probable 
 error. Winch is to be commended for his care in securing equivalence 
 with respect to initial subtraction ability, for efforts to control non- 
 experimental factors, and for the statistical treatment of his results. 
 He is to be criticized for the non-representativeness and smallness of 
 his groups and for the short duration of his experiments. While the 
 
22 Bulletin No. 58 
 
 techniques used in these experiments are in many respects excellent, 
 it seems unsafe to generalize the findings reported. 
 
 Mead and Sears (72) used two second-grade classes of unreported 
 size which were shown to be approximately equivalent with respect to 
 ability in addition. One group was taught additive subtraction for 
 four months while the other group learned the subtractive method. 
 The final test revealed a possibly significant difference in favor of the 
 subtractive method, so far as single-column subtraction was con- 
 cerned. An additional test of three-figure-subtraction examples 
 revealed no significant difference between the groups. Mead and 
 Sears are to be criticized for failure to secure more adequate equiv- 
 alence and for not revealing the size of their groups. They are to be 
 commended for certain precautions taken to secure control of non- 
 experimental factors and for their rather satisfactory interpretation 
 of results. 
 
 In the experiment of Buckingham (16) seven pairs of groups 
 ranging in size from five to twenty-nine pupils were equated in seven 
 schools by means of the Pressey Primary Classification Test. Each 
 of the teachers participating in the experiment taught both groups of 
 a pair for-a period of seven months, at the end of which time the pupils 
 were tested for their proficiency in single-column subtraction. The 
 differences in achievement for six of the seven groups favored the 
 subtractive method as compared with the additive, but in no case was 
 the difference "statistically" significant. Buckingham is to be com- 
 mended for his techniques in securing equivalence, for using children 
 of no initial ability in subtraction, and for certain precautions taken 
 to secure control of non-experimental factors. He is also to be com- 
 mended for using so many different groups and schools. The inter- 
 pretation of his data would seem to exaggerate the effectiveness of 
 the subtractive method. A more conservative interpretation would 
 seem to be required. 
 
 Ballard (5), Beatty (8), Taylor (114), and Johnston (52) have 
 reported the results of investigations in which the data were collected 
 by test from pupils whose method of subtracting had been deter- 
 mined. Ballard (5) administered his test to 18,678 eight- and nine- 
 year-old English school children. He found the achievement in sub- 
 traction in schools where equal addition, or carrying, was taught to be 
 significantly superior to the achievement in schools where decompo- 
 sition, or borrowing, was taught. He is to be criticized for failure to 
 determine more adequately the methods actually used by the pupils. 
 Taylor (114) had teachers of 11,368 fourth-, fifth- and sixth-grade 
 children put a subtraction example on the board and determine, by 
 
Summary of Research Relating to the Teaching of Arithmetic 23 
 
 asking the children what they would say in solving the given example, 
 the methods of subtraction that the children were using. His data 
 showed that only 37.6 per cent were continuing to use the additive 
 equal-addition method which they were supposedly taught, while the 
 balance of the pupils had somehow learned and were using subtractive 
 methods. Beatty (8) has criticized Taylor (114) for concluding 
 that his results showed the inferiority of the additive equal-addition 
 method, since evidence was not secured to prove that no other 
 method was taught. 
 
 Beatty (8) administered the Courtis Research Standard Tests, 
 Series B, to 54 pupils who used the additive methods and 115 pupils 
 who used the borrowing (subtractive?) methods. While his results 
 favor the additive methods for accuracy, they favor the borrowing 
 methods for speed. He is to be criticized for his few cases and for 
 failure to define the methods evaluated. He does contribute the 
 information that 51.8 per cent of one group of eighty-three children 
 actually did abandon the additive for borrowing methods. 
 
 Johnston (52) determined the subtraction methods used by 277 
 normal-school students and tested the students for speed and accuracy. 
 His results are slightly significant with respect to the superiority of 
 equal addition, or carrying, when used both with additive and sub- 
 tractive methods, but are entirely inconclusive with respect to the 
 additive versus subtractive methods. Ruch, Knight, and Lutes 28 
 have criticized Johnston for failure to make adequate allowance for 
 the statistical limitations of his data. A computation by them of 
 probable errors of the differences showed that none of the differ- 
 ences were "statistically" significant. Johnston 29 replied that their 
 computations failed to consider the significant difference in speed in 
 favor of the equal-addition method. When the accuracy means are 
 corrected for speed, Johnston claims the difference is significant. Ruch, 
 Knight, and Lutes 30 replied to this that no differences can be con- 
 sidered "statistically" significant from groups of eight, thirteen, or 
 twenty-three cases. They add that the original report should have 
 contained adequate information with respect to standard deviations 
 and probable errors. 
 
 3. Justified conclusions. The great majority of the investigations 
 favored the subtractive, or take-away, methods rather than the addi- 
 tive methods, and the equal-addition, or carrying, process rather than 
 
 "Ruch, G. M., Knight, F. B., and Lutes, O. S. "On the Relative Merit of Subtraction Methods: 
 Another View," Journal of Educational Research, 11:154-55, February, 1925. 
 
 "Johnston, J. T. "Still on the Relative Merits of Subtraction Methods," Journal of Educational 
 Research, 12:80-83, June, 1925. 
 
 30 Ruch, G. M., Knight, F. B., and Lutes, O. S. "A Rejoinder to Professor Johnston's Criticisms, 
 Journal of Educational Research, 12:83-85, June, 1925. 
 
24 Bulletin No. 58 
 
 that of decomposition, or borrowing. However, the faulty techniques 
 used in these investigations, plus the failure to find truly significant 
 differences in achievement between the different methods, would 
 cause one to question the dependability of a conclusion in favor of the 
 subtractive method in which equal addition is used, although the 
 evidence is in its favor. 
 
 In this connection it is interesting to note that in two summaries 
 of research in the field of arithmetic, Buswell favors the subtractive 
 method in which equal addition is used. 31 This conclusion agrees 
 with that of Irmina, 32 but differs with that of Knight, Ruch, and 
 Lutes, 33 who present certain theoretical considerations in favor of the 
 subtractive method in which borrowing or decomposition is used. 
 Osburn 34 has reported a summary in which he computed the statistical 
 errors of the differences given in the experimental literature. He 
 states that the differences are significantly in favor of the subtractive 
 equal-addition method as compared with the subtractive decompo- 
 sition method, but the subtractive equal-addition method has not 
 been shown to be significantly superior to the additive methods, 
 although the chances are 16 to 1 in its favor. In another recent review 
 of the subtraction experiments the opinion is expressed that "the 
 differences among the rival methods of subtraction must be small; 
 otherwise centuries of observation and a dozen empirical studies 
 would long since have laid down the broad outlines of truth." 35 
 
 DIVISION 
 1. Summary of reported conclusions. There have been only two 
 investigations of the methods of teaching and of learning division. 
 Mead and Sears 36 report that multiplicative division is superior to 
 the traditional method. They illustrate multiplicative division as 
 follows: 
 
 4 
 The .... multiplicative-division class said: "5 | 20, five times what are 
 twenty? Five times four are twenty. 
 
 Conard and Arps (32) reported that in division the most effective 
 results are secured when pupils are taught to "think results only." 
 
 31 Buswell, G. T. and Judd, C. H. "Summary of Educational Investigations Relating to Arith- 
 metic,' Supplementary Educational Monographs, No. 2 7. Chicago: University of Chicago Press, 
 1925, p. 78. 
 
 Buswell, G. T. "A Critical Survey of Previous Research in Arithmetic," Twentv-N inth Yearbook 
 of the !\>ational Society for the Study of Education. Bloomington, Illinois: Public School Publishing 
 Company, 1930, p. 460-61. 
 
 32 Irmina, op. cil., p. 26-27. 
 
 33 Knight, F. B., Ruch, G. M., and Lutes, O. L. "How Shall Subtraction Be Taught?" Journal of 
 Educational Research, 11:168, March, 1925. 
 
 34 Osburn, W. J. "How Shall We Subtract?" Journal of Educational Research, 16:237-46, No- 
 vember, 1927. 
 
 35 Ruch, G. M. and Mead, C D. "A Review of Experiments on Subtraction," Twentv-Xinth 
 Yearbook of the National Society for the Study of Education. Bloomington, Illinois: Public School 
 Publishing Company, 1930, p. 678. 
 
 36 Mead and Sears, op. cit. 
 
Summary of Research Relating to the Teaching of Arithmetic 25 
 
 2. Evaluation of experiments. Two third-grade classes of unre- 
 ported size participated in the experiment by Mead and Sears (72). 
 The initial test, which was in addition, showed some lack of equiv- 
 alence so far as the trait tested was concerned. No other attempt 
 was made to estimate the degree of equivalence. The division prac- 
 tice of both groups was restricted to simple division by fives. At the 
 end of four months a possibly significant difference was found in 
 favor of "multiplicative" division, as restricted in the preceding 
 statement. A final test containing longer examples showed no sig- 
 nificant difference between the groups. Mead and Sears are to be 
 criticized for failure to secure equivalent groups, for failure to report 
 the size of the groups used, for the restricted character of the training, 
 and for attempting to correct for lack of equivalence in an unjusti- 
 fiable manner. The units and zero points of the initial and final tests 
 were shown to be different, and, therefore, correction by subtracting 
 the difference between initial-test means from the difference between 
 final-test means cannot be condoned. 37 Furthermore, there was not 
 adequate control of the non-experimental factors. 
 
 The experiment of Conard and Arps (32) was evaluated under 
 addition. 38 
 
 3. Justified conclusions. The faults of these two experiments 
 make the listing of a justifiable conclusion impossible. It is doubtful 
 whether the conclusion of Mead and Sears (72) should be regarded as 
 indicative or suggestive. 
 
 FRACTIONS, DECIMALS, PERCENTAGE, AND PROPORTION, 
 AND DENOMINATE NUMBERS 
 
 1. Summary of reported conclusions. Collier 39 has reported that 
 children learn to multiply fractions effectively, if addition of fractions 
 is used as a point of departure. For example, a child may be taught to 
 multiply 4 by % through a request to add %, %, %, %. When the 
 result % has been obtained by the child, the teacher should point out 
 that 8 is the product of 4 X 2. Anspaugh 40 has reported that drill on 
 the fundamental combinations is effective in securing greater effi- 
 ciency in handling common and decimal fractions. 41 
 
 37 Monroe, W. S. and Engelhart, M. D. "Experimental Research in Education," University o 
 Illinois Bulletin, Vol. 27, No. 32, Bureau of Educational Research Bulletin No. 48. Urbana: Uni 
 versity of Illinois, 1930, p. 63. (Footnote 14) 
 
 3& See page 18. 
 
 39 Collier, Myrtle. "Learning to Multiply Fractions," School Science and Mathematics, 22:324-29, 
 April, 1922. (30) 
 
 40 Anspaugh, G. E. "Teaching the Number Facts in the Komenskv School," Chicago Principals' 
 Club, Second Yearbook. Chicago: Chicago Principals' Club, 1927, p. 88-89. (2) 
 
 41 Knight and Setzafandt have shown that training in the addition of fractions having certain 
 denominators transfers to the addition of fractions having other denominators. Some inferences might 
 be drawn from their conclusions with respect to effective methods of teaching the addition of fractions. 
 See: 
 
 Knight, F. B. and Setzafandt, A. O. H. "Transfer within a Narrow Mental Function," Ele- 
 mentary School Journal, 24:780-87, June, 1924. (62) 
 
26 Bulletin No. 58 
 
 Clapp, Chase, and Merriman 42 found that practice material so 
 prepared that it focuses the attention of the pupils on the kind of per- 
 centage problem they are attempting to solve is more effective than 
 the ordinary textbook material. In ordinary textbook material prob- 
 lems solved similarly are grouped together, but in the experimental 
 material "the pupil is not aided in solving the second problem (of a 
 group of problems) by having solved the first one, unless he begins to 
 understand the principle that underlies the solution of such problems." 
 The nature of the problem statements is varied in the experimental 
 material, and some problems not involving percentage are included to 
 keep the minds of the pupils alert to the kinds of problems they are 
 solving. 
 
 Monroe 43 concluded that children do not learn to place the deci- 
 mal point in a quotient by a general rule, or as the result of the acqui- 
 sition of a general ability. He contends that the placing of the 
 decimal point in quotients requires several specific abilities. 
 
 Drushel 44 investigated the relative merits of two methods of plac- 
 ing the decimal point in long division by a test administered to college 
 freshmen. In Method A the student used the rule: "There are as 
 many places in the quotient as those in the dividend exceed the 
 divisor." In Method B the rule was: "First render the divisor an 
 integer by multiplying both dividend and divisor by 10 or some power 
 of 10. Then proceed as with integral divisors." The conclusion 
 favors Method B. 
 
 Winch 45 has reported that the "method of unity" is an effective 
 method of teaching proportion. This method is illustrated in the 
 following problem: 
 
 I pay 4 shillings for 2 pairs of boots. What shall I have to pay for 1 pair? 
 What shall I have to pay for 3 pairs? 
 
 The use of the two questions in these problems directs the solution 
 of the problem from the easy to the more difficult. Winch also 
 reported that proportion in its simpler forms may be taught to 
 children as young as seven years of age, that there do not appear to be 
 any clear sex differences in ability to handle proportion, and that 
 vacation seemed to have little effect on the proportion abilities. He 
 states the very interesting conclusion : "The pupils of schools of very 
 low social class — 'slum schools' — cannot, even in the most favorable 
 
 42 Clapp, F. L., Chase, W. J., and Merriman, Curtis. "A Study of the Effectiveness of Two Kinds 
 of Teaching Material," Introduction to Education. Boston: Ginn and Company, 1929, p. 420-24. (.25 ) 
 
 43 Monroe, W. S. "The Ability to Place the Decimal Point in Division," Elementary SchoolJ our nal, 
 18:287-93, December, 1917. (77) 
 
 44 Drushel, J. A. "A Study of the Amount of Arithmetic at the Command of High-School Grad- 
 uates Who Have Had No Arithmetic in Their High-School Course," Elementary School Journal, 
 17:657-61, May, 1917. (35) 
 
 "-Winch, W. H. "Should Young Children Be Taught Arithmetical Proportion?" Journal of 
 Experimental Pedagogy, 2:79-88, 319-30, 406-20; June, 1913; June 5, December 5, 1914; 3:89-95, 
 June 5, 1915. (126) 
 
Summary of Research Relating to the Teaching of Arithmetic 27 
 
 pedagogical circumstances, be expected to undertake the work at as 
 early an age as the others." 
 
 Springer 46 conducted an experiment in which the effectiveness of 
 memorizing tables of cubic and linear measure was compared with 
 the effectiveness of using the facts of these tables in connection with 
 problems. The conclusions favor isolated memorizing of denominate- 
 number facts rather than attempting to learn them in connection 
 with the solving of problems in which they occur. 
 
 2. Evaluation of the experiments. The studies of Collier (30); 
 Clapp, Chase, and Merriman (25); Winch (126); and Springer (109) 
 were experimental in nature. Collier (30) used two groups of fifth- 
 grade children, each of which numbered four individuals. No attempt 
 was made to secure equivalence, and the experiment lasted only five 
 days. It was observed that the experimental pupils learned to mul- 
 tiply fractions more quickly than did the control pupils. It is evident 
 that this was a very crude experiment. Its faults are many: small 
 groups, lack of equivalence, short duration, inadequate measurement 
 of gains, and so on. The conclusion that children should be taught to 
 multiply fractions through addition of fractions seems reasonable, but 
 Collier's evidence in support of this conclusion is of doubtful value. 
 
 Clapp, Chase, and Merriman (25) employed twenty-three pairs of 
 groups of unreported size. Both groups of a pair were taught in the 
 same room by the same teacher. Equivalence was sought with 
 respect to intelligence and initial ability in arithmetic. The duration 
 of the experiment is not stated. At the end of the experiment three 
 tests of eight percentage problems and two other problems each were 
 administered. The results in twenty out of the twenty-three rooms 
 favored the experimental factor — the novel percentage practice ma- 
 terial. Clapp, Chase, and Merriman are to be commended for using 
 so many pairs of groups, for attempting to secure equivalence with 
 respect to two important pupil characteristics, and for the attempt to 
 control non-experimental factors by having the same teacher instruct 
 both experimental and control children. Instruction of a pair of 
 classes by the same teacher, however, does not necessarily insure com- 
 plete control of the non-experimental factors. Since the practice 
 material was novel, it would not be unreasonable if there was some 
 lack of equivalence in the teacher factors of zeal and effort. Further- 
 more, the merit of the experiment is possibly obscured by the method 
 of reporting. One wishes for data relative to the sizes of the groups, 
 to the degree of equivalence secured, and to the differences in gains in 
 
 46 Springer, Isidore. "Teaching Denominate Numbers," Journal of Educational Psychology, 
 6:630-32, December, 1915. (109) 
 
IS 
 
 28 Bulletin No. 58 
 
 achievement along with measures of the "statistical" significance of 
 these differences. 
 
 The report by Winch (126) refers to five single-group experiments 
 and one control-group experiment. The single groups varied in 
 size from 39 to 361. The smallest group was located in a school in a 
 good district; the rest were located in schools in the poorer districts of 
 London, England. There was no attempt in the single-group experi- 
 ments to control non-experimental factors. These experiments lasted 
 from three to five months in the different groups. At the close of each 
 experimental period, informal tests were administered and the im- 
 provement was noted. In the controlled experiment, two groups of 
 twenty-three English school girls averaging nine years of age were 
 equated with respect to initial arithmetical ability, as revealed by a 
 series of preliminary tests. One group was taught in the usual 
 fashion, while the other group was instructed in proportion by the 
 method of unity. After three practice periods of 17, 16, and 22 min- 
 utes' duration, two of the preliminary tests were repeated. The 
 difference in achievement favors the method of unity, but since thi 
 difference is but 2.5 times its probable error, it may not be regarded 
 as "statistically" significant. Winch is to be commended for his care- 
 ful analysis of the method of instruction used, for repeated experi- 
 ments, and for his attempts to allow for the influences of non-ex- 
 perimental factors even where control groups were not used. 
 
 Springer (109) used two sixth-grade groups of fifty pupils each. 
 Equivalence was secured with respect to initial ability in arithmetic 
 as revealed by a test of arithmetical problems and with respect to 
 language ability as shown by a language test. The experimental 
 factor does not appear to have been adequately defined and isolated, 
 and, although the groups were rotated, the control of the non-experi- 
 mental factors was not satisfactory. The experiment is also to be 
 criticized for its short duration — six periods of ten minutes each. 
 The differences in achievement in favor of the isolated learning of the 
 denominate-number facts appear to be fairly significant, although no 
 standard or probable errors are given. The experiment is to be criti- 
 cized for its failure to secure adequate control of non-experimental 
 factors, as well as for its short duration. 
 
 Monroe (77) collected his data relative to the abilities required in 
 placing the decimal point in division by means of four tests lasting 
 one minute each, which were administered to seventy-eight sixth-, 
 seventh-, and eighth-grade pupils. Anspaugh (2) merely reports 
 what happened in a few elementary schools as a result of greater 
 attention to the mastery of the fundamental number combinations. 
 
Summary of Research Relating to the Teaching of Arithmetic 29 
 
 His study may be termed "experimental" only in the sense any trial 
 of a new method is experimental. Drushel (35) collected his data by 
 the administration of his test to 624 entering college freshmen. The 
 test results revealed that the method in which the divisor is rendered 
 an integer by multiplying both dividend and divisor by 10 or by some 
 power of 10 is significantly the better method. Considering the 
 number of cases on which it is based, the "statistically" significant 
 differences in achievement, and the approximate equivalence of the 
 groups in general arithmetical ability, this conclusion seems quite 
 dependable. This investigation, however, is not an experiment and 
 consequently the degree of control of non-experimental factors is 
 unknown. Hence, the superiority of "rendering the divisor an integer 
 by 10 or some power of 10" cannot be said to have been demonstrated. 
 3. Justified conclusions. The crudity of the experiments de- 
 scribed prevent the listing of justified conclusions. 
 

 CHAPTER III 
 
 DRILL IN THE FUNDAMENTALS 
 
 Consideration is given first in this chapter to the experiments 
 which have been conducted for the purpose of revealing the effect of 
 drill in the fundamentals. Attention is given next to the relative 
 merits of systematic and incidental instruction in calculation. This 
 is followed by a summary of the investigations in which the type of 
 learning exercises was made the experimental factor. The chapter 
 closes with an evaluation of the research on methods of distributing 
 practice time in arithmetical calculation and on the influence of 
 requests for speed and for accuracy on achievement in the funda- 
 mentals. 
 
 THE EFFECT OF SYSTEMATIC DRILL IN THE FUNDAMENTALS 
 
 1. Summary of reported conclusions. Studies of the effect of a 
 period of systematic drill on achievement in arithmetical calculation 1 
 have produced evidence in support of the wide-spread belief that 
 ability to add, subtract, multiply, and divide may be increased by 
 systematic drill. Hagen 2 is the only investigator whose findings are 
 not in entire agreement with this belief. 
 
 2. Evaluation of the experiments. Although Brown's study (13) 
 is the earliest of this group, the technique used seems to have been 
 superior to the techniques of any of the later experiments. In the 
 first of Brown's studies, two groups of twenty-five sixth-, seventh-, 
 and eighth-grade children were paired on the basis of their initial 
 ability in arithmetic. The arithmetic instruction of one of the groups 
 differed from that of the other in that five minutes of each of thirty 
 recitation periods were devoted to drill in the four fundamentals. 
 At the end of the experiment, a final test, similar to the initial test by 
 which the groups were equated, was administered. The second exper- 
 
 'Brown, J. C "An Investigation on the Value of Drill Work in the Fundamental Operations of 
 Arithmetic," Journal of Educational Psychology, 2:81-88, February, 1911; 3:485-92, 561-70; November. 
 December, 1912. (13) 
 
 Burton, C. B. "Results of Definite Drill in the Four Fundamental Processes as Shown by the 
 Woody-McCall Mixed Fundamentals," Fifth Yearbook of the Department of Elementary School Principals. 
 Washington: National Education Association, 1926, p. 323-28. (19) 
 
 Kerr, M. A. "Effects of Six Weeks Daily Drill in Arithmetic," Studies in Arithmetic, Indiana 
 University Studies No. 32. Bloomington: Indiana University, 1916, p. 79-95. (56) 
 
 Phillips, F. M. "Value of Daily Drill in Arithmetic," Journal of Educational Psychology, 4:159-63, 
 March, 1913. (100) -vs.. 
 
 Smith, J. H. "Individual Variations in Arithmetic," Elementary School Journal, 17:195-200, 
 November, 1916. (107) 
 
 Wiramer, H. "Experimental Study of the Effects of Drill in Arithmetic Processes under Yarv- 
 mg Conditions," Indiana University Studies No. 32. Bloomington: Indiana University, 1916, 
 p. 96-102. (124) 
 
 2 Hagen, H. H. "A Study of Practice Periods in Arithmetic Fundamentals," Chicago Principals' 
 Club, Second Yearbook. Chicago: Chicago Principals' Club, 1927, p. 93-95. (45) 
 
 30 
 
 

 Summary of Research Relating to the Teaching of Arithmetic 31 
 
 iment was similar to the first with the exception that 222 pupils in 
 four schools participated for twenty recitation periods. Brown is to 
 be commended for the techniques which he used in securing equiv- 
 alent groups, for his care in controlling non-experimental factors, and 
 for his elaborate analysis of the data. He is also to be commended for 
 repeating his experiment with pupils in several schools and in different 
 cities. His differences in gains in achievement, secured in this way, 
 are of sufficient magnitude to support adequately his conclusion with 
 respect to the effect of systematic drill of five minutes per day on 
 achievement in arithmetical calculation. 
 
 The other five studies of the effect of systematic drill are subject 
 to criticism. Kerr (56) used 423 sixth-, seventh-, and eighth-grade 
 children in her single-group experiment. These children received five 
 minutes of drill in addition, daily, for a period of six weeks. The 
 application of an initial and a final test showed a gain in ability to 
 add, but the significance of this gain is obscured by the failure of the 
 experimenter to employ a control group. Phillips (100) used two 
 groups of thirty-four and thirty-five sixth-, seventh-, and eighth- 
 grade children. After these pupils had been paired on the basis of 
 initial ability in arithmetic, the members of the experimental group 
 were given ten minutes of daily drill in the fundamental operations 
 and with reasoning problems (mental arithmetic). At the end of two 
 months the final test showed a "statistically" significant gain for the 
 drill group. The techniques employed by Phillips seem much superior 
 to those employed by Kerr (56) , but his experiment does not seem to be 
 without fault. The size of his groups was small, and the instructional 
 conditions were not entirely normal. Smith (107) used three fifth- 
 and sixth-grade classes of unreported size. No attempt was made to 
 secure equivalence. One class received what amounted to diagnosis 
 and remedial treatment during drill. The second class received extra 
 drill for the inferior pupils. The third class was merely drilled. 
 After three drill periods per week of twenty-five minutes each for 
 four weeks the final tests were administered. The magnitude of the 
 gains in achievement seems to warrant the statement: "All three 
 types of drill produced very large increases in the achievement of the 
 pupils." The conclusions which state that the first type of drill is 
 significantly superior to the other two would seem to be less depend- 
 able. Smith is to be criticized for failure to secure equivalent groups, 
 for evidently poor control of the time factor, and for failure to report 
 the size of his groups. With respect to the comparative value of 
 drill, this must be regarded as a single-group or uncontrolled 
 experiment. 
 
 
32 Bulletin No. 58 
 
 Wimmer (124) employed fifth-, sixth-, seventh-, and eighth-grade 
 pupils. The pupils in the sixth grade were divided into two appar- 
 ently equivalent groups of twenty-two pupils each. The other classes 
 which averaged about thirty-five pupils each were used as single 
 groups. The Courtis Standard Test, Series A, was administered at 
 the beginning of the experiment, at the end of six weeks, and again at 
 the close of the experiment — twelve weeks from the beginning. Com- 
 parisons are made between the gains of the different classes and be- 
 tween the two groups of the sixth-grade class. The classes which had 
 systematic drill made the greater gains, but the magnitude of the 
 differences in gains is obscured by faulty or complete lack of equiv- 
 alence. The gains are large enough, however, for the classes which 
 had drill to justify the conclusion that "it pays to give regular drill 
 work in arithmetic." 
 
 Burton (19) employed 2500 third-, fourth-, fifth-, sixth-, seventh-, 
 and eighth-grade pupils in the white rural schools of a county in one 
 southern state. Systematic drill was administered ten minutes daily 
 for a period of six weeks. Curves are given to show the consistent 
 gains in efficiency made by the pupils. The experimenter is to be 
 commended for the large number of pupils used, but he is to be criti- 
 cized for not using some of the pupils for control purposes. 
 
 Hagen (45) employed twelve pairs of groups of fourth-, fifth-, 
 sixth-, and seventh-grade pupils which were equated on the basis of 
 intelligence test scores. Each teacher participating in the experiment 
 taught a pair of groups. One of the groups of each pair received 
 systematic drill in fundamental problems twice each day, while the 
 other group received the drill once a day. After three months of such 
 insttuction the final test was administered. The difference in achieve- 
 ment, when the gains of all the groups are averaged, slightly favors 
 drill once a day. That this difference is not of much significance is 
 shown by the fact that in six of the twelve pairs of groups the mean 
 differences in achievement slightly favor the use of drill twice a day. 
 The following statement of Buswell relative to the experiment seems 
 justified: "Data might be interpreted differently." 3 
 
 3. Justified conclusions. If the Law of Exercise is accepted, it is 
 obvious that pupils who have not attained their maximum skill in 
 arithmetical calculation will profit from systematic drill, especially 
 when the drill is conducted in a way that stimulates a desire to in- 
 crease achievement in this field. Consequently this group of six 
 studies may be labelled as "attempts to prove the obvious." The 
 
 3 Bus\vell, G. T. "Summary of Arithmetic Investigations,*' Elementary School Journal, 28:705, 
 May, 1928. 
 
Summary of Research Relating to the Teaching of Arithmetic 33 
 
 conclusions, except possibly certain incidental details, are merely 
 what should have been anticipated. 
 
 THE RELATIVE VALUE OF SYSTEMATIC VERSUS INCIDENTAL 
 TEACHING OF CALCULATION 
 
 1. Summary of reported conclusions. Meriam 4 and Collings 5 se- 
 cured results that favored incidental teaching of calculation, but 
 Gates, Batchelder, and Betzner 6 have reported that the differences in 
 arithmetic achievement in their experiment favored the "systematic" 
 rather than the "opportunistic" method instruction. Wilson 7 has 
 reported recently that incidental instruction of the informational type 
 is just as effective as instruction of the traditional type, so far as the 
 first two grades are concerned, and that a combination of both types 
 with more emphasis on systematic drill results in very superior 
 arithmetical achievement in the third grade. 
 
 One of the conclusions of the investigation recently reported by 
 Olander (94) may be interpreted in favor of systematic teaching of 
 calculation: 
 
 Examination of the scores of one group of children who had no formal instruc- 
 tion in arithmetic for twelve out of the seventeen weeks of the experiment and of 
 another group who had no formal arithmetic instruction whatsoever during the 
 entire seventeen weeks shows that, during the time when no class instruction in 
 numbers was being given, the children learned from approximately a third to less 
 than a half as many number combinations as did the children who were being 
 given the regular class instruction. 
 
 2. Evaluation of the experiments. Meriam (73) merely reported 
 a comparison of grades in high school of 362 pupils who had received 
 incidental instruction in arithmetic, in the elementary school, with 
 the grades of those who had had the more traditional form of instruc- 
 tion. The findings of such an investigation cannot be accepted as 
 conclusive, in any sense. There were too many factors unaccounted 
 for which may have influenced the results. 
 
 Collings (31) used forty-one pupils in one rural school as his exper- 
 imental group and sixty pupils in two other rural schools as his control 
 group. The initial arithmetic test revealed the fact that the experi- 
 mental pupils were slightly inferior to the control pupils in ability in 
 the four fundamentals. Collings also presents much evidence relative 
 to the approximate equivalence with respect to reading ability, hand- 
 
 «Meriam, J. L. "How Well May Pupils Be Prepared for High School Work without Studying 
 Arithmetic, Grammar, etc., in the Grades?" Journal of Educational Psychology, 6:361-64, June, 1913. (73) 
 
 5 Collings, Ellsworth. An Experiment with a Project Curriculum. New \ork: Macmillan 
 Company, 1923. 346 p. (31) 
 
 6 Gates, A. I., Batchelder, M. I., and Betzner, Jean. "A Modern Systematic versus an Oppor- 
 tunistic Method of Teaching," Teachers College Record, 27:679-700, April, 1926. (40) . . ,, 
 
 7 Wilson, G. M. "New Standards in Arithmetic: A Controlled Experiment in Supervision, 
 Journal of Educational Research, 22:351-60, December, 1930. (123) 
 
34 Bulletin No. 58 
 
 writing ability, spelling ability, chronological age, number of years of 
 schooling, number of years spent in the experimental schools, 
 school attitudes, community attitudes, social and economic status of 
 the districts, parentage of children, length of school term, course of 
 study, and so on. After four years of the project curriculum in the 
 experimental school and four years of the traditional curriculum in 
 the control schools the final tests were administered. With respect to 
 ability in the four fundamentals, the differences favor, but not sig- 
 nificantly, the informal method. Collings has been criticized for his 
 failure to control important non-experimental factors: 
 
 In the experiment by Collings the children taught by the project method 
 achieved more than those taught by the traditional method, but it appears from 
 Collings' report that these teachers worked much harder at their task than did 
 the teachers in the control schools. In view of this fact, it does not appear 
 justifiable to ascribe the superior achievement of the project-method group 
 entirely to the method of instruction. 8 
 
 Gates, Batchelder, and Betzner (40) employed two groups of 
 twenty-five first-grade children who were approximately equivalent 
 with respect to such traits as sex, chronological age, mental age, 
 general information, speed of reading, oral spelling, and so on. The 
 group subjected to the opportunistic method was somewhat inferior 
 to the other group in initial ability in oral arithmetic. Techniques 
 used to control teacher factors are described in the following 
 quotation: 
 
 Both teachers were interested in the project as an experimental study; both, 
 understanding that the results would in no way reflect upon their professional 
 reputation, taught their pupils as under ordinary circumstances except for certain 
 imposed limitations and regulations which were cheerfully accepted and faith- 
 fully observed. Both teachers followed the same general schedule, the same time 
 assignment to different phases of the work, recesses, lunch periods, assembly 
 music, gymnasium work, and so forth. Neither teacher gave any out-of-school 
 time to individual pupils nor allowed others to do so; neither suggested home 
 work, and each as far as possible, prevented it. Neither was given any assistance 
 in teaching; neither enjoyed any advantage in clerical or other help, in funds for 
 materials, in special demonstrations, and so on. 
 
 It is the opinion of the present writers that the techniques used to 
 control the teacher factors and the other non-experimental factors in 
 this experiment were superior to those used by Collings ( 31 ). "Each of 
 the two methods, 'the modern systematic' and the 'opportunistic,' 
 was followed by an exceptionally able teacher who was experienced 
 in the method and believed it to be, on the whole, the best one." 9 
 
 8 Monroe, W. S. and Engelhart, M. D. "Experimental Research in Education," University of 
 Illinois Bulletin, Vol. 27, No. 32, Bureau of Educational Research Bulletin No. 48. Urbana: Uni- 
 versity of Illinois, 1930, p. 36. 
 
 9 Gates, Batchelder, and Betzner, op. cit., p. 682. 
 

 Summary of Research Relating to the Teaching of Arithmetic 35 
 
 If this was the case, it would seem that the teacher factors, skill and 
 zeal, were rather adequately controlled. 
 
 The difference in achievement, as revealed by the final test in 
 arithmetic at the end of the year, was 2.5 times the probable error of 
 the difference. As such, the difference may be regarded as possibly 
 "statistically" significant. A limitation of this experiment, so far as 
 arithmetic is concerned, is the lack of equivalence in arithmetic 
 ability at the beginning of the experiment. Some of the difference in 
 the final achievement in arithmetic may be attributed to the initial 
 superiority of the systematic group. Hence, it appears that the dif- 
 ference should not be interpreted as more than suggestive. 
 
 Wilson (123) compared the scores of 475 pupils completing the 
 second grade, who had received informal or incidental instruction in 
 arithmetical calculation, with the scores of one group of 174 second- 
 grade pupils and one group of 154 third-grade pupils, who had re- 
 ceived the traditional formal type of instruction. These data support 
 the contention that up to the close of the second grade the informal 
 type of arithmetical instruction results in achievement equal to, and 
 possibly superior to, the achievement resulting from formal instruc- 
 tion. In the later phases of Wilson's experiment over one thousand 
 third-grade children were subjected to a combination of incidental 
 and systematic instruction. One day a week during the third year 
 was devoted to incidental instruction of the informational type, while 
 the other four days were devoted to systematic drill on addition and 
 subtraction. The tables of test results indicate that the pupils at- 
 tained a very high level of achievement in addition and subtraction. 
 W T hile Wilson's conclusion seems well supported by his data, one 
 wonders whether too much emphasis was not placed on the informal 
 aspect of the instruction and too little recognition given to the part 
 played by systematic drill in securing the superior achievement of 
 the third-grade children. 
 
 In dander's experiment (94) one group of one hundred second- 
 grade pupils received no instruction in arithmetic for the last twelve 
 of the seventeen weeks of the experiment. 10 Another group of eighty- 
 six pupils received no formal arithmetic instruction during the entire 
 seventeen weeks. The achievements of these groups were compared 
 with each other and with the achievement of a group of 296 pupils 
 receiving daily instruction. The initial ability of the group of eighty- 
 six was considerably superior, and that of the group of one hundred, 
 slightly superior, to the initial ability of the group receiving daily 
 
 10 See pages 17 to 18 for evaluation of this experiment, which had to do with the effectiveness of 
 generalization instruction. 
 
36 Bulletin No. 58 
 
 instruction. It would seem, therefore, that the differences in favor 
 of systematic daily instruction are rather highly reliable. 
 
 3. Justified conclusions. The conflicting conclusions of the ex- 
 periments evaluated prevent the formulation of a justified conclusion 
 favoring either the incidental or the traditional method of instruction 
 in arithmetic. The question as to which method is superior awaits 
 further experimental investigation. In view of the relatively specific 
 character of calculation abilities and the demonstrated efficacy of 
 systematic drill, it is difficult to conceive of the incidental method 
 alone as highly efficient. It is possible that the best method would 
 be a combination of the two procedures. 
 
 THE RELATIVE MERITS OF CERTAIN GENERAL TYPES OF LEARNING 
 EXERCISES FOR DRILL IN CALCULATION 
 
 1. Summary of reported conclusions. Ten experimental investi- 
 gations are summarized under this heading. Evans and Knoche 11 
 have reported that drill in which Studebaker Economy Practice 
 Exercises are used results in achievement superior to that resulting 
 from the use of learning exercises based on materials devised by the 
 teacher. Kelly 12 compared the effectiveness of the Courtis Standard 
 Practice Tests, the Studebaker Economy Practice Exercises, and 
 "the best methods of drill which the teachers could devise." He 
 reported that the Courtis drill material is superior to the Studebaker 
 material, but that both are superior to drills devised by the teachers. 
 Mead and Johnson 13 compared the Courtis Standard Practice Tests 
 with the Thompson Minimum Essentials and reported a conclusion 
 favorable to the Courtis material. Morgan 14 compared the effective- 
 ness of the Economy Remedial Exercise Cards when used with the 
 Compass Diagnostic Tests to that of Lennes' Pads and reported a 
 conclusion favorable to the former. Newcomb 15 found that drill 
 exercises prepared in such a way that proportionate drill is given on 
 the higher decades are more effective than those ordinarily used. 
 Fowlkes 16 concluded that it is desirable "to teach the one hundred 
 combinations (multiplication) by means of text material alone, the 
 teacher doing as little talking as possible" and "to make remedial 
 adjustments by means of printed directions and devices rather than 
 
 n Evans, J. E. and Knoche, F. E. "The Effects of Special Drill in Arithmetic as Measured by the 
 Woody and the Courtis Arithmetic Tests," Journal of Educational Psychology, 10:263-76, May-June, 
 
 12 Kelly, F. J. "The Results of Three Types of Drill on the Fundamentals of Arithmetic,** Journal 
 of Educational Research, 2:693-700, November, 1920. (55) 
 
 "Mead, C D. and Johnson, C W. "Testing Practice Material in the Fundamentals," Journal of 
 Educational Psychology, 9:287-97, May, 1918. (71) 
 
 14 Morgan, L. D. "Specific vs. General Drill in the Fundamentals of Arithmetic," School Science 
 and Mathematics, 29:528-29, May, 1929. (80) 
 
 15 Newcomb, R. S. "Effective Drill Exercises in Arithmetic," Journal of Educational Psychology, 
 16:127-31, February, 1925. (88) 
 
 16 Fowlkes, J. G. "A Report of a Controlled Study of the Learning of Multiplication by Third- 
 Grade Children," Journal of Educational Research, 15:181-89, March, 1927. (38) 
 
Summary of Research Relating to the Teaching of Arithmetic 37 
 
 oral instruction." Knight 17 has reported a conclusion which favors 
 "drill material carefully constructed as to the distribution of practice 
 in addition, subtraction, multiplication, and division of whole num- 
 bers," rather than drill material "slightly in excess as to sheer amount 
 but so built that certain combinations were slighted." 18 The con- 
 clusions of Newcomb (88),Fowlkes (38), and Knight (60) all favor 
 the contention that the relative difficulty of the number combinations 
 must be accounted for in preparing efficient materials of instruction 
 for use in drill. 
 
 Kulp 19 investigated the relative effectiveness of two types of prac- 
 tice material, the essential difference between the two being that one 
 of the types provided practice in solving reasoning problems in con- 
 nection with computational drill. It is reported that the material 
 which provided practice in arithmetical reasoning was relatively more 
 effective in securing computational achievement, and that its use 
 resulted in a decided increase in arithmetical reasoning ability. A 
 similar conclusion is reported by Rosse. 20 These conclusions seem to 
 agree with that reported by Kirkpatrick 21 several years ago. Kirk- 
 patrick found that use in calculation is a more effective means of 
 learning the multiplication combinations than memorization divorced 
 from use. 
 
 Myers and Myers 22 investigated the problem of whether it was 
 better to find mistakes among a group of examples of addition, multi- 
 plication, and subtraction combinations than to think of the corre- 
 sponding correct associations. Their results are favorable to learning 
 exercises which emphasize correct associations rather than learning 
 exercises which demand the observation of errors. It is interesting to 
 note, among their conclusions, that pupils thought the discovery of 
 errors made by other people much more interesting than the drill in 
 which correct associations were exercised. 
 
 The problem of whether learning exercises should be restricted to 
 one arithmetical operation or should deal with more than one has 
 been studied in three experiments. Buckingham 23 sought to deter- 
 mine whether it is "better to teach subtraction facts in connection 
 
 ^Knight, F. B. "The Superiority of Distributed Practice in Drill in Arithmetic," Journal of 
 Educational Research, 15:157-65, March, 1927. (60) 
 
 isKnight summarizes in this article the report of an experiment conducted by Luse. bee: 
 
 Luse, E. M. "Transfer within Narrow Mental Functions, A Study of the Effects of Distributed 
 versus Non-Distributed Drill in Arithmetic,'* University of Iowa Monograph in Education No. 5. 
 Iowa City: University of Iowa. (61) . 
 
 19 Kulp, C . L. "A Study of the Relative Effectiveness of Two Types of Standard Arithmetic Practice 
 Materials," Journal of Educational Research, 22:381-87, December, 1930. (65) 
 
 2u Rosse, J. C. "An Experiment to Test the Increase in Reasoning Ability from the Use ot lest 
 and Practice Sheets in 6A Arithmetic, '• Journal of Educational Research, 22:210-13, October,1930. (105) 
 
 21 Kirkpatrick, E. A. "An Experiment in Memorizing versus Incidental Learning, Journal of 
 Educational Psychology, 5:405-12, September, 1914. (58) . . . 
 
 22 Myers, G. C and Myers, C E. "Finding Mistakes versus Correct Associations in Simple 
 Number-Learning," Journal of Educational Research, 18:25-31, June, 1928. (86) m 
 
 ^Buckingham, B. R. "Teaching Addition and Subtraction Facts Together or Separately, 
 Educational Research Bulletin (Ohio State University), 6:228-29, 240-42; May 25, 1927. (17) 
 
38 Bulletin No. 58 
 
 with related addition facts than to teach the addition facts first and 
 the subtraction facts afterward." His conclusions favor the teaching 
 of addition and subtraction together. Myers and Myers 24 prepared 
 learning exercises which required the pupils to shift rapidly among 
 the four fundamental operations. Their conclusions are distinctly 
 unfavorable to such mixed exercises. ". . . . rapid shifting by the 
 pupil from one process to another not only causes great confusion of 
 processes, but the pupil so confused also tends to be more confused 
 when he later works on combinations grouped twenty-five to a proc- 
 ess." Repp 25 prepared two sets of drill material the objective of 
 which was the maintenance of skill. Each of the exercises of one set 
 of material dealt with a single topic, such as addition of fractions, 
 while each exercise of the other set of material was of mixed nature. 
 This difference in organization was the only difference in the content 
 of the two sets of drill material. The conclusions are distinctly favor- 
 able to the mixed type of drill material as a basis of learning exercises 
 for the maintenance of skills in arithmetic. "All pupils profited by 
 use of drills furnished them, but those using mixed drills showed 23 
 per cent greater gain than those using isolated drills." 
 
 2. Evaluation of experiments. Evans and Knoche (37) used two 
 groups of sixth-grade children of unreported size. With respect to 
 equivalence they state that "the children in the two rooms were quite 
 similar in ability. The 6A class was one semester in advance of the 
 6B group." The pupils of the 6B class were drilled with the Stude- 
 baker Economy Practice Exercises five minutes each day for forty- 
 three days, the time being taken from their regular arithmetic work. 
 The tests administered at the end of the experimental period yielded a 
 probably significant difference in mean gain for the group using the 
 Studebaker Exercises. The experimenters are to be criticized for not 
 attempting to secure equivalent groups and for utilizing pupils whose 
 arithmetic instruction, other than that inherent in the experimental 
 factor, differed so greatly. It is stated that during the period of drill 
 "the main work for the 6A grade was percentage with a general review 
 of the fundamental processes. The work of the 6B grade was deci- 
 mals." It is possible that the zeal of the teacher for the novel practice 
 material was another uncontrolled factor. 
 
 Kelly (55) used three groups of 133, 146, and 173 fourth-, fifth-, 
 sixth-, seventh-, and eighth-grade children, making no effort to secure 
 equivalence. The groups used the Courtis Standard Practice Tests, 
 
 24 Mvers, G. C and Myers, C E. "The Cost of Quick Shifting in Number Learning," Educational 
 Research Bulletin (Ohio State University), 7:327-34, October 31, 1928. (85) 
 
 25 Repp, A. C "Mixed versus Isolated Drill Organization," Twenty-Xinth Yearbook of the National 
 Society for the Study of Education, Bloomington, Illinois: Public School Publishing Company, 1930, 
 p. 535-49. (103) 
 
Summary of Research Relating to the Teaching of Arithmetic 39 
 
 the Studebaker Economy Practice Exercises, or informal exercises 
 prepared by the teachers for eight to fifteen minutes of drill per day, 
 depending on the grade level, for twenty successive days. The tech- 
 niques used in this experiment are open to criticism. A lack of 
 equivalence is indicated by the unequal representation of the different 
 school grades in each of the groups. For example, there were no 
 fourth-grade children in the group using the Courtis material and no 
 VA or VI B pupils in the group using the Studebaker material. Failure 
 to control important teacher factors is indicated in the statement that 
 "The differences from class to class by the same method suggest that 
 after all the efficiency of any method depends mostly on the teacher 
 who is using it." 
 
 Mead and Johnson (71) used two groups of 105 fifth- and sixth- 
 grade pupils. No attempt was made to secure equivalence, and the 
 preliminary tests reveal some departures from equivalence. The 
 pupils of one group practiced ten minutes a day with the Courtis 
 material, while the pupils of the other group used the Thompson 
 material. No attempt was made to prevent home practice, it being 
 felt by the experimenters that if a practice material stimulated such 
 practice such stimulation should be allowed to operate during the 
 experiment. After ninety days of practice the Courtis Research Test 
 was administered, the results of which were possibly significantly in 
 favor of the Courtis Standard Practice Tests. This experiment is 
 faulty in that no effort was made to secure equivalence or to control 
 practice time. Precision in experimentation demands that pupils of 
 experimental and control groups spend an equal amount of time in 
 learning. Another possible fault is that the Courtis Research Test 
 would be more valid with respect to the Courtis drill material than 
 with respect to the Thompson drill material. 
 
 Morgan (80) used two groups of twenty-eight fourth-grade pupils. 
 The groups were equated on the basis of average scores made on two 
 standardized arithmetic tests. One group used the Economy Reme- 
 dial Exercise Cards and was subjected to the Compass Diagnostic 
 Tests, while the other group merely used practice pads prepared by 
 Lennes. Both groups were taught by the same teacher for a period of 
 twelve weeks. At the end of this period, the other forms of the initial 
 tests were administered, and the average scores, computed. The 
 difference in mean gains significantly favors the group that used the 
 Economy Remedial Exercise Cards and that had the Compass Diag- 
 nostic Tests administered to it. There seems little reason to doubt 
 the reliability of the findings, but it is impossible to ascribe the supe- 
 rior achievement of the group which excelled to the practice material 
 
 
40 Bulletin No. 58 
 
 or to the diagnostic tests. It would seem, therefore, that the chief 
 criticism which may be made with respect to this experiment has to 
 do with the failure of the experimenter to restrict the experimental 
 factor to a single technique. 
 
 Newcomb (88) used an experimental group of fifty-one pupils and 
 a control group of twenty-one seventh-grade pupils. With respect to 
 equivalence he states, "A comparison of the intelligence quotients of 
 the pupils of the several classes did not reveal on the whole any ap- 
 preciable differences." The experimental group was practiced five or 
 six minutes a day for thirty-five days on drill material which provided 
 practice on the higher decades, while the instruction of the control 
 group was conducted "in the usual manner." The administration of 
 the Courtis Standard Research Test at the close of the experiment re- 
 vealed the probably significantly superior achievement of the experi- 
 mental group. Newcomb is to be criticized for not securing more 
 adequate equivalence of groups and for not specifying the type of 
 learning activity engaged in by the control pupils. It is possible that 
 greater zeal was exerted by the teachers in utilizing the experimental 
 drill material, since the failure to mention the type of drill material 
 used by the control pupils would indicate a lack of enthusiasm for it. 
 
 Fowlkes (38) used a single group of thirty-one third-grade pupils 
 whose median I. Q. was 104.5. This group of pupils was drilled on 
 multiplication twenty minutes a day for twenty days by means of the 
 text material alone, "the teacher doing as little talking as possible," 
 and remedial adjustments were made by printed directions and de- 
 vices. There resulted from this instruction achievement which is 
 claimed by the author to be significantly better than that of other 
 third-grade classes. While a single-group technique is not usually to 
 be relied upon, the fact that Fowlkes was able to compare his results 
 with those of other third-grade classes would give his conclusions 
 some dependability. It is possible that he should have allowed for 
 the somewhat superior intelligence of his third-grade class in formu- 
 lating his conclusions. 
 
 Luse, as reported by Knight (60), used two groups of three hun- 
 dred fifth-grade pupils which were equivalent with respect to general 
 arithmetic ability. One of these groups used carefully constructed 
 material, while the other employed material which slighted certain of 
 the number combinations. "All other conditions were held constant." 
 After fifty consecutive drill periods of fifteen minutes each, the final 
 tests were administered. The differences in achievement were prob- 
 ably "statistically" significant in favor of drill material in which 
 practice is carefully distributed over the number combinations. The 
 
Summary of Research Relating to the Teaching of Arithmetic 41 
 
 techniques used in this experiment compare favorably with the best 
 of contemporary experimental research in education. 
 
 In the experiment of Kulp (65) four classes used the practice ma- 
 terial which did not provide practice in arithmetical reasoning, while 
 six classes used the material which did. A total of 113 fourth-grade 
 pupils took the final test. It is evident from the figures given in the 
 report of the investigation that the experimental and control groups 
 were initially equivalent in computational ability, but that the group 
 receiving the training in reasoning was initially superior in reasoning 
 ability. The teacher factor, experience with instructional procedure, 
 favored the practice material which did not provide practice in solving 
 reasoning problems, but it is possible that the influence of this experi- 
 ence was offset by the usually occurring greater zeal for a new method 
 or procedure. The experiment lasted from October to April. The 
 differences in gains in achievement are apparently significantly in 
 favor of the type of material which provided practice in arithmetical 
 reasoning in connection with calculation drill. The investigator is to 
 be criticized for failure to secure more adequate equivalence at the 
 beginning of the experiment, and for failure to indicate more clearly 
 the differences in gains in achievement and the "statistical" signifi- 
 cance of these differences. The investigator is to be commended for 
 his careful description of the compared factors, for measures taken to 
 control non-experimental factors, and for conducting his experiment 
 over a comparatively long period of time. His conclusions would 
 seem to be fairly dependable with respect to the groups used in the 
 experiment. Further experimentation is needed before generalization 
 is justified. 
 
 Rosse (105) used two groups of eighteen sixth-grade pupils which 
 were equivalent with respect to initial arithmetic reasoning ability 
 and with respect to intelligence as measured by the Otis Arithmetic 
 Reasoning Test and the National Intelligence Test. One group used 
 practice sheets which provided drill in reasoning problems, while the 
 other group used an ordinary arithmetic text. At the end of fifty- 
 eight days the same form of the Otis Arithmetic Reasoning Test was 
 administered. The difference in achievement favors, but not signifi- 
 cantly, the method in which the practice sheets which provided drill 
 in reasoning problems were used. While the conclusions do not seem 
 to be highly dependable because of the size of the groups used, be- 
 cause of the lack of control of important non-experimental factors, 
 and because of the unreliability of the difference reported, they may 
 be accepted as evidence supplementing that presented by Kulp (65). 
 
 Kirkpatrick (58) used two groups of ten and two groups of twenty- 
 
42 Bulletin No. 58 
 
 five normal-school students and two groups of twenty sixth-grade 
 pupils, making no attempt to secure equivalence of groups. No men- 
 tion is made of any procedures used to secure control of non-experi- 
 mental factors. The groups were tested at the end of ten days, and 
 the normal-school students, again at the end of three weeks. The 
 differences in achievement in each case favored the method of learning 
 multiplication combinations through use. It is evident that this 
 experiment may not be regarded as other than crude. Since no 
 attempt was made to secure equivalence of groups, or to control ade- 
 quately non-experimental factors, the differences in achievement may 
 not with certainty be ascribed to the method reported superior. 
 Myers and Myers (86) used two groups of one hundred fourth- 
 and fifth-grade pupils which were matched on the basis of initial 
 arithmetic ability. These groups were also matched with other 
 groups of equal size in order to control the practice effect of the 
 initial test. The experiment was conducted just long enough for the 
 pupils of one group to observe errors in the answers of a group of 
 twenty number combinations, while the members of the other group 
 examined twenty combinations and their correct answers. The differ- 
 ence in achievement, as shown by the final test, was probably signifi- 
 cantly in favor of the exercise in which the pupils observed only 
 correct answers. The chief criticism of this experiment is its short 
 duration. It is possible that the confusion caused by the, exercise 
 containing errors might have worn off with more prolonged use and 
 that, in the long run, its use would result in superior achievement. 
 It may be true, also, that this type of exercise is one which would 
 engender the ability to locate mistakes— a well recognized objective 
 of arithmetic instruction. 
 
 Buckingham (17) equated seven pairs of groups of from twelve to 
 twenty-eight second-grade children in seven schools on the basis of 
 scores on the Pressey Primary Classification Test. During a daily 
 period of twenty minutes one of the groups of a pair was taught 
 related addition and subtraction facts together, as for example: 
 1+6, 6+1, 7 — 1, and 7 — 6. The other group of pupils was 
 taught all of the addition facts and then all of the subtraction facts 
 for the same time per day. With the exception of this difference in 
 the learning exercises, the instructional materials and techniques used 
 for each pair of groups were the same. No home work was required 
 and no new topics were introduced in arithmetic during the experi- 
 mental period. The hour of the instruction was alternated for each 
 pair of groups at the end of each week. The statement is made that 
 the experiment lasted about a month for one of the pairs of groups, 
 
Summary of Research Relating to the Teaching of Arithmetic 43 
 
 but nothing is said in this respect about the others. Three of the 
 differences in achievement revealed by the final test are "statistically" 
 significantly in favor of the "together" method, and three more of the 
 differences are in favor of the together method, but not significantly 
 so. One difference favors, but not significantly, the separate method. 
 Buckingham attaches great significance to this "all but unanimous 
 verdict." He states, "When an experiment conducted seven times 
 yields six results all in the same direction, the evidence is rather con- 
 clusive even though some of the differences, when considered individ- 
 ually, are small or lacking in statistical significance." He recognizes 
 the limitation of the short duration of his experiment and the failure 
 to test retention. While the techniques used in this experiment have 
 some admirable features, a question may be raised with respect to the 
 validity of the final test. Was it adapted to the type of learning exer- 
 cises used by the different groups? If its examples were of mixed 
 nature, it is probable that the test was more valid with respect to the 
 mixed learning exercises. If, however, one of the groups, of a pair, 
 had a test in which addition and subtraction were kept separate while 
 the other group had the same items mixed, the results would probably 
 be more valid with respect to each group, but it is difficult to see how 
 they could be considered comparable. In the face of this dilemma 
 of measurement one does not seem justified in accepting the conclu- 
 sions as highly dependable. 
 
 Myers and Myers (85) used fifty fifth-grade pupils, sixty-four 
 sixth-grade pupils, and fifty normal-school girls selected in a random 
 fashion. "The first pupil of a given group was tested with the grouped 
 combinations followed by the mixed combinations; the next pupil was 
 tested with the mixed examples first and then with the grouped ex- 
 amples; the third pupil began with the grouped examples and so on 
 alternating throughout the group." The pupils made their responses 
 orally, and the experimenter recorded the time required. An analysis 
 was made of the results, and a check was made of the practice effect. 
 The results significantly favor the method of grouped, rather than the 
 method of mixed, exercises. 
 
 Applying the two types of exercises to alternate pupils does not 
 insure that they were applied to equivalent groups. 26 Another criti- 
 cism concerns the length of the tests, each of which contained forty 
 items. More dependable results could have been secured by the 
 utilization of a much longer test, or by the utilization of a long period 
 
 2fi This technique is probably j ustified when the groups are very large. For example, Monroe used 
 a similar technique, but with a total of 9,256 pupils. See: , . 
 
 Monroe, W. S. "How Pupils Solve Problems in Arithmetic, University of Illinois Bulletin, 
 Vol. 26, No. 23, Bureau of Educational Research Bulletin No. 44. Urbana: University of Illinois. 
 1929. 31 p. (79). 
 
44 Bulletin No. 58 
 
 of learning prior to a final test. However, if this were done, the exper- 
 imenter would yet be faced with the dilemma of a choice between a 
 doubtfully valid mixed test or non-comparable separate tests. 
 
 Repp (103) used groups of 263 and 267 twelve-year-old pupils 
 which were equivalent with respect to arithmetical ability as shown 
 by an initial test of .97 + .006 reliability. One of these groups used 
 drill material consisting of twenty-six twenty-minute exercises, each 
 of which dealt with one topic. The other group used material of the 
 same total content but of mixed organization. After twenty-six 
 weeks an exhaustive final test, also of .97 ± .006 reliability, and of 
 a mixed nature was administered. The results of this test are "sta- 
 tistically" in favor of the mixed drills. The final test probably was 
 more valid with respect to the abilities engendered by the mixed 
 drills than with respect to the abilities engendered by the isolated 
 drills. It should be mentioned, however, that an analysis of the 
 achievement during practice ultimately favored the mixed drills. 
 The conclusion may be justified, therefore, that mixed drills are 
 superior for maintenance of skill, while isolated drills are superior in 
 the earlier stages of learning. 
 
 3. Justified conclusions. If one accepts the principle that arith- 
 metical ability in the field of calculation is specific, or at least largely 
 so, and that, consequently, ability to calculate consists of a large num- 
 ber of specific abilities, it follows that drill must be provided on each 
 specific ability, unless it is believed that there is essentially complete 
 transfer from one specific ability to another when these abilities are 
 at all closely related. 27 Furthermore, it appears reasonable that the 
 more difficult combinations should receive more drill than the easier 
 ones. Consequently, it is to be expected that learning exercises con- 
 structed with due recognition of the specific abilities to be engendered 
 and of their relative difficulties and interrelations should be more 
 effective than learning exercises not so constructed. This group of 
 investigations supports this general hypothesis and appears to justify 
 the assertion that the hypothesis has been demonstrated. It might be 
 argued that this hypothesis is obvious and, hence, that the principal 
 contribution of these studies is to be found in their details. The more 
 significant of these detailed findings appear to be: 
 
 27 The conclusions of the recent investigations of Beito and Brueckner (9) and of Olander (94) 
 would seem to indicate that there is a large amount of transfer in the case of certain abilities. The 
 conclusions of Beito and Brueckner (9) were referred to in a footnote on page 16. Olander (94) has 
 reported that "The ability gained by children on fifty-five simple number combinations in addition 
 and on fifty-five similar combinations in subtraction transferred almost completely to the forty-five 
 remaining simple number combinations in each of the two processes." This conclusion seems to be 
 reasonably dependable, since Olander used relatively large equivalent groups, controlled non-experi- 
 mental factors rather adequately, and secured measures of achievement which seem acceptably reliable 
 and valid. Such a conclusion would not seem to oppose the contention above that the best ma- 
 terials for drill are those constructed so that the more difficult combinations receive the greater 
 practice. It is commonly accepted as a principle in education that the best way to insure attain- 
 ment is to practice the needed abilities directly rather than to depend on transfer. 
 
Summary of Research Relating to the Teaching of Arithmetic 45 
 
 1. Practice material prepared by experts seems to be more effective than 
 
 learning exercises based on material prepared by teachers. 
 
 2. Learning exercises in which the practice is carefully distributed over the 
 
 number combinations so that none are slighted and the more difficult 
 combinations occur with relatively greater frequency are superior to 
 learning exercises which have not been thus prepared. 
 
 3. Learning exercises to be used in the initial stages of learning calculation 
 
 should probably require the practice of addition, subtraction, multi- 
 plication, and division separately. Learning exercises whose objective 
 is the maintenance of skill should be mixed in character. The pupils 
 should be given some opportunity to practice their calculation abilities 
 in the situations represented by examples varied with respect to the 
 fundamental process called for. 
 
 THE INFLUENCE OF DISTRIBUTION OF PRACTICE TIME ON 
 ACHIEVEMENT IN THE FUNDAMENTALS 
 
 1. Summary of reported conclusions. Three experiments have 
 been reported on the effect of distribution of practice time on learn- 
 ing, and one has been reported on the distribution of practice needed 
 for retention, or maintenance of skill. Kirby 28 compared practice 
 periods in addition of 22J/2, 15, 6, and 2 minutes' duration and in 
 division of 20, 10, and 2 minutes' duration. The gains in achieve- 
 ment, for both addition and division, favored the two-minute inter- 
 val. Hahn and Thorndike 29 compared practice periods in addition of 
 5, iy 2 , 10, 11 34, 15, 20, and 22 minutes' duration. Their results tend 
 to favor the longer periods. YVimmer (124) reported that pupils who 
 were given one fifteen-minute drill per week made greater progress 
 than those who were given five minutes of drill five times per week. 
 Reed 30 compared a single hour of practice in addition with a distribu- 
 tion of twenty minutes a day for three days, ten minutes a day for 
 six days, and ten minutes twice a week for three weeks. The gains in 
 achievement favor the distribution of twenty minutes a day for three 
 days. 
 
 Norem and Knight 31 investigated the distribution of practice 
 needed for retention or maintenance, of skill. They concluded with 
 respect to drill in multiplication that when mastery has been attained 
 "one practice a week is sufficient for maintenance." They state also, 
 however, that one practice a week "is often insufficient practice for 
 maintaining the combinations during the first two weeks following 
 the initial learning of them." 
 
 "Kirby, T. J. "Practice in the Case of School Children," Teachers College, Columbia University 
 Contributions to Education, No. 58. New York: Bureau of Publications, Teachers College, Columbia 
 University, 1913. 98 p. (57) 
 
 "Hahn, H. H. and Thorndike, E. L. "Some Results of Practice in Addition under School Con- 
 ditions," Journal of Educational Psychology, 5:65-84, February, 1914. (46) 
 
 30 Reed, H. B. "Distributed Practice in Addition," Journal of Educational Psychology, 15:248-49, 
 April, 1924. (102) 
 
 3iNorem, G. B. and Knight, F. B. "The Learning of the One Hundred Multiplication Combi- 
 nations," Twenty-Ninth Yearbook of the National Society for the Studv of Education, Bloomington, Illinois: 
 Public School Publishing Company, 1930, p. 551-68. (91) 
 
46 Bulletin No. 58 
 
 2. Evaluation of the experiments. Kirby (57) employed groups 
 of 194, 104, 205, and 229 fourth-grade children in his addition experi- 
 ment. These groups were practiced fifteen minutes in addition as an 
 initial test. They were then subjected to forty-five minutes of prac- 
 tice divided into periods of 22^, 15, 6, or 2 minutes in length. Fi- 
 nally, they were practiced for another fifteen-minute interval, which 
 represented the final test. The experimenter exercised considerable 
 care to prevent the children from practicing outside of the practice 
 intervals and to control other non-experimental factors. He conduct- 
 ed the practice himself in practically all of the classes. The experi- 
 ment with practice divided into periods of 20, 10, and 2 minutes' 
 duration in division was conducted in a similar fashion, using groups 
 of 204, 209, and 193 third- and fourth-grade children. The differ- 
 ences in gains seem possibly significant with respect to addition prac- 
 tice periods of two minutes' duration and certainly significant with 
 respect to division practice periods of the same length. 
 
 Kirby is to be commended for his attempt to secure a representa- 
 tive sample of school children. He checked the performance of 
 thirty-eight of the school classes which were used in this experiment, 
 and which were located in New York City, with results obtained with 
 a class outside of this city. One fault to be found with this experi- 
 ment is that of failure to secure equivalent groups. While the failure 
 to secure equivalence does not invalidate the results, it does obscure 
 their precise significance. The experimenter calls attention to the 
 possible influences of factors not inherent in the short practice period : 
 
 (1) The groups, working in shorter periods, because of the number of days 
 over which the experiments ran, had greater opportunity during the experiment 
 to profit from the regular school work than other classes .... (2) The 
 groups working in shorter periods had a longer time in which to catch the spirit 
 of the experiment and to become enthusiastic over surpassing their previous 
 performance. They had their records read to them more times and had the in- 
 centives to intense effort repeated more often. (3) They also had greater 
 opportunity and incentive to do work outside of the time given to the experiment. 
 
 The experiment of Wimmer (124) was described and evaluated on 
 page 32. His conclusion with respect to the distribution of practice 
 time may not be regarded as dependable. 
 
 Hahn and Thorndike (46) used eight experimental groups varying 
 in size, when approximate equivalence had been secured, from six to 
 nineteen fourth-, fifth-, sixth-, and seventh-grade pupils. These 
 groups were subjected to ninety minutes of practice in addition, 
 divided into periods of 5, 7}^, 10, 11 34, 15, 20, and 22 minutes' 
 duration. While the use of the practice sheets would seem to make 
 
Summary of Research Relating to the Teaching of Arithmetic 47 
 
 negligible the teacher factors, it is possible that an important extra- 
 school factor was uncontrolled. The investigators state: 
 
 It should be kept in mind throughout the reading of what follows that any 
 child was free to write out sums and to practice with them at home, during the 
 course of the experiment .... no attempts were made to prevent practice 
 apart from the specified practice in school. 
 
 The differences favor, but not significantly, the longer practice 
 intervals. More dependence could be placed on this conclusion if 
 larger groups had been used and used with more adequate control of 
 non-experimental factors. 
 
 Reed (102) used four groups of 60, 50, 51, and 42 first- and second- 
 year college students. The scores on the initial test in addition indi- 
 cate that these groups were only approximately equivalent. One 
 group practiced addition for a period of one hour, while the other 
 groups practiced an equal amount of time distributed in periods of 
 twenty minutes a day for three days, ten minutes a day for six days, 
 or ten minutes twice a week for six weeks. It should be mentioned 
 that the initial ten minutes of practice and the final nineteen minutes 
 constituted the initial and final tests. The results favor significantly 
 the distributed practice as compared with the one hour non-distrib- 
 uted practice. With respect to the distributed practice, the results 
 favor, but not significantly, the daily twenty-minute practice periods. 
 The chief criticisms of this experiment are that it was conducted with 
 adults and that the groups having the distributed practice were 
 initially superior. Hence, its conclusions are probably not applicable 
 to school children. The adults but relearned an old skill. Results 
 might be quite different with new learning. 
 
 Norem and Knight (91) used twenty-five third-grade pupils in 
 their investigation of the distribution of practice effective for reten- 
 tion or maintenance of skill in multiplication. The parents of the 
 pupils were requested to refrain from assisting them in drill at home, 
 and the pupils were instructed not to practice except when required 
 to do so by the experiment. After an initial administration of two 
 tests, given a week apart, which disclosed unlearned combinations, 
 each pupil was individually drilled to the point of mastery of his 
 formerly unlearned combinations. The pupil was then tested once a 
 week for a period of six weeks on these newly mastered combinations, 
 and then once a month for three months. The analysis of the practice 
 and test achievements of these twenty-five pupils is a commendable 
 feature of this experiment. It would seem to justify the conclusion 
 that one practice a week is sufficient for maintenance of skill in multi- 
 plication after mastery has been attained, so far as this group of pupils 
 
48 Bulletin No. 53 
 
 is concerned. It is probable that this investigation should be re- 
 peated with larger groups for greater reliability in the findings. 
 
 3. Justified conclusions. The conclusions of Kirby (57) and of 
 Hahn and Thorndike (46) are opposed to each other, while that of 
 Reed (102) tends to agree with that of Hahn and Thorndike (46). The 
 conclusions of Norem and Knight (91) seem reliable for the pupils 
 used in their experiment, but do not seem more than suggestive for 
 pupils in general. The conflicting testimony, plus the obvious faulty 
 techniques of the experiments, prevents the authors from stating a 
 justified conclusion. 
 
 It would seem, however, that until more adequate experimental 
 evidence has been presented, the teacher will be acting wisely in em- 
 ploying intervals approximately twenty minutes in length with a 
 frequency of one a day until mastery has been attained. After this 
 objective has been reached, shorter practice periods distributed at 
 longer intervals will possibly serve to maintain skill. 
 
 THE INFLUENCE OF REQUESTS FOR SPEED OR ACCURACY ON 
 ACHIEVEMENT IN THE FUNDAMENTALS * 
 
 1. Summary of reported conclusions. The influence of requests 
 for speed or accuracy has been studied in three experiments. 32 Wim- 
 mer (124) has reported that "the difference in progress made by the 
 two groups, one being drilled for accuracy and the other for speed is 
 not very large." Messick 33 reports that if speed is the objective of 
 achievement in addition, it makes little difference which is requested, 
 speed or accuracy. However, if accuracy is the objective, it is much 
 better to request accuracy rather than speed. He states, "In teaching 
 addition to pupils of the fourth and fifth grades of the elementary 
 schools it is better to emphasize accuracy rather than speed." 
 Myers 34 concludes that requests for speed are causes of inaccuracy in 
 the fundamentals. "One may conclude that the loss to learning effi- 
 ciency from the strong speed pressure as applied to the simple number 
 combinations in arithmetic under which many school children must 
 work in school today is appalling." 
 
 32 There have been several investigations of the relation of speed to accuracy in the fundamentals 
 of arithmetic; see: 
 
 Bird, G. E. "A Test of Some Standard Tests," Journal of Educational Psychology, 11:275-83, 
 May, 1920. 
 
 Courtis, S. A. "Courtis Standard Research Tests: Third, Fourth, and Fifth Annual Accountings, 
 1913-16," Bulletin No. 4. Detroit: Department of Cooperative Research, 1916. 112p. 
 
 Luderman, W. W. "Speed and Scholarship Arithmetical Accuracy," School Science and Mathe- 
 matics, 25:522-24, May, 1925. 
 
 Monroe, W. S. "A Report of the Use of the Courtis Standard Research Tests in Arithmetic in 
 Twenty-Four Cities," Studies by the Bureau of Educational Measurements and Standards, No. 4. 
 Emporia: Kansas State Normal School, 1915. 94 p. 
 
 Phelps, C L. "A Study of Errors in Tests of Adding Ability," Elementary School Teacher, 
 14:29-39, September, 1913. 
 
 33 Messick, A. I. "Effect of Certain Types of Speed Drills in Arithmetic," Mathematics Teacher, 
 19:104-09, February, 1926. (75) 
 
 34 Myers, G. C "The Price of Speed Pressure in the Learning of Number," Educational Research 
 Bulletin (Ohio State University), 7:265-68, September 19, 1928. (84) 
 
Summary of Research Relating to the Teaching of Arithmetic 49 
 
 2. Evaluation of the experiments. The experiment of Wimmer 
 (124) was described and evaluated on page 32. His conclusion with 
 respect to speed versus accuracy may not be regarded as dependable. 
 Messick (75) used two groups of 136 fourth- and fifth-grade children. 
 No attempt was made to secure equivalence. One group practiced 
 addition four minutes a day for twenty days, with emphasis on speed. 
 The other group practiced addition for the same length of time, but 
 requests were made for accuracy rather than for speed. The final 
 tests revealed a certainly "statistically" significant difference in 
 accuracy in favor of the group for which accuracy was emphasized. 
 The small difference in speed also in favor of this group cannot be 
 regarded as "statistically" significant. This experiment is faulty in 
 that no attempt was made to secure equivalence. There is some rea- 
 son for believing that important non-experimental factors were not 
 adequately controlled. The experiment was rather short in duration. 
 Myers (84) used one group of ten first-grade children. These 
 children, who had been practiced for two months in addition, were 
 administered a test, the results of which indicated almost 100 per cent 
 accuracy. After two years, "The ten who were still in school were 
 studied again. In the meantime, these children .... had been 
 exposed to rapid-fire drills in the simple addition facts and the basic 
 subtraction facts. The test-flash card .... was their torturer 
 almost daily .... They were frequently subjected to games in 
 which the fastest answers won." The children were then subjected 
 to five practice-test periods, after each of which they were told that 
 they had done very well and were urged to go faster. The decrease in 
 accuracy as more and more emphasis was placed on speed is signifi- 
 cantly shown in this experiment. Myers is to be commended for pro- 
 longing his investigation over so long a period of time. He is to be 
 criticized for securing data from so small a group, for failure to employ 
 a control group, and for creating what appear to be abnormal condi- 
 tions. It is possible that the conditions to which these children were 
 subjected are not typical of good, or even usual, school practice. 
 
 3. Justified conclusions. While dependable conclusions must 
 await further controlled experimentation, it seems justifiable to rec- 
 ommend requests for accuracy rather than requests for speed. In any 
 case, it seems justifiable to hold that requests for accuracy should 
 precede requests for speed. After pupils have attained satisfactory 
 accuracy on a given level of difficulty, a teacher is possibly justified 
 in encouraging them to increase their rate. 
 
CHAPTER IV 
 
 METHODS OF TEACHING PUPILS TO SOLVE 
 VERBAL PROBLEMS 
 
 It is commonly assumed that the responses made by pupils when 
 presented with verbal problems in arithmetic are the result of reflec- 
 tive thinking. Consideration is given in the first part of this chapter 
 to investigations of the nature of pupil responses to verbal problems. 
 The experimental factors of the experiments summarized in the 
 second part of the chapter are variations in types of verbal problems 
 and of problem statements, and those in the third and final portion of 
 the chapter are various methods of teaching pupils to solve verbal 
 problems in arithmetic. 
 
 THE NATURE OF PUPIL RESPONSES TO VERBAL PROBLEMS 
 1. Summary of reported conclusions. Three studies have been 
 reported on the problem of the part played by reasoning when pupils 
 attempt to solve verbal problems in arithmetic. Bradford 1 reported 
 from an analysis of test results that "arithmetical work is not done in 
 a critical frame of mind." This conclusion has since been substan- 
 tiated by the more comprehensive investigation of Monroe, 2 in which 
 the conclusion was reached that "a large per cent of seventh-grade 
 pupils do not reason in attempting to solve arithmetic problems . . 
 Many of them appear to perform almost random calculations upon 
 the numbers given. When they do solve a problem correctly, the 
 response seems to be determined largely by habit." Kline and 
 Anderson 3 have reported a laboratory study, the findings of which 
 indicate the nature of the dual role of specific habits and reasoning 
 abilities in solving verbal problems in arithmetic. 
 
 2. Evaluation of the investigations. The data in the investiga- 
 tions of both Bradford (11) and Monroe (79) were collected by means 
 of a single administration of tests. The tests of Bradford (11), which 
 were administered to several hundred pupils in Standards VII and 
 VIII in certain elementary schools in England, were composed of 
 examples impossible of solution, of which the following quoted from 
 the report are illustrative: 
 
 Februa B r rad l f 925' E (\l) °' " Suggestion - Reasoning, and Arithmetic." Forum of Education, 3:3-12, 
 Vni o^ N? ro< V a W u S- " H .°w Pupils Solve Problems in Arithmetic," University of Illinois Bulletin, 
 1929 .31 p ' (7Q) eaU Educatlonal Research Bulletin No. 44. Urbana: University of Illinois, 
 
 M*/JSv e * ™)Xk^ d v A u derSOn \^ K -,c'7 he RoIe of Habit in Reasoning," School Science and 
 Mathematics, 26:156-67, February, 1926. (59) 
 
 50 
 
Summary of Research Relating to the Teaching of Arithmetic 51 
 
 1. If the distance from Aries to St. Brieuc is 500 miles, and from Vire to 
 
 St. Malo is 50 miles, how far is it from St. Brieuc to St. Malo? 
 
 2. If Henry VIII had six wives, how many had Henry II? 
 
 The extent to which attempts were made to solve such problems 
 was taken by Bradford to be indicative of the absence of critical 
 reflective thinking in the solving of arithmetical problems by school 
 children. While this conclusion seems reasonably dependable, it 
 should be remembered that the data refer to the children of English 
 schools and for this reason may be somewhat less applicable to Amer- 
 ican children. It is in agreement, however, with the conclusion of 
 the investigation reported by Monroe (79). 
 
 Monroe (79) secured his data by administering a test to 775 sixth- 
 grade, 5902 seventh-grade, and 2579 eighth-grade pupils in forty-one 
 Illinois cities. These pupils were divided into four groups, and equiv- 
 alence was secured by distributing the tests to the pupils in a random 
 manner. 
 
 In order that each of the tests might be given to a random sample of pupils, 
 the four tests were arranged in alternate order so that when distributed to the 
 pupils in the class, the first, fifth, ninth, thirteenth, and so forth, would receive 
 Test A; the second, sixth, tenth, fourteenth, and so forth, would receive Test B; 
 the third, seventh, eleventh, fifteenth, and so forth, would receive Test C; the 
 fourth, eighth, twelfth, sixteenth, and so forth, would receive Test D. Since the 
 tests were to be given in a large number of classes, it seemed that this plan of 
 sampling would provide equivalent groups 
 
 It is evident that the four groups were equivalent not only in 
 arithmetical ability but also with respect to teachers, textbooks, and 
 other factors. In general each of the four equivalent groups was 
 equally represented in each classroom, and this representation was 
 secured in a random fashion. 
 
 The tests administered to these groups differed only in the termi- 
 nology used in stating the problems. For example, in Test A, the 
 second problem is stated in simple terminology, all of the data given 
 are relevant, and the setting is concrete. In Test B, technical termi- 
 nology is used, all the data given are relevant, and the setting is con- 
 crete. The difference in the statement of the problem in these two 
 tests is the change from simple terminology to technical terminology. 
 In Test C, the problem is stated in simple terminology, the data given 
 are relevant, and the setting is abstract. In Test D, technical termi- 
 nology is used, irrelevant data are included, and the setting is abstract. 
 The problems of the tests are so stated that comparisons are possible 
 with respect to the relative influences on correctness of response of 
 simple and technical terminology, wholly relevant data and data 
 partially irrelevant, and concrete and abstract setting. These com- 
 
52 Bulletin No. 58 
 
 parisons are made for the data of this investigation, and the results 
 are presented in tabular form in the report of the research. 
 
 The techniques used in this study appear to be reasonably free 
 from criticism. There seems to be little question that the sample of 
 pupils was representative, and the groups used, equivalent with 
 respect to all significant factors. The data secured seem to be of 
 sufficient quality to warrant the statement that responses of pupils to 
 verbal problems are usually characterized by absence of reasoning. 
 
 The experiment of Kline and Anderson (59) was conducted with 
 four adults in a psychological laboratory. Time and accuracy were 
 recorded for the responses to four hundred questions, such as "If 
 Thursday is the twelfth, what day is the eighteenth?" The conclu- 
 sions of this experiment are interesting, but may not safely be applied 
 to school children. It would seem, however, that Kline and Anderson 
 have made but another attempt to prove the obvious. It is com- 
 monly recognized that there is close interdependence between specific 
 habits and reasoning. 
 
 3. Justified conclusions. The data secured in these three investi- 
 gations appear to justify the conclusions stated, insofar as they apply 
 to the groups of pupils to which the tests were given and by which the 
 test exercises were used. The generalization of the conclusions may 
 be questioned, especially for all types of problems and for all condi- 
 tions of responding to them. Hence, the generalization should be 
 considered tentative. It should also be noted that these investiga- 
 tions deal with the question of what responses pupils make as the 
 result of the instruction they have received. They do not consider 
 the type of responses that pupils should make. 
 
 THE EFFECT OF DIFFERENT TYPES OF PROBLEMS AND 
 
 PROBLEM STATEMENTS 
 
 1. Summary of reported conclusions. Myers, 4 Hydle and Clapp, 5 
 
 Washburne and Morphett, 6 Bowman, 7 Mitchell, 8 Monroe, 9 Wheat, 10 
 
 and Osburn and Drennan 11 have reported conclusions relative to the 
 
 *Myers,G. C. "Imagination in Arithmetic," Journal of Education, 105:662-63, June 13, 1927. (83) 
 i • 7-.u \ • '.^Sr CIap P' T F; L - ."Elements of Difficulty in the Interpretation of Concrete Prob- 
 1927 1 "m "(50?' aU ° f Educatwnal Research Bulletin No. 9. Madison: University of Wisconsin, 
 
 «Washburne, C. W. and Morphett. M. V. "Unfamiliar Situations as a Difficulty in Solving 
 Arithmetic Problems, Journal of Educational Research, 18:220-24, October, 1928. (118) 
 
 'Bowman, H. L. "The Relation of Reported Preference to Performance in Problem Solving," 
 University of Missouri Bulletin, Vol. 30. No. 36, Education Series, No. 29. Columbia: University of 
 Missouri, 1929. 52 p. (10) 
 
 sMitchell, Claude. "The Specific Type of Problem in Arithmetic versus the General Type of 
 Problem, Elementary School Journal, 29:594-96, April, 1929. (76) 
 
 9 Monroe, op. cit. 
 
 lowheat, H. G. "The Relative Merits of Conventional and Imaginative Types of Problems in 
 Arithmetic Teachers College, Columbia University Contributions to Education, No. 359. New York: 
 Bureau of Publications, Teachers College, Columbia University, 1929. 124 p (1?1) 
 n n !■ ,l\ l { r . n O V - J - TT and D . re " nan - L - J- "Problem Solving in Arithmetic," Educational Research 
 Bulletin (Ohio State University), 10:123-28, March 4, 1931. (95) 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 53 
 
 effect upon pupil responses of certain variations in the statement of 
 the problems. Myers (83) administered two problems to fifth-grade 
 pupils and reported that these pupils were able to solve the "imagi- 
 natively stated" one much more easily. Hydle and Clapp (50) studied 
 the following characteristics of arithmetical problems in an effort to 
 determine whether or not these characteristics were causes of diffi- 
 culty in problem solving: 
 
 1. Objective setting 
 
 2. Size of numbers 
 
 3. Unfamiliar objects 
 
 4. Arrangement in a series 
 
 5. Nonessential elements 
 
 6. Visualization vs. experience 
 
 7. Project vs. problem form of statement 
 
 8. Symbolic terms 
 
 Variations of these characteristics, with the exception of the 
 arrangement of similar problems in series and the presence of non- 
 essential elements were found to be "statistically" significant causes 
 of difficulty. In addition to this conclusion the authors state that 
 problem solving for pupils is largely a matter of visualization. Prob- 
 lems should be formulated with this in mind in the earlier stages of 
 learning, but in order that generalizing ability might be engendered, 
 it is concluded that the pupils should have as learning exercises a 
 considerable number of problems not related to their first-hand 
 experiences. 
 
 Washburne and Morphett (118) report that fifth-grade pupils 
 achieve better results with familiar problems than with those con- 
 taining unfamiliar elements. The following problems quoted from 
 the report are illustrative of those used in his study; the first is in 
 unfamiliar terminology, and the second, in familiar terminology: 
 
 A merchant sold 20 bags of charcoal. Each bag held 35 pieces. How many 
 pieces of charcoal did he sell? 
 
 The girls have to make 30 boxes of taffy. Each of the boxes holds 25 pieces. 
 How many pieces of taffy do they have to make? 
 
 Bowman (10) reported that pupils of high ability, as measured in 
 his study, performed equally well on the following types of problems: 
 
 1. Problems based upon adult activities 
 
 2. Problems based upon children's activities 
 
 3. Problems whose setting is in the field of science 
 
 4. Problems so stated as to take on the nature of a puzzle 
 
 5. Problems of pure computation only, where directions for the right pro- 
 
 cedure are given 
 Pupils of lower ability showed a higher relative degree of perform- 
 ance on problems of the pure computation type. Mitchell (76) re- 
 
54 Bulletin No. 58 
 
 ported that "Problems with definitely expressed numerical quantities 
 seem to be more readily understood and solved than problems of a 
 general nature involving general principles. ' ' The following examples 
 illustrate the types of problems compared in this study. The first is a 
 specific problem, and the second, a general problem. 
 
 The width of a room is 10 feet, and its length is 15 feet. Find its perimeter 
 If you know the length and the width of a room, how can vou find the 
 perimeter? 
 
 Monroe (79) reported as another of the conclusions of his study 
 that "If the problem is stated in the terminology with which they 
 [the pupils] are familiar and if there are no irrelevant data, their 
 response is likely to be correct." Wheat (121) determined the relative 
 achievements of pupils with conventionally-stated problems and 
 imaginatively-stated problems. He reported that differences in 
 achievement are negligible. The first of the examples quoted below 
 illustrates the conventional type of statement; the second of the 
 examples illustrates the imaginative type. 
 
 Margaret spent $3.68 for handkerchiefs at 23 cents each and gave one-fourth 
 oi them to her sister. How many did her sister get? 
 
 Margaret had been shopping all morning for Christmas presents. She had 
 bought presents for her father and mother and brothers but could not decide 
 what to get for her sister and several of her friends— there were so many things 
 to pick from Just then she saw some pretty handkerchiefs which were marked 
 IS cents each. These were just what she wanted, so she counted her money, 
 found she had S3.68, and spent all of it for handkerchiefs. She kept out one- 
 iourtn of the handkerchiefs to give to her sister and gave the rest to her friends. 
 How many did she keep out to give to her sister? 
 
 Osburn and Drennan (95) have reported a recent experiment in 
 which vocabulary difficulty did not appear to be a significant factor 
 in problem-solving achievement. These investigators conclude that 
 their data "seem to indicate that pupils are able to sense the meaning 
 of problems even if they do not understand all the words." The con- 
 clusion is also reported that a few of the most important problem 
 types should be taught thoroughly, with the expectation that transfer 
 of training will take care of the remainder. 
 
 2. Evaluation of the experiments. Myers (83) administered his 
 two problems to 513 fifth-grade children. One hundred and ninety- 
 seven solved the first problem correctly, while 253 correctly solved the 
 second and more imaginatively-stated problem. It would seem, 
 probably, that the difference is due to practice effect rather than to 
 the fact that the second problem was more imaginatively stated 
 than the first. 
 
 Hydle and Clapp (50) constructed tests in which the problems 
 were paired with respect to each of the elements of difficulty investi- 
 
Summary of Research Relating to the Teaching of Arithmetic 55 
 
 gated. That is to say, a problem appearing in one form of the test 
 differed from its mate in the other form with respect to a given ele- 
 ment. For example, in the case of symbolic terms one problem 
 statement would contain symbols, such as X, Y, and Z, instead of the 
 names of objects given in the other problem statement. The tests 
 included five pairs of problems for each of the following elements of 
 difficulty: (1) objective setting, (2) size of numbers, (3) unfamiliar 
 objects, (4) arrangement in a series, (5) nonessential elements, (6) 
 visualization vs. experience, (7) project vs. problem form of state- 
 ment, (8) symbolic terms. The tests were administered to pupils 
 varying in number from 5870 to 7029. These pupils were widely dis- 
 tributed in village and city schools. Those taking the tests were di- 
 vided into two groups of approximately equal ability as shown by a 
 test of twenty-five problems of a concrete character. The statistical 
 interpretation of the data indicated that variations in six of the eight 
 elements investigated might dependably be expected to cause diffi- 
 culty. These elements are (1) objective setting, (2) size of numbers, 
 (3) unfamiliar objects, (4) visualization vs. experience, (5) project vs. 
 problem form of statement, and (6) symbolic terms. Hydle and Clapp 
 are to be commended for their comprehensive and intensive investi- 
 gation. The possible invalidity of their problem tests is adequately 
 recognized in the report of the study. The investigators are to be 
 commended for this and, in the opinion of the present writers, for not 
 contending that the arithmetic curriculum should be so constructed 
 that difficult elements in problem solving be eliminated. 
 
 Washburne and Morphett (118) used a single group of 441 fifth- 
 grade pupils in six different towns. A test of eight pairs 1 * of problems 
 was administered to all of these children. The results appear to be 
 "statistically" significant in favor of the problems containing famil- 
 iar elements. The data collected would seem to be sufficiently reli- 
 able to warrant acceptance of the conclusion. However, this experi- 
 ment would seem to be but another attempt to prove the obvious. 
 A more worth while investigation would be one that would attempt 
 to show whether or not problems containing unfamiliar elements 
 should be used as learning exercises. 
 
 Bowman (10) administered both forms of his test to a total of 564 
 seventh-, eighth-, and ninth-grade pupils of Sedalia, Missouri. Evi- 
 dence is presented to show that the pupils of this group represent an 
 approximately normal distribution of intelligence and are typical of 
 the grades they represent with respect to parentage, parental occu- 
 
 i2An illustration of one of the pairs of problems is given on page 53. 
 
 
56 Bulletin No. 58 
 
 pations, and environment. 13 Each of the two test forms contained 
 twenty-five problems of the types previously referred to. At the 
 bottom of each page of the forms was placed the following statement 
 to be completed by the pupil: "The problem on this page I liked best 
 
 is No. ." This was done to secure data relevant to preferences 
 
 for different types of problems. 14 The coefficients of reliability and of 
 validity for the test as a whole were quite high. The coefficient of 
 reliability was reported as .95 ± .003 in the measurement of perform- 
 ance and .77 ± .01 in the measurement of preference, and the coeffi- 
 cient of validity was reported as .82 ± .01 when the scores secured 
 from an administration of the Stanford Arithmetic Reasoning Test 
 were used as the criterion. The representativeness of the group and 
 the comparatively high reliability and validity of the instrument used 
 constitute strong arguments for the dependability of the conclusions 
 that pupils of high ability perform equally well on (1) problems based 
 upon adult activities, (2) problems based upon children's activities, 
 (3) problems whose setting is in the field of science, (4) problems so 
 stated as to take on the nature of a puzzle, (5) problems of pure 
 computation only, and that pupils of lower ability perform relatively 
 better on problems of the purely computational type. 
 
 Mitchell (76) administered a test containing fifteen quantitative 
 problems and fifteen general problems — problems without expressions 
 of numerical quantities — to seventy eighth-grade and sixty seventh- 
 grade pupils. The mean difference in scores between the two types of 
 problems is sufficiently large to seem to be "statistically" significant, 
 although no standard or probable error is reported. The dependabil- 
 ity of the findings may be questioned, however, because of certain 
 faults in the data. The sample of pupils is too small to be regarded as 
 representative. It may be that the pupils had greater difficulty with 
 the general, or non-quantitative, type of problem because of lack of 
 experience with problems of this type. 
 
 Wheat (121) administered tests containing ten pairs of conven- 
 tional and imaginative problems to approximately two thousand 
 fifth-, sixth-, and eighth-grade pupils in several towns in different 
 parts of the country. The differences in achievement between the 
 conventional and imaginative types of problems were not of sufficient 
 magnitude to be considered "statistically" significant, with the 
 possible exception that the conventional type of problem required 
 much less time. Wheat is to be commended for the size and repre- 
 sentativeness of his sample, but his procedures for handling and inter- 
 
 "While measures of intelligence of some of the pupils are not reported, there is no reason to 
 believe that they were less typical of children in general than those for whom data are reported. 
 
 '^This matter will be referred to again in the summary of research on motivation of learning 
 in arithmetic. See page 81. 
 
Summary of Research Relating to the Teaching of Arithmetic 57 
 
 preting his data have been seriously criticized. Osburn 1 * states that 
 Pearson coefficients of correlation are computed from unsuitable data: 
 In at least two cases correlations are figured which are partly based upon the 
 number of problems solved. The distribution of the number of problems solved 
 I not normal ; in fact it is clearly of the U type. The use of the Pearson coefficient 
 of correlation with distributions of this sort may be justifiab e if the regression 
 lines are rectilinear. This necessary condition is not substantiated, and the use 
 of the Pearson technique is therefore open to question. 
 
 Again the Pearson correlation was originally intended for use with two var- 
 iables onlv. In a number of cases in this study it is used where three and even 
 lour variables are involved. For example, a correlation 1S shown be ween 
 intelligence quotients and indices of similarity scores. In this case four variables 
 are reaW involved, but they appear as two because quotients of respective _ pa r 
 
 are use d This is handy, but hardly justifiable, as a statistical 
 
 procedure. 
 
 Osburn also criticizes the study from other points of view. He 
 states, "In conventional problems, as here defined, the setting is left 
 to the imagination, while in the imaginative problems the setting is 
 made explicit by description but is still not perceptually present. 
 The critic points out that the pupils quite possibly received previous 
 training only on the conventional type of problem. 
 
 In spite of the fact that they had had little or no training in the solution of 
 imaginative problems the pupils did well with them. This might mean the 
 existence of transfer, or it might indicate a marked advantage for the imaginative 
 type when the factor of previous training is properly controlled by acceptable 
 scientific techniques. 
 
 Finally, Osburn contends that Wheat is to be criticized for assum- 
 ing that arithmetic material should be used which can be bought 
 cheaply and taught quickly and easily. Osburn holds that the ob- 
 jectives of arithmetic must be considered here. "The question there- 
 fore is not which problem is most economical to teach, or to buy, but 
 which one will better prepare the pupil for quantitative thinking in 
 real conditions— the sorts of situations which he will meet in Me. 
 Osburn then presents arguments for the imaginative type of problem. 
 The present writers are inclined to grant that most of Osburn s 
 criticisms appear to be justified. It should be pointed out, however, 
 that Osburn is somewhat inconsistent. For example, he holds that 
 the two types of problems are synonymous and then contends that 
 training has been different with respect to each. If they are synony- 
 mous, why should each not be equally well adapted to engender those 
 abilities accepted as the objectives of arithmetic? After all, it would 
 seem that the conclusion that "pupils of the intermediate grades are 
 
 uosburn, W. J. "Two Recent Books on Arithmetic," Educational Research Bulletin (Ohio State 
 University), 9:66-73, February 5, 1930. 
 
58 Bulletin No. 58 
 
 neither hindered nor helped in their problem practice exercises by 
 problems of the imaginative type, when no limits are imposed upon 
 the amounts of time of the practice periods," may be accepted as 
 fairly dependable until better evidence has been obtained experi- 
 mentally which reverses it. 
 
 Osburn and Drennan (95) had teachers of two classes of third- 
 grade children teach a representative list of problems with particular 
 emphasis on the ''cues," or language aspects, of the problems. An 
 examination made up of twenty verbal problems containing new ones, 
 but no additional vocabulary difficulty, was given after six weeks of 
 such instruction. On the next day, another test was administered 
 containing twenty problems which involved vocabulary difficulties, 
 illustrated by such terms as narcissus, gypsum, tortoise, chemist, 
 sulfuric acid, and excavating. The data indicate that the pupils 
 made very acceptable scores on both tests. The investigators suggest 
 that the changes in vocabulary may have been a factor of little 
 significance, because "mainly just 'nouns' were changed, and since 
 the test was given the next day after the first test, that the pupils 
 sensed the similarity of Test II to the test of the day before." This 
 appears to be a very serious limitation of this investigation. The 
 present writers are inclined, therefore, to give little weight to the 
 conclusions of other studies of the influence of terminology on 
 problem-solving achievement in arithmetic. 
 
 3. Justified conclusions. These eight studies of the effect of dif- 
 ferent types of problems and problem statements are not comparable, 
 and, hence, it is difficult to synthesize the findings. Most of them, 
 however, support the principle that pupils make higher scores on 
 tests consisting of familiar problems, or problems stated in familiar 
 terminology. The conclusion that pupils respond more correctly to 
 problems stated in concrete rather than imaginative or abstract form, 
 with irrelevant elements excluded, and related to activities exper- 
 ienced by children is less unanimously supported by the experimental 
 evidence. This generalization is an obvious inference from the 
 psychology of learning, but these studies contribute to our under- 
 standing of what makes a rabblem unfamiliar. 
 
 METHODS OF TEACHING PUPILS TO SOLVE VERBAL PROBLEMS 
 1. Summary of reported conclusions. Newcomb, 16 Stevenson, 17 
 Greene, 18 Clark and Vincent, 19 Washburne and Osborne, 20 Lutes, 21 
 
 JournTnfZ%%%eZfr C ^ PU So) H ° W ^ ^ Pr ° blemS *" Arithmetic '" Elementary School 
 B.* M ll? ,e n e £ a ? n, £<i?" " Inc !;easing the Ability of Pupils to Solve Arithmetic Problems," Educational 
 Research Bulletin (Ohio State University), 3:267-70, October 15, 1924 (112) 
 
 Tnur*nln? C pJ'r H r A- i p ' Direc ^ D/j" in the Comprehension of Verbal Problems in Arithmetic," 
 J ournal of Educational Research, 11:33-40, January, 1925. (42) 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 59 
 
 Washburne, 22 Hanna, 23 and Adams 24 have reported studies on meth- 
 ods of teaching pupils how to solve problems in arithmetic. Newcomb 
 (90) concluded that the pupils in his experiment who were supplied 
 with sheets of general directions for solving verbal problems achieved 
 more, particularly with respect to speed, than the pupils not so sup- 
 plied. Stevenson (112) secured effective results with a large group of 
 pupils who were taught to read and analyze problems by the provi- 
 sion of systematic training in finding the facts pertaining to the 
 problem, in deciding upon the processes to be used, and in rinding the 
 answer in round numbers. Greene (42) reported that training in 
 selecting and recognizing the process involved in the solution of a 
 problem is more effective in securing correct solutions from pupils 
 than when such training is not given. He states in this connection, 
 however, that "This drill, strangely enough, seems to increase the 
 accuracy of problem solution more than the ability to select the 
 correct principle in solving the problem " 
 
 Clark and Vincent (26.) compared the relative effectiveness of the 
 conventional and graphical methods of solving verbal problems in 
 arithmetic. The results are favorable, but not significantly so, to the 
 graphical method. This method is illustrated by the following 
 example quoted from the report: 
 
 A grocer bought 24 bushels of potatoes at SI. 50 per bushel. Four bushels 
 spoiled. The others were sold at S2.00 per bushel. Find his profit. 
 
 Xumber of bushels bought (20) 
 
 Cost 
 
 Price per bushel (SI. 50) 
 
 Xumber of Xumber of bushels 
 
 bushels sold bought (20) 
 
 Selling Price <^ Xumber of bushels 
 
 ^^\^^ spoiled (4) 
 
 " Price per bushel (S2.00) 
 
 The pupil is directed to think of the diagram as illustrating the following: 
 To find the profit, I should have to know the cost and the selling price; to find the 
 cost I would have to know the number of bushels bought and he price per bushel; 
 
 isClark J. R. and Vincent, E. L. "A Comparison . Two Methods of Arithmetic Problem 
 Analysis," Mathematics Teacher, 18:226-33, April, 1925. (26) _ 
 
 "Washburne. C. W. and Osborne, Raymond. "Solving Arithmetic Problems, Elementary School 
 Journal, 27:219-26,296-304; November, December, 1926. (119) cw AriH, mP tir 
 
 "Lutes, O. S. "An Evaluation of Three Techniques for Improving Ability to Soke Arithmetic 
 Problems: A Study in the Psychology of Problem Solving," University of Iowa Monographs m Edu- 
 cation, Series 1, No. 6. Iowa City: University of Iowa, 1926. 42 p. (68) i#-r.hani«i 
 
 "Washburne, C. W. "Comparison of Two Methods of Teaching Pupils to Apply the Mechanics 
 of Arithmetic to the Solution of Problems," Elementary School Journal, 2 '., :758-67 . J U "%J?; ' ; f }X'A 
 
 =3Hanna, P. R. Arithmetic Problem Solving: A Study of the Relative Effectiveness of Three Methods 
 of Problem Solving. New York: Bureau of Publications, Teachers College, Columbia University, 
 
 1929 * 24tdams! 4 R. E. A Study of the Comparative Value of Two Methods of Improving Problem 
 Solving Ability in Arithmetic, Philadelphia: University of Pennsylvania, 1930. 68 p. (1) 
 
60 Bulletin No. 58 
 
 to find the selling price I would have to know the number of bushels sold and the 
 price per bushel. 
 
 Washburne and Osborne (119) compared the relative effectiveness 
 of three methods of teaching pupils to solve verbal problems in arith- 
 metic. 
 
 Method 1 is to train children in the solving of problems by giving them a 
 large number of problems — no special technique 
 
 Method 2 is to train children to analyze problems. It is a definite technique 
 of attacking problems. 
 
 Method 3 is to train children to see the analogy or similarity between difficult 
 written problems and corresponding easy oral problems and thereby to decide 
 what process to use in attacking the difficult problems. 
 
 They state in their conclusions: 
 
 Training in the seeing of analogies appears to be equal or slightly superior to 
 training in formal analysis or the superior half of the children; analysis appears 
 to be decidedly superior to analogy for the lower half; but merely giving many 
 problems, without any special technique of analysis or the seeing of analogies, 
 appears to be decidedly the most effective method of all. 
 
 Lutes (68) compared the relative effectiveness of (1) drilling pupils 
 in computation only (2) drilling pupils in choosing operations, 
 (3) drilling pupils in choosing correct solutions, along with emphasis 
 on reading problems correctly, and (4) the traditional method of 
 teaching pupils to solve verbal problems. The results are significantly 
 in favor of drilling pupils in computation. Washburne (117) com- 
 pared the achievement of pupils who were taught the fundamental 
 processes as applied to verbal problems with the achievement of 
 pupils who were taught fundamental processes and verbal problems 
 separately. The results were not significantly in favor of either 
 method. 
 
 Hanna (47) compared the relative effectiveness of the depend- 
 encies method (graphic or diagrammatical), the conventional-formula 
 (four steps) method, and the individual, or informal, method of 
 teaching pupils to solve arithmetical problems. The dependencies 
 method is similar to the graphic method of Clark and Vincent (26). 
 The conventional formula method consists of the following steps: 
 
 1. What is asked for in the problem? 
 
 2. What is given in the problem? 
 
 3. How should these facts be used to secure the answer? 
 
 4. What is the answer? 
 
 In the individual method the pupils were allowed to use any 
 method of problem analysis which they desired. The conclusions of 
 this study are distinctly unfavorable to the conventional-formula 
 method. The dependencies and the individual methods were found 
 to be approximately equal in effectiveness. 
 
Summary of Research Relating to the Teaching of Arithmetic 61 
 
 Adams (1) compared the relative effectiveness of teaching pupils 
 to solve verbal problems in arithmetic by an analytical method and 
 by one in which no attempt at analysis was made. The analytical 
 method is illustrated by the following quotation concerning a demon- 
 stration of the solution of a one-step problem by the teacher: 
 
 How many apples will Tom need to fill 4 baskets if he puts 6 apples into 
 each basket? 
 
 1. The problem is read. 
 
 2. "What are we asked to find?" 
 
 3. "What do we know that will help us to find the answer?"— that there are 
 
 4 baskets and that Tom puts 6 apples into each. 
 
 4. "W T hat will be the name of the answer?" — apples 
 
 5. "Will he need more or less than 6 apples?" Select the number in the 
 
 problem that corresponds to the name of the answer. This device 
 cannot be used in some division problems. 
 
 6. "What two operations give us more for an answer?"— addition and 
 
 multiplication 
 
 7. "Which shall we use here?" — multiplication 
 
 "Why could we not use addition?"— because you cannot add "apples" 
 and "baskets." 
 The non-analytical method of teaching the solution of verbal 
 problems is illustrated in the quotation below: 
 
 How much will Frank have to pay for 3 cans of peas that are sold for 18 cents 
 a can? 
 
 1. Read the problem carefully. 
 
 2. Teacher asks, "What are we asked to find?" 
 
 3. "What do we know that will help us find it?" 
 
 4. "Shall we add, subtract, multiply, or divide?" 
 
 5. The solution is then performed. 
 
 The conclusions reported by Adams are favorable to the analytical 
 method when used with third-grade children. The evidence does not 
 significantly favor either method for fourth-grade children. Adams 
 states in this connection that possibly insufficient time was devoted 
 to the experiment to permit the breaking down of problem-solving 
 habits previously learned. 
 
 2. Evaluation of the experiments. Newcomb (90) used four 
 experimental and two control groups varying in size from fourteen to 
 thirty-six pupils each. These groups, which were made up of seventh- 
 and eighth-grade pupils, were approximately equivalent in arithmeti- 
 cal reasoning ability as shown by the Stone Reasoning Test. The 
 experimental groups were taught one problem a day for twenty days, 
 by means of sheets of general directions for solving verbal problems, 
 while the control pupils were taught the same problems in the tradi- 
 tional fashion. At the end of the experimental period of twenty days 
 the Stone Reasoning Test was again administered. The results 
 
62 Bulletin No. 58 
 
 showed that the pupils who had used the sheets of general directions 
 were significantly better in speed, but only slightly better in accuracy. 
 The experiment may be criticized from several standpoints. The 
 groups used cannot be said to be representative of seventh- and 
 eighth-grade pupils in general, nor do they appear to have been 
 sufficiently equivalent. No mention is made of any attempt to con- 
 trol important non-experimental factors. One suspects that the 
 experimental method was applied with greater zeal. Newcomb is 
 justified, however, in expressing his conclusion in favor of the sheets 
 of general directions with appropriate limitations. 
 
 Stevenson (112) used a single group of 1027 fifth-, sixth-, and sev- 
 enth-grade pupils in eight localities. These pupils were taught to 
 read and analyze problems and to estimate answers in round numbers 
 for a period of twelve weeks. The gains in achievement are certainly 
 significant. While Stevenson shows that this method is effective, he 
 does not show that it is more effective than other methods. It is 
 unfortunate that control groups were not used. 
 
 Greene (42) used an experimental group of sixty-two pupils and a 
 control group of thirty pupils. These pupils were all in the sixth 
 grade and were attending four schools in one system. The groups were 
 not equivalent in arithmetical reasoning ability, as was shown by the 
 Monroe test. The pupils in the experimental group were given 
 training in recognizing and selecting the process involved in the 
 solution of the problem, while the control pupils did not have the 
 advantage of such instruction. Both groups were practiced ten 
 minutes a day for eight days, at the end of which time the Monroe 
 Standard Reasoning Test was administered again. The investigation 
 sought to correct for lack of equivalence by correcting the gain of one 
 of the groups by proportion, a procedure that may not be sanctioned, 
 unless it is proved that practice has no effect on individual differences. 
 The experiment is to be further criticized for the failure of the investi- 
 gator to continue it for a sufficient length of time to reveal significant 
 differences in achievement. It should be mentioned that the conclu- 
 sions favorable to the instructional method used with the experi- 
 mental group are expressed with appropriate restrictions. 
 
 Clark and Vincent (26) used two groups of forty seventh- and 
 eighth-grade pupils each in one school. These groups were equated 
 with respect to intelligence as measured by the Stanford Revision. 
 One group was taught by the conventional method for seven recita- 
 tions, while the other group was taught by the graphic method. 25 
 At the close of the experiment the relative achievement of the two 
 
 ^See page 59 for an illustration of the "graphic" method. 
 
Summary of Research Relating to the Teaching of Arithmetic 63 
 
 groups was measured by the arithmetic section of the Stanford 
 Achievement Test, Form A. The results appear to be somewhat 
 significantly in favor of the graphic method. These experimenters are 
 to be commended for their care in securing equivalence and for the 
 precision with which they describe the compared factors in the report 
 of their research. They are to be criticized for the short duration of 
 their experiment and for failure to mention the use of procedures to 
 secure control of important non-experimental factors. One wonders 
 whether the graphic method advocated by them would engender 
 abilities compatible with recognized arithmetical objectives. In the 
 opinion of the present writers, it might be responsible for the engen- 
 dering of habits which will later need to be unlearned. 
 
 Washburne and Osborne (119) used three groups of sixth- and 
 seventh-grade children in eighteen schools, in investigating the rela- 
 tive effectiveness of (1) assigning large numbers of problems — no 
 special technique, (2) training in analysis of problems, and (3) training 
 in seeing analogies between difficult written problems and easy oral 
 problems. These groups were of the following sizes: 322, 307, and 
 134 pupils. Equivalence was sought with respect to (1) problem- 
 solving ability, (2) ability with fundamental processes, (3) intelli- 
 gence, (4) chronological age, and (5) judgments of teachers with 
 respect to capacity. The following quotation, from the directions 
 issued to the participating teachers, indicates the precautions taken 
 to control important non-experimental factors: 
 
 All other factors should, therefore, be made equal except for the particular 
 differences in method which constitute the experiment. To this end, the same 
 teacher teaches both groups. She does not know the children in one group better 
 than she knows those in the other. The children who are taught earlier in the 
 day one week should change class periods with the others the next week. The 
 amount of time spent by the two groups should be the same. The amount of 
 time, if any, given to drill in the fundamental processes will be the same and the 
 method the same. The amount of oral work, or work done by the class with the 
 teacher, will be the same. No home work will be permitted. No extra time will 
 be allowed in school with this exception: children who have been absent may 
 make up in school the number of periods they missed and do the problems they 
 missed. If this is done, it must, of course, be done in both groups. 
 
 The experiment continued for six weeks, at the end of which time a 
 specially devised problem test was administered. It is unfortunate 
 that the experimenters did not report measures of the "statistical" 
 significance of the differences in achievement. They do not appear 
 to be of sufficient magnitude to be significantly in favor of any one of 
 the methods. While many of the techniques used in this experiment 
 were excellent, one wonders whether it is not somewhat futile to com- 
 
64 Bulletin No. 58 
 
 pare methods, each of which contains some logically excellent 
 characteristics. 
 
 Lutes (68) used four groups of sixth-grade pupils in twelve ele- 
 mentary schools of Des Moines, Iowa. The following evidence is 
 cited by the investigator relative to the representative character of 
 the groups. 
 
 The twelve schools were scattered widely over the city in such a way as to 
 include groups which were representative of widely diverse elements of the 
 population, a wide range of native intelligence, of social status, and of personality 
 of the teachers involved. 
 
 These groups, which varied in size from sixty to seventy-four 
 pupils, were approximately equivalent with respect to arithmetical 
 ability as measured by the Stanford Achievement Tests, Parts 4 and 
 5, and with respect to intelligence as measured by Scale A, Form 1, of 
 the National Intelligence Test. The pupils in the first group were 
 drilled in computation, those in the second group were trained in 
 choosing operations, those in the third group were taught to choose 
 correct solutions and to read problems, while those in the fourth 
 group were taught by the traditional method. Considerable care was 
 exercised in the control of non-experimental factors: 
 
 The same days of the week were used by each group, the same length of 
 recitation period, and the experimenter spent practically the same amount of 
 time with each group and each teacher. No home study was required in any 
 case .... though of course it is impossible to be certain that some of the 
 pupils did not practice the skills at home in order to make a good showing in 
 the test. 
 
 At the end of twelve weeks the second form of the Stanford 
 Achievement Test, Parts 4 and 5, was administered. The differences 
 in gains appear to be significantly in favor of the group drilled in 
 computation. While the techniques of this experiment are for the 
 most part excellent, one may raise the same question that was raised 
 with respect to the preceding experiment. Each of the methods 
 appears to be logically desirable. Why should the relative effective- 
 ness of computational drill, drill in choosing operations, and drill in 
 choosing correct solutions along with emphasis on reading problems 
 correctly be compared? 
 
 Washburne (117) used two groups of 175 second-grade pupils, two 
 groups of 177 fourth-grade pupils, and two groups of 240 sixth- and 
 seventh-grade pupils of sixteen cities of northern Illinois. Equiv- 
 alence was sought with respect to the following traits: (1) problem- 
 solving ability, (2) ability in arithmetic mechanics, (3) mental age, 
 (4) chronological age, and (5) general ability to work as judged by 
 the teacher. The pupils in one group were taught the fundamental 
 
Summary of Research Relating to the Teaching of Arithmetic 65 
 
 processes in connection with verbal problems, while the pupils in the 
 other group were taught fundamental processes and verbal problems 
 separately. Evidence is presented in the report of the experiment 
 which indicates that considerable care was exercised in the control of 
 important non-experimental factors. At the end of six weeks the 
 final tests were administered. The difference in achievement was not 
 significantly in favor of either method. Many of the techniques used 
 in this experiment are very commendable. It would seem, however, 
 that the tests used were more valid with respect to the group which 
 had practiced verbal problems over the longer period of time. The 
 group which learned the fundamental processes in connection with 
 verbal problems had their practice in verbal problems distributed in 
 a way considered to be more psychologically effective. 
 
 Hanna (47) used three groups of seventy-five fourth-grade pupils 
 and three groups of eighty-four seventh-grade pupils in his attempt 
 to determine the relative effectiveness of the dependencies (graphic 
 or diagrammatical), of the conventional-formula (four steps), and of 
 the individual, or informal, methods of teaching pupils to solve arith- 
 metical problems. The groups were shown to be equivalent (for both 
 grade levels) with respect to intelligence and initial arithmetical 
 ability. The arithmetic tests, the same forms of which were used at 
 the beginning and end of the experiment, were the new Stone Test in 
 Arithmetic Reasoning and the Stanford Achievement Test in Arith- 
 metic Reasoning (Form A, Test 5) . The teachers were given detailed 
 written directions for conducting the experimental instruction. The 
 materials of instruction were also carefully prepared. The pupils 
 were given practice sheets, and during the first seven days of the 
 experimental period, they were requested to work the problems 
 thereon with the help of instructions given by the teacher. On the 
 eighth day and on alternate days, until the close of the experiment, 
 the pupils worked the problems on the sheets independently of the 
 teacher. The experiment lasted six weeks, or a total of twenty prac- 
 tice periods. At the end of this time the final tests were administered. 
 In addition to the differences in mean gains, and the "statistical" 
 significance of these differences, the investigator reports learning- 
 curve data secured by scoring the practice sheets for the days in 
 which the pupils worked independently. 
 
 Hanna is to be commended for the many excellent techniques em- 
 ployed in his experiment. It would seem that he has rather adequate- 
 ly defined his experimental factors, secured equivalent groups, con- 
 trolled important non-experimental factors, and measured achieve- 
 ment. It would seem that the only important adverse criticism to be 
 
66 Bulletin No. 58 
 
 made with respect to the techniques used in this experiment has to do 
 with the somewhat artificial conditions necessitated by procedures 
 employed to control non-experimental factors. It would appear, 
 however, that some sacrifice of usual schoolroom conditions is justified 
 if adequate control of non-experimental factors is thus attained. The 
 conclusion that the dependencies, or graphic, method and the indi- 
 vidual, or informal, method are superior to the conventional-formula 
 method appears to be reasonably dependable. The conclusion that 
 the dependencies method is not significantly better than the individ- 
 ual method also appears to be reasonably dependable. 
 
 In his first experiment Adams (1) taught 834 pupils by the ''meth- 
 od of analysis," 772 pupils by the method prescribed in the Philadel- 
 phia Course of Study in Arithmetic, and 507 pupils "by the methods 
 usual to the teachers in charge." The pupils participating in the 
 experiment were located in the third and fourth grades of ten Phil- 
 adelphia public schools selected in an effort to secure representative- 
 ness and control of school and extra-school factors. The experiment 
 lasted for a period of eight weeks. The analysis of the data showed 
 that while the scores of the experimental classes were highest in only 
 one instance, the greatest gains in achievement were made in these 
 classes. In the second experiment 1033 experimental and 1065 con- 
 trol pupils were used. All of the teachers participating in the experi- 
 ment were paired according to their teaching ability as estimated by 
 supervisors, and other steps were taken in an effort to secure control 
 of important non-experimental factors. The final test was admin- 
 istered at the end of seven weeks. The analysis of the data thus 
 secured was quite inconclusive with respect to the relative effective- 
 ness of the methods compared. 
 
 In the third experiment 1938 experimental and 1836 control 
 pupils were used. The ninety-six school classes participating in the 
 experiment were paired on the basis of class medians on the initial 
 arithmetic test. The investigator contends that since there is a high 
 correlation between intelligence and the problem-solving ability 
 measured by the initial test, the experimental and control group 
 were probably equivalent in intelligence. The argument is advanced, 
 and quite rightly, that the use of experimental and control groups of 
 such great size very probably secures adequate equivalence with 
 respect to pupil characteristics through the operation of chance. 
 The pupils in the experimental group were taught to solve problems 
 by an analytical method, while no attempt at analysis was made in 
 the teaching of the control pupils. Only one of the two methods was 
 taught in any one school or by any single teacher. Data are pre- 
 
Summary of Research Relating to the Teaching of Arithmetic 67 
 
 sented to show that the teachers were approximately equivalent with 
 respect to training, experience, and after-school professional training. 
 Data are also presented to show that supervision of the experimental 
 and of the control teachers was approximately the same. The final 
 test was administered at the end of eight weeks. The data secured in 
 the third experiment were also quite inconclusive with respect to the 
 relative effectiveness of the compared methods, although the method 
 of analysis was shown to be slightly more effective in the third grade. 
 
 Adams is to be commended for the many excellent techniques used 
 in his experiments. He should be criticized, however, for failure to 
 conduct his experiments over a longer period of time. 
 
 3. Justified conclusions. Several of the studies in this group 
 contribute evidence in support of the generalization that systematic 
 and persistent training in a procedure for attacking verbal problems 
 results in higher scores on problem tests. This generalization is a 
 fairly obvious inference from the Law of Exercise and the supple- 
 mentary Law of Intensity. 
 
 With respect to relative evaluation of comparable methods of 
 teaching pupils to solve problems, the findings are probably not 
 highly dependable. It was pointed out in the evaluation of several 
 of the experiments that the non-experimental factors of the zeal and 
 skill of the teacher were inadequately controlled and that differences 
 favoring a given method are possibly more justifiably attributable to 
 these influences than to any merits inherent in the method. It may 
 be concluded, therefore, that several methods of teaching pupils to 
 solve verbal problems in arithmetic are feasible, but the effectiveness 
 of these methods in practice depends to a large extent upon the zeal 
 and skill of the teachers using them. 
 
CHAPTER V 
 
 METHODS OF DIAGNOSIS AND REMEDIAL 
 TREATMENT 
 
 ''Diagnosis" is the term used to designate the methods by which 
 specific disabilities of pupils are discovered. "Remedial treatment" 
 designates the methods used in eliminating these specific disabilities. 
 In the experiments on diagnosis and remedial treatment in arith- 
 metic attempts have been made to determine the effectiveness of a 
 variety of methods of diagnosis and of a variety of methods of remed- 
 ial treatment. The experimental factor in these experiments may be 
 characterized as exceedingly complex. Usually the factor includes a 
 somewhat complicated procedure of diagnosis, still more complicated 
 procedures of remedial treatment, and aspects more properly desig- 
 nated as "motivation devices." In none of the experiments does the 
 experimental factor approach the specificity essential in order to give 
 definite meaning to the findings. 
 
 Summary of reported conclusions. That diagnosis and remed- 
 ial instruction are effective procedures in arithmetic is indicated in 
 the investigations of Merton, and others. 1 Kallom, 2 Morton, 3 Smith, 4 
 Stevenson, 5 Yeager, 6 Buswell and John, 7 Sister Kathleen, 8 O'Brien,' 9 
 Otto, 10 Clemens and Neubauer, 11 Neal and Foster, 12 Brownell, 13 
 Chase, 14 Gabbert, 15 Guiler, 16 Lazar, 17 Soth, 18 and Stone. 19 It does not 
 
 RE««2S^>i5i ^' eta J; "^ e r me u 4 ial Work in Arithmetic," Second Yearbook of the Department oj 
 Elementary School Principals Washington: National Education Association, 1923, p. 395-429 (74) 
 
 calionaTplyThotogy] i(>: Jit jSSSr^Sw. ^T^ " EdUCati ° nal Measurement." Journal of Edu- 
 9:117-2^, Fe bruaryh924 n ^sT) 1 ^ 8 ^ PUPllS ' Em>rS in Fractions '" Journal of Educational Research, 
 Novemb? 1 r th 1916 H '(107 I ) n ' liVidUal Variations in Arithmetic," Elementary School Journal, 17:195-200, 
 
 j?„* ^ey.enfon, P. R. "Increasing the Ability of Pupils to Solve Arithmetic Problems," Educational 
 Research Bulletin (Ohio State University), 3:267-70, October 1924 (112) 
 
 ,«^c 6 u^ a I e Pv Will !f- m - "SiW? 1 S , and £ ernedial Measures,in Arithmetic," Twelfth Annual School- 
 men s Week Proceedings Philadelphia: University of Pennsylvania, 1925, p. 296-305. (127) 
 
 'Buswell, G. T and John, Lenore. "Diagnostic Studies in Arithmetic," Supplementary Educa- 
 tional Monographs, No. 30. Chicago: University of Chicago Press, 1926. 212 p ?ffi menlar > * auca 
 
 Arithm^ fe n / ^r^iJJ : ^° m ? Re 1 U i> ts - 0f Remedial Instruction Following the Use of Diagnostic 
 Arithmetic Tests, Catholic Educational Review, 24:19-27, January, 1926 (54) 
 
 ™„™,?™ ^i ?* £' '' Im T Pr ° Vement T °- f I nstr . u ction in Arithmetic," University of Kansas Bulletin of 
 Education, VoL 1, No. 4. Lawrence: University of Kansas, 1927. 42 p. (93) 
 
 rv» ft K.r!Si7 a J ,' *.' Rernedlal Instruction in Arithmetic," Elementary School Journal, 28:124-33. 
 October 1927. Also in: Journal of the National Education Association, 17:87-89, March. 1928 (96) 
 
 Frf„JSiT P nS ; i ^i an A- N ni lb r\ Uer * P u F ' " A Su P er ™ion Project in Multiplication," Journal of 
 Educational Research, 18:38/-96, December, 1928. (28) 
 
 ri^Z^t a K E u A ,' r and F ? S ^\i£ e o- ^ An Ex Periment with Remedial Work in Common Fractions." 
 Elementary School Journal, 29:280-83, December, 1928. (87) 
 
 Septemb5 W 1929 W (14)" " Remedial Cases in Arithmetic," Peabody Journal of Education, 7:100-107. 
 
 meHr P^hf^'w' "^Jvf* ?* and Treat ment of Some Common Difficulties in Solving Arith- 
 metic Problems, Journal of Educational Research, 20:335-42, December. 1929 (24) 
 
 7:147-5?. a Nov r emberA929.' R (3 m 9f U CaSCS '" Arithmctic '- Case l " Peab <>dy Journal of Education, 
 
 Octobe? U i929 ^(ASi " Improving Computational Ability," Elementary School Journal, 30:111-16, 
 
 68 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 69 
 
 seem worth while to present in detail the reported conclusions of all of 
 these investigations. The conclusions of the single-group experi- 
 ments and case studies contribute to our understanding of the effec- 
 tiveness of discovering the individual arithmetical disabilities of 
 pupils by means of diagnostic tests and by means of first-hand ob- 
 servation of the work of the pupil in which he is requested to think 
 aloud in performing the fundamental operations or in solving prob- 
 lems. 20 The conclusions of these investigations also contribute to our 
 understanding of the effectiveness of intensive and zealous instruc- 
 tion to eliminate the disabilities so discovered, either through the use 
 of practice materials prepared in advance or informally at the time. 
 These conclusions, important as they are, do not contribute mate- 
 rially, however, to our knowledge with respect to the relative effective- 
 ness of the various methods of diagnostic and remedial treatment. 
 The conclusions of the controlled experiments contribute, in some 
 measure, to our knowledge of the relative effectiveness of the various 
 methods of diagnostic and remedial treatment. Smith (107) reported 
 that class drill, supplemented by individual assistance on points of 
 weakness revealed by diagnostic tests, is more effective than class 
 drill with extra drill periods provided for the slow pupils who were 
 drilled in groups rather than individually and class drill in which 
 explanations were made only with respect to the group as a whole. 
 Sister Kathleen (54) reported that remedial treatment is more effective 
 when based on analysis and classification of the errors made on the 
 test than when based only on class medians on the test. Neal and 
 Foster (87) have reported that "organized practice material in the 
 hands of the children, with provision for the diagnosis of difficulties 
 and remedial work, is more effective in economy of the teacher's time 
 and of the children's time and in final results in maintaining skill in 
 the manipulation of common fractions than is the usual practice 
 provided by the teacher." The conclusion of Stone (113) that diag- 
 nostic and practice tests produce "greater gains in ability to reason in 
 arithmetic than does the regular work in arithmetic that the tests 
 may displace in classroom use" agrees with that of Neal and Foster 
 (87). 
 
 "Lazar May. Diagnostic and Remedial Work in Arithmetic Fundamentals for Intermediate 
 Grad 't>sS™M.°£ : "Altudy %Z?fi&2£iJ£ J&itfiStic." Elementary School Journal, 29:439- 
 42 ' FC .S' C 2 W (1 "An Experimental Study in Improving Ability to Reason in Arithmetic, ' 
 Twenty mnt'h Yearbook of the National Society fc* the Study of Education. Bloomington. Illinois. 
 
 P » bUC 0?ou?n^^ ^^n^Sd^f ir™ 1 ^ 
 
70 Bulletin No. 58 
 
 Evaluation of the investigations. The studies reported by 
 Merton, and others, (74) and by Yeager (127) are to be characterized 
 as ''descriptive accounts of what is going on in some school." Some 
 quantitative data are given and some comparisons in achievement of 
 different classes are reported, but it is not possible to justify the 
 labeling of such investigations "experiments." The studies of 
 Kallom (53), Brownell (14), Chase (24), Gabbert (39), and Soth (108) 
 were based on data secured from the following numbers of cases: 
 3, 4, 17, 1, 1. Descriptive accounts of what is taking place in schools 
 and reports of case studies are interesting. They should be very 
 suggestive to teachers in practice. It is impossible, however, to 
 generalize from data so restricted. 
 
 Morton (81), Stevenson (112), O'Brien (93), Otto (96), Clemens 
 and Neubauer (28), Guiler (43), and Lazar (66) conducted single- 
 group experiments. Morton (81) used one group of thirty-six eighth- 
 grade pupils for a period of rive months. He measured the improve- 
 ment of these pupils as a result of diagnostic and remedial treatment 
 by means of tests constructed by himself. The substantial gains 
 shown may not with certainty be ascribed to the experimental factor, 
 because of the failure to employ a control group. The single-group 
 experiment of Stevenson (112) was described and evaluated, rather 
 unfavorably, in the previous chapter. 21 
 
 O'Brien (93) used 357 pupils in the seventh, eighth, ninth, and 
 tenth grades of three small school systems. After an initial program 
 of mental and achievement testing, diagnosis was made with respect 
 to "mental ability, previous schooling, achievement in various phases 
 of the subject, and specific types of errors or difficulties which char- 
 acterized the students' work." The program of remedial instruction 
 was based on the weaknesses discovered by the tests. Pupils were 
 informed of their individual weaknesses, and the teachers were pro- 
 vided with general and detailed suggestions for carrying out the re- 
 medial instruction. They were also provided with advice in confer- 
 ences and with information in the form of abstracts of selected articles 
 in current literature. At the end of five months the final tests were 
 administered. While the increases in achievement are large, it is 
 difficult to ascribe these increases to any specific experimental factor. 
 No control groups were used, and it is evident that the pupils were 
 subjected to a complex of factors. 
 
 Otto (96) used a single group of nine fourth-grade pupils for a 
 period of seven months. Achievement was measured by diagnostic 
 tests, and remedial treatment was provided by means of prepared 
 
 2l See pages 58 to 67. 
 
Summary of Research Relating to the Teaching of Arithmetic 71 
 
 practice materials, but, again, because of lack of control, it is impos- 
 sible to say how much of the improvement found is to be ascribed to 
 the experimental factor. Clemens and Neubauer (28) employed a 
 single group of 425 fourth-, fifth-, sixth-, seventh-, and eighth-grade 
 pupils in twelve elementary schools of one city. Tests were con- 
 structed by the authors which covered forty-two multiplication diffi- 
 culties. Tests were administered four times: (1) at the beginning of 
 the experiment, (2) at the end of a week, (3) at the end of two more 
 weeks, and (4) at the end of three months from the administration of 
 the third test. "Individual help was given to each pupil who failed 
 to obtain a perfect score in the first test. After correcting the child's 
 error and showing him how to work the example correctly, the teacher 
 gave him the drill card designed to meet his difficulty." Substantial 
 gains in achievement were indicated by the test results, but failure to 
 use a control group again makes it impossible to determine how much 
 of this gain is to be ascribed to the experimental factor. Guiler (43) 
 used a single group of ten seventh-grade pupils for one hour a week 
 for twelve weeks. An analysis was made of the errors of these pupils 
 on the diagnostic tests used, and remedial instruction adapted to 
 individual needs was provided. The gains in achievement were meas- 
 ured by several standardized arithmetic tests, but it must be re- 
 peated again that failure to use a control group renders the con- 
 clusions of doubtful dependability. 
 
 Lazar (66) used a single group of forty-three sixth-grade pupils. 
 The initial status of these pupils was determined by means of intelli- 
 gence tests, of standardized arithmetic achievement tests, of a diag- 
 nostic arithmetic test, and by individual observation and oral exam- 
 ination. Ten minutes of the daily arithmetic period were devoted to 
 remedial work characterized by the experimenter as follows: (1) Spe- 
 cific instruction on class or individual weaknesses as determined by 
 diagnosis was given; (2) the Courtis Standard Practice Tests were 
 used for drill on the operations in which deficiencies were shown; 
 
 (3) supplementary material was devised to overcome difficulties with 
 the addition combinations, with long division, and with fractions; 
 
 (4) the pupils were taught how to make records and graphs to show 
 their achievement, and the teacher made graphs of the class achieve- 
 ment; (5) training the pupils to have the proper attitude toward their 
 deficiencies was an important phase of the work. At the end of five 
 months the initial arithmetic tests were again administered. The 
 gains in achievement appear to be "statistically" significant. While a 
 control group was not used, some of the functions of a control group 
 were attained by comparison of the experimental results with test 
 
72 Bulletin No. 58 
 
 norms. The experiment is to be commended for its comprehensive 
 and intensive nature, but Lazar's experiment deserves criticism sim- 
 ilar to that applied to the experiment of O'Brien (93) — the experi- 
 mental factor was exceedingly complex. 
 
 Buswell and John (20) investigated the problem of arithmetical 
 diagnosis by means of two types of laboratory technique and by 
 means of a comprehensive single-group experiment. In the labora- 
 tory study of eye-movements in column addition two fourth-grade, 
 eight fifth-grade, and seven sixth-grade pupils were used. In addition 
 to these groups of children, three adults were used. In the report of 
 this research data are given in graphic form, which are dependable 
 evidence with respect to the nature of eye-movements in column 
 addition. This evidence emphasizes the need for diagnosis in 
 arithmetical instruction. 
 
 The second laboratory investigation, in which thirty subjects 
 were used, was conducted by means of dictaphone and kymograph 
 apparatus. Time analyses were made of the four fundamental 
 operations. What each child was asked to do is described in the 
 following quotation: 
 
 The children who participated in the experiment were seated one at a time at 
 a table on which was a sheet of paper. On this paper were typewritten the ex- 
 amples which they were to work. The only piece of apparatus in the room was a 
 specially constructed telephone transmitter, which was clamped to the edge of 
 the table. The experimenter sat beside the child and instructed him as to his 
 procedure. The child was asked to give his partial answers aloud and also to 
 say the digits which he wrote on the paper at the same time that he wrote them. 
 In the case of an example in column addition the child was instructed to give 
 each of the sums as he proceeded down the column. 
 
 The sound of the child's voice was reproduced by an amplifier in 
 another room and recorded by means of a dictaphone. These records 
 were then ''transcribed on kymograph paper by using an electric 
 time-marker and a telegraph key." The kymograph record may be 
 described as follows. One line broken at regular intervals showed the 
 time elapsed in intervals of fifths of a second. The second line, 
 broken at irregular intervals revealed the time required for each 
 partial answer. To illustrate by data secured from one child it was 
 found that the child in adding a single column of thirteen digits 
 required three-fifths of a second each to add 4 + 9, 13 + 3, and 
 16 + 2. He required 19/5 of a second to add the combination 29 + 3. 
 Data relative to time required to perform the fundamental operations 
 for all the subjects are presented in tabular form in the monograph. 
 An examination of the description of the techniques used gives no 
 reason to doubt the reliability of these data. They are additional 
 evidence of the need for diagnosis in arithmetical instruction. 
 
Summary of Research Relating to the Teaching of Arithmetic 73 
 
 Buswell and John used a single-group of 303 children, in nine 
 classes, in twelve elementary schools. In a preliminary study they 
 used a single group of 250 children in the third, fourth, fifth, and sixth 
 grades. Diagnostic sheets, for each of the fundamental processes, 
 were followed by remedial treatment administered by the teachers to 
 suit the individual needs of the pupils. The Cleveland Survey Test 
 was administered before and after the ten-weeks' period of diagnosis 
 and remedial treatment, and substantial gains were found. Buswell 
 and John hold that these gains may be ascribed to the experimental 
 factor, even though a control group was not used. They state: 
 
 Owing to the lack of a refined technique in carrying on the experiment, a 
 small difference between the actual improvement shown and the normal expected 
 improvement cannot be considered significant. However, if the difference is 
 fairly large, it seems fair to conclude that the difference is due to the diagnostic 
 procedure and remedial instruction given by the teacher. 
 
 If this contention is accepted as correct, the evaluation of the 
 dependability of the conclusions of the other single-group experiments 
 must be modified. The gains in achievement were, without exception, 
 large. The present writers do not feel, however, that the conclusions 
 derived from data secured by single-group experimentation can be as 
 satisfying, other things being equal, as those obtained from controlled 
 experimentation. Obviously, it is impossible to determine how much 
 of the gains in achievement was due to inherent qualities in the meth- 
 ods of diagnosis and remedial treatment and how much was due to 
 additional and zealous instruction and to the mere drill afforded. 
 
 Control groups were used in the experiments of Smith (107), 
 Sister Kathleen (54), Neal and Foster (87), and Stone (113). The 
 experiment of Smith ( 107) has already been described and evaluated 
 somewhat unfavorably. 22 Sister Kathleen (54) used two groups of fifty 
 sixth- and seventh-grade pupils in neighboring schools in her investi- 
 gation of the relative effectiveness of remedial treatment based on 
 analysis and classification of the errors made on a diagnostic test and 
 remedial treatment based only on class medians on the test. She 
 stated with respect to equivalence that the groups were "about the 
 same average mental ability." The differences in gains which are not 
 highly "statistically" significant were measured by the Woody- 
 McCall Mixed Fundamentals Test, Forms I and II. The conclusions 
 of Sister Kathleen seem to be somewhat more dependable than those 
 of Smith (107), but the techniques used in this experiment were not 
 without criticism. There is evidence of failure to control important 
 non-experimental factors, particularly the factor of zeal on the part 
 of the teachers. 
 
 22 See pages 30 to 33. 
 
74 Bulletin No. 58 
 
 Neal and Foster (87) used approximately six hundred experi- 
 mental and approximately four hundred control pupils in the fifth 
 grade. These groups were not equivalent according to the initial-test 
 scores, but allowance for non-equivalence is made in interpreting the 
 results. The pupils in the larger group used "organized practice ma- 
 terial, with provision for diagnostic and remedial work," while the 
 pupils in the smaller group had "the usual practice provided by the 
 teacher." The experiment lasted three months. The differences in 
 gains in achievement, which are possibly "statistically" significant, 
 were measured by the Stanford Achievement Test, Forms A and B, 
 and by an informal fraction test prepared by the investigators. The 
 experimentation deserves commendation with respect to the direc- 
 tions given participating teachers by means of mimeographed sheets. 
 The conclusions stated would be more satisfying to the critical reader 
 if appropriate restrictions had been made in addition to the recog- 
 nition given to faulty equivalence. 
 
 Stone (113) made comparisons between groups of paired fifth-, 
 sixth-, seventh-, and eighth-grade pupils of various sizes. In his pre- 
 liminary trial 175 pairs of equivalent pupils were used. In his main 
 trial comparisons were made between a total of 1 72 pairs. Other com- 
 parisons were made without resorting to pairing. The pupils partici- 
 pating in the experiment were located in twenty-three schools of five 
 school systems. These pupils were paired with respect to arithmetic 
 scores, mental age, chronological age, and school grade. Pairs were 
 located in the same school systems. The pupils in the experimental 
 groups had the benefit of a program of diagnostic and practice tests 
 described by the experimenter as follows: 
 
 The diagnostic tests were designed to accompany the survey tests. Their 
 purpose is to afford more precise means of locating each pupil's difficulties in 
 arithmetical reasoning. They enable each pupil to think, by graduated steps, 
 into and through his individual difficulty. The practice tests were designed to 
 follow the diagnostic tests. Their purpose is to afford needed practice on specific 
 difficulties, as located by survey and diagnostic tests. They enable each pupil 
 to rethink the reasoning involved in his individual difficulty. 
 
 The pupils of the control group had the regular work in arithmetic 
 without the benefit of a program of diagnosis and remedial treatment. 
 The experiment lasted for five weeks. Gains in achievement were 
 measured by the Stone Survey Tests I and II and by the Stone 
 Reasoning Tests in Arithmetic. The differences in gains appear to 
 be "statistically" significant. The chief criticism with respect to this 
 experiment concerns the validity of the measuring instruments used. 
 It seems possible that the tests may have been more valid with re- 
 spect to the abilities engendered by the practice material. If this 
 
Summary of Research Relating to the Teaching of Arithmetic 75 
 
 was the case, some of the differences in gains should be attributed to 
 this cause. The techniques used in this experiment are for the most 
 part very commendable, especially those used in securing a repre- 
 sentative sample and equivalent groups. The conclusions in favor of 
 the diagnostic and remedial methods used with the experimental 
 pupils are stated conservatively and as such seem quite dependable. 
 Justified conclusions. The generalization seems justified that 
 diagnosis and remedial treatment should be recognized as necessary 
 phases of instruction in arithmetic. The conclusions relative to the 
 methods of diagnosis and remedial instruction are less certain. It 
 seems evident from the comprehensive investigation of Buswell and 
 John (20) that individual diagnosis and remedial instruction adapted 
 to the needs of individual pupils are most effective. Other investi- 
 gators obtained good results by means of diagnostic tests and practice 
 material placed in the hands of the pupils, with less individual atten- 
 tion being given. There seems to be no reason to doubt that such 
 methods are effective. Further research is needed, however, before it 
 may be said that such methods are as effective as, or more effective 
 than, methods in which emphasis is placed on direct observation of 
 the pupil engaged in arithmetical learning activity and in which im- 
 mediate provision of remedial instruction for the disabilities is dis- 
 covered. It is quite evident that more attention should be given, in 
 experimental evaluations of diagnostic and remedial methods, to the 
 evaluation of specific aspects of such instruction rather than to 
 evaluation of a complex of factors. 
 
CHAPTER VI 
 
 METHODS OF TEACHING READING OF 
 ARITHMETICAL SUBJECT-MATTER 
 
 It is fairly well known that children differ in their abilities to read 
 various types of subject-matter. The reading of examples and of 
 verbal problems in arithmetic involves the use of abilities quite differ- 
 ent from those used in reading historical description or exposition. 
 The research referred to in the first part of this chapter indicates the 
 necessity of recognizing the significance of unique reading skills as 
 factors in arithmetical achievement. The small number of experi- 
 mental evaluations of methods of teaching the reading of arithmetical 
 subject-matter is an indication that this problem has not received 
 wide recognition among research workers in the field of arithmetic. 
 One of the experiments described deals with the effectiveness of 
 general training in reading. The second experiment deals with the 
 effectiveness of a questioning method. It is also an attempt to 
 evaluate dramatization and story telling as means of teaching the 
 reading of verbal problems. In the third experiment, instructions in 
 reading were included on the problem solution sheets provided for the 
 pupils. There is need for an evaluation of a method which is more 
 likely to engender the specific reading abilities required for arith- 
 metical subject-matter. 
 
 Summary of reported conclusions. The necessity of instruct- 
 ing pupils in the reading of arithmetical subject-matter has been 
 shown in a number of studies. Buswell and John, 1 Brooks, 2 Chase, 3 
 Edano, 4 and Partridge 5 have reported that a technical vocabulary is 
 needed by children engaged in arithmetical learning activity. The 
 conclusion stated by Chase (23) is typical: 
 
 .... the investigation here recorded has shown after careful study of 
 numerous textbooks, that many problems involve conditions that are quite 
 untrue to life; that many of the words used are 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 vocabularies of the various 
 texts whi ch they have used quite unsuited to the capacities of their pupils. 
 
 l/0Ho*™X e N?'™ a rift h ": L T e r n - rC - -! Th f Vocabulary of Arithmetic,- Supplementary Educational 
 Monographs, No. 38. Chicago: University of Chicago Press, 1931. 146 p (21) 
 
 *™„, ,• , I' , ~ A , Stu ^ of the Tech nical and Semi-Technical Vocabulary of Arithmetic" 
 
 Educational Research Bulletin (Ohio State University), 5:219-22, May 26 1926 (12) * Amnmeuc - 
 
 ^hase, S. E. "Waste in Arithmetic" Teachers College Record, 18:360-70, September, 1917. (23) 
 1:81-84; Fe n bruSy, U i r 92°8. (36) Analys,s of Ari thmetic Textual Matter," Philippine Public Schools, 
 26^6^^^^^ Ne ° ds in Childre »' s R ™ di "* Activities." Elementary School Journal, 
 
 76 
 
Summary of Research Relating to the Teaching of Arithmetic 77 
 
 Several studies of errors made by pupils in the solution of arith- 
 metical problems indicate that reading disability is an important 
 cause of errors. 6 Studies of the correlation between arithmetical 
 ability and reading ability seem to indicate that a small but "statis- 
 tically" significant correlation exists. 7 In certain discussions of 
 measurement in arithmetic it has been indicated that arithmetical 
 achievement is in part a function of reading ability. 8 In the opinion 
 of the present writers the most significant evidence relative to the 
 importance of instructing pupils to read arithmetic is to be found in 
 the laboratory studies of Buswell and John 9 and of Terry. 10 The 
 latter investigator has stated some suggestions for instructing pupils 
 in reading arithmetical problems which seem worthy of quotation: 
 
 1. Pupils should be taught to distinguish between the first reading and the 
 
 re-reading phases in their attack on problems. 
 
 2. They should learn to consider numerals and the accompanying descriptive 
 
 conditions as different elements of a problem and separable for reading 
 purposes. 
 
 3. During the first reading, they should devote their attention to the 
 
 conditions of the problem. 
 
 4. At the same time skill should be developed in partial reading of numerals. 
 
 5. While this skill is being acquired, pupils should be apprised of the essential 
 
 similarity between the conditions of the problem and such details of the 
 numerals as are perceived by partial reading. 11 
 Experimental investigations of methods of instructing pupils to 
 read arithmetical subject-matter have been reported by Newcomb 12 
 
 6 H ydle L. L. and Clapp. F. L. "Elements of Difficulty in the Interpretation of Concrete 
 Problems in Arithmetic," Bureau of Educational Research Bulletin No. 9. Madison: University of Wis- 
 
 COnSin john 2 , 7 Lenor?: ^Difficulties in Solving Problems in Arithmetic," Elementary School Journal, 
 
 31: % 1 ort?n°T 1 £ r, ''S A^lV8is of Errors in the Solution of Arithmetic Problems," Educational 
 Research Bulletin (Ohio State University), 4:187-90 Apnl 29 1925 (82) p> 
 
 Stevenson, P. R. "Increasing the Ability of Pupils to Solve Arithmetic Problems, Educational 
 Research Bulletin (Ohio State University), 3:267-70 October 15, 1924. (112) 11 -95-103 
 
 Stevenson, P. R. "Difficulties in Problem Solving," Journal of Educational Research, ll.vs-iuj, 
 
 FebrU 7Hlckl 9 er 5 J M.^'The Relation between Successful Progress in Mathematics and the Ability to 
 Read and U^de^nd, and the Factors that Contribute to Success, or Failure in Mathematics. 
 Unpublished master's thesis in Education. Chicago: University of Chicago, 1921 82 p ;> (44) 
 
 Harlan, C L. "Years in School and Achievements in Reading and Arithmetic, Journal, oj 
 Educational Research, 8:145-49, September, 1923. (48) . .. ~„ „ nf PrnhlMn< , in 
 
 Wheat, H. G. "The Relative Merits of Conventional and Imaginative Types of Problems in 
 Arithmetic," Teachers College, Columbia University Contributions to Education, No. .359. New York. 
 Bureau of Publications, Teachers College, Columbia University, 1929. 124p. 021) >( 
 
 sDawson, C D. "Some Results in Using Starch's Arithmetic Reasoning Test, Journal of 
 
 ^"'ifco^ Rea'sonfn'gTests in Arithmetic," School and Society, 8:295-99, 
 
 32 "?feS£lU G? T^AdThn/Linore. "Diagnostic Studies in Arithmetic - Supplementary Edu- 
 al Monographs, No. 30. Chicago: University of Chicago Press, 1926. 212 p. (2UJ 
 "Terry P. W. "How Numerals Are Read: An Experimental Study of the Reading of Isolated 
 
 Numerals in Arithmetical Problems," Supplementary Educational Monographs, No. 18. Chicago. 
 
 University of Chicago Press, 1922. 110 p. (115) < See also: i 7/ 7„ / .„,v /Mf/ ,i Pwrhnlnev 1 ?-365- 
 
 Terry P. W. "The Reading Problem in Arithmetic," Journal of Educational Psychology, 12.^05 
 
 77 October' 1921. (A Summary of the monograph referred to above.) (116) . -..„,, 
 
 1 "Terry, P. W. "How Numerals Are Read: An Experimental Study of the Reading of Isolated 
 
 Numerals in Arithmetical Problems," Supplementary Educational Monographs, No. 18. Chicago. 
 
 ^^^omb^^'^i^uX &wto Solve Problems in Arithmetic," Elementary School 
 
 Journal, 23:183-89, November, 1922. (90) 
 
 cational Monographs 
 
78 Bulletin No. 58 
 
 Wilson, 13 and Lessenger. 14 The pupils in the experiment of New- 
 comb (90) were given instructions in reading problems on problem 
 solution sheets, while in the experiment of Wilson (122) the pupils 
 were taught to read problems by a questioning method and through 
 dramatization and story telling. Lessenger (67) reported an experi- 
 ment where general reading instruction was the experimental factor. 
 These experiments lead to the general conclusion that reading instruc- 
 tion increases significantly the ability of pupils to solve arithmetical 
 problems. 
 
 2. Evaluation of experiments. The investigations of Brooks (12), 
 Chase (23), Edano (36), and Partridge (99) were analytical rather 
 than experimental in character. The need for instruction in reading 
 was inferred from analyses, of arithmetical materials of instruction. 
 The investigations of Hydle and Clapp (50), John (51), Morton (82), 
 Stevenson (111), and Stevenson (112) were also analytical in nature, 
 but the analysis was made of pupil responses to arithmetical prob- 
 lems. Buswell and John (21) prepared group tests of arithmetical 
 vocabulary and administered them to 1500 fourth-, fifth-, and sixth- 
 grade pupils in several school systems. Their findings are probably 
 the most significant in this group. 
 
 It is evident that the analytical investigations are limited by the 
 inferences which had to be made. One may not be sure from observ- 
 ing a mistake made in a problem whether the cause of the faulty 
 response was lack of reading ability or lack of some other ability. 
 For example, the written performances of two pupils may be identical 
 and thus not indicative of the fact that one of the pupils was handi- 
 capped by arithmetic disability while the other failed to solve the 
 problem correctly because of reading disability. 
 
 Hackler (44) and W 7 heat (121) indicated the importance of reading 
 ability in arithmetic learning activity by typical correlation tech- 
 niques, and Harlan (48) showed that arithmetic and reading ability 
 tend to occur together, indicating his correlation in graphic form. 
 The correlation studies of Hackler (44), Wheat (121), 15 and Harlan 
 (48) are limited in dependability in the sense that all correlation 
 studies are limited when the attempt is made to interpret them in 
 terms of cause and effect. The raw coefficients obtained between 
 arithmetic scores and reading scores are probably due in a large meas- 
 ure to the common factor of intelligence. If an attempt is made to 
 partial out intelligence, the coefficient so obtained may be too much 
 
 13 Wilson, Estaline. "Improving the Ability to Read Arithmetic Problems," Elementary School 
 Journal, 22:380-86, January, 1922. (122) 
 
 u Lessenger, W. E. "Reading Difficulties in Arithmetical Computation," Journal of Educational 
 Research, 11:287-91, April, 1925. (67) 
 
 n See page 57 for unfavorable criticism of Wheat's use of correlation methods. 
 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 7<> 
 
 reduced. Intelligence as represented in the intelligence score usually 
 obtained includes reading ability. Partial correlation would not 
 separate the two effectively, and the partial coefficient would of 
 
 necessity be low. 16 
 
 The laboratory investigation of Buswell and John (20) has already 
 been described and favorably evaluated. 17 Terry (115) used similar 
 techniques. A portion of his data was secured by having his subjects 
 record by means of a telegraph key and kymograph apparatus the 
 time spent in the first reading and in the re-reading of arithmetical 
 problems. The following data secured from one subject on one prob- 
 lem are illustrative: 
 
 7.6 seconds— time required for first reading 
 1.4 seconds— time required to re-read one numeral 
 2.4 seconds— time required to re-read another numeral 
 .2 seconds— time required to re-read last sentence 
 
 Additional data were secured by means of eye-movement appa- 
 ratus. All of Terry's data appear reliable evidence of the important 
 function of reading ability in solving arithmetical problems. The 
 suggestions made by Terry with respect to instruction in reading 
 arithmetical problems may be regarded, however, only as suggestions. 
 Terry has not shown by experimental trial that the method suggested 
 is effective in increasing reading ability with respect to arithmetical 
 
 problems. 
 
 The experiment of Newcomb (90) has already been described and 
 criticized with respect to lack of representativeness of pupils used, 
 lack of equivalence, and failure to secure adequate control of non- 
 experimental factors. 18 Wilson (122) used one group of thirty-four 
 sixth-grade pupils of relatively low intelligence. These pupils were 
 given the Stone Reasoning Test at the beginning of the experiment 
 and were taught to read problems by a questioning method for twelve 
 minutes three times a week for five weeks; at the end of this time they 
 were tested again. The significant increase in achievement may not 
 be ascribed with certainty to the experimental factor. Wilson re- 
 ported similar results for instruction by which the children were 
 directed to convert problems into stories and to dramatize them. 
 One wonders how much of the reading ability so engendered would 
 transfer to ordinary problem-solving activity. 
 
 Lessenger (67) used data collected from a single group of 111 
 
 EdUCa gZ£r ^TdSSS'. 8*2* V&ffilS^tS^J^ of EtucaUenal 
 
 Psychology, 21:657-79, December, 1930. 
 17 See pages 72 and 73. 
 l<i See pages 61 and 62. 
 
80 Bulletin- No. 58 
 
 pupils in Grades III to VIII, inclusive. Analysis of the arithmetical 
 computation scores on the first test administered to the pupils showed 
 a mean loss in arithmetical age of 6.1 months because of faulty read- 
 ing. After a year of intensive general training in reading, analysis of 
 the final-test results showed a mean loss in arithmetic age due to 
 faulty reading of only .7 months. Classification and tabulation of the 
 data secured from the initially good and poor readers revealed a 
 superior gain in arithmetical age for the poorer readers. The investi- 
 gator attributes this superior gain to the general training in reading. 
 It is evident that this study is to be characterized as a rather crude 
 experiment. No control group was used, and for this reason it is 
 difficult to ascribe the improvement noted to the experimental factor. 
 Justified conclusions. The conclusion seems justified that reading 
 ability is an important factor in arithmetical achievement. The 
 magnitude of its influence in arithmetical achievement is not known, 
 but the investigations of Buswell and John (20) and of Terry (115) 
 indicate that it is a very important influence. This being the case, 
 it seems justifiable to say that pupils should receive instruction in 
 reading arithmetical subject-matter. Further research must be 
 conducted, however, before a dependable conclusion may be stated 
 relative to the nature of the most effective instruction. 
 
CHAPTER VII 
 
 MOTIVATION OF LEARNING IN ARITHMETIC 
 
 The assignment of learning exercises which are of immediate 
 interest to pupils is recognized as a basic procedure in securing moti- 
 vation of learning activity in the various school subjects. Attention 
 is given, therefore, in this chapter to research on the stimulating 
 effect of various types of learning exercises in arithmetic. Certain 
 supplementary procedures for securing intensive effort and persist- 
 ence in learning have been shown to be effective in the general 
 research on motivation. 1 Some of these procedures have been em- 
 ployed as experimental factors of experiments in the field of arith- 
 metic. These supplementary procedures are definite goals or objec- 
 tives, knowledge of status or progress, competition, commendation, 
 and reproof. 
 
 Summary of reported conclusions. The conclusions of investi- 
 gations relating to the motivation of learning in arithmetic are sum- 
 marized under the following heads: (1) effect of types of learning 
 exercises, (2) effect of definite goals, (3) effect of knowledge of status 
 or progress, (4) effect of competition , (5) effect of commendation and 
 reproof. As will be noted, several of the investigations involved more 
 than one motivation procedure. Consequently such studies will 
 appear under two or more heads. 
 
 Number games, 2 problems presented in story form, 3 dramatiza- 
 tion of activities that create arithmetical problems, 4 problems relating 
 to the out-of-school life of pupils, 5 and problems which the pupils 
 believe they can solve successfully 6 have been reported as effective 
 in stimulating learning activity. 
 
 The stimulating effect of definite goals is usually involved in the 
 use of standardized tests, especially when the attention of the pupils 
 
 ~^o^Ts y ^dK^SuU: e b. "Stimulating Learning Activity/' University of Minos 
 Bulletin, Vol! 28, No . 1, Bureau of Educational Research Bulletin No. 51. Urbana: University of 
 
 Illin °2Stdnway, L.sSaii Experiment in Games Involving a Knowledge of Number," Teachers College 
 EMe ^^^h&! V& I^^ L Ability to Read Arithmetic Problems," Elementary School 
 Journal, 22:380-86, January, 1922. (122) 
 
 RatvSf'w. C "The Social Motive in the Teaching of Arithmetic," Elementary School Journal, 
 
 lSl2 ^k^^0. 19 "Ati^tic Reasoning Project and the Measurement of Improvement." 
 
 Chicago Principals- Club, Second Yearbook. Chicago: Chicago Principals Club, 192/, p. ■ 86-87. (41) 
 Kulp C L. "A Method of Securing Real-Life Problems in the Fundamentals of Arithmetic, 
 
 Elementary School Journal, 29:428-30, February, 1929. (64) . c„i„:„„ " 
 
 Bfiowman, H. L. "The Relation of Reported Preference to Performance in Problem Solving 
 
 University™/ Missouri Bulletin, Vol. 30, No. 36, Education Series, No. 29. Columbia: University of 
 
 Missouri, 1929. 52 p. (10) 
 
 81 
 
 
82 Bulletin No. 58 
 
 is directed to the norms specified by the tests. Motivation of learning 
 activity by means of administering standardized arithmetic tests has 
 been reported by Ballou, 7 Courtis, 8 Krause, 9 O'Brien, 10 and Werth- 
 eimer. 11 In such cases it is likely that the attainment of a high score 
 was recognized by the pupils as a definite goal. 
 
 In several investigations it is difficult to separate the effect of 
 definite goals from the effect of knowledge of progress. The latter 
 factor, however, has been reported as having a beneficial influence in 
 arithmetical learning activity by Sheerin, 12 Richardson, 13 Anthony 
 and others, 14 Chapman and Feder, 15 Hahn and Thorndike, 16 Kirby, 17 
 and Panlasigui and Knight. 18 
 
 The ease with which arithmetical achievement, especially in the 
 field of calculation, is measured facilitates competition between indi- 
 vidual pupils and between groups. Mailer 19 has reported that indi- 
 vidual competition is the more effective. Hahn and Thorndike (46) 
 have reported that directing each pupil to compete with his own rec- 
 ord was found to be an effective motivating device in learning 
 addition. 
 
 The motivating effects of commendation and reproof have been 
 studied by Hurlock 20 and by Newcomb. 21 The former found that 
 although, in general, commendation is superior to reproof as a moti- 
 vating procedure, girls are more affected by praise than boys, while 
 boys are more affected by reproof than girls. She found also that 
 older and younger children are about equal in responsiveness to 
 praise and reproof, and that inferior children are most responsive to 
 
 /Ballou, F. W. "Improving Instruction through Educational Measurement," Educational 
 Administration and Supervision, 2:354-67, June, 1916. (7) 
 
 8 Courtis, S. A. "The Courtis Standard Tests in Boston, 1912-15: An Appraisal," School Docu- 
 ment No. 15, 1916. Department of Educational Investigation and Measurement Bulletin No. 10. 
 Boston: Boston Printing Department, 1916. 48 p. (33) 
 
 9 Krause, A. K. "Why Monroe Diagnostic Tests in Arithmetic?" Contributions to Education, 
 Vol.2. Yonkers, New York: World Book Company, 1928, p. 15-17. (63) 
 
 10 O'Brien, F. P. "Co-operative Experiment Pertaining to Instruction in Arithmetic," American 
 Education, 31:219-21, February, 1928. (92) 
 
 "Wertheimer, J. E. "Some Results of Monroe's Diagnostic Tests in Arithmetic," Journal of 
 Educational Psychology, 11:109-12, February, 1920. (120) 
 
 12 Sheerin, E. M. "Application of the Dalton Plan to Teaching Arithmetic," Contributions to 
 Education, Vol. 2. Yonkers, New York: World Book Company, 1928, p. 18-22. (106) 
 
 "Richardson, J. W. "The Campaign Method in Elementary Education," Journal of Educational 
 Research, 2:481-92, June, 1920. (104) 
 
 "Anthony, Kate, et al. "The Development of Proper Attitudes Towards School Work," School 
 and Society, 2:926-34, December 25, 1915. (3) 
 
 15 Chapman, J. C. and Feder, R. B. "The Effect of External Incentives on Improvement," Journal 
 of Educational Psychology, 8:469-74, October, 1917. (22) 
 
 16 Hahn, H. H. and Thorndike, E. L. "Some Results of Practice in Addition under School 
 Conditions," Journal of Educational Psychology, 5:65-84, February, 1914. (46) 
 
 17 Kirby, T. J. "Practice in the Case of School Children," Teachers College, Columbia University 
 Contributions to Education, No. 58. New York: Bureau of Publications, Teachers College, Columbia 
 University, 1913. 98 p. (57) 
 
 18 Panlasigui, Isidoro, and Knight, F. B. "The Effect of Awareness of Success or Failure," 
 Twenty-Ninth Yearbook of the National Society for the Study of Education. Bloomington, Illinois: 
 Public School Publishing Company, 1930, p. 611-19. (98) 
 
 19 Maller, J. B. "Cooperation and Competition, An Experimental Study in Motivation," Teachers 
 College, Columbia University Contributions to Education, No. 384. New York: Bureau of Publications, 
 Teachers College, Columbia University, 1929. 176 p. (70) 
 
 20 Hurlock, E. B. "An Evaluation of Certain Incentives Used in School Work," Journal of 
 Educational Psychology, 16:145-59, March, 1925. (49) 
 
 21 Newcomb, R. S. "Securing the Maximum Amount of Work from Every Pupil," Elementarv 
 School Journal, 25:376-79, January, 1925. (89) 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 83 
 
 praise while superior children are most responsive to reproof. New- 
 comb (89) urged pupils to solve supplementary problems, and by 
 commending them when they did so, secured effective results. In 
 addition to informing pupils of their progress and of their goals, 
 Kirby (57) secured motivation by commending the attainment of 
 high scores. 
 
 Evaluation of experiments. Several of the investigations may 
 be termed "uncontrolled" experiments. Graham (41) based his con- 
 clusions on the apparently successful results secured in his school 
 system when the method of relating problems to the out-of-school 
 life of the pupils advocated by him was tried out. He reports little 
 quantitative data. Kulp (64) reports no quantitative data at all but 
 describes the success in his school when a similar method was toed. 
 Wilson (122) did not secure equivalence for the groups used in her 
 experiment, and, hence, the relative merits of the methods of dramati- 
 zation and story telling in connection with teaching verbal problems 
 cannot be determined. She presents somewhat more quantitative 
 data than Graham (41) in favor of the effectiveness of her methods. 
 Steinway (110) used two groups of children in, her investigation, 
 reporting the effectiveness of securing motivation by number games, 
 but here again, the lack of equivalence and failure to use suitable 
 measuring instruments makes it impossible to list this as other than a 
 crude experiment. 
 
 The studies of Reavis (101), Sheerin (106), Richardson (104), and 
 Newcomb (89) were single-group experiments. Reavis (101) used a 
 single group of twenty-one eighth-grade pupils, organized the class as 
 a bank in which such learning activities as exercises with stocks, bonds, 
 deposits, and checks were engaged in, and measured the gain in achieve- 
 ment by means of an informal problem test administered at the close 
 of the experimental instruction and again some months later. Sheerin 
 (106) used a single group of unreported size for a period of four 
 months One aspect of the experimental factor was that of informing 
 pupils of progress. No mention is made of any attempt to measure 
 quantitatively the improvement ascribed to the method by the in- 
 vestigator. Richardson (104) used single groups of indefinitely 
 reported size. In the first "campaign" ten intermediate-grade classes 
 participated for a period of nine weeks. In the second campaign 
 pupils in the fourth, fifth, sixth, seventh, and eighth grades of some 
 fifteen" schools participated. It is stated that the numbers in each 
 grade ranged from 250 in the fourth grade to 150 in the eighth. This 
 campaign lasted for six weeks. In the third campaign the fourth 
 fifth and sixth grades took part, while the seventh and eighth served 
 
84 Bulletin No. 58 
 
 in some measure as control groups. Newcomb (89) used a single group 
 of seventh- and eighth-grade pupils of unreported size. The Courtis 
 and Stone tests were administered before and after the experimental 
 period. Substantial gains in achievement were reported, but because 
 of the lack of control it is impossible to ascribe these gains with cer- 
 tainty to the experimental factor. 
 
 It should be apparent that all of these single-group experiments 
 are open to serious criticism. One cannot determine to what extent 
 the improvement was the result of the application of the experi- 
 mental factor, since many other factors were operating. In Richard- 
 son's investigation (104) teacher zeal probably was an influential 
 factor. Other criticisms may be mentioned. In most of the experi- 
 ments the improvement was inadequately measured, if it was meas- 
 ured at all. For the most part these experiments may be character- 
 ized as "descriptive accounts of what is going on" in a certain school. 22 
 
 Hahn and Thorndike (46), Kirby (57), Chapman and Feder (22) 
 Panlasigui and Knight (98), Hurlock (49), Mailer (70), and Bowman 
 (10) conducted controlled experiments. Those of Hahn and Thorn- 
 dike (46) and of Kirby (57) were quite favorably evaluated in the 
 section on the effect of distributing practice in drill on the funda- 
 mentals. 23 
 
 Chapman and Feder (22) used two groups of sixteen fifth-grade 
 pupils. These groups were exercised ten minutes a day on an addi- 
 tion test, one minute a day on a cancellation test, and five minutes a 
 day on a substitution test. One of the groups was subjected to such 
 motivating influences as the following: 
 
 (1) Each individual's results of the previous dav were published 
 
 (2) On sheets presented for the day's work, the point reached on the last 
 
 occasion by the subject was marked in heavy blue pencil. 
 
 (3) The general improvement of the class was presented in the form of a 
 
 graph. 
 
 (4) Credits were given in the form of stars, .... It was understood that 
 
 prizes of a merely nominal value were to be given at the end of the 
 ten practice periods to the 50 per cent in Group A which had gained 
 the greatest number of stars for efficiency and improvement. 
 Data secured for ten practice periods are presented in tabular and 
 graphic form. The achievement of the motivated group in addition 
 was certainly significantly superior to the achievement of the non- 
 motivated group. The chief criticisms to be made of this experiment 
 have to do with the complex experimental factor described above and 
 the artificiality of conditions. 
 
 "/*n dc l ^°^ ld deny the ,abel "educational research" to such writir.es See- 
 "S^pages ^TfJ 48. ^^ *****' "^ Se P tember ' &™ W«Ut£falO 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 85 
 
 Panlasigui and Knight (98) used a total of 358 experimental 
 pupils and an equal number of control pupils in the fourth grade of 
 ten school systems in nine states. The pupils were paired with re- 
 spect to arithmetic ability shown by the initial test. The degree to 
 which equivalence was attained is indicated by the fact that the 
 means, first and third quartiles, and standard deviations of the two 
 distributions of initial-test scores were identical. The drill materials 
 used by the experimental pupils differed from the materials used by 
 the control pupils in that each pupil could determine his individual 
 progress. Class progress charts were also provided for the pupils in 
 the experimental group. With respect to the control of non-experi- 
 mental factors the authors state that "serious attempts were made to 
 minimize all unusual factors and to approximate normal conditions. 
 It is unfortunate that the authors do not describe what these at- 
 tempts were. The experiment continued for twenty weeks; at the 
 end of this time the final test was administered. The difference in 
 final-test means was 3.93 times its probable error and is an indication 
 that the chances of the true difference having the same sign, or of 
 being in the same direction, are approximately 286 to l. 24 The data 
 are interpreted also for the "top and bottom quarters" of all groups 
 on the initial test, and other comparisons are made. It is evident that 
 Panlasigui and Knight have reported an excellent experiment. What 
 criticism may be made concerns such things as failure to report the 
 reliability of the tests used and failure to state how non-experimental 
 factors were controlled. It is possible that the conclusions are not 
 sufficiently restricted with respect to limitations of the data, but 
 until better evidence is reported to the contrary, it would seem that 
 they may be accepted as dependable evidence of the effectiveness of 
 stimulating arithmetical learning activity by insuring that pupils are 
 aware of their progress. 
 
 Hurlock (49) used four groups of fourth- and sixth-grade pupils, 
 two of twenty-six pupils each and two of twenty-seven pupils each. 
 It is stated with respect to equivalence that "these groups were equal 
 not only in initial ability as displayed on these tests in addition, but 
 also in average age and number of boys and girls within each group." 
 The first group was praised over a period of five days in the presence 
 of other members of their classes. The second group was reproved 
 under the same conditions, while the third group was ignored. It 
 
 24286 to 1 are the chances when a difference is four times its probable error. Chances of at least 
 
 369 t0 M^n?o? 8 W^ in Education,- University of 
 
 lUin^^H^vll^o^^not Educational Research Bulletin No. 48. Urbana: University 
 
 ° f "^cCanfw. A." Vwto Measure in Education. New York: Macmillan Company. 1922. p. 404-05. 
 
 
86 
 
 Bulletin No. 58 
 
 should be mentioned that the pupils of the third group heard the 
 praise and reproof of the others. The fourth group was used as 
 control and was tested in a separate room. Modifications of the ad- 
 dition test of the Courtis Research Tests in Arithmetic were admin- 
 istered each day for five days. The differences in test means are 
 greatest when the praised group is compared to the control, the 
 chances of significance in favor of praise being "10,000 in 10,000." 
 When the reproved group is compared with the control, the chances 
 are "9,382 in 10,000" in favor of reproof, and when the ignored group 
 is compared with the control, "5,338 in 10,000" in favor of hearing 
 praise and reproof of others. The conclusions seem reasonably de- 
 pendable from the standpoint of the conditions of the experiment. 
 One wonders how significant they are for ordinary classroom practice. 
 It is possible that praise and reproof are effective incentives to learn- 
 ing arithmetic in the typical class, but how effective they are must 
 await experiments with less abnormal conditions. 
 
 Mailer (70) used 814 experimental and 724 control pupils in 
 Grades V to VIII. The experimental pupils, alternately stimulated 
 by individual recognition and reward and by group or class recog- 
 nition and reward, solved addition examples. The investigator 
 states in this connection: 
 
 The tests of work for self and work for class were repeated twelve times, two 
 minutes each. The motives of self and class were alternated six times, respec- 
 tively. The problem of practice effect was thus practically eliminated. All 
 conditions of work aside from the motives were identical. 25 
 
 The difference in favor of individual competition when compared 
 with group competition was almost thirteen times its probable error. 
 For the conditions of the experiment there is little reason to doubt 
 the significance of this difference. The experimental conditions may 
 be characterized as abnormal. It is doubtful whether competition 
 would appeal to school children as a continuous diet in ordinary 
 teaching. It is likely that its effectiveness would lessen with con- 
 tinued use. 
 
 The experiment of Bowman (10) has been described and favor- 
 ably evaluated in the section on methods of teaching and learning 
 verbal problems. 26 
 
 The data of the investigations of Ballou (7), Courtis (33), Krause 
 (63), O'Brien (92), and Wertheimer (120) were secured by the admin- 
 istration of such standardized tests as those by Courtis and by 
 Monroe. Increased achievement in arithmetic seem to result 
 through repeated administration of such tests. The authors of the 
 
 
 
 25 MalIer, op. cit., p. 15. 
 ™See pages 52 to 58. 
 
Summary of Research Relating to the Teaching of Arithmetic 87 
 
 reports of these investigations ascribe some of the improvement to 
 the stimulating effect of the tests. There is no means of showing, 
 even by controlled experiments, which these investigations certainly 
 were not, how much of the achievement may be ascribed to this 
 factor. It is doubtful whether a controlled experiment could be set 
 up which would satisfy the law of the single-variable, since it would 
 be impossible to separate the motivating factor from the complex 
 group of factors which systematic testing involves. 
 
 Anthony and others (3) secured their data from intensive case 
 studies of three children. The studies which were conducted over 
 a period of five months revealed the progress of the children by 
 means of learning curves. Their conclusions in favor of the use of 
 learning curves cannot be regarded as other than suggestive because 
 of the small number of cases. 
 
 Justified conclusions. The only conclusion which may be offered 
 as dependable is that knowledge of progress in arithmetical learning 
 is an effective motivating influence. It does not seem to matter a 
 great deal what methods the teacher uses to insure that pupils are 
 aware of their success or failure. Individual learning curves, progress 
 charts, test scores, and the like seem to be effective devices. The con- 
 clusions relative to commendation and reproof are less certain, but 
 research in other subjects with respect to motivation seems to indicate 
 that commendation is most effective, reproof somewhat effective, and 
 both are more effective than no comment at all. 27 
 
 Evidence that certain devices and methods— namely, the project 
 method, the Dalton plan, the use of games involving a knowledge of 
 numbers, the telling of stories in connection with problems, the dram- 
 atization of the stories, 28 and the use of tests— are stimulating to 
 learning activity in arithmetic is to be found in the single-group 
 experiments. It should be noted that the evidence with respect to 
 these methods and devices may not be regarded as highly dependable 
 Formulation and presentation of appropriate learning exercises are 
 possibly the most effective means of securing motivation of learning 
 activity in arithmetic. It should not be inferred that pupil preference 
 is the most important criterion in the devising of learning exercises. 29 
 It should be used as a criterion only after the test of compatibility 
 with recognized objectives has been applied. The data of Bowman 
 (10) reveal that belief in success causes preference. Capable instruc- 
 
 "Monroe W S and Eneelhart. M. D. "Stimulating Learning Activity," University of Illinois 
 Bullet ™\%l 28, No. 1 Bureau of Educational Research Bulletin No. 51. Urbana: University of 
 Illinois^930, Pj«;54. uid ^^^ th . s statement with the conclusions of Wheat (121). See pages 
 
 52 l ° 2 5 9The findings of Bowman (10) should be referred to in this connection. See pages 52 to 58. 
 
88 
 
 Bulletin No. 58 
 
 tion by problems which are desirable from the standpoint of objec- 
 tives should be effective in engendering preferences for such problems. 
 
 As a conclusion to this summary of the experiments on motivation 
 in arithmetic the following statement taken from the monograph on 
 motivation previously referred to seems pertinent: 
 
 If the teacher has a real interest in children and in teaching, if she approaches 
 her pupils with the attitude that doing the exercises assigned is an interesting and 
 challenging activity, the problem of motivation will tend to disappear. Motiva- 
 tion procedures and devices will be needed only to supplement the stimulating 
 effect of other instructional procedures. 30 
 
 30 Monroe, and Engelhart, op. cit., p. 58. 
 
CHAPTER VIII 
 GENERAL SUMMARY AND CONCLUSION 
 
 This chapter begins with a list of the problems of the investiga- 
 tions summarized in the preceding chapters. The statement of each 
 problem is followed by a note with respect to the reported conclu- 
 sions. Where the statement is made that a reported conclusion is 
 undependable, it may be inferred that the conclusion is unworthy of 
 generalization. In some instances the note following the problem 
 statement contains a remark relative to a possible, more correct 
 solution of the problem. In the paragraphs following this list of 
 problems an estimate is presented of the contribution of experimental 
 research up to the present. This estimate is followed by suggestions 
 for further research in this field and with a statement of the require- 
 ments for precise evaluation of instructional techniques in arithmetic. 
 The chapter closes with a discussion of feasibility versus relative 
 effectiveness of instructional techniques. 
 
 The problems studied. The following questions represent the 
 problems of the arithmetic investigations summarized in the preced- 
 ing chapters. While the questions are, for the most part, quite spe- 
 cific in character, it was felt that some synthesis was desirable. Where 
 several investigations were made of practically the same problem, one 
 problem statement was formulated to represent all of them. Where 
 investigations were made of different aspects of the same problem, a 
 compound statement was formulated to represent the aspects 
 investigated. 
 
 1. What is the relative efficiency of upward versus downward addition? 
 
 The reported conclusion that the method of teaching pupils to add in the down- 
 ward direction is superior in effectiveness to the method of teaching pupils to add in 
 the upward direction seems undependable. It appears probable that there is no sig- 
 nificant difference in effectiveness between the two methods. 
 
 2. What is the relative effectiveness of the following methods of teaching 
 addition and subtraction: (1) Showing pupils how to perform the process with no 
 consideration of generalization or of underlying principles; (2) helping pupils 
 to formulate general methods of procedure from specific types taught and em- 
 phasizing these generalizations throughout the teaching; (3) teaching the 
 reasons and principles underlying the specific types taught; (4) teaching both 
 general methods and general principles? 
 
 The reported conclusion favoring (2) appears undependable. It seems reasonable 
 that (4) should be superior in effectiveness to either (2) or (3). 
 
 3. What is the relative effectiveness of three minutes' instruction daily in 
 generalizing groups of addition and subtraction combinations included within 
 
 89 
 
90 Bulletin No. 58 
 
 twenty-minute practice periods in addition and subtraction and twenty-minul 
 practice periods without the generalizing instruction? 
 
 Three minutes' daily instruction in generalizing groups of addition and subtrac- 
 tion combinations, within twenty-minute practice periods in addition and subtraction 
 is reported not to add significantly to the achievement engendered by the twenty- 
 minute practice periods alone. The conclusion as stated is reasonably dependable 
 but it should not be inferred that generalizing instruction is inherently ineffective! 
 
 4. What is the effectiveness in performing the fundamental operations of 
 "thinking results only?" 
 
 The reported conclusion that the method is effective is based on limited experi- 
 mental evidence, but it seems reasonable that this method is effective since it tends 
 toward the establishment of more direct mental processes. 
 
 5. What is the effectiveness of teaching pupils to break long columns into 
 two parts and to add each part separately? 
 
 The reported conclusion that this method is effective is based on very limited 
 experimental evidence. It seems reasonable that the method would engender 
 undesirable addition habits. 
 
 6. What are the relative merits of adding digits in regular serial order and 
 making mental combinations or rearrangements? 
 
 Th ^u rep ° rted F onclusion favoring serial order is based on faulty experimental 
 data. The conclusion, however, appears reasonable since excessive combination and 
 rearrangement is likely to prove confusing to immature pupils. 
 
 7. What is the effectiveness of teaching pupils to check their answers in 
 addition? 
 
 The reported conclusion favoring the effectiveness of this method is not supported 
 by adequate experimental evidence. It appears reasonable, however, that checking 
 is an effective means of securing accuracy in addition, and that the attainment of 
 accuracy is worth the possible sacrifice in speed necessitated by checking. 
 
 8. What are the relative merits of the following methods of subtraction: 
 (1) Subtractive or take-away in which borrowing or decomposition is used; 
 (2)^ subtractive or take-away in which carrying or equal addition is used; (3) ad- 
 ditive in which borrowing or decomposition is used; (4) additive in which 
 carrying or equal addition is used. 
 
 The second of these four methods of teaching or learning subtraction is reported 
 to be superior in effectiveness to the others. It seems reasonable to assume that all 
 four of the methods are feasible and that there is no significant difference in their 
 effectiveness. 
 
 9. What are the relative merits of the multiplicative method of division and 
 the traditional method? 
 
 The multiplicative method is reported to be superior on the basis of inadequate 
 experimental evidence. It seems reasonable to assume that the multiplicative method 
 is not significantly more effective than the traditional method. 
 
 10. What is the effectiveness of using addition of fractions as a basis for 
 teaching the multiplication of fractions? 
 
 The method is reported to be effective on the basis of faulty experimental data. 
 It seems reasonable to assume, however, that the method is an effective one since it 
 conforms to the principle of apperception. 
 
 11. What is the effectiveness of providing drill in the fundamental combi- 
 nations as a means of increasing achievement in common and decimal fractions? 
 
 The evidence supporting the reported conclusion that the above method is effec- 
 tive is independable. It appears reasonable, however, that the method is effective. 
 It is self-evident that pupils are unlikely to have sufficient mastery of the four funda- 
 
Summary of Research Relating to the Teaching of Arithmetic 91 
 
 mentals that further drill will not increase their achievement where these skills 
 are used. 
 
 12. What is the relative effectiveness of practice material so prepared that 
 the type of percentage problem set for solution is apparent to the pupils and of 
 the ordinary textbook material? 
 
 The reported conclusion favoring the prepared material is not supported by very 
 dependable experimental evidence, but the conclusion appears reasonable. It is 
 compatible with other findings respecting prepared practice material. 
 
 13. What is the effectiveness of teaching children to place the decimal point 
 in a quotient by means of a general rule? 
 
 The conclusion which reports that the method is ineffective is based on scanty 
 experimental evidence, but the relatively specific nature of the division abilities jus- 
 tifies the assumption that the conclusion is reasonably correct. 
 
 14. What, in learning division, is the relative effectiveness of the rules: "There 
 are as many places in the quotient as those in the dividend exceed the divisor" 
 and "First render the divisor an integer by multiplying both dividend and divi- 
 sor by 10 or some power of 10. Then proceed as with integral divisors." ^ 
 
 In learning division it is reported that use should be made of the rule, "First 
 render the divisor an integer by multiplying both dividend and divisor by 10 or some 
 power of 10, and then proceed as with integral divisors" rather than of the rule, 
 "There are as many places in the quotient as those in the dividend exceed the divisor. 
 
 15. What is the effectiveness of the "method of unity" in teaching pro- 
 portion? 1 . . 
 
 The experimental evidence supporting the conclusion that the method is effective 
 is not dependable. It seems reasonable to postulate, however, that the method is 
 an effective one. 
 
 16. What is the relative effectiveness of memorizing tables of cubic and linear 
 measure as compared with the effectiveness of using the facts of these tables in 
 connection with problems? 
 
 It is reported that it is more effective for pupils to memorize tables of cubic and 
 linear measure than to learn them through using the facts of these tables in connection 
 with problems. It is a fairly well accepted principle of learning, however that in- 
 formation learned through use is usually better retained than information learned in 
 isolation from use. 
 
 17. What is the effect on achievement in arithmetical calculation of system- 
 atic drill in addition, subtraction, multiplication, and division? 
 
 The conclusion that systematic drill is effective is supported by comprehensive 
 and reasonably dependable experimental evidence. It may be accepted as an 
 established general principle. 
 
 18. What is the relative effectiveness of systematic versus incidental teaching 
 
 of calculation? 
 
 The systematic method of teaching calculation is reported to be more effective 
 than the incidental method. The incidental method of teaching calculation is also 
 reported to be more effective than the systematic. The findings of research in other 
 fields, and logical thinking would favor a combination of both methods, with possibly 
 greater emphasis on the systematic. 
 
 19. What is the effectiveness of a combination of systematic and incidental 
 methods of teaching calculation? 
 
 The conclusion that a combination of systematic and of incidental method of 
 teaching calculation is effective is not based on highly dependable experimental 
 
 ^See page 26 for an illustration of this method. 
 
92 Bulletin No. 58 
 
 evidence. It appears, however, to be a reasonably correct solution of the problem 
 since the incidental method should contribute motivation and the systematic method 
 should insure the distribution of practice compatible with the recognized obiectives 
 of arithmetic. 
 
 # 20. What is the relative effectiveness of various types of drill materials 
 which have been prepared by experts? How do these drill materials compare in 
 effectiveness with those prepared informally by teachers? 
 
 The conclusions reported with respect to the relative effectiveness of the different 
 prepared materials are undependable. The conclusions with respect to the superiority 
 of the materials prepared by experts as compared with those prepared by teachers 
 appear to be reasonably dependable. 
 
 21. What is the effectiveness of drill exercises in addition prepared in such a 
 way that proportionate drill is given on the higher decades as compared with 
 drill materials ordinarily used? 
 
 The conclusion favoring the prepared material is not supported by adequate 
 experimental evidence. The conclusion seems, however, to be reasonably correct. 
 
 22. What is the effectiveness of teaching the one hundred multiplication 
 combinations by means of text material alone with the teacher doing as little 
 talking as possible? 
 
 The conclusion favoring the above method is not supported by sufficient experi- 
 mental evidence. Further research is needed before it may be concluded that the 
 teacher has no function in drill. 
 
 23. What is the relative effectiveness of drill material so constructed that 
 practice is distributed over the number combinations and of drill material in 
 which certain combinations are slighted? 
 
 The reported conclusion in favor of the material which provides distributed 
 practice is supported by rather highly dependable experimental evidence. 
 
 24. What is the effectiveness of drill material so prepared that the amounts 
 of practice provided on the number combinations are proportional to their 
 difficulty? 
 
 The conclusion that drill material should be prepared in this way seems to be 
 supported by reasonably acceptable experimental evidence. 
 
 25. Is it better to have pupils find mistakes among a group of examples of 
 addition, multiplication, and subtraction combinations than to have them think 
 only the correct associations? 
 
 The conclusion that it is better to have pupils think only the correct associations 
 is not supported, in this instance, by dependable experimental evidence. It is a well 
 accepted principle of learning, however, that it is more desirable for pupils to come in 
 contact with that which will engender correct associations, than to come in contact 
 with that which is likely to engender incorrect associations. 
 
 _ 26. What is the effect on computational achievement of drill materials 
 which provide practice in arithmetical reasoning? 
 
 The reported conclusion that such materials increase computational achievement, 
 while not supported by adequate experimental evidence, appears, however, to be 
 reasonably correct since it conforms to the Law of Exercise. 
 
 27. Should addition and subtraction be taught together or separately? 
 It is reported that addition and subtraction should be taught together. It is 
 
 reasonable to assume that there should be separate teaching of addition and subtrac- 
 tion during the initial stages of learning, and mixed teaching for maintenance, or 
 increase, of skill. 
 
 28. What is the relative effectiveness of drill material of mixed nature and 
 drill material in which practice on addition, subtraction, multiplication, and 
 division is provided for separately? 
 
Summary of Research Relating to the Teaching of Arithmetic 93 
 
 The conclusion that drill material of mixed nature is relatively more effective 
 than drill material in which separate practice is provided for addition, subtraction, 
 murtioHcat?on and division is based on reasonably acceptable experimental evidence 
 multipl cation ana ai\ ib u reporte d to be effective for maintenance of skill 
 
 anfincrea': ^^^SSffi-lilW before a certain level of attainment 
 has been reached with each of the four fundamentals. 
 
 29 What is the optimum distribution of practice time in the fundamentals? 
 
 The reported conclusions with respect to this problem are not in close agreement, 
 nor are thev based on adequate experimental evidence. It would seem reasonable 
 however to surest that twenty-minute practice periods at daily intervals until 
 
 retention. 
 
 30. What are the relative effects of requests for speed and of requests for 
 accuracy on achievement in the fundamentals? 
 
 The conclusion is reported that it is preferable to request accuracy of pupils 
 rather than speed in the earlier stages of learning. After mastery has been attained 
 Seed mav be requested. While this conclusion is not supported by acceptably 
 dependS experirnental evidence, it seems compatible with the principle that repe- 
 tition of incorrect response should be avoided. 
 
 31 What are the characteristics of pupil responses to verbal problems in 
 arithmetic? To what extent is the response the result of reflective or critical 
 
 ^^The^conclusion that pupil responses to verbal P™ ble ™ ^ 
 by lack of critical reflective thinking appears to be reasonably dependable. 
 
 32 What are the influences on problem-solving performance of the following 
 characteristics of problem statements: familiar terminology, unfamiliar termin- 
 ology, imaginative elements, irrelevant elements, size of numbers, amount of 
 
 ^Th^X^Supil responses to verbal problems are more satisfactory 
 when tt> arettatTd in fan^ilfar terminology and without i™j«™£*™^ 
 reasonably dependable The conclusion that responses are less likely to be satistac 
 S wfe^ d r e o P blemsare stated imaginatively is less dependably but appears Reason- 
 able The conclusions with respect to other aspects of problem statements are even 
 Lss dependable Further research is needed for determining what is most effective 
 with respect to these aspects. 
 
 33 What is the effectiveness of providing pupils with systematic training in 
 
 finding the facts pertaining to the problem, in deciding the processes to be used, 
 
 and in finding the answer in round numbers? Aof . iA : na th e 
 
 Svstematic training in finding the facts pertaining to the prob em, in deciding the 
 
 dence supporting this conclusion appears reasonably dependable. 
 
 34. What is the relative effectiveness of teaching pupils to solve problems 
 hv the graphic and by the conventional methods? 
 
 > It "reported that" it is more effective to teach pupils to solve Y^ probUms .n 
 
 35. What is the effectiveness of assigning large numbers of problems in 
 teaching children to solve problems? 
 
 It * reported to be effective in increasing problem-solving achievement to ass.gn 
 
94 Bulletin No. 58 
 
 large numbers of problems. This conclusion, while not based on acceptable exoeri- 
 mental data, agrees with the Law of Exercise. 
 
 36. What is the effectiveness of teaching pupils to see the analogies between 
 difficult written problems and correspondingly easy oral problems? 
 
 The conclusion is reported that the method is not effective. Since this conclusion 
 is not supported by very dependable experimental evidence, and since the method 
 would appear to be compatible with the Law of Association, it would seem reasonable 
 to suppose that the method is effective. 
 
 37. What is the value of diagnostic and remedial treatment in arithmetic? 
 Diagnostic and remedial treatment is highly effective in the field of arithmetic' 
 
 Ine experimental evidence in support of this conclusion is comprehensive and 
 reasonably dependable. 
 
 38. What is the relative effectiveness of individual diagnosis in which 
 "first-hand observation is made of the actual work of the pupil" and diagnosis 
 by means of diagnostic tests? 
 
 Conclusions have been reported in favor of both methods of diagnosis. Further 
 research is needed to determine which method is relatively more effective It seems 
 reasonable that both methods are very feasible. 
 
 39. What is the relative effectiveness of remedial treatment in which pupils 
 are given organized drill material affording practice of abilities diagnosed as 
 weak and of informal material prepared by the teacher? 
 
 The conclusion favoring the expertly prepared remedial drill material is not sup- 
 ported by adequate experimental evidence, but it does conform with other con- 
 clusions respecting expertly prepared drill material. 
 
 40. To what extent is reading ability a factor in arithmetical achievement? 
 ii Tha J jading is an important factor in arithmetic achievement seems reasonably 
 
 well established. Further research is needed to show the precise magnitude of the 
 influence of this factor. 
 
 41. What is the effectiveness of general training in reading in engendering 
 greater achievement in arithmetic? 
 
 General training in reading is reported effective in engendering greater achieve- 
 ment in arithmetic. The experiment in which general training in reading constituted 
 the experimental factor was very crude, but the conclusion is supported by the re- 
 search which reveals that reading ability is a factor in arithmetical achievement. 
 
 42. What is the effectiveness of solution sheets containing information with 
 respect to the manner of reading problems and containing spaces for recording 
 of data useful at different stages in the solution of the problem? 
 
 Solution sheets containing information with respect to the manner of reading 
 problems and containing spaces for the recording of data useful at different stages in 
 the solution of problems are reported to be an effective device in teaching pupils to 
 solve problems. While the experimental evidence is not of acceptable dependability, 
 the method would seem to be feasible since more direction is given to the learning 
 activity. It is possibly more desirable for the earlier rather than the later stages of 
 earning to solve verbal problems. 
 
 43. What is the effectiveness of story-telling and dramatization in teaching 
 pupils to read verbal problems in arithmetic? 
 
 Story-telling and dramatization are reported, on the basis of very limited experi- 
 mental evidence, to be effective devices in teaching pupils to read verbal problems in 
 arithmetic. This conclusion appears to be in agreement with the principle that inten- 
 sive effort is secured in learning activity through creating a need. It is likely, however 
 that neither of these devices should be given prolonged use. 
 
 44. What types of learning exercises are most stimulating to learning 
 activity in arithmetic? 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 95 
 
 While the experimental evidence is not of acceptable dependability, it seems 
 
 ^rfnn difficult In order that well motivated learning act.vrty may be secured, 
 ITu^^trp supe^r pupils may be stated ^^^-^00, 
 
 of the purely computational type. 
 
 45. In stimulating learning activity in arithmetic, what is the effectiveness 
 of informing pupils of definite goals to be achieved? 
 
 46 In stimulating learning activity in arithmetic, what is the effectiveness 
 of informing pupils with respect to their status or progress. 
 
 The conclusions in favor of this method of stimulating learnmg activity are sup- 
 ported oy dependable experimental evidence both from anthmettc and from other 
 
 subject-matter. 
 
 47. What is the value of competition as a means of stimulating learning 
 
 * C *1^o£Znt basis of fairly dependable experimental evidence that com- 
 
 mmmmmmm 
 
 SeSevfce to^HnVa cl/sfoTp^pibout of a slump in learning by relievmg the 
 monotony of ordinary learning exercises. 
 
 48. What are the relative merits of commendation and reproof in stimulating 
 fcm S l ^K^S are both reported to be stimulating to learning 
 
 mmmmmtm 
 
 conclusion also conforms to the Law of Effect. 
 
 The contributions of research to the teaching of arithmetic. 
 What constitutes a contribution depends upon the interpretation 
 given to that term. It may be considered a contribution to show that 
 an instructional procedure as applied to a particular group of pupils 
 produces as satisfactory results, or nearly as satisfactory results, as 
 another procedure may produce. Usually, however, a contribution is 
 interpreted to mean the demonstration of the relative merits of two 
 or more comparable procedures not merely for a particular poupot 
 pupils, but for all groups of pupils of a certain intellectual and edu- 
 cational status. If this more restricted interpretation is applied to the 
 conclusions indicated in the preceding list, it is apparent that the 
 dependable contributions of research in the teaching of arithmetic 
 
 are relatively meager. 
 
 Probably the most significant contributions relate to the specific- 
 
96 Bulletin No. 58 
 
 ity of calculation abilities and to the use of practice materials con- 
 structed so that adequate exercise is provided for each specific ability 
 involved. Although research has not yet produced a complete and 
 dependable list of the specific abilities in the field of arithmetical cal- 
 culation, there are tentative lists for certain segments of this field 
 which appear to be rather highly dependable with reference to many 
 of the items. The superiority of practice materials which provide for 
 the exercise of each specific ability in proportion to the difficulty of 
 attaining it has been demonstrated. It is, of course, not unlikely 
 that, as these tentative lists of specific abilities are refined, superior 
 practice materials may be devised, but this possibility does not de- 
 tract from the fact that research has already contributed to the 
 improvement of practice materials. 
 
 Closely related to this contribution is the demonstration of the 
 effectiveness of diagnosis and of remedial instruction, and of 
 systematic practice. 
 
 Research has contributed to an understanding of the nature of 
 pupil responses to verbal problems and of the effect of introducing 
 certain changes in the problem statement. Pupil responses to verbal 
 problems are more satisfactory when they are stated in familiar ter- 
 minology, and it appears that very little reasoning enters into the 
 response of most pupils. Reading ability appears to be an important 
 factor in the ability to respond to verbal problems, but the precise 
 nature of its function has not been ascertained. Systematic training 
 in finding the data given in a problem, in deciding upon calculations 
 to be made, and in estimating the answer in round numbers is an 
 effective procedure for teaching pupils to solve verbal problems. 
 Informing pupils of the status of their achievements in arithmetic 
 is an effective means of securing intensity and persistence of effort in 
 attaining higher levels of achievement. This procedure encourages 
 each pupil to compete with his own past record. Competition be- 
 tween individual pupils and between groups is also effective. 
 
 There is considerable evidence that there is little or possibly no 
 difference in the relative merits of several alternative calculation 
 techniques. For example, the data secured in the studies of down- 
 ward versus upward addition have been interpreted as favoring the 
 latter technique, but the fact that the differences in achievement are 
 so small that their significance is doubtful suggests the generalization 
 just stated. This conclusion is also supported by a priori reasoning. 
 If there is any significant difference in the relative merits of such 
 alternative techniques it is likely that they would not be very appar- 
 ent except on the higher levels of achievement, and since the function 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 97 
 
 of the school is not to produce highly expert calculators, it seems that 
 the generalization stated at the beginning of this paragraph is the 
 most significant contribution of the research attempting to evaluate 
 alternative calculation techniques. Of course this generalization does 
 not apply to cases in which one of the techniques is obviously time 
 consuming or otherwise inefficient. For example, it should not be 
 applied in support of "counting on the fingers." 
 
 Suggestions for research in the field of instructional methods in 
 arithmetic. The evaluation and summary of research relating to the 
 teaching of arithmetic afford a basis for some suggestions for future 
 studies in this field. Although it is difficult to cite much definite 
 evidence, the present writers have been impressed with the need for 
 additional studies of verbal problems and of the nature of pupil 
 responses to them. In the field of arithmetical calculation investi- 
 gators have gone far in identifying the types of examples and the 
 abilities involved in responding to them. It seems reasonable to 
 assume that these are types of verbal problems. Research is needed 
 to identify these types, if they exist. There is also need for more 
 information about the function of reading in pupil responses to verbal 
 problems and the relation of the form and vocabulary of problem 
 statements to these responses. 
 
 Another suggested field of research relates to the instructional 
 procedures employed in teaching pupils to solve problems. Should a 
 method of analysis be employed? Should a complex problem be 
 broken up into a series of simpler problems? Should a pupil be di- 
 rected to compare the problem with ones he has solved and with 
 solutions given in the text? What sort of attention should be given 
 to the vocabulary? What types of learning exercises should be used 
 in connection with verbal problems? Should pupils be taught a 
 variety of problem types simultaneously or should each type be 
 taught separately? To what extent and for what pupils is problem- 
 solving activity stimulated by an occasional problem of the puzzle 
 type? To what extent is the level of intelligence of the pupils a 
 factor in generalization from number combinations specifically 
 taught to those not taught? To what extent are flash cards used for 
 drill purposes likely to engender improper eye-movement habits with 
 respect to arithmetical subject-matter? 
 
 The possibility of evaluating comparable instructional proced- 
 ures. The relatively meager contribution of the research summarized 
 in this bulletin probably has suggested to the thoughtful reader the 
 possibility that comparable instructional procedures cannot be eval- 
 uated with a high degree of precision. The evaluation of a procedure 
 
98 Bulletin No. 58 
 
 by experimentation is dependent upon the control of all factors 
 affecting the learning of pupils except the one being studied. The 
 zeal and skill of the teacher in applying a given procedure affect the 
 achievements of the pupils and these factors are difficult or impos- 
 sible to control in many cases. Consequently it does not appear that 
 precise and highly dependable evaluations of comparable instruction- 
 al procedures should be expected. Attempts to determine the relative 
 merit of certain 'methods of teaching" will show that the procedures 
 are approximately equal in merit, except when one of the procedures 
 is distinctly inferior. In most such cases it is likely that a competent 
 person could accurately predict this inferiority. 
 
 In support of this judgment, the requirements for precise and 
 dependable evaluation of instructional procedures are briefly 
 described. 
 
 REQUIREMENTS FOR PRECISE EVALUATION OF INSTRUCTIONAL 
 METHODS IN ARITHMETIC 2 
 
 1. Equivalent groups. The groups of pupils used in the experi- 
 ment should be equivalent in all respects that will affect their arith- 
 metical achievement during the experiment. This requirement can 
 be approximated by pairing pupils on the basis of intelligence test 
 scores and then comparing the groups thus formed with respect to 
 chronological age, to previous achievement in the school subject, and 
 to measures of arithmetical reading ability. If the differences be- 
 tween the means and the standard deviations of the groups with 
 respect to these three characteristics are relatively small, the groups 
 may be considered approximately equivalent. It is desirable that the 
 groups also be approximately equivalent with respect to personality 
 traits, physical conditions, sex, and race. 
 
 Two other techniques of securing equivalent groups may be sug- 
 gested. The first is particularly adequate for investigations of the 
 relative effectiveness of differing types of learning exercises. It is 
 that of using such large groups that equivalence with respect to many 
 factors is secured as a result of the operation of chance. 3 It should be 
 noted that this procedure is only feasible where the learning activity 
 of the pupils is wholly directed by means of printed or mimeographed 
 instructions. When this procedure is used the different groups are 
 equally r epresented in all the classes participating in the experiment. 
 
 *These requirements have been taken with considerable adaptation from 
 Tin~ n - R n ;^'^>f-o a - 7 n( ir En ^ el l 1 , art - M -,£\ "Experimental Research in Education," University of 
 Illinois Bulletin Vol. 27, No. 32, Bureau of Educational Research Bulletin No. 48. Urbana: University 
 ot Illinois, 1930, p. 77-79. 
 
 3 For a description of this technique, see: 
 ir i ^°£ roe ;„ W ^ S - " H ?w Pupils Solve Problems in Arithmetic," University of Illinois Bulletin, 
 1929 9 °(7 )' eau ° Educational Research Bulletin No. 44. Urbana: University of Illir 
 
 MM.i.-, 
 
Summary of Research Relating to the Teaching of Arithmetic 99 
 
 The second procedure which may be suggested is that used by 
 Olander. 4 This investigator paired pupils chiefly on the basis of 
 growth in arithmetical ability over a period of five weeks during 
 which the pupils were subjected to the same, or similar, instruction. 
 The argument presented for this technique may be quoted here: 
 
 If two groups exhibit similar learning curves under similar instruction until 
 a certain point is reached, it can be assumed that the groups are equal in the 
 function in question. If a variation in the instruction of one group is then intro- 
 duced which causes the learning curve of that group to rise abnormally, whereas 
 the curve of the group under the unchanged technique continues to rise normally, 
 it may be assumed that a difference in scores at any later point on the curve is 
 attributable to the entrance of the variation in instruction. 
 
 2. Specification of experimental factor and control of non- 
 experimental factors. The experimental factor should, if possible, 
 be restricted to a single phase or detail of instructional procedure. 
 The method used with the experimental group should vary from that 
 used with the control group in only this single phase, and if other 
 variations are permitted, their effect must be accurately measured or 
 a plan of neutralization must be devised. 5 The total instructional 
 procedure to be used in both groups should be specified in writing, or 
 at least a detailed record should be kept of what is done. 
 
 Controlled experimentation involves maintaining equal status for 
 all factors in both the experimental and the control groups, except 
 the single phase or detail of procedure which constitutes the experi- 
 mental factor; or if the equal status is not maintained, the non- 
 equivalence must be recognized and its effect on the experimental 
 learning must be determined. The teacher factors whose control in 
 arithmetic experiments appears to be the most important are 
 (1) instructional techniques employed during the recitation period, 
 especially those relating to the assignment, and motivation; (2) skill 
 of the teacher in carrying out instructional techniques and classroom- 
 management procedures; (3) zeal of the teacher; (4) personality 
 traits of the teacher. In addition, care should be exercised to avoid 
 marked differences in the minor teacher factors— physical condition, 
 
 sex, and age. 
 
 The important factors under the head of general and extra-school 
 factors are (1) materials of instruction, (2) environment in which 
 learning activity takes place, and (3) minutes per day devoted to learn- 
 ing activity in arithmetic. The materials of instruction, desks, chairs, 
 light, heat, ventilation, and other aspects of the learning environment 
 should be identical for both groups. Study and recitation periods 
 
 ♦Olander, H. T. "Transfer of Learning in Simple Addition and Subtraction," Elementary School 
 Journal, 31:363, January, 1931. (94) c: n „i„ Variable 
 
 STbis requirement is sometimes designated as the Law of the Single Variable. 
 
100 Bulletin No. 58 
 
 should be of equal length in the experimental and control group. 
 Parents should be urged to refrain from influencing the arithmetical 
 learning activity of the pupils, and, possibly, should be asked to 
 cooperate in restricting the arithmetic learning activity to the 
 classroom. 
 
 It should be noted that the precise prescription of an instructional 
 procedure and the strict control of non-experimental factors is incom- 
 patible with good teaching. A teacher should adapt her techniques 
 to the needs of her pupils as they become apparent. Hence conform- 
 ity to the requirement for precise experimentation will, in many 
 cases, tend to reduce the effectiveness of the teaching, and this in 
 turn will introduce an element of uncertainty in the interpretation 
 of the results of the experiment. 
 
 3. The measurement of achievement. In the consideration of the 
 requirements under this head, the meaning of the validity of a test 
 should be given careful attention. The problem of an experiment, 
 when fully defined, either specifies or definitely implies the achieve- 
 ment to be measured. This achievement may be restricted to certain 
 calculation skills or it may include also certain items of knowledge 
 and certain general patterns of conduct. It may be restricted to the 
 degree of ability possessed at the close of the period of experimenta- 
 tion, or it may consist of the residue after a period during which there 
 is limited exercise of the ability. 
 
 A test that is highly valid for one purpose may be distinctly lack- 
 ing in validity when used for another purpose. Consequently the 
 validity of a test is a relative rather than an absolute characteristic, 
 and this quality of one used in an experimental investigation can be 
 determined only with reference to the specifications or implications 
 of the problem. This means that the experimenter must assume the 
 responsibility for determining the validity of the tests that he uses. 
 The reliability of a test refers to the variable errors in the resulting 
 scores, assuming perfect validity. If the validity is also considered, 
 any variable errors introduced because the achievement measured is 
 not identical with that specified by the problem must be added to the 
 effects of unreliability. Consequently the actual variable errors in 
 the measures of achievement may be considerably larger than is 
 indicated by the coefficient of reliability. 
 
 Finally the measures of achievement may involve constant or 
 systematic errors. 
 
 4. The interpretation of differences in mean gains in achievement. 
 In a typical experiment the treatment of the data results in a differ- 
 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 101 
 
 ence between the mean gains in achievement, or between the means of 
 the final- test scores, of the experimental group and of the control 
 group. If the groups are perfectly equivalent, if all non-experimental 
 factors have been completely controlled, and if the measures of 
 achievement are perfect— i.e., do not involve any errors, either var- 
 iable or systematic— the obtained difference may be accepted as the 
 actual difference in the mean gains of the two groups. These condi- 
 tions are seldom, if ever, completely realized. Furthermore, when 
 interpreting a difference in mean gains, the investigator usually de- 
 sires to generalize— i.e., to make a statement with reference to the 
 probability that the obtained difference has the same sign as the 
 difference which might be obtained from any repetition of the experi- 
 ment. The investigator may also wish to make a statement with 
 reference to the probability that the obtained difference, in addition 
 to having the same sign, is of the same order of magnitude as the 
 difference which might be obtained from any repetition of the exper- 
 iment. Hence, it is necessary to consider also the effect of sampling 
 upon the data secured. In the following paragraphs attention is first 
 directed to the statistical procedures to be employed in making allow- 
 ances for variable errors of measurement and of sampling. 
 
 The statistical procedures outlined in the following paragraphs 
 yield the standard 6 error of the difference in mean gains, or of the 
 difference between final-test means, due to the combined 7 effect of 
 variable errors of measurement and variable errors of sampling. If 
 the difference in mean gains, or final-test means, is equal to, or greater 
 than, 2.78 times the standard error of the difference, or 4.4 times the 
 probable error of the difference, it is customary to recognize the dif- 
 ference as "statistically" significant. The statement may be made in 
 interpretation, that the chances are 369 to 1, or better, that the sign 
 of the obtained difference is not due to the combined effect of the 
 variable errors of measurement and the variable errors of sampling. 
 The chances that the true difference does not differ from the ob- 
 tained difference by more than plus or minus the standard error of the 
 difference are 2.15 to 1, by more than plus or minus twice the standard 
 error of the difference, 21 to 1, and by more than plus or minus three 
 times the standard error of the difference, 369 to 1. This interpreta- 
 tion may be used when the investigator is interested in stating the 
 
 ^Th^robable error may be obtained by multiplying the standard error by the constant, .6745. 
 'For a discussion of the fact that -?- allows for the combined effect of variable errors of meas- 
 
 Vn 
 
 Urem Kel?e n y d T L.'^Note upon Holzinger's Formula for the Probable Error," Journal of Educational 
 
 ^SS? L 7 kn S d P Dou g b ra r ss!H 23 R. "On the Standard Errors of the Mean Due to Sampling and 
 to Measurement," Journal of Educational Psychology, 19:643-49, December, 1928. 
 
102 Bulletin No. 58 
 
 probabilities that the true difference is of the same order of magnitude 
 as well as of the same sign as the obtained difference. 8 
 
 The maximum allowance which needs to be made for the combined 
 effect of variable errors of measurement and variable errors of sam- 
 pling may be determined by means of the following formulae in which 
 <r e and o- c are the standard deviations of the distributions of individual 
 gains of the experimental and of the control pupils: 
 
 °"Mean Gain E — 
 
 °"Mean Gain C 
 
 Vn 
 
 Vn 
 
 ^Difference — Y ^Mean Gain E + ^Mean Gain C 
 Mean Gain E — Mean Gain C 
 
 If equivalent forms of an arithmetic test are not used at the begin- 
 ning and end of the experiment, or if scores are not converted into 
 comparable units, calculation of individual gains is impossible. 9 
 
 The subtraction of a pupil's initial-test score from his final-test 
 score is justified only when the scores are in terms of approximately 
 equal units — a condition approached when equivalent forms of a test 
 are used, or when scores are converted into comparable units. When 
 equivalent forms are not used or conversion has not been resorted to, 
 comparison is restricted to the difference between the final-test means. 
 In this case the standard deviations, a e and <r Cl refer to the distribu- 
 tions of final-test scores of the experimental and of the control pupils. 
 The first two formulae will then yield the standard errors of the final- 
 test means, and the third formula, when the squares of the standard 
 errors of the final-test means are inserted under the radical, will yield 
 the standard error of the difference between the final-test means. 
 It was stated in introducing the formulae given above that they 
 provide the maximum allowance which needs to be made for the com- 
 bined effect of variable errors of measurement and variable errors of 
 sampling. Two reasons may be given in support of this statement. 
 
 8 For a table of these probabilities, see: 
 Monroe and Engelhart, op. cit., p. 66. 
 _ 9 If the pupils start the experiment with zero arithmetic ability, the scores on the final test represent 
 gains. If the tests used at the beginning and end of the experiment are equally valid measures of the 
 experimental achievement, although not equivalent forms, conversion of the initial and final measures 
 into standard scores, T-scores, or grade scores makes possible the calculation of individual gains. 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 103 
 
 The first reason is that -}=?, in addition to measuring the effect of 
 
 variable errors of measurement, measures the effect of chance where 
 the operation of chance in the selection of the groups is not restricted. 
 In the following paragraphs it is indicated that the prodecure usually 
 employed in securing equivalent groups— i.e., pairing pupils with 
 respect to intelligence test scores, or making adjustments so that 
 means and standard deviations of the two groups are equal even 
 though pupils are not paired "pupil for pupil"— tends to reduce the 
 effect of chance. The formulae given above yield a precise allowance 
 for the effect of chance in the selection of the groups, only where the 
 groups are both random with respect to the population from which 
 they were drawn and with respect to each other. 
 
 The second reason for stating that the formulae given above pro- 
 vide a maximum allowance is that these formulae neglect the corre- 
 lation that may exist between the gains of the paired pupils, or be- 
 tween their final-test scores. In other words, the expression, 
 -2r gegc ^MeanGainE - "Mean Gain c, where r ge gc is the coefficient ob- 
 tained by correlating the distribution of individual gains of the ex- 
 perimental pupils with the distribution of individual gains of the 
 control pupils, should also be included under the radical of the 
 third formula given above. Coefficients of correlation are regularly 
 obtained by correlating two distributions of measures of the same 
 individuals. The uncertain conclusions of research on the effect of 
 practice on individual differences would cause one to question the 
 dependability of a coefficient obtained by correlating gains of paired 
 individuals. Owing to the uncertainty of this correlation and owing 
 to the reduction in the operation of chance where procedures are 
 employed to secure equivalence, the standard errors obtained through 
 the utilization of the formulae given above should be interpreted as 
 limits beyond which the true standard errors cannot fall. 
 
 Lindquist has stated with regard to the formulae given above that 
 they are based on "the assumption that the samples used are strictly 
 random selections from the populations they represent." 10 He con- 
 
 tinues: 
 
 This assumption is not applicable to matched groups. The process of matching 
 on the basis of a measure which is correlated with the final measure destroys the ran- 
 domness of the samples with respect to this final measure. The probable amount ot 
 sampling error in the obtained difference, instead of being as large as that indicated 
 
 loLindquist, E. F. "The Significance of a Difference Between 'Matched* Groups," Journal of 
 ational Psychology, 22:198, Mar 
 Advice with respect to the for 
 Dr. Lindquist is deeply appreciated. 
 
 ^n^t^thS^^'Sle^mita'^ven on page !04 received through correspondence with 
 
104 Bulletin No. 58 
 
 by the formulas given above, is usually considerably less, in some cases by more than 
 fifty percent. 11 
 
 The allowance to be made for the combined effect of variable 
 errors of measurement and variable errors of sampling in the case of 
 paired, or matched, groups may be determined by means of the fol- 
 lowing formulae: 12 
 
 °"Mean Gain E — 
 
 °"Mean Gain C — 
 
 Vn 
 Vn 
 
 "Difference — \ OMean Gain E + CM 
 
 Mean Gain E — Mean Gain C 
 
 ean Gain C 
 
 In the formulae given above <j e and <r c are the standard deviations 
 of the distributions of individual gains of the pupils in the experimen- 
 tal and in the control groups. The coefficient of correlation, r iej 
 refers to the relation between the intelligence test scores, or other 
 measures, used in pairing the experimental pupils and their corre- 
 sponding individual gains. 13 The coefficient of correlation, r ic , refers 
 to the relation between the intelligence test scores, or other measures 
 used in pairing the control pupils, and their corresponding individual 
 gains. 14 Lindquist suggests that where the methods compared are 
 unlikely to result in producing a significant difference in r ie and r ic , 
 the statistical technique may be simplified by using the formula: 15 
 
 Ai 
 
 f N,' (1 r) 
 
 Mean Gain E — Mean Gain C 
 
 "Lindquist, op. cit., p. 198. 
 
 12 For rigorous mathematical proof, see: 
 
 P5 3 «AK2f;205.(iT h M^h? d i a 9?i. ErrOr ° f thG MCanS ° f MatCh6d SampleS '" J ° Urnal °f Ed ^ational 
 ^UlZ™J£tt}S ££?* Tu ay , be intelligence test. scores, or they may be the scores of the 
 
 bination, is the appropriate coefficient to use. See, 
 
 Wilks, op. cit., p. 208. 
 
 *nH th^T f eq " ivalent f ° rm f ° f the , Same subject-matter test are not administered at the beginning 
 a nil C ? d °J -5 e % x P enmen ±> or where scores have not been converted into comparable units, cal- 
 l^°rA ^vidual gains is impossible. In using the above formulae. * e and <r c should represent the 
 fnH?JnH e r iat \ 0nS iH f the dlst " b l utlon 1 s of final-test scores of the experimental and the control groups! 
 ?£? I&™ > tI ' ° Uld re P res ? n f th e relationships between the test scores used in pairing and the final- 
 standard error of ?h?H-ff C r ntal a i nd t C ° ntr ? P , U , pils res Pectively. The third formula then yields the 
 standard error of the difference between final-test means for matched grouDs 
 
 15 Lindquist, op. cit., p. 202-03. 
 =*o^ T !T formu f ^ g^en by Lindquist has been slightly modified by the authors to represent the 
 standard error of the difference in mean gains, and the symbols have been changed. 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 105 
 
 N e and N c represent the numbers of pupils in the experimental and 
 control groups, usually the same, a e and <r c , the standard deviations of 
 the two distributions of individual gains, and r stands for the relation- 
 ship existing between the measures of all the pupils used in pairing 
 and their individual gains. 
 
 It should be noted in this connection that Lindquist suggests that 
 the formulae given above "should be valid for use with groups that 
 have not been matched 'pupil for pupil', but in which the means and 
 standard deviations alone have been equated." He adds, however, 
 that "a more rigid mathematical proof of this proposition should be 
 provided before much confidence is placed in it." 16 The following 
 quotation is indicative of just what is allowed for when these formulae 
 
 are used: 
 
 It is also important to note that formula (9) (the one just given) does not indicate 
 how far the obtained difference between two matched samples is likely to deviate trom 
 the difference that would have been obtained had the entire population been meas- 
 ured, but tells only how far the obtained difference is likely to deviate from the 
 difference that would have been found between infinitely large groups showing the 
 same distribution of initial measures as that of the matched samples that were used? 
 
 It will be seen from the statement quoted above that generaliza- 
 tions in which this standard error of difference is used apply to 
 "infinitely large groups showing the same distribution of initial meas- 
 ures as that of the matched samples." If the matched samples are, 
 for example, somewhat superior in intelligence to the average intelli- 
 gence of the general population from which the samples were drawn, 
 strictly speaking, the generalizations apply to similar matched sam- 
 ples. When N is greater than 30, the experimental group is selected 
 in a random fashion from the general population; and the control 
 group is obtained by selecting pupils from the general population who 
 match the experimental pupils, this standard error of difference may 
 be used with considerable justification in formulating conclusions 
 relative to the general population. 18 
 
 Finally, it should be noted that the standard error of difference 
 obtained by the formulae suggested by Lindquist neglects the corre- 
 lation that may exist between the gains of the paired pupils, or their 
 final-test scores. Hence, the standard error of difference so obtained 
 is also to be interpreted as a limit beyond which the true standard 
 error cannot fall. The limit, however, is probably closer to the true 
 
 16 Lindquist, op. cil., p. 202. 
 
 nferailv soeakine "infinitely large groups" would include the "entire population" and thus have 
 
 have distributions differing from those of the "entire population. 
 *Ibid., p. 203. 
 
106 Bulletin No. 58 
 
 standard error than that obtained when —7=^ is used in computing the 
 
 standard error of the mean gains, or the final-test means. 
 
 When the groups of pupils are ordinary school classes, and not ran- 
 dom samples, statistical procedure may be employed in making allow- 
 ance for variable errors of measurement alone. In preceding para- 
 graphs it was indicated that these errors are of minor importance in 
 comparison with the other possible sources of undependability of a 
 difference in mean gains, or in final-test means. Where reasonably 
 reliable tests have been used, and the experimental and control groups 
 are fairly large, the computation of the standard error of measurement 
 of the difference in mean gains, or final-test means, is unlikely to 
 contribute much to the meaning of the findings. If the difference in 
 mean gains is comparatively large the investigator is justified in 
 assuming that the dependability of the difference, so far as the groups 
 used in the experiment are concerned, is not significantly affected by 
 variable errors of measurement. 
 
 The procedures just described constitute a means for calculating 
 the probable effect upon the difference of the mean gains, or final-test 
 means, of only the combined effect of the variable errors of measure- 
 ment and of the error of sampling. Unfortunately it is not possible 
 to calculate the probable effects of the systematic errors of measure- 
 ment in either, or both, the first and second trial scores, the invalidity 
 of the test, as determined by the problem of the experiment, and any 
 lack of control of significant non-experimental factors. In general 
 the experimenter can only estimate the probable effects of these 
 conditions. Usually some circumstantial evidence can be cited in 
 support of his estimate, but the uncertainty of any estimate justifies 
 the assertion that the determination of "statistical" significance by 
 means of the formulae given above should not be treated very seri- 
 ously. The interpretation of a small difference in mean gains, or 
 final-test means, will usually be uncertain even when it is shown to 
 be "statistically" significant. 
 
 The statement just made is important. The use of statistical 
 formulae, especially when somewhat complex, tends to be impressive, 
 and when the difference is shown to be "statistically" significant 
 there is doubtless a suggestion to the uninformed that all limitations 
 of the data have been allowed for. This is not the case. As a matter 
 of fact it is reasonably apparent that, in many cases, the probable 
 error of the difference in mean gains is an index of the least signifi- 
 cant limitations of the data. By way of emphasizing this point it 
 may be suggested that when attempting to interpret a difference an 
 
Summary of Research Relating to the Teaching of Arithmetic 107 
 
 experimenter should focus his attention upon the limitations of the 
 data whose probable effect cannot be calculated by a formula. 
 
 5 Generalization. Consideration of the probable effect of the 
 variable errors of sampling has, of course, related to the generalizing 
 of the data secured in a particular experiment. The formulae pre- 
 sented furnish a statistical basis for generalizing only when both the 
 control group and the experimental group constitute random samples 
 from the larger population, or when one of the two equated groups 
 has been selected at random. 
 
 Generalization appears justified, where random sampling has not 
 been employed, when it can be shown that lack of representativeness 
 of the groups does not seriously limit the dependability of the differ- 
 ence in achievement in favor of a given method. In showing that the 
 groups used in the experiment are sufficiently typical or representa- 
 tive to justify generalization, the experimenter should present all 
 available evidence relative to the traits of the groups concerned. 
 For example, the intelligence test scores will be known, and the 
 experimenter should show how the mean and the standard deviation 
 of these scores compare with corresponding measures of the larger 
 population. If the available evidence indicates that the groups are 
 highly representative of the larger population, he may generalize 
 with considerable confidence; if the evidence indicates that the 
 groups are not reasonably representative of the larger population, he 
 must refrain from generalizing or appropriately limit his statements. 
 Feasibility versus evaluation (effectiveness) of instructional tech- 
 niques. In closing this discussion it seems appropriate to comment 
 upon the demonstration of the feasibility of a procedure versus the 
 determination of the relative merits of two or more specified pro- 
 cedures. The former can be accomplished by a single-group experi- 
 ment. It is only necessary to show that as applied by a certain 
 teacher or group of teachers the procedure resulted in reasonably 
 satisfactory achievements. To determine the relative merits erf two 
 or more specified procedures controlled experimentation is required. 
 The difficulties encountered in controlling non-experimental factors 
 and in securing accurate and valid measures of the achievement 
 specified by the problem of the experiment have been noted in the 
 preceding pages. It is apparent that the expectation of precise 
 evaluation is not justified. 
 
CHAPTER IX 
 
 BIBLIOGRAPHY OF INVESTIGATIONS PERTAINING 
 TO METHODS OF TEACHING ARITHMETIC 
 
 1. Adams, R. E. "A Study of the Comparative Value of Two Methods of 
 Improving Problem Solving Ability in Arithmetic," Philadelphia: Uni- 
 versity of Pennsylvania, 1930. 68 p. 
 
 2. Anspaugh, G. E. "Teaching the Number Facts in the Komensky School," 
 Chicago Principals' Club, Second Yearbook. Chicago: Chicago Princioal's' 
 Club, 1927, p. 88-89. P 
 
 3. Anthony, Kate, et al. "The Development of Proper Attitudes Towards 
 School Work," School and Society, 2:926-34, December 25, 1915. 
 
 4. Arnett, L. D. "Counting and Adding," American Journal of Psvcholoev 
 16:327-36, July, 1905. * 
 
 5. Ballard, P. B. "Norms of Performance in the Fundamental Processes of 
 Arithmetic, with Suggestions for Their Improvement," Journal of Experi- 
 mental Pedagogy, 2:396-405, December 5, 1914; 3:9-20, March 5, 1915. 
 
 6. Ballenger, H. L. "Overcoming Some Addition Difficulties," Journal of 
 Educational Research, 13:111-17, February, 1926. 
 
 7. Ballou, F. W. "Improving Instruction through Educational Measure- 
 ment," Educational Administration and Supervision, 2:354-67, June, 1916. 
 
 8. Beatty, W. W. "The Additive versus the Borrowing Method of Subtrac- 
 tion," Elementary School Journal, 21:198-200, November, 1920. 
 
 9. Beito, E. A. and Brueckner, L. J. "A Measurement of Transfer in the 
 Learning of Number Combinations," Twenty-Ninth Yearbook of the 
 National Society for the Study of Education. Bloomington, Illinois: 
 Public School Publishing Company, 1930, p. 569-87. 
 
 10. Bowman, H. L. "The Relation of Reported Preference to Performance in 
 Problem Solving," University of Missouri Bulletin, Vol. 30, No. 36, Edu- 
 cation Series, No. 29. Columbia: University of Missouri, 1929. 52 p. 
 
 11. Bradford, E. J. G. "Suggestion, Reasoning, and Arithmetic," Forum of 
 Education, 3:3-12, February, 1925. 
 
 12. Brooks, S. S. "A Study of the Technical and Semi-technical Vocabulary of 
 Arithmetic," Educational Research Bulletin (Ohio State Universitv) 
 5:219-22, May 26, 1926. 
 
 13. Brown, J. C. "An Investigation on the Value of Drill Work in the Funda- 
 mental Operations of Arithmetic." Journal of Educational Psvchology, 
 2:81-88, February, 1911; 3:485-92, 561-70; November, December, 1912.' 
 
 14. Brownell, W. A. "Remedial Cases in Arithmetic," Peabodv Journal of 
 Education, 7:100-107, September, 1929. 
 
 15. Buckingham, B. R. "Adding Up or Down: A Discussion," Journal of 
 Educational Research, 12:251-61, November, 1925. 
 
 16. Buckingham, B. R. "The Additive versus the Take-away Method of 
 Teaching the Subtraction Facts," Educational Research Bulletin (Ohio 
 State University), 6:265-69, September 28, 1927. 
 
 108 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 109 
 
 17 Buckingham, B. R. "Teaching Addition and Subtraction Facts Together 
 ' or Separately," Educational Research Bulletin (Ohio State University) 
 
 6:228-29, 240-42; May 25, 1927. 
 
 18 Buckingham, B. R. "Upward versus Downward Addition," Journal of 
 Educational Research, 16:315-22, December, 1927. 
 
 19 Burton C. B. "Results of Definite Drill in the Four Fundamental Proc- 
 " esses as' Shown bv the Woody-McCall Mixed Fundamentals," Fifth \ ear- 
 book of the Department of Elementary School Principals. Washington: 
 National Education Association, 1926, p. 323-28. 
 
 20 Buswell, G. T. and John, Lenore. "Diagnostic Studies in Arithmetic," 
 ' Supplementary Educational Monographs, Xo. 30. Chicago: University of 
 
 Chicago Press, 1926. 212 p. 
 
 21 Buswell, G. T. and John, Lenore. "The Vocabulary of Arithmetic, _ Sup- 
 ' plementary Educational Monographs, Xo. 38. Chicago: University of 
 
 Chicago Press, 1931. 146 p. 
 
 22. Chapman, J. C. and Feder, R. B. "The Effect of Extern allnce ntives ion 
 Improvement," Journal of Educational Psychology, 8:469-74, October, 
 1917. 
 
 23. Chase, S. E. "Waste in Arithmetic," Teachers College Record, 18:360-70, 
 September, 1917. 
 
 24 Chase VE "The Diagnosis and Treatment of Some Common Difficulties 
 ' in Solving Arithmetic Problems," Journal of Educational Research, 20 :33z- 
 
 42, December, 1929. 
 
 25 Clapp, F. L., Chase, W. J , and Merriman, Curtis. "A Study of the Effec- 
 ' tiveness of Two Kinds of Teaching Material," Introduction to Education. 
 
 Boston: Ginn and Company, 1929, p. 420-24. 
 
 26. Clark, J. R. and Vincent, E. L. "A Comparison of Two Methods of 
 Arithmetic Problem Analysis," Mathematics Teacher, 18:226-33, April, 
 1925. 
 
 27. Clark, J. R. and Vincent, E. L. "A Study of the Effect of Checking upon 
 Accuracy in Addition," Mathematics Teacher, 19:65-71, February, 1926. 
 
 28. Clemens, P. B. and Neubauer, P. F. "A Supervision Project in Multipli- 
 cation," Journal of Educational Research, 18:38/-96, December, 1928. 
 
 29. Cole, L. W. "Adding Upward and Downward," Journal of Educational 
 ' Psychology, 3:83-94, February, 1912. 
 
 30. Collier, Myrtle. "Learning to Multiply Fractions," School Science and 
 Mathematics, 22:324-29, April, 1922. 
 
 31. Collings, Ellsworth. An Experiment with a Project Curriculum. Xew 
 York: Macmillan Company, 1923. 346 p. 
 
 32. Conard, H. E. and Arps, G. F. "An Experimental Study of Economical 
 Learning," American Journal of Psychology, 27:^07-29, October, 1916. 
 
 33 Courtis, S. A. "The Courtis Standard Tests in Boston, 1912-1915 : An 
 Appraisal," School Document Xo. 15, 1916. Department of Educational 
 Investigation and Measurement Bulletin Xo. 10. Boston: Boston Printing 
 Department, 1916. 48 p. 
 
 34 Dawson, C. D. "Some Results in Using Starch's Arithmetic Reasoning 
 ' Test," Journal of Educational Research, 2:677-78, October, 1920. 
 
110 Bulletin No. 58 
 
 35. Drushel, J. A "A Study of the Amount of Arithmetic at the Command of 
 High-School Graduates Who Have Had No Arithmetic in Their High- 
 School Course," Elementary School Journal, 17:657-61, May, 1917. 
 
 36. Edano, Tiburcio. "An Analysis of Arithmetic Textual Matter," Philippine 
 Public Schools, 1:81-84, February, 1928. 
 
 37. Evans, J. E. and Knoche, F. E. "The Effects of Special Drill in Arithmetic 
 as Measured by the Woody and the Courtis Arithmetic Tests," Journal of 
 Educational Psychology, 10:263-76, May-June, 1919. 
 
 38. Fowlkes, J. G. ' 'A Report of a Controlled Study of the Learning of Multi- 
 15^1-89 b MaTch r ^7 de ChiIdren '" J ° Urnal ° f Educa tional Research, 
 
 39. Gabbert, M. L "Remedial Cases in Arithmetic; Case 1," Peabody Journal 
 of Education, 7:147-55, November, 1929. 
 
 40. Gates, A. I., Batchelder, M. I., and Betzner, Jean. "A Modern Svstematic 
 
 SfSS ^° P . POrtUnistic Method of Teaching," Teachers College Record, 
 27:679-700, April, 1926. 
 
 41. Graham, V. O. "Arithmetic Reasoning Project and the Measurement of 
 Improvement," Chicago Principals' Club, Second Yearbook. Chicago- 
 Chicago Principals' Club, 1927, p. 86-87. 
 
 42. Greene H. A. "Directed Drill in the Comprehension of Verbal Problems 
 in Arithmetic," Journal of Educational Research, 11:33-40, January, 1925. 
 
 43. Guiler W. S. "Improving Computational Ability," Elementary School 
 Journal, 30:111-16, October, 1929. 
 
 44. Hackler, J. M. "The Relation between Successful Progress in Mathematics 
 and the Ability to Read and Understand, and the Factors That Contribute 
 to Success or Failure in Mathematics." Unpublished master's thesis in 
 Education. Chicago: University of Chicago, 1921. 82 p. 
 
 45. Hagen, H. H. "A Study of Practice Periods in Arithmetic Fundamentals " 
 
 £ u ^£ lnapals ' Club ' Second Year book. Chicago: Chicago Principal's' 
 Club, 1927, p. 93-95. 
 
 46. Hahn, H. H. and Thorndike, E. L. "Some Results of Practice in Addition 
 under School Conditions," Journal of Educational Psychology, 5-65-84 
 February, 1914. 
 
 47. Hanna, P. R. "Arithmetic Problem Solving: A Study of the Relative 
 Effectiveness of Three Methods of Problem Solving." New York: Bureau 
 of Publications, Teachers College, Columbia University, 1929. 68 p. 
 
 48. Harlan, C. L. "Years in School and Achievements in Reading and Arith- 
 metic," Journal of Educational Research, 8:145-49, September, 1923. 
 
 49. Hurlock, E. B. "An Evaluation of Certain Incentives Used inSchool 
 Work," Journal of Educational Psychology, 16:145-59, March, 1925. 
 
 50. Hydle, L. L. and Clapp, F. L. "Elements of Difficulty in the Interpretation 
 of Concrete Problems in Arithmetic," Bureau of Educational Research 
 Bulletin No. 9. Madison: University of Wisconsin, 1927. 84 p. 
 
 51. John, Lenore. "Difficulties in Solving Problems in Arithmetic," Elemen- 
 tary School Journal, 31:202-15, November, 1930. 
 
 52. Johnston, J. T. "The Merits of Different Methods of Subtraction," 
 Journal of Educational Research, 10:279-90, November, 1924. 
 
Summary of Research Relating to the Teaching of Arithmetic 
 
 111 
 
 53 Kallom A W. "The Importance of Diagnosis in Educational Measure- 
 
 " ment "Journal of Educational Psychology, 10:1-12, January, 1919. 
 54. Kathleen, Sister M. "Some Results of Remedial Instruction Following the 
 
 Use of Diagnostic Arithmetic Tests," Catholic Educational Review, 24:19- 
 
 27, January, 1926. 
 
 55 Kelly, F.J. 'The Results of Three Types of Drill on the Fundamentals of 
 Arithmetic," Journal of Educational Research, 2:693-700, November, 1920. 
 
 56 Kerr M A. "Effects of Six Weeks Daily Drill in Arithmetic," Studies in 
 Arithmetic, Indiana University Studies No. 32. Bloomington: Indiana 
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 57 Kirby, T. J. "Practice in the Case of School Children," Teachers College, 
 Columbia University Contributions to Education No. ,8 New York: 
 Bureau of Publications, Teachers College, Columbia University, 1913. 98 p. 
 
 *« TCirknatrick E A "An Experiment in Memorizing versus Incidental 
 J^?Journi of Educational Psychology, 5:405-12, September, 1914. 
 
 59 Kline, L. W. and Anderson, P. K. 'The Role of Habit in Reasoning," 
 ' School Science and Mathematics, 26:156-67, February, 1926. 
 
 60 Knight, F. B. "The Superiority of Distributed Practice in Drill in Anth- 
 metiV' Journal of Educational Research, 15 :157-65, March, 1927. (A sum- 
 mary of 61.) # 
 
 61 Luse E.M. "Transfer within Narrow Mental Functions. A Study of the 
 ' Effects of Distributed versus Non-Distributed Drill in Arithmetic, Uni- 
 versity of Iowa Monograph in Education No. 5. Iowa City: University 
 
 of Iowa, 1925. 
 
 62 Knight F. B. and Setzafandt, A. O. H. "Transfer within a Narrow 
 Mental Function," Elementary School Journal, 24:780-87, June, 1924. 
 
 63 Krause A K. "Why Monroe Diagnostic Tests in Arithmetic?" Contn- 
 butions'to Education, Vol. 2. Yonkers, New York: World Book Company, 
 
 1928, p. 15-17. , . , _ , 
 
 64 Kulp C. L. "A Method of Securing Real-Life Problems m the Funda- 
 mentals of Arithmetic," Elementary School Journal, 29:428-30, February, 
 
 1929 
 
 65 Kulp, C. L. "A Studv of the Relative Effectiveness of Two Types of Stand- 
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 22:381-387, December, 1930. m 
 
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 " lournal of Educational Research, 11:287-91, April, 192x ^ 
 
 68 Lutes, O. S. "An Evaluation of Three Techniques for Improving Ability 
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 Iowa City: University of Iowa, 1926. 42 p. 
 
 69 McClelland, W. W. "An Experimental Study of the Different Methods of 
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 70 Mailer J B. "Cooperation and Competition, An Experimental Study in 
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112 Bulletin No. 58 
 
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 71. Mead, CD. and Johnson, C. W. "Testing Practice Material in the Funda- 
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 72. Mead, C. D. and Sears, Isabel. "Additive Subtraction and Multiplicative 
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 73. Meriam, J. L. "How Well May Pupils Be Prepared for High School Work 
 without Studying Arithmetic, Grammar, etc., in the Grades?" Journal of 
 Educational Psychology, 6:361-64, June, 1915. 
 
 74. Merton, E. L., et al. "Remedial Work in Arithmetic." Second Yearbook 
 of the Department of Elementary School Principals. Washington: 
 National Education Association, 1923, p. 395-429. 
 
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Summary of Research Relating to the Teaching of Arithmetic 115 
 
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