AN ANALYSIS or THE EFFECTWENESS.QF THE ' WORKSHOP AS AN m-smwca Means; spa IMPROVING MAYHEMATICAL UNDERSTANDINQQ; . . _ _i_'. OF ELEMENTARY SCHOOL TEACHERS ' Thssis far fhe Degree of? Ph. D. MICHEGAN S‘M‘FE umvsksm Mildred Jeriine Dossefl' 19634 7 ”“5 l\l\\\\\\\\\\\\\\\\\ . F This is to oertifg that the thesis entitled AN ANALYSIS OF THE EFFECTIVENESS OF THE WORKSHOP AS AN IN-SERVICE MEANS FOR IMPROVING MATHEMATICAL UNDERSTANDINGS OF ELEMENTARY SCHOOL TEACHERS presented by Mildred Jerline Dossett has been accepted towards fulfillment of the requirements for l fl_ degree in W Major professor LIBRARY Michigan State University ABSTRACT AN ANALYSIS OF THE EFFECTIVENESS OF THE WORKSHOP AS AN lN—SERVICE MEANS FOR INIPROVING NIATHENIATICAL UNDERSTANDINGS OF ELEMENTARY SCHOOL TEACHERS by Mildred Jerline Dosset‘t Adviser: Calhoun C. Collier The purpose of this study was to analyze the effectiveness of the workshop as a means of in-service education for elementary teachers for: (l) improving basic mathematical understandings, (Z) changing attitudes toward mathematics, and (3) improving the classroom practices in the teaching of arithmetic. The two workshops used in this study were drawn from the 1963-64 series of workshops in mathe- matics for elementary school teachers, a part of an in-service educa- tion program being sponsored by the N'Iissouri State Department of Education under Title III, National Defense Education Act. The two state-sponsored workshops included sixty-seven participants, of whom forty-five were primary teachers and twenty- two intermediate teachers. These, along with the twenty-two teachers in the control group, were grouped, for purpose of analysis, as follows: (1) school system, (2) level of teaching assignment, (3) completion of in-service program, and (4) testing procedure. The following hypotheses were advanced: Mildred Jerline Dossett There will be a significant difference on a test of mathematical understandings between the post-te st scores of a group of elementary teachers who participated in a mathematics workshop and their pre -test scores or the scores of a similar group who did not participate. There will be a significant difference on an arithmetic attitude inventory between the post-te st scores of a group, of elementary teachers who participated in a mathematics workshop and their pre -test scores or. the scores of a similar group who did not participate in the workshop. There will be a significant difference between the class practices of a group of elementary school teachers who participated in the mathematics workshop and a similar group who did not partici- pate. In addition, two related questions were studied: What effect, if any, does a pre -te st have upon the post ~te st scores of a group of teachers who participated in the mathe- matics workshops? What is the relationship of improvement in basic mathematical understandings or change in attitude toward arithmetic to teacher background factors? The investigation proceeded as follows: Mildred Jerline Dossett l. A form entitled "Teacher Background Information" was dis- tributed to participants in the state-sponsored workshops and the control group. These data were transferred to a code sheet and used in the final analysis. 2. Two parallel forms of a ”Test of Mathematical Understandings" were constructed, tried out in experimental form, revised, and administered to both the experimental and control groups at or near the first and last sessions of the two mathematics work- shOps. 3. The Dutton Arithmetic Attitude Inventory was administered to the experimental and control groups at the same time that the groups took the "Te st of Mathematical Understandings” for the purpose of studying the effectiveness of the workshop in attitude changes toward arithmetic. 4. The Texas Classroom Interview Question Schedule was used to elicit responses from a random selection of two pupils from eight classrooms of teachers in one of the experimental groups and an equal number of pupils from classrooms of teachers in the control group. Such a procedure provided data relative to the classroom practices of teachers in the teaching of arithmetic. The analysis of covariance was utilized in the analysis of the data relative to the effectiveness of the workshop. Pearson Product- Mildred Jerline Do ssett Nioment Correlation Coefficient was used in the relationship analyses reported in the study. Conclusions which were an outgrowth of the findings of this study were: 1. Workshop participants, with the exception of one group of primary teachers, made statistically significant gains between pre- and post-te st on a test of mathematical understandings and on an arithmetic attitude inventory. When scores of a group of elementary teachers who participated in the mathematics workshop were compared with those of a group of teachers who did not participate in the workshop, it was found that, with initial differences allowed for, the workshop had contributed both to the development of mathematical under- standings and to a change in attitude toward arithmetic. Within the limitations of this study, and for the particular group used, a pre -test did not contribute significantly to the develop- ment of mathematical understandings or to a change in attitude toward arithmetic. A significant relationship was found to exist between the pre- and post -test scores on the mathematical understandings.test and semester hours credit in college mathematics; no significant re- lationship was found to exist between either the pre-te st or post— test scores of an attitude inventory and background factors. AN ANALYSIS OF THE EFFECTIVENESS OF THE WORKSHOP AS AN IN-SERVICE MEANS FOR IMPROVING MATHEIVIATICAL UNDERSTANDINGS OF ELE hlENTA RY SCHOOL TEACHERS BY Pvlildred Jerline Dossett A THESIS Submitted to The Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY College of Education 1964 71ml,» 25' Fl '14:! ACKNOW LE DGMENTS Nlany persons have contributed to the planning and preparation of this dissertation. The writer is deeply grateful to all of these people. Dr. Calhoun C. Collier, under whose direction the study was planned and executed, gave generously of his time. His guidance and con— structive criticism were of inestimable value. The writer's guidance committee, Dr. Calhoun C. Collier, Dr. Charles Blackman, Dr. Duane L. Gibson, and Dr. Jean LePere, made valuable suggestions concerning the manuscript. Dr. William L. Allison, Dr. James M. Drickey, Dr. Vincent J. Glennon, Dr. W. Robert Houston, Dr. Lois Knowles, and Mr. Joe L. Wise offered useful criticisms of the testing instrument developed for use in this study. Appreciation is also expressed to Mr. Carleton Fulbright, Director of Title III, National Defense Education Act, Missouri State Department of Education, IVlr. Joe L. Wise, Niathematics Consultant from the lVlissouri State Department of Education, and the superin- tendents of the participating school districts for their assistance. The COOperation of principals, teachers, and pupils: who were involved in this study, is also gratefully acknowledged. Since a large ii part of the study was in the area of human relations, the experience of finding people willing to help in evaluating this iii-service education venture was gratifying indeed. To her mother, the writer is indebted for interest, encourage- ment, and unselfish love. Without this, graduate study would not have been possible. lViiss Jane Grebe, who typed the manuscript, has been a partner and unwavering friend. She has rendered service far beyond mere copying of the study. It is apleasure to acknowledge the part each person had in bring- \ ing the study to a successful completion. M. J. D. Michigan State University East Lansing, Michigan ‘ August, 1964 TABLE OF CONTENTS ACKNOWLEDCINIENTS . LIST OF TABLES LIST OF APPENDICES Chapter I. THE NATURE OF THE STUDY The Need for the Study . The Problem . . . Statement of Problem Hypotheses The Research Design The Limitations of the Study—- . Definition of Terms Summary . II. REVIEW OF RELATED LITERATURE Improving Mathematical Understandings of Elementary School Teachers . Utilizing Iii-service Education Procedures The Workshop Idea . . . . . Iii-service Education for Teachers of Mathematics . . . . . . .i . . Changing the Attitudes of Elementary School Teachers Toward Arithmetic . Improving the Classroom Procedures of Elementary School Teachers in Teaching Arithmetic . Sunnnary. . . iv Page ii vii ix IO 12 12 13 13 17 18 20 7 L. 33 37 44 j] \J O\ N TABLE OF CONTENTS --Continued Chapter III. THE PROCEDURES OF THE STUDY . The State -Sponsored Workshop in Mathematics for Elementary School Teachers . . . Objectives of Elementary School Mathematics (K- 8) In- service Education Program . . Description of the Workshops . . . The Population Sample Used in the Study The Workshop Sites . . . . . Description of Teacher Population Assignment Criteria Utilized The Procedures Used in the Collection ofData............... The Preparation and Use of an Information Form............ . The Construction of, the Administration of, and the Scoring of a Te st of IVIathematical Understandings . . . . . . . . Using the Final Test . . . . The Selection and Use of an Arithmetic Attitude Inventory . . . . . The Procurement of and the Utilization of a Classroom Interview Instrument Procedure for Analysis of Data . Summary. IV. ANALYSIS OF THE DATA Growth in Basic Mathematical Understandings of Elementary School Teachers Hypothesis I . Findings . . . Recapitulation Changes in Attitude of Elementary School Teachers Toward Arithmetic Hypothesis II . Treatment of the Data Findings. . Recapitulation . . . . Changes in Teacher Classroom Procedures with Respect to the Teaching of Arithmetic Hypothesis III \I Page 64 64 67 67 68 69 7O 71 103 106 112 113 116 116 116 123 126 126 127 127 133 136 137 TABLE OF CONTENTS -- Continued Chapter Findings . Recapitulation Summary . . V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS General Summary . Summary of Procedure Summary of Results Major Conclusions . . . . . Implications and Recommendations for Curriculum and Research . Implication for In-service Education Implications for Research BIB LIOGRAPH Y APPENDICES . . . . . . . . vi Page 138 144 145 147 149 149 154 160 163 164 166 169 181 Table 1. 10. LIST OF TABLES Teaching Assignment for Eighty-Nine Teachers Utilized in the Study . Composition of Population with Respect to Years of Teaching Experience Central Tendency and Variability of Mathematics Background for Population Sample Composition of Population by Highest Degree Held . Distribution of Items on Test of Mathematical Understanding According to Test Form and Te st- Content Outline Pre- and Post-Test Results on the Test of Mathe- matical Understandings for the Research Groups in this Study Summary of the Analysis of Covariance for the Scores of Forty-Four Intermediate Grade Teachers on the Test of Mathematical Understandings AnalySis of Variance for Research Group Ben the Test of Mathematical Understandings Correlation Coefficients Between Selected Teacher Background Information and Scores on Test of hiathematical Understandings . Pre— and Post-Te st Results on the Arithmetic Attitude Inventory for the Re search Groups in This Study . vii . Page 73 78 87 118 120 129 LIST OF TABLES -- Continued Table 11. Summary of the Analysis of Covariance for the Scores of Forty-Four Intermediate Grade Teachers on the Arithmetic Attitude Inventory 12. Analysis of Variance for Research Group B on Arithmetic Attitude Inventory. . . . 13. Correlation Coefficients Between Selected Teacher Background Information and Scores on Arithmetic Attitude Inventory . . . 14. Pre- and Post-test Descriptions of Teacher Class- room Practices as Determined by Pupil Interviews............... 15. Summary of Acceptance or Rejection of the Research Hypotheses Tested in the Present Study . viii 130 132 134 142 146 LIST OF APPENDICES Appendix Page A. Teacher Information Form . . . . . . . . . . . . . . 181 B. A Test of Mathematical Understandings, Form A . . . . 182 C. A Test of Mathematical Understandings, Form B . . . . 192 D. Arithmetic Attitude Test . . . . . . . . . . . . . . . 204 E. Texas Elementary School Mathematics Class- room Practices Schedule of Questions . . . . . . . . 205 F. Interpretation of Coding the Texas Elementary School Mathematics Classroom Practices InterviewScale.................. 209 G. Outline of Basic Mathematical Concepts Developed for the Elementary School Mathematics (K-8) In-service Workshops . . . . . . . . . . . . . . . 218 H. A List of Arithmetic Textbooks and Experimental Materials Examined . . . . . . . . . 226 1. Names and Addresses of Persons Used for TeSt ReVieW . O O O O O O O O O O O O O O O O O O 228 J. Outline of Basic Mathematical Understandings . . . . . 229 . K. An Item Analysis on Tryout Test . . . . . . . . . . . i 241 L. Raw Scores, Research Group Identification, and Test Form for Eighty-nine Elementary School Teachers Utilized in This Study. . . . . . . . . . . 245 M. Raw Scores for Responses of Twenty-eight Students to Classroom Practice Schedule of Questions . . . . 248 ix CHAPTER I THE NATURE OF THE STUDY “Education is being brought into the mainstream of national life."1 The American people, operating on the premise that our welfare is dependent, among other factors, upon an enlightened citizenry, have focused a spotlight on public education. This spotlight has brought to- gether groups of professional people representing different backgrounds of experience and training to study educational problems and to suggest directions. As a part of this nationwide attempt to revitalize and up- date the schools, changes have been and are being made in the mathe- matics curriculum of'the elementary school. The position that mathematics education needed revamping probably could have been taken any time in the last seventy-five years. In fact, David NIaxey, writing for £935, said: Today's curriculum reform was forced on us. The Post-Sputnik wailing about our science gap is partly responsible. Computers dramatized the mathematical challenges of automation. Most of all, educators realized that the traditional curriculum was not answering the nation’s need for scientists, teachers, businessmen-- in fact, anybody who could deal with mathematics creatively. 2 1Jerome Bruner, The Process of Education (Cambridge: Harvard University Press, 1960), p. 18. ZDavid Maxey, "Why Father Can't Do Johnny's Math, " Look, XXVII, No. 2 (November 5, 1963), p. 88. 1 This call for new programs in mathematics, programs which meet the changing needs of Society, was also noted by John Kemeny. While Mathematics has always been recognized as one of the cornerstones of our educational system, we are entering an era in which an understanding of mathematics will be of even greater importance to all educated men. As our civilization grows in complexity and science plays a more vital role, the man ignorant of mathematics will be increasingly limited in his grasp of the main forces of civilization. 3 Thus, as a result of recent developments in science and mathe- matics and also better use of our knowledge of how people learn, there is much activity and interest in the development of new elementary school mathematics programs that are different in both content and approach. In experimental programs, such as the School IVIathematics Study Group (SMSG), the Greater Cleveland hiathematics Program (GCIV’IP), and the University of Illinois Project, xnathelnaticians and educators have worked together to set goals and develop new materials, materials which have stimulated new programs and changed existing mathematics curricula in the elementary school. These new programs can neither be ignored nor accepted blindly. One cannot predict with assurance either the mathematics needs of today's learner, or the mathematical ideas required for tomorrow's society, but we do know that the learner must be mathematically literate . . . children cannot be denied the g 3John Kemeny, "Teaching the New Nfatheinatics, " The Atlantic, CCX, No. 4 (October, 1962), p. 90. opportunity to learn the mathematics they need as a citizen or the mathematics that is a necessary stepping-stone to many careers.4 What do these changes imply for elementary school teachers? Since wise decisions about what to teach are dependent upon the teacher's understanding of curriculum reform, it is imperative that: . . . The elementary teacher should know the mathematics necessary to teach the content of the new curriculum. Teachers need a mathematics education which makes it possible for them to understand and appreciate the structure of the mathematics they teach, which makes it possible for them to help children develop a problem-solving technique, and which makes it possible for them to have confidence in their teaching of arithmetic. 5 ‘ Dr. Bernard Gundlach, former Educational Consultant for the Greater Cleveland Mathematics Program, referred to the teacher as the key to progress. He declared that it was up to the teacher to pre- pare the next generation not only to step in and take over, but also to lVIary Folsom, "National Problems and Trends in iV'Iatheniatical Training of Elementary School Teachers, " Ten Conferences on the Traininggf Teachers_o_f_ Elementary School Mathematics, A Seventh Report of the Committee on the Undergraduate Program in Mathe— matics to the. Mathematical Association of Alnerica (Pontiac, Michigan: The Association, 1963), p. 4. 5Clarence Hardgrove, "National Problems and Trends in Mathematical Training of Elementary School Teachers, " Ten @nferences on the Training of Teachers of Elementary School fithematicsIfiA Seventh Report of the Committee on the Under- graduate Program in Mathematics to the Mathematical Association of America (Pontiac, Michigan: The Association, 1963), pp. 30 and 34. 4 make its 'own contribution to the growing storehouse of human knowledge and development. Furthermore, he saw the elementary teacher as a builder. "Not only does she set the pace, " he asserted, "but she has to build the very foundation upon which the mighty superstructure of human knowledge is to rest. "7 Continuing to emphasize the importance of the elementary teacher, Gundlach noted that: . . . It is on the most elementary level that any curricular re- vision makes its strongest demands. . . . the elementary teacher cannot any longer fulfill his or her duties entirely by presenting choice bits of information; he or she must interpret and explain the language in detail for the benefit and future use of the young students. The essential condition for this to take place -- understanding and insight -- is that the teacher herself be thoroughly familiar with the language. . . .8 What mathematical understandings are needed by elementary teachers? What is the present status of elementary school teachers with respect to these important mathematical understandings? What means can be used to help teachers gain these needed mathematical under- standings? These three questions are questions related to elementary school mathematics which have been pursued by many investigators. g 6Berna rd Gundlach, Basic Mathematics for Elementary Teachers (Bowling Green, Ohio: Educational Research Council of Greater Cleveland, 1961, p. 3. 7Ibid., p. 4. 8mm. The first question, that of mathematical understandings needed by elementary school teachers, has been the subject for several re- search studies. While there is a lack of agreement as to specific mathematical understandings, Newsom, 9 Stipanowich,10 Schaaf,11 and others have outlined what they believe should be included in a course dealing with the subject matter of arithmetic, more frequently referred to as elementary school mathematics. The material from the Greater Cleveland Mathematics Program (GClVIP)12 and the volumes on Number Systems13 and Intuitive Geometry” prepared by the School lV'Iathe- matics Study Group (SMSG) are also examples of materials availa- ble to help in planning programs for elementary school teachers. 9C. V. Newsom, ”Mathematical Background Needed by Teachers of Arithmetic, " The Teaching of Arithmetic, Fiftieth Yearbook of the National Society for the Study (if—Education, Part II (Chicago: Univer- sity of Chicago Press, 1951), p. 232.. lOJoseph Stipanowich, "The Mathematical Training of Prospective Elementary Teachers," Arithmetic Teacher, IV (December, 1957), pp. 240-248. 11William L. Schaaf, "Arithmetic for Arithmetic Teachers, " School Science and Mathematics, LIII (October, 1953), pp. 540-543. 12Robert E. Eicholtz and hilartin Emerson, Topics for the Ele- fintary Arithmetic Teacher (Cleveland, Ohio: Educational Research Council of Greater Cleveland, 1961). 13School hiathematics Study Group, Number Syst ems, VI: Studies iiMathematics (New Haven, Connecticut: Yale University Press, 1961). 14School Mathematics Study Group, Intuitive Geometry, VII: Studies in Mathematics (New Haven, Connecticut: Yale University H‘m Press, 1961). Clarence Hardgrove, writing for the Committee on Undergraduate Programs in Mathematics, stressed that the emphasis should be the same for the teachers as it is for children. "The needed under— standings, . . . are not only those understandings which give meaning to mathematics in the modern elementary school but also those which help the teacher tie the subject together as a related whole."l5 Question two, the present status of elementary school teachers with respect to mathematical understandings, also has been the topic for several investigations. Robinson observed that ”elementary teach- n16 ers have at best. only a mechanical knowledge of aritlnnetic. Glennon gave further evidence of the arithmetic inadequacies of teachers. He noted that it is difficult for teachers to help children grow in under- standings which they themselves do not possess. 17 Bean, 18 Orleans,19 15Hardgrove, op. cit., p. 31. 16A. E. Robinson, The Professional Education of Elementary Teachers in the Field of Arithmetic (”Teachers College Contributions to Educatio—n,—"_No. 6727New York: Bureau of Publications, Teachers College, Columbia University, 1936). 17Vincent J. Glennon, "A Study in Needed Redirection in the Preparation of Teachers of Arithmetic,” Mathematics Teacher, XLII (1949), p. 395. 18John Bean, "Arithrnetical Understandings of Elementary School Teachers," Elementary School Journal, LIX (May, 1959), pp. 47-50. ngacob S. Orleans, The Understandings of Arithmetic Processes flConcepts Possessed by Teachers of Arithme—tic (”Office of Research and EvaluationT—Division of T-eacher Education, " Publication No. 12; New York: College of the City of New York, 1952). \l and.a host(fifother'invesnigators also have reported researctistudies which point to the deficiency of elementary school teachers with respect u)rnathennafica1 understandings. How can the mathematical understandings of elementary school teachers beixnproved? Tlns questknn a Hnrd questknirelauxlto elenMHNary'schoolrnafiunnathm has beconae,in Umzeyes ofrnany educators, one ofthe.nM)stcflufllenging and persisfinfl:prokflerns fiused by educathan. Itis a questhan‘which calhsfor study ofinethods or pro- cedures for remedying the situation rather than collecting and pre- senting additional evidence on the inadequacy of elementary school teachers. INurnerous educators see Hus needlkn'professknnrlgrovnh as a need far too pressing to be limited to either the pre-service or the in-service education of teachers. Certainly, as suggested by Bean, Schaaf, Orleans, and Newsom, teachertraineescxuibe preparedthroughzmntqrto-dauaapproachto mathematics. Yet, if an adequate progran'i of mathematics could be inaugurateclal once on a‘pre-serwdce basis, onlyzaboutfive per cent ofthe 900,000 eknrmnnary'schoolteachers wmndd becfirectlyzufected 20 each year. At that rate, except for the spread from individual to individual, the schools would be at least twenty years incorporating presentideas. g 20Hardgrove, op. cit., p. 31. Five years ago, just a few months after the appearance of the 1957 yearbook of the National Society for the Study of Education, which was devoted to in-service education, Ashby, in an editorial for Educational Leadership, wrote: The rapid acceleration of the phenomenon of change in modern society makes in-service education a more significant and challenging problem than ever before in the history of the teaching profession. Today‘s teacher must be alert to keep up with the out-of—school learnings of pupils. At the same time, the revolution in technology and production has provided new knowledges and tools to be used in teaching. As teaching becomes more of a profession and less of a pro— cession, more teachers are removed from their preservice training than ever before. The training they received even a decade ago is inadequate today either as to substance or as to methodology. This underlines the demand for effective in- service education. . . . Without it, our schools cannot adequately . . . . 7 prepare boys and girls for a dynamic soc1ety.“l Bean recommended the following two courses of action for the in- Service program for improving arithmetic instruction: 1. A series of state, regional, and district workshops in arithmetic. Z. A series of regional extension courses under the direction of 7') the colleges in each state. ”“ 21Lynn Ashby, "Today's Challenge to In-Service Education, " Educational Leadership, XV, No. 5 (February, 1958), pp. 270-271. 22Bean, op. cit., p. 450. 9 In the preceding introductory paragraphs it has been indicated that changes have been and are being made in the mathematics cur- riculum in the elementary school. Also, it has been pointed out that these changes have made it imperative for the elementary teach- ers "to understand and appreciate the structure of the mathematics they teach --a means which makes it possible for them to help chil- dren develop a problem-solving technique and a means whi‘ch makes it possible for them to have confidence in their teaching of arithme- tic."‘23 While various forms of iii-service education have been utilized by school systems and state departments of education, few have been the reports which attempted to evaluate objectively the effectiveness of any of the techniques being employed in the iii-service education program of teachers. The lack of such studies pointed toward the need for the present study. Specifically, the present study attempted to evaluate the effectiveness of one series of state-sponsored workshops in arithmetic for elementary school teachers in: (1) improving basic mathematical understandings, (Z) changing the attitudes of elementary school teachers toward mathematics, and (3) improving the classroom practices of teachers with respect to the teaching of arithmetic. Z3Hardgrove, 22' cit., p. 34. 10 I. The Need for the Study During recent years, there has been rather general acceptance among educators of the need for in-service education. Many educators see it as one of the outstanding needs in the field of education. Re search on the effectiveness of various kinds of in—service programs is needed if efforts to improve understanding of teachers are to be efficient and effective in the improvement of educational opportunities for boys and girls. 24 While research studies with respect to arithmetic have increased in number, there has been a paucity of studies in the area of in-service teacher education with respect to arithmetic. Since 19 57, the annual bibliographies of re search, prepared by J. Fred Weaver for the Arithmetic Teacher, list over three hundred research studies and dissertations on elementary school mathematics. However, less than twenty-five studies of which nine were dissertations had teacher education as the main area of emphasis. Only three of the disserta- tions, one by Rudd, another by Boyd, and a third by Houston, attempted to study specific in-service procedures. Several other studies, how- ever, either surveyed present conditions or studied a particular aspect of the pre-service program. 23 24W. Robert Houston, Claude c. Boyd, M. Vere DeVault, "An In-Service Program for Intermediate Grade Teachers, " Arithmetic Teacher, VIII (February, 1961), p. 65. ’? . . . . “SFor further 1nformation concerning these studies, see Chapter II, pp. 44-50. ll Sparks, in an article which summarized recent research with respect to arithmetic understandings needed by elementary school teachers, concluded that the research to date has been "too limited in scope to offer substantial assistance in planning programs of teacher education. "26 Barnett and Jansen, in a book of readings edited by DeVault, referred to the situation as a distinct challenge to those interested in teacher education. They suggested that there is a need today to dis- cover ways of remedying the situation rather than to collect and to present additional evidence of the elementary school teacher's lack of mastery of mathematics. They enumerated the following as possible areas to include in future studies: 1. Experimentation in search of effective ways to improve the teacher's mathematical knowledge and understanding. 2. Clarification of what the elementary school teacher should know. 3. Evaluation of the contribution of pre-service and in-service aspects of the teacher's preparation. 4. Utilization of various techniques to relate certain aspects of 26Jack Sparks, "Arithmetic Understandings Needed by Elemen— tary School Teachers, " Arithmetic Teacher, VIII (December, 1961), P- 393. preparation of improved teaching and learning mathematics in the classroom. 2’ Houston and DeVault, in describing a study carried out in con- nection with the elementary schools of Dallas, Texas, stated that: Studies are needed which investigate the changes brought about in the classroom as a result of in-service education programs. Case-studies, classroom observation, pupil interview technique, and experimental studies involving pre— and post-te sting of both teachers and pupils are but some of the techniques which should be utilized. . . . Such studies must be forthcoming if continued effort devoted to in-service education is to make the greatest possible contribution to the continued growth of teachers. 28 Thus the need for a study which would push further the "knowl- edge claim” that an iii-service education program, with proper content emphasis, can raise the level of understandings of elementary school teachers with respect to basic mathematical understandings has made itself manifest. II. The Probletn Statement of Problem It was the purpose of this investigation to analyze the effective- ness of the "workshop" as a means of iii-service teacher education for 27[Glenn Barnett and Udo Jansen, "Planning for Teaching Competence in Mathematics Education, " Improving Mathematics Program, ed. M. Vere DeVault, (Columbus: Charles E. Merrill Books, Inc.,1961), p. 466. 28 '- Houston, DeVault, and Boyd, gp. cit., p. 6‘). 13 developing needed mathematical understandings of elementary school teachers, for changing the attitudes of elementary school teachers toward mathematics, and for improving classroom procedures. The workshops used in the study were a part of the 1963—64 series of Science and Mathematics Workshops conducted by the Missouri State Depart- ment of Education under Title III, National Defense Education Act. The effectiveness of the "workshop" was evaluated in terms of: (1) growth inmathematical understandings of elementary school teachers, (2) changes in attitudes of elementary school teachers toward mathematics, and (3) improvement in teacher classroom practices. Hypothe ses The type of changes made by elementary school teachers and the effect of these changes upon classroom procedures, such as were examined as part of this study, were grouped into three categories: A. Changes in mathematical understandings of elementary school teachers which were: 1. Related to the ba s'ic structure of mathematics. 2. Related to informational concepts. 3. Related to logical abstractions or generalizations. B. Changes in teacher attitudes toward mathematics which were: 1. Related to over-all reaction to the subject. 2. Related to its importance in the elementary school curriculum. 14 C. Changes in teacher performance in the classroom which were: 1. Related to use of materials, activities, and teaching aids. 2. Related to classroom organization and evaluation. The following three hypotheses were formulated and tested in the present study. Hypothesis Number One. --There will be a significant difference on a test of mathematical understandings between the post-te st scores of a group of elementary school teachers who have participated in an in- service education program and their pre -test scores or the scores of a similar group who have not participated in the iii-service teacher education program. Two other questions directly related to this hypothesis were examined. 1. What effect, if any, does a pre -test have upon the post -test scores of teachers on a test of mathematical understandings? 2. What is the relationship of improvement in basic mathematical understandings to background factors? Hypothesis Number Two. --There will be a significant difference on an arithmetic attitude inventory between the post-te st scores of a group of elementary school teachers who have participated in an iii-service edu- cation program and their pre-test scores or the scores of a similar group who have not participated in the iii-service education program. 15 The two questions concerning test-retest effect and the relation- ship to background factors were noted here. The questions were stated as: 1. What effect, if any, does a pre —test have upon the post -test scores of teachers on an attitude inventory? 2. What is the relationship of improvement in arithmetic attitudes to teacher background factors? Hypothesis Number Three.-—There will be a significant difference be- tween the classroom practices of a group of elementary school teachers who had participated in an iii-service education program and the classroom practices of a similar group who had not participated in the iii-service education program. The Research Design All participants in two of the five mathematics workshops for elementary school teachers sponsored by the Missouri State Depart- ment of Education during the second semester of 1963—64 under Title III, National Defense Education Act, constituted the sample for this study. A third groupjof teachers, who had indicated an interest in participating in a mathematics workshop but who had not begun the workshop experience, was used as a control group. These workshops, specific iii-service activities, a part of a state -wide iii-service teacher education program, were conducted in o 16 the Southeast htlissouri State College Service Area by the same mathe- matics consultant from the Missouri State Department of Education. To test the three major hypotheses of the study and for the evaluation of the data relative to the background factors, a variation of the design referred to by Campbell and Stanley as an expanded "Recurrent Institution Cycle Design" was utilized.29 Such a procedure allowed the researcher to utilize features of both the one-group "Pre -test, Post-test Design" and the "Static Group Comparison De- sign. " For this particular study, the following independent variables were used: 1. Iii-service presentation 2. Pre-testing 3. Post-testing 4. Grade level Dependent variables were: 1. Growth in basic mathematical understandings of elementary schoolteachers. 2. Changes in attitudes of elementary school teachers toward m athematic s . 29Donald T. Campbell and Julian C. Stanley, ”Experimental and Quasi-Experimental Designs for Research on Teaching," Handbook of Research on Teaching, ed. N. L. Gage (Chicago: Rand McNally and— Company,T963), p. 227. l7 3. Changes in teacher classroom procedures. 30 The background information furnished by each teacher included: (1) sex, (2) years of teaching experience, (3) number of credits in high school mathematics, (4) number of semester hours credit in college mathematics, (5) number of semester hours credit in college mathematics teaching methods, and (6) highest degree attained. Instruments selected or developed for use in the collection of data were: 1. Teacher Information Form and Problem Census Questionnaire. 2. A Test of Nlathematical Understandings (two parallel forms). 3. Revised Form of Dutton Arithmetic Attitude Inventory. 4. Interview Question Schedule and Rating Scale. 31 The analysis of covariance statistical technique was utilized in the analysis of the data relative to the effectiveness of the in-service presentation. Pearson Product-Moment Correlation Coefficient was used in the relationship analyses reported in the study. III. The Limitationsgfthe Study It seems pertinent to point out at least three limitations in the present study. First, the length and number of sessions did not 30For a more detailed explanation of the design, see Chapter III, pp. 72-112. 3‘lSee Appendices A, B, C, D, and E for samples of these instru- ments. ’ 18 permit either treatment in depth of new concepts or extensive review of concepts teachers may have had in an arithmetic course for teachers. Second, since the data used in the study were obtained from one series of workshops, any generalizations concerning the results and conclusions drawn were limited by the extent to which the population represented regional and local groups. Finally, the extent to which the instruments used measured adequately the effects of the iii-service program was a limitation. Certainly the instruments used in the study had the inherent limitations of paper-and -pencil tests and inter- view instruments. As pointed outby Glennon, far superior to the pencil-and -paper testnwould be ”the study of the behavior of each person individually through conversing with him and keeping anecdotal . . 32 records of his performances on the test items. " IV. Definition _o_f_ Terms Attitude Attitude was the term used to refer to the position assumed or reaction toward the subject of arithmetic that was or was not re- flected by the participants' behavior pattern. Classroom Procedures or Classroom Practices The use of the term classroom procedures, or classroom 3ZGlennon, op. cit., p. 395. l9 practices, in this study was used to refer to the teachers' performance in the c 1a 5 sroom . Educational Con sultant Educational consultant, as used in this study, referred to a qualified staff member from the state department of education, a person possessing special knowledge and experience, who had been asked by professional groups to work directly with them in providing assistance in connection with an educational problem. Elementary School Teachers The term elementary school teachers was used to identify persons teaching in grades kindergarten through six. In -Service Education The term iii-service education, as used throughout this investiga- tion, referred to those experiences, processes, procedures, and activities on the part of the employed teacher which were designed to contribute to professional growth. Matheniatic al Unde r standing 5 Mathematical understandings, as used in this study, was the term used to refer to those generalizations about the structure of mathe- matics which give significance to computational skills, informational concepts, and logical abstractions. 20 \Vorkshop Workshop was the term used to refer to a particular form of in- service education designed to help teachers secure new or modified points of view and to acquire new knowledge, new understandings, and new techniques for classroom presentation. It involved planning with, active involvement of, and evaluation by the participants. IV. Sumnia ry This chapter's chief concern was to orient the reader to the study. The study was introduced through a discussion on the current significance of teachers understanding the mathematics which they teach. The problem was concisely stated and the nature and scope of the study described. In brief form the procedure for implementing the study was discussed. Chapter II will review the literature pertinent to the study. Chapter III will describe the study setting. the teacher population, and the procedures utilized in pursuing the study. Chapter IV will present, through tables, figures, and explanatory material, an analysis of data relative to the study. In Chapter V, a summary will be presented; conclusions and implications for further research also will be discussed. CHAPTER II REVIEW OF RELATED LITERATURE Extensive research into the various aspects of elementary school mathematics has been reported not only in published and unpublished research but also in journals, monographs, periodicals, encyclopedias, professional books, and publications of the government and learned societies. Several authors have, at various times, presented ex- - haustive reviews of research in The Arithmetic Teacher, The Review 'of Educational Research, and School Science and lvlathematics.1 At least three annotated bibliographies also have been compiled. The review of literature pertinent to this study has been or- ganized under four categories: (1) improving mathematical under- standings of elementary school teachers, (2) utilizing in-service education procedures, (3) changing attitudes of elementary school teachers toward mathematics, and (4) improving classroom procedures of elementary school teachers with respect to teaching mathematics. 1For further information see bibliographical entries under the names of: (l) J. Fred Weaver, (2) Robert L. Burch and Harold F. Moser, (3) Glenadine Gibb and Henry Van Engen, (4) Herbert F. Spitzer and Paul C. Burns, (5) Maurice L. Hartung, or (6) E. G. Summers. ZSee bibliographical entries under Kenneth Brown, William L. Schaaf, or Guy T. Buswell. [u [\a 1. Improving Mathematical Understandings of Elementary School Teacher 5 One of the main points of concern with respect to present-day education has been that arithmetic, a very important branch of mathe - matics, is failing to provide the solid basis of competence needed either for effective citizenship or to move up the path into higher mathematics. Many authorities believe that success in mathematics "rests on the foundation built in the first six grades, and an improve- ment in these grades will allow for a substantial strengthening of the .3 program for the higher grades. Since the quality of mathematical instruction and hence the level V' l of pupil achievement depend, in part, upon the mathematical compe- A“! tence of the teacher, a careful preparation of elementary school teachers in mathematics subject matter becomes an important aspect of an improved program of arithmetic in the elementary school. "A firm grasp of basic arithmetical conceptsand processes is essential to teach arithmetic meaningfully. Teachers cannot teach under- standings that they themselves do not have. " Few research studies, however, can be found which closely re- late to the problem of the mathematical competence of the elementary 3J. Fred Weaver, "A Crucial Problem in the Preparation of Elementary School Teachers," Elementary School Journal (February, 1956), p. 436. 4Bean, op. cit., p. 447. 23 school teacher. The first known study was an investigation made by E. H. Taylor in 1938. Taylor adn‘iinistered a test on meanings in arithmetic to three hundred and thirty—three freshmen at Eastern Illinois State Teachers College. He found that the group tested was deficient in both the mechanics and the understandings of arithmetic. 5 Vincent J. Glennon, in his pioneer study on the growth and ,mastery of certain basic mathematical understandings, brought to light serious deficiencies in the mathematical background of teachers and also began a needed redirection in the preparation of teachers of arithmetic. He had difficulty locating a suitable instrument for evalu— ating the mathematical understanding of teachers and subsequently developed, as part of his study, an eighty-item test. for measuring basic mathematical understandings. Since that time, this multiple- choice test, referred to as a "Test of Basic Mathematical Under— standings," has been used by numerous other investigators. For his study, which attempted to obtain an index of prevailing conditions within the groups being studied, Glennon used the results from administering the test to persons at seven different educational levels (pupils in grades seven, eight, nine and twelve, freshmen, college seniors, and teachers-in-service). He found that there was no 5E. H. Taylor, "kilathematics for a Four-Year Course for Teachers in the Elementary School," School Science and Mathematics, XXXVIII (May, 1938), pp. 409—503. 24 significant difference in achievement of basic mathematical under- standings between teachers' college freshmen and teachers' college seniors. He also found that the teachers—in-service understood about fifty-five per cent, or slightly more than one-half, of the total items. From his data, he concluded "that significant growth was not being accomplished by persons at any step of the educational ladder. "6 Glennon‘s findings supported two other hypotheses: 1. There was no significant difference in achievement of basic mathematical understandings between a teachers' college senior who had taken a course in the Psychology and Teaching of Arithmetic and a teachers' college senior who had not taken such a course. 2. There was no significant difference in achievement of basic mathematical understandings between teachers-in -service who had done graduate work in the Psychology and Teaching of Arithmetic and those who had not taken such graduate work. 7 In his summary, Glennon suggested that one aspect of a needed redirection in the training of teachers seemed to lie in the professional training offered in teachers colleges and schools of education. 6Glennon, op: cit., pp. 392-393. 71bid. # 25 This training, as it is usually set up at the present time, con- sists of a single course in the methodology of teaching arithmetic as a ”tool" subject. Little emphasis is placed upon the pro- fessional study of arithmetic as a science of number, as a system of related ideas, or as a series of number relation- ships. His findings also seem to suggest several aspects of needed redirection in the program of iii-service education of teachers of arithmetic. Curriculum revision of the professional courses must be con- cerned with emphasizing the subject matter as well as with the principles of teaching the subject-matter. Many investigations have substantiated Glennon's major findings and conclUsions. For example, J. Fred Weaver administered the Glennon test as a pre-test to four groups of upper-classmen prior to their taking the course entitled "Methods of Teaching Arithmetic." The average score made by the students was forty-four per cent, a 'score which did not vary greatly from the average score as noted by Glennon when he administered the same test.10 For one of the four groups taking the methods course, Weaver used the test both as a pre -test and as a post-test. When the average level of student understanding was raised from slightly more than forty-five per cent to seventy per cent, he noted the gain as significant and concluded that: 81bid. 9Ibid. ————_ 10Weaver, op. cit., p. 261. 26> A methods course with proper content emphasis . . . can result in significant improvement in the students' level of mathematical understanding. Bean, working at Stanford University, used the Clennon test for a comparative study of the mathematical understandings of four hundred and fifty Utah teachers. The mean score for the Utah teachers was 52.46, or 65. 58 per cent of all items, answered correctly. However, when questioned, 64. 57 per cent of the teachers did not, think the test adequately measured the extent of their understandings while 35. 43 per cent thought it did. 12 Rodney saw the improvement of pre-service preparation of teachers as one approach to the improvement of mathematics in the elementary school. In 1951, he carried out an investigation which attempted to evaluate the pre-service preparation for teaching mathe- matics in the elementary schools as offered by the State College for Teachers at Buffalo, New York. He developed a forty-item test of mathematical understandings and administered the test to over three hundred students at the State College for Teachers at Buffalo. Then, using the normative survey methods of research, he attempted to: (l) analyze the content of the teacher preparation program at that institution, (2) compile a list of “Ibid. 12John Bean, "The Arithmetic Understandings of Elementary School Teachers," (unpublished Ph. D. dissertation, Stanford Univer- sity, 1958). 27 major objectives for the teaching of elementary school mathematics, and (3) study the conditions existing in three groups (freshmen, seniors, and classroom teachers) with respect to mathematical understandings. His findings, which did not vary significantly among the three . .. . . 13 . . . groups, supported preVious investigations. In summarizing his study, Rodney saw the need for organizing a program for the pro- fessional preparation of teachers which would "alleviate, to the _ . . . . . ._ ”14 greatest extent p0551ble, apparent def1c1enc1es. He also endorsed a program of iii—service education when he said: An iii-service program for the study of the meanings of the content of the elementary school mathematics should contribute to an iniproven’ient of teaching elementary school mathematics. 13 That the present condition of teacher understanding in arithmetic leaves much to be desired also was noted by Orleans in a study pub- lished by the Office of Research and Evaluation, College of the City of New York. He made "a systematic effort to ascertain the extent to which teachers and prospective teachers of arithmetic understand the arithmetic processes and concepts represented by the short cuts they ,16 teach.‘ l3Cecil T. Rodney, "An Evaluation of Pre-Service Preparation for Teaching the lVfathematics of the Elementary School," (unpublished Ph. D. dissertation, University of Buffalo, Buffalo), 1951, p. 105. 14Ibid., p. 108. 15ibid., p. 107. 16Orleans, op. cit., p. 1. 28 For his first test, Orleans administered a free—answer type test to seven hundred twenty-two persons from five different levels: (1) undergraduates from four colleges, (2) upper classmen who were doing their student teaching, (3) students in graduate courses in education, (4) classroom teachers, and (5) other persons. Then, using many of the responses given, he constructed an eighteen-item test which was of the multiple -choice type. This test, designed to measure teachers' understanding of the processes they teach, included items concerning long division and multiplication, the meaning of dividing by a fraction, reducing and raising fractions, remainders, subtraction, and per cent. After administering the test to three hundred twenty-two teachers (fifty-three primary teachers, seventy-six upper grade teachers, sixty-seven junior and senior high mathematics teachers, and one hundred twenty-six other teachers), he noted that. there were "few ’ processes, concepts, or relationships which were understood by a large percentage of teachers. "17 While the test was a relatively short one and no attempt was made to establish validity, Orleans concluded that "it was difficult for this group of people to verbalize their thoughts when they attempted to ex- plain arithmetic concepts and processes. "18 He contributed this lack of 17Ibid. , pp. 1-59. 18Ibid., p. 37. understanding of arithmetic to the rote process which has been so prevalent in the past. 19 Other individuals have carried out studies directed to the pre- service level. Of particular interest were Fulkerson's findings with respect to teaching experience and college classification. Since students with teaching experience did significantly better than those without experience and performance became increasingly better as the level of college classification increased, findings which somewhat contradicted previous research, Fulkerson suggested further study of this phase of teacher training. 20 Carroll's study attempted to deal directly with the problem of teachers' background in mathematics. An evaluation instrument, which was designed to assess the mathematical background of a group of prospective teachers, was the primary research tool. The initial part of the study sought to establish a list of mathe- matical understandings which elementary teachers should know. To devise such a list, five series of elementary arithmetic textbooks, published between 1946 and 1952, were analyzed. A panel of sixteen authorities in the field of mathematics education evaluated the list 19Ibid. 20Elbert Fulkerson, "How Well Do 158 Prospective Elementary Teachers Know Arithmetic?" The Arithmetic Teacher, VII (March, 1960), pp. 141-146. 30 of understandings and the questions which were subsequently devel- 21 oped. Using questions on which there was almost unanimous agreement, Carroll assembled two eighty-one ite_-n tests which she administered to teacher-training classes at Wayne. State University. After the re- sulting data were statistically analyzed, Carroll selected items for a test which she "hoped would aid in the problem of diagnosing and remedying inadequate teacher background in mathematics. "22 A study of university students' comprehension of arithmetical concepts was carried out by Dutton in 1961. His study was designed to measure students‘ understanding of arithmetical concepts as they pro- gressed through courses designed to teach these processes. Two classes, composed of fifty-five prospective teachers, were tested at the beginning of the semester and again at the end with the University of California Achievement Test for Sixth Grade, a test covering basic arithmetical concepts that students are expected to know at the com- pletion of grade six. 23 21Emma Carroll, "A Study of the Mathematical Understandings Possessed by Undergraduate Students Majoring in Elementary Educa- tion, " (unpublished Ed. D. dissertation, Wayne State University), 1961. 22Ibid. 23The appendix of Evaluating Pupils' Understanding gimmi- metic, pp. 143-148, contains a copy of this test. For full bibliographic reference, see entry under Wilbur H. Dutton in the bibliography. 31 The conclusions drawn were that prospective elementary school teachers adhered to many arithmetical concepts and procedures learned in elementary and junior high school. While there were many basic arithmetical concepts understood by these prospective teachers, the students often adhered to traditional methods and mechanical procedures when attempting to explain the following concepts: (1) partial products in multiplication, (2) placement of quotient figures in long division, (3) placement of the decimal point in problems involving decimal fractions, and (4) understanding of and use of denominate numerals. Augustine P. Cheney, a graduate student at the University of California, constructed a fifty—two item test using thirty items from the instrument prepared by Dutton to measure understanding of basic mathe- matical concepts. After administering the test to one hundred twenty teachers in eight selected elementary schools in Ventura and Los Angeles Counties, he reported that these teachers understood some aspects of arithmetic but had difficulty with other phases such as place value, remainders in division, fractions involving multiplication and division, comparison of decimal fractions, and use of denoniinate numbers. He also compared the mean scores of the primary teachers with those of the intermediate grade teachers. With a mean score of 29.91 24Wilbur H. Dutton, Evaluating Pupils' Understanding ofArith- metic, Englewood Cliffs, N. J.: Prentice-Hall, 196—1, pp. 54-55. til i I ..|llli|llll\llll.l for the primary group and 38. 74 for the intermediate grade teachers, he reported the difference as significant: well beyond the one per cent level. 25 Dutton, in a recent publication, called the study of Cheney important for the following reasons: . . (1) His careful review of research studies dealing with the evaluation of teachers‘ understanding of arithmetic points out the paucity of studies and valid instruments in this area. (2) He accentuates the important fact that teachers have difficulties with certain aspects of arithmetic and that wide differences exist among teachers. (3) Primary-grade teachers seem to have an understanding of the simple concepts taught to young children and have neglected more advanced concepts taught in inter- mediate grades. 26 Thus, in study after study, the need for teachers to have a better .understanding of the basic concepts of mathematics has been stressed. Elementary teachers have "neither the facility in the computation processes which they are expected to teach nor a firm grasp of the basic mathematical concepts underlying the processes. “ Weaver, in reviewing what he termed a "crucial problem" in education, noted that: 7 “SAugustine B. Cheney, "Evaluation of Elementary School Teachers‘ Understanding of Basic Arithmetic Concepts," (unpublished M. A. thesis, University of California, Los Angeles, 1961). 20Dutton, op. cit., pp. 55-56. 27Doyal Nelson and \Valter Worth, “Mathematical Competence of Prospective Elementary Teachers in Canada and United States, The Arithmetic Teacher, VIII (April, 1961), p. 147. 33 Two things stand out clearly from the conditions discussed thus far: (1) the general level of arithmetic on the part of the undergraduate in representative teacher—training institutions is inexcusably low; (2) all too few teacher—training programs provide appropriate work in background mathematics that could be definitely helpful in raising the undergraduates' level of arithmetic scholarship. II. Utilizing Iii-service Education Procedures Increasing standards of pre-service education have not lessened the need for continued iii-service education. New chapters in the history of education have been written. Experimental programs have been tried out in all parts of the country. Some of these experiments have been pronounced worthless and discarded; others were deemed good and, therefore, widely adopted, thus influencing aims, methods, curricula,and course content. As a result, teachers, whether "neophytes" or "old-timers, " found themselves needing to acquire new knowledges, new points of view, and new methods. . . Our rapidly changing culture and its implication for cur- riculum change, the continuing increase in pupil enrollments and numbers of teachers, the need for improved school leader- ship, and the continuous additions to our knowledge about children and youth and the learning process . . . mean that professional school people need to work continuously to keep abreast of what they must. know and must be able to do. Z8Weaver, op. cit., p. 258‘. 2‘QStephen ‘4. Corey, "Introduction," Chapter I, Fifty—sixth Yearbook of the National Society for the Study of Education, Part I (Chicago, 111.: University of Chicago Press, 1957), p. l. 34 That there is a need for programs of in-service education is rarely contested. Baker, in commenting on the urgency, not only for adequate and up-to -date preparation of teacher trainees, but also for the development of means for today's teachers to be cognizant of the new content and new tools which the technological revolution has pro- vided for education, said: Whether a teacher is new to a school system or is an "old- H timer, whether he is a beginning teacher or one of long ex- perience, there is need for an effective iii-service education progran'i. . . . The teacher who does not have the opportunities afforded by an in-service program, in too many cases, soon becomes antiquated. 30 Actually, in-service education, defined as those experiences engaged in by the employed teacher during her service which are designed to contribute to professional growth, does not represent a new idea. “Such activities have been pa rt and parcel of American education for more than a century. "31 Programs, designed for contributing to the in-service stimu- lation. and growth of the teacher, have included: (1) the teachers‘ institute, (2) the reading circle, (3) the summer school, (4) university extension, (5) supervision, (6) the teachers' meeting, (7) voluntary associations, and (8) the workshop. 30'1‘. P. Baker, "What Is An Effective In-Service Education Pro- gram?" Proceedings of The Thirty-Eighth Annual Convention, National Association of Secondary School Principals (Washington: The Associa- tion, March, 1951), p. 46. 31Corey, op. cit., p. 2. 35 Historians are in general agreement that the teachers‘ institute, wholly American in origin, was the earliest form of iii—service edu- cation. Originating at a time when there was a shortage of well-trained teachers, the institute was said to have been designed to provide O O 0 n I Q 0 ’) intenSive additional training for the inadequately prepared teacher. The nature and purpose of the early institute as described by Barnard, Page, Sweet, and others was summarized by Horace Mann when, in 1845, he wrote: It is the design of a Teacher's Institute to bring together those who are actually engaged in teaching Common Schools, or who propose to become so, in order that they may be formed into classes, and that these classes, under able instructors, may be exercised, questioned and drilled, in the same manner that the classes of a good Common School are exercised, questioned, and drilled. 33 Thus the institute, teaching content which the prospective teachers would later teach, served to provide, when well conducted, instruction in the approved methods of teaching. Then, shortly after 1900, the institute began to disappear. Its value as an agency of in-service education, however, had been recog- nized. Today it is acknowledged as a forerunner of other improved practice 5 . 32Willard S. Elsbree, The American Teacher (New York: American Book Company, 1939), p. 135. 33Teacher Institutes (_)_i_' Temporary Normal Schools, pp. 45-46, Cited by Herman Richey, Iii-Service Education (Chicago: University of Chicago Press, 1957), p. 39. 36 As normal schools and the evolving teachers‘ colleges came to require high school graduation for admission to their programs and as many of the college graduates moved into teaching, reading circle courses, conducted by various related state organizations, ca ne to be recognized as a source from which lists of books on the new movements in education and on various subjects relating to the profession could be obtained. Even today, in some states, reading-circle services continue to be maintained and used widely by teachers. Other forms of iii-service teacher education--the summer school, university extension, supervision, teachers' meetings, and voluntary associations-~have all aided in the upgrading of the teaching staff. 34 It was, however, the Eight-Year Study of the Commission on the Relation of School and College in the Progressive Education Asso- ciation which brought significance to the workshop movement and gradually changed the concept of in-service education from that of p—- "upgrading" to “keeping abreast of a changing world. “33 At one time the main emphasis in the iii-service education of teachers was on bringing up to "standard" persons who had been employed with what was deemed to be inadequate prepara- tion. The re was a certain implication that, once this level was reached, and suitably recognized either by a degree, a higher 34Elsbree, Op. Cit-, pp- 135-137- 35c. Glen Hass, "In-Service Education Today,“ cn. II. In- Service Education, Fifty-sixth Yearbook of the National Society for the Study of Education, Part I (Chicago. 111.: University of Chicago Press, 1957). p. 16. 37 type of certificate, or an automatic increase in salary, the education of the teacher would be complete. . . . The newer emphasis is on every teacher continuing to give a certain amount of time to experiences calculated to lead to personal and professional growth. 30 The Workshop Idea Since its inception in 1936, when the first workshop for teachers was organized at the University of Ohio under the leadership of Ralph W. Tyler, Chairman of the Department of Education, University of Chicago, the workshop idea has not only gained acceptance but has been widely used as one means for the iii-service education of teachers. 37 W'hile often differing in type of organization, kind of facilities, length of operation, frequency of meetings, source of leadership, and sponsoring agencies, the workshop is thought normally to consist of opportunities for teachers, under expert guidance, to work on individual problems while acquiring new knowledge, new understandings, and new techniques for classroom presentation. 38 This type of iii-service education has appeared in the form of courses offered for credit, seminars for studying particular problems, clinics, field trips, testing programs, discussion groups, and curriculum studies. 36Commission on Teacher Education, Teachers for Our Times (Washington: American Council on Education, 1944), p. 19. 37Mary A. O'Rourke and William Burton. Workshop for Teachers (New York: Appleton-Century-Crofts, Inc., 1957), p. l. 38Ibid., pp. 3-4. 38 Earl C. Kelley, in his book, The Workshop Way of Learning, referred to the workshop as an educational method which placed the responsibility for learning upon the student-~a possible means for "putting into practice the truths that have become known about how people learn. “39 He also called the workshop an adventure "on the growing edge of the learning process. "40 Mitchell, in a report for the North Central Association of Colleges and Secondary Schools which attempted to identify the charac- teristics of the workshop, contended that the workshop idea was a widespread and generally accepted form of iii-service education. However, he found only nine dissertations which were devoted to some phase of the workshop experience. Three of these studies investigated the workshop in general, two others considered the contribution of the workshop experience to classroom practices, another evaluated an organized program of workshops conducted by a state department of education, and the remainder described workshops conducted in ) spec1f1c areas of education or in selected Situations. 39Earl C. Kelley, The Workshop Way_o_f_ Learning, (New York: Harper and Brothers, 1951), p. ix. 401bid., p. 1. 4‘lJames Mitchell, “The Workshop as an Iii-service Education Procedure,“ Report of the Sub-committee on Iii-service Education of Teachers of the North Central Association of Colleges and Secondary Schools, North Central Quarterly, XXVIII (April, 1954), pp. 423-457. 42lbid., pp. 424—425. .39 An early study concerned with the contribution of the workshop experience to classroom practices was based on observation of participants upon their return to their classrooms. Heaton stated that: As near as sincere professional appraisals cotwld tell by using observation guides, there was valid evidence that the participants return to their classrooms with: (l) recognizable and notable changes in attitude toward their work and (2) a strong drive toward doing something about their curricular and in- structional problems. 43 Improvement in use of materials and in instructional practices also was observed and attributed to workshop presentation. 44 Forest Mitchell investigated the effect of participation in a work- shop upon classroom practices. The purpose of the study was to identify changes that were made in selected classroom practices and to discover any contributions that participation in a summer workshop might make. 43 Another of the early studies devoted to the workshop idea was the investigation carried out by Mode Lee Stone at George Peabody College for Teachers. Often referred to as the first comprehensive study on workshops, this research presented an analysis of the total faculty 43Kenneth L. Heaton et a1, Professional Education for Ex- perienced Teachers (Chicago: University of Chicago Press, 1940). 44lbid. 4jFor a full report on this study, see page 60. 40 workshop technique as it related to the Florida Program of Curriculum Improvement. The researcher sought to explore the consistency with which the procedures and products of the total faculty workshop technique: (1) met the criteria of a good learning experience, (2) contributed to the in-service education of teachers, and (3) resulted in better oppor- tunities for learning on the part of children. ‘ First, from professional books, current bulletins, dissertations, and periodicals, Stone assembled a list of sixteen techniques used in in-service education. Then state departments of education were asked to respond to a questionnaire on the value of each technique. In addition, data from two school systems which had participated in the in-service program for more than one year were also examined. From this, Stone concluded that there was direct evidence demon- strating the benefits of this type of workshop. Another intensive study on workshop participation, according to many sources, is the one prepared by lVlary O‘Rourke. In this re- search, O‘Rourke, after reviewing more than 200 references on the in- service workshop, traced the history of the workshop movement. The historical account was supplemented with a body of original data gathered from a questionnaire sent to 261 administrators in 43 states. 46Mode Lee Stone, “An Analysis of the Total Workshop Technique,“ (unpublished Ed. D. dissertation, George Peabody College for Teachers, 1941). 41 In addition, a group of fifty-one elementary teachers in New England was interviewed and observed in the classroom. Seeing the iii-service education workshop as: (1) an effective vehicle for cooperatively attacking a problem, (2) a valuable means of inducting new teachers, and (3) an opportunity for administrators and teachers to evaluate and retrain their skills, O‘Rourke concluded that the “in-service workshop, one current type of professional.education, was a positive force in raising the level of teacher education as testified by teachers and administrator-participants. “48 While it has been nearly. twenty years since Henderson reported a study on the Evaluation of the workshop program directed by the Ohio State Department of Education, the investigation continues to hold the unique position of being one of the few known investigations relative to a workshop program conducted by a state department of education. Collecting the data over a period of three years (1944-47), Henderson attempted to evaluate the effectiveness of the workshop program for elementary school teachers in Ohio by determining the degree and extent to which the workshop program contributed to the development in teachers of democratic attitudes toward teaching. 47Mary O‘Rourke, "The Iii-service Workshop in Elementary Education: Its Effect Upon Participants," (unpublished dissertation, Cambridge, Massachusetts: Harvard Graduate School of Education, 1954). 481bid., p. 9. 42 Approximately 1600 teachers, 155 principals, and 26 county, village, and city superintendents participated in the program. Also, nearly two hundred consultants, who had served the workshops, were asked to evaluate the effectiveness of the workshops. Questionnaires, attitude inventories, and letters of inquiry were used. The following were included in the findings: 1. Superintendents, in most cases, had initiated the workshops in their school systems; few teachers had participated in planning and evaluating the workshops. 2. Inadequate pre-planning of each project constituted a major weakness. 3. Most consultants and teachers indicated that the length of time devoted to each workshop was inadequate. 4. In most instances, the workshop group was too large for effective work. 49 Hempel and Engle conducted investigations pertaining to the attitudes of teachers toward in-service programs. Henipel reported the relationship between the attitude of a selected group of teachers and their knowledge of the agreement of educational psychologists toward 49Clara Henderson, “An Evaluation of the Workshop Program for the In-service Teacher Education directed by the Ohio State Depart- ment of Education, 1944-1947, " (unpublished Ph. D. dissertation, Ohio State University, 1948). 43 the learning process. 50 Engle's study noted the relationship between teachers who were identified as “more open" and their response to new ideas in educationally significant ways. F Robert Anderson reported a three -year cooperative staff study in the LaCrange (Illinois) Public Schools. The study was concerned with the influence of the iii-service program on teacher test behavior and classroom procedure. From a representative sample of the total staff participating in the workshop program, the investigator selected thirty-three class- room teachers and measured sonie of the influences of the in-service program upon the group through a series of tests and questionnaires given to the teachers and their pupils. Data were also gathered through personal observations in the classrooms and through evaluation questionnaires completed by the participating teachers. As a basis for comparison of the differences noted, a control group of thirty teachers was selected from neighboring school systems in compa rable conimunitie s . Anderson summarized his findings as follows: 50Carl Hempel, "Attitude of a Selected Group of Elementary / School Teachers Toward Iii-service Education," (unpublished Ph. D. dissertation, University of Connecticut, 1960). 51Harry Engle, “A Study of Attitudinal Change in Teachers and Administration during a Summer Workshop," (unpublished Ed. D. dissertation, Auburn University, 1960). 44 l. The experimental group of teachers made numerically higher but not particularly significant scores on all instruments in the post—te st administration. 2. Two of the seven measures of teacher progress indicated sig- nificant gains for the experimental group. 3. Teachers who had participated in several years of child study made slightly higher gains over other LaCrange teachers on the Purdue Test. Anderson suggested that teacher attitude and practices can be improved through cooperative and continuous iii-service procedures.52 Now, after nearly thirty years of use, the workshop continues to be one of the most used forms of iii-service education. Its history has been fast moving; its variety has been intriguing. While not a panacea for all the ills of teacher education, investigators continue to view the workshop as a useful agency for the iii-service education of the professional staff. Iii—service Education for Teachers of Mathematics Rare indeed were the research studies which the writer was able. to locate with respect to iii-service education programs for teachers of elementary school mathematics. A study by Rudd which included an in-service aspect, a cooperative iii-service. study in arithmetic by the 52Anderson, op. cit., pp. 205-215. 45 New York Council, an investigation by Procunier on the impact of Title 111, National Defense Education Act, a report of research opera- tions at the University of Texas by DeVault, Houston, and Boyd, and an investigation by Ruddell and Brown on the effect of three different programs were the only investigations of this nature which the writer was able to locate. Rudd‘s study had to do with the effectiveness of in-service pro- ‘/ cedures. For measuring teacher growth in mathematical under— standings, Rudd utilized: (1) the Glennon test, (2) teacher conferences, (3) teacher questionnaire, (4) teacher summary, (5) classroom visitation, and (6) teacher opinion check list. After administering the Glennon test to fourteen groups of teachers, he asked one group to participate in an eight-session course devoted to the development of seventy-two arithmetical understandings. A group who did not participate in the course served as a control group. Rudd concluded that, while teacher growth may be produced by means of such practices, the modest gains made by teachers in mathematical understandings were not large enough to produce a wide . H53 margin of improvement on test scores. He also stated that: 53Lonnie Edgar Rudd, "The Growth of Elementary School Teachers in Arithmetical Understandings Through Iii-service Proce- dures," (unpublished Ph. D. dissertation, Ohio State University, Columbus, 1960), p. 109. 46 Perhaps the most conclusive evidence of teacher growth was to be found in the results of teacher summaries. 54 A cooperative iii-service study in arithmetic was conducted by the \/ ‘6 Central New York School Study Council during the school year of 1949- 50. The objectives of the study, as stated in the report, were: (1) to explore the place of mathematical meanings in the teaching and learning of arithmetic and (2) to develop some teaching techniques through which pupils could be led to gain essential mathematical understandings and some test items through which pupil mastery of these understandings could be evaluated. 53 The study involved teachers, principals, supervisors, and con— sultants in a program which utilized committees at both local and district level. Meetings were scheduled during school hours, thus helping teachers feel that the work of the council was a part of their school work and not an extra burden. In describing the results of the study, Norem stated: Teachers and supervisors are understanding arithmetical processes for the first time and consequently are more interested in teaching them. 56 54Ibid., p. 109. 55Richard C. Lonsdale, “Preface, “ Developing Meaningful Practices in Arithmetic, A Third Report to the Central New York School Stud—gr-Council prepared by the Coni‘mittee on Flexibility (Syracuse, N. Y.: Bureau of School Services, School of Education, Syracuse University, 1951), pp. viii—ix. 56Evelyn Norem, "Some Results of Our Working Together to Improve Learning in Arithmetic, " Developifi Meaningful Practices in Arithmetic. A Third Report of the Committee on Flexibility, Central—l— New York School Study Council (Syracuse, N. Y.: Bureau of School Service, School of Education, Syracuse University, 1951), p. 106. 47 DeVault, Houston, and Boyd investigated the relative effectiveness of television, television supplemented by classroom consultant services, face-to-face lecture -discussion, and face—to-face lecture discussion supplemented by consultant services as methods of in- service education for elementary school teachers. These four methods of iii-service education were evaluated in terms of change in teacher achievement, change in teachers‘ classroom practices, and change in H pupil achievement and interest. 5‘ Eighty-nine teachers of elementary school mathematics in grades four, five, and six in one school system volunteered to participate in the study. The content for the iii—service education program was pre- pared by the research team from the University of Texas. The same professor of mathematics served as the. instructor for all groups. Conclusions drawn from the results of the study were: 1. Television was as effective as face-to -face lecture discussion in changing the mathematics and methods understanding of teachers, in the reaction of teachers to the iii-service edu- cation program, in changing all but one of the nine components of the classroom practices of teachers, and in changing the 57M. Vere DeVault, Robert Houston, and Claude Boyd, Television and Consultant Services as Methods of Iii-Service Education for Elementary School Teachers of mthematics—(Bureau of Laboratory Schools Publication N"; 15. Austin, Texas: University of Texas, 1962), pp. 1-102. q. v“ 48 mathematics achievement and interest of pupils in classes of the participating teachers. 2. Consultant services as a supplement to television and face-to— face lecture-discussion made a significant contribution in some situations. 58 Tests and conferences were the means used to obtain this in- formation. Both Houston and Boyd completed dissertations related to aspects of the study. Houston's study had to do with pupil achieve- ment;59 Boyd studied teacher achievement and reaction. In their suggestions for further research, these researchers not only suggested educational television as a promising means of in- service education, but also advised school systems to study the effect of written materials and test-rete st effect upon teacher change in achievement and in classroom practices. 61 Among the questions sug- ge sted for further research were: 58The Bureau of Laboratory Schools, University of Texas, pub- lished the findings of the major study, Television and Consultant Services as Methods of In-service Education for Elementary School Teachers_c)_:f_ Mathemaffcs. ' 59W. R. Houston, "Selected Methods of Iii-service Education and the Mathematical Achievement and Interest of Elementary School Pupils, " (unpublished Ph. D. dissertation, University of Texas, Austin, 1962). 60C1aude C. Boyd, “A Study of Relative Effectiveness of Selected Methods of Iii-service Education for Elementary School Teachers, “ (unpublished Ph. D. thesis, University of Texas, Austin, 1962). 61DeVault,l—louston, and Boyd, op. cit., pp. 101—102. 49 1. Would written materials and a testing program . . . be as effective as television or face-to-face lecture -discussion series in bringing about hypothesized outcomes? 2. Is in-service education by television without testing likely to be as effective in bringing about specific changes as when testing is included? 3. What anxieties are produced as a result of the testing program and to what extent are the outcomes of an iii-service problem restricted by the nature of the test instruments?92 Procunier, working under the direction of Maurice Stapley, carried out an investigation which had to do with the impact of Title III, National Defense Education Act of 1958, upon the arithmetic pro- grams in the public schools of Illinois. This researcher, using interviews and an opinion questionnaire, found that the state supervisory and consultant service, developed under Title III, had become an integral function of the State Super- intendent's office. The consultant staff, so he reported, had been of value to participating school districts through demonstrating worth- while experiences and assisting in the in-service training of local staffs. He did, however, note that, even though participants and supervisory staff found the Title III workshop to be extremely valuable, there was an over-all lack of utilization. 621bid., p. 100. 63Robert Wilford Procunier, “The Impact of Title III, NDEA Programs in the Public Schools of Illinois,“ (unpublished Ed. D. dissertation, University of Illinois, 1962). 50 A recent investigation by Ruddell and Brown studied the effect that three different programs of iii—service education had on teachers and pupils. Mathematics consultants were used in different ways in an attempt to improve instruction in elementary school mathematics. One group met with the consultant for one six-hour session during orienta- tion week. Two other groups were involved in long range programs. One group participated in ten meetings scheduled at spaced intervals throughout the year. The third group also had ten meetings but were served by an intermediary. Both a pre -test and post—test were used with teachers in all three groups. None of the groups, so the investigators said, made impressive gains although the second group made significantly greater gains than did the third group. 64 III. Changing the Attitudes o_f_ Elementary School Teachers Toward Arithmetic It has been said often that one of the factors which may limit the effectiveness of any teacher is his attitude toward the subject he is attempting to teach. Indeed, there are those who say that fear, dis- like, and frustration toward a subject build up because of insufficient challenge, too difficult work, or poor presentation of the subject. 64Arden Ruddell and Kenneth Brown, “In-service Education in Arithmetic: Three Approaches," Elementary. School Journal (April, 1964), pp. 377-382. 51 . . . The teacher‘s attitude is a more important factor than his formal preparation in his effectiveness. . . . If he is flexible, willing to try new things, and exercises critical judgment, he is likely to do well. If he is rigid, resistant to change, or uncritically for or against, he is likely to do poorly.63 Could not an elelnentary teacher with an inadequate understanding of mathematics and a genuine dislike of the subject infest a large number of boys and girls with an enduring fear and hatred to mathe- matics? While much of the current literature concerning arithmetic in the elementary school has indicated that arithmetic is a much dis- liked subject, there is not complete agreement on the effect of the teacher's attitude. Dyer, Lakin, and Lord reported a study in which nearly three- fourths of the elementary teachers interviewed expressed a long standing hatred of arithmetic. They suggested that: . . . Elementary: school teachers pass through the elementary school learning to detest mathematics. They drop it in high school as early as possible. They avoid it in college because it is not required. They return to the elementary school to teach a new generation to detest it. 66 Poffenberger and Norton also placed strong emphasis upon the attitude of the teacher toward the subject. They stressed that one 65Paul C. Rosenbloom, “Mathematics K-l4," Educational Leadership, IX, No. 6 (March, 1962), p. 361. 66Henry Dyer, Robert Lakin, and Frederick M. Lord, "The Teacher, " Problems in Mathematical Education (Princeton, N. J.: Educational Testing Service, 1956), pp. 7-12. 52 teacher‘s dislike of the subject could destroy favorable attitudes to- H . . . 6r ward arithmetic and mathematics. Any teacher who fears mathematics, who teaches the subject in a climate of distrust--or who believes that elementary arithmetic is a set of arbitrary mystical rules to be swallowed like a distasteful medicine, is bound to transmit some of these attitudes to the student. McDermott, for his doctoral dissertation, used structured inter- views and the case-study approach to obtain data relative to factors that cause fear and dislike of mathematics. After interviewing forty- one students (seven proficient in mathematics and thirty-four who had been recommended for remedial help by their teacher), he concluded that "most students who have a fear and dislike of mathematics met with sonie frustration in the elementary grades. “69 As early as 1951, Wilbur H. Dutton, after observing the overt behavior of university students as he taught methods courses in arith- metic, secured data pertinent to attitudes toward arithmetic. These data were obtained through written statements from two hundred and eleven students. He examined the statements and separated them into 67 Thomas Poffenberger and Donald Norton, “Factors Deter- mining Attitudes toward Arithmetic and Mathematics, " The Arithmetic Teacher, III (April, 1956), p. 114. 68Ibid. 69Leon McDermott, “A Study of Factors that Cause Fear and Dislike of .Mathematics, " (unpublished Ed. D. dissertation, Michigan State University, 1958). 53 ‘ one of two groups: (1) factors responsible for favorable attitudes or (2) factors causing unfavorable attitudes. After studying the findings, four conclusions were drawn: 1. There was a tremendous outpouring of unfavorable attitudes toward arithmetic. 2. There was a clustering of unfavorable responses around: (a) lack of understanding, (b) teaching unrelated to life, (c) too many pages of word problems, (d) boring drill, (e) poor teaching, (f) lack of interest, and (g) fear of making mistakes. 3. University students often came to methods classes with antagonistic attitudes toward arithmetic. 4. Student reaction often was so charged emotionally that learning could be affected. 70 Dutton has continued to study the attitudes of prospective teachers toward arithmetic. In 1954, he reported a second investigation for which he constructed and evaluated an instrument designed to measure the attitude of prospective teachers toward arithmetic. The responses which nearly six hundred students had written over a five -year period were grouped around forty-five statements. Using a technique de- veloped by Thurstone and Chave, students were asked to sort the state- ments using a scale of one to eleven (extreme like to extreme dislike). Twenty-two statements then were selected for incorporation into the evaluation instrument. The instrument was used to test two hundred and eighty-nine students. No attempt was made to develop a total score or an average 7OWilbur H. Dutton, "Attitude of Prospective Teachers Toward Arithmetic, “ Elementary School Journal, LVI (October, 1951), pp. 85-87. 54 score for each student. The data, however, were organized into tables in such a way that positive liking and pronounced unfavorable feeling could be noted. Obtaining a reliability of . 94 through a re-test procedure, Dutton concluded that attitudes can be measured objectively and significant data can be obtained which will be helpful in the education of prospec- tive elementary school teachers. A later study by Dutton, recently reported in The Arithmetic Teacher, included a revision of the earlier instrument. The original - scale was reduced to fifteen items and five new sections were added. A reliability of . 84 was obtained for this revision by test-retest procedure. ‘ Dutton administered the revised attitude scale to one hundred twenty-seven prospective teachers enrolled in classes at the University of California (Los Angeles). Both favorable and unfavorable attitudes were expressed by the students. Further study of the findings revealed that: . . . Liking or disliking arithmetic is an individual affair. Diag- nosing students‘ feelings about arithmetic and planning corrective measures must be directed toward individual pupils. . ’2 71Wilbur H. Dutton, "Ivleasuring Attitudes Toward Arithmetic," Elementaiy School Journal, LV (September, 1954), pp. 24-31. 72Wilbur H. Dutton, ”Attitude Changes of Prospective Elementary School Teachers Toward Arithmetic, " The Arithmetic Teacher, IX (December, 1962), p. 424. 55 Also, while he noted that attitudes toward arithmetic of students responding to the revised attitude inventory were ambivalent, he con- cluded that they were, when compared with those of the 19.54 sample, almost identical. 73 Perhaps the most outstanding facet of the Dutton studies is the conclusion that attitudes toward arithmetic, once developed, are tenaciously held. "Continued efforts to redirect the negative attitudes of students into constructive channels have not been very effective. . . . The best antidote is probably improved teaching in each elementary school grade. . . . ”74 O'Donnell made a study of attitudes of one hundred and nine elementary education seniors in Pennsylvania. His findings, however, were not in total agreement with those of Dutton. Using the Remmer's "Attitude Toward Any Subject Scale, " he obtained scores which were inconsistent with either the students' expressed like or dislike of arithmetic or achievement scores. 7 Stright also conducted an attitudinal study which involved 1,023 students and 29 teachers. The Dutton Attitude Scale was revised for v“ 73Ibid. 741bid. 75.1. R. O'Donnell, "Levels of Arithmetical Achievement, Attitudes Toward Arithmetic and Problem Solving Behavior Shown by Prospective Elementary School Teachers, " (unpublished Ed. D. thesis, Pennsylvania State, 1958). 56 use in this study. This revised form, including twenty-five items for students and thirty-five items for teachers, asked the subjects of the study to either "agree" or "disagree" with the items presented. From the data relative to the replies of the teachers, the following con- clusions were cited as significant: 1. Ninety per cent of all teachers said that no matter what happens, they fit arithmetic into their schedule each day. 2. Ninety-three per cent stated that they really enjoy teaching arithmetic, while 97 per cent indicated that they thoroughly enjoy teaching arithmetic. (Several questions were re- peated in different form as a check; this was one which varied slightly.) 3. Ninety per cent of the teachers felt that a good teacher should keep up with modern methods, but twenty-one per cent felt that they teach arithmetic well without reading periodicals and methods books. 4. Seventeen per cent felt that methods of teaching arithmetic had not changed in the past thirty years. ' 5. All of the teachers agreed that arithmetic is a great‘ value . 7 6 Thus the assertion that arithmetic, when taught by teachers who dislike the subject, produces some undesirable attitudes has been open to dispute. The investigations of Dutton and McDermott supported the assertion while the data from studies by O'Donnell and Stright tended to disagree. Perhaps a statement in the preface to the material pre- pared by the School Mathematics Study Group summarizes rather well the present thinking with respect to the effect of attitudes. 76Virginia Stright, "A Study of the Attitudes Toward Arithmetic of Students and Teachers in the Third, Fourth, and Sixth Grades, " The Arithmetic Teacher, (October, 1960), pp. 280—286. 57 If mathematics is taught by people who do not like, and do not understand the subject, 'it is highly probable that pupils will not like and will not understand it as well. 77 Certainly it is hoped. that the vicious circle can be broken. Certainly it is hopedrthat a new generation is not being taught to detest mathematics. Has it not been said that the teacher who enjoys his sub- ject has the be st start in the world? . . . A good teacher, a challenging experience and numerous practical application of arithmetic are highly significant factors in the development of favorable attitudes toward the subject. 7.8 IV. Improving the Classroom Procedures of Elementary School Teachers in Teaching Arithmetic Stephen M. Corey said that the acquiring of new understandings and new attitudes is but a means to an end. "Improvement of pro- fessional behavior is the main objective. "79 He stressed this ob-‘ jective when he wrote that one of the real values of any in-service education is the possibility of helping people to change and grow. . . . Reading something in a book, discussing it intelligently, or even memorizing it is completely inadequate. The test of in-service education is whether it results in better 77School Mathematics Study Group, Number Systems, VI: Studies_i_r_i Mathematics (New Haven, Connecticut: Yale University Press, 1961), p. i. 78Dutton, op. cit., p. 104. 795tephen M. Corey, "Introduction," Ch. I, Fifty—sixth Yearbook o_f_the National Society for the Study 9_f_ Education, Part1 (Chicago, Ill.: University of Chicago Press, 1957), p. l. 58 learning and in better living experiences for boys and girls.8 Yes, the true test of an in-service program is how it affects the behavior of the participants. Did the teachers who participated change in their classroom procedures? Did these teachers relate some of the new ideas learned to actual classroom instructional pro- cedures? Were they able to present new content in a way that could be. easily understood? These basic questions-~to what extent did teache rs' increase in understanding of the subject and a more favorable attitude toward the subject result in a change in classroom procedure--are questions open for investigation. ‘ In the comprehensive study, Television and Consultant Services, by DeVault, Houston, and Boyd, an attempt was made to evaluate the classroom practices of eighty-seven teachers through the use of pupil interview technique. Six pupils from one class of each of forty-five teachers were randomly selected. An interview instrument, a twenty- seven item questionnaire which was patterned after one used by Shannon and Wishard for an Elementary School Science Program, was used to elicit pupil response on the following components: (1) materials, (2) classroom activities, (3) organization, (4) teaching aids, and (5) evaluation. 80Robert Gilchrist, "Highway to Quality Teaching," NEA Journal, XL (May, 1959). Pp. 18—19. 59 The investigators noted significant changes in the classroom prac- tices of elementary school teachers involved in the program. In addition, they observed that teachers in the face-to-face lecture dis- cussion group changed more in their manner of using materials than did teachers in the television group. 81 Can teachers grow through in-service education? This question was asked by Ned Flanders in Educational Leadership as he reported on two recently completed in-service training projects which attempted to measure change in teacher behavior. The purpose of the study carried out by Flanders and others at the University of Niichigan under Title VII, NDEA, was to increase the flexibility of the teachers' influence and also to increase the use of those teacher behavior patterns which support pupil participation. Using tests to select teachers whom they thought would benefit from such an in-service education program, they hypothesized that a compatibility between preferred patterns of learning and in—service education procedures would affect the progress of teachers. In their conclusions they noted that "very few in-service educa- tion programs are evaluated with enough care to tell whether or not the quality of classroOm instruction has been, affected. "82 81Houston, Boyd, and DeVault, "An Iii-Service Mathematics Education Program," p. 68. 82Ned Flanders, "Teacher Behavior and In -Service Programs, " Educational Leadership, XXI, No. 1(October, 1963), pp. 25-29. 60 In the same article, Flanders reported a project in human rela— tions which was undertaken by Bowers and Soar. The purpose of the study was to help teachers achieve their own preferred degree of democratic classroom management by increased sensitivity to their own behavior, increased sensitivity to causes of pupil behavior, and greater self-direction by pupils working in groups. In their con- clusions they indicated that "not all teachers can benefit from this kind of training. "83 Two studies previously mentioned, one by Anderson and another by A‘IitChCll, inve stigated the effect of participation in a workshop upon classroom practices. Anderson's study, a three-year cooperative staff study, was concerned with theinfluence of the in-service edu- cation program on teacher test behavior and classroom procedure. Forest C. hiitchell investigated the effect of participation in a workshop upon classroom practices. The purpose of the study was to identify changes that were made in selected classroom practices and to discover any contributions that participation in a summer workshop might make. His data consisted of case -study material obtained through . 83N. D. Bowers and R. S. Soar, Studies in Human Relations in Teaching-Learning Process, Final Report of Cgoperative Re search— Project, No. 469, 1961, cited by Ned Flanders, "Teacher Behavior and In-Service Programs, " Educational Leadership, XXI, No. 1 (October, 1963), p. 25. 84Anderson, op. cit., pp. 205-215. 61 interviews, classroom observation, and daily logs collected on twenty— seven elementary school teachers. He concluded that the workshop was a rich experience with many opportunities for teachers to get the kind of help solicited. The greatest change he detected was in the use of materials. 85 Other individuals, such as Turner, 86 who studied the probleni- solving proficiency among elementary school teachers, and Mork, 87 who investigated the effects of an in-service education program on the science knowledge of fifth- and sixth-grade pupils, have reported closely related studies. However, the idea of a teacher's own learning being related to effective teaching is a question which must be an- swered by additional research. Studies are needed which investigate the changes brought about in the classroom as a result of in-service education programs. Case studies, classroom observation, pupil interview technique, and experimental studies involving pre- and post-testing of both teachers and pupils are but some of the techniques which should be utilized. 85Fore st C. Mitchell, "The Effect of Participation in a Summer Workshop Upon Selected Classroom Procedures, " (unpublished Ed. D. dissertation, University of California, 1951). 86Richard L. Turner, Problem Solvglg Proficiency Among Elementary School Teachers, Part II (Bloomington, Indiana: Institute of Educational Research, School of Education, Indiana University, 1960). 8'Gordon M. A. Mork, "Effects of an Iii-service Teacher Training Program on Pupil Outcomes in Fifth and Sixth Grade Science," (unpublished Ph. D. dissertation, University of Alinnesota, 1953). 88DeVault,Houston, and Boyd, Television and Consultant Services, pp. 99-100. 62 Have not good teachers always been recognized as those who were in the process of searching for new ideas, new understandings, and interesting ways to present their subject? DeVault, Houston, and Boyd said that, while the extent to which this ”searching and experi- rnenting" is related to increased understanding and a more. favorable attitude yet remains to be determined, educators may continue to plan in-service education activities in the belief "that change does beget change”'--that professional growth of teachers can be carried 89 over into classroom procedures. V. Summary Investigation after investigation has stressed the need not only for teachers "to grow on the job, " but also the need for studies which investigate the effectiveness of various types of in-service education. . . . An in-service education program just makes it possible for the teacher to make better use of pre-service instruction and to keep up with the demands and problems of a changing society. There is a need for the critical examination and evaluation of ‘/ in-service programs planned to meet the need of teachers if more 891b1d. 90DeVault, Houston, and Boyd, Television and Consultant Services, p. 102. 63 efficient in—service education is to contribute to the continued success of the American education effort. 91 This chapter on related literature has sought to organize and summarize the major research development which was considered relevant to the present study. This review abstracted major studies which considered the following topics: (1) improving the mathematical understandings of teachers, (2) utilizing iii-service education pro- cedures, (3) changing the attitudes of elementary school teachers toward mathematics, and (4) improving the classroom procedures of teachers with respect to arithmetic. ’ Various sources indicated that an inexcusably large number of prospective and experienced elementary school teachers simply do not know as much arithmetic as they should in order to teach it effectively. They saw the teacher's attitude toward the subject as a factor which. might limit his or her effectiveness. The various states, the many colleges and universities, and the local school districts were urged to continue to study and evaluate the many forms of in-service teacher education. ”Only continuous study and growth will provide teachers sufficiently up -to -date to cepe with the task at hand. "92 91C. W. Phillips, ”What are the Characteristics of an Effective In-service Program?" Issues presented at Thirty-sixth Annual Con- ference, National Association of Secondary School Principals (February, 1952), pp. 360-361. 92A. S. Barr, ”Teacher Personnel III, In-service Education," Encyclopedia o_f Educational Research. A project of the American Re- search Association (New York: Macmillan and Company, 1950), p. 1421. 64 The following chapter will describe the setting for the study, an in-service education program conducted by a mathematics consultant from the Missouri State Department of Education, and the procedures utilized in carrying out the study. Chapter IV will present through tables, graphs, and explanatory materials the findings from the study. In Chapter V, findings will be summarized and implications from the study discussed. CHAPTER III THE PROCEDURES OF THE STUDY This chapter is concerned with the procedures which were followed in carrying out the present investigation. It has been divided into four sections. The first part of the chapter describes the state- sponsored workshop in mathematics for elementary school teachers, the in-service education program which was investigated in this study. The second section is a report on the population sample used in the study. It includes a description of the sample as well as an explanation of the procedures used for assignment of the sample to groups for purpose of analysis. The third pa rt of the chapter is concerned with the collection of data. It includes a discussion of the instruments used, the administration of the instruments, and the preparation of the data for analysis. A brief summary concludes the chapter. I. The State -Sponsored Workshop i_1_1 Mathematics for Elementary School Teachers Fred Weaver, in describing an in-service education program in mathematics for elementary school teachers, said: Rare indeed today is the school system that does not' recog- nize a need for in-service education in mathematics for its 65 66 elementary teachers. Rare indeed, too, is the school system that is not seeking some means to satisfy this need effectively. The state of Missouri, through an in-service teacher education program financed by Title III, National Defense Education Act, has been able to assist nearly one. hundred school systems in satisfying such a need. This study attempted to analyze the effectiveness of the 1963-64 series of state -sponsored workshops in mathematics for ele- mentary school teachers. Specifically, this study attempted to evalu- ate the effectiveness of one series of state-sponsored workshops in mathematics for elementary school teachers in: (l) improving basic mathematics understandings, (Z) changing the attitudes of elementary school teachers toward mathematics, and (3) improving the classroom practices of teachers with respect to the teaching of arithmetic. Since the inception of the Title III, National Defense Education Act Program in Missouri in 1959, the state-sponsored workshops in mathematics for elementary school teachers have been a major aspect of Missouri's plan for administering this type of financial assistance. Viewed as one means for helping teachers to "modernize the content and upgrade their teaching of arithmetic, "2 these state -sponsored \Vorkshops have been and continue to be important elements in the x 1J. Fred Weaver, "Focal Points, " The Arithmetic Teacher, X, N0. 6 (October, 1963), p. 359. 2"Elementary Mathematics (K-8) In-service Program, " (Depart- ment of Education, State of Missouri, 1963), p. 2. (\‘Iimeographed.) 67 iii-service teacher education program. Through these mathematics workshops, teachers have had Opportunities to learn of new 'develop- ments, to examine new materials, and to observe and study new methods of pre sentation. Objectives 2f Elementarl School Mathematics 55-8) In-service Education Program A bulletin issued by the State Department of Education sets forth the following as objectives for the state-sponsored workshops in mathematics for elementary school teachers: 1. To provide teachers with the opportunity to raise their level of understanding of the concepts of mathematics and to lay the foundation for further self-improvement. To illustrate through demonstration teaching some of the new approaches to, and methods of, presenting mathematical ideas to elementary students. To create an atmosphere of enthusiasm around mathematics so that teachers and students alike may enjoy the pursuit Of its excellence. To acquaint the teachers with new teaching aids, manipula- tive devices and the laboratory approach to the teaching of elementary school mathematics. To provide experiences and materials whereby teachers may broaden their horizons relative to experimental prOgrams, pertinent literature, and extracurricular activities in the field of mathematics. 3 Descriptiongf the Workshops These state-sponsored workshops, directed by competent and Well-trained mathematics consultants from the state department, were V 3’lbid. , pp. 3—4. 68 organized at the request of local school systems or adjoining school districts. I They were open to all elementary school teachers. The re was no fee and no college credit was given. Each group met for a period of ten weeks, one session per week. The weekly sessions, approximately two hours in length, were scheduled either from 4:30 to 6:30 or from 7:00 to 9:00 in the evening. The workshop sessions were, for the most part, built around the lecture -demonstration method of presentation. At least two sessions were devoted to the construction of and the demonstration of exploratory materials deemed vital to the presentation of a well- rOunded mathematics program in the elementary school. In addition, each participant received printed materials which supplemented the Oral presentations. The mathematics consultant also visited the classrooms of teachers participating in the workshops and, upon request, taught demonstration classes using modern concepts and some of the newer approaches to the teaching of elementary school mathematics. II. The Population Sample Used i_n_, the Study Permission was obtained from the IVlissouri State Department of Education to evaluate the effectiveness of the 1963—64 series of state- sponsored workshops in mathematics for elementary school teachers, 69 the in-service teacher education program set up by the Missouri State Department of Education under Title III, National Defense Education Act. In order to control as. many variables as possible, the popu- lation from which a sample of two workshops was drawn consisted of only those state-sponsored workshops in mathematics for elementary school teachers conducted in the Southeast IVIissouri State College Service District by one and the same mathematics consultant during the 1963-64 school year. The Workshop Sites The sample for this study utilized workshops conducted in two school systems located in suburban areas of a metropolitan city. 4 One school system was the site for the first workshop; the other was the site for the second workshop, as well as being the school system from which a third group (used as a control group) was drawn. 5 In this metropolitan area, often referred to as a large diversified 1Ildustrial center, are located a number of industrial plants, some the 4In accordance with an agreement between personnel from the i\llissouri State Department of Education and the superintendents in the participating school districts, no mention will be made of the name of any particular school system. 5When volunteers from one of the school systems exceeded the rhaximum number, it became necessary to divide the group and a third workshop planned for a later date. This third workshop group, Since it also was made up of volunteers, was used as a control group. 70 largest of their kind in the world. Many employees in these industrial plants live in suburban areas surrounding the city. Both school systems utilized in this study were a part of a network of public schools which had been set up to care for an expanding population. The findings of Rogers with respect to a high degree of similarity in public education in iVIissouri, irrespective of differing geographical, cultural, and economic environment, was considered by the inve sti- gator as sufficient evidence that this sample did not differ signifi- cantly from normal. During the school year of 1963-64, the first school system, later to be referred to as School System X, had seven elementary schools with an enrollment of 3500 pupils (grades K—6) and an in- structional staff of 148 persons assigned to work with children in the elementary grades. The second school system, referred‘to-as School System Y, had nine elementary schools with an enrollment of 7300 —9 ( students and 349 elementary school teachers. Description of Teacher Population During the first semester of the 1963-64 school year, teachers in both school systems were given the opportunity to volunteer to 6William R. Rogers, ”Public Opinion Rtigarding Selected Public Education in Niissouri, " (unpublished Ed. D. dissertation, University Of i\-'Ii350t1ri, 1949). 7Data obtained from i\*lissouri School Directory, 1963-64. (Depart- lVient of Education, State of Missouri, 1963), pp. 171-172; pp. 181-183. (Mimeographed.) 71 participate in a workshop in mathematics for elementary school teachers. . One hundred twenty teachers, or approximately thirty per cent, of the nearly five hundred and fifty elementary school teachers in the two school systems indicated an interest in participating in the mathematics workshops. Of the teachers who originally indicated interest in participation, ten withdrew, four attended only the first session, six did not attend sufficient sessions to obtain iii-service credit, five did not take the post-test, and six returned their papers with incomplete data. The remaining eighty-nine teachers constituted the population sample for this study. Henceforth, all descriptions and analyses will be made relative to this group. As signment C rite ria Utilized For purposes of analysis, the data concerning the participants were grouped according to the following four criteria: (1) school sys- tem, (2) level of teaching assignment (primary or intermediate), (3) completion of in—service education program, and (4) testing procedure. School system. --The assignment of teachers to either Group A, B, or C was determined by the first assignment criterion, school system. All teachers in Group A were from School System X. The participants in Groups B and C were from School System Y. Level of teaching assignment. --The second assignment criterion utilized had to do with the grade level at which the teachers were 72 teaching. The workshop participants in Group A were divided into two _ groups, primary and intermediate. The primary teachers were identi— fied as Group Al; the intermediate teachers were designated as A2’ Group B was made up of teachers who were teaching at the primary level. A third group, Group C, included teachers whose assignment xvas at the intermediate level. Te sting procedure. --A third assignment criterion was testing proce- dure. Groups A1, A2, B1, and C were tested both before and after the Second semester in-service education program; those in Group B2, a randomized sample of primary teachers from School System Y, were te Sted only at the end of the mathematics workshop. The number of teachers initially assigned to each group is shown in Table 1, page 73. Tlme of participation in the iii-service education program. --A fourth \— —_ g, as Signment criterion was completion of the in-service education program during the time of this study. Group C, the control group, did not even begin the workshop experience even though they had vOlunteered to participate in a workshop in mathematics for elementary SC hool teachers. III. The Procedures Used i_n_the Collection _o_f_Data The specific procedures used in gathering the data for this study 11‘iCluded: (l) the preparation and use of an information form, (2) the 73 TABLE 1 ASSIGNMENT FOR EIGHTY-NINE TEACHERS UTILIZED IN THE STUDY Level of Iii-service Education Control School Teaching Group Group Total SystemaL As signment Pre -te st Pre -te. st Yes No Yes X Primary (Group Al) 10 10 Inter- mediate (Group AZ) 22 22 Sub -total 32 32 Y Primary (Group B1) 20 20 Primary (Group 82) 15 15 Inter- mediate (Group C) 22 22 Sub -total 20 15 22 57 Total 52 15 22 89 aIn accordance with an agreement between personnel from the Missouri State Department of Education and the superintendents in the participating school districts, no mention will be made of the name, of any particular school system. 7—} construction of, the administration of, and the scoring of a test of mathematical understandings, (3) the selection and use of an arithmetic attitude inventory, and (4) the procurement of and the utilization of a C lassroom interview instrument. The Preparation and Use of an Information Form An information form entitled "Teacher Background InformationH was prepared for distribution at the first session of each workshop. This information form requested each teacher to furnish the following data relative to background information: (1) sex, (2) years of teaching experience, (3) grade level assignment, (4) highest degree attained, (5) number of semester high school credits in mathematics, (6) number of semester hours credit in college mathematics, and (7) number of semester hours credit in methods of teaching arithmetic. After the teacher had completed this information form, the data for each teacher were transferred directly from the information sheet Completed by the teacher to a code sheet and subsequently punched into IBM cards and used in the final analysis. Years of teaching experience. --The eighty-nine teachers utilized in the M~ L4 Study varied widely in terms of teaching experience. The years of teaching experience for all teachers ranged from none to thirty-five \_‘ 8 . . . . . . For examinatlon of this registration form, see Appendix 1, Po 228. .11 U1 years. For the total group, the mean was 11.11 and the standard devia- tion was 8. 7178. Table 2, page 76, presents a summary of the popu- lation with respect to years of teaching experience. Niathematics background. --The data concerning the mathematics back- ground of the eighty-nine teachers has been organized under three headings: (1) high school credit in mathematics, (2) college credit in mathematics, and (3) college credit in methods of teaching arithmetic. Mean, standard deviation, and range were obtained on each item for each group. Table 3, page 77, shows this information. Highest degree attained. --The professional training of the eighty-nine elementary school teachers was described in terms of one of the following five classifications: (1) less than a Bachelor's degree, (2) a Bachelor's degree, (3) more than a Bachelor's degree but less than a Master's degree, and (5) a Master's degree or more. Table 4, page 78, shows the number of teachers by group in each of the classifications. The Construction of, the Administration _o_f, and the Scoring_ of a T__e__st of hiatliematical Undei standings Even though most educators agree that paper-and -pencil tests are Ollly one pa rt of a well-rounded program of evaluation in mathematics, SUCh tests always have been and probably will continue to be one of the Ihost valuable means of obtaining evidence of achievement. ' The! [the paper-and-pencil testflcontribute much useful data to the total program 76 TABLE 2 COMPOSITION OF POPULATION WITH RESPECT TO YEARS OF TEACHING EXPERIENCEa School System X School System Y Years of Teaching Experience Group Group Total 32 - 35 O O 1 0 1 2 28 - 31 0 0 2 0 1 3 24 - 27 0 O 3 l 0 4 20 - 23 l 3 l l 2 8 l6 - l9 0 l 0 5 3 9 12 - 15 0 2 3 2 4 ll 8 - ll 3 6 0 l 5 l5 4 - 7 3 5 2 3 3 l6 0 - 3 3 5 8 2 3 21 N : 10 22 20 15 22 89 Mean: ' 8.2 8.68 12.50 12.60 12.54 11.1 Stand. Dev. = 5.20 8.15 11.00 8.14 8.31 8.7 aThe decision to use nine intervals was based on a statement by N011 in which he said that the interval used for frequency distribution ShOuld be of "such a size as to give a distribution containing not less than eight nor more than sixteen intervals. " Since the years of teaching experience ranged from none to thirty-five years, an interval of four established nine categories. For further information, see Victor Noll, Educational IVIeasurement (Boston: Houghton Mifflin c:Ornpany, 1957), pp. 393-402- 77 TABLE 3 CENTRAL TENDENCY AND VARIABILIT Y OF MATHEMATICS BACKGROUND FOR POPULATION SAMPLE Background Standard NIean . . Ran e Facto r Dev1ation g High School Credit in Niathematics Group A1 2. 8 4.40 2 - 3 Group A2 2.8 1.09 1-4 Group B1 1.7 2.61 l —4 Group B2 2. 5 6. 60 ‘ 1 - 4 GroupC 3.0 1.07 1 -4 College Credit in Niatherna tic sa Group A1 2.1 l 2 - 3 Group A2 6. 0 30 O -14 Group B1 3.4 3.50 0 - 6 Group B2 2.4 1.3 0 -6 GroupC 3.0 1.09 0 -6 College Credit in Methods of Teaching Arithmetica Group A1 2.9 1.8 2 -3 Group A2 3.1 1.46 O -6 Group B1 2.8 3. 94 0 - 6 Group B2 2.4 .71 O - 6 GroupC 2.13 3.10 0 - 6 aThe mathematics preparation in college is expressed in Semester hour credits. 78 TABLE 4 COMPOSITION OF POPULATION BY HIGHEST DEGREE HELD School System X School System Y Degree Held Group Group Total More than Ma ster's degree 0 2 1 0 5 8 Master's degree 0 1 0 0 0 1 More than Bachelor's degree but less than Master's degree. 9 ‘3 18 10 4 46 Bachelor's degree 1 13 l 5 12 32 Le s s than Bachelor's degree 0 1 0 0 1 2 Total 10 22 20 15 22 89 79 of evaluating mathematical learnings. " 9 With these. ideas in mind, the writer, for the initial part of this investigation, sought to find a paper-and-pencil instrument for evaluating the mathematical under- standings needed by elementary school teachers. For guidance, the investigator turned to the major reference work on tests and test reviews, the series of Mental .=\"leasurenients Year- books, and a publication of the National Council of Teachers of Mathe- . . . . . 10 matics, Nlathematics Tests Available in the Unlted States. The first reference, one of the most important sources of in- formation with regard to tests, is, up to now, a five volume publica- tion. Each yearbook includes reviews of tests published during a specified period, thereby supplementing rather than supplanting earlier yearbooks .11 An examination of the arithmetic tests cited in any of the pre- viously mentioned references revealed that no test commonly used at the present time contained more than a few items designed to measure mathematical unde r standing 5 . ‘ 9Noll, SE cit., p. 3. 10Sheldon S. Myers, Ntathematics Tests Available iithe United States. A bulletin to the National Council of Teachers of Mathematics, prepared by the Secondary School Curriculum Committee (Washington: The Association, 1959). 11 For further information concerning these yearbooks, see the blbliographical entries under Oscar Buros. 80 This was not surprising since, in recent years, many writers have stressed the lack of adequate testing instruments for evaluating mathematical understandings. As early as 1932, Butler wrote: It is certain that, in the domain of published tests, the specific testing for master of mathematical concepts understandings has received scant attention. Brownell, an outstanding authority in the field of elementary school mathematics, noted that: . . Exceedingly little has been done either informally or systematically to find practical and valid procedures for evaluating the outcomes under the heading above [ mathematical unde r standings] . Collier, in his doctoral dissertation at Ohio State University, stated that: Evaluation of important mathematical outcomes has not kept pace with instruction in arithmetic. Standardized tests commonly used measure mainly computational skill. There is a definite need for more systematic and comprehensive research relative. to evaluating the newer outcomes of arith- metic instruction. 14 12C. H. Butler, "Alastery of Certain Nlathematical Concepts at the Junior High School, " Mathematics Teacher, (March, 1932), pp. 117 -172. 13William A. Brownell, "The Evaluation of Learning in Arith- mEtic, " Arithmetic in General Education. A Report of the National Council Committee on Arithmetic, Sixteenth Yearbook of the National COuncil of Teachers of Mathematics (New York: Bureau of publications, Teachers College, Columbia University, 1941), p. 247. l4Calhoun C. Collier, ”The Development of and the Evaluation of a Non-Computational Mathematics Test for Grades 5 and 6, " (Unpublished Ph. D. dissertation, Ohio State University), p. 43. 81 Van Engen contended that much attention has been given to teaching arithmetic meaningfully, but not nearly enough research has been carried out in the area of testing for meaning. Glennon, while conducting one of the most significant studies in the field of mathematics education, found it necessary to develop his evaluation instrument.16 Since that time a number of researchers have used the Glennon test when conducting investigations pertaining to mathematical understandings. However, when the writer sought information and permission to use the test, Glennon wrote: If I were building the test today, it would be a somewhat modified test and, I think, a somewhat improved test. You may wish to use some of the items and replace others with itens of your own to obtain a better cove rage of the present—day elementary school mathematics program.1 When the review of literature and testing instruments indicated that, at the present time, there we re no published tests available for measuring many of the important mathematical concepts or under— standings, it became necessary to develop a pencil-and -paper test which could be used for evaluating the mathematical understandings of teachers both at the pre-service and in-service levels. 15Henry Van Engen, "A Summary of Research and Investigation and Their Implications for the Organization and Learning of Arithmetic," Mathematics Teacher, XLI (October, 1948), pp. 260-265. 1()Glennon, EB. cit., p. 7. 7 . . . . . Letter from Vincent J. Glennon, D1rector, Arlthmetic Studies Center, Syracuse University, Syracuse, N. Y., October 8, 1963. 82 It was Lindquist who wrote that preparation of an educational achievement test includes five major steps: (I) planning the test, (2) writing the test items, (3) trying out the test in preliminary form and z assembling the finished test after tryout, (4) preparing the directions for administering and scoring the test, and (5) reproducing the test.18 Other writers, even though their ideas were somewhat similar, have organized these steps in various ways. Noll proposed four steps: (I) planning the test, (2) constructing the test , (3) using the test, and (4) evaluating the testing instrument. 19 The procedure, however, followed in the construction of this paper-and -pencil test was a modification of the procedure as proposed by Lindquist. In this particular study, planning involved not only the preparation of an outline or blueprint specifying the content to be Cove red by the test, but it also involved careful attention to types of items, to provision for review, to arrangements for tryout, to item difficulty, to problems of test reproduction, and to interpretation and use of test results. Determining test objectives and preparing an out- line of content, however, were the first steps to receive consideration. £S‘rmining test objectives. --Preliminary to actual test construction, the person attempting to develop an evaluation instrument must decide upon the purpose of objectives of the test. \ 18E. F. Lindcjuist (ed.), Educational Measurement, (Washington, D' C.: Anuerican Council on Education, 1951), p. 19. 19Non, op. cit., p. 3. 83 No single test . . . can measure all objectives; he, the testmaker, must choose from an adequate list of objectives those which he will attempt to measure . . . and then formulate his teaching and measurement program on the basis of the. objectives selected.20 Often a statement of objectives has been worked out by com- mittees or groups and the resulting formulation generally represents the best and most forward-looking ideas that the group can produce at that time . Support for using such a list to serve as a basis upon which to build a test was discussed by Collier. He quoted a statement from Greene, Jorgenson, and Gerberich which read as follows: "Reports of national committees and the writings of subject and test specialists Often serve as good guides to content in educational test construction."21 One list which the writer found most helpful was a list of unde r- Standings in elementary school mathematics prepared by Brownell. 2'2 Though prepared several years ago, this list continues to be ‘Videly quoted by authorities. And, while changes have been and are ZONoll, op. cit., p. 96. 21Calhoun C. Collier, "The Development and Evaluation of a Non- Computational Mathematics Test for Grades 5 and 6, " (unpublished ph- D. dissertation, Ohio State University, 1956), p. 47, quoting Harry A. Green, Albert Jorgensen, and J. Raymond Gerberich, ki’egzurement and Evaluation_i_ri the Elementary School, (New York: LOhgmans Green and Company, Inc., 1953), p. 68. 22William A. Brownell, ”The Evaluation of Learning in Arithmetic, " Arithmetic_i_n General Education, The Sixteenth Yearbook of the National Council of Teachers of l\-'Iathematics (New York: Teachers College, Columbia University, 1941), pp. 231-232. 84 being made in the mathematics program in the elementary school, there is almost unanimous agreement that today‘s program is made up of more mathematics new to the elementary school than new mathe- matics. The objectives which were grouped under the heading of mathe- matical unde r standing 5 we re: 1. 7. Meaningful conceptions of quantity, of the number system, of whole numbers, of common fractions, of decimals, of per cents, of measures, etc. A meaningful vocabulary of the useful technical terms of arithmetic which designate quantitative ideas and the rela - tionships between them. Grasp of important arithmetical generalizations. Understanding of the meanings and mathematical functions of the fundamental operations. Understanding of the meanings of measures and of measurements as a process. Understanding of important arithmetical relationships, such as those of which function in reasonably sound estimations and approximations, in accurate checking, and ingenious and resourceful solutions. Some understanding of the principles which govern number ‘ relations and computational procedures. .Iktirmining test content. --The second step in planning the test was the outlining of the content to be covered by the test. Actually, the problem of selecting the content to be covered in the test was chiefly One of determining the arithmetical concepts or mathematical under- Stfindings most commonly taught ‘in the elementary school. This was aCComplished through an examinatiOn of recently published arithmetic \ “use, p. 231. 85 textbooks and experimental materials. 24 Detailed study of articles or books recently published by Schaaf, 25 Newsom, 26 Mueller, Z7 and Brumfiel also prOved helpful. 28 After developing the list of important mathematical concepts, it became necessary to organize these understandings under appropriate headings. The writer found a summary which Grossnickle had adapted from a section of Growth of Mathematical Ideas, the Twenty—fourth Yearbook of the National Council of Teachers of Mathematics, very helpful for developing a framework around which to organize the list of mathematical understandings.29 m 24A list of the materials used for this examination is included in Appendix H, page 226. 25Jack N. Sparks, "Arithmetic Understandings Needed by Elementary-School Teachers," The Arithmetic Teacher, VIII (December, 1961), pp. 395-400, quoting w. L. Schaaf, "Arithmetic for Arithmetic Teachers, ” School Science and Mathematics, III (October, 1953), pp. 537-555. 26 C. V. Newsom, ".N-Iathematical Background Needed by Teachers of Arithmetic, " The TeachingBiArithmetic, Fiftieth Yearbook of the National Society for the Study of Education, Part II. 7Francis J. Mueller, ”Arithmetic and Teacher Preparation, " Ti: Mathematics Teacher, LII (November, 1959), pp. 572—573. 28Charles F. Brumfiel, Robert E. Eicholtz, and Merrill Shanks, wamental Concepts oiElementary Mathematics (Reading, xiiissachusetts: Addison-Wesley Publishing Company, 1962). 29A list of the important mathematical understandings may be £0klnd in Appendix J, page 228. 86 The following are key ideas or important strands in mathematics included in the summary by Grossnickle. 30 l. Numerals and numeration systems 2. Principles underlying numbers 3. Relationships and generalizations 4. Nleasurement and approximation 5. Estimation and proof 6. Symbolism 7. Rational numbers 8. Geometry The next step was the formulation of a blueprint or outline of the specific content to be covered by the test. Many authorities believe that content outline and statements of objectives represent two dimensions into which a test plan or blueprint must be fitted. "These tWO dimensions needed to be fitted together in order to give a complete framework and to decide upon the relative importance given to the seVeral content areas and objectives. "31 The distribution of items for the tests of mathematical under- Staarldings, both according to content and test form, is shown in Table 5. \ 3OFoster E. Grossnickle and Leo J. Brueckner, Discoveriilgg Nieaningsirl Elementary School Mathematics (Philadelphia: Holt, Rlnehart and Winston, Inc., 1963), pp. lO-l4. 31Noll, 22. cit., p. 98. 87 TABLE 5 DISTRIBUTION OF ITEMS ON TEST OF MATHENIATICAL UNDERSTANDINGS ACCORDING TO TEST FORM AND TEST-CONTENT OUTLINE Test-Content Outline Number Of Items Form A Form B 1. Numerals and Numeration System 10 10 Z. Symbolism (including introduction to sets) 2 Z 3. Concept of Sets (including mathematics sentences) 6 6 4. Principles Underlying Number Operations 6 6 5- Relationships and Generalizations 5 5 6. Operations with Whole Numbers 11 ll 7 . Estimation and Proof 1 l 8 . Measurement and Approximation l l 9- Geometry 4 4 1 0- Rational Numbers 9 9 Total 95 55 X 88 Developing the test instrument. --A review of the literature was made in order to gain information concerning the construction of good tests. It was found that various writers in the field of tests and measurements have issued statements or developed criteria of a general nature for the construction of tests. The suggestions for item writing which Lindquist included in Educational Measurement proved most helpful as the writer began to consider the construction of objective test items. The following we re included in a list of suggestions made by Lindquist. 10. Express the item as clearly as possible. Choose words that have precise meaning wherever possible. Avoid complex or awkward word arrangements. Include all qualifications needed to provide a reasonable basis for response selection. Avoid unessential specificity in the stem or the responses. Avoid irrelevant inaccuracies in any part of the item. Adapt the level of difficulty of the item to the group and purpose for which it is intended. Avoid irrelevant clues to the correct responses. In order to defeat the rote -learner, avoid stereotyped phraseology in the stem or the correct response. Avoid irrelevant sources of difficulty. 3‘2 ZLindquist, 33. cit., pp. 213—227. 89 Even though there are many forms of test items in general use-- essay, true -false, short answer, matching, and multiple -choice-- most authorities indicate that the multiple-choice item, consisting of a stem, which may be an incomplete statement or a direct question, followed by two or more alternatives or possible answers, is the "most valuable and the most generally applicable of all types of test _ ’ g'33 BACICISQS. Since the purpose of this particular test was to determine the extent of understanding in certain areas of elementary school mathe- matics and these understandings called for inferences, judgments, gene ralizations, and noting relationships, it was felt that these under— Stalldings fitted into categories which Hawkes said were tested by the m ultiple -choice items . The multiple -choice type of test can be made particularly effective in requiring inferential reasoning, reasoned under- standing, sound judgment and discrimination on the part of the Student; it is definitely superior to other types for these Purposes. Several sources suggested specific principles for test con- Struction with respect to multiple -choice items. Some, of these prlnCliples were: \ 33 . Noll, op. c1t., p. 130. 34E. F. Lindquist, Educational Aleasurement (Washington: The Ame rican Council on Education, 1951), p. 249, citing The Construction m Use BfAchievement Examinations, p. 138. 10. 90 The stem of a multiple -choice item should clearly formulate the problem. All the options should be possible and plausible answers. The stem should not be loaded down with irrelevant material. One choice should be a be st answer, but the others should appear plausible to the uninformed. The writer should beware of clues from the length of the option. Irrelevant grammatical cues should be avoided. The best or correct answer should be. placed equally often in each possible position. Choices should be in parallel form whenever possible. The choice of an item should come at or near the end of the sentence. The number of choices in multiple-choice items should be at 35 least four. W the test items. --Using the list of understandings, test items Of the multiple -choice type, items which Inight be used in evaluating the mathematical understanding of elementary school teachers, were COnStI'ucted. As these items were constructed, they were recorded 0 . . . . n CaI‘ds and filed under one of the eight categories around which the 1' . . . . . ISt of understandings was organized. Since permissmn had been ObtaLlned to use any or all items in the Glennon test, some twenty-two _\ 35Noll, op. cit., pp. 129-134- 91 o f his items were included in the pool of items which was placed on c: ards. The objective here was to construct a pool of test items from \v hich one hundred and ten of the best questions could be selected and a ssembled into two experimental tryout forms of fifty-five items each. S electing the best items. --When it was felt that a sufficient number of items had been formulated, attention was turned to the problem of selecting the best items which could be assembled into the two equiva- lent forms. 36 Each test was to contain one or more items for each of . . . 37 the ten categories in the Test Content Outline. Form A and Form B Of the test, each form consisting of fifty-five items, were developed. Except for the first few items in each test, no effort was made to ar- range the items in order of difficulty. A set of test directions using Simple, clear language was also prepared. After items for the two comparable forms of the test were a ssembled, several sets were reproduced for review by persons con- . . 38 . SldCI‘ed to be competent Judges. One person was asked to rev1ew the items from an editorial standpoint, two test technicians were asked to review the items from the technical point of view, and five persons \ 36 plained on page 96. The procedure used in test for equivalence of form is ex- 37The distribution of items both according to test form and test O1.1.t1ine is shown in Table 5, page 87. ‘ 38The names and addresses of the judges are listed in Appendix I. 92 \v ere asked to review the subject-matter. Comments and suggestions of the judges were utilized in revising many of the items in the try-out forms. Administering the tryout forms. --After the test items had been writ- ten, reviewed by a team of experts, and revised on the basis of their Suggestions, it became necessary for the test items to be tried out experimentally on a sample of examinees "as nearly like the popu- lation with whom the final form of the test was to be used as reasonably possible. ”39 As a pre -tryout, the preliminary form of the test was ad- ministered to approximately fifty elementary school teachers enrolled in a graduate course in ”Problems in Teaching Arithmetic." These teachers, whose experience ranged from none to over thirty years, \vorked, two or three to a group, and informally discussed the various items as they took the test. Their questions and comments con- Cerning each. item were recorded by the test constructor. Ambiguous and cumbersome items were noted. Some respondents crossed out “Words and rewrote the questions in what they believed to be clearer and more concise language. Next, on the basis of the data obtained in this pre -tryout, items ‘Vere revised, deleted, or left unchanged. Two parallel forms of the \ 9Lindquist, op. cit., p. 250. 93 test were prepared. Lindquist says that if, ”in developing equivalent forms of a test, each form is built to conform to a detailed set of s pecifications, while at the same time care is taken to avoid identity 0 r overlapping of content, the two resulting test forms should be truly equivalent. "40 A further check of equivalency was made by examining item correlations within and between the two test forms.41 For the trial administration, each form of the test was given to a group of fifty students who were majoring in elementary education and enrolled in a college course, Technique of Teaching Arithmetic, at Southeast Missouri State College, Cape Girardeau, Missouri. Form A was given during one class period and Form B on the day following. This trial administration, as Lindquist states, served to 42 "indicate how the test would function in actual use. " Since the test constructor administered the test, it was not necessary to prepare detailed directions for the test examiner. On the first page of each test, however, were directions for taking the test and a sample test item.43 There was no time limit to the test; however, each student was asked to record the time he completed the test on his answer sheet. \ 40Lindquist, 92. cit., p. 575. 4 . . . . , 1For details concerning this test for equivalency, see page 90. 42Lindquist, 32' cit., p. 250. 3See Appendix B and c, pp. 183—203. 94 T his information was helpful in determining appropriate time for the finished test. In the scoring of the test, one point was allowed for each correct answer. If the student marked more than one choice, even though one of the choices was the right answer, the item was considered to be answered incorrectly. All work was checked by a person other than the original scorer in ‘order to minimize errors. I H Correction was made for 'guessing or chance success through the use of the following formula: 5 = R - _.__L 11 " 1, Where S = the score, R = the number of correct responses, W = the number of incorrect responses (excluding omitted items), 11 = the number of choices. . . . Because correction for chance success often makes test scores easier to interpret, the writerLLindquist} is inglined to urge its use for almost all educational purposes. . . The data obtained through the administration of the tryout forms of the test were recorded on IBM work sheets and punched into IBM CEirds for later use in: (l) computing the coefficient of reliability, (2) determining equivalency of test forms, (3) studying difficulty index, and (4) determining the discriminating power of the individual items. \ 44Lindquist, 33. cit., p. 365. 45mm. 95 An arithmetic mean and standard deviation were computed for both forms. S tudying the tryout tests. --It has been said that all good measuring instruments have certain qualities in common. While such character- 1 stics as objectivity, ease of administration, ease of scoring, and ease of interpretation are important, the two primary qualities most agreed upon are reliability and validity. A test which lacks a known and satisfactory degree of these principal qualities is not a measuring instrument in any true sense, and little or no dependence can be placed upon results obtained by its use. Reliability. --Often, it is customary to think of validity as the most important quality of a test and to discuss it first. It was Noll, how- ever, who said that ”since reliability is essential to validity and the opposite is not so, there is something to be said for placing reliability at the head of the list. ”47 Reliability of measuring instruments is usually determined by One of three methods: (1) self-correlation, (Z) correlation of equivalent forms, and (3) split -halves correlation. Since two equivalent forms of the test were available, the reliability for the t‘vvo measuring instruments developed for use in connection with the present study was determined by correlating the scores made by the \ 46N611, 9_p_. cit., p. 66. 471bid., p. 67. 96 fifty college students on the two equivalent forms of the test. For purpose of analysis, Form A was referred to as X and Form B re- ferred to as Y. Pearson Product-Moment Correlation Coefficient \\-'as used. The correlation coefficient between the two forms of the test was found to be . 87. This correlation coefficient, indicative of a high positive relationship between the two sets of scores, also was sig- . . . 48 . . . . 111f1cant at the five per cent (.05) level. Such a degree of reliability 1 5 considered adequate by mest test specialists. Desirable reliabilities differ according to purpose. Where a test is intended only for use in studying groups, a lower re- liability and coefficient (around . 75) may be sufficient to make fairly accurate comparisons. Where individual difference is the goal, reliability of . 95 or higher is very desirable. 4 Equivalency of forms. --The problem of preparing truly equivalent forms was, according to Lindquist, ”a problem in the logic and . . 5 practice of test construction. " He stated that the best guarantee for equivalency of two test forms is the. preparation of a complete and detailed set of specifications for the test in advance of any final test . 51 construction. k 48When a table of the ”Values of the Correlation Coefficient for Different Levels of Significance" (df = n-Z) was consulted, it was noted that, with fifty cases, any correlation of .280 or higher was Significant at the . 05 level and any correlation of . 363 or higher was Significant at the . 01 level. For further information, see Allen L. Edwards, Statistical Methods for the Behavioral Sciences, (New York: Holt, Rinehart and VVinstonTTgol), p. 147. 49 Noll, op. cit., p. 73. 1J1 SOLindquist, op. cit., p. 575. 511bid., p. 75. 97 If each test form is built to conform to the outline while at the same time care is taken to avoid identity or detailed overlapping of content, the two resulting forms will be truly equivalent. 3 A further check upon the degree of equivalency was made by examining item correlation within and between the two test forms. 53 0e, these should be equal." Lindquist says that "on the averag Both Form A and Form B of the tests used in connection with this study were sub -divided into two halves (odd-numbered items and even-numbered items); alternate halves were used to obtain correla- tions among the resulting six scores. NIcNemar suggested that the following t-test, which takes into account the relationship between the pairs of r's, be. used: ..g .— -flfl-C-- u.—--——.- (N - 3) (1 + ) t = (r12 - r13) 1‘23 2 2 Z 2(l-r12 -r13 -r23 +Zr12r13r23) where r12 2 odd and even-nurnbered items, Form A, with odd and even-numbered items, Form B, r13 = even-numbered items of both forms and odd-numbered items of both forms, and r23 = even-numbered items, Form A, and odd-numbered items, Form B, with odd-numberedsiiems, Form A, and even-numbered items, Form B. 521bid., p. 575. 53Ibid., p. 576. 54N. M. Downie and R. W. Heath, Basic Statistical‘ Methods. New York: Harpers and Brothers, 1959, pp. 145-146, quoting Q. lVIcNema r, Psychological Statistics, New York: John Wiley, 2nd ed. , 1954. 98 The resulting t (1.02), with 47 (N-S) degrees of freedom, was not significant at the . 01 level. 55 Hence, since there was no significant difference in the correlation coefficients, the two test forms could be Viewed as equivalent. Validity. --It has often been said that a test can be adequately reliable and still not be valid. Bean, in distinguishing between validity and reliability, said: The validity of a test refers to its accuracy with reference to the particular trait it is intended to measure, whereas reliability refers to the consistency with which it does so. 5 Noll has said that the validity of a test may be elucidated by such questions as: 1. What does this test actually measure? 2. To what extent does it measure this particular ability or quality? 3. In what situations or under what conditions does it have validity? 57 There are several widely accepted methods of determining the validity of measuring instruments. Many authorities Classify them 55: = 2.81 at .01 level. 56Kenneth L. Bean, ConstructionpfiEducational and Personnel Tests (New York: McGraw Hill Company, Inc., 1953), p. 15. 5’Non, pp. cit., pp. 73-74. 99 LlndCI‘ three categories: (1) curricular validity, (Z) logical validity, and (3) statistical or empirical validity. 58 The validity of the tests used in this study was established in terms of their curricular validity. Since the tests were: (1) designed around a specific list of understandings, (Z) analyzed by a jury composed of mathematicians, psychologists, and professors of ele- mentary education, (3) examined and studied by experienced class- room teachers, and (4) scrutinized as to form of items, curricular or content validity could be assumed. Noll said that such a method of designing the test was, in itself, a validating procedure. 59 Item analysis. --Another technique employed as a measure of validity was the determining of the difficulty index and the discriminating power of individual questions or items. The first of these, the difficulty of the item, is the proportion of individuals who answer the item correctly while the discrimination index is a measure of how well the item separates two groups. . . . When item-discrimination is used as a means of establishing validity, the test, which is in this case the criterion of validity, is assumed to have curricular validity because it measures what it purports to measure. The second assumption that the scores on a particular valid item agree with scores on the whole test is tested by comparing results on a given test item with scores on the whole test. Since. the test is assumed to be valid, an item which agrees with the test is valid. That is, an SSNOIL 9p,3cfiucd30 i: .m 3:2 3.0 No.3. mesa S .3 H83. 3. No. max 2.x om.m mm.» coflaigm mm .2 oo 4 3 .N me .0 so .N 3.5 $3. mcfioaoe *5. .m 3 .N 5 .N we .2 S .N cm .2 552.830 i: .m mms 2 .m 8:: 3...“ max. 33333. Mama oo.N $3 3.2 2 a. 3.: 23832 o>flduflc30 Seasons Impemxmv Tam QDOHO 32:32 Om Smog Om smog mmoflowhnm QSOHO nu OlwcmLO . umehupmonm $an mhnm EoonmmgO Loammwmm mmBMH>mHHZH dHannw wm QMZHEmMHHQ m< mMOHHOénm 200mmm<40 MMEOwm20£0pmgoo whoa: fin 143 oo. oo. Hm; hm; Nm .H mum; 33.8me xoofimeusoZ MO HOD—#532 me. my . No. me. 3. cm. weir/Sod. xoofonnsoz MO HOOESZ ppm. «.04 oq.w >¢.mm Hog: mmgvm ocoflmmdomwm CO 35056. e>3mfipcm30 mo. 5. Ame 6.0.3.. mwgl mice 1309 on. .. me. .. om; mod Nv.m 00.0 :OEmgmc/m mm .N- S... S .N 3 .m :4 2 .m mesa. “£589. 9... am ._ m: .m 3. cl 5 .N we.» :osmficmwgo :4 co; ow; NM.NH Nn.N 3W: maid/«90¢. vim; do; fiche ww.w ow.v min mfimwneumg 9/332me SOBCOOV O QSOMO NEIHE Om cam: Qm a532, meofiomnm ODOMO no «@230 “mmHuumom omen... mum EoonmmmHO gunmemmm poscflzoO i E MAm < ) (i) ( > Since (c) is correct, the space under c is blackened. *Those items with an asterisk (3‘) are items from Glennon's "Te st of Basic Mathematical Understandings." Permission was granted by the author to use any or all of the items on the "Test of Basic Mathematical Understandings." Copyright 19 64 182 A 183 1. When you write the numeral "5" you are writing a. b. #C- d. the number 5. a pictorial expression. a symbol that stands for an idea. a Hindu-Babylonian symbol. 2. Bill discovered that > means "is greater than" and < means "is less than. " In which of the following are these symbols lit used correctly? d. The number of states in the United States < the number of United States Senators. The number of states in the United States > the number of stripes in the flag. 23>3Z 3+a<5+a 3. When two Roman numerals stand side by side in a symbol, their values are added a. always. / b. sometimes. c. never. d. if the base is. X. 4. Which of the following describe/describes our own system of f numeration? a. additive b. positional c. subtractive (1. introduces new digits for numbers larger than 10 1) a and b are correct. 2) a and c are correct. 3) a and d are correct. 4) a, b, and d are correct. 5. Zero may be used. a. b. c. .41- as a place holder. as a point of origin. to represent the absence of quantity. in all of the above different ways. 200. 02 is shown by 2000 + 200 + 20. 2000 + 20 + 2/10. 2000 + 200 + 2/100. 2000 + 200 + 200. 10. ll. 12. 13.. 184 5840 rearranged so that the 815 200 times the size of the 4 would be a. 5840. b. 8540. c. 5048. d. 5403. Which of the following does not show the meaning of 423ten? a. (4x100) +(2x10) + 3(1) 2423 b. 42 tens + 3 ones = 423 c. 423 ones 2 423 d. 4 hundreds + 42 tens + 23 ones I 423 A numeral for the X‘s in this example can be written in many different bases. Which numerals are corregt3_ Y——.—— -—-—-v-— a. loofour é XX X XX XX 13' 14twelve : X X X X c. 16ten 1) X XX x X d. 31 five “ ‘ 1) a and c are correct. 2) b and c are correct. 3) a, b, and c are correct. 4) all four are correct. 43 wt A "2" in the third place of a base twelve number would represent a. 2 x 123 b. 12 x 23 C. 12 x 212 d. 2 x 12 2 In this addition example, in what base are the numerals written? a. base two b. base three 1%8: c. base four + ' (:1. none of the above 200? About how many tens are there in 6542? a. 6540 b. 6541 c. 65:2 d. 6. :3 Place or order in a series is shown by a. book no. 7. b. three boxes of matches. c. a dozen cupcakes. d. two months. 14. 15. 16. '17. 18. 19. 20. 185 Which of the following indicates a group? a. 45 tickets . b. track 45 c. page 54 d. apartment No. 7 The sum of any two natural numbers a. is not a natural number. b. is sometimes a natural number. c. is always a natural number. (1. is a natural number equal to one of the numbers being added. The counting numbers are closed under the operations of a. addition and subtraction. b. addition and multiplication. c. addition, subtraction, multiplication, and division. (1. addition, subtraction, and multiplication. If a and b are natural numbers, then a + b = b + a is an example of a. commutative prOperty. b. associative property. c. distributive property. d. closure. Ifaxb=0then a. a must be zero. b. b must be zero. c. either a or b must be zero. d. neither a nor b must be zero. When a natural number is multiplied by a natural number other than 1, how does the answer compare with the natural number multiplied? a. larger b. smaller , c. the same d. can't tell from information given Which of the following is the quickest way to find the sum of several numbers of the same size? a. counting b. adding c. subtracting d. multiplication 21. 22. 23. 24. 26. 27. 186 How would the product in this example be affected if you put the 29 above the 4306 and multiplied the two numbers? a. The answer would be larger. b. The answer would be smaller. 4306 c. You cannot tell until you multiply both ways. x29 d. The answer would be the same. An important mathematical principle can be helpful in solving the following example. Z8+659+7Z.:D What principle will be of most help? a. the associative principle b. the commutative principle c. the distributive principle d. both the associative and distributive principles The product of 356 x 7 is equal to a. (300 x 50) x (6 + 7). b. (3x7)+(5x7)+(6x7). c. 300x50x6x7. d. (300x7) +(50x7)+(6x7). Which of the following is not a prime number? a. 271 b. 277 c. 281 d. 282 Which of the following numbers is odd? a. 18 x 11 b. 11 x 20 c. 99 x 77 (1. none of the above The inverse operation generally used to check multiplication is a. addition. b. subtraction. c. multiplication. d. division. The greatest common factor of 48 and 60 is a. 2 x 3. b. 2 x 2 x 3. c. 2x2x2x2x3x5. (1. none of the above. 28. 29. 30. 31. 32. 33.‘ 187 Look at the example at the right. Why is the "4" in the third partial product moved over two 157 places and written under the 2 of the multiplier? x £6 ‘ a. If you put it directly under the other partial 942 products, the answer would be wrong. 628 b. You must move the third partial product two 314 places to the left because there are three 38622 numbers in the multiplier. c. The number 2 is in the hundreds column, so the third partial product must come under the hundreds column. d. You are really multiplying by 200. Which of the fundamental properties of arithmetic would you employ in proving that (a + b) + (a + c) = 2a + b + c? a. Associative property ._ b. Commutative property c. Associative and distributive properties d. Associative and commutative properties If N represents an even number, the next larger even number can be represented by a. N + l. b. N + 2. c. N + N. d. 2xN+l. Every natural number has at least the following factors: a. zero and one. ' b. zero and itself. 0. one and itself. d. itself and two. It is said that the set of whole numbers has a natural‘order. To find the successor of a natural number, one must a. add 1. i b. find a number that is greater. c. square the natural number. d. subtract 1 from the natural number. The paper below has been divided into 6 pieces. It shows _' l a. sixths. b. thirds. c. halves. d. parts. 34.‘ 35. 36. v37. 38. 39. 188 A fraction may be interpreted as: a. a quotient of two natural numbers. b. equal part/parts of a whole. c. a comparison between two numbers. d. all of the above. When a common (proper) fraction is divided by‘ a common _ fraction, how does the answer compare with the fraction divided? a. It will be larger. b. It will be smaller. c. It will be twice .as large. d. There will be no difference. -Which algorithm is illustrated by the following sketch? 1 ; /- / 1 3 /- ’x ‘ a. —X—= ? I / y, I 1/2 2 4 ///// .1 3 z ‘ ///1 // _l b. —+—= ? 2 4 1/2 1 1 1 _ A c. _+ _+ — '? 2 2 '2’ 1/4 1/4 1/4 1/4 d.'i __.3_ : ? 4 2 Another name for the inverse for multiplication of a rational number is the a. reciprocal. b. opposite. c. reverse. d. ze r0. Examine the division example on the right. 3 Which sentense best tells why the answer 5 .. — = 6 is larger than the 5? 3 4 a. Inverting the divisor turned the - upside down. b. Multiplying always makes the answer larger. c. The divisor 3 is less than 1. d. Dividing by p oper and improper fractions makes the answer larger than the number divided. WIN The value of a common fraction will not be changed if a. we add the same number to both terms. b. we multiply one term and divide the other term by that same number. c. we subtract the same amount from both terms. d. we multiply both terms by the same number. 40. 41. 42. 43. 44. 189 The nearest to 45% is a. 44 out of 100. b. .435. c. 4. 5. d. .405. The principal of a school said that 27 per cent of the pupils went to the museum. Which statement best describes the expression "27 per cent of the pupils went to the museum? " a. It means that 27 children out of every 100 children went to the museum. b. It means that you must multiply the number of children in the school by 27/100 to find the nmnber who went to the museum. c. If the children were divided into groups of 100 and those who went to the museum we re distributed evenly among them, there would be in each group 27 who went to the museum. d. 27 per cent is the same as . 27 -- a decimal fraction written in per cent form. Sally completed 2/3 of the story in 12 minutes. At that rate how long will it take her to read the entire story? a. 18 minutes b. 12 minutes c. 6 rminutes d. 24 minutes There were 400 students in the school. One hundred per cent of the children had lunch in the cafeteria on the first day of school. On the second day 2 boys were absent and 88 children went home for lunch. Which of the following equations can be used to find the per cent of the school enrollment who went home for lunch? a. 400 - 88 :2 X b. i - 88 100 400 C- j; = 400 88 d. 400-9ozx What can be said about y in the following open sentence if x is a natural number? a. Xy c. x=y do Xry A 45. 46. 47. 48. 49. 50. 190 Which one of the following fractions will give a repeating decimal? a. 1/2 b. 3/4 c. 5/8 (1. 6/11 Which of the following is not an Open sentence ? a. 7+2=|:] b.h—5=9 c.c/1-30=6 d. n-3 For a mathematical system consisting of the set of odd numbers and the operation of multiplication, a. the system is closed. b. the system is commutative. c. the system has an identity element. d. all of the above are correct. Measurement is a process which a. ccrnpares an object with some known standard or accepted unit. b. tries to find the exact amount. c. is never an exact measure. d. chooses a unit and then gives a number which tells how many of that unit it would take. 1) a and b are correct. 2) a and c are correct. 3) a, b, and d are correct. 4) a, c, and d are correct. The set of points sketched below represents a l L \ 7 a. line. b. ray. c. line segment. (1. none of the above. How many triangles does the figure contain? a. 4 b. 6 c. 8 d. 10 51. 52. 53. 54. 55. 191 The set of points on two rays with a common end-point is called a. b. c. d. a triangle. an angle. a vertex. a side of a triangle. If a circle is drawn with the points of the compass 3 inches apart, what would be 3 inches in length? 8.. b. c. d. circumfe rence diameter area radius The solution set of an open sentence may consist of two or more numbers. no numbers. only one number. any or all of these. Consider a set of three objects. How many sub-sets or groups can be arranged? nine eight seven six If two sets are. said to be equivalent, then a. b. C. every element in the first set can be paired with one and only one element in the second set. every element in one set must also be an element in the second set. they are intersecting sets. one must be the null set. APPENDIX C A TEST OF MATHENIATICAL UNDERSTANDINGS * Form B Prepared by hllildred Jerline Dossett Southeast Missouri State College Cape Girardeau, Missouri With the assistance of: Calhoun C. Collier, James M. Drickey, Vincent J. Glennon, W. Robert Houston, Lois Knowles, and Joe L. Wise Directions: This test is designed to measure your understanding of mathematics. Many of the items relate to the new content in present programs of mathematics for elementary school pupils. Each of the fifty-five questions is of the multiple -choice type and includes four possible answers. Read each question carefully and decide which answer best fulfills the requirements of the statement. Then mark the space on the answer sheet to indicate your choice. Mark only one answer for each question. If you change your choice, erase your original mark and mark the correct one. Sample Que stion: Which of the following shows the decimal form of the fraction % ? a. 125 c. 1. 25 b. 12. 5 d. .125 Answer Sheet: b d (a) < > (i) < ) Since (c) is correct, the space under c is blackened. *Those items with an asterisk (=3) are items from Glennon's "Test of Basic Mathematical Understandings. " Permission was granted by the author to use any or all of the items on the "Test of Basic Mathematical Understandings. " Copyright 1964 192 193 Which of the underlined words or signs in the following sentences refer to symbols rather than the things they represent? a. 4 can be written on the blackboard. b. Pegardless of what symbol we use, we are thinking about the number 2. c. A pencil—is used for writing. d. The number _1_6_is the same as the nulnber 7 + 9. When we use the = sylnbol between two terms (as 2 + 2 = 4) we mean that both terms represent the same concept or idea. Which of the following is not correctly stated? a. 3 + 4 -_~ 5 + 2 _ b.5+2:7 and 7=5+2 c.(5+2)x3=7x3 d. 7 = 7 l) a and b are correct. 2) a and c are correct. 3) a, b, and c are correct. 4) a, b, c, and d are correct. If the Roman system of numeration were a "place value system“ with the same value for the base as the Hindu-Arabic system, the first four base symbols would be a. I, X, C, and M. b. I, V, X, and L. c. X, L, C, and M. d. X, C, L, and D. Which of the following does not describe a characteristic of our decimal system of numeration? a. It uses zero to keep position when there is an absence of value. b. It makes ten a standard group for the organization of all numbers larger than nine. c. It makes 12 the basis for organizing numbers larger than eleven. (1. It uses the additive concept in representing a number of several digits. In the numeral 7,843, how does the value of the 4 compare with the value of the 8? a. 2 times as great . b. 1/2 as great C. 1/10 as great d. 1/20 as great 10. 194 In the numeral 6, 666 the value of the 6 on the extreme left as compared with the 6 on the extreme right is a. 6, 000 times as great. b. l, 000 times as great. c. the same since both are sixes. d. six tlIT;€‘S as much. Which of the following statements best tells why we write a zero in the numeral 4, 039 when we want it to represent "four thousand thirty-nine? " a. Writing the zero helps to fill a place which would otherwise be empty and lead to misunderstanding. - b. The numeral would represent "four hundred thirty-nine" if we did not write the zero. c. Writing the zero tells us not to read the hundreds' figure. d. Zero is used as a place-holder to show that there is no number to record in that place. 1) a and b are correct. 2) a and c are correct. 3) a and d are correct. 4) a, b, and d are correct. Below are four numerals written in expanded notation. Which one is not written correctly? 21. —4—(ten)) + 9(ten)1 + 3(ones) = 493ten b. 3(5 even)i + 6(seven)1 + 1(one) = 363seven c. 4(twe1ve)2 + 5(twelve)1 + e(one) = 459twe1ve d. 21(1' ve)2+ 2(five)1 + 4(one) 2 224five If you are permitted to use any or all of the symbols 0, l, 2, 3, 4 and 5 for developing a system of nume ration with a place value system of numeration similar to ours, a list of all possible bases would include: a. base one, two, three, four, five, and six. b. base two, three, four, five, and six. c. base two, three, four, and five. (1. base one, two, three, four, and five. About how many hundreds are there in 34,870? a. 3 it», b. 35 c. 350 d. 3, 500 195 11. In what base are the numerals in this Inultiplication example written? , 34¢ a. base five 23'} b. base eight .___. c. base eleven 184? d. you can‘t tell ’ ? 1024? 12. Which of the following are correct? In the symbol 53, 5 is the base and 3 is the exponent. In the symbol 53, 3 is the base and 5 is the exponent. 33: 51:5115 5323x3x3x3x3 l) a and d are correct. 2) b and c are correct. 3) a and c are correct. 4) b and d are correct. 13. In the series of numerals l, . . .17, 18, 19, 20, 21,. . . , what term best applies to 19? a. nominal ordinal composite cardinal l4. Examine the following illustration: 1 2 3‘74 5 (5 > Which of the following does the above best illustrate? a. b. c. d. The idea of a cardinal number The use of an ordinal number A means for determining the cardinal number of the set by counting with ordinal numbers None of the above 15. The quotient of any two whole numbers a. b. c. d. is not a natural number. is sometimes a natural number. is always a natural number. is a natural number less than one of the numbers being divided. 16. l7. l8. 19. 20. 196 The integers are closed under the operations of a. addition. b. subtraction. c. multiplication. d. division. 1) a and b are correct. 2) a and c are correct. 3) a, b, and c are correct. 4) a, b, c, and d are correct. A student solved this example by adding down; then he checked his work by adding up. Add 34 34 T 52 T 52 86 Check 86 It could be classified as an example of a. the distributive principle. b. the associative principle. c. the commutative principle. (:1. the law of compensation. The statement ”the quotient obtained when zero is divided by a number is zero" is expressed as a o —:O a o b. 9 :0 a o C. —:a 0 d. i =0 a When a natural number is divided by a natural nmnber other than 1, how does the answer compare with the natural nmnber divided? a. larger b. smaller c. one -half as large d. can‘t tell from information given If you had a bag of 350 marbles to be shared equally by 5 boys, which would be the quickest way to determine each boy‘s share? a. counting b. adding c. subtracting d. dividing 21. 22. 23. 24. 25. 26. 197 If the multiplier is x, the largest possible number to carry is a. x. b. x +1. c. 0. d. x - 1. Which one of the following methods could be used to find the answer to this example? 17 f 612 a. Nlultiply 17 by the quotient. b. Add 17 six hundred times. c. The answer would be the sum. d. Subtract 17 from 612 as many times as possible. The answer would be the number of times you were able to subtract. Which one of the following would give the correct answer to this example? 2. 1 x 21 a. The sum of 1 x 2.1 and 21 x 2.1. b. The sum of 10 x 2.1 and 2 x 2.1. c. The sum of 1x 2.1 and 20 x 2.1. d. The sum of 1x 2.1 and 2 x 2.1. Which would give the correct answer to 439 x 563? a. Multiply 439 x 3, 439 x 60, 439 x 5 and then add the answer. b. Multiply 563 x 9, 563 x 3, 563 x 4 and then add the answer. c. Multiply 563 x 9, 563 x 39, 563 x 439 and then add the answer. d. Multiply 439 x 3, 439 x 60, 439 x 500 and then add the answer. Which of these numerals are names for prime numbers? a. 3 b. 4/2 C' 12five (1. 9 - 2 l) a is correct. 2) a and c are correct. 3) a, b, and d are correct. 4) a, b, c, and d are correct. Let x represent an odd number; let y represent an even number. Then x + y must represent a. an even number. b. a prime number. c. an odd number. d. a composite number. 27. 28. 30. 31. 198 The inverse operation for addition is a. addition. b. subtraction. c. multiplication. d. division. The least common multiple of 8, 12, and 2015 a. 2x2. b. 2x3x5. C. 2x2x2x3x5. d. Zx2x2x2x2x2x2x3x5. Which statement best tells why we carry 2 from the second column? a. If we do not carry the 2, the answer would be 20 less than the correct answer. 251 b. Since the sum of the second column is more 161 than 20, we put the 2 in the next column. 252 c. Since the sum of the second column is 23 271 ‘ (which has two figures in it), we have room for the 3 only, so we put 2 in the next column. (1. Since the value represented by the figures in the second column is more than 9 tens, we must put the hundreds in the next column. 935 The operations which are associative are a. addition. I b. subtraction. c. multiplication. d. division. 1) a and b are correct. 2) a and c are correct. 3) a, b, and c are correct. 4) a and d are correct. Which of the following is an even number? a. (lomthree b. (100)five C’ “Omseven d. (zoolfive 32. 33. 34. 35. 36. 199 The fact that a + (b + c) is exactly equal to (c + b) + a is an example of a. distributivity. b. commutativity. c. closure. (1. associativity. When the circle is cut into equal pieces, the size of each piece Observe the drawing on the right. a. decreases as the number of pieces increases. / b. increases as the number of pieces decreases. ,/ c. increases as the number of pieces increases. (1. decreases as the number of pieces decreases. // l) a and b are correct. ~ _ ,/ 2) a and c are correct. 3) b and c are correct. 4) b and d are correct. The symbol 3/4 may be used to represent the idea that a. 3 is to be divided by 4. b. 3 of the 4 equal parts are being considered. c. 3 objects are to be compared with 4 objects. d. all of the above. When a whole number is multiplied by a common (proper) fraction other than one, how does the answer compare with the whole number? a. It will be larger. b. It will be smaller. c. There will be no difference. d. You are not able to tell. Which of the addition examples is best represented by the shaded parts of the diagram below? 1 l a — + _ , ° 2 3 / , , I ‘ //'/'--V:’ /' - /1/ is b 2 + 3 , [1. 1 7 z, . .3. 4— - L - 41,! l ‘ c 2 + 1 l + W1) .--_-_ 4 . 3— E J r73?” 1 l d. _1_ + _1_ -——--~-~-~---.-. 3 3 B 37. 38. 39. 40. 41. 200 We can change the denominator of the fraction to the number "1" without changing the values of the U1|»Ls’w|l\) fraction by a. adding 5/4 to the numerator and denominator. b. subtracting 5/4 from the nume rator and the denominator. c. multiplying both the numerator and the denominator by 5/4. d. dividing the numerator and the denominator by 5/4. \Vhat statelnent best tells why we "invert the divisor and multiply" when dividing a fraction by a fraction? a. It is an easy method of finding a common denominator and arranging the numerators in multiplication form. b. It is an easy method for dividing the denominators and multiplying the numerators of the two fractions. c. It is a quick, easy, and accurate method of arranging two fractions in multiplication form. (1. Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. If the denominator of the fraction 2/3 is multiplied by 2, the value of the resulting fraction will be a. half as large. b. double in value. c. unchanged in value. d. a new symbol for the same number. 45% may also be written as a. .45 b. 45/100 C. 45 X 100310 d. .450 l) a and b are correct. 2) a and c are correct. 3) a and d are correct. 4) a, b, and d are correct. . 5 and . 27 are illustrations of "decimal fractions. " They could be written as "common fractions" in the form of 1/2 and 27/100 respectively. What is a decimal fraction? a. It is another way of writing percentage. b. It is an extension of the decimal number system to the right of one's place. c. A number like . 37,—. which has both a decimal and a fraction as parts of it. d. A number like . 2 which is a fraction and has a decimal as . 56 either the numerator or denominator or both. 42. 43. 44. 45. 46. 47. 201 The ratio of x's in Circle A to " in Circle B can be shown by a. 16/4. a b. 1/4. 4* x c. 1/2. ( XX 1 d. 4/16. ‘\ XX \\\ A B Sue paid 20¢ for 4 apples. Which of the equations below could be used to find the cost of 1 apple? 4 1 a. —— =— 20 x b. x+4=20 C,_Z‘;:zo 4 d. x-4=20 The decimal for the nmneral 6/17 will a. be a repeating decimal. b. not repeat or end since 17 is prime. c. repeat in cycles of less than 23 digits. 1) a is correct. 2) a and b are correct. 3) a and c are correct. 4) a, b, and c are correct. Which of the following statements is not correct? a. (-9) + 6 = -3 b. (-5) + (-5) = ~10 c. -8 + 0 = -8 d. (41) + (9) = - Which of the following is a list of all of the factors of 12? a. 1, 2, 3, 4,8&12 b. 1, 2, 3, 4, 6&12 c.1,2,3,4&6 d.2.,3,4,6&12 Niodular arithmetic is i a. commutative. b. associative. c. distributive with respect to multiplication over addition. . d. all of the above. 48. ‘49. 50. 51. 52. 202 Which of the following is an approximate measure? a. 35 farms b. 12 buttons C. 71/2 inches d. 15 beads Which of the following does the sketch below represent? \. / a. line b. ray C. line segment (1. set of points 1) a is correct. 2) a, b, and d are correct. 3) a, c, and d are correct. 4) b and d are correct. Which of these triangles are right triangles according to the length of the sides given? \\ e \O \\ a. \9 b. ,x W \‘ 5” C. fl“ Cl. \ q” A distinct point is a. a point you can see. b. a sharp object. c. the intersection of two lines. d. a dot. A clerk sold a lady a round tablecloth that had a radius of 14 inches. Which of the formulas can she use to determine the length around the cloth? a. A=1rr2 b. C=7Td c. C=21rr d. A=C/d B 53. 54. 55. 203 Which of the following best defines a solution set? a. A solution set is a set which includes each and every member that gives a true statement. b. A solution set is a single sentence which identifies a variable that will give a true statement. c. A solution set is a set containing all the positive integers, zero, and the negative integers. , d. A solution set is a set containing rational numbers. Examine the following illustration. S : {0:1, ('1): 2: (’2): 3:00-10} Which one of the following is not a subset of S? a. )+9, +10) b. (o, (—2), 5) c. {3, (-3), to) d. p, (-1), 6, 101 If we use the set concept to define the operations for the counting numbers, addition would be defined in terms of a. the intersection of disjoint sets. b. the union of intersecting sets. c. the intersection of sets with common elements. d. the union of disjoint sets. _APPENDIX D DUTTON ARITHMETIC ATTITUDE INVENTORY Directions: (Check (x) only the statements which express your feeling toward arithmetic. Agree Disagree 1. I avoid arithmetic because I am not very good with figures. ( ) ( ) 2. Arithmetic is very interesting. ( ) ( ) 3. I am afraid of doing word problems. ( ) ( ) 4. I have always been afraid of arithmetic. ( ) ( ) 5. Working with numbers is fun. ( ) ( ) 6. I would rather do anything else than do arithmetic. ( ) ( l 7. I like arithmetic because it is practical. ( ) ( ) _ 8. I have never liked arithmetic. ( l ( ) 9. I don‘t feel sure of myself in arithmetic. ( ) ( ) 10. Sometimes I enjoy the challenge presented by an arithmetic problem. ( ) ( ) 11. I am completely indifferent to arithmetic. ( ) ( l 12. I think about arithmetic problems outside of school and like to work them out. ( ) ( ) l3. Arithmetic thrills me and I like it better than any other subject. ( l ( ) 14. I like arithmetic but I like other subjects just as well. ( l ( ) 15. I never get tired of working with numbers. ( ) ( ) 204 APPENDIX E TEXAS ELEIVTENTARY SCHOOL IVIATHENIATICS CLASSROOM PRACTICES SCHEDULE OF QUESTIONS Tell me about your arithmetic class. Where do your arithmetic lessons come from? TEXT How many arithmetic textbooks do you use? 1. How do you work through your arithmetic textbook? EPg : SkPg : Skpt : SkMst : SkAll 2. Are all of you working in the same place in your textbook? AllSm : : 2-3 Grps : : AllDf WORKBOOK . *Why do you think you use your arithmetic workbook? How Inany different kinds of arithmetic workbooks are used in your class? 1. How do you work through your arithmetic workbook? RegAll : IrgSmtAll : TerNd : PplNd 2. How often do you use it? Daily : Weekly : Te rNd : PplNd 3. Are all of you working in the same workbook? . All : DfntGrps : IndfntGrps : Indiv 4. Are all of you working in the same place? All 7: Dfnthrps : IndfntGrps : Indiv TRADE What other kinds of books about arithmetic have you used? How Inany? 1. Who chooses these books for you to use? TerTxthstd : Ter : Ter-Chld : Chld 2. How have you used them? ReRpts : Terplnsgrp : RptTer—Ppl : IndivInt IndvrptTer ° 205 206 APPENDIX E -— Continued 3. How often do: you use these books? Over Nionth : lVlonth : Week : AlmstDly FREE —TI1\/IE AC TIVIT Y 1. What do you do when you finish your assignments in class? NxtAsgnmt : TerChs ': ArithChs : FrChs 2. What kinds of special activities or projects have you done in arithmetic class? Text : TerElabText : Teacthstd : Ter-ChlfDrv : Chld HOME WORK 1. What kind of homework do you have? Txt : WkBk : DittoCde : AsgnExpPrb :ChldExp Prb FILMS *Why do you see films or movies about arithmetic? How many have you seen this year? 1. What did you do afterthe film was over? T st : Te rSmy : RptDsc s : IndvExplr INT ECRAT ION 1. In what other subjects do you most often use arithmetic? None : Isubj : ZSubj : 3Subj : AllSubj 2. Do you also have a regular arithmetic lesson that day? Yes : Usually : Not Often : Never TEACH INC AIDS *What teaching aids and supplies do you think are important in the classroom for learning arithmetic? Which of these have you used in your classroom? 1. Tell me how you used the to learn a new idea? Skills : Abst : Expl : Appl Prep 207 APPENDIX E -- Continued v' 2. Where did these aids come from? Comm : Teacher : Teach -Child : Child 3. How does your class use arithmetic charts and graphs (posters and bulletin boards)? Alkl s -Ab st : Prep : Appl : Expl Alks-Abst : Prep : Appl : Expl 4. Where do they come from? Comm : Teach : Teach-Child : Child ORGANIZATION 1. Do you spend the same amount of time studying arithmetic every day? Yes : FwExcep _: SmExcep : Frquxcep : ersni 2. Do all of you study the same lesson every day? No : 2 Crps : 3 Grps : 4 Crps : Indiv 3. Do some pupils have longer or harder assignments in arithmetic ? No : Length : : Level : Lgtlfovl 4. How do pupils get individual help from the teacher? Little : Terrexpcls : Grpllp : TerChsIndv : Hal -Indiv or None ° ° ' ° DISC USSION How much of your arithmetic is spent talking about ideas in arithmetic? 1. Do most pupils take part in the discussion? Ter : Ter : Ter : Joint : Child *Do you ever have problems without pencil and paper? That is, mental arithmetic tests? 2. Where do your mental arithmetic problems come from? Text : TextEpnd : Terdretd : : Ppl -Pp1 EVALUATION 1. How do you know how well you are getting along in arithmetic ? RptCrd : Tests-DiagAch : Tlertes : TerCthonf : ChlPrgCut 2. Where do your tests come from? Txt : )VkBk : Terh'lade : Ter-Child : Child 0.) *Tell me about your favorite day in arithmetic this year. 208 APPENDIX E -- Continued . How often do you have tests? Dly : ka : 6 wks : Tpk . Who usually grades your papers? Ter : . : Otherl—Ter . Where are your grades usually written or kept? Ter : : : Joint Slf Slf Ppl APPENDIX F INTERPRETATION OF CODING THE TEXAS ELEMENTARY SCHOOL NIATHEMATICS CLASSROOM PRACTICES SCHEDULE OF QUESTIONS A. TEXT (Number) -- If use out of adopting text or other text to supplement, indicate. total number of kinds. 1. Use =3 (1) Every page --if progress through text from front to back (with the exception of practice sets) and for all practical purposes cover every page (2) Skip page -- if a few pages are skipped frequently --at the first of year skipped but not now (3) Skip part --if skip is from part of part, or parts omitted; not (1) or (2) ‘ (4) Skip most --if the text is used primarily as a reference or only as a source of practice problems (5) Skip all --if a textbook is not used at all or rarely if ever 2. Assignment (1) All the same --if everyone in the class has the same assignment in the textbook (2) Only one or two persons have different assignment or textbook (3) 2-3 groups --have different assignments or work at different places in the textbook (4) (5) A11 different --for all practical purposes assignments are made on an individual basis B. WORKBOOK (Number) -- Count tear sheets, ditto practice sheets as workbook type materials 1. Use (1) None (2) Regular assignment for all --work daily, no differentiation in assignments 209 3. 210 APPENDIX F -- Continued (3) Irregular assignment sometimes for all Irregular assignments for all Same but permitted to work at own rate One person has a different assignment (4) Teach ident. —-with groups or individual assignments (5) Pupil selects needed material Frequency (1) None (2) Daily (3) Weekly--used regularly once a week, or seemingly without need --as "to keep us busy while he has other work to do. " Also for extra credit. (4) Teacher identifies need of individual or group -—when the teacher sees the. need for practice, etc. and assigns to either the class, group, or individual as the case may be (5) Pupil selects needed material --the pupil selects which material he will work on in the workbook Same workbook (1) None (2) All --all workbooks and materials are the same for all members of the class (3) Definite groups --one group uses different materials than the other; one or two students may have a different workbook or related materials (4) Same place -- (1) None (2) All --all the class is working in the same place in the workbook (3) Definite groups --where one group is in a different place in the workbook than the other group/s. Or one or two students may be working in a different place than the ren'iaining group (4) Indefinite groups --the groups change from time to time; groups have different assignments. Also if the class has no common assignment, but individually permitted to work at own rate (5) Individual -—for all practical purposes, assign— ments are made on an individual basis 211 APPENDIX F -- Continued C . TRADE (Number) -- includes library books, pamphlets, books in the 1. teacher's personal library, trade books brought from home, encyclopedias used, etc. Choice (1) None (2) Teacher chooses text suggested --includes books ordered from textbook company (adopted text) (3) Teacher —- she chooses the books herself, either from the library or other sources. If teacher has child check out specific book from library ‘ (4) Teacher-child --teacher selects the books and the children select from her selection; teacher selects some and children select some from library, home, etc. (5) Child --the child selects the books to be used, and may bring them to the classroom Use (1) None (2) Regular reports, drill, tests --if they are required to report at regular intervals on reading; if the books are a source of drill or practice materials; if a test is the only follow-up on the reading of the books. If teacher uses books and children Inention that she lectures on materials (3) Teacher plans groups of individual's report --assigned report (4) Report as a result of teacher-pupil planning --allows for pupil choice in topic, but the teacher has a hand in planning, assigning, or suggesting. Also includes extra- credit reports (5) Report as interest to individual --may report if he wants to, strictly voluntary --however, usually evidence needed that someone has or evidence that there is ample opportunity to read and not report if he chooses to Frequency None Over month --once or twice in a six months period Month --average of once a month Week -—not every week yet more Often than monthly A A A A A U] +1-1- W (V r-‘ v v V v V Daily --evidence that some pupils use the books daily and that there is opportunity for many students to use them 212 APPENDIX F -- Continued D. FREETIME ACTIVITY 1. 2. Finish text -workbook assignment (1) Don‘t finish Next Assignment --when they say they go on to homework, etc. Teacher choice -- teacher either tells them what to do or "to sit and be quiet until the others finish. ” Arithmetic choice -- students can do anything in arithme - tic such as go to the arithmetic table, read trade books, etc. -- students choice Free choice -- students choose to do anything Special arithmetic activities (l. 3. AA [‘v‘ r—I V (5) List all. 2. Record total number of special activities. record number of project related activities.) None Teacher elaboration on text --teacher takes a topic from the text and initiates some activity suggested there —- such as Inaking a chart, graph, etc. Teacher suggested --or assigns something outside of the text, discusses a topic outside of the text, initiates an activity not in the text (or related materials) Teacher-child developed --teacher makes broad assignment or gives broad suggestion and the children have the freedom to choose and develop an activity themselves Child on his own --child has freedom to choose what he does and how to develop it; evidence that he has originated and developed the activity brought to class E. HOME WORK 1. Source (1) (2) (3) Text --almost solely the text Workbook --about half from the workbook and half from the textbook some ditto, some text and/or workbook Ditto ~Cha1kboa rd * Not from textbook Sometimes experiential problem and sometimes text or workbook IVIake up problems similar to textbook problems 213 APPENDIX F -- Continued (4) Assigns experiential problem -- source outside of text where child has to find facts as well as to work the problem (5) Child brings experiential problem --brings something from his own experience to class; brings a "real" problem to class -- voluntary on the part of the student F. FILMS (Number) -- includes filmstrips, slides, etc. 1. Follow-through (1) None (2) Text --only follow through (3) Teacher summary —-teacher asks questions and students answer or teacher simply summarizes (4) Report or discussion --children report on film; discussion must be a 50-50 one where pupils take an active part -- not merely a teacher questions—child answers session (5) Individual exploration --where the children try some of the things shown in the film or do some additional activity or reading G. INTEGRATION -- Omit -- Not applicable to Dallas because of platoon organization H. TEACHING AIDS -- list all mentioned. tabulate total number of manipulative and graphic aids. (Number) -- charts, etc. (felt board) manipulative, etc. Use (1) None (2) Skills --games, etc. for drill or practice (little use, or teacher used) (3) Abstraction --when aid is used concurrently with pencil and paper, etc. , when we get mixed up (4) Exploration --use of teaching aid before they work problems --children actually use it (5) Application, preparation --preparation: used in intro- ducing a topic; application: after abstraction to relate the abstract back to the concrete 29 3. 214 APPENDIX F —- Continued Source (1) None (2) Commercial (3) Teacher --either the teacher makes or some source (4) (5 other than the students in this particular class Teacher-child --half and half Child -- children make practically all Charts, use 2f (1) None Skills, abstraction --to develop skill through practice or use --of a concept already learned Preparation --to prepare students for the new concept -- stimulate interest Application --whe re the children (or teacher) make a chart to show application of concept learned Exploration --actually explore when making the chart; or use a chart to explore -- for example, making a Base 5 chart. I. ORGANIZATION 1. 2. Equal time daily -- not applicable to Dallas Grouplng (1) (Z) (3) (4) (5) None 2 groups --two major groups or if only one or two persons have a different assignment 3 groups 4 plus groups Individual J. DISCUSSION -- Record per cent of class period spent in discussion 10 Who talks p—l ) ' ) AA [\J AA (Teacher) -- virtually does all the talking (Teacher) --children talk a little -- teacher asks some questions (Teacher) --teacher questions, children answer (Joint) --both contributing equally and asking questions Z. 215 APPENDIX F -- Continued Mental a rithmetic (Topics from text)--if notflmental arithmetic or text used only for mental arithmetic problems (Text expanded)--text plus teachers personal books or plus some made up and estimation (Teacher directed) --she makes most of them up or some source other than the text, workbook or teachers manual (Pupil and teacher directed)--pupils bring some and teacher brings some (Pupil-pupil)-—whe re pupils take initiative and administer them K. EVALUATION 1. Progress (1) Report card only ‘ (2) Tests (diagnostic or achievement tests) -—teacher sends tests or notes home to parents (3) Teacher tells --The child --or daily papers she keeps -- progress chart up for everyone. Include notebook or folder if just keep it, and no evidence of real use. Sometimes the class averages their grades, sometimes class keeps some grades and sometimes individuals keep all grades ' (4) Teacher-child conference -—ta1ks with child privately (5) Child's own progress chart --child keeps a record of his own grades, texts or otherwise Evaluation de vie e s (1) Test --if directly, or from teacher manual (2) Workbook --if sometimes text, sometimes made up, sometimes other tests (3) Teacher-made --prima rily made up by teacher (4) Teacher-child (5) Child Frequency (1) Daily (2) V’Veekly--set day (3) 6-weeks (4) Topic (5) As needed 216 - ..l.|1.En_ 1.. . ..1 4.1.w41.-1.1o a 1 anwoh Be... .853 1—1 Ea Qua—“Mu Bu u Eu tone E a @3023 841.39 cough. 88 889 50a an; o._. fix: .3, «*8 3E ufl «wagon. .8 so: nausea“. 3:3 BB a flu: 2:333 an o v.55 3 . .n 1. I 030:0 32 .. oumoamv nificant. 39A 3.20 n33 .812 8:03... one: .6 383 Sun 123213 v D «>0 so u wanna: .au «Saga m 1mm;— w n :2 08a- 06 U>o 89. 098 «so 1 38 98 o 086 :6..— ro 313 g" n h. 1 . 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H—u wovoon 3933 2155 now ASE 5:03P . .1 23.695 new 1 .mod on: Eowmvflflir mayo—0 mum «an... 05 =< . 1 . 1 n l . :« mmxw 1 1M1 1 MOE 9.3 tag QCm «mum 93w .1.an io>m 0-D u hufih 1%) 1 :1 3w ,1 ruidrofis 1, 1 6539 5.6 Edens has 76.2.33 .6 15:.“me 9.5.. lilllHII 1i" .1 111011 1 1 11i11 1111 1141.111 1111111111111 mq<1Um12m4>Mmbé meP 93E émmnfiu mg<<fi11ixF<2 48mg .Cmchu sud nah 3:53:00 11 h NHQZMnHAw/w. 217 APPENDIX F -- Continued 4. Grading and checking Teacher --all Teacher and helper Self and teache r (1) (3) (3) Other pupil and teacher (4) (5) Self U1 0 Recording ) Teacher ) Occasionally otherwise (3) Folder or notebook ) Joint-group and teacher ) Self '3‘ Numerals in parentheses: (1) "Text" col. TESMCPIS ) "Text-Teacher" TESAICPIS ) "Teacher" TESMCPIS (4) "Teacher-Child" TESMCPIS ) ”Child" TESMCPIS APPENDIX (3 OUTLINE OF BASIC INIATHEIVIATICAL CONCEPTS DEVELOPED FOR THE ELENIENTARY SCHOOL MATHEMATICS (K-8) IN-SERVICE WORKSHOPS I. Introduction to Sets A. Set vocabulary and operations with sets The set concept has a language and symbolism that has applications in all mathematics. The operations on sets are unique ways of thinking of two sets to get one set. B. Applications of sets to elementary mathematics 1. Sets in the elementary grades will not be a radical de- parture for most teachers. The set concept may be used to develop the concept of whole numbers. The set concept and set operations may be used to give meaning to the operations of whole numbers. We classify whole numbers into sets such as even, odd, composite and prime numbers, factors and multiples of numbers, etc. We classify numbers by sets such as sets of natural, whole, integers, rational and real numbers. II. Number and Numeral A. Historical background of the number concept 1. 2. Number is an invention of man. Number is an abstract idea; it is something that exists in the mind; a number has many names. 218 B. C. 219 APPENDIX G -- Continued Pre-Numeral recordings (matching) 1. Primitive number concepts probably were limited to a one- to -one matching of objects with fingers, pebbles, etc. 2. lV'lan learned to count before he. learned to symbolize the counting numbers. Ancient systems of numeration 1. Systems of numerations are man made. 2. Scientists have made it possible for us to study the numeration systems developed by the Egyptians, Greeks, Babylonians, Mayan Indians and other ancient civilizations. Characteristics of numeration systems 1. Most early numeration systems had single symbols for groups, the principle of repetition of symbols, and the principle of addition. 2. The development of the concept of place value and a symbol for zero gives us the numeration system that we use today. III. Structure of a place value system of numeration A. Using base other than 10 1. In a place value system each symbol has an absolute. value and a place value depending upon the base. 2. Ten is probably the most common base because man has ten fingers. 3. Any whole number greater than one may be used for a base. Fundamental operations with bases other than 10 1. In our every day activities we sometimes use other bases. Z. 220 APPENDIX C -- Continued Working with other bases should give greater insight into the structure of our base 10 system. Hindu -Arabic system of nume ration (decimal system) Our decimal system has place value, a base of ten, and a symbol for zero. The Hindu-Arabic system of numeration was brought to Europe by the Arabs. The present form of the digits have for the most part evolved in Europe. IV. Addition and multiplication of whole numbers A . Addition and multiplication defined 10 Addition may be considered in terms of the union of disjoint sets. Multiplication may be considered as repeated addition or in terms of sets. Properties of addition and multiplication: Closure, commuta- tive, associative, distributive and the identity elements 10 U1 When we add or multiply two whole numbers we are guaranteed an answer and that answer is a whole number. We have the right to change the order of addends of factors without affecting our answer. When adding or multiplying more than two numbers, we may group or associate different pairs of numbers to- gether without affecting our answer. When we multiply numbers, it is possible to use the opera- tion of addition to simplify the process. Addition of zero to any number gives that number as an answer. Multiplication of any number by one gives that number as an answer. C. 221 APPENDIX C -- Continued Algorisms for addition and multiplication l. The algorisms may be developed by using sets of physical objects and the characteristics of the numeration system. 2. The algorisms may be developed by using the commutative, associative, and distributive properties of numbers. 3. The algorisms that we teach today are not the only possible algorisms for the whole numbers. V. Subtraction and division of whole numbers A . Subtraction and division defined 1. Subtraction and division may be considered as unique ways of separating or partitioning sets. 2. Subtraction and division are secondary operations in arithmetic. Subtraction and division as inverse operations 1. Addition and multiplication may be considered basic operations in terms of which the inverse operations, subtraction and division, are defined. 2. Subtraction may be considered as finding a missing addend when one addend and the sum is given. 3. Division may be considered as finding a missing factor when one factor and the product is given. Algorisms for subtraction and division 1. The algorisms may be based on the numeration system and sets or the inverse properties of addition and multiplication. 2. A method of successive subtraction is the most simple division algorism. 222 APPENDIX G -- Continued VI. Equations and inequalities A. Operation, relation and place holder symbols 1. hiathematicians have defined many symbols to indicate number ope rations. 2. To indicate that two expressions name the same thing we use the symbol "2." 3. To indicate that two expressions do not name the same thing we use the greater than > , less than < or # unequal to symbol. 4. In mathematics a place holder is a symbol (0 , A , X, A, ? , etc.) for which one substitutes the names of numbers. Number sentences and solution sets 1. The language of mathematics has a grammar and symbolism of its own just as the English language does. 2. It is possible to have true, false, or neither true nor false sentences in mathematics. 3. There may be a set of none, one, two or many nulnbers that will make an open number sentencetrue. Picturing number sentence 5 1. Numbers may be associated with the points on a line. 2. A pair of numbers may be associated with a point on a grid. Ordered pairs, ratio and rate 1. The concept of ordered pairs of numbers may be structured to produce a useful tool for solving number problems. 223 APPENDIX C1 -- Continued VII. Ra tional nunibe rs A. Historical developments of fractions 1. A new kind of number resulted in a need to describe, measurements, parts of a whole or a part of a group of physical Objects. Structural development of rational numbers 1. To insure closure under the operation of division new numbers have to be invented. Algorisms for rational numbers 1. The algorisms for fractions may be developed from physical models such as the number line, parts of circles, squares, etc. 2. The algorisms for rational numbers may be developed by inventing new numbers and assuming that they obey the basic properties of the whole numbers. 3. A decimal fraction is a different symbol for a rational number. VIII. Inte ge r s A. Physical models for the integers l. The number line may be extended to the left of zero to picture negative numbers. ' 2. Number stories may be used to develop the operations for the integers. For example, profit and loss stories or stories Of receiving and taking away bills and checks. Structural development of the integers 1. New numbers (the integers) may be invented to insure closure under the operation of subtraction. 224 APPENDIX C1 -- Continued 2. The rules of operations for the integers may be developed by assuming that the integers obey the basic properties .1 of the whole numbers. IX. Geometry A . Introduction to the language and symbolism of intuitive geometry 1. We cannot see a point, line or geometric figure; we only draw pictures to represent the idea. 2. Unique symbols have been defined to represent a point, line, line segment, ray, angle, etc. Simple constructions l. The straight edge, compass, and protractor are the tools of geometry. 2. Teaching devices and Inethods should lean heavily on the senses of sight and touch. Measurement 1. Standard units of measurement are arbitrarily chosen. 2. Numbers obtained by measuring represent approximations of the quantity. X. Patterns in Arithmetic A . B. Finite number systems 1. Clock arithmetic or modulo number systems have many properties in common with our real number systems. 2. Children have had some experience with modular or clock arithmetic. Mathematical games and number puzzles 1. Many of the concepts of arithmetic may be discovered from patterns of numbers. 225 APPENDIX C -- Continued 2. hiatheniatical games using grids, magic squares, number lines, number arrays, number puzzles, etc. have many applications to number concepts. XI. Expe rimental programs School Mathematics Study Group Nladison Project University of Illinois Arithmetic Project Stanford Elementary Mathematics Project APPENDIX H A LIST OF ARITHAIETIC TEXTBOOKS AND EXPERIIVIENTAL NIATERIALS EXAIVIINED A. TEXT BOOKS Buswell, Brownell, Sauble, and Weaver. Arithmetic W3 Need (Grades 1-8). Boston: Ginn and Company, 1963. Clark, Junge et al. Growth i_n_Arithmetic (Grades 3-6). Harcourt, Brace and World, 1962. Deane, Kane, and Osterle. I\'Iodern Mathematics Series (Grades 1-8). Cincinnati: American Book Company, 1963. Fehr, et al. Learning to Use Arithmetic (Grades l-b’). Boston: D. C. Heath and Company, 1962. Hartung et a1. Seeing through Arithmetic (Grades 1-6). Chicago: Scott, Foresman and Company, 1963. I\/1CS\V8I11, Brown, Gundlach et al. The Laidlaw .s\'[.'-1thematics Series (Primer through Grade 8). Rive r Forest, Illinois: Laidlaw Brothers, 1963. Merton, Brueckner, and Grossnickle. Moving Ahead i_n Arithmetic (Grades 1-6). New York: Holt, Rinehart and Winston, 1963. Morton et al. Modern Arithmetic through Discovery (Grades 1-6). Morristown, N. J.: Silver Burdett Company, 1963. B . EXPE RIMENTAL IVIATE RIALS Eicholtz, Robert et a1. Elementary School Mathematics (outgrowth of Ball State Experimental Project) Reading, .\vlass.: Addison- Wesley Co., 1963. Davis, Robert B. Discovery in hiathematics (Madison -Webster Project) Syracuse, N. Y.: Syracuse University, 1963. 226 227 APPENDIX H -Continued Greater Cleveland Mathematics Research Council. Elementary Mathematics Series (now SRJX). Cleveland: The Council, 1961. Page, David. "University of Illinois Project in Elementary Mathe— matics." Urbana, Illinois: University of Illinois, 1960. (Mimeographed.) School Mathematics Study Group. Mathematics for the Elementary School (Grades 4-6, Student Texts and Teacher's Comtnentaries). Suppes, Pat. Sets and Numbers. New York: Blaisdell Publishing Co. , l962. APPENDIX I NAMES AND ADDRESSES OF PERSONS USED FOR TEST REVIEW Dr. William Allison, Assistant Superintendent Cape Girardeau Public Schools Cape Girardeau, Missouri Dr. Calhoun C. Collier, Professor of Education Michigan State University East Lansing, IVfichigan Dr. James M. Drickey, Professor of Psychology Southeast Missouri State College Cape Girardeau, Missouri Dr. Vincent J. Glennon, Director Arithmetic Studies Center Syracuse University Syracuse, New York lVIr. Neal Holmes, Science -Mathematics Consultant Parkway Public Schools St. Louis, .\1issouri Dr. W. Robert Houston, Associate Professor of Education lV’Iichigan State University East Lansing, Michigan Dr. Lois Knowles, Professor of Education University of Missouri Columbia, .\«lissouri Mr. Joe L. Wise, Mathematics Consultant 1\/Iissouri State Department of Education Jefferson City, Missouri 228 APPENDIX J OUTLINE OF BASIC MATHEMATICAL UNDERSTANDINGS I. Numerals and Numeration System A. B. History of number and numerals l. 2. Numerals are man-made, created to stand for concepts which men have. hlan had some sense of number long before he was able to count or use numerals to stand for number. Numbers are the same, regardless of the symbols used to represent them. lV'lan now has both symbols (1, 2, 3, . . .) and words (one, two, three, . . .) which may be used to represent numbers. Some instruments, like the abacus, were used by ancient people to do computation. The Roman numerals were used for recording numbers and not for computation. Many of the early systems, such as the Roman, Greek, and Egyptian, were based upon 10 but did not use place value or have a symbol for zero. The Babylonians are said to have developed the first sylnbol to use for zero. A decimal system of numeration is used in most of the. world today because it is less cumbersome. and more concise than earlier systems. Place value and bases 1. 2. Our numerals are called Hindu-Arabic numerals. Since the Hindu-Arabic system uses groups of ten, it is called a decimal system; values decrease in a tenfold ratio to the right and increase in a tenfold ratio toward the left. Since grouping is by ten in the decimal system, its base is ten. The decimal system of numeration uses the idea of place value to represent the size of the group; the number that the numeral represents depends upon the position of the symbol and the value of the digit used to represent it. The decimal point indicates a change in the character of the digits. All numbers to the left are integral; all numbers to its right are fractional. 229 II. Symbols. A . B. 10. ll. 12. 13. 14. 15. 16. 230 APPENDIX J -- Continued The individual must have a clear understanding of place value in order to carry out the operations of addition, multiplication and division. ‘ To understand the meaning of a number represented by a numeral such as 123, the individual must add the numbers represented by each symbol (1x100) + (2 Ci?) + (3 x l) or 100 + 20 + 3. There is no limit to the size of the number which can be represeiigd by the decimal system of numeration. The numeral "0" has the special property of always representing zero; there is no number to record in that place; zero is used to fill a place which would otherwise be empty and might lead to misunderstanding. In an expression as 10 , the number 10 is called the base and the number 2 is called the exponent. A number such as 10 is called a powergften. In many everyday activities and in modern electronic computing machines, bases other than ten are used; the most frequently used bases other than ten are two, twelve, and sixty. The base in any system of numeration is the number of units which must be reached before a change is made in the pattern used to denote the number. In counting, we can group in any way; the selection of the base is arbitrary. A number can be represented by different numerals. Numeration systems, such as base four, do not represent a different number system, but only a different way of writing the same number. A symbol is a mark, an object, a sign, or a word that represents another object or idea. 1. 2. Usually there is little resemblance between a symbol and the object, or idea, which it represents. NIany symbols are used in mathematics. a) Some symbols represent numbers, some represent operations, and some compare numbers. b) In using a syn’ibol, the individual must know the object or idea which it represents. There is a difference between a number and a numeral. 1. Z. A numeral is a symbol which is used to stand for a number. The number is the idea for which the symbol stands. III. IV. 231 APPENDIX J -- Continued The individual characters we use in writing a numeral are called digits. A symbol for a number is different from the name of the number. Introduction to Sets A. The concept of sets is one of the unifying concepts of niathe- matics. l. The word ”Bit” can be used to mean any particular collection or group of objects. 2. The objects contained in a set are called the elements or the members of the set. a) “fee can describe sets in words (the set of all cities in the U. s. with a population greater than 100, 000). b) We can also list or tabulate the elements of a set within braces—:Tl, Z, 3, 4, 5} or [the set of counting numbers from one through five}. 3. The set which contains no elements is called an em )ty set or null set and is represented by the symbol 1} When it is written that Set A = Set B (A : B), this means that "A" and ”B" are different names for the very same set; the symbol "2" is used to mean "the same as. " If two sets belong to the same family, they are equivalent. The number of elements in a set is often called the cardinal number of the set. Subsets of a given set are those sets whose elements are members of the given set. 1. Every set is a subset of itself. 2. The empty set is a subset of every set. 3. Subsets which are different from the original set are called proper subsets. A solution if: for the sentence contains those elements which give us a true statement. An equation is a sentence which states that two expressions represent the same number. In dealing with sets, it is often necessary to consider the set composed of the two sets lumped together, or combined. 1. We can form a set by combination-union of two sets (U). 2. The intersection of two sets (writtenn) is the set con- sisting of all objects in both sets. Principles underlying Number Operations A. Addition and multiplication are commutative. 232 APPENDIX J -- Continued 1. The order in which two numbers are added does not affect the sum (.1 + b = b + c). 2. The order in which two numbers are multiplied does not affect the product (a x b = b x C). Addition and multiplication are associative. l. Grouping numbers differently does not affect the sum (a+b)+c=a+(b+c). Z. Grouping numbers differently does not affect the product (axb)xc =ax(bxc). Multiplication is distributive with respect to addition —- 3(a+b) =(3xa) +(3xb). Property of closure 1. The set of counting numbers is closed under addition and multiplication. Z. The set of counting numbers is not closed under division-— (20-2—6 = 31/3). Properties of zero and one 1. The number one is a special number. a) It is the smallest of the counting numbers. b) Because the product of any counting number and one is the original counting number, the number one is called the identity element for multiplication. c) The sum of a counting number and one is always the next larger counting number. 2. The number zero is also a special number. a) The number zero is the number of elements in the empty set; it is used to indicate the absence oi quantity. b) Zero may also be used as a point of origin from which one Inay move in either direction. c) The sum of a counting number and zero is always that whole number; zero is an identity element for addition. d) The difference between the same two rational numbers is the special number zero. e) A counting number cannot be divided by zero (be- cause 0 x 7 = O). f) If the product of two or more whole numbers is zero, then one of the numbers must be zero. Addition and multiplication are binary operations; we operate with just two numbers at a time. There is exactly one whole number which is the sum of every two whole numbers. This is referred to as the principle of uniqueness. 233 APPENDIX J —- Continued V. Relationships and Generalizations A. B. Inve r se ope rations 1. 2. Addition and subtraction are related processes. Sub- traction is the inverse of addition. kiultiplication and division are also related processes. The inverse operation of multiplication is division. Betweeness on the number line 1. 2. The manner in which whole numbers are related may be shown with a number line 012345678 I 4L1. J \ J1] l ‘ 7 a) Any whole. number is smaller than any of the numbers on the right side of it. b) Any whole number is greater than any of the numbers on its left. The number line can be used to determine how many whole numbers the re are between any two whole numbers. The sum or difference is not altered if one number is de- creased or increased provided the other number is increased or decreased by the same amount. This is referred to as the principle of compensation. VI. Operations with Whole Numbers A. Whole or counting numbers 1. 2. 3. a The numbers first used by man were the whole numbers. The whole or counting numbers are the numbers used to answer the question "how many? " Primitive man developed the idea of number by matching objects or things in one set with objects in another set; two groups are said to be in one -to-one correspondence when each item in the first group corresponds to one and only one item in the second group. The re is a standard set of counting numbers which can be 1 used to note that there are ”just as many’ in one set as in the other set. Counting numbers and zero are called the set of whole. numbers. aBrumfiel says that the whole numbers are counting numbers and includes zero; the SMSG materials state that the counting numbers and zero make up the whole numbers. 234 APPENDIX J -- Continued 6. There is no largest counting number. 7. When a number is used in its serial (or order) meaning, it is referred to as ordinal. 8. When a number is used to indicate the size of a group (collected meaning), it is referred to as cardinal. B. Basic combinations and algorisms 1. Addition of whole numbers a) Addition is a binary operation which assigns a number called the sum to an ordered pair of numbers called addends. b) The sum of any ordered pair of numbers also can be determined by joining two disjoint sets and then counting the members in their union. c) The plus sign (+) is the sign of Operation for addition. d) The sum may be recorded in various ways. (1) The equation 3 + 4 2 7 is one way to state the sum of an ordered pair. (2) Sums of ordered pairs of numbers may also be written in vertical form. 3 _+__4_ 7 (3) The sums of the ordered pairs of numbers may be summarized in table form. e) An algorism is a method of arranging numbers so as to reduce the number of steps in determining the correct result; it produces answers through numeral manipulation based on the use of number properties and definitions. f) All "carrying" or regrouping in addition is based on a combination of place value notation; associativity and commutativity. g) The commutative and associative properties also are utilized in checking a sum by adding "up" a column after adding "down. ” Z. Subtraction of whole numbers a) Subtraction, also a binary operation, assigns to an ordered pair of numbers a number that is called the difference. b) Subtraction is related to addition; it ”undoes additive operation. V! the c) The difference may be written in either equation or column (vertical) notation form. 235 APPENDIX J -- Continued d) Differences of whole numbers do not always exist as whole numbers; the set of integers makes it possible to perform such an operation as 3 - 8 2 -5. e) A thorough understanding of the addition algorism makes clear the meaning of a subtraction algorism such as: 47 = 40 + 7 = 30 +17 -19:10+9:lO + 9 ""7 20 + 8 The computer did not borrow; he has regrouped. Multiplication of whole numbers a) Multiplication is a binary operation -- a regrouping process through which equal measuring units are combined in terms of the number system. b) thiltiplication of whole numbers can be interpreted as repeated addition when the addends are the same. c) Multiplication of whole numbers is both commutative and associative. d) Both multiplier and multiplicand may be given the name factors. The product of any two whole numbers (other than ('5 v zero and one) is a whole number. f) The number one is the identity number for multipli— cation. g) The number zero, when used as a factor, always gives a product that is zero. h) Multiplication is distributive with respect to addition -- 2 (a + b) = 2a + 2b. i) Multiplication combinations can be summarized in a multiplication table or a matrix. j) The multiplication algorism can be explained in terms of the properties of multiplication and addition; it can be used to find the product of any two whole numbers. Division of whole numbers a) Division is a binary operation -- a regrouping process which finds either of two factors when their product and the other factor are known; it assigns to an ordered pair of numbers a number called the quotient. b) Division is the inverse of multiplication. c) Division is sometimes thought of as repeated sub- traction. 236 APPENDIX J -- Continued d) The divisionlalgorism may be written as 12 {’3 or .1;— or 3r—12L e) The quotient of some whole numbers (as 8 4'3) does not exist as a whole number. f) .V'Iathematicians avoid division by zero because there is no number other than zero to satisfy the. equation Dx0=8. VII. Rational Numbe rs A. History of fractions 1. D. 6. 7. hian has not always known about fractions; it is said that he began to use fractions when he began to measure as well as count. Unit fractions, which were used first by the Egyptians, are fractions with a numerator of 1. Our present fractional notation (e. g. 3;) came into general use in the 16th century. A symbol "E." where a and b represent numbers, with b not zero is called a fraction; it represents a fractional, number; the numerator tells how many parts of the whole are to be considered and the denominator tells the number of parts into which. the whole has been divided. A number which can be represented by fractional l 5 3 —a symbols is called a rational number ( _, _, etc. ). A fraction can have different names (.632 z 9 = 5:); it is 21 3 l the measure or value of one or more equal parts of a unit expressed as a relation of the parts to the whole. Two fractions which represent the same number are called equivalent fractions. Properties of rational numbers 1. The set of rational numbers is closed with respect to the ope ration of addition and multiplication. The operations of addition and Inultiplication for the rational numbers are both associative and commutative. The operation of multiplication is distributive over addition for the rational numbers. Zero is the identity element for addition of rational numbers; one is the identity element for multiplication of rational numbers. VIII. 237 APPENDIX J -- Continued 5. If the numerator and denominator of a fraction are multiplied or divided by the same counting number, the number represented is not changed. 6. 1f the numerator is multiplied by a number, the value of a fraction will be changed; if the denominator of a fraction is multiplied by a number, the value of a fraction also will be changed. To write a fraction in the simplest form, find the \) greatest common factor of both numerator and denomi— nator and multiply both terms by that: number. 8. A fraction in which the numerator is greater than the denominator often is called an improper fraction. Reciprocals 1. If the product of two numbers is 1, the numbers are called reciprocals of each other (51.x E = 1 if neither is zero). 1) a The number line can be very useful in working with fractions (i. e. , divide the line segment with endpoints O and 12 into four equal parts); equivalent fractions can be illustrated with the number line. Fractions also can be ordered. A fraction may express the ratio of one number to another. Per cent may be written as a fraction or as a decimal. Decimal Notation A. B. Differences between common fractions and decimal fractions are differences in notation only; decimal fractions may be expressed as common fractions. 1. When written in decimal notation, each digit represents a certain value according to its place in the numeral. a) The form for place value in base ten shows that the value of each place immediately to the left of a given place is ten times the value of the. given place; each place value immediately to the right of a given place is one -tenth of the place value of the given place. b) The places to the right of one's place are usually referred to as the decimal places. c) The decimal point locates the one's place. You can operate with decimals in Inuch the same way you handle whole numbers. 1. Numbers can be carefully written directly under one another for addition and subtraction. IX. 238 APPENDIX .1 -— Continued 2. To find the number of decimal places in the product when two numbers are multiplied, add the number of decimal places in the numerals. 3. The usual form of division may be used; then the same number of decimal places may be pointed in the quotient as the dividend. 4. You can also divide by a decimal fraction by multiplying the dividend and divisor by such a power of ten as will make the divisor a whole number. Real Num be r s A. The entire set of numbers represented by decimals is called the set of real numbers. 1. A real number is rational if its decimal representation repeats. Z. An irrational number is represented by non-repeating decimals. Close approximations of square roots of numbers can be made; a table of square roots may also be helpful. Numbers Inay be represented by many different kinds of numerals. 1. The decimal repreSentation is an important one. a) Some decimal symbols "come out even. " (terminating e. g. 2. = 0. 5000) H b) Other decimals never "come out even. (repeating mg. g). = 0.333) 2. Repeating decimals represent rational numbers. 3. Numbers represented by non-repeating decimals are called irrational numbers. (e. 0 "\l 2 : 1. 4142.13) D. The decimal terminates if and only if the rational number can be expressed as the quotient of two natural numbers in which the decimal is the power of 10. These will be only 2 or 5 (or both). .\/Iea sure ment A. B0 iVIeasurenient, or the process of nieasurii'ig, involves assigning a number to some physical quantity; the number is called the measure of the quantity. A measure can be assigned to the quantity on the basis of either direct or indirect comparison of the object to be measured with a standard unit. XI. 239 APPENDIX J -- Continued 1. In direct measurement, the number assigned to the quantity is determined by direct comparison of the object to be measured with a standard unit of measuretnent -- choose a unit of measure and then give a number which tells how many of these units it will take. 2. In direct measure (as ) the quantity being measured cannot be compared directly with the unit of measurement. C. The choice of a selection of a unit of measure is arbitrary; the selection of a unit of measure is made in terms of the physical quantity to be measured and in terms of the pre- cision desired. D. A given quantity can be measured in terms of different standard units. E. When a physical quantity is described by a number, it is essential that the unit of measure be designated. F. When measuring continuous properties, one can never be assured of arriving at an exact measure (all measures are an approximation). G. A special symbol ( I:1) read as “the same amount as” has been adopted to use with units of measure; it has been found that the symbol "1:1 H avoids considerable confusion if used until both the idea of unit measurement and the idea of equality are, well established. Geometry A. Points, lines, planes, and space 1. A geometric point is thought of as being so small that it has no size; it is represented by a dot. Z. The set of all points is called space; there is an un- limited number of points in space, 3. Mathematicians think of a plane as a special set of points. 4. A line is a set of points in space; it extends without limit in each of two directions; it contains an unlimited number of points. ’3. Through any two different points in space there is exactly one line. 6. Every line segment has exactly two endpoints. . A ray has only one endpoint; the endpoint is named first. a) A ray is the union of the endpoint and all points on a line in one direction from the point. b) A ray is always a part of a line. 240 APPENDIX J -- Continued Geometric figure 5 l. 2. 4. A geometric figure is any subset of the set we call space. The set of points consisting of two rays from a single point is called an angle; the rays are called the sides of the angle. . The. set of points consisting of three points not all on one line and all the points between the pair of points is called a triangle. 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