SCHOOL TEACHERS IN WESTERN STATE NIGERIA};:;;x‘:f:iigj THE DEVELOPMENT OF A MODEL PROGRAM FOR A52; ' INSERvIcE MATHEMATICS EDUCATION OF PRIMARY? ;;;;*%;"'?:;s:~:§_s'fi; DIssertatIon IOT the Degree Of Ph D MICHIGAN STATE UNIVERSITY ‘BUKUNOLA MABOGUNIE OSIBODU ETERARY Michigan State University .2 - ' This is to certify that the \ thesis entitled é THE DEVELOPMENT OF A‘MODEL PROGRAM 2,-0R INSERVICE MATHEMATICS EDUCATION OF PRIMARY SCHOOL TEACHERS INj'NESTERN STATE, NIGERIA. presented by Bukunolq Mabogunje Osibodu has been accepted towards fulfillment of the requirements for Pth0 degree in Education :46 Major professor ‘ 1F 0-7639 3 1293 10205 3240 ‘ PICKUP WTR 1983 WM 5 It, ABSTRACT “’1 5’2. ’7 THE DEVELOPMENT OF A MODEL PROGRAM FOR 0L INSERVICE MATHEMATICS EDUCATION OF 01 PRIMARY SCHOOL TEACHERS IN / WESTERN STATE, NIGERIA BY 'Bukunola Mabogunje Osibodu This study had two major purposes. One was to survey the provision of inservice mathematics education for primary school teachers (grades l-6) in the Western State of Nigeria, in order to identify the needs of primary teachers for mathematics teaching, and to investigate the views and preferences of teachers, headmasters and organizers on inservice mathematics education. The second was to develop a model for a systematic inservice mathematics education program based on the survey findings and other research results. Utilizing a framework for inservice mathematics components, developed through a review of literature, three sets of ques- tionnaires were designed and sent to a sample of Western State, Nigeria, teachers, headmasters and organizers respectively. The results of the questionnaire study, the review of litera— ture on inservice mathematics education patterns and practices, and the Nigerian governmental publications on mathematics cur- riculum reforms gave direction for the development of a model for inservice mathematics education. A return of 95 percent, 100 Percent and 40 percent of responses was obtained from 400 teachers,80 headmasters and 5 organizers respectively. 'Bukunola Mabogunje Osibodu Upon analysis, the data supported the followingconclusions: 1. Elementary school teachers in many countries can and do benefit from a variety of inservice programs in mathematics and in other aspects of the curriculum. A systematic long—sustained inservice program is likely to be more effective than a concentrated once— a—year program. A majority of teachers (85%) in the Western State sample had never participated in inservice mathe- matics education programs. Teachers realized their inadequate mathematics back- ground, due mainly to poor preservice training, and were willing and ready to participate in an inservice mathematics program in order to upgrade their mathe— matical knowledge and for better mathematics teaching. The most commonly expressed views on inservice mathe- matics training by all respondents were that: (a) the program should extend to all teachers, (b) it should be long-sustaining, (c) learning experiences should be through a practical approach with adequate provision for instructional materials, (d) appropri- ate evaluation should be planned, and (e) teachers should be remunerated accordingly. The differences in the responses of different sub— groups of teachers were not significant except in: (a) the responses of lower and upper primary teachers in their choice of the topic -— Solid geometry —— as “~*“““““‘jfiifi:fiii-IIII” 'Bukunola Mabogunje Osibodu a mathematical topic for inservice training, and (b) male and female teachers on "family responsibility" as a possible deterrent to participation in inservice mathematics programs. A model for inservice training of primary school teachers A two-week concentrated course during the long was developed. vacation followed by seven one—day course during the school Major components of the model included: year was proposed. (b) the selec— (a) the establishment of inservice objectives, tion of inservice learning experiences, (c) the organization of inservice programs, and (d) a plan for inservice evaluation On the basis of the findings and the requirements for model implementation, the following recommendations were made: A Central Inservice Coordinating body with representa~ l. tives from the government, the teacher training insti- tutions, the professional organization, and the com— munity should be established in the state. Decision-making on the evaluation of systematic in- service programs and on the promotion procedures of primary teachers based on performance criteria was suggested. Material production centers should be established in the state for the design, testing and mass production of instructional materials. Quality inservice education should be an integral part of primary school mathematics teaching, as well as other aspects of the primary curriculum. THE DEVELOPMENT OF A MODEL PROGRAM FOR INSERVICE MATHEMATICS EDUCATION OF PRIMARY SCHOOL TEACHERS IN WESTERN STATE, NIGERIA BY 'Bukunola Mabogunje Osibodu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1975 @Copyright by 'Bukunola Mabogunje Osibodu 1975 ii Dedicated to All Nigerian Primary School Teachers who, with the support of the whole community, must face the challenge of quality instruction for Nigeria's Children |'—————__——f ACKNOWLEDGMENTS The preparation of this dissertation has benefitted im— measurably from the assistance of many people. Only a few can be mentioned here, but I express my sincere appreciation to all of these people. I wish to sincerely thank Dr. Calhoun C. Collier, the chairman of my doctoral committee, for his interest in the study, the guidance and suggestions which he gave throughout the duration of the study. I am grateful to Dr. John W. Hanson, who as a member of my doctoral committee shared his in—depth understanding of Nigeria with me. Sincere thanks is also due to Dr. Howard W. ickey and Dr. Jean Lepere for serving as members of my doc- oral committee, and for their counsel and warmth. My appreci— tion goes to Dr. Sheldon Cherney and Dr. Sheila Fitzgerald or serving on my oral examination committee. I would like to express special thanks to the Dean of the ollege of Education, Dr. Keith Goldhammer, and the Chairman f the Department of Elementary and Special Education, Dr. ames E. Snoddy, for giving me the opportunity of a field ex- erience in elementary school mathematics teaching through a raduate assistantship. The experience added great insight to y study. I am indebted to the Dean of Faculty of Education, Univer— ty of Ife, Nigeria, Professor A. Babs. Fafunwa; to my col— agues, Dr. Stella A. Olatunji and Dr. Adeniji Adaralegbe; and other staff members and students of the faculty, without om the collection of data for this study would not have been ssible. The cooperation of the headmasters, teachers and ganizers, who were involved in the study is invaluable. I express my gratitude to the University of Ife, Nigeria r granting me the study leave and for providing the financial pport. My deep appreciation is expressed to my parents, Mr. and s. J. O. Mabogunje; to the entire Mabogunje family, especial— Professor and Mrs. Akin L. Mabogunje, and Dr. and Mrs. A. Oyejide; and to my children, Omobolaji, Somuyiwa and intomiwa. Without their interest, encouragement, understand- g and continued support, graduate study in Michigan would not ve been a reality. The use of the Michigan State University computing facili— as for this study was made possible through support, in part, am the National Science Foundation. .:--:v TABLE OF CONTENTS er Page INTRODUCTION .................................. 1 THE PROBLEM AND ITS SIGNIFICANCE ........... 2 PURPOSE OF THE STUDY ................... .... 4 PROCEDURE AND SOURCE OF DATA ............... 5 Literature Review ................... .... 5 The Survey Procedure .................... 5 Treatment of Survey Data ................ 12 Developing the Model .................... l4 BASIC ASSUMPTIONS .......................... l6 EXPLANATIONS OF TERMS ...................... l6 LIMITATIONS OF THE STUDY ................... l7 ORGANIZATION OF THE STUDY .................. l8 WESTERN STATE, NIGERIA AND ITS EDUCATIONAL SYSTEM .................... 20 HISTORICAL SYNOPSIS ........................ 22 The Beginning ........................... 22 Early Teacher Training Programs ......... 24 TEACHER EDUCATION AND PRIMARY SCHOOL INSTRUCTION IN WESTERN STATE ............... 26 Teacher Training Programs.... ........... 26 Primary School Instruction .............. 33 CURRENT MATHEMATICS PROGRAMS: Primary SCHOOLS AND TEACHER TRAINING COLLEGES ...... 35 Primary School Mathematics Program ...... 36 The Teacher Training Mathematics Program ................................. 39 Evolving Programs for the Primary Schools ................................. 41 PROBLEMS OF IMPROVING INSTRUCTIONAL QUALITY .................................... 43 vi r-‘wr" Y w'JI-I— - D 1 (h n. In ll) 2:: I‘ n- .. .. ’11 V IN T CHAR TEAC INSE —r- r-x—d‘w Lpter Teacher Preparation ..................... Materials of Instruction ................ Supervision and Retraining .............. Special Needs of Mathematics ............ PROMISING DEVELOPMENT ...................... SUMMARY .................................... I. THEORETICAL BACKGROUND AND RELATED RESEARCH... SOME LEARNING THEORIES AND MATHEMATICS INSTRUCTION IN THE ELEMENTARY SCHOOL ....... Studies in Cognitive Development and Mathematics Learning .................... Instructional Process in Mathematics.... THEORY AND PRACTICE IN INSERVICE EDUCATION. Purpose and Philosophy .................. Forms of Inservice Mathematics Education Selected Practices in the United States ............................... The British Teachers' Centers ........ Some Patterns in Africa .............. The Mass Media and Inservice Mathematics Education ................ THEORY AND RESEARCH ON PLANNED EDUCATIONAL CHANGE ..................................... SUMMARY .................................... ANALYSIS AND INTERPRETATION OF SURVEY DATA ................................... PARTICIPATING SCHOOLS ...................... Findings ........... . .................... Discussion ................... . ........... PROBLEMS OF ARITHMETIC TEACHING IN THE SCHOOLS ............................. CHARACTERISTICS OF TEACHERS RELEVANT FOR CONSIDERATION IN INSERVICE DESIGN ...... TEACHER PAST PARTICIPATION IN MATHEMATICS INSERVICE TRAINING PROGRAM ................. How Much Have Teacher Participated? ..... Who Organized the Courses Attended by Teachers? ................... What Main Mathematics Topics Were Covered in the Courses? ................. Page 54 56 63 68 68 73 74 86 89 92 95 99 103 104 104 106 108 112 116 116 117 119 r u: ‘k’ (I m :1 (I) ”V. DEA?“ .. "r A b v “513. {\‘1 s I; In. COXC RICK Ipter What Were the Effects of Courses on Classroom Teaching?... The Non-Attenders ....... ....:.:.:::::::: TEACHER ATTITUDE TOWARD MATHEMATICS AND CLASSROOM PRACTICES .................... Rank of Arithmetic Teaching ............. Teachers' Attitude Towards Mathematics Activities.... ....... The Materials of Instruction ............ Teachers' Classroom Practices ........... VIEWS AND PREFERENCES ON INSERVICE MATHEMATICS PROGRAMS: TEACHERS, HEADMASTERS AND ORGANIZERS ............................. Inservice Learning Experiences .......... Organization of Inservice Programs ...... Evaluation of Inservice Effectiveness. 0 Demand for Future Courses ............... Discussion .............. . ............... SUMMARY OF MAJOR FINDINGS.. ................ A MODEL FOR INSERVICE MATHEMATICS EDUCATION. RATIONALE FOR MODEL OBJECTIVES. ............ A MODEL FOR INSERVICE EDUCATION ............ Establishment of Inservice Objectives. Selection of Learning Experiences. ...... Organization of Inservice Programs.... A Plan for Inservice Evaluation....... SUMMARY OF THE CHAPTER ................. .. SUMMARY, CONCLUSION AND RECOMMENDATION...... 0 SUMMARY OF FINDINGS ........................ CONCLUSIONS ................................ RECOMMENDATIONS AND IMPLICATIONS ........... Recommendations for Implementation ...... Recommendations for Primary Teacher Training................ ........ Implications for Primary Education ...... Implications for Further Research ....... viii Page 120 122 129 130 132 136 142 145 145 151 160 162 165 170 172 173 173 176 189 204 210 211 211 213 215 217 219 220 220 " ‘ I voo~‘-r _ n:~:....-.'. .. ..-.. “' I...:.. ‘ K...‘ -.. C T.-.-.I.'.‘. .3 2"“ '--v:..',__. --..-.~_--..-.,. . _ Page endix A List of Organizers and/or Sponsors ....... 222 :ndix B Instruments Used in Gathering Survey Data .............................. 223 andix C List of Towns in Which Participating Schools are Located by Classification.... 244 andix D Test of Basic Mathematical Understandings ........................... 245 .iography ......................................... 262 ix .u-»- _ “av-sans 2‘. 1' :c-u\....._. ‘~~ :3. 5.2.93“..- N "u— v ... A33»- 5... I Re53>OFISP Nathemat 8' SUENary Teaching CatEgor: PETCEDEI Stateme: PerCent IOclude ChOice by Teac ' NUmber t0 Math LIST OF TABLES Page Participating Teachers and Headmasters .......... 106 Problems of Arithmetic Teaching as Identified by Headmasters in Urban and Rural Schools, by Percentage Count of Responses............... ... 108 Percentage of Teachers Who Had Attended or Not Attended Any Previous Inservice Mathematics Program by School Location ...................... 117 Percentage of Teachers Who Had Attended or Not Attended Any Previous Inservice Mathematics Program by Teacher Qualification ................ 117 Rank of Arithmetic as a Choice Subject by Teachers in Categories ........................... 132 Percentage of Teachers' Responses on Mathe— 134 matical and Non—Mathematical Activities......... Responses of Attenders and Non-Attenders on Two 135 Mathematics Interest Items ......... ............. Summary of Pupils‘ Books Used for Arithmetic 137 Teaching by Categories ..................... ..... Relating Teachers' Class Levels to the 139 Categories of Pupils' Textbooks...., ........ .... Percentage of Teachers‘ Responses to 144 Statements on Classroom Practices... ........... . Percentage of Teachers Who Would Like Topics 147 Included in Inservice Programs ......... . ....... . Choice of Inservice Learning Experiences 150 by Teachers in Order of Priority. ..... .......... Number and Percentage of Responses in Regard 152 to Mathematical Content of Inservice Programs... Table 1L ‘Lv A. ~- ~.' .--£‘.~»-- \ - A.‘ .L—.:..‘ Maw-g...“ M. v... n. J»...- - o ‘. --v. a... n-3c-‘.‘u. “v3.2..t . Nae»--- :- .. 2... .. -..S- .‘ vvo~v -- t-h‘u‘ '5 VW‘» .. . ..at..€..IaC q - . " n -. *-a "lay A Segue: matics I L7. L8. 20. 21. 22. E3. Preference for Inservice Locations...., ........ Location Preference of Urban and Rural Teachers in Percentage.... ........ . ............ Distribution of Responses Concerning Time and Duration of Inservice Training Programs ........ ...... ......... . ............... Distribution of Responses on Whether Inservice Mathematics Programs Should be Made Compulsory or Not..... ...... ..... .................... ..... Distribution of Responses Concerning the Credits to be Given for Successful Attendance at Inservice Programs .......................... Preference for Inservice Instructors by the Teachers... ............ . ................ Factors Affecting Teachers' Wish or Ability to Attend Inservice Programs.... ............... Distribution of Responses on How Inservice Programs Should be Evaluated ................... Mathematics Topics for Inservice Training Programs ........ . ...... . .............. A Sequential Design Chart of Inservice Mathe— matics Program Procedure... ..... . .............. xi 154 161 182 198 :r Aka" ..I..\ The exe SDDjSCT The cat & I eachi books LIST OF FIGURES igure Page 1. Western State, Nigeria: Divisional Headquarters .................................... 8 2. Location of Western State, Nigeria .............. 21 3. Total teacher population in 80 schools by qualifications ............................... 105 4. Categories of teachers by years of teaching experience ...................................... 113 5. Distribution of teachers by classes they are presently teaching ..................... 114 6. Organizers of courses attended by teachers by percentage distribution of total courses attended ........................................ 118 7. The exact ranking of arithmetic as a teaching subject by all teachers ......................... 131 8. The categories of teachers by years of teaching experience and the type of reference books they use .................................. 140 Di ever: ~52“ 1‘ Wu“. The ca . 'A' L A V: 83C L \- DOORS LIST OF FIGURES igure Page 1. Western State, Nigeria: Divisional Headquarters .................................... 8 2. Location of Western State, Nigeria .............. 21 3. Total teacher population in 80 schools by qualifications ............................... 105 4. Categories of teachers by years of teaching experience ...................................... 113 5. Distribution of teachers by Classes they are presently teaching ..................... 114 6. Organizers of courses attended by teachers by percentage distribution of total courses attended ........................................ 118 7. The exact ranking of arithmetic as a teaching subject by all teachers ......................... 131 8. The categories of teachers by years of teaching experience and the type of reference books they use .................................. 140 e: 33‘... w) .. .6 - ”Ems“ ., "5. L2..- ..\.uv-\. ' '-‘-~¢-<:nu: ‘ C‘ E..ut....-c :ries whose r lizites. The ref: :eachers, es; grams sugges need for Pro “5 meet the CQIIIItries cc 0f inservicE led to inc“ wueational CurriCulum I CONNectiOHS tion in Net ideas into CHAPTER I INTRODUCTION The contemporary movement for the reform of both content and methodology in the teaching of mathematics at all levels of education affects the developing countries just as much as countries of Europe and North America. Indeed, in many ways, the reform raises more acute problems for the developing coun- tries whose resources, in particular human resources, are very limited. The reform has presented various challenges to school teachers, especially at the elementary level. These teachers need to acquire sufficient background in mathematics to enable them to provide effective instruction in the mathematics pro— grams suggested for the elementary school. Recognizing the Need for programs which would help the elementary school teach— ers meet the challenge of re—education in mathematics, many :ountries continue to initiate and implement different forms )f inservice programs in this discipline. These attempts have Led to increasing awareness of greater needs for the continued educational growth of teachers in the area of mathematics. Zurriculum workers continue to have deeper insight into the :onnections between the course of study or a program of instruc— :ion in mathematics and the human teacher who translates their Ldeas into reality. A pronouncement made jointly by four 1 crofea‘vsional C :teasir-9 of curre tasks as ahead. classro: II LA‘ ..5F.- -=€ Level, I an: tea: he :s C‘» institutions training, in Of inservice with the con :unity servi The la: in mathemat: interests, New curricu QAUCation. . The N IStréltive R (Nash III ington fitics. 196 professional organizations in the United States succinctly summarized this awareness: Planning for a changing mathematics curriculum should provide for continuous inservice education of teachers both in mathematics content and in methods of mathematics instruction. The mathematics program not only has changed but will continue to change. The changing nature of the program results in a need for continuous inservice education. Teachers are in- creasingly recognizing that change is in the nature of current curriculum development, and that their tasks as teachers will constantly change in the years ahead. Ample time and help should be provided so that classroom teachers can remain alert to these changes. Many teachers, particularly at the elementary school level, have fears about their ability to understand and teach mathematics. An adequate inservice program ' helps overcome this fear. The implications of this situation are vast for educational institutions, in particular institutions engaged in teacher training, in every country. The initiation and implementation of inservice training programs for teachers who are unfamiliar with the contemporary concepts of mathematics is a prime com— Nunity service. THE PROBLEM AND ITS SIGNIFICANCE The last few years have witnessed some programs of reforms in mathematics education in Nigeria. Both national and states' interests, including those of the Western State, have produced new curribula in mathematics for all levels of pre—university education. In describing its work, the mathematics group of 1The National Council of Teachers of Mathematics, Admin— istrative Responsibility for Improving Mathematics Program (Washington, D.C.: The National Council of Teachers of Mathe— matics, 1965), P. 11. TReNatiofial ,;c~-'I'£'_ Hofks Ne -.'.. r; 2 !\;je-“‘ --C £63539~ Raining hax UR new cur: new reforms Yet, U RhOOl leve andnationa Conference P perman consun \ 2 1. Nigel matics GrOI the National Education Research Council, Primary School Cur— riculum Workshop stated:2 We are building a curriculum not only for the Nigeria of the 1970's but for a Nigeria of future decades changing and developing fast towards the let century. However, the report of the group further indicated that:3 There is an acute shortage of knowledgeable teachers. This is a problem which in the main must be resolved by every state and every Govern— ment, helped where necessary and possible, by the Federal Government. Some attempts are being made to reform the teaching of mathe- matics in primary schools at the preservice teacher training level. Meanwhile, there is a whole generation of primary school teachers now in the schools whose basic education and training have neither prepared them to teach the content of the new curriculum nor use the methodology called for by the new reforms in mathematics teaching. Yet, the effective teaching of mathematics at the primary school level is crucial to the attainment of educational goals and national aspirations. The Nigerian National Curriculum Conference of 1969 stated:4 Primary school curriculum must aim at functional permanent literacy to ensure better producers and consumers of goods. It should provide a sound basis 2Nigeria Educational Research Council, Report of Mathe— matics Group, National Workshop on Primary School Curriculum (Lagos: Federal Ministry of Education, 1971), p. 4. Ibid., p. 4. 4National Curriculum Conference in Nigeria, 1969. "Rec— ommendations of the 1969 National Curriculum Conference." In: .B. Fafunwa, History of Education in Nigeria, London: George llen and Unwin, Ltd., 1974, p. 233. for scxer GETELO; 2 1.21215? :.y.,.‘.av-rvg _ A. ”.C.—.- ~l -:.‘.~-;~ - na‘ .a...C_~_a.u A n... -u- ~~-’ I-... we Ge“. 1... -2 ..... -... \ V-:- .. .13..- .. ..; ..,. .. -' .. yu-¢‘ v - I'~. L..eachers' SUC mat tee mat frI Se for scientific and reflective thinking....It should develop in children mechanical, vocational and manipulative skills and competencies. Furthermore, most educators predict that a very good grip on mathematical skills and concepts will be essential for coping with the complicated world in which children now in school will be obligated to live. There is evidence that many of the most important of these concepts take root most easily during the early childhood years. However, there is a contradiction between these facts and the mathematics instruction now given in the Western State primary schools. The problem lies main— ly in the inadequate preparation that teachers have for mathe— matics teaching. Programs of inservice training and adequate guidance can be directed towards the correction of the defects in teachers' background knowledge. PURPOSE OF THE STUDY The purpose of this study was twofold: l. to survey the existing state of mathematics inservice education for primary school teachers in Western State of Nigeria, in order to deter— mine the degree of involvement of teachers in such programs, teachers' needs for the knowledge of the content and methods in the teaching of mathematics, and the views and preferences of teachers, headmasters and organizers on mathe- matics inservice training programs; and 2. to develop a model for a systematic inservice mathematics education program based on the survey findings and other research findings from a review of literature relevant to in— service training and the learning of mathematics. L—_ l The pro“ (ll ‘3 l) (D K) ad- (1. .. ‘ «u- . " "' 5.... .aqa: -u“ End—u .. . 1 In} .-.....-v ..‘..,’U‘S v I , insI ervice prog literature c Other 1 of learning research on GOVermnenta SiVely. The s w Many a tional (HON \ Sherbg Teacher Tr; PROCEDURE AND SOURCE OF DATA The procedure used in gathering data for this study in- cluded: (l) the review of literature on inservice mathematics education and other related theories, (2) the construction and administration of three sets of questionnaires, (3) the exami- nation of governmental publications and documents, and (4) correspondence with governmental officials. Literature Review An extensive review of literature on inservice mathematics education in some countries was an essential tool of the study. Because of the changes in the elementary school mathematics program, inservice education for teachers has been recognized as a necessity by teachers, administrators and university per- sonnel in many communities. Many patterns and practices of in— service programs have evolved. Information was gathered from literature on some of these patterns. Other literary works used in the study were in the area of learning theories and mathematics instruction, theory and research on teacher education and on planned educational change. Governmental publications and documents were also used exten— sively. The Survey Procedure Many authors have recommended teacher-determined educa— tional growth programs. Thelen5 asserts that teachers are the 5Herbert A. Thelen, “A Cultural Approach to Inservice Teacher Training,“ Improving Inservice Education: Proposals best judges 0- Dictation- 5‘ bye cer.a1n I unions. as R111 the to: assertions. consideratio: involved in ‘ The majI teachers' ’(J D) classroom pr needs for ma Preferences service natr teachers ant t0th randor transPortat: Postal SYS‘CI \ Nd Procedu and B con, II.N.R Developin 9 W ‘ See A and lnstitu best judges of what should go on in programs of continuing education. Hawes6 stressed that the curriculum planner must have certain basic information in order not to operate in a vacuum. This information, he added, should include conditions under which teachers work, their academic, linguistic and pro- fessional backgrounds, their attitudes towards the present curriculum and towards the proposal of change, their relations with the community they serve, their morale and future aspir— ations. The survey in this study was predicated on the above assertions. At the same time, the survey procedure put into consideration the level of academic sophistication of teachers involved in the study. The major aim of the survey was to explore and describe teachers‘ participation in previous inservice training, their classroom practices related to arithmetic instruction, their needs for mathematical content and methods, and the views and preferences of teachers, headmasters and organizers7 on in— service mathematics training. In selecting the sample of teachers and headmasters for the survey, many constraints made total randomization difficult. Among them were the cost of transportation to the primary schools, the uncertainty of postal system to all schools, and the non—availability of a and Procedures for Change. Rubin, L.J. (ed.) Boston: Allyn and Bacon, Inc., 1971, pp. 71-103. H.W.R. Hawes, Planning the Primary School Curriculum in Developing Countries. (Paris, Unesco: International Institute for Educational Planning, 1972), p. 25. See Appendix A, p.222, for individuals, associations and institutions designated as Organizers and/or Sponsors. ’I". I- “. s 1‘58 CQZ’QIEIIEn ~...- -eac'r.e: I v Dung. - -. ' . ,. A .. - -,.._- _I ‘—a.‘\v-‘—- C... \v . . ~1-V~"~"~tcz . Ll .. ‘u-6 ..- v iajority of fire in this Sthtistics < 16,292 totaj H 0r Grade 50rce of 26 the total 9 was that th teachers Ca Only Offer‘ \ Annual Di 6 M Ler' comprehensive directory of the entire population of primary school teachers in the state. ‘ The main sampling strategy, however, was to reach repre- sentatives of teachers in as many administrative divisions and geographical areas of the state as practicable within thegiven :onstraints. A two—stage cluster sampling was therefore de— signed. Schools were purposefully selected according to their location in urban or non—urban areas in the main administrative iivisions of the state.8 In each school, a stratified, simple random selection of Grade II and Grade III teachers was made. All headmasters of schools from which teachers were sampled were included in the survey. The decision to limit the sample of teachers to a group of Grade II and Grade III teachers was based upon the fact that majority of the teachers trained for the state's primary schools re in this group. The 1971 Annual Digest of Educational tatistics of Western State, for example, showed that out of 6,292 total trained teachers, 16,035 (about 98%) were Grade I or Grade III. When examined on the total primary teaching orce of 26,609 trained and untrained teachers, about 60% of he total group were Grade II or Grade III.9 A second reason as that the basic mathematical training of this group of eachers can generally be identified. Their basic training nly offered them a course in Arithmetic Process. g___._____ See Map on page 8. Figures culled from Western State of Nigeria, Ministry f Economic Planning and Reconstruction, Statistics Division: nnual Digest of Education Statistics, (Ibadan: Government rinter, 1971), Vol, XI, p. 43. /\ I "‘\ N r I I'd ) ’ . \ KWARA ShoLKL )\ STATE ‘\ oflbomQSEQ fiu\l‘__ ‘ 1"" I kip“ n I 97%. U \\ 0 05"} b oLk' ‘ IKOLQ 0‘19 03 0 IjeuuLjeshq 'Ekiti \ . . . 0 'RL E , , (ya >-I SIM/o I‘éshq C0 KIT; H E E“. . I o ) 4‘ uwa 1&3 IKarq, 53. 1nd Ftkirre . / g b“ “It . - Owe ) AbéoRR-H. 00343 f”, I o Sqéomq OlJebu Ode ‘ 61/ Ilaro q _———-.__ QM“ ”P“ fLATGOS STATE 7 03’ A) MIDWEST V L“ °5 \qf‘ STATE 6. ) L 89 169an ATLANTIC OCEAN 0 50 100ml Figure 1. Western State, Nigeria: Divisional Headquarters Source:- Ministry of Lands and Housing, Survey Division: Road Map of Western State, Nigeria, Second Edition, (Ibadan, Western State, 1969). 1': was c forces 1.151319 -‘~,e organiza: éizicns SOUL: i::eres:e;‘ i: the coordina' neationnairr m: the sc teaching and Organizers a had offered state or the soring such Three 5 survey: (1; organiZers. to three seI Nation on 9 ing, In de made t0 con Conducted W importanCe 1h SuI-VeY S l\0 One: and Organ; . H AgreSSOr I 1’5 I W It was of interest to the study to investigate other forces inside and outside the schools which might influence the organization of inservice mathematics programs. Such con— ditions would give prerequisites for cooperation among bodies interested in inservice training and suggest the approach to the coordination of available resources. The headmasters' questionnaire was designed to collect pertinent information about the schools, general problems of and need for arithmetic teaching and their views on mathematics inservice training. Organizers and/or sponsors that were contacted are groups who had offered refresher courses on mathematics teaching in the state or those who have the potential for organizing or spon- soring such inservice programs. Three sets of questionnaires10 were designed for the survey: (1) the teachers', (2) the headmasters' and (3) the organizers' or sponsors'. Three different letters were written to three senior governmental officials to seek further infor- mation on governmental plans and policies on inservice train— ing. In designing the teachers' questionnaire, attempts were ade to control for acquiescent responses. Several studies conducted with African subjects have called attention to the importance of controlling for acquiescent response patterning 'n survey studies with less sophisticated respondents.ll In 1 . . 0Questionnaires and Letters for Teachers, Headmasters and Organizers are found in Appendix B, pages 224 to 244. llM.H. Segall, "Acquiescence and 'Identification with the gressor' Among Acculturating Africans," The Journal ofSocial sycholog , 1963, 61:247—262. ‘-'-e ozesent S‘. . _., ., "era :25: ”.9; I nan—.- ...v s ' n 1",- ‘="‘”" 5.. U..- ........-- ‘ n L‘s 5;... -..- I F! ‘es' “s t e retain among". - _:‘L11‘\" 5C; . ...... .3 SETS; findi' Althog :,,‘__ ‘ --L1... the POP due to the O Jaire was pr fouhd in the the exPerier of theSe hac‘ AeStern StaI The qu, Cation, Uni' teachers an AssOciatesh trained in a FaCulty IT Lahts WEre 10 the present study, questionnaire items were stated in a way that they were not psychologically threatening to teachers. The introductory statement of each section presented the issue explored as a common problem for all teachers rather than a problem for a specific teacher. A balance was kept between open-ended and closed—ended questions as a means of capturing more of the complexities involved in the problem of study. This balance was also reflected in both the headmas— ters' and the organizers' questionnaires. Respondents were to remain anonymous, but they were given an option of indicating a mailing address if they were interested in a summary of the survey findings. Although the pretesting of the questionnaires on a sample from the population of the research subjects was not possible due to the overseas distance involved, the teachers' question- naire was pretested on a small group of West African students found in the Michigan State University community, who have had the experience of teaching in African primary schools. Some of these had taught as Grade II teachers until recently in Western State, Nigeria primary schools. The questionnaires were reproduced at the Faculty of Edu- cation, University of Ife, Nigeria and later administered to teachers and headmasters in the selected schools by fifteen Associateship Diplomaeokuta in 1867. It later began its third phase when it pened at Oyo in May, 1896 expressly to produce workers for 1e Yoruba mission. Lack of suitable candidates proved an astacle to the expansion of the program, but by 1904, the :incipal reported that 29 teachers in all had been sent out 1d were influencing at least 700 children. He also reported iat students came from areas as far apart as east of the .ver Niger, the Gold Coast (now Ghana) and the Hausa Mission 1 Northern Nigeria.23 The need for trained teachers was already acute by the rst decade of this century, and in addition to the provision teacher training institutions in the western part of Nigeria, a missions in the east and the north also provided teacher iining programs. The curriculum of the early training in— .tutes combined theology with teaching methods. At the end the two—year course, the teacher trainees took a prescribed chers' certificate examination and were certified if they sed the examination. In appraising the early elementary >ol instruction in one of the areas, Inspector Cummings, 23T.T. Solaru. Teacher Training in Nigeria. Ibadan: an Univeristy Press, 1964. p. 5. reverting 1n 1,. g H in; :he two- served as p: mil-teach teachers. ' about fourt hour daily teacher, Wh During the the head te their exam amount to s misSIOn Re; training as teachers W curriculum w m 25 aorting in 1908, as cited by Solaru, revealed the need for acher training on a higher level:24 The work of schools is fairly satisfactory up to Standard III. Beyond this stage when in- dependence of thought is required, the results are not satisfactory -— not because the native child is incapable of thinking for himself, but the teachers are to blame to a great extent... Lessons are haphazard, no definite aim or system, and correct method is wanting. Too much talking to the child, hardly testing questions to train habits of correct thought. The teacher trainees for the early training programs re drawn from Standard VI (about Grade VII). Before start— g the two—year training course, they were expected to have rved as pupil—teachers for two years and to have passed the oil-teacher examination and then to have acted as assistant achers. The pupil-teachers were selected pupils, aged aut fourteen, who had passed Standard V. They received one 1r daily instruction from the head teacher or other approved tcher, while they taught some classes during other hours. 'ing the two years of apprenticeship, grants were paid to head teacher who successfully prepared pupil—teachers for ir examinations. The pupil teachers received a token unt to supplement their allowance. The Phelps-Stokes Com- sion Report of 1922 also criticized this system of teacher— Lning as being unsatisfactory, partly because the pupil- rhers were overworked and underpaid and partly because the iculum was poorly conceived and the supervisory system 24Ibid., p. 17. . h 1".» PI. .. » . - we “.33. —-\ -7... r N- h vvvvvvv - ~..-1 v; ’ C a... -- ...... = "5 :..»~:- -«—. v- -‘nc .. - a“..- 2.... yo..-. .:-- :...-.‘. ..-.... -Skrvb. ~sv- an we ...;. rumba ..\.- ~ 1 ‘n'w .333 and - :ence of 3‘. cation, es imbalance has become in? Nigeri TEAC Te % The t most PErs: The 1929 ; SYStem re ZSFO m9nt in N Ukeje (19 26 was inadequate. With the amalgamation of the two departments of education in the northern and southern provinces of Nigeria in 1929, the control of education became centralized. The new Director of Education then set himself to the task of the re—orienta— tion and re—organization of the educational system along the line suggested by the Phelps—Stokes Report. Meanwhile, the gap in the educational development between the north and the south continued to widen. Through the years, other contribu— ting factors such as the creation of free and universal pri- mary education in both the western and eastern regions between 1955 and 1958, but not in the northern region; and the reti- cence of Moslem parents about the blessings of western edu- cation, especially for girls had intensified the problem of imbalance in the country's educational system. This imbalance has become one of the major educational challenges confront— ing Nigeria today.25 TEACHER EDUCATION AND PRIMARY SCHOOL INSTRUCTION IN WESTERN STATE Teacher Training Programs The training of competent teachers continues to be a most persistent educational problem beseting Western State. The 1929 re—orientation and re—organization of the education system referred to earlier led to the evolution of two types 25For more detailed descriptions of Educational Develop— ment in Nigeria see: Adetoro (1966), Fafunwa (1974), and Ukeje (1966). ’PT, I‘- IIIII \uL ':": -a- n “a-.. - .H - - ~ A = ‘C C. an. 0“e:ec the “Wilt at 1 These 1930's and rate Colle< 'l'iEWed all has more p while that “one the English, A curriCula Volthary The w 1119 progra 27 of teacher—training programs for elementary schools. These Were the Elementary Training Centres (E.T.C.) for the lower primary school teachers and the Higher Elementary Training Colleges (H.E.T.C.) for teachers of the upper classes. The E.T.C. course lasted for two years and culminated in the Grade III Teachers' Certificate, while the H.E.T.C. course, also of two years duration, led to the Grade II Teachers' Certificate. A teacher trainee moved in succession through these courses. To go into the Grade III course, he must have been a pupil—teacher for at least two years. On the success— ful completion of the Grade III course, he had to teach for another two years before proceeding to the H.E.T.C. which offered the Grade II course. Some of the H.E.T.C. graduates taught at the E.T.C.'s. These teacher training programs survived through the 1930‘s and most of the 1940's in this form. There were sepa- rate colleges for men and women, and the curricula were re- viewed all through the l950's. The curriculum of the E.T.C. as more practical and more suitable for the rural community, hile that of the H.E.T.C. was more academically oriented. ong the prescribed subjects for both types of colleges were nglish, Arithmetic, Simple Accounts and School Methods. Both urricula reflected religious education since most of the oluntary educational bodies were missionary groups. The war time economy of the 1940's affected teacher train— ng programs in Western State. Salaries of experienced and mined teac ‘ ',... ”I :lace: .; u of books an: :ezbers lez'. 3::er educa :5 "20:1". the ::Lieges am By the agencies ha. :he educati Vissionary Coarse for in the Stat Concerted e deficient b fresher cou primary edu Dance 0f 19 Wk for ed initiated I Xigeria. r] +.he miSSioI Regional G( assumed th( al facilit. 28 trained teachers could no longer be met and some were re— placed by untrained teachers for lower salaries. Shipments of books and equipment were lost. Many expatriate staff members left for good. Teacher training problems in general tended to receive less sustained attention than those of sec- ondary education which had attracted the financial support of both the government and the voluntary agencies. As a re— sult of this situation, staffing at both the teacher training colleges and the primary schools was unsatisfactory. By the end of the 1940's, both government and voluntary agencies had produced proposals aiming at the improvement of the educational system. One of the proposals of the Church Missionary Society, for example, was to provide a one—year course for uncertificated teachers at three different centers in the state. This move may be regarded as one of the early concerted efforts towards the retraining of teachers with deficient background training other than through short re- fresher courses. Of great importance to later development in primary education in Western State was the Governmental Ordi- nance of 1948 which in addition to the provision of a frame— work for educational develOpment for the following six years initiated regionalization of educational administration in Nigeria. The responsibility for education thus shifted from the missionaries to the regional governments. The Western Regional Government, like the other regional governments, aSSumed the reSponsibility for the provision of all education— 1 facilities up to the pre—university level in its region. :55: culmi: eiucatio: i 2: :a2 :2“ a 13:; 5-75 PICjecz enrollIenz. . ‘ H ‘zer- :LCC“ regiorial 9, meter of < large mm) in the Sch‘ Standard 0 In 19 of 37,544 and withou At th the fed6ra bysirEriC would matc of indeper 29 A major effect of the regionalization in the Western State was a great surge of energy for an interest in primary education that culminated in the establishment of universal primary education in the State which began in 1955. To make the universal primary education program a real- ity, a large number of new teachers had to be supplied for the projected increases in the number of schools and in pupil enrollment. The introduction of such a vast scheme of pri— mary education posed immediately the problem of the training and supply of large numbers of teachers. To meet this exi— gency, local authorities and voluntary agencies established several Grade III colleges with financial support from the regional government. As a result of the scheme, a large (number of children have received primary education, but a large number of untrained or half—trained teachers were left in the school system by the scheme and thereby lowered the standard of education. In 1959, of a total teaching strength in primary schools of 37,544 persons, there were 23,979 or 64% uncertificated and without training, while 920 or 2% were on probation.26 At this point, one year before Nigeria's independence, he federal government set up a nine—man commission, chaired yrsirEHjI:Ashby, to recommend a pattern of education which ould match Nigeria's aspirations over the first twenty years f independence. Teacher training for primary schools formed 26T.T. Solaru, op, cit., p. 102. a Significan CDZiSsiOv-er ..~' -ic.".€r € :36 nv‘. (J Y Lev l]! (1,00! (1 ’ .v '(.1 (I) D) (l . 'r) L‘" M the earlier Ashby Corni heart of th cational in and qualif i atioh, the mission mad canoe to tr W i E Fe ede n d % leate and IY of Edu< WES APPOinteclsi W 30 a significant part of the commission's report, although the commissioners were primarily concerned with post—secondary and higher education. Ashby Commission stated in its report submitted in 1960:27 Education is a seamless web; although the primary and secondary schools lie outside our terms of reference we cannot disregard the fact that some 80,000 teachers are seriously deficient in general education, especially in spoken and written English. Accordingly, our recommendations include not only proposals for increasing the numbers of teachers, but also proposals for improving the quality of teachers already in Nigerian schools. The problems which face the country over teacher—training are very formidable indeed. Still more specific to the present study is the report of the Banjo Commission appointed to review the educational system of Western Nigeria in December, 1960 as a result of the earlier nationwide review. As it was the case in the Ashby Commission Report, teacher training was also at the leart of this review. "Evidence shows that all types of edu— :ational institutions in the region lack adequate, stable and qualified academic staff. To remedy this alarming situ— "28 The Com— ltiOh, the most urgent measures must be taken. fission made many recommendations and suggestions of signifi— =ance to the improvement of primary education. In particular, Federal Ministry of Education, Nigeria. Investment n Education. Report of the Commission on Post—School Certi— icate and Higher Education in Nigeria (Lagos: Federal Minis- ry of Education, 1960), (Ashby Commission Report), p. 15. 28Western Nigeria Government. Report of the Commission ppointed to Review the Educational System of Western Nigeria Ibadan: Government Printer, 1961), (Banjo Commission), p. 49 , h rf l" . ,- Fn n- ,. (1 f. :1' o In to: precede a‘Rircpriate an already iiSSion‘s ; leather tr.- ular the m erS accord In the fir and it Wou teachers w retommema liShment C m 31 it emphasized that:29 All these things, however, would be value— less without an attack upon the greatest of all weaknesses in the schools of the Region, namely, the use of ill—educated and unqualified people as teachers. Primary schools in which two—thirds of the teachers are unqualified and in which many have nothing more in the way of education than their own years as primary school pupils, are scarcely worthy to be called schools at all. To remedy this situation, the Commission's recommendations included the complete replacement of untrained teachers by :rained personnel, the introduction of a system of promotion for conscientious teachers, the improvement and reorganization )f the system of supervision. No doubt the rapid expansion of the mid 1950's which was [0t preceded nor accompanied by a corresponding increase in ppropriate teacher—training programs contributed to lower n already low standard of education. Some of the Banjo Com— ission's recommendations resulted in the initiation of some eacher training improvement programs, while others, in partic— lar the recommendation on the replacement of untrained teach— rs accordingly received many criticisms for being unrealistic. n the first instance, the trained teachers were not available nd it would have been unethical to dismiss the untrained eachers who served the society when they were needed. The ecommendations of the Ashby Commission also led to the estab- ishment of some refresher courses aimed at improving the 29Ibid., p. 11. caality 0: -r 215:2- ,_;- pritary sc Grace Ill, but uncert handicraft was better certificat primary te Percent 01 for Primal refleet 01 tionefi ea is becomi \30 An training 31 . Ca thSI Op‘ 32 {uality of teaching in the primary schools of the state. As a result of the Banjo Commission's recommendation, Lhe Grade III teachers' course was abolished in the early 960's and a two—year residential training program to up— rade this group of teachers to Grade II has been mounted ince then. Though the Grade III teacher training programs ave been terminated, the products of these early programs till linger on in the teaching force and their poor quality 5 still well reflected in the primary classrooms of Western tate. The 1971 Annual Digest of Education Statistics, for xample, showed that out of a teaching force of 27,016 in its rimary schools, 34 percent were Grade II, 15 percent were fade III, 38 percent were untrained, 10 percent were trained 30 1t uncertificated, 15 percent were special vernacular or indicraft teachers, and one percent had qualification that is better than Grade II status.31 The Grade II Teachers' artificate is the official minimum qualification for the rimary teacher in Nigeria. In effect, a total of about 63 arcent of all the teachers have inadequate qualifications >r primary school teaching. These 1971 figures seemed to eflect only a slight difference from the 1959 figures men— oned earlier. The adequacy of the Grade II teachers course becoming more and more questionable especially when one 30An uncertificated teacher went through the program of aining but failed the Teachers' Certificate Examination. 31Calculated from: Annual Digest of Education Statis— cs, op. cit., p. 43. considers t1 :0 C318 . —. i— a. ...._= .. ~--..\. - "' C?“ u a .. '34: ..... v ~.. éard in th( port state: i the e( SChoo ised . the p‘ ConCej a lit Educa of le the t team 33 siders the changing curriculum of the 1970's and the years come . nary School Instruction The statistics on the categories of primary teachers ad in the last section has great bearing on the type of :ning environment that exists in the Western State primary ssrooms. Evidence abounds in writing, speeches and dis- ;ions that point to the dissatisfaction which many people :he state have with the present primary education system. 'ly thirteen years after the introduction of the six—year : primary education in the region, the Taiwo Commission set up to review the educational system. In unequivocal s, the Taiwo Commission put the cause of the high wastage and the serious deterioration in primary education stan— in the poor quality of teachers. The Commission's re- states:32 We are in no doubt that two major reasons why the educational objectives implicit in the primary school curriculum in the west are imperfectly real— ised are deficient knowledge of subject matter on the part of too many teachers and an inadequate conception of the teaching function....We were not a little disappointed to find how little primary education has been touched by findings on psychology of learning, the mental development of children, the teaching of languages, the formation of skills, team teaching, etc. 32Western State of Nigeria: Report of the Committee on {eview of Primary Education System in the Western State .geria (Taiwo Commission Report), Ibadan: Government .er, 1968, p. 11. '7er ‘ '. _ u..- we : -:S.:.. H.- — a. ‘f‘ ere-'2 1;;- ._.).. 5- .. - 1...-.-.:- —..1 .1..-- » vh- ...‘.. A : I---“ 3, -.. -v: .. a- \ un‘ ........ - “Ana...” C ~:.~-u. ‘l _ u.“ t ,. .»~-~. _. “M.V. —;_ .......:. __.: u- .« :“~ -usc ...... .- —“Y: .:- ‘r- “11 ‘32“... hot Challem o'er. Whel blmary Edul ”‘70“? Other llSCipline Y 33 in hes - C :d tern ”1%, 34 A closer look at the instructional program shows specific es related to the learning environment. The instructional ram places emphasis on English language, Yoruba language Arithmetic. Other subjects taught include religion, hand— ing, history, geography, nature study, health and physical hing. Social studies, as a substitute for history and faphy, science and mathematics are now taught in a few )15. English is the medium of instruction in the upper -s. The first two or three years are usually taught in la language. The First School Leaving Certificate, a a1 examination given by the state, marks the end of the ear course. Much of the teaching is done with an eye on the syllabus he First School Leaving Certificate Examination. The 31 learning environment is very formal. The relationship an the teacher and the child can be indicated by the word )ritarian." The teacher demands a formal and automatic :t from the child. The children are hemmed in by admoni— They are passive and not active learners. They are allenged to reason reflectively, to observe and to dis— When Calcott33 investigated the causes of wastage in y education in one of the provinces in 1967, he found, other causes, that "Some children do not like the severe Line which is often administered by teachers -- particu— 3D. Calcott. "Some Trends and Problems of Education :ern Nigeria, 1955—1966.” West Africa Journal of :23, Vol. XI, No. 3, Oct., 1967, pp. 128-135. I :er- " 1... lCOHJ -... 5 -‘.;~ 5 ’5: in...“ --__ u. 5- An ex in acme i ackmwledq Subject, 5 \34Eril Developmez 35 arly to new enrollments -— in order to get the children to :onform'. These children then beg their parents to let iem stay at home." Furthermore, most instruction is characterized by routine zaching and rote learning. The common teaching aids are the .ackboard and chalk, the latter being in short supply in some 'hools. Materials of instruction, including textbooks for pils and resources for teachers, are inadequate. Someteach— s attempt to improvise teaching aids. Unfortunately, some dels improvised by teachers are distorted to such an extent at pupils are led to incorrect concepts. In general terms, e poor quality of teaching in these primary classrooms had an succinctly expressed in the 1967 report of a study by 3 Education and World Affairs Committee on Education:34 There is a gross efficiency in the primary educational system. Sixty—three percent of those pupils who entered the primary cycle which terminated in 1965 did not complete that cycle. The quality of instruction at the primary level is notoriously poor. CURRENT MATHEMATICS PROGRAMS: PRIMARY SCHOOLS AND TEACHER TRAINING COLLEGES An examination of the school mathematics programs shows acute instructional problem. Mathematics is universally nowledged in the Western State as being a very important ject, and yet it is a subject that many pupils dislike 34Education and World Affairs, Nigeria Human Resources alopment and Utilization, New York: EWA Committee on :ation/USAID, 1967, p. 130. and fear. "00!, 37.3 I a the Teacher university with Primary sc‘ mathematic Indeed, it hetic, YEt hrimary sc The p *5 (if SECOnda Gary Level We: SYllabuS 36 id fear. The results in external examinations are generally >or, and very few students pursue the subjects through to agree level. Indeed, given thewholepopulathm10fthestate, 11y 83 graduate mathematics teachers were reported as teach- .g in the secondary schools in 1970.35 Of these, a substan— al number did not hold a degree in mathematics. Many hold grees in agriculture or economics, but since they studied thematics up to Higher School Certificate level and since e schools lack any teacher of mathematics or have too few ch teachers, the graduates in these other disciplines are ed as substitutes. The situation is particularly acute in a Teacher Training Colleges where only a few teachers are iversity graduates. -mary School Mathematics Program With insignificantly few exceptions, many of the state's ,mary schools are still teaching from the same or similar hematics curriculum which they taught from two decades ago. eed, it is doubtful if mathematics, as opposed to arith— ic, yet exists at all in the vast majority of the state's nary schools. The present arithmetic syllabus36 shows two striking 35A. Adaralegbe. "Western State," Report of the Supply Secondary Level Teachers in English—Speaking Africa, Secon— 7 Level Teachers: Supply and Demand in Nigeria, Hanson, (Project Director), East Lansing: Institute for Inter- .onal Studies in Education and African Studies Center, ‘, pp. 36—54. Western Nigeria Ministry of Education, Primary School abus (Ibadan: Government Printer, 1954), pp. 36—48. @J—i .— ..m-n~vu 1 pence-.. basic com: 50: 6: bett. 90 a long x the Primarj Ihel’more’ functional Primary ed It is learning 1 PIOgram, m Tables are ding to In almoSt fro 37 ficiencies: narrowness of scope and lack of a clear state— nt of objectives. Arithmetic is mathematics, but it is a aset of the body of knowledge known as mathematics. There— :e, the narrowness of the arithmetic program is a matter of icern if the state's primary school mathematics program is offer a solid foundation for the type of mathematics edu— ion which is essential for a society aiming at scientific technological development. The second defect is a matter pedagogy. The role of clear instructional objectives is amount in any learning—teaching situation. It is par— Jlarly needed when teachers are inadequately prepared for ir jobs. An enriched mathematics program, including some Lc concepts of algebra and geometry, will not only make a better understanding of geometric patterns and objects , are present in the child's everyday experience, but also long way to provide the hitherto missing link between primary and secondary school mathematics programs. Fur— nore, such an enriched program would provide a more :ional mathematics education to those children for whom iry education is terminal. It is in arithmetic, more than any other subject, that ,ing is a little more than rote memorization. In this am, mathematics is taught in a purely mechanical way. 5 are memorized and operations are carried out accor— to rules which are seldom explained. Children embark, : from the start, on learning the two times table up to 7' lleal and Piper need is totally some Of UT §IEat Shec rectly wor 0f matheme the demant’ ducts of t and learne to have (if 38 x 12 and then proceed to the three times table and so on. e relationship between 2 and 4, 5 and 10, and among l2, 2, 4 and 6 are seldom pointed out or made use of. Sums are [ually mechanical and it is usual for all the four fundamen— ll operations to be learned and practiced mechanically before 1y of them is used for any practical purpose. When they are zarned, they are applied, equally mechanically, to length, :ea, volume and money. Until recently, the Nigerian money rstem made the process laborious, but the recent adoption of decimal system should eliminate this difficulty if teachers 1ke the necessary transition. Finally, after the long course of mechanical learning and )rking, the "Problems“ are introduced. Most of these pro- .ems are still mechanical sums wrapped up in words rather an being practical problems. They are often concerned with real and irrelevant situations such as finding the cost of per needed to cover the walls of a room, a situation that totally absent in the child's environment. In general, e of the children learn these mechanical processes with eat success and produce exercise books of neatly and cor- Ltly worked sums without ever acquiring much understanding mathematics or seeing the link between the subject and 2 demand for correct change in the market place. A few pro— :ts of the system have progressed beyond the primary school i learned some real mathematics. However, a good many seem have developed a mathematical phobia which may last for ~ 4 ""6 [657. O. -eache:s :2 . - “I hi‘ .I. r... imam The syllabx 10 have b6! :etic to lg Sk'llabus h, Colleges. COnsiderab ComtempOra: the need f( The m tion Counc 37Uke LagOS: Mal . Wes affllnatiOnS 0f Ni‘JEria 39 :he rest of their lives. It must be said that although a few :eachers have done some good work of arithmetic teaching, yet :he description of the learning climate above generally holds. ’rofessor Ukeje's comment on this learning climate almost ten ears ago seems to hold presently. He wrote:37 The schools impart knowledge with little re— flection; they teach the students what to think but not how to think; they tell them what to know but not how to know, and the students acquire knowledge with little understanding; they learn to memorize but not to digest, to repeat but not to reflect, and to adopt rather than to adapt. 1e Teacher Training Mathematics Program The mathematics programs of the Teacher Training Colleges ffer similar deficiencies as that of the primary schools. e syllabus is an "Arithmetic Process" program and appeared have been designed to give the teacher just enough arith— tic to "give" to his pupils. A new "Basic Mathematics" Llabus has been recently introduced in the Teacher Training Lleges. This new syllabus is wide in scope and contains a Isiderable amount of the new materials called for by the [temporary reforms in mathematics. There is evidence of :need for a continual revision and adaption of the new labus. The new syllabus, published by the West African Examina— n Council,38 is an optional alternative to the "Arithmetic ‘ 37Ukeje, B.O. Education for Social Reconstruction, bs: Macmillan and Company (Nigeria) Ltd., 1966, p. 79. 38West African Examination Council conducts central ex— ations on behalf of the Federal and States' Governments igeria and some other West African countries. Process" In . a... .-... M" -- -x. .H - ‘ ~7QA‘: ‘ -. C-Gu _ .‘d-A-n .- -4 k "a u ......c.. a ‘. “a \ -.. “mt, - -~-. . -:»-.S as ( if this 51 :eachers t he éeficie Ram at th :etic as a mu1&8 are applied Wi tion of 017 the Comple preOCCUPat 4O >cess" in the annual Grade II Teachers' Examination as from 'l.39 It is planned that the new syllabus will become a lpulsory program for all Teacher Training Colleges in the .ntry probably from 1976. However, less than five of the Grade II Teachers' Colleges in the Western State sent in didates for examination in the new mathematics syllabus June, 1973.40 One or two other colleges planned to intro- e the Basic Mathematics syllabus in September, 1973, while t of the others had no definite plan at that time as to n they would offer courses to teacher—trainees in mathe— ics as opposed to the arithmetic course. An implication :his situation is that for some years to come the new :hers trained in these colleges will most probably still leficient in their mathematical background. There are also grave pedagogical limitations in the pro— 1 at the teachers' colleges. The trainee is taught arith— c as a series of facts to be learned independently. For- 8 are learned with or without deriving them and rules are ied without understanding why they work. The demonstra— of one or two algorithms on the blackboard followed by completion of similar examples by the trainees is a usual ccupation. The main approach is through "talk and chalk". 39West African Examination Council, Teachers' Higher entary (Grade II) Certificate Examination: Regulations Syllabuses (Lagos: Nigernews Publishing Company, Ltd., )1 pp. 20—35, l8—l9, 47—55. Privileged source. ‘ '5 S: I - ~v , 2": :-:.. .- -‘-- -..- ,._._ ‘.~-‘y.. \ firs..- .. V—‘n .. n M... a hQ British T , matefials SChools a Oblectiye ““5. it \ 41 J B. Ournal N 41 However, even with the type of new program described, little will be gained unless the teacher training procedures used are also modified. Evolving Programs for the Primary Schools Although the present school mathematics program in the Western State is still very traditional in nature, some attempts have been made during the last decade aimed at re— forming the mathematics curriculum. Significant among these is the African Mathematics Program of the Education Develop- ment Center, an American—based research foundation. The program was inaugurated in Accra, Ghana, in 1961 with a policy of bringing together Africans, Americans and British Educators in English—speaking African countries in order to influence mathematics education in Africa.41 To achieve its objectives, it organized writing workshops in Africa which produced the Entebbe Mathematics series. Between 1962 and 1969, the African Mathematics Program conducted an— nual eight—week writing workshops in Entebbe and Mobasa for resource persons and produced over 80 volumes of textual materials. Among these are experimental textbooks for primary schools and teacher training colleges. Although the primary )bjective of the workshops was to produce good mathematics exts, it was hoped that participants at the writing workshops 41B.O. Ukeje. "The Entebbe Mathematics," West African 23£E§i_9f_gducation, Vol. 9, No. 1, February, 1965, pp. 15—18. primary 1 “P a seri labuses c The Maths mathematj as HaWES 0“ Paper. learning her relat 42 E( D t w 42 >uld develop interest in writing and use their experience > assist in national curriculum change. The program, however, left its impact on Western State L the form of the produced texts for schools and some ad hoc service training programs which were carried out through e Institutes of Education in Ibadan and Ife. Most of the achers in these inservice programs were secondary school teacher training college teachers and only a few were 'mary teachers. A 1970 report of the foundation stated, 1 of the participating countries have accepted the idea at 'modern mathematics' must be introduced into their 1001 curricula, although just how this should be done may : be clear.”42 This is true of the Western State as it is some of the other countries. Cognizant of the need for a curriculum reform at the mary level, the Western State government, by 1972, had set a series of syllabus committees to review the existing syl— uses of the primary school and suggest a new curriculum. mathematics curriculum panel had completed work on a new iematics syllabus proposed for the primary school. However, iawes asserted, curriculum change depends on people, not >aper.43 New approaches to mathematics, involving the ning of new concepts and an entirely new approach to num— relationships will have to be learned or relearned by many 42Education Development Center. African Mathematics ram: A Report (Newton, Mass., June, 1970), p. 3. 43H.W.R. Hawes, op, cit., p. 21. RI-i‘AF ‘ S "5.4.... F "V; ”(2.”... - v 1 M..- 3“: :c-.. ' a - .: ~~n~c~.- C y ........ wizh tion a: t instructi ? :1) teach \ rials of teachers . One in the p, qualifiec educatior Was made trained 1 \44 £ 43 teachers. Even in situations where teachers' general edu— cation is regarded as being adequate, such a new mathematics program necessitates that teachers' knowledge of mathematics be brought up to date. In a situation where teacher's gen- eral education and professional training are inadequate, as it is in the Western State, the need to train teachers in the content of mathematics and teaching methods becomes greater, if quality mathematics teaching is desired. The problems of training and retraining of teachers and other factors affec— ting quality of instruction are highlighted in the nextsection. PROBLEMS OF IMPROVING INSTRUCTIONAL QUALITY With few exceptions, the problems of mathematics instruc- tion at the primary school level are also problems of general instructional quality. Among these fundamental problems are: (1) teacher preparation and conditions of service, (2) mate— rials of instruction and (3) supervision and retraining of teachers. Teacher Preparation One reason for the large number of untrained teachers 'n the primary schools was the lack of an adequate number of ualified personnel to meet the demands of universal primary ducation in the 1950's. Through the years, some progress as made as Taiwo Commission reported a good 83 percent rained teachers were in the primary schools in 1966.44 It 44Taiwo Commission Report. op. cit., p. 8. ~ mar-i c ; cC:.......- ers" 0- - -':,‘ . ZECEIel--x :reineé :5 335, there Your Teac} school tea are some 6 Cess Whic} Just mary level the lack ( ing COlleg do not re: USe of ne‘ COHceptS prOgrammeC hive the \ 45 . Ed: “IQ, N( __ ,_.-~r- :-.\.. 44 must be recognized, however, that the fact that a teacher was deemed trained does not guarantee an individual with a broad academic background. In addition, many of the "trained teach— ers" of 1966 were Grade III teachers, while some others were uncertificated Grade II or Grade III teachers. A trained but uncertificated teacher is one who, although completed a course of training at a teachers' college, failed to obtain a certi— ficate because of unsatisfactory performance on the leaving examination. Thus,the actual progress made in the quality of trained teachers in the mid 1960's was in fact minimal. It was, therefore, not surprising that in a 1973 "Project Know Your Teachers" the state reported that 52 percent of primary school teachers were untrained.45 It seems clear that there are some embedded problems in the teacher preparation pro- cess which should be reorganized. Just as the quality of teaching is impaired at the pri— nary level because of poor quality of the teachers, so does the lack of qualified teachers create problems at the train- Lng colleges. The courses offered in the training colleges lo not reflect the changes in the syllabus which involve the se of new teaching approaches. "The sweeping changes and oncepts in modern language learning, the new mathematics, rogrammed learning and elementary science all of which in— olve the use of Audio-Visual materials, are unknown in our 45Editorial Comment. "Sending Nigeria to School," West rica, No. 2962, March 25, 1974. .. ',. ,. CE:C..€-S. ' .n ” r... .. c 3 '7‘”« H 3 L... V:— The p the social Children 5 Curiosity, instrucmc are needed acute Shor of the sta This Short agreat ma ideas. 463. Teacher T1 ment print ea Results 8} Calculate( 1.’ ' ' -._\\:—- 45 training institutions," wrote a Nigerian teacher.46 The training colleges themselves need a corps of properly trained teachers. In addition to the problem of the teacher trainer, other problems plaguing teacher preparation for primary’schools are those of poor recruits, poor training programs, high failure rate47 and poor rewards. The problem of reward, coupled with some other complex causes in the society have contributed to the general low morale of the teachers, which in turn affects the quality of teaching in the classroom. Materials of Instruction The physical surrounding for learning is as important as the social climate provided by the teacher and a child's peer. Children need a stimulating environment, designed to pique curiosity, to entice and to challenge. Several materials of instruction including textbooks and various teaching aids are needed for such supportive physical environment. The acute shortage of these teaching aids in many primary schools of the state is a great constraint on instructional quality. This shortage is a handicap for mathematics teaching, where a great many concrete aids are needed in simplifying abstract ideas. 468.0. Olukoya, "The Functions of Audio-Visual Aids in Teacher Training Colleges," Teachers' Forum, (Ibadan: Govern— ment Printer, 1974), Vol. 2, p. 5. 7Teachers' Certificate Examination Grade II, June, 1973. Results showed: 53.6% pass, 35.3% referred, ll.l% failed. Calculated from Teacher Forum, Ibid., p. 12. 3. . A ._ "CD_.:L. - -. _‘: - K': Van. sum; _ ‘vu “:1 ‘ ..-.,..:-.,.: in; “M... n'..‘..,. 3 v x..-» ‘;:e:vwcgo; Ihnv “1: shoes leade tiona mest Offic. educa The Commis Should be itthe pri usiSt Cla 4\e UNE matiCs Con 49 . Tal 46 Attempts are being made to improve the present curriculum for the primary schools. However, the issue of the central and external examination at the primary level will have to be resolved in order to remove a formidable obstacle to instruc- tional quality. An examination does not have to foster rote- learning, as some writers maintained.48 To serve its purpose, examinations should be used to check if the objectives of in— struction have been achieved, not to check memorization. Any design of a new primary education must also plan for a new way of assessing the program. Supervision and Retraining i_____.____________________ Many authors have written copiously on the shortage of qualified supportive administrative personnel for the primary . . . . 49 schools. The report of Taiw0 CommiSSion in 1968 stated that: The Inspectorate for reasons not of its own Choosing is ill—equipped to provide the kind of leadership which an efficient and dynamic educa- tional system demands. In this connection, the most glaring omission is the lack of a senior officer whose full-time preoccupation is primary education. The Commission added, "We strongly urge that the position ¥hould be remedied." In order to aid quality of instruction t the primary level, the school inspector must be able to SSist classroom teachers, in particular the unqualified ones, . 48UNESCO/UNICEF. The Development of SCience and Mathe— atics Concepts in Young Children in African Countries. airobi: UNESCO/UNICEF, 1974, p. 70. 9Taiwo Commission Report. op. cit., p. 21. (.116 ”E l “u, i‘.‘ \. to m nan; Still aim both spec as Englis have reac matiCal t \50 J . January, * —_( while he is at work in the schools. He should be able to assist in the development and introduction of new teaching techniques and related teaching aids. Unfortunately, the cadre of school inspectors that can effectively perform such roles are still being developed. The need for retraining of teachers has been emphasized all along, though the lack of it continues to be one of the obstacles for instructional quality. As far back as 1959, J.O.A. Herrington, then of the Education Serivce of Western . . 50 Nigeria, remarked: Refresher courses are not new things, but we feel that many of the refresher courses that have been held in the region are not particular- ly valuable. They are too diffuse, and they try to do too much in a short time. We want to over- haul the system. The Ashby Commission in 1960 also recommended vigorous efforts to raise the quality of work in the schools, in particular in the primary schools. This recommendation led to the be- ginning of an annual refresher course for teachers. The Ashby Courses, now known as Teachers' Vacation Courses, aimed and still aim, at giving participants short refresher courses in both Special and general methods of teaching in subjects such as English, Arithmetic and Science. These refresher courses have reached only a few teachers and do not cover many mathe— matical tOpics. Although modest attempts are still being 50J.O.A. Herrington, "Problems of the Inspectorate in Western Nigeria," Overseas Education, Vol. XXX, No. 4, January, 1959, p. 179. .......... 1- in (1' m 1' I" " (D U!” () r m I D; In. {2"(1 & m (D (I) () 0 (D (I. r) (J. (D r'-.() keelel 3' K Stood and First, sc are relex and dynar group to l)’ to ma‘ needs to SitUatio‘ in the 1 knowledg \ 51A 48 ,de to reorientate teachers, the task of upgrading the know— dge of teachers who have insufficient education is still itically needed. After a lengthy appraisal of the teaching tuation, the Lagos Curriculum Conference made this recom— ndation:51 Inservice training and retraining of teachers at all levels must be embarked upon on a continuous basis with a mind to improve teacher/classroom ef— fectiveness and to encourage him through further incentives for additional experience gained. Pro- spects for further training should be built into the teacher-education programme and this should be adequately compensated for or remunerated as an additional incentive. acial Needs of Mathematics While much of the above is common to all subjects, there a certain features which make mathematics a peculiarly sen~ Live subject and this nature of the subject should be under- ‘od and appreciated by all teachers teaching the subject. st, social and cultural experiences, including language, relevant to the reform of mathematics teaching. Spatial dynamical experiences, for example, are different from up to group. Yet, these experiences contribute tremendous- to mathematics learning that some form of each experience is to be provided for in order to enhance learning in some lations. Secondly, mathematics has developed rapidly with— :he last ten years that most teachers need to upgrade their 'ledge of it. Thirdly, contemporary research has shown 51A.B. Fafunwa (1974), op. cit., p. 238. that child certain sf :a‘:'ne:.ati< 5:325. a: effective: teasing : results i ation of ject. ye 0f matherr It is, th and Viabl Sim educat 101 ment in (- “tional trends i Nigeria 49 :hat children are ready to learn mathematical concepts at :ertain stages of development. This is an important area of lathematics learning—teaching which all teachers must under— ;tand. Lastly, there is the need to use and consider the effectiveness of simple teaching materials and apparatus in :eaching mathematics and teachers need to be trained in the se of some of these. These special needs of mathematics make the retraining f teachers for mathematics teaching more significant, espe— ially teachers with deficient mathematical background. The roblem of mathematics instruction in the state tends to be elf—perpetuating. A shortage of good qualified teachers esults in poor teaching, which in turn results in the alien— tion of the next generation of school children from the sub- act. Yet, no major impact may be made on the whole standard 3 mathematical education without involving the primary schools. : is, therefore, necessary to effect change in the present *imary school mathematics through both viable preservice .d viable inservice courses for primary teachers. PROMISING DEVELOPMENT Since the possibility of any fundamental improvement in ucation and training of teachers (like any marked improve- nt in any segment of the plant, plan and personnel of edu— :ional system) is closely interrelated with large scale ands in all sectors of a nation, recent development in reria holds some promise for primary educational development izbalance ‘i‘ - 3—.£-Lcj c-r,..- ache-We al-»~ educatioz Ano: plans to primary Service, been inc three ti Udoji Co quality Schools. will be of the p in the F \\\\33‘ F Nigeria. EView C \\\\33‘ 50 and improvement. In its attempt to rectify the different imbalances in education, the Nigerian Federal Government has committed itself to nation—wide universal primary education scheduled to begin in September, 1976. The plan for expan— sion has resulted in another appraisal of the existimgprimary education system in every state of the country. Bearing in mind the lesson from the past period of expansion in some parts of the country, the government is making plans for qualitative improvement amidst quantitative expansion in order not to further lower an already inadequate level of primary education. Among the measures being taken in this direction are >lans to upgrade the academic and professional standards of >rimary school teachers and to improve their conditions of :ervice. The financial remuneration of primary teachers has >een increased beginning with the current fiscal year to about hree times the old salary after the recommendation of the doji Commission.52 These new developments are necessary if uality instruction is to be provided in the primary chools. In the words of Professor Beeby, "Reformers' efforts ill be largely wasted if salaries and conditions of service E the primary teachers are not such as to retain good people 1 the profession."53 Federal Republic of Nigeria: The Public Service of .geria. Government Views on the Report of the Public Service :View Commission (Udoji Commission), December, 1974, p. 20. 53C.E. Beeby, The Quality of Education in Developing untries. Cambridge, Mass: Harvard University Press, 1966, 130. the stand ‘. "“1 In Jaimng Struction :orale of eaotiona‘. pation "n training phases wi Particula Anot for teach ducatior The State which is the unive the Teacf forum fox tion of E educatim Operatiw AfriCa.51 \S4AU Spring 0.: (A'A-A.) 51 It is, however, necessary but not sufficient to raise the standard of intake into preservice programs in teacher training colleges in order to maintain quality in the school ystem. A systematic retraining and retention of those eachers already in service is also essential to quality in- truction. The new developments are needed to raise the orale of practicing teachers and provide both financial and motional incentives which are conducive for their partici— ation in inservice training programs. These systematic re— :raining programs could be organized in different stages or bases with emphasis on a particular subject matter or a articular teaching—learning skill at a given stage. Another development of significance that holds promise or teacher retraining is the growing communication among iucational bodies both in and outside the Western State. 1e State University at Ife houses an Institute of Education, ich is governed mainly by three educational bodies, namely: e university, the Western State Ministry of Education and e Teacher Training Colleges.‘ This group could provide a rum for coordinated effort and plans aimed at the re—educa- on of primary school teachers. Similarly, Western State ucational bodies are participants in an international co— erative effort oftheAssociation for Teacher—Education in ica.54 Among the objectives of the organization are:55 54Association for Teacher—Education in Africa is an off— ing of The Afro-Anglo—American Teacher Education Programme .A.A.), an association established in 1960 and financed by educatio In educatic ing of 1 in scho< Of the X Primary program Structi m irePrESe COIUmbi Ministr College 55 52 l. to strengthen the teacher—education programme in Africa through regular annual conferences of directors of institutes and others inter— ested in African teacher—education programmes, for example, the Ministries of Education, teachers colleges in Africa, the United Kingdom and the United States; 2. to conduct research and promote the exchange of information among the participating members from the three continents. The activities of such an organization, especially the pro— motion and exchange of ideas on teacher education programs could further indicate the needed professional cooperation among educators and educational planners in periods of rapid educational changes. SUMMARY In this chapter, the setting for in—service mathematics education has been described. Since the professional train- ing of teachers is closely connected with what is happening in schools and what is planned for schools, the description of the Western State setting covered: (1) the development of primary education, (2) the development of teacher training programs and their inadequacy, and (3) the problems of in- structional quality. the Carnegie Corporation of New York. Educational bodies represented in the association include: Teachers College, Columbia, the University of London Institute of Education, Ministries and Institutes of Education and Teacher Training ‘Colleges in some thirteen African countries. ‘ 55A.B. Fafunwa, (1974), op. cit., p. 202. Spec problens 53 Specifically, mathematics programs and instructional problems in mathematics were highlighted. Finally, the cur— rent social and economic events in the society that are con- ducive to primary teacher retraining programs were mentioned. *7. z: the ; school : :erhena: is rat‘ne helping concepts mathemat general allY! il’ they ma) the org; node of of SOCii Th from ea, from th tary Sc -—;:,v— 1W ' ¢ CHAPTER III THEORETICAL BACKGROUND AND RELATED RESEARCH There are three main areas of theory and research related to the problem of inservice mathematics education for primary school teachers. First, this study fits into the area of mathematics learning since the subject matter for retraining is mathematics. Secondly, the study deals with the issue of helping to shape teachers' understandings of mathematical concepts, their attitudes towards and their techniques of mathematics teaching. Therefore, the study fits into the general framework of inservice education of teachers. Fin- ally, in order for teachers to demonstrate these competencies, they may need to change not only their own behavior, but also the organization and management of their classrooms, their mode of interaction with the learner, and perhaps the system of social norms in the school as well. This chapter will discuss the related theory and research from each of these three areas, including any significant work from the setting of study. SOME LEARNING THEORIES AND ELEMENTARY SCHOOL MATHEMATICS INSTRUCTION Extensive research into the various aspects of elemen— tary school mathematics has been reported not only in 54 oublisheé monograp! of goven sides of on learn: teaching :athenat as well-1 which he Theories tion in has prov ties inv teacher 0t P5ych the part In ChOlOgiC \ 56 E Under ti (3) BUSV (6) Cro: 57. 58 In: E( 0f the} 55 published and unpublished research but also in journals, monographs, periodicals, professional books, and publications of government and learned societies. Several authors on both sides of the Atlantic have, at various times, presented views on learning theories for the study of mathematics learning, teaching and understanding.56 "The days when a teacher of mathematics could shut his eyes to psychology, and dismiss it as well—meaning advice of which he had no need, or as opinion which he did not share, are gone," observed Fletcher.57 Theories of learning have influenced curriculum and instruc— tion in mathematics. However, no single theory of learning has proved robust enough to encompass the range of complexi- ties involved in mathematics instruction. Consequently, a iteacher of mathematics must borrow selectively from a variety of psychological theories those principles most relevant to the particular instructional decision he has to make. "58 In "Psychology and Mathematics Education and in "Psy— chological Controversies in the Teaching of Science and 56For further information see bibliographical entries under the names of: (1) McConnell, T.R., (2) Brown, K.E., (3) Buswell, Guy T., (4) Wallace, J.C., (5) Weaver, J.E., or (6) Crosswhite, F.J., et al. 57 ' T.J. Fletcher, Some Lessons in Mathematics, Cambridge niversity Press, 1964, p. 2. Lee s. Shulman, Psychology and Mathematics Education, n: E.G. Begle (ed.), Mathematics Education, 69th Yearbook f the National Society for the Study of Education. Chicago: Diversity of Chicago Press, 1970, pp. 23—71. Vathemat: i sents tht :atics a: matics 1' 0f maths when tre COnstam On the ; ies hav: moSt of thEorie W in w \59 Teachin (Septem 56 59 Shulman has presented and analyzed what repre— Mathematics," sents the current Western thinking on the psychology of matae- matics education. The literature review at this point centers around some theories and studies on cognitive aspects of mathe— matics learning and instructional process, especially on the issue of their applicability in non—Western culture. Studies in Cognitive Development and Mathematics Learning "We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of develop— ment....The task of teaching a subject to a child at any particular age is one of representing the structure of that subject in terms of the child's way of viewing things." Bruner, 1963, p. 33. Although the universality of the subject matter of mathe— matics is generally accepted, the universality of the theories of mathematics education calls for a re-examination in an era when transplantation of ideas and theories in education is constantly being questioned. The great bulk of the research on the development of cognitive skills and the derived theor- ies have been carried out with Western children. Similarly most of the mathematics curriculum building based on the theories of Piaget, Bruner, Gagne, Ausubel and others has gone on in Western educational systems. Sometimes such curricula _______________ 5 . . . _9Lee S. Shulman, "Psychological ControverSies in the Teaching of Science and Mathematics," The Science Teacher (September, 1968): pp. 34-38. teach: stand: PSYCh< much ( the d had 9 learn gener mOre conee methc Part1 are 1: Piage and I 57 have been transferred, almost unaltered, to non—Western soci— eties; in other cases, they have been adapted to meet local needs. However, very seldom, if ever, has their fundamental, psychological, logical basis been questioned. Is it the case that the theories of cognitive development on which they rest stand up in non-Western contexts? Do the primary school children in Western State, Nigeria, for example, develOp cog- nitive concepts in the same way as the American children? If not, what are their modes of concept formation and what are the implications for mathematics curriculum and for the teaching of mathematics? Much progress has been made in this century in under— standing the learning of mathematics. The educational and psychological research into the growth of pupils' thinking, much of it carried out and inspired by the Geneva School under the direction of Piaget, Inhelder and their co—workers, have had great influence on the current thinking about mathematical learning of children. The work of Piaget not only embraces studies of children's general intellectual and logical development, but also the more specific studies of the growth of number and mathematical concepts. There have been several major publications on the method, results and underlying thinking in this field. Of particular interest to the teaching of mathematics among them are Piaget (1952a), Piaget and Inhelder (1956), Inhelder and Piaget (1958), Piaget, Inhelder and Szminska (1960), Piaget and Inhelder (1964), and more recently, Copeland (1974). .l . #_.__i. In E imortance he asserts which are called sex :‘omal ope in the sa; Pour Each “Y are 1 libration In t‘ (1952b) f at the in then the Cardinal Finally, Classes a vestigate (Piaget a first apt appears : clasS am ing of se 58 In The Origin of Intelligence, Piaget subscribes to the importance of heredity in intellectual development. However, he asserts that such inherited structures alone cannot ex- plain development. Other factors of development are found in the interaction of the endowment and experience in the environ- ment. Intellectual development proceeds through major stages which are continuous and fixed in order. These stages are called sensorimotor, preoperational, concrete—operational and formal operational. All persons develop through the stages in the same way, though not necessarily at the same rates. Four factors are necessary for transition between the stages. Phey are maturation, experience, social interaction and equi— Libration. In the study of children's concept of number, Piaget f1952b) first investigated the way in which children arrive m the invariance of wholes and conservation of quantity, and hen the problem of one-to—one correspondence leading to erdinal and ordinal meanings of number was carefully probed. inally, in the same study, theiway in which children combine lasses and numbers additively and multiplicatively were in— estigated. Piaget's work on the Child's concept of space Piaget and Inhelder, 1956), showed that topological space is irst appreciated by the child and that Euclidean space only Qpears later. While the study of the relationship between Lass and number is very appropriate nowadays with the teach— lg of sets in primary schools becoming widespread, the findings on great impli level. .Vany ( concept of :ated with rate ted 5 :he findin As do Lite the c occurrence it appears ever, the: because it EXiSting valuable when he e addI‘ESSQd 0f the Yc Etuk ted Yorul intervieh the the \60 entific W 61 cholo(lie 59 indings on the appreciation of the topological space have reat implication for geometry teaching at the primary school level. Many of Piaget's experiments on his theory of children's :oncept of number, material, space and time have been repli- :ated with Children in many countries. In England, Lovell :epeated some of the experiments and substantially confirmed :he findings of the Geneva School.6O As documented by Judith Evans, 61 experimental studies .nto the cognitive process of the African child are recent >ccurrences. Only a few studies had been done before 1960 and .t appears that the validity of most of them is doubted. How- :ver, there is a growing interest in this field, especially ecause it provides information about the universality of the xisting theories in educational psychology, and also provides aluable understanding of the child's background and abilities hen he enters school. A few of these recent studies have ddressed themselves to some aspects of cognitive development f the Yoruba child of Western State, Nigeria. Etuk (1967) conducted a Piaget-oriented study with selec- ed Yoruba-speaking children of Western State. A standardized nterview schedule based on Piaget's tasks was used to investi— ate the theory that conservation, seriation, andcflassification 6OK. Lovell, The Growth of Basic Mathematical and Sci— ntific Concepts in Children, London: London University Press, 61. 61Judith L. Evans, Children in Africa (A Review of Psy- ological Research), New York: Teachers College Press, 1970. sex diffe formed at tional hc 0f g ment and on the de young oh: and Afri ing cons reviewed sons sh 60 develop concurrently. Etuk also examined the relationship of the development of number concepts to intelligence, the ef— fect of sex differences on the development of number concepts, and differences in performances of children from modern and traditional homes. Her results generally supported Piaget's theory. Etuk found only a slight relationship between intel- ligence and performance on number tasks. She found slight sex differences and that the children from modern homes per- formed at a higher level than their counterparts in tradi- tional homes. Of great significance to the issue of cognitive develop- ment and the African child is a recent Unesco sponsored seminar on the development of science and mathematics concepts of young children in African countries.62 Emphasis was on "Piaget and Africa." Many published and unpublished studies involv— ing conservations and classification with African childrenwere eviewed and analyzed. Some studies of intercultural compari- ons showed that African children were behind Europeans in heir performance on the operatory tasks, but in no case were fforts made to check on the subjects' understanding of the Other intercultural studies, in which the method f indirect comparison were used found little or no difference etween African and European performance.64 The results 62UNESCO/UNICEF, Op. cit., p. 8. 63See bibliographic entries under: (1) Beard (1968), (2) eron & Simonson (1969), (3) Vernon (1969), and (4) Goldschmid t al. (1973). 4See bibliographic entries under: (1) Prince-Williams 1961), (2) Lloyd (1971), and (3) Ohuche (1971). interviews culture as Over their 0011. mung erl ethnograph erlle can solving, m conclude t faring use Anum 61 suggest that intercultural differences are least when the re- search involves: (1) subject from the elite class, (2) simpler task, (3) a flexible clinical method, rather than standardized interviews, and (4) an accomplished interviewer from the same Lulture as the third—world sample. Over a period of more than ten years Gay and Cole and heir collaborators have carried out series of investigations mong erlle children of Liberia.65 Their major tool has been thnographic analysis through which daily activities of the elle can be related to such cognitive activities as problem- lving, memorization, and rule learning. The researchers ‘nclude that cultural differences in thinking indicate dif- 2ring uses of general cognitive skills in specific situations. A number of studies have investigated mathematical con- pts development of the African child. Hill (1964) worked th Ghanian and American first grade children to look at the y they developed mathematical concepts. A trend in her data licated that initially American children responded correctly ordered sets, while Ghanian children responded correctly identical sets. Hill hypothesized that each culture has iuced a preferred concept and that knowing the preferred Ept of a cultural group is important in designing the rele- mathematical experiences. Traditional mathematical con— 3 have also been investigated in Nigeria (Taiwo, 1968), 5John Gay and Michael Cole, The New Mathematics and An Yulture (A Study of Learning Among the Kepelle of Liberia), ork: Holt, Rinehart and Winston, 1967, p. 30. Sierra Leone (0 Although most 0 fro: them shoul of mathematical in Africa are also enable te In all of matical concep ved problems p. sive. Among t1 and unfamiliar uaqe of commun the classroom is still the data, present wrOte in 1908 62 erra Leone (Ohuche, 1973), and in Tanzania (Mmari, 1974). though most of these studies were very exploratory, findings om them should be of-great value in investigating the types ‘mathematical concepts with which rural and urban children Africa are equipped before they start school. They would so enable teachers to determine how to extend the social and {sical experiences of children in directions which will help 2m to cope with the demands of formal mathematical thought. In all of the studies of cognitive development and mathe- :ical concepts related to the African children many unresol- ! problems present themselves and many data are not conclu— 'e. Among the unresolved problems are the issues of familiar , unfamiliar materials used for the investigation, the lang— e of communication, the determination of subjects‘ age and classroom application of some of the findings. While there still the need for further research to verify some of the I, present evidence seems to support what Inspector Cummings e in 1908 and what was further confirmed by Abiola,66 that African child has great potential if only he can be accordai 2r learning experience through better and adequately train- achers. Abiola added:67 It is not ability but inadequate environmental and educational factors that are responsible for the iifferential performance between the African and the :uropean children on European-imported skill patterns. 6E.T. Abiola, "Understanding the African School Child," Erican Journal of Education, Vol. 15, No. 1, February, >p. 63—67. Ibid., p. 54. fl Furthermore, if child, like any for that matter 'vay of viewing After a re‘ some non-Hester concluded that ‘ lowing in the c l. Curric method based; 2. The 1e of eac fully teachi take a 3. The 11' shoulc‘ Gator} Philp's conclus which aims in 1 change, but a1: strategies. M provided by th Instructional The focus rise to new ai --——_____.._._—— 68Hugh P tries-—Some P Howson (ed.) , ings of the E Education), C P- 178. 63 .hermore, if one assumes Bruner's assertion, the African d, like any child, should learn mathematics or any subject that matter if instruction can be presented to him in his of Viewing things." After a review of mathematical concept development in non-Western cultures, including some of Africa, Hugh Philp luded that particular attention should be given to the folv ng in the construction of mathematics curriculum:68 1. Curriculum should be process oriented, and methods should be heavily discovery learning based; 2. The learning processes and preferred strategies of each particular culture group should be care— fully investigated and the curriculum built and teaching-learning methods devised in order to take account of them; 3. The linguistic structure of the mother tongue should be analysed to determine the classifi— catory system. )‘S conclusion is especially relevant to the present study L aims in the long run not only to facilitate curriculum 'e, but also to help teachers change their instructional egies. More information on this type of curriculum is ded by the theory and research on instructional process. uctional Process in Mathematics The focus on how children learn mathematics has given to new attempts to create a more appropriate learning )8Hugh Philp, "Mathematical Education in Developing Coun~ --Some Problems of Teaching and Learning," in: A.G. 1 (ed.), Developments in Mathematical Education (Proceed— >f the Second International Congress on Mathematical :ion), Cambridge: Cambridge University Press, 1973, fl Furthermore, if child, like any for that matter Way of viewing After a re some non-Hester concluded that loving in the c l. Currie method based; 2. The 1e of eac fully teachi take a 3. The 11' shoulc‘ catory Philp's conclus which aims in 1 Change, but als strategies. Ma provided by th< Instructional The focus rise to new at M 68Hugh P1 tries-—Some P: Howson (ed.) , ings of the S Education) , C P. 178. 63 :hermore, if one assumes Bruner's assertion, the African .d, like any child, should learn mathematics or any subject that matter if instruction can be presented to him in his of viewing things." After a review of mathematical concept development in non-Western cultures, including some of Africa, Hugh Philp luded that particular attention should be given to the fol~ mg in the construction of mathematics curriculum:68 1. Curriculum should be process oriented, and methods should be heavily discovery learning based; 2. The learning processes and preferred strategies of each particular culture group should be care- fully investigated and the curriculum built and teaching-learning methods devised in order to take account of them; 3. The linguistic structure of the mother tongue should be analysed to determine the classifi- catory system. u's conclusion is especially relevant to the present study aims in the long run not only to facilitate curriculum e, but also to help teachers change their instructional egies. More information on this type of curriculum is ied by the theory and research on instructional process. actional Process in Mathematics the focus on how children learn mathematics has given :0 new attempts to create a more appropriate learning )8Hugh Philp, "Mathematical Education in Developing Coun- --Some Problems of Teaching and Learning," in: A.G. L (ed.), Developments in Mathematical Education (Proceed- »f the Second International Congress on Mathematical .ion), Cambridge: Cambridge University Press, 1973, environment to a workable th lated to four tent. That is viable theory personality, sequence and materials and examined teac characteristi tional modes The gene approaches to books: 11133} Instruction. 7] should have f: l. a 2. a o: 3. a 4. a o 69M. 0f Mathematic Sixty-Ninth Y Education, E. Chicago Press 7oJer Vintage Books 71 York: w.w."fiE 721m 64 vironment for children. According to DeVault and Kriewall,69 workable theory of instruction must account for variables re- ted to four sources: learner, teacher, instruction and con— nt. That is, in addition to the diverse learner needs, a able theory must account for teacher differences such as rsonality, education and roles; content variables, such as quence and structure; and instructional variables such as terials and equipment. The earlier part of this writing has amined teacher education and personality, and the learners' aracteristics. This section deals with the issue of instruc— onal modes relevant to mathematics teaching. The general psychological foundation of many of the new proaches to mathematics instruction can be found in Bruner's oks: The Process of Education70 and Toward a Theory of struction.71 Bruner states that a theory of instruction ould have four major features:7 1. a plan for motivating the learner; 2. a plan for helping the learner grasp the structure of a body of knowledge; 3. a plan for sequencing learning activities; 4. a plan which designates the types and distribution of rewards and punishments. 69M. Vere DeVault and T.E. Kriewall, "Differentiation Mathematics Instruction," in Mathematics Education, The ty-Ninth Yearbook of the National Society for the Study of cation, E.G. Begle (ed.), Chicago: The University of cago Press, 1970. 70Jerome Bruner, The Process of Education, New York: tage Books, 1973. 71 Toward a Theory of Instruction, New I r: W.W. Norton and Co., Inc., 1968. 2Ibid., p. 41. An examination tion has been assxmptions. assumptions an developed is " leaminq by di its origin dat however, captu ies curriculm sional and la involves the learner, and the questions No other relevance for discovery and as opposed to own strategie: as needed. I: answered exte the problem a Lectures and The guic‘ Gagne's them and how a bel the task. I‘ 65 examination of the literature shows that mathematics instruc- ion has been evaluated to a great extent against these major ssumptions. In addition, the term around which many of the ssumptions and much of the new psychology of learning has eveloped is "learning by discovery." Although the idea of earning by discovery has been greatly associated with Bruner, ts origin dates as far back as the days of Plato. Bruner, owever, captured the spirit of discovery in the new mathemat— cs curriculum and communicated it effectively to both profes- ional and laymen through his books. Learning by discovery volves the formulation of questions or hypotheses by the arner, and the development of his own strategies to answer he questions. Two other major theories of instructional mode that have elevance for primary mathematics instruction are the guided Lscovery and expository theories. Guided discovery approach : opposed to pure discovery is when the child develops his In strategies for answering the questions with teacher's help a needed. In the expository mode the problem is posed and swered externally and the child is expected to replicate e problem and solution or to work others which are similar. ctures and explanations are expositorily presented. The guided discovery approach is closely associated with gne's theory (Gagne, 1965). For Gagne, learning is a goal, d how a behavior or capability is learned is a function of a task. It may be by discovery, by guided teaching, by practice, by d: ing' and disco for Bnmer, it one cannot ms the fundamenta imortanoe of the terminal 0 its many ramif tional action matics . For t very much ass Ausubel' body of theo pository appr ery learning : ingful verbal great imports tion. Evidence as to which < est contribu for students is precious “995 teachir \T‘.‘ . Davi< looming in 1968), pp. #4; 66 practice, by drill, or by interview. The focus is on "learn- ing" and discovery is but one way to learn something. Whereas for Bruner, it is learning by discovery. Gagne believes that one cannot master more complex material learning tasks until the fundamental elements have been learned. He advocates the importance of doing task analysis which begins by specifying the terminal objective. His idea of learning hierarchy and its many ramifications for analysis and development of instruc- tional action is important for sequential learning in mathe- matics. For this reason, the method of programmed learning is very much associated with Gagne. Ausubel's expository theory stands in counterpoint to the body of theory which advocates discovery approach over the ex— pository approach. He maintains that the opposite of discov— ery learning need not be "rote learning," it ought to be mean- ingful verbal learning.73 Like Gagne, Ausubel emphasizes the reat importance of systematically guided process of instruc- tion. Evidence from empirical studies seems to be inconclusive s to which of the instructional approaches would make great— -st contribution to facilitate the mathematics learning process for students. Cronbach (1966), for example, stated: There -s precious little substantiated knowledge about what advant- lges teaching through discovery offers, andsunder what 73David P. Ausubel, "Facilitating Meaningful Verbal earning in the Classroom," The Arithmetic Teacher (February, 968), pp. 126—132. auditions thes other studies h reviewed some 0 roots on the fi tory sequences :ediate leami approach seems pared task pre expository app learning appro of transfer of findings sugge concepts and t valued outcome an integral pa eral, it appea discovery app: immediate lea: The them the mode of i1 1earning proct instruction 1 tea<3her to ca ”1“ eXpositc HOWeVQr l muc] 67 onditions these advantages accrue. Since that time, some ther studies have investigated the issue. Shulman (1968) eviewed some of these studies, and made some general state- ents on the findings. In general, guided learning or exposi- ory sequences seem to be superior methods for achieving im— ediate learning. On the issue of long-term retention, neither pproach seems to be consistently better. Worthen (1968) com— ared task presentation through a discovery approach and an xpository approach. One of his conclusions was that discovery earning approaches appear to be superior when the criterion f transfer of principle to new situations is employed. His indings suggest that if pupil ability to retain mathematical oncepts and to transfer the heuristics of problem solving are alued outcomes of education, discovery sequencing should be n integral part of the methodology used in teaching. In gen— ral, it appears from the published studies, that the guided iscovery approach has achieved best both at the level of nmediate learning and of later transfer. The theories and related studies examined here show that 1e mode of instruction is an important variable in teaching- earning process; so is the teacher. Success or failure of istruction is highly dependent upon the ability of the aacher to carry it out. Ausubel,74 for instance, points out lat expository instruction can be meaningful instruction. tWever, much of expository instruction in many classrooms is ______________ 74Ibid., p. 130. not meaningful the teacher, n can be said th the bands of 5 their role as which requires just skill de‘ need enjoyabl instill natur ularly mathem velop these 5 an inservice be most usef training pro Vice patterns THE( W BY inse Which teache fessionally 0f literatu: thmuqh the purPoses. and accordi d1 institut —‘—‘ 68 not meaningful. Ausubel maintains that that is the fault of the teacher, not of the mode of instruction. Similarly, it can be said that guided discovery can become rote learning in the hands of a less competent teacher. Therefore, teachers in their role as instructional facilitators assume a responsibility which requires some skills. The responsibility is more than just skill development for mathematics instruction. Children need enjoyable experiences which foster positive attitudes and instill natural curiosity about learning in general and partic- { ularly mathematics learning. Some teachers can probably de— velop these skills on their own. For a majority of teachers an inservice program that focuses on such needed skills would be most useful. More discussion on this type of inservice training program is provided next under the heading of inser— vice patterns and practices. THEORY AND PRACTICE IN INSERVICE EDUCATION Purpose and Philosophy By inservice education is meant, all of the activities in which teachers may engage in order to improve themselves pro- fessionally while they actually are teaching. An examination of literature shows that inservice training has been used all through the history of formal education to serve a variety of Purposes. These purposes vary from one society to the other, and according to the needs of the society, which the education— al institutions have to respond to at a particular time. 10. Acqua resez The above lisa indicates the While th Purposes for Purpose for j ITS??- weilth and 1: ”ing and Lil 16~2e. 76 69 Johnston75 identified twenty different purposes some of which are relevant to this study. These purposes are as follows: 1. Extension of knowledge. 2. Consolidation and reaffirmation of knowledge. 3. Regular acquisition of new knowledge. 4. Acquaintance with curricular developments. 5. Acquaintance with psychological developments. 6. Repetition or extension of original preservice education after intervals. 7. Acquaintance with new aids. 8. Introduction to new methods. 9. Understanding the new relationship between teacher and taught. 10. Acquaintance with and participation in education research. The above list of objectives is by no means exhaustive but it indicates the wide range of inservice general objectives. While these purposes may be regarded as societal or group Jurposes for inservice training, Corey76 identified another >urpose for inservice education--the purpose of the teacher 75D.J. Johnston, Teachers' Inservice Education (The Common- ealth and International Library of Science Technology Engine- ring and Liberal Studies), Oxford: Pergamon Press, 1971, pp. 6-28. 76Stephen M. Corey, "Process of Inservice Education," in eading in Inservice Education, Patel, I.J. and Buch, M.B. eds.), Gamdi-Anand, India: Anand Press, 1968, p. 123. to improve hi: purpose in or: itself in the undertakes to of his teachi of the need 5 indicated rea inservice tra A great for inservice cations on th Greased,77 0 0f the detail the United St Perhaps the fiftY-six of EducatioH, tains everyth outlining Spe zation ' Evalu entitled ’ II Te anther makes Program fOr Ir k See w (3) Johnston 78N Nel SuperVie lQS SOT CiCagoo one C 70 to improve his teaching. It is important to understand this purpose in order to place the whole process of improvement itself in the proper context, Corey added. Before a teacher undertakes to improve his teaching there must be some aspects of his teaching which he is dissatisfied with. The awareness of the need for improvement on the part of the teacher is an indicated readiness which is needed for the kind of help that inservice training can give. A great deal has been written on theory and philosophy for inservice training of teachers, and in recent years publi- cations on the improvement of inservice training have also in— creased.77 Owing to the immense variety of publications most of the detailed references are tailored to the conditions in the United States. Perhaps the most complete work on inservice education is the fifty-sixth yearbook of the National Society for the Study of Education, In-Service Education.78 This publication con— tains everything from justification for inservice education to outlining specific programs. History, roles, programs, organi— zation, evaluation, and so on, are discussed. In the chapter entitled, "Teachers and Inservice Education Programs," the author makes a point that inservice education should not be a Program for making up teacher deficiencies, although this See bibliographic listing under the following names for examples: (1) Bessent (1967). (2) Harris and Bessent (1969), (3) Johnston (1971), (4) Rubin (1971), and (5) Thornbury (1974). Nelson B. Henry (ed.) In-Service Education for Teachers, upervisory and Administrators, Chicago: The University of vhicago Press, 1957. could be one 0 main theme For rent in mees about by encou attack on prof More rece is the discuss thirty-second Mathematics (b of “new" mathe institutes, a Inservice Edu< inservice pro with total in: suPport recei‘ Weaver8O training in 1 be taught? ( ll or should time be provi SW3 Of these thoush Weaver for the dEVel W hat - . c 1970, hemati‘ Th 80 .J.F. o W 71 could be one of the functions of inservice education. The main theme portrayed in the publication is that great improve- ment in professional practices of teachers may be brought about by encouraging and assisting them to make a cooperative attack on professional problems of common concern. More recently and closely related to the present study is the discussion of inservice mathematics education in the thirty—second yearbook of the National Council of Teachers of Mathematics (NCTM).79 The discussion includes the development of "new" mathematics curricula, discovery teaching, the summer -. institutes, academic year institutes, establishment of the Inservice Education Committee of the NCTM, and state and local inservice programs. The chapter gives a general familiarity with total inservice movement for mathematics and the strong support received from the National Science Foundation. Weaver80 raises some basic issues concerning inservice training in 1963. Among them are: (1) What content should be taught? (2) Should subject matter be dealt with exclusive- ly or should methodology be incorporated? (3) Should released time be provided? and (4) What about the physical set up? Some of these issues are still unresolved even today (1975), though Weaver's presentation of them provides a good checklist for the development of new inservice programs. 79 E.G. Gibbs, et al., "Inservice Education," A History f Mathematics Education, 32nd Yearbook, NCTM, Chapter 16, 970. 80J.F. Weaver, "Inservice Education and the Teacher,“ he Arithmetic Teacher, Vol. 10, November, 1963, pp. 456-457. The gener fective classr Fox.81 These involvement in that, whenever identifying th to him, but a1 should assist direction at i out individual fessional deve of a group pr< development . In their present day i: tw0 relevant manacJEment of Someone who i ”hid! the tea bEtter teachi With somethln e ' . r 5 behav1or 72 The general organization of inservice programs for ef— fective classroom learning is the main target of Lippitt and Fox.81 These authors emphasize the need for the teacher's involvement in his own learning. Not only do they suggest that, whenever possible, the teacher should participate in identifying the growth experiences which will be most useful to him, but also they suggest that the growth activities should assist the teacher to develop his capacity for self- direction at the onset. Although Lippitt and Fox do not rule out individualized learning, they strongly hold that the pro- fessional develOpment of teachers must be seen in the context of a group process, of team relationships, and of total staff development. In their argument against some of the weaknesses of some present day inservice approaches, Lippitt and Fox underscore two relevant issues. They emphasize that the design and management of an inservice education program must be done by someone who is intimately acquainted with the environment in which the teachers operate. Alsp, they stress the fact that etter teaching requires more than familiarizing the teacher ith something new: it requires fundamental changes in teach- r's behavior. 81Ronald Lippitt and Robert Fox, "Development and Mainten- nce of Effective Classroom Learning," in Improving Inservice ducation, L.J. Rubin, (ed.) 1971, op. cit., pp. 133-169. Forms of Inse Many met‘ vision of ins in many count in two ways. vidualized an om initiativ education mad needs. A tea books and jou travel or by hilt a few of devoted mostl Cation refere they are Ofte ir15ervice on Among th tiCed in SOme by the School sity 0r C0116 and academic prOgrams; and selected inse tion of the E gram effE‘Ctiv‘ carried Out _______‘ 73 Forms of Inservice Mathematics Education Many methods and means have been employed for the pro+”/\\ vision of inservice education to elementary school teachers in many countries. In general, these activities are initiated in two ways. One kind of inservice education is highly indi— vidualized and is undertaken by a particular teacher on his own initiative and not as part of a "program" of inservice education made available to a number of teachers with similar needs. A teacher might improve himself on his own by reading ‘ books and journals, by getting help from his colleagues, by travel or by individual classroom experimentations, to mention but a few of the many possibilities. Although this study is devoted mostly to a consideration of programmed inservice edu- cation reference is made to the self-initiated approaches as they are often useful in enhancing and sustaining the group inservice work. Among the common approaches to inservice education prac:\W‘ ticed in some countriesgaret late—afternoon classes offered by the school district using its best—qualified staff; univer- sity or college extension classes; week—end, summer, one—term and academic year institutes; conferences; films; television programs; and correspondence courses. An examination of selected inservice programs in some details gives an indica- tion of the patterns and practices. Related research on pro— gram effectiveness is described where such evaluation has been arried out. sive report 0 education, SC ticed in some six professic Office of Ed: and some core: blens include instruction, to conduct Ct Inservi teachers hav 01' the busy tents who le two categori periOd Of tw early in the eXtends for (l) % ated a Show bY a School nine’CY‘minu‘ give“ Year 74 Selected Practices in the United Statesz—In a comprehen— sive report of promising practices in inservice mathematics education, Schult and Abell (1963) documented approaches prac- ticed in some school districts of twelve states, involving six professional organizations and departments of the U.S. Office of Education. The authors reported the achievements and some common problems on inservice organization. The pro— blems include the selection of course content, materials of instruction, the issue of credit and non—credit courses, how to conduct correspondence courses and many others. Inservice education in mathematics for elementary school teachers have, in general, been the workshop type. Because of the busy schedules of the school teachers and the consul- tants who lead the workshops, the workshops are usually of two categories: the "one—shot" type carried out over a short period of two or three weeks during the summer vacation or early in the fall term; and the "slowly-paced" type, which extends for a long period of about the school year. 82 (1) Short Term Coursesz—Hunkler described and evalu- ated a short—term or "oneflshot" inservice course sponsored by a school district. The course was organized through five ninety-minute sessions, closely—spaced, and one course in a given year. The specific content of the teachers' course 2Richard Hunkler, "An Evaluation of a Short-Term In- service Mathematics Program for Elementary School Teachers,“ School Science and Mathematics, Vol. 71, 1971, pp. 650-655. included ways mastery of CO] system. Seco: mathematics w pupils who ha short-term co a order to d program. Hunkler nithmetic cc Emir grade 6 ms results ] “Ce mathemai SiXth‘grade 1 ting a modem means Of pre] metic. ‘2) Lin service prOg 1961! inVolv was reverted pI‘Ogram was W W . R. Educatior1 ‘ Ar ‘ ithme . \t_1c\r] 75 included ways by which teachers can help children towards the mastery of computational skills, number system and algebraic system. Secondly, the use of some teaching aids in modern mathematics were presented to the teachers. Sixth grade pupils who have been taught by teachers with or without the short-term course over a period of three years were studied in order to decide on the effectiveness of the shortvterm program. Hunkler collected and analyzed data on the students' arithmetic concepts and problem—solving scores in terms of their grade equivalents on the Iowa Test of Basic Skills. His results led him to conclude that while shortrterm inser- vice mathematics programs are effective means of preparing sixth-grade teachers to teach arithmetic concepts when initia- ting a modern mathematics textbook, they are not effective means of preparing teachers to teach problem—solving in arith— metic. (2) Long-Sustained Part-Time Courses:-A "slow-paced" in- service program in Dallas, Texas, during the school year 1960- 1961, involving some 89 teachers and 1,977 of their pupils was reported by Houston and DeVault.83 The l3—hour inservice program was organized via four methods of inservice education: (1) television, (2) television and consultant to teachers, 83W.R. Houston and M.V. DeVault, "Mathematics Inservice Education: Teacher Growth Increases Pupil Growth," The Arithmetic Teacher, Vol. X, May, 1963, p. 243. (3) face-to-i face lecture- tent of the 3 matics conce; cal develops In eval as pretests pupils. Eff standing of researchers ' teacher and Effect of t} service proq Program Was both for te; Dutton pmaches th The pUrpose of tw0 inst the new mat Standing of 76 (3) face-to-face lecture-discussion alone, and (4) face—to— face lecture—discussion and consultant to teachers. The con- tent of the program was primarily related to important mathe- matics concepts for the elementary school. Aspects of histori— cal development of mathematics and the relationship between mathematics and other selected areas of general education were included. In evaluating the program, various tests were used, some as pretests and others as posttests for teachers and the pupils. Effort was made to control for the mathematics under— standing of teachers prior to the inservice education. The researchers' main interest in this report seemed to be on the teacher and pupil variables and there was no reference to the effect of the different media of instruction used in the in— service program. However, they concluded that the inservice program was effective in increasing mathematics achievement both for teachers and their pupils. Dutton and Hammond84 made a study of two inservice ap- proaches that were both paced out through the school year. The purpose of the study was to determine the effectiveness of two instructional plans for helping teachers understand the new mathematics. Emphasis was placed upon teacher under— standing of the basic mathematical concepts and upon attitudes 84Wilbur H. Dutton and H. Reginald Hammond, "Two Inservice Mathematics Programs for Elementary Teachers," California Journal of Educational Research, March, 1966, Vol. 17, pp. 63-67. of teachers t grams used in professor of planned insei tion. It was I professor, a about more c matical conc workshops. teachers tor. Structured E Showed that Cantly more of Clain in I tured distr 9I0up. The Nice might trict~staff for individ the Structt The re in reVerse‘ t0 post‘te1 teaChers i: pECted and group r ein 77 of teachers toward the subject.‘ The two instructional pro— grams used in the study were: (1) a workshop using a college professor of mathematics as the instructor, and (2) a district planned inservice workshop using their own staff for instruc— tion. It was hypothesized that the workshop using the college professor, a structured program, and a textbook would bring about more change in teacher understanding of basic mathe- matical concepts than would the informal district planned workshops. It was further hypothesized that the attitudes of teachers toward arithmetic would be more favorable in the structured program than in the informal program. The results showed that teachers in the informal program gained signifi— cantly more than teachers in the formal program. The amount of gain in mean score on post-test of teachers in the unstruc— tured district program was almost double that of the structured group. The authors remarked that the reason for the differ— ence might be found in the fact that the unstructured, dis— trict-staffed program, probably provided many opportunities for individual teachers to work on specific difficulties than the structured program did. The results of attitude measures in the two groups was in reverse, with the structured district gaining from pretest to post-test, while there was a drop in attitude measure of teachers in the unstructured district. This result was unex- pected and the authors wondered whether this was due to the group reinforcement found in the structured district. Nevertheless Teacher (or mat ,5 use 0. teacher own att mrthermore, permeate ins ' C teaching 0; (1) pr (3) g has been ca Brown87 is deScribed 5 Stage is or instruCtioI baSis, The tary SChooj themSelVes an admini 85Iii 6 Ibi 8777‘ . A.K ArlthmEtic (April, 19 78 Nevertheless, Dutton and Hammond maintained:85 Teachers' attitudes towards new mathematics (or mathematics) are important. Through the use of meaningful work in teacher workshop, teachers have a good chance to improve their own attitudes toward mathematics. Furthermore, they emphasized that two major concepts should permeate inservice education to prepare teachers for the teaching of mathematics:86 (1) provision of instruction which is meaningful and which will enable teachers to present mathematics so that children understand the structure of the number system; and (2 V use of appropriate evaluation instruments throughout the workshop so that teachers will become involved in the same kind of appraisal techniques which should be used with children. (3) The "Intermediary" Approach:-A third approach which has been called the "intermediary" approach by Ruddell and Brown87 is also practiced. Unlike the first two approaches described so far, this approach is in two stages. The first stage is one in which a group of inservice leaders receive instruction from a consultant or consultants on a short-term basis. They, in turn, conduct inservice training for elemen— tary school teachers, as a second stage. These leaders are themselves outstanding elementary school teachers, supervisors and administrators with mathematical interests and aptitude. 85Ibid., p. 67. 6Ibid., p. 67. 87A.K. Ruddell and G.W. Brown, "Inservice Education in Arithmetic: Three Approaches," Elementary School Journal (April, 1964), pp. 377—382. They are refs the consultax Izzo anc this pattern the summer 0 suner insti he followin l. lea sub 2 pre gra W 3 lec aPl The th: acquiring in Which they ; ”E inserVi The in was similar Participant fOr about 1 The main or hen510n and CUP Mathemat i C ,ii _—_4 i 79 They are referred to as intermediaries since they act between the consultant and the classroom teachers. 88 described an actual situation in which Izzo and Izzo this pattern was demonstrated in the State of Vermont during the summer of 1963 and the school year that followed. The summer institute for the intermediaries had as its objective the following: 1. learning the essential school mathematics subject matter recommended by CUPM,89 2. preparation for conducting an inservice pro— gram in mathematics, and 3. learning about the knowledge of effective approaches at the elementary level. The thirty-one participants spent an intensive summer acquiring knowledge of the basic concepts of mathematics after which they returned to their school districts and carried out the inservice program. The inservice program conducted by the intermediaries was similar to a class or a course with the summer institute participant as the instructor. Usually, the session lasted for about 1% to 2 hours in a week all through the school year. The main objective was to help teachers develop better compre— hension and appreciation of the structural aspects of mathe- 88J.A. Izzo and Ruth Izzo, "Re—Education in Mathematics for Elementary School Personnel: Inservice Programs-—One Way to Solve the Problems," The Arithmetic Teacher, Vol. XI, October, 1964, pp. 413—417. 89CUPM-—Committee on the Undergraduate Program in Mathematics. matics. The topics in th Emphasis was ledge to imp materials of service educ The eve pupils taugl' program. Ir reactions 01 gram. In gs Ruddell cussed so f,- and the "im taught by d: Year's time and Six, Si< at every 1e. Pils of teacj Spread OVer Wpe of dir sary to bri and unders t mathematica as muCh fro W Rudd 80 matics. The content of the inservice covered a good many topics in the basic concepts of elementary school mathematics. Emphasis was placed on how the teachers could use the know— ledge to improve their teaching. Analysis of textbooks and materials of instruction was part of the activity of the in— service education. The evaluation of this program did not extend to the pupils taught by teachers who participated in the inservice program. In fact, the basic evaluation reported comments and reactions of the participants at different stages of the pro— gram. In general, they were all favorable comments. Ruddell and Brown90 evaluated the three approaches dis— cussed so far, that is, the "one-shot," the "slowly—paced" and the "intermediary" approaches. The achievement of students taught by different groups of teachers were measured over a year's time. It was found that in grades three, four, five and six, significant differences between mean gains were shown at every level and in each instance the gain favored the pu- pils ofteachers whose inservice sessions--ten half—days-—were spread over the year. The researchers concluded that some type of direct contact between consultant and teacher is neces- sary to bring about change in teachers' mathematical knowledge and understanding. Furthermore, they claimed that teachers' mathematical knowledge and understanding can be changed just as much from an intense "one—shot" program as from a slowly 90Ruddell and Brown, op. cit., pp. 377—382. paced, long-1 ted in the c service educi growth, then preferred to Another dence that d the achievem volved in su (Ruddell and The evi increase in serviCe prog ried out. A in the proce Change 01‘ gr (4) E @,-91Whil similarities tiVe feature inservice pr bilities of be used in a 81 paced, long—range program, but that the change is not reflec- ted in the children's achievement. If a basic premise of in— service education is that teacher's growth increases pupils' growth, then it appears that a slow—paced program should be preferred to a one—shot course based on the above conclusion. Another study at the first grade level gives added evi- dence that direct, well—spaced inservice program increases the achievement in mathematics understanding of teachers in— volved in such a program as well as of the students they teach (Ruddell and Barlow, 1963). The evidence presented so far suggests that the resulting increase in achievement of students and teachers from an in— service program depends somewhat on the type of program car— ried out. Also, the evidence suggests that teachers who are in the process of changing are more likely to effect similar change or growth in the pupils with whom they work. (4) The Multi~Jurisdictional Behaviorally-Based Pro— grgmz—nghile the approach described in this section has some similarities with the earlier approaches, it has a distinc— tive feature that is worthy of attention in planning a new inservice program. The program indicates how combined capa— bilities of the different human resources in a community can be used in attacking the problem of teacher retraining for 91University of Maryland Mathematics Project, Multi— Jurisdictional Behaviorally-Based Inservice Program for Ele— mentary School Teachers in Mathematics, Final Report, 1969. (College Park: University of Maryland, 1969). mathematics Concern and the rate led to the d velopment, a service educ competency c conducted b) and the Mar} ation with I The pr< grams can bt teachers ca] Si\- >\‘ '\ .V!;§\i; 1-5 6—10 ll—15 16—2021—33 Years of Teaching Experience. Figure 4. Categories of teachers by years of teaching experience. 120Alice Huettig and John N. Newell, "Attitudes Towards Introduction of Modern Mathematics Program by Teachers with Large and Small Number of Years' Experience." The Arithmetic Teacher (February, 1966), Vol. 13, No. 1, pp. 126—129. 114 120. 3 100]- a) .5. 3 80" a) [-4 u-4 60-4- 0 U) E 40—- E 2 20+ . 1 x 1 1 #1 P1 P2 9‘3 9'4 P5 P's Primary Classes Figure 5. Distribution of teachers by classes they are presently teaching Figure 5 shows the distribution of teachers by the classes they are presently teaching. A majority of teachers, 83 per- cent, had taught more than three different classes in their teaching career. The trend in the data of higher number of upper class teachers continued when teachers were categorized into lower (P1 to P3) and upper (P4 to P6) levels. One ex— planation of this trend may be due to the questionable practice by which the untrained teachers are usually put in the lower classes, and since this study is concerned with the trained Grade II and Grade III, they are probably found in general at the upper level. Nonetheless, the data indicate a distribution Of 60 percent and 40 percent respectively for upper and lower levels teachers in this study. Another characteristic examined had to do with teacher qualification. The study sample included 90 percent Grade II and 10 percent Grade III. This is an indication that fewer 115 Grade III teachers are left in the school system. This trend was confirmed in the total population of primary teachers in the state by a recent governmental digest of statistics refer- red to earlier in this writing. However, the number of un— trained teachers seems to be on the rise as shown by the data. There is a fair distribution of male and female teachers in the sample, 58 percent and 42 percent respectively. The views expressed by this natural categorization of teachers aresigni— ficant in particular because of the growing tendency of women teachers remaining longer in the state's primary schools than the men teachers. In addition, the brunt of the problems of mathematics teaching in the primary schools will have to be shared by both male and female teachers. A consideration of teachers' family involvement is essen— tial to the design of any inservice program, in particular if the program is a long—sustaining one that may demand some out— of-school hours from the teachers. Because of this, teachers were asked to indicate their marital status in the present survey. Ninety—two percent of teachers are married. This is not surprising as it is in keeping with social and cultural setting of the society. However, inservice program design should consider such teacher characteristics related to family and domestic responsibilities which may be a source of obstacle for attendance at inservice programs. 116 TEACHER PAST PARTICIPATION IN MATHEMATICS INSERVICE TRAINING PROGRAMS A major aim of the survey in this study is to find out to what extent teachers participated in inservice or refresher course in mathematics. An analysis of teachers' response to the relevant section of the questionnaire was carried out.121 Participation in previous courses, organizers, main topics dealt with and the effect of this participation on arithmetic teaching are discussed. Non-participants are discussed in terms of awareness of any program in mathematics inservice, willingness to attend future courses, and conditions under which they would participate. In analyzing the responses in this section, major issues are raised and the analyzed data provide some answers. How Much Have Teachers Participated? As Tables 3 and 4 show, teacher participation in previous mathematics inservice is very low. Of the whole group of teachers, only 58 teachers or 15 percent have had any previous participation. A breakdown of data in Table 3 shows that urban teachers had attended programs more than the rural teachers. The percentage of attenders by teachers‘ qualifi— cation is low. Although the percentage of attenders among Grade III teachers appears higher than that of Grade II teach- ers, when the group of 58 attenders was examined, 48 or 83 percent of them were Grade II teachers. 121See questions 5 to 7 of Section 1 in Primary Teachers‘ Questionnaire, p.224. 117 TABLE 3 Percentage of Teachers Who Had Attended or Not Attended Any Previous Inservice Mathematics Program by School Location Urban Rural N9, Pct. N9, Pct. Total Attenders 55 17.4 3 4.8 58 Non-Attenders 261 82.6 60 95.2 321 No response: 1 TABLE 4 Percentage of Teachers Who Had Attended or Not Attended Any Previous Inservice Mathematics Program by Teacher Qualification Grade II Grade 111 N2, Pct. N9: Pct. Total Attenders 48 14.1 10 26.3 58 Non-Attenders 293 85.9 28 73.7 321 No response: 1 A further examination of data revealed that some of the Grade II teachers had indicated their two-year retraining pro- gram from Grade III to Grade II status as attendance at an in— service mathematics program. In effect, the acutal partici— ) pation of total group of teachers in a mathematics inservice training is lower than the indicated 15 percent. This find— ing reaffirms the great need for inservice training elabor- ated upon in Chapter II. Who Organized the Courses Attended by Teachers? The teachers reported organizing bodies for the courses they attended and indicated the duration of the courses. 118 Figure 6 summarizes the distribution of courses among organi— zers. Any government program ranging from a one—hour lecture by the School Inspector to the two-year retraining program for Grade III teachers have been classified as those organized by the government. Local school board short courses were also regarded as governmental programs. Programs organized by unis versities include those of the two Institutes of Education at Ife and Ibadan. Other organizing bodies identified by teach- ers included: the British Council Officer, publishers such as Oxford Press, Ilesanmi Publishing Company and Onibonoje Company. There was no report of any course organized by a pro- fessional teacher organization. Others Schools 2% Universities Government Figure 6. Organizers of courses attended by teachers by percentage distribution of total courses attended About 39 percent of courses lasted from about two hours to three days. Thirty-five percent of courses were the two- year courses, while the duration of the remaining 26 percent of courses was from one week to four weeks. Strictly speaking, the two-year course is not within the consideration of this study, but it was indicated by teachers since it was also a 119 form of inservice program for them. The course, however, covered the arithmetic course of the Grade II colleges dis- cussed earlier. On the whole, it can be said that the dura— tion of most courses is about one to three days and only about 15 percent of total courses offered lasted for three or four weeks, while 11 percent lasted for one or two weeks. Very few courses had been held in the school environment. A closer look at data shows five short lectures on metric system in five different schools. What Main Mathematics Topics Were Covered in the Courses? Teachers were asked to list the main mathematics topics in the courses they attended. Among the most frequently listed topics are: number, set, the basic operations, frac- tions and factors. A few teachers listed metric system and “method of teaching lower classes." Conspicuously absent from teachers' lists are topics deal- ing with shapes, problem—solving and applications of mathe- matics, which are important andyfunctional mathematical topics. The topics related to the metric system and "method of teaching lower classes," indicate the awareness of two of the problems currently confronting the primary schools. Nigeria is in a transitional period of changing from the imperial units of mea— surement to the metric units. However, there is no systematic plan for the modification of classroom instruction at the pri- mary level to meet the needs of this change. It is encourag- ing that some short courses - though rather too short to be 120 effective — were organized for some teachers on metric system. Traditionally, the Grade III colleges taught "infant methods.“ However, with the closure of the Grade III colleges, this area of school methods of teaching had been neglected. The Grade II colleges did not turn attention to this problem soon enough and as a result, it became an area of teacher training which many Grade II teachers are deficient in and, therefore, the need for re-orientation in infant methods. What Were the Effects of Courses on Classroom Teaching? One way by which the impact of the courses attended by the teachers on their classroom teaching was investigated was by asking teachers to list the topics that have influenced their teaching and how they did. Of the 55 teachers that had listed topics earlier, only 47 responded to this question. Topics on metric system, numbers and operations were most frequently listed as those that have helped classroom teaching. The other topics listed included sets, fractions, least com— mon multiple and highest commonyfactors. It is significant that about 28 percent of those who responded actually wrote: ”none," implying that the topics of the refresher course had not changed their teaching. Responses to the question of how the topics have changed teachers' lessons varied from change in teaching approach, greater knowledge of mathematics content to observed change in the children. Some of the responses are presented below: 121 l. The experience I had during my inservice course helped me to approach some topics in a practical way. 2. The children have more understanding of the topic when done practically. 3. They led us to field work. Lastly, a teacher who participated in a 45—minute workshop on metric system wrote: "Pupils seem to understand this system better than the former ones and I as a class teacher find it easier to teach my pupils." Respondents were asked to indicate how the courses they attended could have been improved by their organizers. An analysis of the responses to this open-ended question points out some major weaknesses in the present inservice approach to teaching by the inservice instructors and the lack of teaching aids for such programs. Other comments included the need for content improvement, organization of better evalua- tion process of the inservice programs, the need to make the duration of courses longer and to spread the programs out to more teachers, not only those in the state capital, and the need for a follow-up. Some of the teachers' specific comments point directly to needed improvement that any future inservice plan should consider. Among the specific comments are the following: 1. They could improve the program by conducting more of it to improve teacher's skill in the subject. 2. Those who taught the program could have improved upon it if there had been regular seminars and sufficient teaching aids. 122 3. They could plead to the government to provide the necessary books and materials needed for the teaching of mathematics. That there is an interest in and desire for inservice mathe- matics programs are clearly portrayed in the reSponses of these teachers. The teachers seemed aware of the need for a long-sustained program to help them in developing the needed skills for mathematics teaching. It appears that both short and long courses will achieve some success if well-planned, though long courses will achieve more and most likely will be more effective. The Non-Attenders In an attempt to further assess the impact of the exist- ing programs on all teachers in the sample, the non—attenders gave responses to a few questions related to the organization of inservice mathematics programs. The respondents were asked to indicate if they had heard of any inservice programs even though they had not participated. About 75 percent of 315 non- attenders were not aware of any inservice mathematics programs for primary teachers. Some teachers who had heard about some programs did not know the organizing body. Among the organ- izers identified by non-attenders were the government, the universities and the British Council, in that order. The re— sponse here was very consistent with that of the teachers who had attended former programs. Again, these teachers had never heard of any program initiated by a professional organization or a school. The identified locations included university L__ i 123 campuses, teacher training colleges and the British Council building. In identifying locations, some teachers named the towns. The trend in this identification shows that the big towns of Ibadan, Abeokuta and Ondo were frequently named as locations where inservice training programs had taken place. The evidence so far indicates that teachers who have had some type of participation in inservice programs seem to appre- ciate the need for such programs and would like to see them improved and intensified. However, this fact cannot be assumed for those teachers who have had no participation. Therefore, the non—attenders were also asked to indicate their willingness to attend mathematics inservice programs, the significance of such programs to them and the conditions under which they would like to attend. A majority of the non—attenders, 95 percent or 79 percent of total sample of teachers, indicated willingness to attend mathematics inservice programs. About 85 percent of these teachers indicated their belief that such programs would up- grade their mathematical knowledge and improve their teaching ability. Other items of mathematics inservice training impor— tance that were commonly cited include: "better position to help children for nation building," "mathematics is useful in everyday life," "it will promote the falling standard of mathe— matics teaching in the schools," and "it will help teachers to teach better." The conditions under which non-attenders would participate were first considered in general, and specific conditions were 124 later investigated under four main categories. The categories are: (1) conditions related to time and place, (2) cost of in- service training to teachers, (3) availability of reward, and (4) others. Many teachers indicated "any condition" and some would like a full-time one—year course for such retraining programs. A trend in the data showed that teachers with many years of teaching experience (25 to 33 years) indicated they wanted, "just a short refresher course." However, about 36 percent of the non—attenders would participate if time and place were made convenient. Teachers‘ convenience as coded from the responses means during the long (summer) vacation and in "nearby town." Other convenient locations mentioned by teachers included "a school in town" and "the divisional head- quarter;" and for time, evening, after school hours, and dur— ing school hours were all frequently indicated as convenient times. Only 32 percent of these teachers indicated desire for attendance if it was at no cost to them —- that is, if free lodge, board and cost of transportation to inservice location were provided. ‘Similarly, only a small proportion of these non-attenders, 25 percent, indicated the availability of a reward as a condition for attendance. The types of reward asked for by those who did included: recognized certificate, encouragement from the Ministry of Education and promotion. A variety of conditions were classified under the heading "others." The number of respondents for each of these 125 conditions ranged from one to 25. When summarized, they were grouped as follows: 1. Teaching Aids: some teachers would attend under the condition that inservice programs would help them in producing some teaching aids which they could later use in their classes and in helping them to use some teaching aids appropriately. 2. Organization: some respondents mentioned they would like "experts" to handle the courses, and would be willing to pay a little fee. Others indicated that the course should be non-residential and that partici— pants should be grouped by ability. 3. Content: some respondents indicated willingness to participate under the condition that they could teach the new topics in their classes, and that the course content would be helpful to them. The responses of the non-attenders have been elaborated upon not only because they form a majority or 83 percent of total sample of teachers, but also because the issues involved are relevant to the successful organization of inservice pro- grams. Although the views and preferences of the total group of teachers in the study are considered later in this analysis, the responses of the non-participants raise some problems of arithmetic teaching that merit discussion at this point. The crippling effect of the lack of teaching aids is again seen in the conditions for attendance stipulated by some teach— ers. If classroom teaching should be enriched by an increase in the use of teaching aids, then retraining programs for teachers should help them in this direction as well as in other areas of needs. Evidence that justifies such an objective of inservice training had been provided earlier as in the case of the different workshops in the United States and the British teachers' centers, for example. 126 The issue as to whether the teacher could teach the topics learned in inservice program seems to suggest a problem related to the administrative control of school syllabus and secondly to the lack of basic mathematics understanding of teachers. On the administrative level, the syllabus provided for arithmetic teaching is still the 1954 syllabus mentioned ear— lier, and arithmetic examination in the First School Leaving Certificate examination is still based on this syllabus. In effect, topics that are not explicitly stated in this syllabus appear to some teachers as topics they cannot teach in their classes. For example, if an inservice training program covers the concept of set, as it has been indicated in some of the previous programs, and if the program does not relate this topic to the many number topics and other topics taught in the primary classes and included in the present syllabus, some teachers would regard the concept as "new" and outside the syl— labus. They, therefore, believe that it cannot be taught in their classes. Clearly, one other complex factor contributing to this type of confusion, in addition to the administrative control of the syllabus, is found the the low level of mathe- matical background and understanding of the teachers themselves The concern of teachers in keeping rigidly to the content of the syllabus is probably with some justification. The in— vestigator had been part of a short course in which teachers had complained of their inability to introduce some of the new ideas for fear they might be reprimanded by the school inspec~ tor for teaching a topic that was not stated in the scheme of L___ 11 127 work. The validity of such a statement cannot be assumed, but it calls for further probing and modification if inservice work should serve its purpose. A more valid and more crippling problem of administrative rigidity in connection with mathematics curriculum of the pri— mary school that is not conducive to the organization of ef— fective inservice mathematics programs is in the well-known plight of the "early innovators" of curriculum change in this discipline. As a result of the work of the African Mathematics Program in the 1960's, mentioned in Chapter II, a few head— masters of primary schools initiated and followed through a program in the "new" mathematics or the Entebbe Mathematics Program in their schools. At the end of the program, these school heads petitioned the State Ministry of Education to pro— vide examination related to the new program for their pupils as an alternative to the Arithmetic Examination of the First School Leaving Examination.122 The petition was denied and the schools affected made great effort to cover missing grounds, if any, before the children sat(for the same arithmetic exami— nation as all other children sitting for the examination in the state. It was reported by headmasters that the children in this "converted" program performed better in the arithmetic examination than children in the regular arithmetic program. 122Much of this information is well—known in the State. However, specific information was also gathered during school year 1972—73 in an informal interview with three headmasters of primary schools. 128 The headmasters believed that their gain in performance was due to the "new" mathematics approach used with the children. Consequently, a pattern of three or four years of "new" mathe- matics, followed by three or two years on the approved Arith— metic Syllabus with concentration on the examination require- ment has developed in a few primary schools of the state. The effect of a development of this nature on teacher re- training is negative, since some teachers could look on it as a disapproval of the government towards the introduction of mathematics teaching in the primary schools. Yet, the develop— ment of the more recent years has also shown that Western State government is acutally planning to introduce mathematics teach- ing in all the primary schools. There seems to be enough evi— dence pointing to the fact that one major reason why the new mathematics curriculum is still being shelved is the lack of adequately trained teachers in the primary schools. One way of solving this problem is in the initiation of a systematic inservice program for primary school teachers. In summary, the findings of this section seem to suggest that some attempts are being made to re-educate teachers for the teaching of mathematics in the primary schools. The at- tempts, however, are too short, too haphazard, and they reach only a handful of teachers. The findings further suggest that motivation for inservice can be both intrinsic and extrinsic; that teachers are willing and ready to participate in inservice mathematics programs; that a local arrangement would be more 129 convenient for teachers; and that teachers and administrators should be part of these programs in order that the goals and objectives might be shared in common. Furthermore, there is a suggestion for a more clearly stated and modified governmen- tal policy on mathematics teaching in the primary schools than what now exists, so that creative practices might be en- couraged through inservice training and translated into better teaching in the primary schools. Lastly, findings suggest that there is a grave need for the supply of teaching aids needed for mathematics instruction. TEACHER ATTITUDE TOWARD MATHEMATICS AND CLASSROOM PRACTICES The fifth major area of this analysis deals with the issue of teacher attitude and classroom practices related to mathe- matics teaching. That favorable attitudes towards mathematics maximize the possibility of learning and teaching being effec- tive has been expounded by many authors, among them was Neale.123 Although the survey was not intended to give final answers on attitudes or instructional practices, yet this sec- tion was designed to probe into teachers"practices so that information gathered from such probing might further suggest directions for the design of inservice programs. There are four parts in this section. They are: (1) teachers' ranking of arithmetic teaching, (2) teachers' attitude towards 123D.C. Neale, "The Role of Attitude in Learning Mathe- matics," The Arithmetic Teacher, Vol. 16, No. 11, 1969, pp. 631-640. 130 mathematics activities, (3) the materials of instruction and (4) teachers' classroom practices. Rank of Arithmetic Teaching A rank—ordered scale consisting of the nine main subjects of the Western State primary school curriculum was developed.124 Teachers were to rate each subject in order of the subject they liked best to teach. The data were analyzed in two ways. First, an examination of the exact rank accorded arithmetic teaching by each of the total group of teachers was examined. Secondly, the ranking was dichotomized into three categories, namely: first to third positions, fourth to sixth positions, and seventh to ninth positions. A test of differences between the following groups was carried out: urban and rural teach— ers; male and female teachers; and lower and upper level teach— 8138.125 Findingsz-As shown in Figure 7, 36 percent of teachers ranked arithmetic as the subject they liked best to teach, 24 percent ranked it second. The percentage of teachers who ) chose subsequent positions decreased, except for the rise to five percent of teachers who ranked arithmetic in the ninth position. 124 p. 225 . 125The .01 level of significance for the rejection of statistical hypothesis being investigated was selected for these and subsequent tests in the study. That is, all dif- ferences between reported categories of responses mentioned in the study are statistically significant to the one percent level of confidence. See Primary Teachers' Questionnaire, Section II, 131 Scale: 1mm to 1 percent 15t 2” 3rd 4 5% 9*” 367.: 247. 147. 8% 6% 57. Figure 7. The exact ranking of arithmetic as a teaching subject by all teachers It was expected that there would be differences in the ranking of arithmetic as a teaching subject between urban and rural teachers; male and female teachers; and lower and upper primary level teachers respectively. An examination of Table 5 indicates that teachers' responses were generally the same. To confirm this trend, a chi-square analysis of the data in each category was carried out. There was no significant dif- ference in each case (p <.Ol). 5 Discussion:-It appears that most teachers in the study generally rated arithmetic teaching high in their choice of teaching subjects. In interpreting these data, however, at— tention should be paid to the fact that there might be a pos— sibility of Hawthorne Effect —- that is, teachers putting their priority on arithmetic teaching because they were aware of participating in a study involving the teaching of arithmetic. Secondly, the study of Goodlad and Associates (1970) mentioned 132 earlier suggests that some of the teachers might in fact not be teaching arithmetic best of all the subjects, even though they believe they do. In general terms, however, the find- ings suggest that there is a favorable atmosphere for the motivation needed for an inservice program. TABLE 5 Rank of Arithmetic as a Choice Subject by Teachers in Categories (Rank 1 to 3) (Rank 4to 6) (Rank 7 to 9) N9. Pct. N9. Pct. N9. Pct. Teachers by Locations Ikbmn . .. . 234 76.7 53 17.4 18 Rural.. . . . 49 77.8 12 19.0 . Teachers by Sex Male .. . . . 164 77.0 37 17.4 12 Female .. . . 119 76.8 28 18.1 8 . Teachers by Class Level Inwer.. . .. 116 80.6 25 17.4 3 . Imper.. . .. 165 74.7 39 17.6 17 7.7 No response: 3.1 percent to 3.9 percent Teachers' Attitude Towards Mathematics Activities » The eight forced-choice items found in Section IIB of the Primary Teachers' Questionnaire were used to measure teachers' attitude towards mathematics activities. This scale was adap— ted by the investigator for use in the present study from the 126 Teacher Mathematics Attitude Test which was based on the 126The Teacher Mathematics Attitude Test was used in an Evaluation Program of Mathematics Teaching in the Elementary School in California. Cited in: Larsen, E.P. (director) The Mathematics Improvement Program (A Study of Educational EffEC- tiveness with Planned Program Variation), Oakland: Oakland Uni- fied School District, 1970, pp. 135-136. 133 arithmetic-interest items of Kuder Inventory of Occupational Interests. The teacher was asked to choose one of two activi- ties. His choice showed whether he preferred mathematical activities or non—mathematical activities. A teacher's total score indicated whether his interests were similar or dis- similar to the interests of those engaged in occupations in- volving arithmetic or mathematics teaching. For the purpose of this analysis, a scale score of one was considered positive for a mathematical activity, and zero considered negative for a non-mathematical activity. Findings and Discussionz—Thirty-one percent of teachers scored eight points each on the scale, while each of four teachers scored zero on the total scale score. The average score of teachers was 5.8 with a standard deviation of 2.1. These scores suggest that 68 percent of teachers scored be— tween 3.7 to 7.9 on the attitude scale. However, when the paired items were considered separately, the reaction of teachers to each item was better appraised. Table 6 shows the percentage of teachers' positive and negative responses to each item. Whereas the choice between watering maize and solving arithmetic problems was quite distinct, it was a close choice between being a secretary or being treasurer. v—v 134 TABLE 6 Percentage of Teachers' Responses on Mathematical and Non-Mathematical Activities Percentage of Percentage of Positive Response Negative Response Mathematics Interest Item a. Help a child with his spelling lesson. b. Help a child with his arithmetic 78'5 21.5 problem. a. Collect figures of what is happen— ing in trade and industry. 76.6 23.4 b. Weave cloth on a hand loom. a. Do typing and shorthand. b. Do work that requires mental 82.9 17.1 arithmetic. a. Study methods of supplying maize with water. 6.4 13.6 b. Study rapid methods of solving 8 arithmetic problems. a. Take a course in mathematics. 63 8 36.2 b. Take a course in English language. 3. Be the treasurer of your local club. 53 7 46.3 b. Be the secretary of your local club. a. Make tables of figures on the costs of food and clothes. 62.5 37_5 b. Write compositions on your favour— ite games. a. Estimate the cost of equipment for your school. 74-2 25’8 b. Decorate the school hall for a play. No response in 1 percent of cases on each item. A third dimension of the analysis at this point was a determination of whether or not there was any difference in attitude of teachers who had previously attended any refresher course in mathematics and those who did not. A chi-square an— alysis of differences in the frequency of occurrence ofpositive 135 and negative reactions on the eight forced—choice items among attenders and non-attenders did not yield any significant difference. Table 7 shows a breakdown of positive and nega- tive responses on two of the eight items of the mathematics interest scale. The two items chosen were those with the highest and lowest positive responses respectively when the total group of teachers was considered. TABLE 7 Responses of Attenders and Non~Attenders on Two Mathematics Interest Items Attenders Non—Attenders Mathematics Interest Item Positive Negative Positive Negative Study methods of supplying maize with water/Study rapid method of solving arithmetic problem. 48 10 277 41 (83%) (17%) (87%) (13%) Be the treasurer of your club/ Be the secretary of your club. 34 24 168 150 (59%) (47%) (53%) (47%) No response in 1 percent of cases on each item. The general findings here also suggest that teachers have a positive inclination towards mathematics activities. It is interesting to note the high positive percentage of responses in the two instances where teachers had to choose between a mathematics activity and a farming activity and also between a mathematics activity and being a typist. These high ratings of mathematics activities in these cases might in fact be due partly to some unfavorable social values attached to being a farmer or being a typist. Nevertheless, the reactions on the 136 whole provide a favorable stage for a mathematics retraining program. The Materials of Instruction In order to access the availability and use of materials of instruction, teachers were asked to list both the pupils' and the teachers' books used for the teaching of arithmetic. In addition, they were to list the teaching aids they use in the teaching of arithmetic. Pupils' books were categorized into three groups, namely: Modern, Transitional and Traditional, for the purpose of analysis.127 The teachers' use of reference books was investi— gated by a consideration of whether a teacher used no refer- ence book at all, whether he used the pupils' textbooks or an advanced book as his reference book. Lastly, the teaching aids were considered as to whether they were manipulative or non—manipulative. Findings and Discussion:-Most pupils' books indicated by the majority of teachers (58% of teachers) were traditional books. Their content is still mainly the arithmetic that was taught three or four decades ago. They contain different types of how-to-do-it introductions to topics, and step—by— step solutions to problems. Consequently, problem—solving becomes mere exercise and stereotyped routine procedures 127This classification is based on the investigator's knowledge of the content of the books listed. 137 characterize arithmetic lessons. Table 8 shows a summary of 128 the pupils' books listed by teachers. As suggested by some of the titles, some of the books encourage routine learn- ing and rote memorization. Forty percent of teachers cited books classified as transitional. These are new editions of the traditional books which had been metricated to meet the current change in the society. TABLE 8 Summary of Pupils' Books Used for Arithmetic Teaching by Categories Modern Textbooks New Primary Mathematics for Nigeria Oxford Modern Mathematics Entebbe Mathematics Series Preparatory Modern Mathematics Onibonoje Modern Mathematics Progressive Arithmetic Modern Primary Mathematics Ilesanmi Modern Mathematics Transitional Textbooks Oxford Arithmetic (metricated edition) Larcombes Primary Arithmetic (metric and decimal edition) Traditional Textbooks . » Larcombes Primary Arithmetic (middle and upper standard) Scholarship Speed Sums and Mathematics Revision Arithmetic for Common Entrance Examination Ways to Success —- Step 1 Scholarship Arithmetic Tutor Speed and Accuracy Test New Nation Arithmetic 128Authors' names are left out for reasons of anonymity, although some titles still reflect the publishers. Besides, not all books were named by authors. 138 Although only eight percent of teachers indicated text— books classified as modern textbooks, the indicated variety is encouraging. The Western State of Nigeria, as well as the other states, seem to be witnessing currently an upsurge of the activities of indigenous Nigerian textbook writers. This upsurge is well reflected in the production of primary school textbooks, including mathematics textbooks. The quality and effectiveness of most of the books still have to be evaluated before their impact on primary mathematics teaching can be fully assessed. In addition, teachers still have to learn the mathematics content of some of the books before they can be effectively used for the teaching of mathematics in the primary schools. A second approach employed in the analysis of data on pupils' textbooks was to investigate whether teachers at the lower or upper level differed in their use of the different categories of pupils' textbooks. It was expected that, since new books for primary mathematics are beginning to increase in the book market, and since these books are written in series beginning with primary one, two, and so on, lower pri— mary classes might be using more of the modern books than the upper primary classes. A chi—square analysis of the data in— dicates that the difference in the use of modern textbooks is significant beyond the .01 level in the direction of lower primary teachers. Table 9 summarizes the data. 139 TABLE 9 Relating Teachers' Class Levels to the Categories of Pupils' Textbooks Class Levels Modern Transitional Traditional N_o. P_c_t. 12'2- gg. IE- 133:. Lower Level 22 14.6 55 36.7 73 48.7 Upper Level 10 4.5 91 50.0 121 54.5 No response: 8 X2 = 11.8, significant at .01 level The teachers‘ reference books vary slightly in content from the pupils' textbooks. The findings show that most teach— ers (56percent)usedtflmapupils‘ textbooks and/or the teachers' guides, where such guides existed, as their reference books. Forty—two percent of the teachers used books with more ad— vanced content than the pupils' books, while two percent of teachers indicated they did not use any reference book. Fig- ure 8 shows a distribution of the categories of teachers by years of teaching experience and the type of reference books they use for arithmetic teaching. It is important to note that as many as 56 teachers (about 15 percent) of the total group of 380 teachers did not respond to this question. The suggestion of the investigator is that either these teachers do not see the need for a refer- ence book in arithmetic teaching or they teach from the pupils' books. Suitable reference books for arithmetic teaching seem to be scarce in the society's book market, and unlike the pupils' textbooks, book writers seem not to have turned atten- tion to this need yet. This scarcity is reflected in the types 139 TABLE 9 Relating Teachers' Class Levels to the Categories of Pupils' Textbooks Class Levels Modern Transitional Traditional 1312. P__ct. 119. 13c_t_. N_o—. ECL- Lower Level 22 14.6 55 36.7 73 48.7 Upper Level 10 4.5 91 50.0 121 54.5 No response: 8 X2 = 11.8, significant at .01 level The teachers' reference books vary slightly in content from the pupils' textbooks. The findings show that most teach- ers (56percent)usedtflmapupils' textbooks and/or the teachers' guides, where such guides existed, as their reference books. Forty—two percent of the teachers used books with more ad— vanced content than the pupils' books, while two percent of teachers indicated they did not use any reference book. Fig- ure 8 shows a distribution of the categories of teachers by years of teaching experience and the type of reference books they use for arithmetic teaching. It is important to note that as many as 56 teachers (about 15 percent) of the total group of 380 teachers did not respond to this question. The suggestion of the investigator is that either these teachers do not see the need for a refer- ence book in arithmetic teaching or they teach from the pupils' books. Suitable reference books for arithmetic teaching seem to be scarce in the society's book market, and unlike the pupils' textbooks, book writers seem not to have turned atten— tion to this need yet. This scarcity is reflected in the types 140 of books teachers listed. Some of the books are those tradi— tional textbooks that had been rejected in the secondary schools to give way to modern mathematics teaching at that level. There are, however, a few contemporary books such as Ilesanmi Modern Mathematics Teachers' Note, Entebbe Mathematics Series and The Teaching of Arithmetic in Primary Schools, list- ed by teachers. On the other extreme are some outdated books such as A Shilling Arithmetic Book and Durell Arithmetic Book still being used as reference books by primary school teachers. 70- m 60 )\ w - / .\ o /' \ / 1% 50... v/ 3'2 40- w 0 m 30- CW 3 2 C1 0" m 8 a) 10" m ... :.......‘..:...'.00....“......g...t(v'.‘... 1 1—5 yrs 6—1oyrs 11—1syrg 16—20yrs 21 and above Years of Teaching Experience Key Advanced Books: Pupils' Textbooks: — - - - No Reference Books:. . . . Figure 8. The categories of teachers by years of teaching experienceandiflmatypeof referencebooksthey use _\1 ‘2. 141 The findings on the teachers' teaching aids from the responses given by teachers support the same gloomy picture presented by the headmasters' responses which were discussed earlier. Most teaching aids used in arithmetic teaching are non—manipulative. Only about 35 percent of teachers reported teaching aids that could be classified as manipulative. Among these were seeds and stones for counting, counting sticks, card—board clock-face, and spinning tops for sums. Many teachers indicated as their teaching aids: blackboard sketch- es, drawing board and chalk. Most of the teachers' aids in- f3 volved children's participation in a passive way rather than an active way. In the case of teaching aids used by teachers, as in the case of their reference books, 52 teachers gave no response to this question. In general, the findings further suggest the great scarc— ity of teaching aids for mathematics teaching. Yet, if a pri- mary mathematics class should be a place of inquiry and activ- ity, where children learn mathematics rather than absorb ideas passively, then adequate provision should be made for appro— priate teaching aids. Furthermore, the provision of teaching aids is only a means to an end, teachers would need to use the aids or learn to use them appropriately for effective teaching. Some of these teaching aids can and should be developed from materials in the child's surrounding. However, such adaptations must be skillfully done in order not to present wrong models and faulty concepts. Inservice training 142 programs can be a forum for the creation of such locally pro— duced teaching aids. There is a definite need, however, for commercially pro- duced teaching aids for the Western State primary schools. Hugh Hawes made the case for teaching aids needed for the country's primary education when he wrote on Nigeria's plan for universal primary education:129 The new teachers entering the schools for UPE are going to need support of two main kinds. First, it will be essential that they have pro— fessional guidance and supervision; and secondly, they will require adequate material support... It will further entail at least one major decision: whether some or all of the materials are to be mass produced by the government (and if so, where and how?) or whether commercial publishers are to be in partnership with the government. The findings here suggest that material support is needed for quality teaching of all school subjects including mathematics. Teachers' Classroom Practices The analyses in this section so far have focused on teachers' attitude towards mathematics activities and their instructional materials. But teachers' daily classroom prac— tices are also reflections of their attitudes toward teaching. Therefore, a scale of ten statements related to mathematics teaching was developed to provide some measures of teacher classroom practices. The statements are found in Section IVB of the Primary Teachers' Questionnaire. There are five 129Hugh Hawes, "UPE in Nigeria, 3: Logistics," West Africa, October 7, 1974, p. 1213. % 1 1 .‘< 143 positive statements and five negative statements. Each item in the scale had Lickert—type response alternatives ranging on a five—point scale from "completely agree" to "completely disagree." The dimensions and items covered in the scale are some of the characteristics generally related to mathematics teach— ing. They include issues related to individualization, plan- ning for instruction, group work, discovery approach, applica— tion and evaluation. There was no definite expectation for the reactions of teachers, the main purpose of the scale was to probe their practices in order to determine the experiences which inservice training should establish, reinforce or enrich. Table 10 shows the distribution in percentage count of teach- ers' responses on the five-point scale. Teachers seemed to have agreed with positive statements and disagreed with negative ones in general. However, the data show a special problem in the case of children being too young to discover their own rules. It appears that there is the need to help teachers whether at the lower or upper level in the use of the discovery approach, so that all children can discover their own rules in the classes. Approximately 20 percent of teachers assigned homework for misbehavior, another 18 percent agreed they could teach arithmetic without the use of teachers' guides or a method book; and still another 18 percent agreed that arithmetic has little practical application in Nigerian life. If arithmetic teaching should 144 emphasize application and help pupils enjoy the subject, the type of reactions expressed by these 18 to 20 percent of teachers, as suggested by these findings, need to be discour- aged. The experiences provided in an inservice program for teachers could incorporate activities that could help teachers in establishing more favorable practices. TABLE 10 Percentage of Teachers' Responses to Statements on Classroom Practices Statements CA _A U 2 _CD It is necessary to give individual help to 70.2 25.3 1.8 1.3 1.4 pupils in arithmetic lessons. Pupils learn from one another so I encourage 39.6 42.8 4.5 8.7 4.4 group work in my arithmetic class. Working with arithmetic games and puzzles 39.9 50.5 5.1 3.4 1.1 in the classroom can be a worthwhile activity. A field trip to the market place is a valu— 32.2 47.6 10.3 5.1 4.8 able resource for arithmetic teaching and learning. Teachers should change their approach to 50.3 39.7 6.1 2.7 1.2 classroom teaching to make use of some of the newer techniques and teaching ideas. I can teach arithmetic without reading 6.1 11.4 5.6 41.5 35.4 teachers' guides and method books. Arithmetic is a skill with little practical application in Nigerian life. 6.6 11.4 7.7 34.3 40.0 Children in the class I teach are too young 16.5 27.9 11.4 29.3 14.9 to discover their own rules for solving arithmetic problems. I assign extra arithmetic problems as 5.8 14.1 7.1 31.2 41.8 homework when pupils misbehave. One final test at the end of term is enough 4.5 5.6 4.1 28.9 56.9 to find out how well my pupils understand their arithmetic lessons. CA = Completely Agree U = Uncertain CD = Completely Disagree A = Agree D Disagree No response: 4 in each item ——--—-——-~—' V _7 _" 145 This section of the analysis discussed the attitude of teachers towards mathematics activities and arithmetic teach- ing, teachers' materials of instruction, and some of their classroom practices. VIEWS AND PREFERENCES ON INSERVICE MATHEMATICS PROGRAMS: TEACHERS, HEADMASTERS AND ORGANIZERS An important part of the survey was concerned with the establishment of teachers', headmasters' and organizers' views and preferences on mathematics inservice training. This sec— tion of the data analysis reports the following aspects of the survey: (1) the learning experiences required by teachers in an inservice program, (2) the views of teachers, headmasters and organizers on inservice organizations, (3) their views on inservice program evaluation, and (4) the demand for future inservice mathematics courses. Inservice Learning Experiences In order to investigate what learning experiences teach— ers wanted from inservice programs, data were analyzed from ) responses of teachers to related questions in the Primary Teachers' Questionnaire.130 The main data were those from teachers' selection of mathematics topics for inservice pro- gram and the list of mathematics topics they found difficult to teach. A list of mathematical tOpics was build up from mathematics 130See Primary Teachers' Questionnaire, Section I, No. 4; Section IIIA; and Section V, Nos. 9 and 10. 146 curricula related to primary mathematics teaching in the Western State.131 The selected topics covered concepts in the areas of number, measurement, geometry and application. The teachers were asked to check "yes" for tOpics they would like included in an inservice mathematics program according to their needs. When the group of teachers was considered as a whole for analysis, eight tOpics were requested by over 80 percent of teachers, another eight topics were requested by 70 to 79 percent of teachers, while the remaining five tOpics were requested by 55 to 69 percent of teachers. The priority topic selected by teachers seemed to be "metric system" selected by 96 percent of teachers, while the tOpic "mean, mode and median" was requested least by 57 percent of teachers. Table 11 shows the percentage distribution of selected tOpics first by all teachers, then by teachers classified into lower and upper levels. A further analysis showed that there were no differences between the responses of lower level and upper level teachers, except in the selection of the topic "Solid Geometric Figures" where the analysis of data yielded a significant difference. The data indicated that upper primary teachers more than lower primary teachers requested the topic on solid geometry. 131This list was culled from Primary School Mathematics Guidelines suggested by the Nigerian Education Research Coun- cil (1971); Teacher Training College Mathematics Program sug— gested by NERC (1972); and the Basic Mathematics Syllabus for Grade II Teachers Colleges in Nigeria. 147 TABLE 11 Percentage of Teachers Who Would Like TOpics Included in Inservice Programs IIIIllIlllIIIIII:T—————————————————_______________*T:~fi“ Base 10 numeration system. . . . . . . . Four basic operations with whole numbers . Common and Decimal Fractions . . . . . . . . Four Basic Operations with Fractions . Simple Idea of Set . . . . . . . . . . Adding and Subtracting Positive and Negative Numbers. . . . . . . . . . . . . . . . . Numeration System.with Bases Other than 10, e.g., Base 5 . . . . . . Factors, Primes and Composite Numbers. Properties of Points, Lines and Planes . Solid Geometric Figures (e.g., Cubes, Cdboids, Pyramids, Prisms, etc.) Idea of Area and Perimeter of Regions. Idea of Symmetry . Idea of Congruence . Metric System of Measurement . Interpreting and Constructing Line, Circle and Bar Graphs . . . . . . Simple Geometric Constructions (e.g., Perpendicular Bisector of a Line, Bisector of an Angle, etc.). MEan, Median and Mode. . . . . . . . N . . . Making up and Solving Mathematical Sentences. . . . . . . . . . . . . . . Solving Mathematical Problems. . . . . . . Mathematical Ideas Used in Trade and Business (e.g., Discount, Interest, Profit and Loss) . . . . . Simple Idea of Probability . All No Teachers Lower Upper Response 75.4 77.0 74.4 13 70.0 72.9 68.1 13 80.9 80.4 81.2 9 73.8 75.0 72.9 10 87.7 89.9 86.2 13 73.7 79.1 70.1 15 77.9 80.7 76.1 18 79.6 77.2 81.2 13 68.0 70.1 66.7 17 79.5 69.8 85.7 10 78.1 77.0 78.9 14 62.9 63.3 62.7 16 62.1 55.8 66.4 16 95.7 93.9 96.9 9 81.5 77.5 84.3 17 80.1 76.7 82.3 19 57.3 58.1 56.8 24 80.1 84.2 77.4 18 86.2 86.9 85.7 17 83.2 82.5 83.6 12 62.8 63.5 62.3 20 148 In all cases, more than 50 percent of teachers requested all the tOpics. A suggestion from the findings is that these topics should form the needed guide for tOpics to be included in mathematics inservice training. Only 24 of the 380 teach- ers indicated additional topics in the section provided for these. All the additional topics indicated were subsets of the major tOpics already requested. For example, eleven teachers indicated "metric system-conversion" as their addi— tional topic even when they had already checked the tOpic "metric system." Another source of mathematical topics suggested for in- clusion in the inservice program was the response of teachers to the question: List the topics in arithmetic you found most difficult to teach. The tOpics listed by teachers were classi- fied for analysis under the headings: numeration and nota- tion; operation on whole numbers; fractions and decimals; operations with fractions; measurement, including metric sys- tem; simple business arithmetic, including simple and compound interests; and geometry. The two most difficult tOpics listed by many teachers were metric system and fractions. These two tOpics were also rated as highly needed as the earlier data had shown. When the data were further analyzed, some other problems that an inservice program can help teachers with were found. These problems include those of topic placement at a particu- lar class level and those dealing with the application of 149 mathematical concepts to other areas of learning. As an ex— ample, a Primary 2 teacher listed "long—division" as a diffi— cult teaching tOpic in his class. It is most unlikely that many Primary 2 children, most of them seven years old, have the readiness and the prerequisites for long-division. Other problem areas indicated with a high frequency by teachers were those related to "addition with carrying" and "subtraction with changing." The data here suggested the need to help teachers, eSpecially lower primary teachers, with these basic but important arithmetical concepts. In addition to the mathematical topics for inservice pro- gram, teachers were asked to rate a set of inservice learning experiences in order of their greatest needs. The chosen priority by teachers was "learning the subject matter of mathe- matics covered in the syllabus." Table 12 shows other in-' service experiences rated by teachers in order of priority. The learning experiences were rated by an indication of l for the first priority, 2 for the next and so on. The data were analyzed by taking the average index for each learning ex- perience, hence the statement with the least average has the highest priority. It might be suggested from the data that teachers' first priority is a combination of learning the subject matter of mathematics and learning about child development. It could be argued that the six topics offered for priority selection did not necessarily include one or more suggestions that 150 teachers might have suggested for inclusion in future train- ing program. The teachers were, however, given a chance to add any new topics that might be uppermost in their minds. There were no additional topics or comments on inservice learn— ing experiences requested by the teachers. TABLE 12 Choice of Inservice Learning Experiences by Teachers in Order of Priority 143.2 S.D. Learning the subject matter of mathematics covered in the school syllabus. . . . . . . . . . . . . . . . . 2.52 1.6 Study of child development, learning problems of pupils and methods of dealing with them . . . . . . . . 2.57 1.5 Teaching methods, aids and materials. . . . . . . . . . . 3.09 1.2 Planning and Preparing Lessons. . . . . . . . . . . . . . 3.12 1.3 Method of dealing with large classes including pupils with varying abilities . . . . . . . . . . . . . 4.41 1.4 Setting, marking and interpreting teacher—made tests and school examinations . . . . . . . . . . . . . . . . 5.58 1.2 On the whole,the data seem to suggest that the teachers would want an inservice mathematics training program to include not only the subject matter of mathematics, but other learning experiences which include the study of child development, teach- ing methods, aids and materials. It might further be suggested that teachers would like to have learning experiences in the other areas such as planning and preparing lessons, setting and making examinations, and how to organize learning environment. 151 Organization of Inservice Programs This sub-section of the data analysis deals with re- sponses of teachers, headmasters and organizers on organiza- tional aspects of the inservice training programs.132 The issues investigated include the following: (1) mathematical content sc0pe and method, (2) location, duration and time, (3) related administrative questions, (4) inservice instruc- tors, and (5) factors that affect attendance at inservice programs. Mathematical Content and Method:-Each group of respon- dents was asked to respond the question, "What mathematical content should be included in the program?" Three alternatives were given with space provided for respondents to name other alternatives. The analyzed responses are presented in Table 13. The content of the primary school mathematics syllabus appears to be the unanimous choice of teachers, headmasters, and an organizer. In response to the question dealing with method of approach for inservice mathematics program, a majority of the respondents indicated that the program should deal with the subject matter, methods and materials of instruction. A total of 75 percent of teachers, 83 percent of headmasters and 100 percent of the organizers who responded suggested this 132Responses analyzed here were those made to the ques- tions found in (1) Primary Teachers' Questionnaire, Section V; (2) Headmasters' Questionnaire, Section II; and (3) Organizers' and Sponsors' Questionnaire, Section II. trail."- 152 approach. Only seven teachers felt that inservice mathematics program should deal exclusively with the subject matter of mathematics. TABLE 13 Number and Percentage of Responses in Regard to Mathematical Content of Inservice Programs Teachers Headmasters Organizers Np, Pct. Np, Pct. N9. Pct. Content of the primary school mathematics syllabus. . . . . . . 186 49.2 42 52.5 1 100 Content that goes beyond the primary syllabus. . . . . . . . . 61 16.1 20 25.0 0 0 Content of both the primary and secondary school mathematics syllabi . . . . . . . . . . . . . 130 34.4 18 22.5 0 O Other (specify) . . . . . . . . . . 2 0.3 O O 0 0 TOTAL. . . . . . . . . . . . . 378 100 80 100 l 100 No response: . . . . . . . . . 2 O 4 Location, Duration and Timez—The questions of location, duration and timing are curcial to the inservice design. A definite trend in data showed a high percentage of respondents would like locations close to their school or town. Table 14 shows the summary of the responses. A teacher training college near the teachers' town or a secondary school in town was most frequently chosen by teachers and headmasters, about 71 percent of each group. An organizer, 33 teachers and six headmasters indicated a primary school in town or the village community center as possible location for inservice training. 153 TABLE 14 Preference for Inservice Locations Locations Teachers Headmasters Organizers Np. Pct. Np. Pct. Np. Pct. Teacher Training College near Town. . . . . . . . . . . . 136 35.8 29 36.3 0 O A Secondary School in Town . . . . 133 35.1 28 35.0 0 0 University Institute of Education. . . . . . . . . . . . 77 20.3 17 21.3 0 0 Others . . . . . . . . . . . . . . 33 8.8 6 7.4 1 100 TOTAL . . . . . . . . . . . . 379 100 80 100 1 100 No response: . . . . . . . . 1 0 4 A further analysis of data, as given in Table 15, showed that there were no differences in the responses of urban and rural teachers. Although an Institute of Education was chosen in the third place by about 20 percent of teachers, it might be suggested that it could be used as an inservice location for teachers in its neighboring schools in addition to its other inservice functions. TABLE 15 Location Preference of Urban and Rural Teachers in Percentage All Urban Rural Teachers Teachers Teachers % % % Teacher Training College near Town. . 35.8 35.4 38.1 A Secondary School in Town. . . . . . 35.2 35.1 34.9 University Institute of Education . . 20.3 20.2 20.6 Others. . . . . . . . . . . . . . . . 8.7 9.3 6.4 154 Preferences for the time and duration for inservice pro— grams were investigated by the question: When should inser- vice program be conducted? Respondents had four alternatives to choose from and a fifth section for them to add their own alternatives, if any. One alternative was an example of a short concentrated course period, while the other three were long—sustained examples suggesting the use of a radio course or a correspondence course for a follow-up. The distribution of responses among the three groups is indicated in Table 16. The majority of teachers or 39.2 percent, chose to attend the three-week course during the long vacation. About 26 percent of teachers would attend a two-week short courses followed by a correspondence course during the school year. TABLE 16 Distribution of Responses Concerning Time and Duration of Inservice Training Programs Teachers Headmasters Organizers Np, Pct. Np, Pct. Np, Pct. Three weeks during the long vacation. . . . . . . . . . . . 148 39.2 28 35.0 0 O ) Two weeks during the long vacation ‘ followed by a correspondence course during the school year . . 98 25.9 28 35.0 0 0 Two weeks during the long vacation followed by a radio course during the school hours throughout the school year . . . . . . . . . . . 73 19.3 15 18.8 0 0 Three—hour sessions fortnightly throughout the school year. . . . 54 14.3 7 8.7 O O Other (please state). . . . . . . . 5 1.3 2 2.5 1 100 TOTAL. . . . . . . . . . . . . 378 100 80 100 l 100 No response: . . . . . . . . . 2 O 4 155 Headmasters were evenly divided on the first two alter— natives on timing for inservice program, with 35 percent say- ing it should last for three weeks during the long vacation and another 35 percent opting for two weeks followed by a correspondence course during the school year. Two other alter- natives were suggested under others. Four teachers suggested one and a half hours meeting weekly throughout the school year; and one teacher, two headmasters and one organizer suggested three weeks over the long vacation followed by a correspondence course. A test of difference on data yielded no significant difference in the responses of urban and rural teachers. It is to be noted that three—hour sessions fortnightly throughout the school year was not favorably received as an alternative by many teachers and headmasters. Related Administrative Questionsz-The findings on four administrative issues related to the organization of inservice programs are discussed here. They include: the issue of com- pulsion, reward for successful inservice attendance, the in- structors of inservice courses'and the participation of head- masters and inspectors of schools. The respondents varied on the issue of whether inservice mathematics programs should be made compulsory for all teachers or not. Table 17 shows that nearly 33 percent of teachers, 49 percent of headmasters and 100 percent of the organizers who responded said that inservice should be made compulsory for all teachers. 156 TABLE 17 Distribution of Responses on Whether Inservice Mathematics Programs Should be Made Compulsory or Not Statements Teachers Headmasters Organizers N9, Pct. Np, Pct. Np. Pct. Compulsory for all primary school teachers. . . . . . . . . 124 32.8 39 48.8 1 100 Optional for teachers. . . . . . . 65 17.2 5 6.2 0 O Compulsory for some teachers, who would become qualified to be special mathematics teachers in the primary school? (They may teach some other subjects as well.) . . . . . . . . . . . . . 189 50.0 36 45.0 0 0 Other (explain). . . . . . . . . . 0 0 O O O 0 TOTAL . . . . . . . . . . . . 378 100 80 100 l 100 No response: . . . . . . . . 2 0 4 On the other hand, 50 percent of teachers and 45 percent of headmasters indicated that the program should be made com— pulsory for some teachers, who would become qualified to be special mathematics teachers in the primary schools and they might teach other subjects as well. Only 17 percent of teach— ers and six percent of headmasters felt that the program should be optional to teachers. A suggestion from these findings is that the use of specialist teachers should be investigated for mathematics teaching in the primary schools. In the case of incentive or reward, four suggestions were given and respondents were to add any additional rewards. About 30 percent of teachers and 34 percent of headmasters in— dicated a promotion prospect as a reward, while 32 percent of 157 teachers and 33 percent of headmasters asked for grades to- wards a certificate. There was a number of teachers, 43, who wanted both types of credits as a reward for successful atten- dance at inservice program. Only about eight percent of teach— ers, five percent of headmasters and an organizer indicated a certificate of attendance as the required reward. Table 18 gives a breakdown of the responses. TABLE 18 Distribution of Responses Concerning the Credits to be Given for Successful Attendance at Inservice Programs Teachers Headmasters Organizers N_o. £92 in I’LL Fa re- A certificate of attendance. . . . 31 8.2 4 5.0 l 100 A reward towards promotion prospects. . . . . . . . . . . . 114 30.1 27 33.8 0 O A grade towards the attainment of a teaching certificate. . . . 120 31.6 26 32.5 0 0 An increment in salary . . . . . . 71 18.7 20 25.0 0 O Other (please state) . . . . . . . 43 11.4 3 3.7 0 0 TOTAL . . . . . . . . . . . . 379 100 80 100 l 100 No response: . . . . . . . . _ ,1 0 4 There was a unanimous agreement on the question of whether headmasters and school inspectors should participate in mathe— matics inservice program. Ninety-five percent of teachers, 95 percent of headmasters and an organizer responded that head- masters and school inspectors should participate. 158 Inservice Instructors:-Teachers were asked to rate four groups of people as inservice instructors in order of prefer- ence. The teachers of Teacher Training Colleges were the most preferred; followed by university lecturers and the well— qualified secondary school teachers in the neighborhood; and lastly by the_school inspectors. Table 19 shows the mean dis— tribution of the choice of instructors for inservice program. TABLE 19 Preference for Inservice Instructors by the Teachers Inservice Instructors Mean S.D. Tutors from Teacher Training College. . . . . 1.90 0.80 University Lecturers. . . . . . . . . . . . . 2.40 1.26 Well-qualified Secondary School Teachers in the Teachers' Neighborhood . . . . . . . 2.41 1.01 School Inspectors from the Ministry of Education. . . . . . . . . . . . . . . . 3.20 0.95 No response: 3 in each case Factors Affecting Wish or Ability to Attend Inservice Program:-In order that individual circumstances of teachers might be put into consideration as much as possible in de- signing inservice programs, teachers were asked to identify factors that could stOp them from participating in inservice mathematics programs. Teachers checked some or all of five given factors that might stop them from participating, and were given a chance to indicate any other factors. These factors can be roughly classified into professional, domestic and financial factors. 159 On the professional factor, very few teachers indicated that inservice training was not helpful as shown in Table 20. However, the list of other factors given by teachers included a few professional issues such as non-ayailability of reward, lack of provision for follow-up and "inservice training not interesting." The financial factors such as cost of trans- portation to inservice center and fees charged for training programs would stop many teachers from participating. Sixty- three percent of teachers would not attend if they were to pay transportation cost that was too high, while 59 percent would not attend if fees were charged for the training program. TABLE 20 Factors Affecting Teachers' Wish or Ability to Attend Inservice Programs Y_e:s_ __ E9. 2 Cost of Transportation . . . . . . . . . . . . . 238 63.0 140 37.0 Fees Charged for Training Program. . . . . . . . 223 59.0 155 41.0 Family and Other Domestic Responsibilities . . . 155 41.0 223 59.0 Dislike of Travel from Home. . . . . . . . . . . 72 19.1 306 80.9 Inservice not Helpful. . . . . . . . 1 . . . . .. 33 8.7 345 91.3 Others . . . . . . . . . . . . . . . . . . . . . 32 8.5 346 91.5 No response: 2 in each case The proportion of teachers who disliked travelling from home was smaller than those who would not attend due to finan- cial reasons. Some other personal factors indicated included illness, mentioned by five teachers and seven teachers stated “-.~__,._ fl“ .. _._’ 160 motor accidents on the road as a factor that would stop them from attending inservice programs. Women teachers more than men teachers mentioned "family and other domestic responsibilities" as a factor that could stop them from participating in inservice programs. The data showed that 34 percent and 50 percent of men and women respec— tively felt this factor could stop them. The differences were significant beyond the .01 level. Similarly, an analysis of differences between the subgroups of men and women showed sig- nificant difference on the item of "cost of transportation" as a factor that might impede attendance. There was no sig- nificant difference between men and women on other factors. A further analysis of differences between urban and rural teachers on factors that might affect attendance at inservice program was carried out. There were no differences in respon- ses on all indicated factors. Evaluation of Inservice Effectiveness The need for an evaluation of inservice training program had been highlighted through the review of literature. The absence of such evaluations is one of the problems plaguing inservice practices in Africa (Trevaskis, 1969). In order to find out what were the views and preferences of respondents on the process of evaluation in inservice program, they were asked to respond to the question: How should we decide on the effectiveness of an inservice mathematics program? fl "u..."- w 161 Preference for evaluation through the use of tests of mathematical understanding for the inservice participants and theirpupilsandthroughanievaluathmmoftheirclassroom prac— tices was expressed by 45 percent of teachers and 64 percent of headmasters. sis. ing alone. issue. TABLE 21 Distribution of Responses on How Inservice Programs Should be Evaluated Table 21 gives further details of the analy- Only four teachers wanted evaluation by practical teach- There was no response from the organizer on this Teachers Headmasters Organizers Np, Pct. Np, Pct. N9, Pct. Through an evaluation of partici— pant's understanding by giving him/her some formal and inform— al tests. . . . . 124 32.9 18 22.5 0 0 Through formal and informal tests for the participant, and other tests for the pupils he/she teaches . . . . . . . . . . 80 21.2 11 13.8 0 0 Through tests for participant and pupils, and through an evaluation of his/her class- 3 room practices in teaching , mathematics . . . . . 169 44.8 51 63.7 0 0 Other (please state). . . . . . . 4 1.1 O 0 O 0 TOTAL. . . . . . . . . . 377 100 80 100 O 0 No response: . . . . . 3 162 Demand for Future Courses The different sections of the analysis so far seem to have indicated that although the periodic workshops and the occasional inservice programs organized for primary teachers are helpful, they are not enough. Many teachers and head— masters recognize this and they are eager to make up for their own inadequacy. Appropriately, many additional comments came from teachersin thelastsectrn1ofthequestionnaire. One hundred and eighteen teachers or 31 percent of total number of teachers made additional comments. When the open-ended reSpon- ses were analyzed, they were classified into four different but overlapping categories, all representing some forms of de— mand for future inservice mathematics programs. These cate— gories are: (1) further suggestions on planning and organi- zation, (2) call for governmental policy on inservice mathe- matics program, (3) the importance of the inservice program and (4) the need for appropriate remuneration for teachers. The following are examples of the demands in the words of the teachers under the specified categories: 1. Further suggestions on planning and organization-- The inservice programme should be given a wide scope and attention. It should be made available to all teachers. It should be made interesting in order to remove the dislike that both teachers and pupils now have for mathematics. The course should be organized in every area both rural and urban. The government should support the programme. If the centre of the inservice training is within fifty miles to the stations of the participants, Tier: ”3‘56?” ’ ..- . -/ ‘ ’fi . 163 many teachers will attend the course. Well qualified teachers should handle the teaching of mathematics. It should be made practical and linked with everyday events. I suggest we have regular refresher courses on mathe— matics. It should be regularly organized for teach— ers on new methods of teaching mathematics. A call for governmental support and policy—— This course should be successful if the government or authority should not mind to run the expenses that it may cost. In order to get a good result on this programme, the government should provide qualified teachers for the course. It should also provide useful teaching aids and create many centres in the state for the course. To be more effective, I wish the Federal Government takes over the organization of the inservice mathe- matics programmes throughout the country. Importance of the Inservice Program—— If this type of inservice mathematics program is intensified, it will go a long way to help many teachers who have no first class knowledge of this subject. The earlier the programmes start the better. My comment is that this scheme should start quickly so that teachers can be in a better position to help the young ones who are to build the nation. 3 The Need for Remuneration to Teachers-- Future participants in inservice mathematics pro- grammes whould be encouraged by the provision of (a) transportation allowance, (b) comfortable living accommodation for those who live far away from the centre, (c) an award of certificate leading to promotion and increment of salary, and (d) they should not be transferred too often from school to school. I whole—heartedly support this programme but to make it a success please make arrangement for incentive to participants. Otherwise, teachers will be attend- ing the course reluctantly and grudgingly and they 164 may not benefit from it at all. They will leave the training centres and leave behind everything they have learnt. The teachers' comments have relevance not only for the organizational aspects of inservice mathematics programs but also for some fundamental issues that are necessary for effec— tive inservice programs. The issues touched upon by the. teachers' comments include those related to cost of the pro- gram, adequate material and human resources, and above all, the need for definite governmental policy on inservice mathe— matics program. It is interesting to note that one of the teachers considered the need for mathematics retraining pro— gram great enough to demand not only a state government policy but a national government policy as well. Another source of data on the demand for future inservice mathematics programs came from the responses of the headmasters in their answers to the questions on how the Institute of Edu- cation at Ife and the Ministry of Education can help them with their problems of arithmetic teaching. The headmasters' indi- cated needs ranged from the demand for the organization of inservice programs, the demonstration of the use of teaching aids, to the reward for successful attendance at inservice programs by the Ministry of Education. The demand for future mathematics inservice program was well—expressed in the final comment of an organizing body who wrote: 165 There is a genuine interest among primary school teachers. Take the help to them where possible -- make it available for all inter- ested teachers. Selection of a few "favoured" ones does not encourage others to make an ef— fort. Almost all primary teachers are mathe— matics teachers. Discussion This section of the analysis had dealt with views and pre- ferences on mathematical content and other learning experien- ces, the organization of program and its evaluation, and the demand for future inservice mathematics programs. Just as classroom procedures and materials are important for success- ful mathematics teaching and learning, so is an understanding of the basic concepts of mathematics on the part of the teach- ers. As O'Daffer succinctly expressed this idea:l33 The good mathematics teacher: knows the children, knows mathematics, is skillful at involving children with mathematics, and makes use of his knowledge of the types of learning involved. Mathematical topics for inservice program is in keeping with the international consensus on the mathematics content that elementary teachers should know (Howson, 1973). There is, however, no general agreement on the depth to which these topics should be treated in any teacher training programs. The findings of this survey seem to suggest that teachers should be helped to see mathematics as concerned with formu~ lating and solving problems, through activity-oriented approach, 133Phares G. O'Daffer. "On Improving One's Ability to Help Children Learn," Arithmetic Teacher, Vol. 19, No. 11 (November, 1972), pp. 519-526. —~‘h 166 covering at least the scope of work in the primary school syllabus. Although teachers' preferences might not all be realiz— able in practice because of lack of resources and shortage of leaders, nevertheless, it is worthwhile that the preferences were established as a basis for planning. It is clear from these findings that teachers would like inservice centers located near their homes or towns. The preference of teachers on timing and duration is, however, not very clear. Majority of the teachers indicated preference for the one-shot, three- week course. This contradicts the earlier View expressed by teachers that refresher courses should last for a longer period and should have a follow-up. Furthermore, in the final addi— tional comments, many teachers repeated their belief that a long-sustained program would be of greater help to them than the short courses. It appears from these data that a follow— up by a correspondence course is preferred to a follow-up by a radio course. Perhaps the non-availability of radio in some areas is a problem here. The choice of a correspondence course has both its advantages and disadvantages; Clearly, it offers the teacher an opportunity to learn and relearn the materials at his own rate, but for it to be effective, it must be well- planned, well—coordinated and it needs capital costs for equip- ment and materials. Whether by radio, correspondence or other— wise, follow-up activities undertaken regularly are crucial to the effectiveness of inservice programs. 166 covering at least the scope of work in the primary school syllabus. Although teachers' preferences might not all be realiz- able in practice because of lack of resources and shortage of leaders, nevertheless, it is worthwhile that the preferences were established as a basis for planning. It is clear from these findings that teachers would like inservice centers located near their homes or towns. The preference of teachers on timing and duration is, however, not very clear. Majority of the teachers indicated preference for the one-shot, three- week course. This contradicts the earlier view expressed by teachers that refresher courses should last for a longer period and should have a follow-up. Furthermore, in the final addi- tional comments, many teachers repeated their belief that a long-sustained program would be of greater help to them than the short courses. It appears from these data that a follow- up by a correspondence course is preferred to a follow-up by a radio course. Perhaps the non—availability of radio in some areas is a problem here. The choice of a correspondence course has both its advantages and disadvantages; Clearly, it offers the teacher an opportunity to learn and relearn the materials at his own rate, but for it to be effective, it must be well— planned, well—coordinated and it needs capital costs for equip— ment and materials. Whether by radio, correspondence or other- wise, follow-up activities undertaken regularly are crucial to the effectiveness of inservice programs. 167 A trend in the analyzed data seemed to suggest that some teachers would prefer to have special teachers of mathematics in the primary schools. Although the use of special teachers of elementary mathematics is receiving increasing attention in some developed countries, yet such a practice will be unreal— istic at this stage of educational development in Western State, Nigeria. First, the supply of mathematics teachers for the secondary level is still being met precariously. Secondly, research studies in societies where such special teachers have been used failed to show the superiority of the achievement of pupils taught by these special teachers of elementary school mathematics. Spitzer134 reported a recent study based on special—teacher plan specifically set up to bring out superior achievement in elementary mathematics. The study found no greater achievement on the part of the pupils taught by these special teachers. In addition, such plans imply departmental- ization of instruction at the primary school level and this might create other administrative and curricular problems. As mentioned in the last section of this analysis, both intrinsic and extrinsic motivations are crucial to the success— ful practice of an inservice program. There is evidence from the data that teachers are intrinsically ready to participate in mathematics inservice programs. Their views on the extrinsic 134Herbert F. Spitzer, Teaching Elementary School Mathe- matics: What Research Says to Teachers, No. 2 (Association of Classroom Teachers Material Education Association, Wash— ington, D.C.), June, 1970, p. 17. 168 rewards were clearly made in terms of a recognized certificate or a reward towards a promotion prospect as Opposed to the pre- sent practice of the award of a certificate of attendance. There is a growing dialogue among the Nigerian educators on how to reward a teacher's successful participation in an in— service program. The problems stem from the fact that most inservice programs are ad hoc, not systematic, not based on clearly defined goals, and they lack systematic evaluative process. A well-planned program should suggest evaluative measures by which the performance of the participants can be measured and adequately remunerated. As Rubin135 confirmed not all incentives have to be mone— tary. The recent economic development in the country, referred to in Chapter II, showed that a review of salaries have led to an increase in salary of primary school teachers since these data were collected. Nevertheless, there is still a place for the type of incentive that comes as a reward for superior teaching. Rubin added that this reward can be, "an honorific title like Master Teacher, if it carries sufficient distinction, might do as well as a monetary reward."' This type of reward for teachers would necessarily be built on a performance- based criteria for superior teaching which could be establish— ed by the inservice organizers, school administrators and the teachers. The establishment of such a reward should give 135Louis J. Rubin, op. cit., p. 247. 169 teachers added motivation to participate in inservice pro- grams and sustain their interest. The overwhelming choice of mathematics tutors of teacher training colleges as a preferred group of inservice instruc— tors above the other groups is perhaps an indication of a long—standing need for a closer link between preservice and inservice training of primary teachers. Because of the scarc— ity of such teachers, however, there would be the need for a collaboration and coordination with other mathematics teachers at all other levels of education. Be that as it may, the establishment of inservice centers at the Teacher Training Colleges, the Advanced Teachers College, and the Institute of Education in the state should go a long way in providing some solutions to the problems of inservice training, especially because both physical and human resources could be shared for preservice and inservice programs. In general, one can make the conclusion from the survey findings that the primary teachers have the necessary readiness for the inservice mathe- matics education programs. Perhaps a further indication of this readiness and the willingness to be identified with a program of this nature were very well demonstrated in the responses to the last question in the questionnaires. Although the questionnaires were to be completed anonymously, 93 percent of teachers, 98 percent of headmasters and an organizer indi— cated their names and addresses in the optional Space provided in order to have a summary of the survey findings. 170 SUMMARY OF MAJOR FINDINGS The analysis of the survey data presented in this chapter revealed the following: 1. There are many problems of arithmetic teaching in Western State primary schools. Among them are: lack of qualified teachers, lack of teaching aids and other materials of instruction, and the lack of adequate provision for inservice training. Some short inservice courses have helped teachers, while others have not. Most inservice mathematics programs have been organ— ized by the government and the universities. A majority of teachers have never participated in inservice mathematics programs, but are willing and ready to participate in such programs in order to upgrade their mathematical knowledge and teach arithmetic better than they do. Teachers' attitude and classroom practices in arith— metic teaching are generally favorable, but there is need for improvement. They lack adequate and appropriate materials of instruction including teachers' reference books and manipulative materials for the pupils. Inservice learning experiences should include not only mathematics content but also content related to child development, use of materials of instruction, and classroom organization. A majority of teachers would like inservice programs to take place close to their schools or towns. Many teachers are willing to travel a short distance to inservice centers, if financial and physical supports are provided. The most common suggestions given by teachers, head— masters and organizers for the successful organization of inservice programs are (a) the programs should be extended to all teachers, (b) they should be long- sustaining, (c) instruction should be through prac— tical approach with adequate provision for teaching aids, (d) appropriate evaluation should be planned and (e) arrangement should be made to remunerate teachers adequately. 171 The differences in the responses of different sub— groups of teachers are not significant except in (a) the responses of lower and upper primary teachers in the choice of the topic —— Solid Geometry, (b) male and female teachers on "family responsibility" as a possible reason for non-participation in in- service mathematics programs. CHAPTER V A MODEL FOR INSERVICE MATHEMATICS EDUCATION This chapter contains a description of the proposed in— service mathematics education model for primary school teachers in Western State, Nigeria. The model is based on a synthesis of findings from the review of literature, the survey find— ings, and the social and economic factors in the State. RATIONALE FOR MODEL OBJECTIVES As mentioned in Chapter I, the basic strategies for the design of the inservice model are flexibility and adaptability. Because of the differences between and among teachers, quali— tative improvement in education never takes place on a solid front. There will always be individuals and institutions that adventure and explore and others that lag behind. This means that two opposing demands are made on a school system by any plan for curriculum improvemenf. First,,it must be flexible enough to give the ablest teachers freedom to experiment and blaze the trail, and secondly, it must be ready to give the weaker teachers the guidance and support they need. Achieving the proper balance between these two demands, and adjusting the balance steadily as new generations of teachers who could take advantage of the new freedom arises, form important 172 173 basis for the rationale of inservice model objectives. In gen— eral terms, broad statements of objectives are given. Where specific outcomes are stipulated, the methods of approach to such outcomes are not rigidly specified and they could be modi- fied to suit the conditions of different groups of teachers and school administrators. The rationale for inservice model objectives is further based upon the fact that primary mathematics teaching in West— ern State is in a transitional stage. The inservice model pro- grams should therefore help teachers to teach better what they are now teaching and at the same time help them towards the requirements of the future curriculum A MODEL FOR INSERVICE EDUCATION The inservice model presented here consists of four major processes. They include: (1) establishment of objectives for inservice training, (2) selection of learning experiences, (3) organization of inservice programs, and (4) a program of in— service evaluation. Establishment of Inservice Objectives T Criteria for Establishing ObjectivesE-The first step towards meaningful, worthwhile inservice activity is the identification of the needs of teachers and other staff members (Harris and Bessent, 1969). To bypass this step, Harris and Bessent asserted, is like attempting to build a house without providing first a proper foundation. Therefore, a major criterion for the establishment of inservice model objectives was found in the 174 identified needs of teachers for the teaching of mathematics, which were analyzed in the preceding chapter. When stated in broader terms, the following criteria were considered in form— ulating the inservice model objectives: 1. To what extent has the proposed model identified teachers' needs? Teachers should be involved in defining and clarifying training needs. The design of training model must rest on this teacher involvement and attempt should be made to offer the training opportunity to all teachers. To what extent does the training activities link cognitive and affective objectives of mathematics learning? A major concern of learning should be to activate and link the cognitive, affective and application aspects of the learner. To what extent does the training program offer parti— cipants the opportunity to learn through the new approaches of mathematics teaching? The total sequence of inservice learning activities should be organized as problem—solving efforts in which the participants take initiative for inquiry. To what extent does the training program allow for individual differences in teachers? The inservice program design should allow for and plan for indivi- dual differences in readiness, sophisticated and content needs of teachers. To what extent does the training program plan for a follow—up? Provision should be made in the in— service model for apprppriate continuing support of efforts the inservice participants make to use the training experiences. To what extent does the training activity give pro— mise of helping members with more than one aspect of their behavior? Effective inservice work in mathematics should focus not only on teachers' understanding of the structure of mathematics, the methods and materials of instruction, but also on theories on how children learn mathematics. To what extent does the training activity support on-the—job experimentation? Inservice training is 175 not of much help unless participants feel they can try out new ideas in the setting where the problems which stimulated the training arose. To what extent does the training activity contain provision for its own evaluation and self-correction? Unless members can discuss and criticize training activities, it is very difficult to improve them. Such evaluation helps to uncover new training needs for the future. A Suggested List of Objectives for Inservice Mathe— matics Modelz—The following are suggested general objectives for the inservice mathematics education model program of pri- mary school teachers in Western State of Nigeria- 1. To promote a well—organized and carefully planned continuing inservice education program designed to upgrade the mathematical competencies of teachers towards the improvement of classroom instruction. To suggest instructional and supervisory techniques and practices which have been found effective and calculated to meet local needs. To stimulate a centrally—coordinated inservice pro— gram plan that can tap all resources, i.e., state and local governmental education bodies, teachers colleges and universities, teachers' professional organizations, mathematics associations, and civic organizations, in carrying out a continuing program of inservice education. To stimulate the educational field in the preparation and production of instructional materials that may be produced on local or statewide basis. To stimulate teachers and other staff personnel of the inservice centers in conducting action research especially those related to how the Nigerian child learns mathematics, and in preparing evaluative in— struments and tests. To develop evaluative criteria for appraising the outcomes of the inservice education activities. 176 Selection of Learning Experiences Objectives for Inservice Learning Experiencesz—The model for inservice programs is designed primarily to help teachers improve their mathematical understanding and their classroom performance in the teaching of mathematics. The discovery or the activity approach required by the new approaches to mathematics teaching demands that the teachers play new roles in the classrooms. These roles include being a guide or a counselor whose main task is to stimulate pupils' activity ‘ and learning. Dienes136 remarked that this is a very difficult i role, since it requires the teacher to change personality, to change firmly fixed attitudes towards learning, students and objectives of teaching, and to refrain from being too much of a distributor of pre-organized knowledge as he may have been accustomed to be. The objectives of inservice learning ex- periences therefore include helping teachers develop behaviors that deal with the knowledge of mathematics as well as the techniques and procedures used in providing and managing in— struction through the use of'adtivity—oriented approaches. The following are representative examples of the objectives of inservice learning experiences stated in the form of the ex— pected behaviors of the teacher: Objectives related to the mathematics content —- 136Z.P. Dienes (ed.) Mathematics in Primary Schools. Hamburg: UNESCO Institute for Education, 1966, p. 102. 177 The teacher demonstrates mastery of mathematical concepts being studied by his pupils. The teacher exhibits an improvement in his arith- metic skills and teaching methods by learning fresh approaches to old subject matter. The teacher selects activities that help his pupils to achieve the objectives of the mathematics cur— riculum. Objectives related to Teacher-Pupils Interaction -— 1. 3. The teacher does not criticize negatively a student's contributions to a group discussion or to other group work. The teacher responds to student statements by asking for validation or justification of the mathematical ideas expressed. The teacher asks questions and leads discussion, rather than "telling." Objectives related to Class Practices -— 1. Given an activity that requires pupils to work in— dividually, in small groups or in large groups, the teacher organizes the students in the appropriate mode. The teacher allows pupils to move purposefully about the room to obtain materials, to consult with others, or for other task-oriented reasons. The teacher allows pupils to interact verbally while working on an activity. Using the appropriate assessment instruments, the teacher assesses pupils' performance and completes performance record. When presented with a pupil who has not mastered a mathematical objective, the teacher can choose an activity that will help the pupil reach that objective. When given the appropriate information on pupil achievement, the teacher can classify pupils into two groups--those that have sufficient mastery of prerequisite behaviors to start a new topic, and those that do not. 178 Objectives related to the use of materials of instruction -- l. The teacher arranges the materials of instructions and furnishings in the classroom in a way that fosters learning. 2. The teacher makes use of concrete materials to simplify abstract ideas. Consideration of Breadth and Depth of Inservice Mathe- matics Content:-A comparison of the list of mathematical 137 topics desired by the primary school teachers of Western State with two elementary school mathematics inservice pro— grams showed its adequacy when the scope of topics covered were considered. Houston,138 for example, suggested the following as the mathematics topics common to many inservice mathematics programs: Elementary Set Theory Distinction between conceptual notion and symbolization of that idea (number—numeral, operation-algorithm, line-representation of a line, etc.) Number Systems, Numeral Systems Equality, equivalence and inequalities Properties of arithmetic Operations with integral and rational numbers Measurement and simple geometry Problem solving and mathematical models Similarly, the African Mathematics Program139 suggested the following major mathematics topics for the inservice 137See pagelAJ'of Chapter IV: Analysis and Interpretation of Survey Data. 138W.R. Houston "The Challenge of Inservice Education," in Evelyn Weber and Sylvia Sunderlin (eds.) New Directions in Mathematics, Washington, D.C.: Association for Childhood Education International, 1965, pp. 65—70. 139 African Mathematics Program, Education Development Center, Mathematics Syllabus Sourcebook for African Training Colle es (Inservice Courses), Newton, Mass.: Affican Mathe- matics Program, 1970, pp. 2-5. 179 program of teachers without a secondary education: Set Modular Arithmetic Number Multiples and divisors Numeration Primes and Composite Fundamental Operations Divisibility Tests of Arithmetic Fractions Use of Properties to Integers perform operations Rational numbers Fundamental Operations Ratio and proportion with fractions Parallel lines Decimal Fractions Measurement Applications of Frac— Scale Drawing tions to problems Non-metric Geometry Symmetry Undoubtedly these different inservice programs had fol- lowed to a great extent the recommendations of the Cambridge Conference on Teacher Training of 1966.140 However, even though there are several recommendations in literature as to what mathematics topics should be covered in an inservice pro— gram, there is little or no consensus as to the depth to which the topics should be covered. Of great importance to any consideration of the content of elementary school mathematics today is the new movement of critics of the new mathematics. In the United States, for example, this movement has been so strong in some areas that it has been called "the Second Revolution in School Mathe- matics," the first being the new mathematics movement itself. Among these critics are parents and some teachers who are 140Education Development Center, Inc., Goals for Mathe- matical Education of Elementary School Teachers (A Report of Cambridge Conference on Teacher Training), Boston: Houghton Mifflin Co., 1967, p. 26. 180 calling for the return to such basics as the multiplication tables. Professor Morris Kline, a notable mathematician who Opposed the new mathematics from its inception, urges a new direction, "diametrically Opposite to that taken by the new mathematics."l4l He maintains that the reason why the new mathematics is a failure is because it is directed towards the minute fractions of students who will one day become pro- fessional mathematicians. The rest of the students, he claims, are left with scarcely enough mathematical capability to per— form simple arithmetic operations, and certainly not enough 142 to fill out an income tax form. Kline recognizes that the old methods of teaching mathematics were imperfect, but argues that rigorous theories and heavy reliance on terminology and symbolism are not the solutions. He calls on teachers to teach mathematics not for its own sake but as an integral part of the culture. Appropriately, the mathematics educators are continually reviewing the programs for better mathematics teaching. An evidence of this continual search for improvement is in the recommendations of the Snowmass Conference on school mathe- 143 matics curriculum in 1973. The recommendations covered issues on goals, teacher education, evaluation, and the 141Morris Kline, Why Johnny Can't Add: The Failure of the New Math, New York: St. Martin's Press, 1973, pp. 144- 170. 142 143Mathematics Education DevelOpment Center, Report of the Conference on the K-12 Mathematics Curriculum, June, 1973, Bloomington, Indiana: Indiana University (mimeographed). Ibid., (frontpiece). 181 teaching of mathematics for a computer era. A very great emphasis was placed on the teaching of problem solving and on the applications of mathematics in the recommendations. A logical direction for action was well expressed by the New York teacher who stated:144 We realize that part of the new math is very good and some of it is not necessary and some has to be phased out. We found that we can use the best of the new and the Old. The implication of the current movement for developing societies such as the Western State of Nigeria is far-reach— ing. Reforms in mathematics curriculum in these societies should borrow from the experiences of reformers of the last two decades in the more developed nations. The selection of mathematics content and methods for inservice training of teachers was, therefore, guided not only by the expressed needs of teachers for the understanding of mathematics but also by the utilitarian aspects of the topics, both to the teachers and the pupils. Suggested Mathematics TOpics for Inservice Mathematics Programz-The proposed list of mathematics topics for inservice program (Table 22) is designed in accordance with findings from the survey and the literature review; and the guidelines suggested by curriculum improvement programs in Nigeria. It encompasses ten units Of work with an eleventh unit which 144Cited in: Fred M. Hechinger, "Math: Integrating the New and the Old," The New York Times, Section 13, Sunday, May 4, 1975, p. 1. 182 consists of additional topics. The additional topics may be regarded as Optional topics though they are recommended where and when teachers have the prerequisites for them. The order of presentation Of units is not rigid. However, the measure— ment unit is deliberately named as unit one to reflect the need of teachers, and also that it might receive priority in program implementation. It is suggested that application to real life situation and problem-solving form important part of each topic treatment and that the technique of spiraling the curriculum will be used in carrying out inservice instruc- tion covering the topics. TABLE 22 Mathematics Topics for Inservice Training Programs Unit 1. Measurement Non—Standard and Local measures Metric System and Conversion-—Length, perimeter, area, capacity, weight, and volume Time, Temperature and Money Unit 2. Numeration Ancient numeration systems3 Different numeration systems and place values Local Nigerian number names and the grouping patterns Unit 3. Set and Logical Games Matching, Joining and Separating with limited use of notations and terminologies Set Operations Unit 4. Whole Numbers Counting and One-to-One Correspondence Order and Cardinality The Number Line The fundamental operations with whole numbers 183 Table 22 (cont'd.) Unit 5. Common and Decimal Fractions Meaning and Notations Equivalent fractions Decimal fractions Fundamental Operations with fractions Ratio, Proportion and Percent Application of fractions to problems Making Estimations Unit 6. Number Theory Prime and Composite Factors, Multiples, Divisors Divisibility Rules Exponents and Exponential notation Unit 7. Basic Concepts of Geometry Points, lines and planes Parallel lines Solid shapes Simple Geometric Constructions including scale drawing Symmetry and Congruence Motion Geometry Unit 8. Business and Civic Arithmetic Discount, Profit and Loss Simple and Compound Interests Simple Cash Discount Taxation, Electricity Bills, Rates, etc. Unit 9. Statistics, Probability and Graphs Pictorial Representation CT data-— _ Bar chart, pictogram, line graph, circle graph Interpreting graphs Mean, median, and mode Simple ideas of probability (experiments with coins and dice) Unit 10. Problem Solving and Applications Making up and solving mathematical sentences Solving mathematical problems Unit 11. Additional Topics Topological Ideas Equations and symbols Formulae and Subsituations Integers and Real numbers Modular Arithmetic . —~‘_ 184 Other Learning Experiencesz—The International Congress on Mathematical Education identified three main strands as essen- tial in primary school mathematics teaching. They are: mathe~ matical knowledge, insight into children's learning and its 145 The in- goals, and classroom procedures and materials. service training program should therefore include other learn— ing experiences in addition to the mathematical content dis- cussed above. Other learning experiences to be included in the in- service programs are outlined as follows: 1. The study of child development, learning problems Of pupils and methods of dealing with them. 2. Teaching methods, aids and materials. 3. Planning and preparing for the lessons. 4. Methods of dealing with large classes including pupils with varying abilities. 5. Setting, marking and interpreting teacher-made tests and school examinations. 6. All other learning experiences required by teachers according to their immediate needs. Whatever the choice of topics for a particular inservice program series, the content shouldlmfl$> the participants with their teaching or other related assignments. Suggestions of various ways of teaching a mathematical topic should be given in the same lecture-discussion in which the topic itself is considered. This integrated approach to content and method in mathematics learning and teaching has proved to be effective 145Howson, A.G., Op. cit., p. 46. 185 in helping teachers improve their mathematical competencies as evidenced by the studies of McLeod (1965) and Hunkler and Quast (1972). Furthermore, to insure the success of the programs, learning experiences must involve the participants in active- ly learning by listening, preparing notes, studying together, engaging in informal discussion, experimenting in workshop or laboratory setting, and reading and criticizing texts or other instructional materials. In other words, if activity—learning approach is to be transferred into the teacher's classroom, the inservice programs should give him such learning experien— ces. Specific instructional strategies that have increased mathematics achievement of the learners should also form part of the learning experiences in the inservice programs. Refer- ence was made in Chapter III to the method of guided dis- covery and the research findings on its effectiveness in mathe~ matics teaching. However, if the method is badly handled it can result in a waste of time and even chaos. If well handled, it enables pupils' motivation to be discovered and accustoms them to purposeful activity, to team work and independent work, and to seeking and using knowledge. Again, to insure the advantages of the discovery approach in the primary class- rooms, inservice experiences for teachers should include learning by discovery. Two other instructional strategies that have effectively contributed to mathematics learning deserve to be considered 186 for inservice instructional strategies. They are the mastery learning approach and the programmed learning approach. Many studies carried out under school conditions indicate that mastery learning strategy has marked effects on the learner's cognitive and affective development and their learning rate. In its simplest form, the strategy, as prOposed by John B. Carroll,146 was based on the theory that if each student was allowed the time he needed to learn to some level and he Spent the required learning time, then he could be expected to attain the level. However, if the student was not allowed enough time, then the degree to which he could be expected to learn was a function of the ratio of the time actually Spent in learning to the time needed: . _ (time actually Spent) Degree of learning — f ( time needed ) A corollary of this theory is that mastery learning strategies are designed to take into account individual dif— ferences among learners in such a way as to promote each learner's fullest cognitive and affective development. Fur- ther studies have also shown that mastery learning approach seems to help most students overcome feelings of defeatism and passivism brought to the learning situation.147 The powerful affective consequences of mastery learning approach may be attributed to many factors, the most important SECTION V In this section, we would like to know your needs for and your views on the organization of mathematics inservice programmes. Please put a circle around the letter for the statement that best describes your View in each case: 1. What mathematical content should be included in the programme? Content in the primary school mathematics syllabus Content that goes beyond the primary syllabus Content of both the primary and secondary school mathematics syllabi d. Other (explain) 229 Should the inservice training programme deal with: a. Subject matter of mathematics only? b. Newer methods of teaching mathematics only? c. Subject matter and methods interwoven? d. Subject matter, methods and materials of instruction? e. Other (explain) When should inservice programmes be conducted? a. Three weeks during the long vacation b. Two weeks during the long vacation followed by a correspondence course during the school year c. Two weeks during the long vacation followed by a radio course during the school hours throughout the school year d. Three—hour sessions fortnightly throughout the school year e. Other (please state) Where should inservice programmes be held? a. In the Teacher Training College near your town b. In a Secondary School in your town c. In an Institute of Education of a University d. Other (please state) Should the inservice programme be made: a. Compulsory for all primary school teachers b. Optional for teachers c. Compulsory for some teachers, who would become qualified to be special mathematics teachers in the primary school? (They may teach some other subjects as well.) d. Other (explain) What kind of credits should be given for successful participation in an inservice programme? a. A certificate of attendance b. A reward towards promotion prospects c. A grade towards the attainment of a teaching certificate d. An increment in salary e. Other (please state) I How should we decide on the effectiveness of an inservice mathematics programme? a. Through an evaluation of participant's understanding by giving him/her some formal and informal tests b. Through fromal and informal tests for the participant, and other tests for the pupils he/she teaches c. Through tests for participant and pupils, and through an evalu- ation of his/her classroom practices in teaching mathematics d. Other (please state) Should headmasters and school inspectors participate in the inservice programme for teachers? Yes No 10. 11. 12. l3. 14. 230 The following topics represent a sample of the content of inservice programmes in mathematics. Please consider your own needs for mathe— matics teaching and rate the topics in order. For example, put (1) in front of the topic of your greatest needs, then (2) for the next, and so on. The number 6 is for the topic you need least. a. Learning the subject matter of mathematics covered in the school syllabus ..................... . ...... b. Setting, marking and interpreting teacher—made tests and school examinations ..................... c. Planning and preparing lessons .................... d. Teaching methods, aids and materials .............. e. Study of child development, learning problems of pupils and methods of dealing with them ...... .... f. Methods of dealing with large classes including pupils with varying abilities ..................... List other ideas you have for inservice activities in mathematics: Please rate the following groups in order of your choice of teachers to teach you in an inservice mathematics programme. Put (1) in front of the group you would choose first as teachers, (2) for the next, and so on. The number 4 is for your last choice. a. School inspectors from the Ministry of Education.. Well-qualified secondary school teachers in your neighborhood ...................................... c. Tutors from the Teacher Training College .......... d. University lecturers. ................... . ....... .. Which of the following will stop you from participating in aninservice mathematics programme? (Check as many as are applicable to you) a. Cost of transportation ............ . ........ . ...... b. Fees charged for the training programmes .......... c. Dislike of travel away from home .................. d. Family and other domestic responsibilities........ e. Inservice training not helpful ................... . f. Other (please state) Hlll Are you currently taking any tuition in mathematics or any other subjects for GCE O'Level? Yes No Do you have any teaching method that you have successfully used in arithmetic and would like to share with other teachers in an inservice programme? Explain briefly: 15. 16. 231 Please feel free to add any additional comments you may have on in- service mathematics programmes below. (use the back of this page, if necessary.) If you are interested in receiving a summary of this questionnaire result, please indicate your name and a permanent address at which you can be reached. Name: Address: THANK YOU VERY MUCH. WE SINCERELY APPRECIATE YOUR HELP AND HOPE WE CAN OFFER YOU BETTER SERVICE IN FUTURE TOWARDS QUALITY EDUCATION IN OUR PRIMARY SCHOOLS. 232 Inservice Mathematics Education for Primary School Teachers in Western State, Nigeria. Headmasters' Questionnaire SECTION I 1. How many teachers are there in your school? 2. Please indicate the number of teachers in each of the following groups: a. Grade II: b. Grade III: c. No teaching qualification: d. Others: 3. How many years have you been headmaster in this school? 4. How many of your teachers have participated in an inservice mathe- matics programme for primary school teachers since you have headed the school? 5. Please name your District Headquarter: 6. How far is your school from this Headquarter? (Check one.) Less than 50 miles Between 50 and 100 miles More than 100 miles 7. How far is your school from the nearest Teacher Training College: (Check one.) Less than 50 miles Between 50 and 100 miles More than 100 miles 8. Is there a radio or a rediffusiop in your school? Yes ; No 9. What are some of the problems you have with arithmetic teaching in your school? 10. How can the Institute of Education at Ife be of most help to you and your teachers with these problems? 11. How can the State Ministry of Education be of most help to you and your teachers with these problems? 233 SECTION II We would like to know your views on the organization of inservice mathematics programmes for primary school teachers. Please put a circle around the letter that best describes your View in each case: 1. What mathematical content should be included in the programme? a. Content in the primary school mathematics syllabus b. Content that goes beyond the primary syllabus c. Content of both the primary and secondary school mathematics syllabi d. Other (please explain) Should the inservice training programme deal with: Subject matter of mathematics only? Newer methods of teaching mathematics only? Subject matter and methods interwoven? Subject matter, methods and materials of instruction? Other (please state) mmoo‘m When should the inservice programme be conducted? a. Three weeks during the long vacation Two weeks during the long vacation followed by a correspondence course during the school year c. Two weeks during the long vacation followed by a radio course during the school hours throughout the year. Three—hour sessions fortnightly throughout the school year e. Other (please state) Where should the inservice programme be held? In the Teacher Training College near your town In a Secondary School in your town In an Institute of Education of a University a. b. c. d. Other (explain) Should the inservice programme be made: a Compulsory for all primary school teachers? b. Optional for teachers? ‘ c. Compulsory for selected teachers, who would become special mathe— matics teachers in the primary schools? (They may teach some other subjects as well.) d. Other (explain) What kind of credits should be given for successful participation in the inservice programme? A certificate of attendance A reward towards promotion prospects A grade towards the attainment of a teaching certificate An increment in salary Other (please state) (DQDO‘ID 10. “MEI..-1¢:~V , 234 How should we decide on the effectiveness of inservice mathematics programme? 3. Through an evaluation of participant's understanding by giving him/her some formal and informal tests b. Through formal and informal tests for the participant, and other tests for the pupils he/she teaches c. Through tests for participant and pupils, and through an evalua— tion of his/her classroom practices in teaching mathematics. d. Other (please state) Should headmasters and school inspectors participate in the inservice ro ramme for teachers? “ p g Yes ; No Please feel free to add any additional comments you may have on inservice mathematics programmes below. (Use the back of this page, if necessary.) Please indicate your name and a permanent address at which you can be reached, if you so desire: Name: Address: THANK YOU VERY MUCH. WE SINCERELY APPRECIATE YOUR HELP AND HOPE WE CAN OFFER YOU BETTER SERVICE IN FUTURE TOWARDS QUALITY EDUCATION IN OUR PRIMARY SCHOOLS. 235 Inservice Mathematics Education for Primary School Teachers in Western State, Nigeria Questionnaire for Organizers or Sponsors of Inservice Training Please respond to each question as fully as possible. Continue at the back of the page, if desired. SECTION I 1. Please list the programme of inservice mathematics training that have been offered to Primary School Teachers by your institution/organi— zation. In each case, indicate the time, duration of programme, place and number of participants. a) b) C) 2. What do you consider to be the most important contribution of these inservice programmes to the primary schools? 3. How could the programmes be improved for maximum benefit to the teachers and the primary schools? 4. What are some of your future plans for inservice mathematics training of primary school teachers? Questions 5 and 6 in this section are for the Research and Training Division of Ministry of Education, Western State, only: 5. In your opinion about what percentage of the primary teachers in the state would need the upgrading of their mathematical background in order to teach the content of the proposed syllabus for primary school adequately. (Check one) (a) Below 25% ............... (b) Between 25% and 50% ..... (c) 50% and above ........... (d) Other (please state).... Ill? 6. Please indicate the number of Teacher Training Colleges in Western State that presented candidates for each of these alternative com— pulsory papers in June, 1974 Grade II Teachers' Examination: a) b) SECTION “mm—_M‘n. ‘I ~. ~- _- 236 Arithmetic Process and Method Papers, No. of Colleges: Basic Mathematics and Method Papers, No. of Colleges: II In this section we sould like to know your views on the organization of inservice mathematics. Please put a circle around the letter that best describes your View in each case. I. What mathematical content should be included in the programme? a. b. c. d. Content in the primary school mathematics syllabus Content that goes beyond the primary syllabus Content of both the primary and secondary school mathematics syllabi Other (please explain) 2. Should the inservice training programme deal with: (DQOU‘CD Subject matter of mathematics only? Newer methods of teaching mathematics only? Subject matter and methods interwoven? Subject matter, methods and materials of instruction? Other (please state) 3. When should inservice training be conducted? a. b. Three weeks during the long vacation Two weeks during the long vacation followed by a correSpondence course during the year Two weeks during the long vacation followed by a radio course during the school hours throughout the year Three—hour sessions fortnightly throughout the school year Other (please state) 4. Where should inservice programme be held? a. b. c. d. In the Teacher Training Colfege near the teacher's town (where he teaches) In a Secondary School in the town where he teaches In an Institute of Education of a University Other (explain) 5. Should the inservice programme be made: a. b. c. Compulsory for all primary school teachers? Optional for teachers? Compulsory for selected teachers, who would become special mathematics teachers in the primary schools? (They may teach some other subjects as well.) Other (explain) uh... 10. 11. 237 What kind of credits should be given for successful participation in the inservice programme? A certificate of attendance A reward towards promotion prOSpects A grade towards the attainment of a teaching certificate An increment in salary Other (please state) (DD-07m How should we decide on the effectiveness of inservice mathematics programme? a. Through an evaluation of participant's understanding by giving him/her some formal and informal tests b. Through formal and informal tests for the participant, and other tests for the pupils he/she teaches c. Through tests for participant and pupils, and through an evaluation of his/her classroom practices in the teaching of mathematics d. Other (please state) Should headmasters and school inspectors participate in the inservice programme for teachers? Yes ; No What is the feasibility of using any mass media for a follow-up of inservice programme for primary teachers in your opinion? Please feel free to add any additional comments you may have on inservice mathematics programmes below. (Use the back of this page, if necessary.) If you would like a brief summary of this survey, please indicate your name and address below: Name: Address: 1 THANK YOU FOR YOUR ASSISTANCE. I SINCERELY APPRECIATE YOUR HELP AND HOPE WE CAN CONTINUE TO WORK TOGETHER TOWARDS QUALITY EDUCATION IN OUR PRIMARY SCHOOLS. 238 557 West Owen Hall Michigan State University East Lansing, Michigan 48824 U.S.A. December 27, 1974. The Chief Inspector of Education Western State of Nigeria c/o. The Ministry of Education, Ibadan, Nigeria. Dear Sir/Madam, In partial fulfilment of a doctoral degree programme in Elementary Education, with emphasis on mathematics and teacher training, I an1 conducting a survey on inservice mathematics education for primary school teachers in Western State, Nigeria. I plan to develop a model for a systematic inservice mathematics education for primary school teachers based on the findings of the survey, the basic mathematics knowledge required of elementary teachers —— based particularly on the content of the prOposed primary mathematics syllabus for Western State schools, the available facilities for such re—education, and on research findings related to the learner and to mathematics learning. The need for a systematic inservice training for primary school teachers, particularly in the area of mathematics and science has been frequently echoed in our society. The Nigerian National Education Research Council at its workshop of 1971 gave recommendations as to the need for the re—education of teachers in order that they may teach the new concepts and ideas proposed in the new curriculum adequately. At the invitation of the Nigerian Federal Ministry of Education, through the British Council, Mr. B.J. Wilson of CEDO, London, visited Nigeria in 1971 for the purpose of studying the state of mathematics teaching in Nigeria. Although his study was to centre on secondary schools, be found it necessary to examine the situation in both the primary schools and the teacher training colleges as well, as stated in his report. Among his recommendations was that permanent inservice centres under the Ministries be established whereever possible. No doubt the different Western State Primary School Curriculum Panels, including the mathematics panel, might also have implications for inservice training in their recommendations. The implementation of the new curriculum calls for a significant and systematic inservice training of teachers. Furthermore, it seems imperative that while we are working hard at the problems of an expanding primary education system, we should upkeep the quality of the system at the same time. Since quality education depends very much on the quality of our teachers, a continuing inservice training seems to be a necessity. "Inn—_— 239 It is with these instances in mind that I am humbly soliciting your comments on inservice mathematics training for the primary school teachers. I would especially appreciate your comments on the following and any other issue related to inservice mathematics training: a. Existing programmes and/or future plans aimed at upgrading teachers' inadequate background for the implementation of the new curriculum; b. The possibilities of granting incentives to teachers for successful participation in the inservice training programs; c. The use of the mass media for inservice training of primary teachers. I would like to ask for your permission to quote you in my writing, if necessary, and would greatly appreciate your response to this letter on or before the end of January, 1975. While I await to hear from you, I thank you in advance. ReSpectfully yours, B. Mabogunje Osibodu (Mrs.) Faculty of Education University of Ife, Nigeria. (Currently on study-leave at Michigan State University, USA.) 240 557 West Owen Hall Michigan State University East Lansing, Michigan 48824 U.S.A. December 27, 1974. The Chief Federal Advisor of Education c/o The Federal Ministry of Education Lagos, Nigeria Dear Sir, In partial fulfilment of a doctoral degree programme in Elementary Education, with emphasis on mathematics and teacher training, I am con- ducting a survey on inservice mathematics education for primary school teachers in Western State, Nigeria. I plan to develOp a model for a systematic inservice mathematics programs for primary school teachers based on the findings the survey, the requirement of the new syllabus —— in particular the Nigeria Educational Research Council (NERC) guide- lines, the available facilities for such re—education, and on research findings related to the learner and to mathematics learning. Although my survey is limited to sample of teachers from selected towns and villages of Western State, it is hoped that the findings can be generalized to other states, to some extent at least, and that the model will be applic- able in other parts of the country. The need for a systematic inservice training for primary school teachers, particularly in the area of mathematics and science has been frequently echoed in our society. The NERC at its curriculum workshop in 1971 gave recommendations as to the need for the re-education of teach— ers in order that they may adequately teach the new concepts and ideas proposed in the new curriculum. At the invitation of the Nigerian Federal Ministry of Education, through the British Council, Mr. B.J. Wilson of CEDO, London visited Nigeria in 1971 for the purpose of studying the state of mathematics teaching in Nigeria. Although his study was to centre on secondary schools, be found it necessary to examine the situation in both the pri- mary schools and the teacher training colleges as well, as stated in his report. Among his recommendations was that permanent inservice centres under the Ministries be established whereever possible. These two instances reaffirm, very encouragingly, the interest and concern of the Federal Government in the teaching of mathematics in Nigerian schools. It is with this in mind that I am humbly soliciting your comments on Inservice Mathematics training for primary teachers. *The report was presented to the professional division of the Federal Ministry of Education in June, 1971, and a copy was sent to the Mathe— matics Association of Nigeria, while I was a Council member of that organization. 241 I do realise that different states have plans for the organization of their primary schools. It seems, however, that the problems of mathematics and science teaching at the foundation stage is such a great task that calls for the coordination and aid of the federal government. Again while we are working hard as a nation on the call and pro— blems of quantity in primary education, it seems imperative for us to upkeep quality at the same time. The quality of our primary schools depends on our primary teachers. Some attempts are already being made toward better preservice training of new teachers in this subject area. It seems, however, that there is a definite need to retrain and upgrade the levels of the thousands of primary teachers whose basic education is inadequate for the demands that the curriculum of the 1970's are making on them. I would, therefore, appreciate receiving from you some state— ments or comments regarding the Federal Government plans and/or policies toward the inservice training of primary school teachers. This state— ment may be specifically in relation to mathematics teaching, or the teaching of mathematics and science, or to the teaching of the curriculum as a whole. It may also be in terms of aids planned or given to the states. I would like to ask for your permission to quote you freely in my writing, where necessary and would greatly appreciate your response to this letter on or before the end of January, 1975, as I am planning to complete the project around June, 1975. While I await to hear from you, I thank you in advance. Respectfully yours, B. Mabogunje Osibodu (Mrs.) Faculty of Education, University of Ife, Nigeria. (Currently on study—leave at Michigan State University, USA.) 242 557 West Owen Hall Michigan State Univeristy East Lansing, Michigan 48824 U.S.A. December 27, 1974. The Chairman, Nigeria Educational Research Council (NERC) P. O. Box 8058 Lagos, Nigeria. Dear Sir , In partial fulfilment of a degree programme in Elementary Education, with emphasis on mathematics and teacher training, I am conducting a survey on inservice mathematics education for primary school teachers. I plan to develop a model for a systematic inservice mathematics educa- tion based on the findings of the survey, the requirement of the new mathematics curriculum —— in particular the Nigeria Educational Research Council guidelines on Teacher Training mathematics curriculum —— and on research findings on the learner and the learning of mathematics. AlthOugh my survey is primarily based in the Western State of Nigeria, it is hoped that the proposed model will be generally applicable in other parts of the nation. As a national education body, the NERC has done great work in pro— ducing new curriculum guidelines for the nation's schools. These at— tempts toward curriculum improvement in the schools are praise worthy. However, a crucial factor affecting curriculum change so deeply is the capacities of the teachers. Discussions and recommendations from the different NERC curriculum workshops have emphasized the need for better preservice and inservice training, in particular, for the primary school teachers. My purpose of writing this letter to you is to humbly solicit your comments and/or plans on inservice mathematics training for primary school teachers. This may be in terms of inservice training for mathe— matics alone, for mathematics and science, or for mathematics and some other subjects. I would especially appreciate your comments on the following and any other issue related to inservice training: a. Existing programmes and/or future plans aimed at upgrading primary teachers with inadequate background knowledge for the implementation of the new curriculum; b. Any plan to coordinate programmes of inservice mathematics or mathematics and other related subjects on the nation-wide basis. 243 I would like to ask for your permission to make reference to your comments in my writing, if need be, and would greatly appreciate your response to this letter on or before January ending, 1975. While I thank you in advance, I remain, Yours respectfully, B. Mabogunje Osibodu (Mrs.) Faculty of Education University of Ife, Nigeria (Currently of study-leave at Michigan State University, U.S.A.) 244 APPENDIX C List of Towns in Which Participating Schools are located by Classification. Urban Towns Ibadan . Okitipupa Abeokuta Ijebu~Ode Ilaro Sagamu Ado-Ekiti Ile—Ife Ikole-Ekiti Ilesha Ijero—Ekiti Oshogbo Akure Oyo Owo Ogbomosho Ondo Shaki Rural Towns Fiditi Yekemi (via Ondo) Ikereku Ode-Aye Itapa~Ekiti Oru-Awa Ilupeju—Ekiti Osu—Ilesha Iju—Akure Edunabon Bagbe—Ondo Ago—Are. * The terms "urban" and "rural" in this classification are used as follows: A divisional headquarter is termed urban, while small towns and villages are classified as rural. APPENDIX D * TEST OF BASIC MATHEMATICAL UNDERSTANDINGS Form A (Pre—test) and Form B (Post—test) Directions: This test is designed to measure your understanding of mathematics. Many of the items relate to the content in present programs of mathe— matics of elementary pupils. Each of the fifty questions is of multiple—choice type and includes four possible answers. Read each question carefully and decide which answer fulfills the requirements of the statement. Then circle the response on the answer sheet to indicate your choice. Circle only one answer for each question. If you change your choice, erase your original mark and circle the correct one. Sample Question: 1. Which of the following shows the decimal form of the fraction 5/4? a. 125 b. 12.5 b. 1.25 d. .125 Answer Sheet: 1. a b (:> d Since 1.25 is the correct answer, the letter (c) is circled. *The original Test of Basic Mathematics Understandings was prepared by Dr. Mildred Jerline Dossett, Michigan State University, East Lan51ng, Michigan, 1964. (See Bibliographic listing under the name, Dossett, for reference). The revised form of t this study by the investigator. he test presented here is adapted for use in 245 246 FORM A (PRE—TEST) 1. When you write the numeral ”5" you are writing a. the number 5 b. a pictorial expression c. a symbol that stands for an idea d. a Hindu-Babylonian symbol Bola discovered the > means "is greater than" and < means ”is less than.” In which of the following are these symbols not used correctly? a. The number of states in Nigeria < the number of states' capitals. b. 3 + a < 5 + a c. 33 > 42 d. The number of teachers in the school < the number of pupils in the school. 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 Zero may be used a. as a place holder b. as a point of origin c. to represent the absence of quantity d. in all of the above different ways 2,200.02 is shown by QOU‘W 5840 @0793 Which of the following does not show the meaning of 423ten. QAOU‘GJ 2000 + 200 + 20 2000 + 20 + 2/10 2000 + 200 + 2/100 2000 + 200 + 200 , rearranged so that the 8 is 200 times the size of 4 would be 5840 8540 5048 5408 9 (4 x 100) + (2 x 10) + 3(1) = 423 42 tens + 3 ones = 423 423 ones = 423 4 hundreds + 42 tens + 23 ones = 423 10. ll. 12. l3. 14. 15. 16. 247 II ll 0 . A 2 1n the th1rd place of a base five number would represent a. 2 x 52 b. 5 x 23 c. 5 x 25 d. 2 x 53 In this addition example, in what base are the numerals written? a base two 120 b. base three ? c. base four +10? d. none of the above 200 ? About how many tens are there in 6542? a 6540 b. 654 c 65% d. 6.5 Place or order in a series is shown by a. book no. 7 b. three boxes of matches 0. a dozen cupcakes d. two months Which of the following indicates a group? a 45 tickets b. track 45 c page 54 d house no. 7 The sum of any two natural numbers is not a natural number is sometimes a natural number is always a natural number is a natural number equal to one of the numbers being added QOUN The counting numbers are closed under the operations of a. addition and subtraction , b. addition and multiplication c. addition, subtraction, multiplication and division d. 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 If a x b = 0 then 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 l7. l8. 19. 20. 21. 22. 23. 24. 248 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. b. c. d. counting adding subtracting multiplication How would the product in this example be affected if you put the 29 above the 4306 and multiplied the two numbers? DUO-3 D. O The answer would be larger 4306 The answer would be smaller x29 You cannot tell until you multiply both ways The answer would be the same The product of 356 x 7 is equal to 0400‘!” Which Cl-DU‘OJ Which DuOU‘D-l The inverse operation generall CLOUD) (300 x 50) x (6 + 7) (3 x 7) + (5 x 7) + (6 x 7) 300 x 50 x 6 x 7 (300 x 7) + (50 x 7) + (6 x 7) of the following is not a prime number? 271 277 281 282 of the following numbers is odd? 18 x 11 11 x 20 99 x 77 none of the above y used to check multiplication is addition subtraction multiplication division The greatest common factor of 48 and 60 is CLOUD) 2 x 3 2 x 2 x 3 2 x 2 x 2 x 2 x 3 x 5 none of the above 25. 26. 27. 28. 29. 30. 31. 249 Look at the example at the right. Why is the "4” 157 in the third partial product moved over two places x246 and written under the 2 of the multiplier? 942 628 a. If you put it directly under the other partial products, the answer would be wrong. 314 b. You must move the third partial product two places to the left because there are three numbers in the multiplier. c. The number 2 is 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 showing that (a + b) + (a + c) = 2a + b + c? . associative property commutative property associative and distributive properties associative and commutative properties O‘DO‘CD If N represents an even number, the next larger even number can be represented by a. N + 1 b. N + 2 c N + N d 2 x N + L Every natural number has at least the following factors: a. zero and one b. zero and itself c. 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 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 a. sixths b. thirds c. halves d. parts 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 32. 33. 34. 35. 36. 37. 38. 250 When a common (proper) fraction is divided by a common (proper) fractlon, how does the answer compare with the fraction divided? it will be larger it will be smaller . it will be twice as large there will be no difference Q-OU‘QJ Which algorithm is illustrated by the following sketch? a. 1/2 x 3/4 = ? b. 1/2 + 3/3 + 2 §Q§ 1/2 c. 1/4+1/4+1/4=? \\& § d. 4/4 — 3/2 = 9 1/2 Another name for the inverse for multiplication of a rational number is the a. reciprocal b. opposite c. reverse d. zero Examine the division example on the right. Which 5 + 3/4 = 6 2/3 sentence best tells why the answer is larger than 5? a. Inverting the divisor turned the 3/4 upside down. b. Multiplying always makes the answer larger. c. The divisor 3/4 is less than 1. d. Dividing by proper fractions makes the answer larger than the number divided. The value of a common fraction will not be changed of 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. The nearest to 45% is 44 out of 100 a. b. .435 c. 4.5 d. .405 Sola 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. d. 0" minutes 24 minutes 39. 40. 41. 42. 43. 44. 251 There were 400 students in the school. One hundred percent of the children had lunch in the dining hall 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 sentences can be used to find the percent of the school enrollment who went home for lunch? a. 400 - 88 = X b. x/lOO = 88/400 c. x/88 = 400 d. 400 - 90 = X What can be said about y in the following open sentence if x is a natural number? X + x + 1 = y a. x < y b. x > y c. x = y d. x # y Which one of the following fractions will give a repeating decimal? a. 1/2 b. 3/4 c. 5/8 d. 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 , compares an object with some known standard or accepted unit tries to find the exact amount is always an exact measure chooses a unit and then gives a number which tells how many of that unit it would take a. b. c. d. l. 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 252 45. The set of points sketched below represents a \# > a. line b. ray c. line segment d. none of the above 46. How many triangles does the figure contain? a. four b. six c. eight d. ten 47. The set of points on two rays with a common end-point is called a. a triangle b. an angle c. a vertex d. a side of a triangle 48. If a circle is drawn with the points of a compass 6 cm. apart, what would be 6 cm. in length? a. circumference b. diameter c. area d. radius 49. The solution set of an open sentence consists of . two or more numbers no numbers only one number . any or all of these _ ‘ moan: 50. Consider a set of three objects. How many sub—sets or groups can be arranged? a. nine b. eight c. seven d. six ‘ . W, '7 I” 4’ » ,. .. 253 FORM B (POST—TEST) 1. When we use the = symbol 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? b. 5 + 2 = 7 a =5+2 C. (5 + 2) x 3 d. = 7 x 3 a and b are correct a and c are correct . a, b, and c are correct a, b, c, and d are correct wal—J \l 2. 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 a 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. d. It uses the additive concept in representing a number of several digits. 3. In the numeral 7,843, how does the value of the 4 compare with the value of the 8? 2 times as great . 1/2 as great C. 1/10 as great d. 1/20 as great 4. In the numeral 6,666 the value of the 6 on the extreme left as com— pared with the 6 on the extreme right is U‘SD . 6,000 times as great . 1,000 times as great , . the same since both are‘sixes a b c d. six times as much 5. Below are four numerals written in expanded notation. Which one is not written correctly? 1 a. 4(ten)2 + 9(ten) + i(ones) = 493ten b. 3(seven)3 + 6(seven)l + 1(one) = 361seven 2(three)2 + 2(threi) + 1(one) = 221three d. 2(five)2 + 3(five) + 4(ones) = 234five 6. About how many hundreds are there in 34,870? a. 3% c. 350 b. 35 d. 3,500 10. 11. 254 . If you are permitted to use any or all of the symbols 0, 1, 2, 3, 4, and 5 for developing a system of numeration 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. d. base one, two, three, four, and five. In what base are the numerals in this multiplication example written? a. base five 349 b. base eight 23; c. base eleven -—‘-L d. you can't tell 124? 70? 1024? Which of the following are correct? a. In the symbol 53, 5 is the base and 3 is the exponent. b. In the symbol 53, 3 is the base and 5 is the exponent. 5 x 5 x 5 3: 3 #WNH U1 Ln and and and and U‘NU‘N = 3 x 3 x 3 x 3 x 3 d are correct c are correct c are correct d are correct Examine the following illustration: 1 2 3 4 5 6 Which of the following does the above best illustrate? a. The idea of a cardinal number. b. The use of an ordinal number. c. A means for determining the cardi counting. d. None of the above. nal number of the set by The integers are closed under the operations of a. addition b. subtraction c. multiplication d. division 1. 2. 3. 4 a and b are correct a and c are correct a, b, a, b, and c are correct c, and d are correct 12. 13. 14. 15. 16. 17. 255 A student solved this example by adding down; then be checked his work by adding up. Add 34 34 $12.12 86 Check 86 It could be classified as an example of a. the distributive principle b. the associative principle c. the commutative principle d. the law of compensation The statement "the quotient obtained when zero is divided by a number is zero" is expressed as a. a/O = O b. O/a = 0 c. 0/0 = a d. a/a = 0 When a natural number is divided by a natural number other than 1, how does the answer compare with the natural number divided? a. larger b. smaller c. one—half as large d. can't tell from information given If you had a bag of 350 walnuts to be shared equally by 5 boys, which would be the quickest way to determine each boy's share? . counting adding subtracting dividing 9400‘!» If the multiplier is x, the largest possible number to carry is a. x b. x + l c. 0 1 d. x - l ’ Which of the following methods could be used to find the answer to this example? 17 612 Multiply 17 by the quotient Add 17 six hundred times The answer would be the sum Subtract 17 from 612 as many times as possible. The answer w0uld be the number of times you were able to subtract. 0 QaOU‘fiJ 18. 19. 20. 21. 22. 23. 256 Which one of the following would give the correct answer to this example? 2.1 x21 The sum of l x 2.1 and 21 x 2.1 The sum of 10 x 2.1 and 2 x 2.1 The sum of 1 x 2.1 and 20 x 2.1 . The sum of 1 x 2.1 and 2 x 2.1 DahU‘SD Which would give the correct answer to 439 x 563? Multiply 439 x 3, 439 x 60, 439 x 5 and then add the answer. Multiply 563 x 9, 563 x 3, 563 x 4 and then add the answer. Multiply 563 x 9, 563 x 39, 563 x 439 and then add the answer. Multiply 439 x 3, 439 x 60, 439 x 500 and then add the answer. Which of these numerals are names for prime numbers? o 0400‘“) a. 3 b. 4/2 C' 12five d. 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 The inverse operation for addition is a. addition b. subtraction c. multiplication ‘ d. division The least common multiple of 8, 12, and 20 is 2 2 x 5 x 2 x 2 x 2 x 3 x 5 0.00:9: x 2 x 3 x 5 2 x 2 x 2 x 3 2 x 2 x 2 x 2 257 24. Which statement best tells why we carry 2 from the second column? a. If we do not carry the 2, the answer would be 20 251 less than the correct answer. 161 b. S1nce the sum of the second column is more than 252 20, we put the 2 in the next column. 271 G. Since the sum of the second column is 23 (which has two figures in it), we have room for the 3 only, so we put 2 in the next column. d. 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. 25. Which of the following is an even number? a. (100)three b. (100) , f1ve c (100) seven d (200) _ f1ve 26. The fact that a + (b + c) is exactly equal to (c + b) + a is an example of distributivity commutativity closure . associativity DaDO‘FD 27. Observe the drawing on the right. When the circle is cut into equal pieces, the size of each piece decreases as the number of pieces increases increases as the number of pieces decreases increases as the number of pieces increases decreases as the number of pieces decreases OuOU‘FD a and b are correct a and c are correct 3. b and c are correct . b and d are correct” 28. T e symbol 3/4 may be used to represent the idea that :r‘ 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 29. When a whole number is multiplied by a common (proper) fraction other than one, how does the answer compare with the whole number? it will be larger it will be smaller there will be no difference you are not able to tell CLOU‘N Wen-:he 11 ohm—... « 258 30. Which of the addition examples is best represented by the shaded parts of the diagram below? a. 1/2 + 1/3 ‘\:::\;:f:::§:ftt b. 2/3 + 3/4 \ C- 2/3 + 1/4 d. 1/3 + 1/3 31. We can change the denominator of the fraction 3 to the number "1" without changing the values of the fraction by 3 . adding 5/4 to the numerator and denominator subtracting 5/4 from the numerator and the denominator multiplying both the numerator and the denominator by 5/4 dividing the numerator and the denominator by 5/4 Q-OU‘CD 32. What statement 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. d. Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. 33. If the denominator of the fraction 2/3 is multiplied by 2, the value of the resulting fraction will be . half as large double in value unchanged in value a new symbol for the same number ) O‘DU‘DJ 34. 45% may also be written as a. .45 b. 45/100 c. 45 x 100% d. .450 l. a and ba are correct 2. a and c are correct 3. a and d are correct 4. a, b, and d are correct 35. 36. 37. 38. 40. 259 .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/.56 which is a fraction and has a decimal as either the numerator or denominator or both. The ratio of x's in Circle A to x's in Circle B can be shown by a. 16/4 b. 1/4 c. 1/2 d. 4/16 A B Olu paid 20k for 4 oranges. Which of the sentences below could be used to find the cost of 1 orange? 3. 4/20 = l/x b. x + 4 = 20 c. x/4 = 20 d. x — 4 = 20 Which of the following statements is not correct? a (-9) + 6 = —3 b (—5) + (~5) = —10 c —8 + 0 = -8 d (-8) + (9) = —1 Which of the following is a list of all of the factors of 12? a. 1, 2, 3, 4, 8 & 12 b. l, 2, 3, 4, 6 & 12 c. 1, 2, 3, 4, & 6 ‘ d. 2, 3, 4, 6 & 12 Which of the following is an approximate measure? a. 35 farms b. 12 buttons c. 7% meters d. 15 beads 260 41. Which of the following does the sketch below represent? \> a. line b. ray c . line segment d. 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 42. Which of these triangles are right triangles according to the lengths ofthe sides given? a. b. 6" 10” 4H 4H 5 c d. 6H 7H 5" H 4 7H 43. A distinct point is a. a point you can see b. a sharp object c. the intersection of two lines d. a dot 44. A woman bought a round mat that had a radius 120 cm. Which of the formulas can she use to determine the length around the mat? a. A = W'r b. c=TTd c. C = 2Trr d. A = C/d 45. Which of the following best defines a solution set? 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. 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. 261 46. 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 47. If two sets are said to be equivalent, then a. every element in the first set can be paired with one and only one element in the second set b. every element in one set must also be an element in the second set c. they are intersecting sets d. one must be the null set 48. Look at the picture below: 20 Ade Tola Oke Sidi Peju Who got the most words right? a Oke b. Ade c Sidi d. Peju 49. How many words did Tola get right? a 10 b. 20 c. 15 d. 5 50. The picture shown in number 48 is called: a word picture a spelling test a bar graph an arithmetic picture DaOU‘iD BIBLIOGRAPHY /4// BIBLIOGRAPHY Books Abernethy, D.B. The Political Dilemma of Popular Education: An African Case, Stanford, California: Stanford Uni- versity Press, 1969. Adetoro, J.E. The Handbook of Education in Nigeria, Ibadan: African Education Press, 1960. 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