A Study of Textbook Material in the Field of Arithmetic

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A STUDY OF TEXTBOOK MATERIAL IN THE FIELD OF ARITHMETIC

by Jean Frances Hamilton

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, in the Department of Education, in the Graduate College of the State University of Iowa June 19%0

ProQuest Number: 10902169

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ACKNOVMIGMENTS

The writer wishes to express her appreciation for the encouragement and assistance given by Dr. Herbert F. Spitzer, who directed this study. The writer also wishes to express her appreciation for the cooperation given by teachers and school administrators who participated in this study.

TABLE OF CONTENTS

Chapter I

Page Introduction and Statement of theProblem . . .

1

Introduction.........................1 Statement of the Problem............... 3 II

Review of the Literature in the Field of Arithmetic.......................... 5 Representative Examples of Theory and Review of Related Studies.......... 5

III

Procedure and Materials................... 25 Introduction ....................... 25 Investigation I .....................25 Part A ................... 25 Part B ........................ 28 Investigation I I .....................31

IY

Results................................ 38 Investigation I .....................38 Part A ........................ 38 Analysis of Observation Reports for Part A of Investigation I . . . 1+3 Part B ........................ 53 Interpretation of Table 1 ......... 5U Analysis of Table 1 .............. 85 Investigation I I .....................92 Interpretation of Tables 2— 5 . . . . 9li Analysis of Tables 2— 5 ........... 99 Detailed Analysis of Question 8 Summarized on Table 2 . ....... 101 Results of Individual Questionnaires.................101; Interpretation and Analysis of Tables 6 and 7 .................105

iii

Page

Chapter

Interpretation and Analysis of Tables 8, 9 and 1 0 ............ 109 Interpretation and Analysis of Tables 11, 12 and 1 3 ...........H U Interpretation and Analysis of Tables lit, 19 and 1 6 ...........120 Interpretation and Analysis of Tables 1?, 18 and 1 9 .......... 129 Interpretation and Analysis of Tables 20, 21 and 22 ..........129 V

Summary, Conclusions and Recommendations

. . . . 137

Investigations I and I I ................ 137 Investigation I ...................... 137 Conclusions.....................138 Investigation I I ................... 11+9 Conclusions.....................lU6 Limitations......................... ll*9 Recommendations...................... 193 Bibliography............................. 195 Appendix I ..................... 198 Reports of Observations Made in Fiftjr Elementary Arithmetic Classrooms for Part A of Investigation I ............ 198 Appendix II ................ 199 Guide Sheet Questionnaire Submitted to Teachers for Part B of Investigation I .....................199 Appendix III ........................... 199 Arithmetical textbook materials submitted to teachers forevaluation . . . 199 Appendix I V ........................... 2h9 Teacher reactions to Item 8 on Guide Sheet Questionnaire II, classified by grade level and preference (Text or Experimental)..................... 2k9

iv

Chapter Appendix V .......................... Part 1. Teacher reactions to Question i| on the individual questionnaires included with each set of arithmetical materials submitted for evaluation. Classified by lesson, grade level, and stated preference. (Textbook or Experimental) ........ Part 2. Other suggestions made by the teachers for changes or improvements in arithmetic textbook materials. Classified by grade level..........

TABLE OF TABLES

Table 1

Page Results Obtained from Teacher Questionnaire for Part B of Investigation I ............... 56 Table Table Table Table Table Table Table Table

1, 1, 1,

question.1 ................... 56 question 2 ...................... 57 question 3 ................... 60 question I4 ................... 6l question.5 ...................... 63 question 6 ................... 65 question 7 ................... 67 question 8 ................... 69

Table 1,

question 9 ...................... 70

Table Table Table Table Table Table

1, 1, 1, 1, 1,

1, question 1 0 ................. 72 1, question 1 1 ......... 75 1, question 1 2 ................. 77 1, question 1 3 ................. 78 1, question l U ................. 80 1, question 1 5 ................. 81

2

Analysis of Questionnaire Guide Sheet II for Investigation II . .................. 97

3

Analysis Question 9 Guide Sheet I I ............ 97

1+

Analysis by Grade Level of the Number of Teachers Expressing a Preference on Guide Sheet II for the Textbook Materials . . . . . .

98

Analysis by Grade Level of the Number of Teachers Expressing a Preference on Guide Sheet II for the Experimental Materials . . . .

98

5

6

Analysis of Questionnaire for Grade I Lesson A .............................. 106

7

Analysis of Questionnaire for Grade I Lesson B .............................. 107

vi

Page 8

Analysis of Questionnaire for Grade II Lesson A ............................... 111

9

Analysis of Questionnaire for Grade II Lesson B ............................... 112

10

Analysis of Questionnaire for Grade II Lesson C ............................... 113

U

Analysis of Questionnaire for Grade III Lesson A ............................... 116

12

Analysis of Questionnaire for Grade III Lesson B ............................... 117

13

Analysis of Questionnaire for Grade III Lesson C ............................... 118

Hi

Analysis of Questionnaire for Grade IV Lesson A ............................... 121

15

Analysis of Questionnaire for Grade IV Lesson B ............................... 122

16

Analysis of Questionnaire for Grade IV Lesson C .............................

123

17

Analysis of Questionnaire for Grade V Lesson A ............................... 126

18

Analysis of Questionnaire for Grade V Lesson B ............................... 127

19

Analysis of Questionnaire for Grade V Lesson C .............................

128

Analysis of Questionnaire for Grade VI Lesson A .............................

131

20

vii

Table

Page

21

Analysis of Questionnaire for Grade VI Lesson B ................................132

22

Analysis of Questionnaire for Grade VI Lesson C ................................133

viii

Chapter I INTRODUCTION AND STATEMENT OF THE PROBLEM Introduction The textbook has long played an important role in American education.

This viewpoint was clearly expressed by a committee of

educators in the introduction to the Thirtieth Yearbook of the National Society for the Study of Education. The significant position of the textbook in the program of American education is so generally recognized that the Society seems to be fully justified in sponsoring a yearbook on the theme "The Textbook." It is the textbook that in thousands of classrooms determines the content of the instruction as well as the teaching procedures. This statement may not be in accord with the usual theory but it is supported by the facts reported by supervisors and state inspectors of schools. In view, therefore, of the important place of the textbook in our educational practice, the preparation and the selection of textbooks is a problem of major importance.! Literature^ on the history of arithmetic indicates that throughout the years the textbook has reflected the feeling of the times regarding the aims and methods of instruction and has served as a constant guide and reference to teachers and pupils in planning the work of arithmetic. 1. National Society for the Study of Education, Thirtieth Yearbook, Part II. "The Textbook in American Education." Public School Publishing Company, Bloomington, 111. 19319 p. 1. 2. Smith, H. E., Eaton, M. T., Dugdale, Kathleen, One Hundred Fifty Years of Arithmetic Textbooks, Bulletin of the School of Education, Indiana University, Vol. 31, No, 1. 19U5-

In most school systems the textbook serves as the course of study or as the instrument for putting the course of study into practice.

Since the teaching of arithmetic is in a large measure

dependent on the textbook its importance cannot be overemphasized. Even though some teachers deviate from the pattern of the book and tiy to individualize their teaching the textbook still provides the framework for the course and sets the pattern for the type of instruction. In view of these facts it would seem that one effective way to improve instruction in arithmetic would be to place superior textbooks in the hands of pupils and teachers.

Textbooks that

present adequate content and suggest methods and procedures that enable a teacher to make arithmetic meaningful to children. While the authors and publishers of most current arithmetic textbooks claim to meet these requirements there is apparently still room for improvement. Despite the advances made in methods of teaching arithmetic in the last twenty-five years, surveys and studies of instructional procedures in typical classrooms^ still report that classroom practice lags far behind methods advocated in more recent writing. 1. Brueckner, L. J., "Analysis of the Instructional Practices in Typical Classes in Schools of the United States,11 Tenth Yearbook, National Council of Teachers of Mathematics. Teachers College, Columbia University. 1935> pp. 32-50.

3

Under our present classroom conditions, a textbook while important to teachers cannot stand alone.

The teacher’s role in

interpreting and translating textbook material into daily classroom practice is of vital importance. Therefore, to be used effectively, a textbook’s aims and purposes must be acceptable to teachers. The methods and procedures suggested for use in a textbook for carrying out a program of work must be understood by teachers.

The lessons

must be developed to serve as guides for teachers in planning day by day work in the classroom.

From the foregoing it would seem

evident that a study of the textbook and the uses of the textbook in the classroom would be a profitable educational experience. Statement of the Problem The primary aim of this investigation was to make a study of textbook material in the field of arithmetic for the purpose of evaluation and improvement.

In order to keep within the limits of

the time allotted, the investigation was limited to the study of two aspects of textbook materials.

These aspects are listed in the two

statements of purposes below: 1. To study teacher use of arithmetic textbooks in selected classrooms. 2. To determine teacher reaction to methods of presenting arithmetical textbook materials that are more nearly in haimony with

4

the best theory than are the methods of presenting material now used in published textbooks. The second purpose listed above requires the identification of best theory, k careful review of the literature in the field of arithmetic was the procedure used to determine, in so far as possible, best theory and method.

5

Chapter II REVIEW OF THE LITERATURE IN THE FIEID OF ARITHMETIC Representative Examples of Theory and Review of Related Studies To carry out the second purpose of this study a careful review of the literature in the field of arithmetic was made in an attempt to identify best theory. The primary purpose of this review was to determine, in so far as possible, suggested principles to use as guides in creating learning situations for children. After this review of representative examples of theory and related studies in the field of arithmetic it is evident that within the last quarter of a century students of the teaching of arithmetic have been increasingly aware of the role of numbers in situations other than those uses which require computation of the type used in finding the cost of several articles or the average number of miles per hour. During this time representative groups of educators and mathematicians have turned away from the idea that arithmetic should be taught for disciplinary values toward the objectives gained through a more functional, informational and simplified arithmetic. This trend has taken shape in (1) attempts to analyze the uses made of arithmetic in business and social life, (2) atteupts to analyze types of quantitative terms met in some subject matter areas in the schools, and (3) attempts to analyze the mathematical elements involved in quantitative situations met in adult life.

6

The changing concepts of arithmetic instruction have been expressed by educators and mathematicians in many different ways. Although there has been a divergence of opinion as to the aims of instruction, nevertheless, there seems to be a clearly discernible trend toward what has became known as the meaning theory# In 192U Young-*- set forth the following specific objectives of instruction in arithmetic: (1) to teach children the mathematical type of thought; (2) to arouse their interest in the quantitative world around them; (3) to give accuracy and facility in simple computations; (U) to impart a working knowledge of a few practical applications of arithmetic; and (5) to prepare the way for further mathematics. Wilson^ has consistently pointed out that the major aim of the teaching of arithmetic is to equip children with the useful skills of business. David Eugene Smith^ describes arithmetic as a social subject, and he contends that it should be taught as such. Smith uses the term social in the same sense as other writers use the word cultural# He believes if this aspect of arithmetic is not stressed children will be deprived of a heritage which is their right and privilege. 1. Young, J. W# A#, The Teaching of Mathematics in the Elementary School and Secondary School# Longmans Green and Co. N. Y., 1921;. 2. Wilson, Guy M., Stone, Mildred B., and Dalrymple, Charles 0., Teaching the New Arithmetic. McGraw-Hill, 1939. p* 73. Snith, David E., "Arithmetic for Intermediate Grades," Classroom Teacher, Vol. 7, Chicago, p. 39k*

Schorling^- combines the social, cultural, and mathematical objectives into a comprehensive program. Iheat^ states that the primary aim in teaching arithmetic is to provide children -with methods of thinking, ideas of procedures, -with meanings inherent in number relationships, with general principles of combinations and arrange­ ments in order that the quantitative situations in life may be handled intelligently. Thiele^ advocates teaching arithmetic from -what he terms the mathematical point of view.

In doing this he would not neglect

the social aims. Thiele^ places great reliance upon the child's discovering for himself effective solutions and upon his seeing relationships. The viewpoint that both the mathematical aim (to develop expertness in quantitative thinking ty teaching arithmetic as a closely knit system of related ideas, facts and principles), and the social aim (to provide number situations and number ideas which enable a child to see the various ways number functions in experience), are important to a well rounded program of instruction 1. Schorling, Raleigh, The Teaching of Mathematics. Ann Arbor Press. Michigan, 1936. 2. Hlfheat, Harry G., The Psychology and Teaching of Arithmetic. D. C. Heath Company. Boston, 1937, p. UO. 3* Thiele, C. L., "The Mathematical Viewpoint Applied to the Teaching of Elementary School Arithmetic," Tenth Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University. 1935, p. 213. 1*. Thiele, C. L., Sixteenth Yearbook of the National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University. I9l±l, pp. 1*6, 53, 55-57.

finds expression in the writings of Buckingham-1* and Brueckner.^ From this type of writing and thinking the meaning theory of arithmetic evolved.

This theory was given its first major

publicity in 1935 by the National Council of the Teachers of Mathematics.^ Morton later defined the position of the committee

and explained what the theory implies: A kind of arithmetic in which both the mathematical and the social aims are clearly recognized— and clearly recognized as interdependent and mutually related. Attainment of the mathematical aim is regarded as possible only if meaning, the fact that children shall see sense in what they learn, is made the central issue in arithmetic instruction. Arithmetic is conceived as a closely knit system of understandable ideas, principles and processes, and an important test of arithmetical learning is an intelligent grasp upon number relations together with the ability to deal with arithmetical situations with proper comprehension of their mathematical significance.4 1. Buckingham, R. B., "Significance, Meaning and Insight— These Three," Mathematics Teacher, Vol. 31, January 1938, P« 26. 2. Brueckner, Leo. J., "The Social Phase of Arithmetic Instruction," Sixteenth Yearbook of the National Council of Teachers of Mathematics! 1911, p. ll|l« 3» National Council of Teachers of Mathematics, Tenth Yearbook. Bureau of Publications, Teachers College, Columbia University. 1935, p. 19. 1. Morton, R. L., "The National Council Committee on Arithmetic," Mathematics Teacher, October 1938, Vol. 31, No. 6, p. 269.

Ten years later Spitzer re-emphasized the viewpoint expressed by Morton in 1938, but he goes a step farther than most educators and mathematicians by showing how content and method are affected by the purposes of arithmetic. He not only recognizes the mathematical and social aim of arithmetic and the important part it plays in clarifying quantitative concepts, but he also suggests that once these concepts and their interrelationships are understood they can be used to open up new fields of thought. By simplifying or clarifying quantitative concepts, arithmetic does its greatest service. If the mind is relieved of cumbersome ways of thinking about concepts, then it is in a position to do something with the concepts. Thus numbers become an instrument of thought. Helping children to an understanding and to a realization of the mathematical and social aims of instruction are major objectives to those interested in the teaching of arithmetic. Almost all writers verbally accept the theories advanced here and almost all textbooks state these aims in their prefaces. However, there is no indication in the literature that books and teachers are approaching these ideals. An analysis of teachers1 use of textbooks, their reactions to the teaching suggestions and a summary of their opinions for change should be one step forward in bringing the purposes of arithmetic into closer contact with the classroom. 1. Spitzer, H. F., "The Teaching of Arithmetic. Houghton Mifflin Company, Boston, Mass"! I9I1B, p* 2jT!

The preceding section has been concerned with the changing viewpoint and concepts of the aims and purposes of arithmetic instruction.

This review indicates rather general agreement by-

educators and mathematicians that arithmetic instruction should stress the mathematical and social phases of the subject. Authoritative committees point out that the mathematical aim can be achieved, only if meaning the fact that children see sense in what they do is made the central issue of instruction.

The social aim

helps to tie the interrelationships inherent in the number system to everyday life. It is evident from a review of the literature that there is general agreement that arithmetic instruction must be meaningful.

However, there is no general agreement on how to make it so. Brownell states this viewpoint clearly when he says: In general, it may be said that at present there is agreement that arithmetic must be taught meaningfully, though there are wide variations both in theory and practice with respect to the meanings which should be taught and to the procedures by which they may be developed. * Much of the research done in this area has been devoted to evaluation of ways of performing certain specific skills in arithmetic^

1. Brownell, William A., "Teaching of Mathematics in Grades I Thru VI," Review of Educational Research, Vol. 15, October 19h$* pp. 276- 2W !---------------------------------------------2. Monroe, Walter S. and Engelhart, Max D., A Critical Summary of Research Relating to the Teaching of Arithmetic, Bulletin No. 58. Bureau of Educational Research, College of Education, Urbana, the , University of Chicago. 1931, pp. 95-96.

or to vindicating specific positions as to learning theory.■** It should be kept in mind that there are large gaps in research in this area when a study of textbook materials is made. Literature in the field does offer suggestions for procedures to use in teaching arithmetic with meaning and understanding. However, research as yet appears to be inadequate to use as a guide for charting a clear course.

In 1931 Schorling and Edmonsen described rather thoroughly

the job which confronts the author of an arithmetic textbook series. Authors of arithmetic textbooks obviously are confronted with the problems of choice of material, sequence of topics, and manner of presentation— what, when, and how. Numerous investigations throwing light on the curriculum and learning have been utilized by authors in the solution of these problems. The general outcome of the research studies dealing with the choice of materials is a fairly definite trail, whereas a sifting of studies relating to grade placement and method yields conclusions that are fragmentary, conflicting, and incomplete. Indeed, the progress made in recent years in the teaching of arithmetic rests to a considerable extent on a psychological basis rather than upon research of fundamental, basic, and inclusive types. Apparently an author, when he wishes to glorify oldfashioned insight, intuition, and "horse sense," assures us that he follows ’The principles of a sound educational psychology.1* The viewpoint held by many educators on the relationship between methods and subject matter is expressed by Horn in this 1. McConnell, T. R., "Introduction," "The Psychology of Learning," Forty-First Yearbook, Part II. National Society for the Study of Education, The University of Chicago Press. 191+2, p. 10. 2. National Society for the Study of Education, "The Textbook in American Education," Thirtieth Yearbook, Part II. Public School Publishing Co., Bloomington, 111! 1931, P* 35*

manner, The particular method or methods to he used in any teaching situation must be determined partly by the nature of the students and partly by the nature of the subject matter to be taught. In discussing meaningful teaching or what and how to think he makes this statement: What to think and how to think should not be set in opposition, for knowledge and thinking are correlative rather than antagonistic. Knowledge in any real sense cannot be acquired without thinking and fruitful thinking is impossible without knowledge. In analyzing nine studies in methods of arithmetic Wheat sets forth the same ideas expressed by Horn. Wheat^ states that the authors of these nine studies pursued a common procedure.

They

produced a plan of methods to be used in teaching a particular topic in arithmetic, undertook to teach the topic according to plan and modified and developed the plan as the work was carried along. Note was taken of pupil responses.

The reactions of the authors of these

plans are perhaps more significant than the reaction of pupils in the experiments to the plan.

The authors looked for the reaction of

their pupils to the methods employed, but found themselves observing 1. Horn, Ernest, Methods of Instruction in the Social Studies. Charles Scribners Sons, New York. 1937, p. 6. 2. Horn., ibid., p. 105. 3. Wheat, Harry G., Studies in Arithmetic. Morgantown, Office of the President, West Virginia University. 19i|5, P* 16.

reactions to the content as it developed in the minds of their pupils. As Wheat says, "In short, their studies of how to teach were from first to last studies of what to teach."'*' Research in arithmetic was analyzed hy Monroe in 1919 from the viewpoint of determining from scientific studies principles that should be stressed in arithmetic instruction. His review of the studies in an article in the Eighteenth Yearbook of the National Society for the Study of Education states each principle briefly, and discusses its meaning and justification. Many of the principles advocated at that time by Monroe have been strengthened by more recent research. Most textbooks written after 1919 used these principles as guides for instruction. Monroe placed heavy emphasis upon the need and place of drill in arithmetic.

This emphasis was a

natural outgrowth of research of the time, as reflected in studies such as Kelly1s.^ Only in the last fifteen years has arithmetic instruction broken away from major emphasis on drill toward a more meaningful approach. 1. Wheat, Ibid., p. 16. 2. Monroe, Walter S., Eighteenth Yearbook, Part II, of the National Society for the Study of Education. Public School Publishing Company, Bloomington, 111. 1919, pp. 78-95. 3- Kelly, F. J., "The Results of Three TYPes °f Drill on the Fundamentals of Arithmetic," Journal of Educational Research, Vol. 11. November 1920, pp. 693-700.

Research in arithmetic to 1925 was summarized fay Buswell and Judd.'*' This summary includes a study of 320 investigations, few of which have bearing on the problem of method.

Since 1925 research

in arithmetic has been reviewed periodically in the Review of Educational Research and the Elementary School Journal. Kipp in 1931 noted a trend toward combining research in reasoning processes with research in the fundamental processes.

She investigated fifty-

seven experimental studies published between 1911 and 19l0 which compared method of teaching.

In 1931 Monroe and Englehart^ listed a

critical review of 128 studies related to the teaching of arithmetic. The studies dealing specifically with method did not emphasize any particular method, but were concerned primarily with short drill. In 193^ McConnell** reported the results of an approximately eight months investigation where one large group of second grade 1. Buswell, Guy T., Judd, Charles H., Supplementary Educational Monographs, No. 27, Summary of Educational Investigations in Arithmetic. University ofChicago. 1925. 2. Kipp, Minnie B., "An Investigation of Experimental Studies Which Compare Methods of Teaching Arithmetic," The Journal of Experimental Education, Vol. 13. 19hh, pp. 23-30. 3* Monroe, Walter S., and Engelhart, Max D., A Critical Summary of Research Relating to the Teaching of Arithmetic, Bulletin No.*38. Bureau of Educational Research, The University of Chicago. 1931* It. McConnell, T. R., "Discovery Versus Authoritative Identification in the Learning of Children," Univ. of Iowa Studies in Educ., Vol. 9j No. 5. Sept. 15, 193k) pp. 13-52^ McConnell, T. R., A Controlled Experiment in Learning of the One Hundred Addition ”* and the One hundred Subtraction Facts, Doctor’s dissertation., U. of Iowa. “I935. '

pupils learned the addition and subtraction combinations by engaging in activities which stressed authoritative identification, mixed practice and specific drill. Another like group followed procedures which stressed discovery, organization and generalization.

In a

series of tests of transfer to untaught processes the group using the meaningful procedures In learning the addition and subtraction facts showed differences that were favorable. At the end of a controlled experiment on learning one hundred additional facts Thiele^ compared the results obtained from two second grade groups on a transfer test of thirty addition examples. One group was taught by a method of specific repetition while the other was taught by a method which emphasized discovery and use of relationships existing among addition facts. At the end of the fifteen week experimental period the group taught by meaningful generalization surpassed the group taught by specific repetition. p "Wheat*- reviewed twelve studies in an attempt to find out whether an indirect method of instruction, characterized by a minimum amount of direct explanation and demonstration of processes, 1. Thiele, Carl L., "The Contribution of Generalization to the Learning of the Addition Facts," Contribution to Education, No. 763. Bureau of Publications, Teachers College, Columbia. 1938? p. 81*. 2. "Wheat, 0£. cit., p. 8-ll*.

could be employed successfully with pupils beginning new phases of instruction.

These studies as Wheat points out, give evidence that

pupils in the elementary school are capable of learning methods of self-instruction in arithmetic and of using such methods to teach themselves much of the arithmetic that is set down for them to learn. Brownell-** cites evidence of favorable results obtained in teaching borrowing in subtraction by using a meaningful method as opposed to a mechanical one. Two groups of third grade children were used in this experiment.

One group was exposed to experimental

situations needing a meaningful method of teaching. The other was taught by a mechanical approach.

Superior achievement was made by

the group using the meaningful approach. These representative studies give evidence that a meaningful approach in teaching the basic addition and subtraction facts and borrowing in subtraction gives favorable results over a mechanical approach. However, analysis of these and similar studies still leave grave doubts as to the best method of presenting these phases of arithmetic to children in a textbook. Another question raised often in the field of arithmetic is whether the apparent or increase-by-one method is best in teaching division to children. Two representative studies will serve to give some idea of the research done on this phase. 1, Brownell, William A., "An Experiment on Borrowing in Third Grade Arithmetic," Journal of Educational Research, Vol. 1*1. November 191*7, pp. l6l^L7r:

Morton-*- in a report on the relative accuracy of estimates of the quotient when the apparent and increase by one method were used with two figure divisors recommended the increase by one method for two place divisors ending in six, seven, eight and nine. Morton’s data show that when the apparent method is used for divisors ending in six, seven, eight and nine correct estimates will be obtained in only thirty-six per cent of the cases. Morton gives evidence that by using the increase by one rule estimates will be correct in seventy-nine per cent of the cases. Morton would use the apparent method with divisors ending in one, two, three, four and five and combine it with the increase by one method in divisors ending in six, seven, eight and nine. Osburn^ on the other hand recommends that only the apparent method be taught to beginners. He makes this recommendation on the basis of a summary of a series of studies which present an analysis of difficulties in long division. For a type of division using a one-digit quotient and a two-digit divisor he reports success of the apparent method in sixty-one per cent of the cases. 1. Morton, R. L., ’’Estimating Quotient Figures "When Dividing by Two Place Numbers,” Elementary School Journal, Vol. 1*6. November 191*7,

pp.

m-iw:

2. Osburn, Worth J., ”Levels of Difficulty in Long Division,” Elementary School Journal, Vol. 1*6. April 191*6, pp. llil-lltf.

The studies on method cited here show that there is a divergence of opinion by research workers on how various phases and processes of arithmetic should be taught meaningfully.

The findings

of these and similar studies on method do not outline a clear approach for textbook writers to follow in creating meaningful learning situations for children. As a result students of the teaching of arithmetic are forced to form their own judgments on methods to be used in creating learning situations on the basis of the relatively few existing studies in this area, or by making a logical analysis of the subject itself.

These sources along with a

review of the writings of educators, mathematicians and authoritative groups will yield some general principles that can be followed with a measure of confidence by authors of textbooks in creating learning situations for children.

The lessons developed for this study and

submitted to teachers for evaluation were built on principles suggested in these sources for creating learning situations for children. Two reports on trends in teaching done by Cooke and Stevens point out what they regard to be general principles utilized in the field today. Cooke-*- lists six principles inherent in a philosophy of arithmetic.

They are:

1. Cooke, Dennis H., "A Usable Philosophy in Teaching Arithmetic,” Mathematics Teacher, Vol. 1*1. February 19i*8, pp. 70-7it*

1. Introduce topics and concepts of arithmetic only when the child is psychologically ready for them. Eliminate socially useless material. 2. Practice and drill to maintain skills taught on topics that need practice and drill. Regulate amount of drill to worth of topic. 3- Recognize and provide for individual differences among students through diagnostic and remedial procedures. 1*. Approach problem-solving logically. Use a simple vocabulary in arithmetic. Word problems should have a vocabulary one grade below the arithmetical level. 6. Socialize and configure word problem material. Stevens-*- makes note of five procedures in reviewing trends in teaching.

These trends are:

1. There has been a marked trend toward greater emphasis on developing number concepts. 2. More time is devoted to concept building programs. 3. Children’s immature ways of dealing with number are recognized and accepted as valuable learning situations. 1*. There has been an increase in the amount of oral work given. 3>. The emphasis today is on individual or group teaching as opposed to class or mass instruction. A survey of the writings of educators and mathematicians and authoritative groups such as that done by Cooke and Stevens indicates a number of seemingly different principles proposed to be used as guides for building learning situations for children. An analysis of these sources reveals the fact that there is a large amount of repetition and overlap. Most of these can be combined into several major statements.

Spitzer has stated and applied to

1. Stevens, Marion Paine, "Teaching Arithmetic. Some Important Trends in Our Schools Today," Grade Teacher, Vol. 62. April 191*5, p. £1*. 2. Spitzer, Herbert F., The Teaching of Arithmetic. HoughtonMifflin, Boston. 191*8, pp. 2f-28*

learning situations five general principles that seem to cover the recommendations of most writers in this area. As stated by Spitzer the principles are as follows:^* 1. "...the first and most important principle governing the selection of learning experiences is that children must be given the opportunity to learn through experience." (Spitzer, ibid., p. 27) Applying this principle in its fullest sense Spitzer would have children participate in experiences which enable them to see that the methods they use are the best methods.

For children to

understand that numbers make for exactness and clarity of thought they must participate in experiences which demonstrate this exactness and clarity. In introducing a new process Spitzer does not advocate telling the child what to do. Rather would he set up a problem situation and allow the child time to figure out a solution to the situation by various methods understood by him even though immature. The exact or most efficient method of solution to a problem situation can therefore be discovered by the child with the teacher’s help through experiences of this type. The correct or most efficient process for dealing with a number situation discovered by a child using this method makes it the child’s own and therefore more meaningful to him. 1*. Statements numbered one through five and enclosed in quotation marks are taken from pages 27-28 in The Teaching of Arithmetict by H. F. Spitzer, ibid.

T McConnell describes the same principle and its application in this manner: Meaningful learning emphasizes discovery and problem solving. In fact from this point of view learning is thinking. Instead of learning "facts" and then using them in thinking, we can learn "facts" by thinking. This doctrine means that learning should be characterized by insight, and it sharply condemns the traditional practice in arithmetic of having children memorize certain operations in abstract form, in order to apply them in verbal problems afterwards. ...Instead of authoritatively identifying correct responses for children, courageous teachers are now encouraging active exploration and discovery and self-directed learning. Buckingham^ follows this principle by pointing out that

the essential thing in teaching arithmetic is to give a child first hand experiences out of which he may build number ideas for himself. 2. "...Basing of the selection of part of the content of arithmetic on the nature of the number system is the second principle to be used in the selection of learning experiences." (Spitzer, o£. cit.) Brownell^ speaks for most writers in the field when he

says, ...the test of learning is not mere mechanical facility in "figuring." The true test is an intelligent grasp upon number relations and the ability to deal with arithmetical situations with proper comprehension of their mathematical as well as their social significance. 1. McConnell, T. R., "Recent Trends in Learning Theory," Sixteenth Yearbook, National Council of Teachers of Mathematics. Bureau of Publications, Teachers College, Columbia University. 191*1, p. 281*. 2. Buckingham, B. P., "Arithmetic as a Contribution to a Liberal Education," Elementary School Journal, Vol. 39. April 1939, pp. 577-583. 3. Brownell, William A., "Psychological Considerations in the Learning and the Teaching of Arithmetic," Tenth Yearbook, National Council of the Teachers of Mathematics, op. cit. 1935, p. 19.

Thiele agrees with this principle when he says, ...Arithmetic is more than a set of specific skills and facts— it is a mode of thought resulting from an appreciation of and an awareness of the many interrelated elements within the system of number. 3. "...The principle of familiarity. This principle states that familiar situations shall be used for the settings in which are presented the phases of number to be taught. Familiar situations are recommended because they offer the maximum possibilities for the child to gain understanding." (Spitzer, oj>. cit.) Stroud in discussing the topic of making arithmetic meaningful to children emphasizes this point: The only way to make instruction meaningful is to make the presentation in terms of what the pupil can understand. On this point there is nothing to be added to the second formal step of Herbartianism— instruction proceeds from the familiar to the unfamiliar. All understanding is predicated on previous understanding** U. "A fourth principle recognizes that each number fact has many significant relationships with other number facts." (Spitzer, ibid.) Spitzer stresses the fact that efficient learning of these many relationships requires a relatively long period of instruction with provision made for essential experiences for the teaching of each number fact. Use of concrete materials is Important to a correct interpretation of this principle. 1. Thiele, C. L., "The Mathematical Viewpoint Applied to the Teaching of Elementary School Arithmetic," ibid., p. 230. 2. Stroud, James Bart, Psychology in Education. Longmans Green and Company, New York. 19u6, p. ±9117

Brownell^ states that many of the difficulties which children have in habituating combinations such as 7 + 5 * 12 are due to causes which relate to initial stages of learning.

The pace set

for the learner is too fast to allow time for understanding. As stated earlier Stevens in reviewing trends in teaching notes the fact that there is a marked increase in time and attention given to concept building programs.

Cooke says that concepts and

problems should be introduced only when the child is psychologically prepared for them. 5>. MA fifth principle states that generalizations grow out of experience. A child must have experiences out of which he can construct generalizations." (Spitzer, op. cit.) In discussing classroom methods Thiele^ states that experiences of manipulation, observation and recording will culminate with the making of deductions or the discovery of generalizations which represent insight into the number system. Spencer made the observation that, "Children should be led to make their own investigations and to draw their own inferences. They should be told as little as possible and induced to discover as much as possible."3 1. Brownell, William A., "Psychological Considerations," Tenth Yearbook, op. cit., pp. 2l;-25>. 2. Thiele, C. L., "Arithmetic in General Education," Sixteenth Yearbook, National Council of Teachers of Mathematics, og. cit., p. 72. 3* Spencer, Herbert, Education. D. Appleton Century Co., New York. 1920, p. 120.

Anderson points out that, "...the principles to be observed in learning arithmetic are discovery, generalization, and testing. The pupils should make for themselves the discoveries which lead to generalizations."-** It is difficult to draw many definite conclusions from articles of the type reviewed here. However the trend evidenced by these educators and mathematicians toward acceptance of major principles for use as guides in building learning situations leads to the assumption that they are worthwhile.

For that reason the

writer used these principles as an identification of the best theory and as guides in building experimental lessons to be submitted to teachers for comparison and evaluation with lessons in the textbook used in their school system.

1. Anderson, Lester G., "Recent Trends in Learning. Their Relation to the Psychology of Arithmetic," National Education Association, Addresses and Proceedings, Vol. 78. 19i*0, p. 373-

Chapter III PROCEDURE AND MATERIALS Introduction The primary aim of this investigation was to make a study of two aspects of textbook materials in the field of arithmetic for the purpose of evaluation and improvement.

In order to accomplish

this purpose two separate investigations were conducted.

These were

Investigation I. To study teacher use of arithmetic textbooks in selected classrooms. Investigation II. To detenaine teacher reaction to methods of presenting arithmetical textbook materials that are more nearly in harmony with the best theory than are the methods of presenting material now used in published textbooks. Investigation I Part A In order to study teacher use of arithmetic textbook materials Investigation I was broken down into two parts called Part A and Part B. For Part A the investigator visited fifty selected elementary arithmetic classrooms in six different school systems in Iowa. The school systems visited were selected on this basis:

(a) Permission for the investigator to visit the schools was given willingly by the superintendent,

(b) The schools fell within a

radius of one hundred miles of Iowa City. Each superintendent of schools in the various systems visited was contacted first by telephone. During this telephone conversation the purpose of the investigation was explained and permission was requested for the investigator to visit arithmetic classrooms in that school system. If the superintendent gave permission willingly, he was asked to set a date for the observer to visit in the system. The date for the visit and the telephone discussion were later confirmed by letter. On the day set the investigator arranged for a conference with the superintendent approximately one hour before school opened. During this conference the purpose of the study was again explained to the superintendent and he outlined to the observer the list of schools to be visited during the day. The choice of schools was left entirely up to the superintendent and the observer followed exactly the program outlined by him. The same procedure was followed in each of the schools visited. The observer first contacted the principal and followed exactly the program of classes he suggested visiting. Each teacher had been told by the principal prior to the visit that the observer was interested in seeing how arithmetic was taught in that grade.

During each of the observations made in the fifty classrooms the observer was seated at the back of the room.

The observer took

notes during the lesson and these notes were later written in the fonn of a report to show the procedures used during the class period. Special attention was given to three factors by the investigator: (a) How the textbook was used by the teacher, was used by the pupils,

(b) How the textbook

(c) "What equipment was used during the

lesson to interpret the textbook.

An example of the observations made of the report of a lesson observed in a sixth grade arithmetic class is presented here: Grade 6

11:10— 11:25

20 children

The teacher asked the class to turn to page 58 in their arithmetic textbooks. This page contained examples of long division with two place divisors. She placed the model problem shown in the book on the blackboard. The teacher then asked one child to came to the blackboard and work the problem while the other members of the class listened. Mien the pupil had completed the problem the teacher asked the class to compare his work with that in the textbook to see if it was correct. The attention of the class was then called to the model problems worked out in the book on page 59* The teacher asked several children to come to the blackboard and re-work these model problems for the rest of the class. Their work was compared with the textbook when they had finished. All model problems on page 58 and 59 of the textbook were completed in this manner. A complete record of the fifty observations made in arithmetic classrooms is included in Appendix I. These observations are grouped according to grade level and numbered consecutively. An analysis of them is made in Chapter IV.

^8

At the end of each observation the investigator talked "with the teacher for a few minutes. During this time the purpose of the study was explained to the teacher and she was asked if she would be willing to take time during the next week to express her opinion of the arithmetic textbook materials she used. If the teacher was willing to do this, the investigator left with her the questionnaire guide sheet developed for Part B of Investigation I. The teacher was asked to express her opinion of the textbook materials she used on this questionnaire and mail it to the investigator unsigned in a stamped addressed envelope provided for that purpose. Part B For Part B of Investigation I the writer developed a questionnaire in the form of a guide sheet to aid teachers in expressing judgments about the arithmetic material used in their classrooms. A copy of the guide sheet appears in Appendix II. The guide sheet was developed by the writer after visiting and observing twenty-five elementary arithmetic classes in the schools of Akron, Ohio in the fall of 19h9* At the end of each of these observations the writer had a conference with the teacher about the textbook used during the lesson and took notes regarding the teacher’s opinion or judgment of its worth. An analysis of the notes taken during these conferences led the writer to make the following assumptions:

1. Most teachers need some guidance in the form of questions before they can express clearly their opinion of textbook material. 2. Most teachers are willing to take time to express opinions and judgments about textbook materials.

However, material presented

to them will be given more consideration if it is concise. 3. Most of the teachers interviewed discussed specific points about textbook material they used.

These included:

a. How the textbook introduced new facts and processes. b.

The amount and kind of practice exercises and

material included in textbooks. c. The teaching suggestions advocated by the textbook for developing understanding. The questionnaire guide sheet (see Appendix II) developed from an analysis of these conferences, to aid teachers in expressing their opinion of arithmetic textbook material, takes into account these assumptions. An analysis of this guide sheet will show that the questions are formed to guide the teachers into thinking about various aspects of textbook materials as well as to get their opinion of the points under consideration. The guide sheet questionnaire was limited to two pages in order to keep within reasonable time limits. An analysis of the individual questions shows that the points discussed by the majority of teachers, as brought out in the test conferences, are included in the questionnaire.

For example: Question six asks the teacher’s

rro opinion about the way the textbook introduces new facts and processes. Question seven asks about the adequacy of the practice material. Question eleven asks about the worth and use made of the teaching suggestions advocated in the textbook. The questionnaire guide sheet was left with the fifty teachers the investigator visited while conducting Part A of Investigation I. Each teacher was requested to read the guide sheet carefully, to think about the questions for a few days, and to base her judgment and opinion on the arithmetic textbook material she was most familiar with through its use with children.

These instructions

were given because in many schools the fourth, fifth, and sixth grade arithmetic classes are taught by one teacher. "When that was the case, that teacher was asked to judge the material for the grade level she worked with most. The questionnaire guide sheet for Part B of this investigation provided no opportunity for the teachers to name the arithmetic textbook being judged. The name of the textbook series was omitted to insure a more objective judgment by the teachers. However, the various arithmetic series used in the classrooms visited ranged in publication date from 1938 to 191*8. The majority of the books judged by the teachers were published after 191*0. The results of the questionnaire were tabulated and a comparison made between those suggestions given by teachers of the different grade levels in the elementary school.

The results of

*s

this tabulation and comparison appear in Chapter IV. Investigation II The purpose of Investigation II was to determine teacher reaction to methods of presenting arithmetical textbook materials that are more nearly in harmony with the best theory than are the methods now used in published textbooks.

This purpose requires the

identification of best theory. The procedures and materials used by the writer to carry out this purpose are described in the following section of this chapter. In order to present a clear picture of Investigation II a step by step account of the methods and procedures employed by the writer to develop and carry out the various phases of the investigation are described. 1. A careful review of the literature in the field of arithmetic was the procedure used to determine, in so far as possible, best theory. An analysis was made in the review of the purposes of arithmetic, statements of theory, examples of method, and suggested principles for use as guides in creating learning situations for children advocated for use in the writings and research done by a representative group of educators, mathematicians, and authoritative committees* The results of this review are presented in Chapter II of this study.

2« As a second step the writer surveyed several typical school systems in Iowa to find those which used an arithmetic textbook series published after 191*0. As a result of this survey the writer found that three large school systems within a radius of one hundred miles of Iowa City used books published after that date. These systems followed the same plan, in as much as,they used one textbook series published in 191*1* in grades one and twoof the elementary school and another series published in 191*8 in grades three, four, five and six. Davenport, Iowa was among the school systems surveyed and it was selected as the system in which the investigation would be conducted. Davenport was chosen because the Director of Elementary Education and the Coordinator of Elementary Education expressed an interest in the problem and said that they felt the teachers would be interested in the study and willing to cooperate in conducting it. 3.

The 191*1* textbook series used in grades one and two and the

191*8 series used in grades three, four, five and six in the Davenport schools were then compared and analyzed by the writer with two other arithmetic textbook series published in 191*6 and 191*8. The purpose of this comparison and analysis was to determine specific phases of arithmetic content presented in recent textbooks at the various grade levels. This analysis showed that the following phases of arithmetic were among those commonly presented at the indicated grade levels in all the textbooks:

tr*

Grade I Reading Numbers One Through Six Reading Number Names Grade II Recognition of the Teens Numbers 11— 19 Understanding the Meaning of Ten Recognition of When to Add and When to Subtract Grade III Addition Combinations of Eight Subtraction of a One-Place Number from a Two-Place Number Multiplication of a Two-Place Number by a One-Place Number Grade IV The Meaning of Larger Fractions Learning to Use Groups of Six Reman Numerals Grade V Adding Like Fractions Multiplying by Two-Place Numbers Subtracting a Fraction from a Mixed Number Grade VI Adding Decimals Multiplying Whole Numbers by Fractions Dividing a Whole Number by a Fraction 1;. The seventeen aspects of arithmetic listed in the preceding paragraph constituted the arithmetical content presented to teachers for suggestions and evaluation.

To do this the specific pages at

each grade level on -which this content was presented in the arithmetic textbooks used in Davenport was determined. For example: The text­ book used in the second grade In Davenport presents a lesson on The

Teens Numbers 11— 19 on page

The fourth grade textbook presents

a lesson on Working With Groups of Six: on pages 15U-15& • Each of the pages on which the seventeen aspects of arithmetic were taught in this series at the various grades levels were identified in this manner. To serve as a basis for teachers to use in judging the methods of presenting arithmetical textbook material and this content the writer developed seventeen experimental lessens to be contrasted at each grade level with similar ones in the textbook.

These

lessons were built, in so far as was reasonably possible, on the principles suggested for use in creating learning situations for children and stated in the review of literature in Chapter II of this study. Because the lessons are based on these principles they are assumed to be more nearly in harmony with the best theory than the methods used in published textbooks. These seventeen lessons are included in Appendix III of this study. An examination of the lessons will show that there were two developed for the first grade, and three developed for grades two, three, four, five and six of the elementary school. It will be noted that each experimental lesson at each grade level is matched to a similar lesson in the textbooks published in 19i^ and 191$ and used in the Davenport schools. 5* The primary purpose of these lessons was to serve as a basis of comparison for teachers in judging methods of presenting textbook

material in published textbooks as against experimental methods more nearly in harmony with best theory.

To determine teacher reaction

to the textbook and experimental lessons and to get teachers’ opinions and recommendations for change or improvement in arithmetical material directions, guide sheets and questionnaires were developed by the ■writer and included with each of these lessons. An examination of the experimental lessons in Appendix III will show that each set of experimental lessons for the six grade levels is accompanied by green guide sheets labeled I and II.

Guide sheet I contains directions to

teachers for comparing the textbook and experimental lessons and suggests ways they can make recommendations for change or improvement in textbook materials in light of their classroom experience. Guide sheet II on each of these sets of lessons contains a questionnaire to be filled in by teachers after judging the textbook and experimental lessons for the grade level at which they teach.

This

guide sheet questionnaire asked the teachers to state their preference for either the textbook or experimental lessons in terms of use in the classroom and in regard to adaptability, motivation, interest, needs, techniques and ability to get arithmetic concepts across to children.

The recommendations of the teachers and the results of

this questionnaire have been tabulated and analyzed in Chapter IV of this study. An examination of the seventeen experimental lesson in Appendix III will also show that a short questionnaire was included

at the end of each lesson.

The teachers were asked on this

questionnaire to state their preference for the textbook or experimental lessons in terms of four specific questions.

These

questions point out differences in the textbook and experimental lessons.

The teachers were also asked to tell what they liked and

disliked about each of these lessons, to state briefly the reasons for their decision, and to recommend changes or improvements in the material they feel would help them teach arithmetic more meaningfully. The results obtained on these short questionnaires have been tabulated and analyzed in Chapter IV. 6.

These arithmetical materials and questionnaires were

submitted for evaluation to the teachers of grades one through six of the Davenport, Iowa schools.

The teachers of grades one and two

judged the lessons at their grade level.

The fourth, fifth and

sixth grade lessons were submitted to the arithmetic teachers at those levels. Davenport like many large school systems in Iowa has on the average only one arithmetic teacher in the upper grades. In these cases the teacher was asked to judge all the lessons for the grade levels taught. The writer did not come in contact with any teacher in the Davenport system during the time the lessons were being considered and evaluated.

The procedure was explained by the writer and

Director of Elementary Education to the principals of the eleven Davenport schools participating in the study. The principals

distributed all of the material to the teachers. The teachers were given two weeks in which to consider the materials and state their opinions and recommendations. At the end of that time the principals collected the material and returned it to the central office. Approximately 115 sets of lessons were distributed to the teachers in these eleven schools. A total of ninety-eight questionnaires and lessons were returned to the writer.

The

tabulation and analysis of these data appear in Chapter IV of this study.

Chapter IV RESULTS

Investigation I

To study teacher use of arithmetic textbook materials In selected classrooms was the purpose of Investigation I of this study.

This investigation was conducted in two parts called Part A

and Part B.

Part A is summarized in the following section of this

chapter.

Part A In order to make a study of textbook materials in the field of arithmetic for the purpose of evaluation and improvement it is necessary first of all to determine in so far as possible the facts concerning the use of those materials under existing classroom conditions.

To determine the prevailing conditions in regard to the

use of arithmetic textbook materials the observational type of survey was used by the writer as a part of the normative-survey method of research. For part A the investigator visited fifty elementary arithmetic classes in six different school systems in Iowa.

All

data for this section were obtained by direct observation of actual classroom practice. of three factors:

During each observation special note was taken (a) How the textbook was used by the teacher.

(b) How the textbook was used by the pupils,

(c) What equipment was

used during the lesson to interpret the textbook.

In addition to

these factors notations were made of the procedures used to initiate, to develop and to summarize instruction.

Samples of questions asked

by the teachers and responses made by the pupils were also noted. These data appear in the form of reports of the work observed in the first six grades of the elementary school and are numbered consecutively and presented in Appendix I of this study. Six representative examples of the reports, one from each grade level, are presented here to show typical findings for the three factors noted during each observation by the investigator. These factors were:

(a) How the textbook was used by the teacher.

(b) How the textbook was used by the pupils,

(c) What equipment was

used during the lesson to Interpret the textbook.

Grade 1

2:30— 3:00

31 children

The teacher wrote the number 3 on the blackboard and the number word three beside it while the class watched her. Then she said, "John pick out three boys and bring then to the front of the room. Mary pick out three girls and bring them to the front of the room." When the children were standing in groups of three at the front of the room the teacher asked the class to help her count the boys and then the girls. The teacher then asked, "Did John and Mary bring the correct number of boys and girls to the front of the room?" The class answered, "Yes." The teacher called on several children and asked them to count various things in the room. She said, "Bill, will you go to the back of the room and touch three chairs? Count them aloud as you do this." When a child completed this exercise by himself the teacher would ask the rest of the class to count them again with him.

The attention of the class was then directed to the blackboard and the number 3 and the word three. The teacher asked the class to watch as she pointed to the number three. She asked them to say the word three as she pointed to it. As the next step the teacher wrote the following numbers on the blackboard: 3 1 2 3 1 3 2. She asked three different children to come to the blackboard and draw a circle around one of the three's written there. Following this exercise she wrote several number wrords on the blackboard. The words were: three one three one three. She asked 3 of the children to come to the blackboard and draw a line under the word that said three. When they had finished she asked the class if they had drawn lines under the correct words. The class answered, "Yes." The teacher then wrote this exercise on the blackboard:

Three Two Cne

1 3 2

She asked different children to come to the blackboard and draw a line between the word and the number that said the same thing, then the exercise was finished the teacher asked the class to count the numbers and say the words she pointed to on the blackboard.

Grade 2

11:10— 11:30

35 children

The teacher wrote the number 118 on the blackboard. She asked the children to look at the number she had placed on the blackboard and find that page in their textbooks. The teacher asked the class to read each sentence in the number story carefully. She called on individual children to read each sentence and to ansYfer the question it asked. The problems were addition and subtraction and Ydien a child had difficulty figuring out the ansvrer or didn't know it the teacher would help him. She helped him by figuring out the answer on a home made number board made of rope and clothes pins. She would show six on the board and then take away three of the clothes pins and explain to the class that three were left. Therefore, said the teacher, "Six take av/ay three leaves three." The class completed the page Y/ith the teacher's help in this manner.

The teacher asked the class to take out a piece of paper and their pencils. She asked them to turn to page 121 in their

textbooks and read each problem carefully. When they had read the problem they were to work out the solution on their papers. She asked the class to raise their hands if they needed help.

Grade 3

9:05— 9:35

23 children

The teacher placed the following examples on the blackboard 12

Ik

16

13

-6

-7

-8

-9

She used flash cards stacked in piles on the chalk tray to illustrate each of the examples for the class. For example— she placed 12 cards on the tray and then took away six. She counted those remaining and made the statement to the class that, "12 take away 6 leaves 6 ." When she had finished illustrating each example she asked for questions. The class had none. The teacher then asked the class to turn to page 88 in the textbook. She told the class that since today was the first time they would meet entirely new problems they should try hard to complete them. She asked them to do all the story problems on pages 88-90 in their textbook. She asked those children who might have difficulty to raise their hands and said that she would come to their desks to help them. She told the class that when this work was finished they should try the examples at the bottom of page 90 in their textbooks.

Grade ij.

9:30— 10:00

31 children

The teacher asked the class to turn to page I48 in their arithmetic textbooks. She called upon one child to read each problem and another to give the correct answer. The problems involved multiplication by eight. The page was completed in this manner. The teacher spent a few minutes of the class period calling out numbers and then asking individual children to multiply them by eight. The teacher asked the class to turn to page i;9 in their textbooks. She sent seven children to the blackboard and asked them to work the first seven problems on the page. The page contained examples of two place multiplication. Vvhen the children had finished these problems the teacher reviewed each example and corrected the mistakes made. When she had finished she asked if

any pupil had a question about the work. from the group.

There were no questions

The teacher then asked the class to take out clean sheets of paper and their pencils and work the remainder of the problems on page h$ of their textbooks. She moved about the room giving help when it was needed.

Grade 9

9:00— 9 •k!?

28 children

The teacher asked the class to open their textbooks to page 63. She placed the first problem (uneven long division) on the blackboard and worked it for them explaining each step as she Trent along. She then asked the class to work the first eight examples on the page. She instructed them to look at the blackboard if they had any difficulty. Ihen the majority of the class had completed this assignment she called on individual children to state the examples and give the correct answer. The remainder of the class checked their own papers. The teacher asked the class to turn to page 3h in their textbook and explained that they were going to discuss measures. She asked a child to name as many different measures as he could think of. He named measures of length and measures of music. The teacher then replied that they could find out other measures by looking at the abbreviations given on page 3h of their textbooks. She called on different children to tell what each abbreviation (lbs., ounces, inches, etc.) listed on the page meant. She asked the class to work the problems listed on thatpage and when they had finished to start the work on page 176 in their textbook.

Grade 6

9:20— 9:50

18 children

The teacher asked the class to turn to page 133 in their textbooks. They were asked to look over the problems on this page very carefully. The teacher took three problems from this page and placed them on the blackboard. She worked them orally for the class explaining each step as she went along. The problems were review work in addition and subtraction of fractions. The class was then directed by the teacher to page 3h in their textbooks. This page had 6 model problems on it involving 6ths and lOths and showed how to change these problems to a common denominator. The teacher went over each model problems as worked out in the textbook and explained them step by step to the class.

"When she had completed the explanation for the model problems she asked three different members of the class to re-explain the last three problems. All three children explained how they could arrive at the answer shown in the book. The teacher then placed the following fractions on the blackboard. 16/30 28/30 6/3O 3/30 She made this statement, "Since we have gone over the model problems in the book we should be able to change these fractions I have placed on the blackboard into l5ths. Divide each of the numbers on the blackboard by a common denominator so that you can change them into l^ths." When the majority of the class had completed this work the teacher asked them to do Row 1 on the next page of their textbook.

Analysis of Observation Reports for Part A of Investigation I

Grade I An analysis of the four reports of observations made in the first grade arithmetic classes shows that only one of the teachers used a textbook during the lesson. Appendix I.)

(See report I4. in

The book used was a large number readiness book and

only one copy was available.

This copy was handled by the teacher

who used the illustrations in it to teach the children to recognize groups of five.

Recognition of numbers ’was part of the other first

grade lessons observed but no textbook was used in these classes. The limited use of the textbook in the first grade reports may be accounted for by time of year.

These observations were made during

the first semester of the 19l|9-195>0 school year.

Most of the

teachers reported that they began use of a textbook with their first grade classes at the beginning of the second semester.

A large amount of active pupil participation in the learning activities was evident in each of the first grade lessons. Each lesson observed had a number of different activities of short duration for the children to take part in.

Most of these activities

involved the use of concrete objects such as books, blocks or chairs with which the children were familiar.

In only one case, lesson U,

were illustrations substituted for concrete objects to teach the cardinal aspects of the numbers under consideration. All of the first grade teachers made use of the blackboard and chalk in these lessons.

The other equipment used to make the

lessons more meaningful consisted of concrete objects found in most first grade rooms.

These objects were:

books, blocks, chairs,

groups of pictures on bulletin boards and coins.

Grade 2 The reports of observations made in second grade class­ rooms (see reports 5 - 12 in Appendix I), show that six of the eight teachers used an arithmetic textbook at some time during the lesson. Reports $, 6, 8 and 12 show that four teachers began the arithmetic lesson by asking the class to turn to a specific page in the textbook.

The teachers in reports,5 and 12 also ended the

lesson by asking the children to work problems from the textbook. In reports 10 and 11 the teachers ended the lesson by assigning a page from the textbook.

Two of the second grade teachers did not

use a textbook during the class period observed. An analysis of these reports indicates that in the majority of the classrooms either the teacher or a child chosen by the teacher read the problems in the textbook aloud to the class and then different children were asked to give an answer.

If the children had

difficulty giving a correct response, the teacher usually went to the blackboard and demonstrated the correct process to use.

In some

cases individual children were asked to do this demonstrating instead of the teacher.

In report 5> the teacher used a home made number

board to demonstrate the process rather than the blackboard.

In

report 11 the teacher spent some time developing an introduction to a lesson in the textbook which the class was going to work.

This

introduction was the teacher's idea and was not a part of the text­ book lesson.

In most cases the teachers deviated very little from

the textbook, and the lessons consisted of a question and answer type of recitation.

In reports 7 and 9 the teachers did not use a

textbook but made use of problem

situations requiring the use of

concrete objects to arrive at number solutions. The reports show that in most instances

the children used

the

textbook as a guide to follow in working with the teacher on

the

development of a lesson.

On the ’whole their work consisted of

reading the problems in the textbook

and figuring out solutions for

these problems under the teacher’s direction. The children had little opportunity to figure out

solutions by themselves as in most cases either the textbook or the teacher told them how to think through a problem to arrive at the correct solution.

More active participation by children in learning

situations was evident in reports 7 and 9 where a textbook was not used than in those reports which mention the use of a text.

These

reports seem to indicate that children are given little opportunity to engage in first hand experiences with number when the textbook is being followed closely. Analysis of the reports show that most of the teachers made use of the blackboard and chalk to demonstrate processes.

In those

classrooms making full use of the textbook the teacher and pupils relied heavily on the use of paper and pencils to solve problems. Concrete objects such as books, blocks, and clocks were used in a few classes to demonstrate processes.

Report 5 points out that a

home made number board was used by the teacher to show the meaning of subtraction.

Grade 3 An analysis of the eleven reports of observations made in third grade classrooms (see reports 13 - 23 in Appendix I), reveal the fact that eight of the teachers used an arithmetic textbook during the lessons.

Three teachers did not make use of a textbook.

Reports 15>, 1?, 21 and 22 show that in these lessons the teachers made use of the illustrations found in the textbook.

In

lesson 22 the teacher asked the class to use colored blocks to make pictures of 2 and 9 like those shown in the textbook.

In reports 17

and 21 the teachers asked the class to use the illustrations in place of concrete objects. do you see in the picture?"

The teacher in report 15 asked, ,rWhat In this case the class did not respond

and the teacher explained the illustration to the group. In the majority of these reports in which a textbook was used the teacher read the introductory material to the children and demonstrated each process the textbook lesson presented.

Most of

the class time was taken up by the teacher in reading, explaining, and demonstrating how to solve the examples and problems in the textbook.

In many cases the lesson began and ended with a page

assignment from the text.

Very little deviation from the textbook

by the teacher was evident in the lessons.

Even though a textbook

was not used in reports 19 and 20 the teachers used the same type of teaching techniques as those teachers did who made use of a textbook.

Lesson 23 reveals an attempt on the part of the teacher

to correlate arithmetic with social studies. Where textbooks were used the reports indicate very little active participation on the part of the children in solving problems.

In most of the reports the children read the problems

given in the textbook and then tried to work out solutions by following step by step the directions given in the text.

Little

opportunity was provided for the children to discover for themselves

through experiences in working with numbers, relationships and generalizations. Most generalizations were stated for the children in the textbook lessons and the examples and problems they worked served as explanations or practice material. as:

Since assignments such

"Work all the story problems on pages 88 - 90," (report 13) or

"Work the first two rows of examples on page 9 $ n (see reports lit., 17 and 22), were common it is evident that the children and the teachers use the textbook in many cases as a workbook. The reports show that the teachers used the blackboard and chalk as part of each lesson.

When assignments from the text­

book were given to the class the teacher usually asked that the work be done with paper and pencils.

In reports 16 and 18 the teachers

used a clock to demonstrate time. were used in two of the classrooms.

Colored blocks and flash cards Report 21 reveals that the

teacher set up a play store and provided paper money and articles such as books, bells and balls for the children to buy.

These

reports seem to indicate that the teachers of the third grade classrooms did not use much special equipment to interpret the textbook.

When concrete objects were needed those available in the

room were used.

Grade k The reports of observations made in the fourth grade classrooms (see ??i-33 in Appendix I) point out the fact that nine of the ten teachers used an arithmetic textbook at some time during the period. An analysis of the fourth grade reports indicates that the textbook was used as a guide by the teachers for introducing new material, for developing concepts and as the source of assignments for written work. These reports reveal that on the whole the teachers deviated little from the pattern set by the textbook. Report 25 indicates that some teachers do make use of situations involving number that arise in the classroom.

The techniques used

by this teacher to explain multiplication of the two place numbers were of the same type as those suggested in current textbooks. The pattern followed by the fourth grade teachers observed seems to be, to present a problem or example to the class, to explain in detail the steps to follow to arrive at a solution, to drill on the steps and then to give the pupils an opportunity to ask questions and to work through several examples. Despite the fact that in most of the classrooms every child was provided with a textbook the teachers on the whole repeated orally material that was presented in each text. In report 31 the teacher attempted to provide for individual

differences. However, it is evident from these reports that on the whole each pupil in the roam was exposed to the same material presented in the same way. Little attention was given to caring for individual differences in most of the classrooms observed. Analysis of these reports show that the blackboard, chalk, paper and pencils constituted the bulk of equipment used to interpret the textbook. Flash cards were used for drill on multiplication facts in one room and a clock was used in another to aid in telling time.

The teaching suggestions in the textbooks used evidently did

not show teachers how to use concrete materials to enhance under­ standing or did not convince the teachers that materials of this type had a place in the classroom.

The reports point out the fact that

the supply of equipment for interpreting the textbook was limited or non-existent in most of the classrooms. Grade 5 An analysis of the nine reports of observations made in fifth grade arithmetic classes (see 3U-U2 Appendix I) shows that eight of the teachers used the textbook during the class period.

In

report 37 no textbook was used. In each case in which the textbook was used the teacher either read the introductory material to the class or asked that the children read it. Report I4I illustrates a typical lesson in which the teacher and pupils used the textbook to master the use of a new

process.

In this lesson the children did not show a clear under­

standing of the process presented.

These reports reveal that most

of the teachers in an effort to clarify children's understandings followed up textbook explanations with more of the same type. These explanations did not seem to meet pupils' needs. When the teaching suggestions advocated in the textbook fail to guide teachers in directing children's learning activities it is evident that no better type of instruction in arithmetic than that shown in report Ijl will be used in classrooms. These reports point out that teachers use the textbook and depend on it to clarify concepts and develop understanding in the use of processes on the part of pupils. One textbook is used in most arithmetic classes and these reports indicate that it sets the pattern for instruction. From these reports it is evident that the children were given little opportunity to have first hand experience with number. In most cases the steps to use to arrive at the solutions to various types of problems were explained to the children either by the textbook or the teacher.

Inherent relationships that exist between

processes were pointed out to the children and they were expected to make use of them in their thinking. Very little provision was made to insure a maximum amount of understanding of these relationships in terms of various processes.

Pupil participation consisted on the

whole of responses to questions asked by the teacher or to attempts to solve with paper and pencil assigned problems. The blackboard, chalk, paper and pencils, as was true in other grades observed,made up the bulk of equipment used to interpret the textbook. situations.

In most cases the children dealt with abstract number In report 36 the teacher provided paper money for the

class to use in making change. Reports 38 and i>2 show that pictures and illustrations were used by the teacher in an attempt to clarify concepts. Grade 6 An analysis of the eight reports of observations made in sixth grade arithmetic classes (see reports k3-5>0 in Appendix I) reveal that all of the teachers used a textbook at some time during the lesson. The reports of the observations show that on the whole the teachers followed closely the work as outlined in the textbook. In reports

h73 h9 and 50 some explanation of the work to be done

during the class period was given by the teacher before using the textbook.

In reports i;3 and Iji* the model problems used as examples

in the textbook and worked out completely for the children were re­ worked by the teacher or at her suggestion by the pupils. Pupil use of the textbook consisted mostly of the class working out solutions to problems in the textbook which had been

assigned by the teacher.

In a few cases children were asked to go

to the blackboard and show the others the steps they had used to arrive at a solution to a given problem. In all instances the children were given an opportunity to ask questions about textbook problems. If a child had a question the teacher either referred him to the explanation given in the textbook or demonstrated the correct process to use on the blackboard. The blackboard, chalk, paper and pencils constituted the equipmentr used in these classes to interpret the textbook. Part B For Part B of Investigation I the writer developed a fifteen item guide sheet questionnaire to aid teachers in expressing judgments about the arithmetic textbook material used in their classrooms.

The questionnaire is included in Appendix II of this

study. At the end of each of the fifty observations made in elementary arithmetic classrooms for Part A of this investigation the writer discussed this questionnaire with the teachers and asked that they use it to express their judgments and opinions of the arithmetic textbook material used in their classrooms. The teachers were asked to circle either yes or no in answer to each question. They were encouraged to make suggestions for changes in the arithmetic materials they were judging that they felt would help them improve

their teaching of arithmetic.

The results of the tabulations are

reported in Table 1. Interpretation of Table 1 Table 1 represents the results obtained from the questionnaire (see Appendix II) left with the fifty teachers participating in Part A and Part B of Investigation I. Each item on the questionnaire was tabulated and reported separately. Teacher comments and suggestions made under the various items were copied verbatim from the data and compiled under e^ch question by grade level.

Table 1 is read as

follows: Question 1 asked the teachers, "Is each child in your room provided with a textbook?" Reading Table 1 from left to right the tabulations show that answers to this question were obtained from teachers in grades one through six. Fourteen per cent of the teachers, seven in all, answered no to the question. Eighty-six per cent, or forty-three in all, answered yes. Examination of the tabulations for each grade level reveals that of the forty-three teachers answering yes to the question four were first grade teacher, eight second grade, nine third grade, eight fourth grade, nine fifth grade and five sixth grade teachers. Hone of the teachers reporting commented about question 1. Table 1, question 2 is read in the same manner as question 1 with one exception.

Teacher comments are summarized by grade level

at the bottom of the tabulation.

These comments are the actual

statements made by each teacher at the various grade levels. Every statement is separate from the others by a double space. For example Eight teachers commented on question 2 in Grade 1.

Results Obtained from Teacher Questionnaire for Part B of Investigation I Table 1

Question 1

Is each child in your room provided -with a textbook? Grades

1

No

1

Yes

1+

2 +

3 1

8

0 9

1* 1 8

£ 0

6 1

9 5

Omit

No teacher comments on this question.

Total

Per Cent

7

Hi.

1+3

86

0

0

Table 1

Question 2

Do you as a teacher follow the sequence of subject matter in your textbook? If you do not follow the textbook, how do you deviate? Grades

1

2

3

h

5

6

Total

Per Cent

No

5

2

2

3

1

3

16

32

Yes

3

7

7

6

8

3

3h

68

Ctoit

0

Teacher Comments Grade 1 By adding the teaching of time, etc., which is not included. Numbers can be more meaningful if used incidentally. We use lessons as they best fit in with our daily experiences. We do not use a textbook in grade 1. We use the Iowa Teachers handbook VIII on arithmetic for first grade. We do not have arithmetic the 1st semester. We do use a workbook the second semester. The student textbook is their workbook. Second semester. Second semester. Grade 2 It is much too hard reading material for beginning of second grade. They need simple reading and much more practice material. I use sticks, squares of paper, toys, children and other devices to illustrate the meaning of the concept I am introducing. This I follow with a written lesson.

Have to provide extra drill as book is vexy limited in this. Grade 3 Whenever an opportunity arises I seize the opportunity to have the children dramatize or (figure out) some specific problem or question. Example: measuring with objects to find pints, qts., etc. Something in measuring or time cernes up, we take it up then, and not wait until we come to it in the book. Much of the first half of the book should be used in second grade. The textbook is hard for the children toread at this level. The reading should be kept in the background in an arithmetic book. Too much time is used teaching words. The only time I deviate from the sequence ofsubjectmatter is when first hand experiences arise in the room. By using a meaning chart.

Grade h I have no standard deviation from year to year but as a general thing the first 10 or 15 minutes of class are spent in reviewing tables or the process of long division etc.— the remainder I use text. At the beginning of the year it is not possible to follow the textbook. In some cases, subject matter is taught when there is a need for it. To meet current needs and interests. Grade 5 I add more review. We always do the self-testing drills a second time— after a 6 weeks or 2 months interval and mark the charts with a different color. Most of the time I follow the text as I feel the authors have arranged the subject matter according to difficulty. Occasionally according to special needs of an individual class, some deviation.

Grade 6 Take more important points first; fundamentals on state tests. Supplement the textbook with the Iowa Course of Study in arithmetic in order to help children to meet the Iowa Every Pupil tests successfully. I introduce new material before the text, if the occasion arises. It often does. For instance; if the text presents fractions of this type 7 1/2 - 5 l/3 and someone asks what to do, if the 1/3 is on top— we go into borrowing right then.

Table 1

Question 3

Do you have a teacher's manual? (Not the answer book) Grades

1

2

3

k

$

6

No

3

k

0

0

1

0

8

16

Yes

k

5

9

9

8

6

ill

82

1

2

Omit

No teacher comments on this question.

Total

Per Cent

Table 1

Question U

Do you find the manual helpful in planning your lessons? lhat changes would you recommend in the teacher's manual? Grades

1

2

3

h

5

6

Total

Per Cent

No

1

3

2

2

3

It

is

30

Tes

k

k

6

7

5

2

28

56

Omit

3

2

1

7

lit

l

Teacher Comments Grade 1 I would prefer a manual on the order of the reading manuals put out by companies where the material used by the pupil is in the same volumn as the suggestions for teaching. More practical ideas for concrete illustrations of basic principles. Plans are too involved.

Suggestions require too much preparation.

The (Iowa Teacher's Handbook) is good. The teacher has to supplement a lot of material. Grade 2 The manual we have is for use with an old edition. I'm not satisfied with the book so find manual not very helpful either. Fits present book— changes recommended in book. The manual I use is "Making Sure of Arithmetic." It contains many suggestions for introducing number concepts as well as follow up activities.

Grade 3 More explicit explanations. ideas.

Related practice and supplementary

I would recommend that more interesting ways of introducing new material be suggested in the manual. Ways that will hold the child's attention long enough to introduce the material. More specific suggestions as to how a child can be led to discover or figure out arithmetic relationships. I believe the manual should be in with the teacher's desk copy. Grade 1; Helpful in details such as "The teacher should see that the child name his answers for all problems." I do not use the manual daily. Very infrequently. Grade $ Either: (1) put the explanation of new processes in the manual so that there will be more space for practice material in the text or (2) at least— if that is not done— enlarge the manual to include practice material. Textbook is largely self-explanatory. I suggest that a special teacher edition be available so that the manual and text be in same book. I feel this would be a great convenience. Grade 6 No suggestions.

Table 1

Question 5>

Do your classes find the textbook material interesting? Have you any suggestions that might make the textbook material more interesting to children? Please state. Grade

1

2

3

k

$

6

No

0

1

1

2

2

1

7

lit

Yes

5

8

8

7

7

*

UO

80

Omit

3

3

6

Total

Per Cent

Teacher Comments Grade 1 Have always found the vocabulary rather difficult. Especially the workbook. Grade 2 More real life problems for each new combination etc. They like the stories if I read to them, but when they struggle with difficult words they lose interest in problems. I’d suggest simplifying stories. My good group find the textbook very interesting. The slower group find seme of the work difficult, although I don't think they should. Not interesting to class as a whole. Reading vocabulary too difficult for second grade. The textbook Number Stories II-Scott Foresman has many interesting stories for introducing concepts, but does not give enough drill or follow up activities. The vocabulary in Book I is much too difficult for first grade. It does not cover a broad enough field. It can be used as a reader in second grade.

Grade 3 More colored illustrations. They look at the pictures and want to skip over much of the reading (which is a lot) and find the examples. If the reading is simplified the numbers would be more interesting. Grade h I would suggest a workbook rather than text— a new book each year utilizing current prices and problems. Pupils cannot understand problems presented in out-dated texts i.e. bread at 7$ a loaf when they go shopping for mother they know that bread is 17$ plus tax. Grade 5 Timed tests on the four processes such as are found in the StraverUpton Social Utility Arithmetics published by the American Book Company I have found to be interesting to children. The fifth grade like marking their charts and comparing progress on the self-testing drills. I don't believe they find the rest of the text particularly interesting. Some of the thought problems are not within the experience of a midwest group living in a factory city. However a teacher may always make the problems suit the child experiences and needs. Grade 6 Yes and no. They like the self-testing drills. Putting more practical written problems in the book. More illustrations.

Table 1

Question 6

Do you like the way your textbook introduces new facts and processes? INhat suggestions, if any, for change do you have that would help in planning your lessons? Grades

1

2

3

U

5

6

Total

Per Cent

No

2

1

h

2

1

1

n

22

Yes

2

8

5

7

8

5

35

70

Omit

h

h

8

Teacher Comments Grade 1 I think there is so much reading involved that sometimes the number facts become almost secondary. Grade 2 More number game Challenges" I follow a good manual for introducing number concepts and use various texts for additional practice. I also like a workbook. Grade 3 Like meaning chart-more meaningful to most children. Yes in every way except long division. Not enough drill. For instance, subtraction is introduced and checking, in the same lesson. It seems the mechanics of subtraction with borrowing should be mastered before teaching how to check. Too much is told to the child. The children do not experience enough in solving problems. Everything must be explained.

I’d like to know that the text is In accordance with the state course of study without my having to check. I never use the text when introducing new facts. I use the manual and past experience has taught me the methods I use. Each class presents a new outlook and a different approach and method is necessary with each. Grade $ I do not like the way the text used introduces multiplication of fractions and whole numbers 12 3/It x6 Grade 6 Problems to illustrate new skills are not practical. Examples or steps could be more clear.

Table 1

Question 7

Does your textbook provide adequate practice material for your classes after introducing new facts and processes? Grades

1

2

3

h

5

6

Total

Per Cent

No

1

6

8

2

3

3

23

k6

Yes

2

2

1

7

S

3

20

ho

Omit

5

1

7

lh

1 Teacher Comments

Grade 1 I think there could be more practice for the more difficult facts and processes. The number workbook helps in this respect. Grade 2 I would like more problem material, or very simple number stories using the number facts as they are presented. I think several pages could be devoted to tall, taller, tallest and so forth. Comparison is important. For new combinations— I like that. Not always— Sometimes needs to be supplemented with more practice lessons. Grade 3 Practice material adequate in addition and subtraction and thought problems. Not so in multiplication, division and measurement. I think it is impossible for any textbook to provide enough practice material. I prefer making a certain amount myself as the need arises. The 3^d grade are required to absorb a great deal more than they should due to the fact that the 1st and 2nd grades are not taught enough in the relationship of numbers and number concepts.

Grade 1; It all depends on the class. Last year the practice material adequate. This year it is not. Grade £ More drill work would be helpful; shorter review lessons, if necessary, to do this. I give my classes more practice material. Grade 6 More drill work would be helpful; shorter review lessons, if necessary to do this. Yes and no. Very little given.

Table 1

Question 8

Does your textbook provide sufficient review on new processes that have been introduced? Grades

1

2

3

h

5

6

Total

Per Cent

No

3

5

6

1

l

2

18

36

Yes

1

3

3

8

8

h

27

ft

Ctait

k

1

5

10

No teacher comments on this question.

Table 1

Question 9

Do you like the way the review work is spaced in the textbook? What suggestions for change do you have on this point? Grades

1

2

3

k

*

6

No

1

3

k

0

1

0

9

18

Yes

2

3

h

9

8

6

32

6k

Omit

5

3

1

9

18

Total

Per Cent

Teacher Comments Grade 1 No suggestions. Grade 2 More review. All at back of book. Seme of the tests are suitable but not comprehensive enough. Not always— sometimes need more review than is presented. Grade 3 "Progress Tests" should come oftener. Should be more frequent reviews. Not enough drill on new work and not enough review. There is a workbook that can be used for the text but we do not have it. The book does not have nearly enough review material therefore it is spaced very poorly. So much is introduced between reviews that the children forget what was previously learned.

Grade i; More review of addition and subtraction processes, learned in Grade 3? in the first part of Book 1±. Grade 5 The practice sets are of mixed examples with only one or two of a kind represented in each one. This review I don’t believe is as effective as a review of one-or-two types of examples at a time. Grade 6 Yes and no— I’ve had classes who needed much more review and drill than others. It would be unreasonable to expect a book to be adaptable to too many individuals.

PM*'* *

Table 1

Question 10

Do you follow the suggestions for review and practice your textbook advocates? If you deviate from the textbook, please state in what manner. Grades

1

2

3

k

$

6

Total

Per Cent

No

2

1

2

2

2

h

13

26

Yes

3

8

7

7

6

2

33

66

Omit

3

h

8

1 Teacher Comments

Grade 1 I attempt to add more material to make numbers from 11 to 100 more meaningful. Grade 2 By having the pupils make up their own number stories using the number facts with which they are having difficulty. They usually write them on the board. Much on creative number story. We use number story for English story writing. We have "play store." Individual number cards made by children and used to "play games" thereby get drill. Sometimes I use games and drills on the board or with flashcards, or other devices to create interest. Have to provide extra practice. I supplement the lessons in the book when I feel it necessary. I like a workbook for additional drill. Also other texts. Grade 3 Give much extra practice by hectographing examples.

L

We use additional games, time tests, flash cards, plus the discussion and use of various ways of study. I use numerous present day texts and workbooks for continued practice and review. They have proven very helpful to me. As I mentioned the book does not give enough practice and review so I give the children outside work such as worksheets, story problems, combinations etc. Grade U I can’t use the review section in the back of the text. At the beginning of the year, I cannot follow the textbook. are not ready for the material suggested in the textbook.

They

I use Book 3 during the early part of year in Grade h to give practice in addition and subtraction. I give little quick quizzes frequently— such as "Multiply these numbers by 8" or "Divide these numbers by 711 etc. It stimulates thinking and the pupils like the satisfaction of getting most of them correct. Grade 5 The students do the practice sets only once unless they do very badly and not two or three times as the book suggests. I used timed tests from another book and the children do work at the board (also from another book) to serve as review and practice. I don’t use the review section in the back of the book all at one time as suggested. We use it several times at irregular intervals over the entire year. Again some difference in needs of individual classes. Try to make review have appeal to child. Child should see and know his progress. Most of the time.

4>>

Grade 6 We all try to prepare for the state tests in January, and rush things a bit. I give outside practice and drill. If a check test shows need for more teaching in an area, we give it.

Table 1

Question 11

Do you think your textbook provides sufficient teaching suggestions? (Especially help in building basic understandings and concepts) What suggestions do you have for additions or changes? 1

No

2

3

k

Tes

2

Omit

2

5

5

1

5

6

Total

Per Cent

CM

Grades

2

3

19

38

7

7

3

29

58

2

h

Teacher Comments Grade 1 More meaningful pictures and illustrations. It does not lay much of a foundation for the addition and subtraction facts. Too many. Grade 2 Good as it goes, but not enough teachers must do much of their own planning. No suggestions given in text. The use of games and interesting devices could be included in the text to make it more interesting. Also more thought provoking problems. Grade 3 But too much is •written and drawn out. The children grasp a concept quicker by showing them on the board, than they do by wading through the reading material. I use both ways.

More stress on “proof” and the “discovery of errors." I think that requiring proof and getting the child to see his own errors is very important to understanding. I believe the 3rd grade text I use skims the surface and does not allow the slow thinker enough practice material. The suggestions never change, and there are very few to begin with. I would like to see more suggestions and have them brought up to date. Grade h Really— I think the teacher has to do this. Perhaps visual aids rather than printed page. Grade $ I think its teaching suggestions are the texts only good point and I feel that most of the explanations can just as well be made by the teacher without having the students look at the text. The teacher must guide them through each explanatory page anyway— else she has misconceptions to clear up. In my case yes— I wonder about one with less experience. Need for more teaching suggestions to develop a better understanding of fractions. Grade 6 The textbook introduces too many new concepts for its grade level. No book can do this too well; teacher must supplement. I have never been able to teach either area or perimeter using just the suggestions in the book.

Table 1

Question 12

Do you find the illustrations in your textbook helpful? Grades

1

2

3

U

£

No

0

0

1

I4.

2

2

9

18

Yes

5

9

8

5

7

3

37

7h

Ctait

3

I;

8

No teacher comments on this question.

6

1

Total

Per Cent

Table 1

Question 13

Do you use these illustrations in your teaching? What suggestions can you make that would enable you to use the illustrations more effectively in your teaching? Grades

1

2

3

h

5

6

No

0

0

0

3

2

2

7

lit

Yes

5

9

9

h

7

2

36

72

Cmit

3

2

7

lit

2

Total

Per Cent

Teacher Comments Grade 1 They should be larger, clearer and more colorful and more of them. I'd like illustrations showing a whole tree when one is teaching high, higher, highest etc. I believe it would make the concept clearer. To use illustration effectively each child needs a book. Too much time is consumed taking small groups, as is necessary in our program. Grade 2 I believe number concepts should first be introduced in concrete form then semi-concrete or by picture method and then in the abstract. Grade 3 By using real objects to work out the problems such as a store to buy and sell. The first half of the book are addition and subtraction combinations, which the children have learned in second grade, so many pictures are skipped.

I would suggest that more illustrations be put in the book so that they may be used. What illustrations are there are very helpful. Grade U Our text is out-dated. I do use some of the illustrations but more times than not I present my own. Some of the newer books have much more appeal in the way of illustrations. Grade 5 Give illustrations involving fractions that are usable. illustrations in my book are of no value.

The

Textbook has practically no illustrations and the illustrations are of practically no value. Grade 6 To seme extent they are helpful. More of them.

Table 1

Question XU

Would you like to use a textbook that would give your classes an opportunity to figure out for themselves arithmetical relationships rather than telling them step by step what to do? Grades

1

2

3

h

5

6

Total

No

1

h

k

3

6

3

21

bZ

Yes

7

5

5

6

3

3

29

£8

0

0

Quit No teacher comments on this question.

Per Cent

Table I

Question 1$

What changes, if any, in your textbook do you feel would enable you to teach arithmetic more meaningfully? Teacher Comments Grade 1 I would like a first grade book with many pictures to illustrate number concepts. Suggestions for teaching arithmetic would be welcome. I would like 1. Simplified vocabulary in arithmetic books. 2. Lessons so planned containing the object (picture) the word and the number 000 apples three 3 3. New ideas in the use of number games. i|. Concrete materials other than spools, marbles, bottle caps, etc. Market something in a durable material that would be lasting and worth while buying. How about plastic. 5. Large wall charts so planned to be used for summary. I would like a first grade text of some kind that had many illustrations showing numbers in groups so that children would get the concepts. A textbook with simpler vocabulary, with many more illustrations, and one that could be used in the first semester of 1st grade also. There are minor changes I would suggest, but the most important one is to give more time to work on numbers. I should also like to see the foundation laid (as far as understanding goes) for the easier addition and subtraction facts. Our group of first grade teachers voted not to use a basic textbook for the teaching of numbers. Many are using sheets from hectograph workbooks. Others use regular number workbooks. We try to utilize every opportunity to make numbers more meaningful by daily application, and in first hand situations.

Grade 2 Since we are asked to follow the state Course of Study in arithmetic, the book we are using is inadequate. The material for the second semester must be supplied entirely by the teacher. Too much time is spent on developing concepts of size, quantity, etc., if we want to learn the addition and subtraction facts through nine and nine. Need more provision for "gifted child." This would help a busy teacher as she must think out the challenging work needed for their interest. In general not more facts, processes, etc., but more work on those presented. I think workbook is very fine— maybe not enough drill on basic facts. I believe I've expressed my feelings on the text. I feel that if my class gets everything in the book we use their arithmetic learning has been very good. More simple vocabulary. More drills on fundamentals. More exercises. Number stories II teaches by means of the picture method instead of beginning with the concrete. Many children do not understand by the picture method alone. This book does not contain enough material to challenge the thinking of the bright student. I prefer "Making Sure of Arithmetic" put out by Silver Burdette. It has an excellent teacher's manual and workbook. Grade 3 There is one general complaint that I have of all third grade arithmetic books and, that is, there is too much material to cover. There are too many new processes introduced. I think multiplication and division should be left out of the third grade text. Let that go until the l*th grade. We could very nicely spend the whole year on addition, subtraction, time and measurement, etc. I like the textbook we use very much. Sometimes I feel that I don't drill enough because there is plenty of material to cover. With a good average group I do well, but sometimes we get groups that need so much attention individually that I find it hard to cover all of the work.

Many pages are given on thought problems which are good for those who can read well. Need more pages of addition, subtraction, multiplication, and division drill, and not so many pictures. I'd like a textbook that provides an opportunity for a child to work out a new process before it is explained to him. My opinion is that too many textbooks work out new steps for the child and consequently do not allow the child to "think it out" for himself. Too the textbooks place a great deal of emphasis on demonstrating and memorizing. I'd like a textbook that provides a child with more experiences, and one that puts more emphasis on helping the child see the relationship between the new phase of arithmetic and the arithmetic he already knows and understands. I would like a clearer teaching of number concepts to begin with, then a lot of work in showing number relationships, such as the re­ grouping idea in multiplication, addition, subtraction and division. I have learned how to teach by re-grouping and by showing number relationships but I should like a well-illustrated book in the hands of every pupil along this line. ■When new material is introduced I would like to see it followed up with the same kind of material rather than introducing new material again. Also I would like more review and practice material. The teacher's manual could be helpful if it were revised. More review on new processes. Not so much review of second grade work. Additional drill in measurements. Grade U In the fourth grade text a pupil who is doing excellent work is unable to meet the standards of achievement tests or state course of study. I think that a book that would help children figure out relationships would be the best way to teach, but it would take a genius to make a text efficient enough to "get somewhere" in case of large classes. Of course if the administrators would agree to accept a new, more slowly arrived at skills that would be o.k. We have to meet our deadlines too. The fourth grade never seems to be quite ready for the textbook. I know the trend is to introduce new processes at lower grade levels and In some cases arithmetic or numbers are not taught in first and

second grades. But in achievement tests and other tests, they are notable to meet the standard set for fourth grade. It would be helpful to me if there could be pages with standards of achievement in computation, such as the pages and standards in problem solving. These serve as incentives both to teacher and pupils. More practice. As I mentioned previously— I do not care for the textbook. I prefer a workbook with practice work. I have used the Lemes Work Pad in the upper grades and found it very satisfactory. Grade $ I like the way the text has the student figure out the relationships between and among fractions. I think textbooks need: 1. Much more practice and review material. 2. Leave out the lengthy explanations to make room for this material. This text we use presumes, apparently, that the teacher is incapable of making a clear explanation. 3* More emphasis on accurate computation and less on problem solving. Perhaps my attitude is a mistaken one, but I believe one's ability in solving written problems is dependent on three factors— intelligence, reading ability, and accuracy in computation] nothing can be done about the first, the second is attacked forcibly in reading classes; the third is the realm of the arithmetic teacher. Concrete examples should certainly be included at frequent intervals but not so much so that accuracy of computation is sacrificed because of lack of practice. If I had the knowledge necessary to answer this question I should be much more satisfied with my teaching. I can't tell you how this text should be changed. I just know it could be improved greatly. An experienced teacher can direct the children to figure out for themselves arithmetic relations. However there is a question in my mind if the situation could be handled by an unskilled teacher. Most children have to be guided step by step. I wonder if valuable time is sometimes lost in children figuring out relationships. I think with any textbook we have to make material fit the child's problems.

I think onr text covers the field. Thought problems based on children's own activities. Grade 6 I feel that the self-help charts in our textbook are too easily accessible, and therefore they become a "crutch" rather than an aid. I just don't feel qualified to say. The first part of the book should go through with a step by step process, but the latter part the child should be able to do this by himself. Analysis of Table 1 Examination of Table 1 Question 1 reveals the fact that eighty-six per cent of the teachers in the first six grades of the elementary school report that each child in their room is provided with a textbook.

These data indicate that in most schools in Iowa

almost every child in grades two through six has access to a textbook. Most of the first grade teachers contacted reported that when textbooks were provided for each child they were used during the second semester of the school year rather than during the first semester. Table 1 Question 2 shows that sixty-eight per cent of the teachers in the schools participating in the study follow the sequence of subject matter outlined in the textbook. These data reveal the fact that the teachers in grades two through six follow the textbook more consistently than do the teachers of grade 1.

Thirty-two per cent of the teachers report that they deviate from the sequence of subject matter in the textbook in same way. The comments made by the first grade teachers show that a textbook is not used in most first grade classrooms until the second semester of the school year.

The second grade teachers say that much of the

reading material in their textbooks is too difficult and that they would like more drill material on the whole. The third grade teachers deviate from the textbook only when an opportunity arises in the classroom to make use of a number situation. The fourth and fifth grade teachers also make use of number situations arising in the classroom to deviate from the textbook.

Two of the sixth grade

teachers are concerned with the achievements of their pupils on arithmetic tests and they deviate by teaching first the facts and points they feel will help their pupils do a good job. Table 1 Question 3 shows that eighty-two per cent of the teachers had access to a manual for the arithmetic textbook series being used. Sixteen per cent reported that no manual was available. Table 1 Question b summarizes teacher opinions of the manuals provided for their use. Fifty-six per cent report that the manual is helpful in planning lessons.

Thirty per cent of the teachers

feel that the manual is not helpful to them. Fourteen per cent of the teachers did not report on this question. Analysis of the teacher comments indicates that on the whole the teachers would welcome a manual which would give them more specific teaching

suggestions especially in regard to introducing new processes.

Two

teachers suggested that the teacher's edition of the textbook should include a manual. Table 1 Question 5 shows that eighty per cent of the teachers feel that the textbook material used is of interest to the children in their classes. Fourteen per cent state that it is not interesting. All grade levels reporting seem to show the same degree of interest. Analysis of the teacher comments points to the fact that the teachers of grades one, two and three feel that much of the reading vocabulary is too difficult for their children.

They

feel the textbook would be more interesting to the pupils if the vocabulary was simplified.

The teachers of grades four, five and

six state that the self-testing drills and progress charts provided in seme books are especially interesting to the children. Two teachers point out the need for up to date material or the use of subject matter more applicable to the middle-west. Table 1 Question 6 reveals that seventy per cent of the teachers like the way their textbook introduces new facts and processes while twenty-two per cent do not. Analysis of the teacher comments shows that some teachers feel that the textbook tells the children too much and does not provide sufficient experience to enable a child to solve problems for himself. One teacher does not like the way the textbook used introduces subtraction, another does not like the introduction to division and another does not like the

way multiplication of fractions by whole numbers is taught. Table 1 Question 7 summarizes the opinions of teachers regarding the amount of practice material inluded in their arithmetic textbooks.

Forty-six per cent of the teachers feel that the text­

book they use does not provide sufficient practice material.

Forty

per cent feel that it does. Fourteen per cent of the teachers did not comment on the question. More teachers in grades one, two and three said that the textbook did not provide enough practice material than did the teachers in grades four, five and six. An analysis of the teacher comments shows that on the whole the teachers would like more practice material in arithmetic textbooks.

Two of the teachers

feel that more story problem material would be helpful. Table 1 Question 8 reveals that fifty-four per cent of the teachers feel that the textbook provides sufficient review on new processes that have been introduced while thirty-six per cent feel that it does not. Table 1 Question 9 shows that sixty-four per cent Of the teachers like the way that the review material is spaced in their textbook. Eighteen per cent of the teachers did not like the spacing. The teachers in grade four, five and six seemed to like the way that review work was spaced in their textbook better than the teachers of grades one, two and three.

In general the teachers

seemed to feel that review work should come after the process taught rather than at the back of the book. Table 1 Question 10 reveals the fact that sixty-six per

t . «/

cent of the teachers follow the suggestions advocated in the textbook for review and practice work*

Twenty-six per cent of the teachers

do not. An analysis of the teacher comments indicates that most of the teachers provide extra review and practice material for their classes.

These teachers make use of games, timed tests, flash cards

and Yfork sheets that they develop themselves. Several teachers report that they use practice material provided in the textbook but not in the manner suggested.

These teachers seem to use only parts

of exercises rather than complete sets of examples. Table 1 Question 11 shows that fifty-eight per cent of the teachers feel that the textbook they use provides sufficient teaching suggestions.

Thirty-eight per cent feel that it does not. These

data reveal that more teachers in grades four, five and six answered yes to this question than in grades one, two and three. Some of the teachers in the first three grades suggested that the textbook provide more foundational work in addition and subtraction and more opportunity for children to correct their own errors in thinking. Most of the teachers felt that the textbooks left teaching suggestions up to the teacher. Two teachers in grades five and six mentioned that they would like the textbook to suggest more ways to each fractions and measurement. Table 1 Question 12 indicates that seventy-four per cent of the teachers find the textbook illustrations helpful. Eighteen per cent do not. Table 1 Question 13 shows that seventy-two per cent

of the teachers use the illustrations provided in their teaching. Fourteen per cent of the teachers do not make use of the textbook illustrations. Most of the teachers asked for more illustrations in the textbooks. They said that the ones provided were helpful to them in teaching.

The first grade teachers suggested larger and

more colorful pictures.

Two teachers thought that more use should

be made of concrete material before illustrative material was presented. Table 1 Question li* reveals the fact that fifty-eight per cent of the teachers would like to use a textbook that would give their classes an opportunity to figure out arithmetical relationships for themselves rather than telling them step by step what to do. Forty-two per cent of the teachers did not approve of this approach. No consistent pattern of responses is evident between grade levels. Table 1 Question 15 summarizes by grade level the suggestions for change in arithmetic textbook material teachers feel would be helpful.

Thirty-six teachers gave suggestions in response to this

question. Most of the first grade teachers thought that a textbook should be used in the first grade.

Those teachers using a textbook

at the present time thought that the vocabulary in it should be simplified.

The teachers would like the textbook to provide a more

adequate concept building program in addition and subtraction. Several of them mention the need for teaching suggestions in the use

of concrete objects.

In general first grade teachers like books with

large illustrations. The second grade teachers suggest the need for a simplified reading vocabulary, more drill work on fundamental processes and more provision for gifted children. Two third grade teachers felt that too much material was presented in the textbook for that grade level. A few thought that textbooks needed to provide more number experiences for children and more opportunities for pupils to work out new processes for them­ selves.

These teachers felt that pupils should be given the

opportunity to work out number relationships and that the books should not spend so much time telling the children step by step how­ to think.

Several mentioned the need for more review work on new

processes that were introduced. In general the fourth grade teachers feel the need of more practice work.

They suggest that the textbook should set up standards

of achievement for computation in the basic processes. The fifth and sixth grade teachers also mention the need for more practice material.

Several of these teachers felt that the

problems presented should be within a childrs experience. On the whole the fifth and sixth grade teachers seem more satisfied with their present textbooks than are the teachers of the other grades.

Investigation II The purpose of Investigation II was to determine teacher reaction to methods of presenting arithmetical textbook materials that are more nearly in harmony with the best theory than are the methods now used in published textbooks.

This purpose required the

identification of best theory. A careful review of the literature in the field of arithmetic constituted the first part of Investigation II and was the procedure used to determine, in so far as possible, best theory. An analysis was made in the review of the purposes of aritimetic, statements of theory, examples of method, and suggested principles for use as guides in creating learning situations for children adovcated for use in the writings and research done by a representa­ tive group of educators, mathematicians, and authoritative committees. In order to conserve space and prevent repetition the principles advocated by this group for use as guides in creating arithmetic learning situations for children and determined as a result of this review are not repeated in this chapter. They are presented and summarized in Chapter II of the study. The methods used by the writer to develop questionnaires and a series of seventeen experimental lessons which were matched with similar ones in a current textbook and presented to the elementary teachers in the Davenport, Iov/a schools for evaluation are described

in Chapter III under steps two, three, four, five and six in the outline of procedures for Investigation II.

The procedures described

in Investigation II resulted in the development, by the writer, of one set of directions for teachers labeled Guide Sheet I, one nine item questionnaire labeled Guide Sheet II, six sets of experimental lessons, seventeen in all, classified by grade level and a series of short four item questionnaires on each individual lesson.

These

materials appear in Appendix III of this study. The questionnaires and experimental lessons were submitted to approximately 115 teachers in grades one through six in eleven schools in Davenport, Iowa. Ninety-eight teachers responded. Each of the experimental lessons at each grade level was matched with a similar lesson, covering the same arithmetic content, in the current arithmetic textbook series used in the Davenport Schools.

The

teachers were asked to compare the experimental lessons with lessons in their own textbooks and to evaluate them in terms of their class­ room experience.

The teachers were asked to judge the worth of each

of the lessons and to express their opinions of the approaches and techniques and to state a preference for either the textbook or experimental lessons in terms of Questionnaire Guide Sheet II. They were asked to state briefly the reasons for their decision and to point out specific things they liked or disliked about these two types of lessons. The teachers were asked to make suggestions for

change or improvements in textbook materials that they felt would help them make arithmetic more meaningful to their pupils. Each set of materials judged by the teachers were accompanied by four questionnaires.

The Green Guide Sheet II Included with each set of

lessons for the six grade levels was to be answered by each teacher after she had judged the individual lessons prepared for her grade level.

The other short questionnaires were to be answered on the

basis of the judgments made after examination of the individual lessons.

Tables 2, 39 ^ suicl 5 are an analysis of the teacher

reactions and judgments of the textbook and experimental lessons as expressed on Guide Sheet Questionnaire II. Interpretation of Tables 2 - 5 Tables 2 and 3 are a tabulation of the total responses and reactions of the ninety-eight teachers participating in Investigation II to the textbook and experimental lessons. Reading from left to right Table 2 is interpreted as follows: Ninety-eight different teachers expressed a judgment about Question 1 on Guide Sheet II. Of this number nineteen per cent, or nineteen teachers in all, expressed a preference for their own textbook approach. Eighty-one per cent, or seventy-nine teachers, expressed their preference for the experimental approach.

The percentages given in these tables

are rounded to the nearest whole number and fractional parts are not shown. Table 3 is a tabulation of the responses of teachers to

Cl /

question 9 on Guide Sheet II. A separate table was made for question 9 because it required a yes or no answer. The number of teachers answering either yes or no to this question are tabulated by grade level on Table 3. It will be noted that thirty-six per cent of the teachers gave a negative answer to this question while sixty-four per cent answered in the affirmative. Tables h and 5 are an analysis by grade level of the responses of teachers to Questionnaire Guide Sheet II. Table it shows the number of teachers at each grade level preferring the textbook approach and Table 5 shows the number preferring the experimental lessons. To insure ease of reading the nine questions asked on Guide Sheet Questionnaire II and tabulated on Tables 1 - 5 are presented here. The original questionnaire appears in Appendix III. Key for Tables 2 - 5 Items that teachers were asked to respond to on Guide Sheet Questionnaire II. Text

Experimental

___

_________

2. Which lessons would be most adaptable to your classroom?________________________ ___

_________

1. If textbooks were published with lessons of both types in them which would you prefer?

3. Dfhich lessons do you think would best motivate and hold the interest of your class?

rv-r»

Text U* Which lessons would help you most in teaching like processes to your class?

Experimental

___

5. Which lessons do the best job of getting arithmetical concepts across to children? 6. Which lessons do the best job of creating a problem situation to stimulate thinking? 7. Which lessons best illustrate teaching for meaning and understanding? Meaning— seeing the reasons for or sense of a process. Understanding— seeing the relationships between a fact or process and other facts or processes? 8. Which of these two types of lessons best meet your needs in teaching arithmetic? Please state briefly the reasons for your decision. 9. Would you recommend that the experimental lessons be followed in an arithmetic book with lessons of the type found in your current textbook?

Circle One Yes

No

Table 2 Analysis of Questionnaire Guide Sheet II for Investigation II Total N

Question

N

Terfc Per Cent

1 2 3

19 23 20 20 19 16 17 20

19 23 20 20 19 16 17 20

98 98 98 98 98 98 98 98

h

5 6 7 8

Experimental! N Per Cent 79 75 78 78 79 82 81 78

81 77 80 80 81 8k 83 80

Table 3 Analysis Question 9 Guide Sheet II Answer

Grades 3 k

5

6

Total

Per Cent

10

6

h

3

5

35

36

16

15

Hi

6

7

5

63

6h

23

25

20

10

10

10

98

100

1

2

7

Yes Total

No

Table k Analysis by Grade Level of the Number of Teachers Expressing a Preference on Guide Sheet II for The Textbook Materials

Question 1 2 3 h

5 6 7 8 9

Total M 19 23 20 20 19 16 17 20

^ ■■g 6 6 6 5 5 It 5 6

9 10 9 9 9 9 8 9

Grades , k 3 5 h h h

2 3 h

1 1 1 2 1 1 1 1

g-

y-

0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0

Table 5 Analysis by Grade Level of the Number of Teachers Expressing a Preference on Guide Sheet II for the Experimental Materials

Question 1 2 3 h

5 6 7 8

Total H

^

79 75 78 78 79 82 81 78

17 17 17 18 18 19 18 17

Grades 2 - ■y 16 15 16 16 16 16 17 16

17 15 16 16 16 18 17 16

9 9 9 8 9 9 9 9

--g 10 10 10 10 10 10 10 10

g10 9 10 10 10 10 10 10

Analysis of Tables 2 - 5 The total reactions and opinions of the ninety-eight teachers participating in Investigation II to the textbook and experimental lesson material submitted to them for evaluation are summarized and tabulated in Tables 2 and 3. Examination of these tables reveals the fact that on every question regarding the two types of materials the teachers indicate a consistent and decided preference for the experimental approach. Analysis of Table 2 Question 1 points out that eighty-one per cent of the teachers participating in the study would prefer to use textbooks that have lessons of the experimental type in them as against the nineteen per cent who prefer the type of lessons found in their current textbooks. Questions 2 and 3 tabulated on Table 2 asked the teachers to choose the types of lessons that would be most adaptable for use in their classrooms and would, in their opinions, best motivate and hold the interest of their pupils.

Seventy-seven per cent of the

teachers said that the experimental lessons would be most adaptable for use in their classrooms and eighty per cent said the experimental approach would motivate and hold the interest of their pupils better than the textbook lessons judged. Questions U and 5 summarized on Table 2 are concerned with children understanding arithmetical concepts and the grasp they have of number relationships between processes. On both of these points

eighty per cent of the teachers thought that the experimental lessons would do a better job of giving the children these understandings than the lessons in their present textbook. Only twenty per cent of the teachers favored the textbook lessons on these points. Examination of the reactions of the teachers to question 6 reveals the fact that eighty-four per cent of the teachers thought that the experimental lessons did a better job of creating a problem situation to stimulate thinking.

Sixteen per cent of the teachers

favored the textbook lessons on this point. Question 7 tabulated on Table 2 asked the teachers to decide which of the lessons, the experimental or textbook, did a better job of illustrating teaching for meaning and understanding. Analysis of Table 2 shows that eighty-three per cent of the teachers favored the experimental lessons on this point as against seventeen per cent who favored the textbook lessons. Examination of Question 8 on Table 2 reveals the fact that eighty per cent of the teachers thought that the experimental lessons would meet their needs in teaching arithmetic to children better than lessons of the type found in their current textbooks.

Twenty per

cent felt that present textbook lessons meet their classroom needs. Table 3 shows the reactions of the teachers to question 9 on Guide Sheet II. This question asked the teachers if they would recommend that the approach used in the experimental lessons be followed in an arithmetic book with lessons on the type they were

101

now using.

Sixty-four per cent of the teachers answered in the

affirmative while thirty-six per cent gave a negative response. Tables U and $ indicate the number of teachers at each grade level preferring the textbook or experimental lessons. Examination of these tables shows that the teachers of grades 5 and 6 almost unanimously chose the experimental lessons in preference to those in their textbook.

For grade k these tables reveal that

nine of the ten teachers responding at this grade level chose the experimental approach. Examination of the responses of the teachers at the first, second and third grade levels indicates that the majority of the teachers preferred the experimental lessons. Detailed Analysis of Question 8 Summarized on Table 2 It will be noted that Question 8 on Guide Sheet II asked the teachers to state briefly their reasons for preferring either the textbook or experimental lessons. To conserve space their answers are compiled, classified by grade level, and presented in Appendix IV of this study. The comments made by each teacher were copied verbatim from the original data, classified by grade level and tabulated under the heading of experimental or textbook preference. Analysis of these data compiled in Appendix IV indicates clearly that most of the teachers participating in this study preferred the approaches, procedures and techniques used in the

102

experimental lessons to those used in their present textbooks.

They

have pointed out in these data the reasons they prefer the experimental approach.

The teachers choosing the textbook lessons

in preference to the experimental ones have also given their reasons. Examination of these teacher comments brings to light many points of interest in both types of lessons.

These comments point out the

approaches, techniques and methods teachers feel should be included in textbooks to help them make arithmetic meaningful to children. Examination of these data show that at every grade level the teachers feel that the problem situation created for children at the beginning of each experimental lesson would stimulate thinking and make the work meaningful to them. One teacher said that the experimental material, "Created a desire for learning." Other comments made by the teachers indicate that they felt that the experimental lessons made more use of number situations within children’s experience and provided more opportunities for active pupil participation in solving number situations. This active participation the teachers felt made for better understanding of arithmetical processes.

The teachers preferred the experimental

materials because they did not tell the pupils how to think, but led them step by step through number activities which would enable them to see and understand inherent relationships between processes. They liked the fact that the children stated generalizations about numbers

103

only after they had many experiences using them. The teachers felt that current textbook lessons tend to tell children too much and in doing so lose their power to motivate and sustain interest.

On the

whole the teachers felt that the experimental lessons would have a high interest value for children because they provided more opportunities for children to participate in figuring out solutions to number problems and situations. These data show that the teachers would like to use material of the experimental type to help them plan lessons.

They

liked the ideas they got from the variety of approaches used in the lessons, and they also liked the fact that the lessons gave them an opportunity to use their own ideas. Examination of these data reveal that relatively few teachers gave reasons for preferring the textbook lessons. Almost all of the teachers in grades four, five and sixrth preferred the experimental lessons, and the one or two at these grade levels who did not, failed to give ary reason for their choice of the textbook approach.

In grades one, two, and three the teachers pointed out that

the textbook materials made more use of pictures than the experimental materials.

They liked the use of pictures and suggested that more

be included in textbook material. A few teachers said that the experimental lessons were complicated and required too much reading on the part of the children.

These teachers had the same criticism

to make of the textbook lessons.

Despite the fact that both the textbook and experimental lessons covered the same content and presented the same arithmetical concepts to children several second grade teachers said that the experimental lessons were too difficult.

They thought the

experimental lessons would place too heavy a reading load on low ability groups.

This point was not brought out at the other grade

levels. One teacher liked the fact that the textbook lessons told the children how to think.

She felt that her group would become

discouraged if they were required to figure out too many solutions for themselves. Results of Individual Questionnaires Tables 6 - 2 2 which follow summarize by grade level teacher reaction to methods of presenting arithmetical textbook materials that are more nearly in harmony with the best theory than are the methods of presenting material now used in published textbooks. The percentages given in these tables have been rounded to the nearest whole number and fractional parts are not shown. Short four item questionnaires were included with each experimental lesson submitted to the teachers of Davenport for evaluation and comparison with a similar lesson from the textbook used in their classrooms.

These questionnaires were placed on the last

page of each experimental lesson (see lessons in Appendix III). teachers were asked to judge these arithmetical materials, to

The

indicate the one they preferred, and to state the reason for their decision.

The following tables summarize the results of these

questionnaires.

The reasons given by the teachers for their

preference of the textbook or experimental lessons were copied verbatim from the original data, compiled by grade level, and tabulated under specified preference.

These data appear in Appendix

V. Analysis is made in connection with each of the tables which follow. Interpretation and Analysis of Tables 6 and 7 Tables 6 and 7 summarize the reactions and opinions of first grade teachers to the arithmetical materials submitted to them for evaluation. Reading table 6 frcm left to right it will be noted that of the twenty-three first grade teachers participating in the study, six teachers or twenty-six per cent preferred their own text­ book lesson approach on the points asked on the four item question­ naire.

Seventeen of the teachers or seventy-four per cent favored

the experimental lesson approach on these points. Examination of this table shows that the majority of the first grade teachers thought that the detailed teaching suggestions given in the Experimental Lesson A would be more helpful to them than the directions given in the textbook lesson. They also thought that the experimental lessons gave children more of an opportunity to have first hand experiences with number and to become familiar with the relationships .

10S

Table 6 Analysis of Questionnaire for Grade I Lesson A Total N

Question

23

Experimental N Per Cent

N

Texrt Per Cent

A-l

6

26

17

7U

23

A-2

6

26

17

Ik

23

A-3

6

26

17

7k

23

A-ii

6

26

17

7k

Key Questions A-l. Experimental lesson A gives detailed suggestions for presenting the lesson to your first grade class. The textbook lesson on page 8 also gives same direction. Which approach do you prefer? A-2.

To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better?

A-3*

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

A-I*. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

107

Table 7 Analysis of Questionnaire for Grade I Lesson B Total N

Question

23

Experimental N Per Cent

N

Text Per Cent

B-l

h

17

19

83

23

B-2

h

17

19

83

23

B-3

3

13

20

87

23

B-U

h

17

19

83

Keg; Questions B-l. As shown in experimental lesson B each exercise for the children to work is preceded by a page addressed to the teacher* This page contains suggestions for working out the lesson with the children. In the textbook lesson on page 5 the directions to the teacher appear in a separate manual. Which technique would you find most helpful in planning your lessons? B-2« To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? B-3.

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

B—It* In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

which exist between various arithmetic facts and processes than did the textbook lesson. Table 7 reveals the fact that over eighty per cent of the first grade teachers preferred Experimental Lesson B to their textbook on the four item questionnaire.

This table shows that the teachers

prefer that teaching suggestions accompany lessons rather than that they be included in a separate manual. The teachers feel that the experimental lesson does a better job than the t extbook lesson in giving children first hand experiences in working with numbers and in enabling them to become familiar with relationships which exist between various arithmetic facts and processes. Question ii on these individual questionnaires asked teachers to state their reasons for preferring the textbook or experimental lessons.

To conserve space the reasons given by teachers for their

preferences have been compiled by lesson and grade level and appear in Appendix V. Analysis of these data appears in the following paragraphs. In general first grade teachers liked these experimental lessons because: 1. Number concepts are being built up in them and not just taken for granted. 2. Provision was made for individual differences through a variety of approaches in the introductions.

3.

They provide clear problem situations which induce and

stimulate good thinking. U* They provide the opportunity for first hand number experiences for children. The lessons show the child how to figure out number relationships and are built on previous knowledge and, therefore, hold the children’s interest. The teachers favoring these experimental lessons thought that they would be more meaningful to children than the textbook lessons with which they were compared. In general the teachers preferring their textbook lessons to the experimental said that the textbook was better because: 1.

They preferred to have children identify groups of marbles

rather than letters. 2.

The directions on the experimental lessons were more

complicated than those of the textbook and therefore would not hold the children’s interest. 3. There were too many problems on the experimental pages. Interpretation and Analysis of Tables £, £ and 10 Tables 8, 9 and 10 show the reactions and opinions of second grade teachers to the arithmetical materials submitted to them for evaluation.

These tables are read in the same manner as Tables 6 and.

7. It will be noted on Table 8 that fifty-six per cent of the teachers favored the Experimental Lesson A to forty-four per cent

favoring the textbook lesson on question A-l.

This question asked

the teachers if they preferred the specific directions on how to proceed given to children on the experimental lesson as compared with their textbook lesson which gave no directions to the children. On Table 9> question B-l, seventy-six per cent of the teachers indicated that they favored the experimental approach of Lesson B which set the stage for learning with a problem situation the children were asked to solve with a drawing as compared to fortyfour per cent who favored the textbook approach on the same content. Table 10, Question C-2 shows that sixty per cent of the teachers thought that Experimental Lesson C did a more effective job of giving children an understanding of the nature of the number system than did the matched textbook lesson. Forty per cent of the teachers favored the textbook lesson on this point. These three tables show that the majority of the second grade teachers thought that the three experimental lessons gave children more opportunity for first hand experiences with number than the textbook lessons and also more opportunity to see and understand inherent relationships between processes. The data compiled and presented in Appendix V on question ij. give the second grade teachers reasons for preferring the experimental lessons to those in their present textbook. Analysis of these data shows that in general the second grade teachers preferred the experimental lessons because:

ill

Table 8 Analysis of Questionnaire for Grade II Lesson A Total N

Question

N

Text Per Cent

Experimental N Per Cent

25

A-l

11

Ul

lU

56

25

A-2

8

32

17

68

25

A-3

11

hh

lh

56

25

A-li

9

36

16

6k

Key Questions A-l*

In problems 1 and 2 in experimental lesson A the children are given specific directions on how to proceed. The textbook lesson on page 1+ does not include directions to the children. Which approach do you prefer?

A-2.

To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better?

A-3*

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

A-li* In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

t

Table 9 Analysis of Questionnaire for Grade II Lesson B Total Question N

N

Text Per Cent

Experimental N Per Cent

25

B-l

6

2k

19

76

25

B-2

7

26

18

7k

25

B-3

3

12

22

88

25

B-U

7

26

18

7k

Kgr Questions B-l* Experimental lesson B sets the stage for learningwith a problem situation which the children are asked to solve with a drawing. The textbook lesson does not use this technique. Which approach do you prefer? B-2. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? B-3*

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

B-U. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

f try,

±

Table 10 Analysis of Questionnaire for Grade II Lesson C Total N

Question

25

N

Text Per Cent

C-l

13

52

12

1*8

25

C-2

10

i*o

15

60

25

C-3

12

1*8

13

52

25

c-JU

11

1*1*

lh

56

Experimental N Per Cent

Key Questions C-l. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? C-2. An understanding of the nature of the number system is important in teaching arithmetic meaningfully. In these lessons which approach is better? C-3. Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better? C-l*. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

4

1.

They provided the children with opportunities for first

hand experiences with number. 2. Children through use of the experimental lessons could feel, see, know and understand number relations. 3. More real life situations within the children’s experience were used. 1*. The lessons and number situations presented a challenge to the children. 5.

The lessons teach by doing instead of telling. Because of

these reasons the teachers felt the experimental lessons were more meaningful. In general the teachers preferring the textbook lessons to the experimental said that the textbook lessons were better because: 1.

The teacher’s manual gave more specific directions.

2. The experimental lessons used too much reading material and the directions were complicated. 3. The textbook left explanations up to the teacher. 1*. The textbook reasoning was better. Interpretation and Analysis of Tables 11, 12 and 13 Tables 11, 12 and 13 show the reactions and opinions of third grade teachers to the arithmetical materials submitted to them for evaluation.

It will be noted on Table 11, that ninety per cent

of the teachers favored the Experimental Lesson A problem situations

and. the suggestions that the children try to figure out a solution with drawings and a diagram. Only ten per cent of the teachers preferred the textbook approach which told the children how to solve the introductory problems. Table 12, Question B-l shows that eighty per cent of the teachers favored the Experimental Lesson B in subtraction which guided the children into figuring out for themselves the steps to use in borrowing in subtraction. Twenty per cent of the teachers preferred the textbook lesson which told the children how to think and worked out the steps to follow in borrowing in subtraction for them. Table 13, Question C-l reveals the fact that eighty per cent of the teachers felt that the experimental lesson gave the children a better understanding of the nature of the number system. Only twenty per cent of the teachers felt that the textbook lesson was superior on this point. Tables 11, 12 and 13 show that over eighty per cent of the teachers thought that the three experimental lessons gave the children more opportunity to have first hand experiences with numbers and more opportunity to become familiar with relationships which exist between various arithmetic facts and processes. The data compiled and presented in Appendix V on question ]*, and summarized on these tables, are a tabulation of the third grade teachers' reasons for preferring the experimental lessons to

•fi . * .'P> ll..o

Table 11 Analysis of Questionnaire for Grade III Lesson A Total N

Question

N

Text Per Cent

20

A-l

2

10

18

90

20

A-2

2

10

18

90

20

A-3

2

10

18

90

20

A-i|

2

10

18

90

Experimental N Per Cent

Kejr

Questions A-l.

The textbook lesson solves for the children each problem presented on page 38* Experimental lesson A asks the children to make drawings and from these drawings to figure out a written number solution. Which approach do you prefer?

A-2. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? A-3*

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

A-l|. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

117

Table 12 Analysis of Questionnaire for Grade III Lesson B Total N

Question

20

N

Text Per Cent

B-l

k

20

16

80

20

B-2

3

15

17

85

20

B-3

2

10

18

90

20

B-l*

1*

20

16

80

Experimental N Per Cent

Ke£ Questions B-l. In experimental lesson B, questions 5 - 11 guide the children in how to figure out for themselves the steps to use in borrowing in subtraction. The explanation in the textbook lesson on page 135 tells the children how to think and works out the process for them. "Which approach do you prefer? B-2* To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. "Which approach does this better? B-3. Children should became familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better? B-1*. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

.13

Table 13 Analysis of Questionnaire for Grade III Lesson C Total N

Question

20

N

Text Per Cent

C-l

k

20

16

80

20

C-2

k

20

16

80

20

C-3

h

20

16

80

20

C-l;

k

20

16

80

Experimental N Per Cent

Key Questions C-l. An understanding of the nature of the number system is important in teaching arithmetic. "Which approach does this better? C-2. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. "Which approach does this better? C-3.

Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better?

C-2;. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

those in their present textbook. Analysis of these data shows that third grade teachers prefer the experimental lessons because: 1. The experimental lessons give the children a chance to solve problems for themselves, therefore they feel that they have a clearer understanding of the process used.

One teacher said, "They do by

doing." 2. The diagrams and use of pictures to introduce problems or figure out solutions stimulate thinking. 3. The number situations used deal with experiences common and familiar to children. Ij. The lessons guide and show instead of telling. 5. Conclusions and generalizations are formed on the basis of first hand experiences with number. 6. Relationships are discovered through number experiences and the children see sense in them. 7.

The vocabulary is simple and easily understood. One teacher preferring the textbook lessons felt that the

vocabulary in them was easier for the children to read than that used in the experimental lessons. Another teacher felt that the directions on the experimental lessons were complicated.

Interpretation and Analysis of Tables 1U, If? and 16 The reactions of fourth grade teachers to the arithmetical materials submitted to them for evaluation are summarized on Tables Hi, 15 and 16. It will be noted on Table 11; that ninety per cent of the teachers favored the experimental lesson on question A-l because it guided children through number experiences and into seeing the purpose of the numerator and denominator of a fraction rather than telling them the purpose of these terns as the textbook did. Table 15, question B-l shows that seventy per cent of the teachers favored the introduction to the experimental lesson as opposed to the thirty per cent who favored the textbook introduction. The experimental lesson created a number situation for the children to solve while the textbook presented a problem and then solved it for the children. Table 16, question C-l reveals the fact that sixty per cent of the teachers favored the experimental lesson on this question. Forty per cent preferred the textbook approach. Question C-l is concerned with the fact that the experimental lesson guides the children into figuring out relationships between numbers while the textbook points out the relationships which exist. Tables 11;, 15 and 16 show that between seventy per cent and ninety per cent of the teachers thought that the three experimental lessons rather than the textbook lessons gave the

121

Table ll; Analysis of Questionnaire for Grade IV Lesson A N

Text Per Cent

A-l

1

10

9

90

10

A-2

1

10

9

90

10

A-3

1

10

9

90

10

k-k

2

20

8

80

Total N

Question

10

Experimental N Per Cent

Key Questions A-l. Question 2 on page 273 of your textbook tells the purpose of the denominator and numerator of a fraction. Questions 9 and 10 on experimental lesson A uses a different approach for the same concept. Which do you prefer? A-2.

To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working "with number. Which approach does this better?

A-3. Children should became familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better? A-l;. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

i;;

Table 15 Analysis of Questionnaire for Grade IV Lesson 8 ? C IT

Question

N

■ £e:i* , Per Cent

N

Per Cent

10

B-l

3

30

7

70

10

B—2

2

20

8

80

10

B-3

3

30

7

70

10

B—U

3

30

7

70

Questions B-l. The introductory problem on page 15k of your textbook, and the introductory problems 1 and 2 in experimental lesson B both set the stage for learning one of the 6 tables. Which approach do you prefer? B-2.

To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in ■working with number. Which approach does this better?

B-3. Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better? B-1+. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

Table 16 Analysis of Questionnaire for Grade IV Lesson C Total N

Question

N

Text Per Cerrb

Experimentai N Per Cent

10

C-l

h

1*0

6

60

10

C-2

2

20

8

80

10

C-3

1

10

9

90

10

C-1+

3

30

7

70

Kgr Questions Question 2 on page 201+ of your textbook and questions 7 and 8 on the experimental lesson C teach the same concept. "Which approach do you prefer? To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach do you prefer? In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

children more opportunity to have first hand experiences with numbers and more opportunity to become familiar with relationships which exist between arithmetic facts and processes* The data compiled and presented in Appendix V for question 1; are a tabulation of the fourth grade teachers* reasons for preferring the experimental lessons to those in their present text­ book. Analysis of these data reveals that fourth grade teachers prefer the experimental lessons because: 1. The experimental lessons create a problem situation to stimulate thinking on the part of children. 2. The number activities are interesting to children and the varied approaches in the lessons give the children a better under­ standing of the use of processes. 3. The experimental lessons guide the child into doing his own thinking instead of telling him how to solve number situations. 1+. The lessons provide more first hand number experiences for children to participate in than the textbook. The teachers preferring the textbook lessons like them because: 1. The directions are simple. They felt that too many approaches to problem solving would confuse children. 2. They felt that the textbook lessons would be easier for slow children to grasp.

Interpretation and Analysis of Tables 17, 18 and 19 Tables 17> 18 and 15 are a summary of the reactions of fifth grade teachers to the arithmetic materials submitted to them for evaluation.

It nri.ll be noted on Table 17 question A-l that

seventy per cent of the teachers preferred the approach used in the experimental lesson to guide the children into figuring out steps in a process by themselves.

Thirty per cent of the teachers favored

the step by step telling in the textbook lesson. Table 18, question B-l shows that ninety per cent of the teachers preferred the experimental lesson because the children were asked in it to state a rule or generalization after figuring out solutions to several problems. Only ten per cent of the teachers chose the textbook lesson and said they liked it because it stated the rule for the children. Question C-l on Table 19 reveals the fact that ninety per cent ofthe teachers prefer the experimental lesson because it gives specific directions to start the children to work. Ten per cent like the textbook lesson because it did not give directions to the children. These three tables point out the fact that the majority of the fifth grade teachers felt that the experimental lessons gave the children more opportunity to have first hand experiences with number and more opportunity to understand inherent relationships between processes.

Table 17 Analysis of Questionnaire for Oracle V Lesson A Total N

Question

10

Experimental N Per Cent

N

Text Per Cent

A-l

3

30

7

70

10

A-2

3

30

7

70

10

A-3

h

iiO

6

60

10

A-U

3

30

7

70

Key Questions A-l.

The explanation for problem 1 in the t extbook lesson on page 60 tells the children the steps to follow to solve the problem. Questions 3 - 10 in experimental lesson A guide the children into figuring out for themselves the steps to follow for a solution. "Which approach do you prefer?

A-2. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? A-3.

Children should become familiar with the relationships which exist between various arithmetic facts and processes. "Which approach does this better?

A-li. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

\2'7

Table 18 Analysis of Questionnaire for Grade V Lesson B Total N

Question

N

Text Per Cent

10

B—l

1

10

9

90

10

B-2

1

10

9

90

10

B-3

1

10

9

90

10

B—ij.

1

10

9

90

Experimental N Per Cent

Key Questions B-l.

In experimental lesson B the children were asked to state a rule after figuring out the process. In the textbook lesson the rule was stated for them. Which approach do you prefer?

B-2. To understand arithmetic and use it effectively children must be given an opportunity to have the first hand experiences in working with number. Which approach does this better? B-3. Children should become familiar with the relationships which exist between various arithmetic facts and processes. Which approach does this better? B—14..

In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

Table 19 Analysis of Questionnaire for Grade V Lesson C N

Text Per Cent

C-l

1

10

9

90

10

C-2

1

10

9

90

10

C-3

2

20

8

80

10

C-ii

2

20

8

80

Total N

Question

10

Experimental N Per Cent

Xegr

Questions In experimental lesson C specific directions were given to start the children to work. The textbook lesson on page 186 does not give directions to the children. Which, approach do you prefer? To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? Children should become familiar with the r elationships which exist between various arithmetic facts and processes. Which approach does this better? In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

The data compiled and presented in Appendix V for grade five are a tabulation of the fifth grade teachers reasons for preferring the experimental lessons to the ones in their current textbook. Analysis of these data reveals that fifth grade teachers prefer the experimental lessons because: 1.

The children reason out their own steps to follow instead

of having someone give them steps to follow. The experimental, therefore, forces more thinking upon the students. 2. The children formulate their own rules on the basis of number experiences in many different situations. This makes the generalization more meaningful to them. 3. The lessons show relationships between processes which the children can figure out for themselves instead of having to be told. The teachers preferring the textbook lessons said that the explanations in them were concise and that they preferred the use of a zero in the partial products in multiplication. Interpretation and Analysis of Tables 20, 21 and 22 The r eactions of sixth grade teachers to the arithmetic materials submitted to them for evaluation are reported in tables 20, 21 and 22. Question A-2 on Table 20 points out that the sixth grade teachers prefer this experimental lesson because it provides opportunities for the children to figure out steps in adding decimals by themselves. There was one hundred per cent agreement by the

teachers on this question.

The matched textbook lesson told the

children step by step the process to follow in adding decimals. Ninety per cent of the teachers thought that Experimental Lesson B gave the children a better understanding of the nature of the number system than the textbook lesson.

These results are shown

on Table 21, in question B-2. Table 22, question C-l reveals the fact that seventy per cent of the sixth grade teachers thought that the experimental lesson did a better job of showing number relationships than did the corresponding textbook lesson. Tables 20, 21 and 22 point out the fact that the majority of the sixth grade teachers felt that the experimental lessons gave children more opportunity to have first hand experiences with number and more opportunity to understand and see inherent relationships between processes. The data compiled and presented in Appendix V for grade six are a tabulation of the sixth grade teachers reasons for preferring the experimental lessons to the ones in their current textbook. Analysis of these data reveals that sixth grade teachers prefer the experimental lessons because: 1, The lessons show relationships between processes and give the children opportunities to figure out these relationships for themselves. 2. The lessons build on previous knowledge and go frcan the

Table 20 Analysis of Questionnaire for Grade VI Lesson A Total N

Question

10

N

Text Per Cent

Experimental W Per Cent

A-l

0

0

10

100

10

A-2

0

0

10

100

10

A-3

0

0

10

100

10

A-li

0

0

10

100

Key Questions To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? The explanation for problem 1 on page 1*1 of the textbook lesson tells the children how to add decimals- Questions 6 - 11 in experimental Lesson A guide the children into figuring out for themselves how decimals are added. Which approach do you prefer? Children should become familiar with the relationships which exist between various arithmetical facts and processes. Which approach does this better? In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

Table 21 Analysis of Questionnaire for Grade VI Lesson B Total N

Question

N

Text Per Cent

10

B-l

1

10

9

90

10

B-2

1

10

9

90

10

B-3

2

20

8

80

10

B-l*

1

10

9

90

Experimental N Per Cent

Key Questions B-l. To understand arithmetic and use it effectively children must be given an opportunity to have first hand experiences in working with number. Which approach does this better? B-2. An understanding of the nature of our number system is important in teaching arithmetic meaningfully. In these lessons which approach is better? B-3* Children should become familiar with the relationships which exist between various arithmetical facts and processes. Which approach does this better? B-lj.. In comparing these lessons which approach to this specific phase of arithmetic do you prefer? Please state briefly the reasons for your decision.

JL«.-' Vol. 1*7, No. 1. 3. Brownell, William A., "An Experiment on Borrowing in Third Grade Arithmetic." Journal of Educational Research, Vol. 1*1, November 191*7. !*• Brownell, William A., "Teaching of Mathematics in Grades I Thru VI." Review of Educational Research, Vol. 15, October 191*5. 5. Buckingham, R. B., "Significance, Meaning and Insight— These Three." Mathematics Teacher, Vol. 31, January 1938. 6. Cooke, Dennis H., "A Usable Philosophy in Teaching Arithmetic." Mathematics Teacher,Vol. 1*1, February 191*8. 7*

Horn, Ernest, Methods of Instruction in the Social Studies. Charles Scribners Sons. New York, 1937*

8. Kelly, F. J., "The Results of Three Types of Drill on the Fundamentals of Arithmetic." Journal of Educational Research, Vol. 11, November 1920. 9.

Knipp, Minnie B., "An Investigation of Experimental Studies Which Compare Methods of Teaching Arithmetic." The Journal of Experimental Education, 19l*l*. Vol. 13.

10. McConnell, T. R., "Discovery Versus Authoritative Identification in the Learning of Children." University of Iowa Studies in Education, Vol. 9> No. 5> 193l*» McConnell, T. R., "A Controlled Experiment in Learning of the One Hundred Addition and the One Hundred Subtraction Facts." Doctor's dissertation, University of Iowa. 1933* 11. Monroe, Walter S., and Engelhart, Max D., A Critical Summary of Research Relating to the Teaching of Arithmetic. College of Education, Urbana, The University of Chicago. Bulletin No. 58 Bureau of Educational Research, 1931*

if'o

12. Monroe, Walter S., Eighteenth Yearbook, Part II. National Society for the Study of Education. Public School Publishing Company, Bloomington, 111. 1919* 13* Morton, Robert L., “Estimating Quotient Figures When Dividing by Two Place Numbers." Elementary School Journal, Vol. 1*8, November 19U7* XU. Morton, Robert L., "The National Council Committee on Arithmetic." Mathematics Teacher, Vol. 31, No. 6, October 1938. 15* National Council of Teachers of Mathematics, Tenth Yearbook, "The Teaching of Arithmetic." Bureau of Publications, Teachers College, Columbia University, 1935- Chapters by Brownell, Brueckner and Thiele. 16. National Council of Teachers of Mathematics, Sixteenth Yearbook, "Arithmetic in General Education." Bureau of Publications, Teachers College, Columbia University, 191*1. Chapters by Brueckner, McConnell and Thiele. 17* National Society for the Study of Education, Thirtieth Yearbook, Part II, "The Textbook in American Education." Public School Publishing Company, Bloomington, 111., 1931* 18. National Society for the Study of Education, Forty-First Yearbook, Part II, "The Psychology of Learning." University of Chicago Press, Chicago, 19h2. McConnell, T. R., "Introduction." McConnell, T. R., "Reconciliation of Learning Theories." \

19. Osburn, Worth J., "Levels of Difficulty in Long Division." Elementaiy School Journal, Vol. 1*6, April 19h&* 20. Schorling, Raleigh, The Teaching Of Mathematics. Press, Michigan, 193&-

Ann Arbor

21. Smith, David E., "Arithmetic for Intermediate Grades." Classroom Teacher, Vol. 7> Chicago. 22. Smith, H. E., Eaton, M. T., Dugdale, Kathleen, One Hundred Fifty Years of Arithmetic Textbooks♦ Bulletin of the School of Education, Indiana University, Vol. 31> No. 1. 191*5. 23. Spencer, Herbert, Education. D. Appleton Century Company. New York. 1920.

1F7 2l*. Spitzer, Herbert F., The Teaching of Arithmetic. Houghton Mifflin Company, Boston, Mass. 25.

Stevens, Marion Paine, "Teaching Arithmetic: Some Important Trends in Our School Today." Grade Teacher, Vol. 62, April 1916-

26.

Stroud, James Bart, Psychology in Education. Longmans Green and Company. New YorkI 191*6.

27.

Supplementary Educational Monographs Number 27• Buswell, Guy T., and Judd, Charles H., Summary of Educational Investigations in Arithmetic, Chicago, University of Chicago. 1925*

28. Thiele, Carl L., "The Contribution of Generalization to the Learning of the Addition Facts." Contributions to Education, No. 763* Bureau of Publications, Teachers College, Columbia University. 1938. 29. Wheat, Harry G., The Psychology and Teaching of Arithmetic. D. C. Heath Company, Boston, 1937* P. 1*0. 30. "Wheat, Harry G«, Studies in Arithmetic. Morgantown, Office of the President, West Virginia University. 191*5. 31. Wilson, GuyM., Stone, Mildred B., and Dalrymple, Charles, Teaching the New Arithmetic. McGraw-Hill. 1939* 32. Young, J. W. A., The Teaching of Mathematics in the Elementary and Secondary School. Longmans Green and Company, New York. 1921*. Textbooks 1.

Brueckner, Leo J., Grossnickle, Foster E., Merton, Elda L., Arithmetic We Use. Books. Three, Four, Five and Six. John C. Winston Company. New York. 19U8T

2. Buswell, G. T., Brownell, W. A., John, Lenore, Jolly Numbers Primer, Jolly Numbers, Book One. Jolly Numbers, Book Two, Ginn and Company, New York. 19lfl*.

if;8

APPENDIX I Reports of observations made in fifty elementary arithmetic classrooms for Part A of Investigation I.

Grade 1

2:30— 3:00

31 children

Report 1

The teacher ■wrote the number 3 on the blackboard and the number word three beside it while the class watched her. Then she said, "John, pick out three boys and bring them to the front of the room. Mary, pick out three girls and bring them to the front of the room." Wien the children were standing in groups of three at the front of the room the teacher asked the class to help her count the boys and then the girls. The teacher then asked, "Did John and Mary bring the correct number of boys and girls to the front of the room?" The class answered, "Yes." The teacher called on several children and asked them to count various things in the room. She said, "Bill, will you go to the back of the room and touch three chairs? Count aloud as you do this." When a child completed this exercise by himself the teacher would ask the rest of the class to count them again with him. The attention of the class was then directed to the black­ board and the number 3 and the word three. The teacher asked the class to watch as she pointed to the number three. She asked them to say the word three as she pointed to it. As the next step the teacher wrote the following numbers on theblackboard: 3 1 2 31 3 2. She asked threedifferent children to come to the blackboard and draw acircle around one of the three's written there. Following this exercise she wrote several number words on the blackboard. The words were: three one three one three. She asked 3 of the children to come to the blackboard and draw a line under the word that said three. When they had finished she asked the class if they had drawn lines under the correct words. The class answered, "Yes." The teacher then wrote this exercise on the blackboard: Three Two One

1 3 2

She asked different children to come to the blackboard and draw a line between the word and number that said the same thing. When the exercise was finished the teacher asked the class to count the numbers and say the words she pointed to on the blackboard.

Grade 1

1:10— 1:25

21; children

The teacher supplied each child in the room with cut in the form of a house. She asked each child to write paper the number of their home. She walked about the room various children their house number and checking to see if written it correctly on the paper house.

Report 2

a paper on the asking they had

When this exercise was completed the teacher asked these children to raise their hands who had brought dimes for a toy show the school was going to see on the following day. Several children raised their hands. The teacher explained that they would need to pay only nine cents to get into the show so they should have some money left over. She explained that she would take the money the children had brought and give them back their change, but that they would have to help her count it. She asked the class to watch her write the number 9 on the blackboard and she asked one child to locate nine on the number chart on the bulletin board. Then she asked, "How many pennies do you have when you have a dime?" The class answered, "10 pennies." Then the teacher asked, "How many pennies does the show cost?" The class answered, "9 pennies." "How many pennies will you get back if you brought a dime for the show?", asked the teacher. The class answered, "One penny." The teacher asked one child to bring his dime to a table at the front of the room. She gave him ten pennies and asked that he lay them out on the table. He did so and then the teacher asked him to take away the 9 pennies the show would cost. She asked him to count out loud. The teacher then asked how many pennies were left and the child answered, "One." The teacher asked all the children to come to the front of the room who had brought dimes and she gave them back their change. She counted aloud each transaction and frequently asked the class to help her.

Grade 1

1:10— 1:25

32 children

Report 3

The teacher started the lesson by asking how many children were absent this afternoon. The class answered that 3 children were absent. The teacher asked them how they knew that three were absent. One child spoke out and said, "By counting." The teacher

asked, "What is counting good for?" answers. 1. 2. 3. U.

To To To To

Different children gave these

tell how old you are. count your money. count books. find out how many days until your birthday.

The room contained several bulletin boards and various groups of pictures were on each board. The teacher asked individual children to go to the groups of pictures, to count them and then to place the number that told how many on the blackboard. The numbers the children placed on the blackboard were— 3, I4, 6 , 9 and 7* The teacher then asked, "Who can arrange these numbers in order?" One child went to the blackboard and began to arrange the numbers in order. The teacher suggested that he leave spaces for numbers that were not there. The child placed the numbers on the blackboard in this manner 3 I4. 6 7 9* The teacher then asked different children to go to the blackboard and fill in the empty spaces. One child wrote in 5 and 8 and another wrote in 1 and 2. The teacher and class then counted aloud all the numbers written onthe blackboard in order. The teacher directed the attention of the class to a small table at the front of the room on which books were piled. She asked the class to count the books with her as she pointed to them. There were six books in all on the table. The teacher re-arranged the books into two piles with 3 in each pile. Then she asked one child to come to the front of the room and count the first pile. He counted to three and she asked him to write the number on the black­ board. Another child counted the 3 in the second pile and placed that number on the blackboard. The teacher asked, "How much are three and three?" The class all called out, "Six." The teacher suggested that they count with her to be certain. She then asked, "Can we say that three and three make six?" The class answered, "Yes."

Grade 1

1:30— 2:00

30 children

Report Ij.

The teacher asked the class to come to the front of the room and sit in a circle on small chairs. She reviewed for them a game called the "Shoemaker's Dance" they had played the day before. She asked if the class remembered how many children had played in

the game. They answered that two children had played. The teacher called one child to the front of the room and asked if that child could play the game by herself. The class answered no, and one boy replied that another child would be needed as it took two children to play the game. The teacher said that he was correct and pointed out that one was an odd number but two was an even number because it took two children to play. The teacher then called the attention of the class to a list of numbers written on the blackboard. She said, "Look at these numbers I have written on the blackboard and tell me which are odd numbers and which are even. Yi/hen you tell me I will write the words odd and even after each number." She pointed to the first number and the class called out, "Odd." This technique was used for all the numbers on the blackboard. 1 2 3

h 5

odd even odd even odd

6 7 8 9 10

even odd even odd even

The teacher called the attention of the class to the fact that every other number on the blackboard was an odd number. She asked one child to read all the odd numbers and another to read all the even numbers she had written on the blackboard. The teacher then took out a large book and placed it on the chalk tray of the blackboard. She explained to the class that they were going to work again today in their number readiness book. She asked if anyone remembered the number they had. talked about yesterday. One child raised his hand and said the number was I;. The teacher called the attention of the class to a new picture in the book and pointed out that it showed a grocery store. She asked the class to look carefully at the picture and pick out groups of 5 things. She pointed out a group of five cans of fruit to show what she meant. Different children pointed to various groups of 5 shovm in the picture. VJhen each group of 5 shown in the picture had been identified the class was asked to return to their seats. The teacher started the class in the singing of the song, "One Little, 'Two Little, Three Little Indians." One group had an opportunity to play musical chairs while the rest of the class sang the song.

Grade 2

11:10— 11:30

35 children

Report 5

The teacher wrote the number 118 on the blackboard. She asked the children to look at the number she had placed on the blackboard and find that page in their textbooks. The teacher asked the class to read each sentence in the number story carefully. She called on individual children to read each sentence and to answer the question it asked. The problems were addition and subtraction and when a child had difficulty figuring out the answer or didn't know it the teacher would help him. She helped him by figuring out the answer on a home made number board made of rope and clothes pins. She would show six on the board and then take away three of the clothes pins and explain to the class that three were left. 'Therefore," said the teacher, "Six take away three leaves three." The class completed the page with the teacher's help in this manner. The teacher asked the class to take out a piece of paper and their pencils. She asked them to turn to page 121 in their textbook and read each problem carefully. When they had read the problem they were to work out the solution on their papers. She asked the class to raise their hands if they needed help.

Grade 2

1:25— 1:1|0

28 children

Report 6

The teacher asked the class to turn to page 13 in their arithmetic workbooks. This page contained a number chart with numbers to 100 written in it. Several of the numbers were omitted and the directions asked the children to write in the missing numbers. The teacher started with the first row of numbers and called on several children to supply the missing ones. Each child in the class had an opportunity to give one number. The teacher then wrote similar exercises on the blackboard such as: 21; - - 2?. She asked different children in the room to supply the missing numbers. The children gave all the numbers orally. No paper or pencils were used during the lesson.

t m

Grade 2

2: £5— 3 **30

32 children

Report 7

The teacher asked the children to listen carefully to the number, story she was about to tell. She made up this story. "Two of the Brown girls came to visit the three Black girls. They spent the afternoon playing with dolls. How many girls were playing together?" Before the class could answer the teacher called five girls to the front of the room and asked them to act out the number story. The girls tried to do this. Vtfhen they had finished the teacher asked, "How many girls were playing?" The class answered, "Five." The teacher went to the blackboard and placed this example on it. 3 + 2 = £ She asked the class to repeat after her this statement, "Three plus two equals five." The teacher made up a similar~story using boys. This too was acted out and the statement 2 + 3 = 5> was placed on the black­ board. The teacher then called a child to the front of the room, gave him £ flash cards and asked him to make up a story showing that 3 plus 2 equals 5. He made up this story, "I have 3 cards and 2 cards. Therefore I have 5 cards. The teacher asked the class to line up along the black­ board and told them that they were going to play train. She gave one child a stack of flash cards with number combinations written on them. This child moved down the line showing a different card to each child. 1/i/hen a child missed a card he went to the foot of the line. If a child went down the whole line and no one missed a card the teacher told the class that they had had a good ride.

Grade 2

1:U5>-- 2:00

15> children

Report 8

The teacher asked the class to turn to page 20 in their textbooks. She explained that they were going to learn about two arithmetic words called tall and short. She asked individual children to reach each sentence on the page and then to tell the other children how they would answer the question asked in the sentence. After the page was completed in this manner the teacher called the attention of the class to a picture on the same page. This picture showed a tall child and a short child. She asked a child to point to the taller child and another to point to the shorter child.

The teacher then called a tall child to the front of the room. She asked the class to pick out a short child to stand with him.One small second grader was chosen by the group and she went to the front of the room to stand beside the taller child. The teacher asked the class to make up two sentences about these children. She placed the sentence on the blackboard as individual children stated them. John is taller than Sue. Sue is shorter than John. The teacher then asked the class to name things that were tall or short. Several children named the following things: 1. 2. 3. I*.

Pencils People Apples (much discussion on this point) Buildings

The lesson ended when the teacher made this statement, "Some things are taller than others while some things are shorter than others."

Grade 2

1:00— 1:20

37 children

Report 9

The teacher passed out a number of colored sticks to each pupil in the room. They were told to lay out 10 sticks on their desks and to hold the others in their hands. Then the teacher instructed each child to lay one of the sticks in his hand down beside the other 10. She asked this question, "Larry, how many sticks do you have now?" Larry replied that he had eleven sticks. The teacher repeated this process for several number combinations. The combinations were, ten and two, ten and seven, ten and five, and ten and ten. Different children were called upon to count the sticks and to state the number on their desks. The usual answer was, "Ten sticks and seven sticks make seventeen sticks.11 The teacher then told the class that since their arithmetic textbook did not have the right kind of exercise in it she had made up one. She asked them to look carefully at the paper she was giving each of them. She said, "On this paper I have given you there are a number of pumpkins. Who can tell me how many there are?" One child raised his hand and answered, "20 pumpkins." The teacher asked the class to write this number on their papers. She then

ih in their number work­ books. She called upon a child to read the first problem and on another to give the answer. Then she asked a child to go to the blackboard and draw a picture showing what the problem told. The page was completed in this manner.

Grade 2

1:25— 2:00

28 children

Report 11

The following number story was written on the blackboard in this room. Number Story One day I went to the store and got seven toys. The next day I got three more toys. How many toys did I have? I had ten toys. 7 +3

10

The teacher introduced the lesson by telling the class a story about a clock. She held up a clock and made up a story about its hands. She called one hand Mr. Slow and the other Mr. Fast. When she had finished she drew a clock on the blackboard. She asked the class this question. "When does school start?" The class answered in a chorus, "Nine o ’clock." She called one child to the front of the room and asked him to make nine o'clock with the hands. When he had finished the teacher set the hands at six o'clock and asked a child to name the time. The teacher then drew two clocks on the blackboard. She asked one child to make five o'clock and another to show seven-thirty o'clock with chalk. When this work was completed the teacher held up the clock she had and asked the class what the small dots between the hours meant. The class answered, "Minutes." The teacher then asked, "How many dots are between the hours?" The class answered, "Five." The teacher asked the class to take out their arithmetic workbooks and turn to page 19. She asked them to read the directions on that page and to make the time asked for on the clocks with crayons. The children spent the remainder of the period doing this work.

Grade 2

9*.00— 9:15

23 children

Report 12

The teacher asked the class to turn to page 86 in their textbooks. She chose one child to act as teacher. That child read a question from the textbook and called upon another child to answer it. The teacher then asked different children to go to the black­ board and write down the answer that was given. The page was completed in this manner. The teacher asked several children to make up number problems similar to those they had read in their textbooks. Each child who made up a story called upon another to give the answer. When the children had tired of this exercise the teacher told them to take out their arithmetic workbooks and do all of the problems on page 86. She moved about the room helping each child, who had difficulty.

Grade 3

Grade 3

9:05— 9:35

23 children

Report 13

The teacher placed the following examples on the blackboard 12

Ik

16

18

-6

-7

-8

-9

She used flash cards stacked in piles on the chalk tray to illustrate each of the examples for the class. For example— she placed twelve cards on the tray and then took away six. She counted those remaining and made the statement to the class that "12 take away 6 leaves 6 ." When she had finished illustrating each example she asked for questions. The class had none. The teacher then asked the class to turn to page 88 in the textbook. She told the class that since today was the first time they would meet entirely new problems they should try hard to complete them. She askedthemto do all the story problems on pages 88— 90 in their textbook. She asked those children who might have difficulty to raise theirhands and said that she would come to their desks to help them. She told the class that when this work was finished they should try the examples at the bottom of page 90 in their textbooks.

Grade 3

3:00— 3:30

i|0 children

Report lU

The teacher asked the class to turn to page 180 in their arithmetic textbooks. The teacher read aloud the introductory material and the first problem. She called the attention of the class to the fact that Joe (boy in book), when adding the problem 37 added the 7 and 5 first. She added the problem orally for +hS the class explaining each step as she went along. The teacher took example 2 from thetextbook and placed it on the blackboard. She asked the class to tell her how many tens and ones werecontained in each number. These she wrote out after the example. It took this form:

f/'

3U 25 36 "95

3 2 3 " 8

tens lj. ones tens 5 ones tens 6 ones tens15 ones

The teacher explained to the class that they learned that numbers were macle up of tens and ones by doing exercises such as these. She asked the class to turn to page 9$ in their textbooks and work the first two rows of addition and subtraction examples.

Grade 3

textbooks. answer it. manner.

10:lj.5— -11:00

30 children

Report 15

The teacher asked the class to turn to page 78 in their She asked one child to read each problem and another to The class covered each exercise on the page in this

The class was then instructed to turn to page 8I4. in their textbooks and look carefully at the picture on that page. The teacher asked, !tWhat do you see in the picture?" The class did not respond so the teacher explained the picture to them. The teacher called upon a child to read the first problem on that page. She asked another to give the answer. The child could not figure out the answer so the teacher placed the problem on the blackboard and worked it for the class explaining each step as she went along. These problems all involved subtraction. The same procedure was followed for the three other problems on the page.

Grade 3

10:30— 11:00

26 children

Report 16

The first part of the lesson consisted of a flash card drill on the subtraction facts. The teacher held up various cards and the class as a whole gave the answer. The teacher then asked the class to turn to page 66 in their arithmetic textbooks. The class and teacher read in a chorus a rhyme about telling time found on that page. The teacher spent several minutes explaing to the class how to tell time after the hour. For example: She explained that 2:l5 o ’clock could be read as ’’two fifteen,” "a quarter past two," or "fifteen minutes past two."

Page 66 in the textbook suggested that the problems on that page be illustrated with a toy clock if one was available. This teacher had a clock in the room so as a child answered one of the problems on the page the teacher set the clock to show the time stated. Each problem on the page was completed in this manner. A child read the problem, gave the answer and then the teacher showed the time on the clock. The teacher passed out work sheets to each child in the room with clock faces showing various times. She asked the children to write with figures the time shown on these clocks. She walked about the room helping individual children.

Grade 3

1:20— 2:00

3b children

Report 17

The teacher asked the class to turn to page 53 in their arithmetic textbooks. This page contained an illustrated showing groups of pencils tied into bundles of ten. The teacher asked the class to count with her the groups of ten shown in the illustration. The class counted ten groups with ten pencils in each group. The teacher asked, "How many pencils have we counted?" The class replied, '’One hundred pencils." The teacher pointed out that one hundred meant ten groups of ten. The and called on completed the of ten would

teacher then read aloud each problem in the textbook various children to give the answers. "When she had page she asked, "If I had 50 pencils, how many groups I have?" The class answered, "Five groups of ten."

The teacher asked several questions of this type: "If I had 92 pencils, how many groups of ten would I have? Would I have any pencils left over?" When this exercise was completed the teacher wrote a series of three place numbers on the blackboard such as: 3^7 > kh$She asked the class to tell her how many groups of hundreds, tens and ones were contained in these numbers. The teacher then instructed the class to turn to page 55 in their textbooks and read carefully all problems on that page. She asked them to begin to solve them when they had finished the reading.

Grade 3

9:50— 10:30

17 children

Report 18

The teacher asked the class to turn to page l5l in their textbooks. She called nine children to the front of the room. These children took turns reading the questions or statements on page l5l to the rest of the class. They called on the children sitting at their desks for the answers. The page was completed in this manner. The page the children were working on involved time. The teacher held up a clock and asked several children to read the time shown on the clock. She set the hands at 11:30, 1:00, 2:l5, and 5:50. When a child read the clock incorrectly the teacher called on another to help him out. Ten minutes of the class time was devoted to this exercise at the end of the period.

Grade 3

10:30— 11:00

19 children

Report 19

The class began with an oral arithmetic exercise. The teacher asked the class to add the numbers she gave in their heads and then raise their hands when they had the answer. She gave numbers such as these, "6 plus 3 minus 7 plus 8 ." About ten minutes of the period was used for this exercise. The teacher asked several children to go to the blackboard and write two figure numbers and three figure numbers. The teacher took two of these numbers and used them to show the class how to add large numbers. She worked two problems aloud for the class. One problem used two figure numbers and the other three figure numbers. She explained the process step by step. The teacher asked the class to get out paper and pencil and add the numbers she gave. She sent one child to the blackboard to work. The teacher dictated nine addition problems involving two and three figure numbers. She would dictate a problem and the children would then copy and work it. The first child completing the problem raised his hand and gave the answer to the rest of the class. When this work was completed the teacher asked the class if they could add 1,000’s as easy as 100's. The class said, "Yes.” The teacher placed this example on the blackboard and added it aloud for the class.