UNIVERSITY OF IBADAN LIBRARY UNIVERSITY OF IBADAN THIS THESIS SUBMITTED BY ........ WAS ACCEPTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE FACULTY OF EDUCATION OF THIS UNIVERSITY THE EFFECTIVE DATE OF THE AWARD IS 8 Cc+oherm ?9C7 ..... t DATE L Sc Ec Cn Ro Ec TT A Ro vY 0 .05). (4) There will be no significant difference in attitude to 'wards mathematics and calculators scores of those groups of high, average and low mental abilities. The null hypo­ thesis four was not rejected entirely because there was no significant difference in the mean post-attitude scores of those groups of high, average and low mental ability lsvels (F (2,123) = 2.147, p > .05). (5) There will be no significant relationship between the attitudes of pupils towards mathematics and calculator-use in mathematics. The null hypothesis five was not rejected because there was no significant relationship between pupils' attitudes towards mathematics and calculator-use CFC1, 124) = 1 .57, p > .05). (6) There will be no significant relationship in pupils' mathematics achievement scores and post-attitude scores. The null hypothesis six was rejected because there was significant relationship in the post-test scores of the groups and the post-attitude scores (F (1,124) = 4.84, p < .05). Generally, the results showed that there UNIVERSITY OF IBADAN LIBRARY I V wore attitudinal eh-mgas between pre- and post-attitudes among all the groups, and that the calculator groups per­ formed better than the non-calculator groups. The results have also shown that pupils within the same ability levels who uce calculators will perform better than those who do not use calculators, Most studios on the use of calculators including this one have not found calculators to have debilitating effects rather it has computational advantage and promotes high achievement gains in mathematics. Teachers and pupils in secondary schools should te encouraged to utilize the advantage of : .iculators in algorithmic computations, so as to reduce those computational chores which often led to loss of interest in learners. However, further research could be done into the effectiveness and efficiency of calculators in concept formation, and problem-solving in secondary schools. In addition, research could be done to find out its effects at primary school level in Nigeria. UNIVERSITY OF IBADAN LIBRARY - V D E D I C A T I O N This work is dedicated to my wife, AyooT.3, children: Omcwumi, Olufemi, Abiola and Oluwadara, rr.jy parents and to the greater glory of God. UNIVERSITY OF IBADAN LIBRARY •„ i f C K r. 0 W L E D G E M G fJ T S I am 1.10st grateful t ; my supervisor and mentor, Froft = sor T. A. 3 a 1 o g u n , Head, Department of Teacher Education who had guided me through this project. His advice, suggestions and corrections hod been vary much useful for the execution of this project. I since.ely appreciate his commitment and dedication to the course of education in Nigeria. fiy profound appreciation also gees to my motivator 3nd inspirer Dr. (Pastor) Ayo Ogunranti, Institute of Education, University of Ibadan far his advice, words of encouragement during H: cficul4- times in this project. He also took keen interest in this work through his suggestion and corrections of the work. I will like to express my deepest than ko to Dr. 'i.Q. Ceunniyi, Department of Teacher Education far his brocoerly advice, and immense help in sharpening my knowledge in research methodology and statistical analysis, He offered me tremendous assistance and suggestions throughout tne course of this study. Dr. R. A. Fakuade, formerly of the Teacher Education Department, University of Ibadan b"t now at Ondo State University, Ado-Ekiti has been instrumental ard inspiration UNIVERSITY OF IBADAN LIBRARY zo my further studies. I am most grateful to him for his personal assistance. Ply wormest thanks go to Professor 2. A. Akinboye, Department of Guidance and Counselling, University of Ibadan and Dr. Titi Hassan, formerly of the Department of Guidance and Counselling, University of Ibadan, but now at the Ogun State University, Agc-Iwoye for thoii initial support. They both helped to give this study a direction. My thanks also go<- to Dr. Wole Folayajo, Institute or Education, University of Ibadan for his useful sugge scions. I like to thank tremendously my colleagues at the Polytechnic, Ibadan: Dr. O.A. Akinkunis and Dr. Soji Sodamade for their encouraging =nd very useful sugg~3tions throughout the duration of this study. My appreciation and thanks go- .. to the Principals, Vice-Principals and most importantly the punils of the schools involved in this project: (i) Ibadan City Academy, i Eleta, Ibadan. C i i} Holy Trinity Grammar School, Old Ife-Rcad Ibadan; (iii) Islamic High School, Grita Basorun, Ibadan* (iv) An-war Islam (formerly Ahrnadiyya) Grammar School, Eleyele, Ibadan. I like to thank the cooperating mathematics teachers from these schools who denoted time and energy to assist me UNIVERSITY OF IBADAN LIBRARY \/111 on execution of this project: (i) Mr. 0. L. Ra j i, Ibadan City Academy, Eleta, Ibadan. (ii) Mr. S. A. Adeyemo, Holy Trinity Gramma: School, Giu It'e Road, Ibadan, (iii) Mr. I. A. Glowuda, Islamic High School Or Lta Qasorun, Ibadan. I like to thank some of my students at the Polytechnic, Ibadan who assisted me in one way or another during the course of this s'udy. The computer programming of this study' data analysis would have been very difficult, but for the personal and brotherly assistance of Mr. S. A. Odewunmi of the Computing Carter, University of Ibadan. I really thank nim. My thanks .o to Mr. C. S. Dada of the Polytechnic, Ibadan who helped to type the proposal of this project. Most importan:ly, I like to thank Mr. T. A. Oke of the University cf laadan library for his patience and endurance i n taping of this thesis. I like to t.iank my wife, children and relations who, with the Grace of God,have endured the strain of this project with me. Their » individual prayer :nd enocuragement have always given me high spirit. Finally I wish to thank those who had, by one way or another,contributed to the successful completion of this work. hay God bliss you all abundantly. Amen. UNIVERSITY OF IBADAN LIBRARY X TITLE PA DF ABSTRACT . . i DEDICATION V ACKNOULE DGEMENTS . . vi CERTIFICATION ix TABLE OF CONTENTS , . X LIST qf FIGURES LIST OF TABLES CHAP TER ONE: INTRODUCTION 1 1 . 1 Background to the study .. 1 1 .1 . 1 The Electronic Calculator ■ • 3 1 .2 Statement of the problem • • 10 1.3 The Hypotheses • • ■ ■ 1 2 1 .4 Significance of the study 13 1.5 Limitations of the study 21 CHAPTER TWO: REVIEW OF RELATED LITERATURE AND RESEARCH 2 . 1 Electronic Calculator and the School System 25 2 .1 . 1 Elementary level . . •• 27 2 .1 .2 Secondary level • • 29 2.1.3 Higher Education level • . • • 30 2.1.4 Learning outcomes in mathematics instruction 31 2.1.5 Computational effects of Calculator 34 UNIVERSITY OF IBADAN LIBRARY nu [ Conceptual evfocts of Calculator 39 Attctudinal effects of Calculator 46 Use c-r' Calculators in Tests 46 Integrating Calculator into Mathematics Curricular in Schools 51 Definition: Concepts and Concept Learning 55 Criteria for Conceptual Learning 58 Concent formation and attainment Concept Learning 60 Associative Versus Mediations! theories of Concent Learning 62 Mathematical Concept Formation 65 Researches in Concept Learning 68 Facilitating Concept Learning 70 Attitude: Attitudes toward mathematics 73 Mathematics learning and attitudes 76 Researches in attitudes 78 ft Instructional design and attitude 80 Calculator and other Instructional Devices 82 Computer-based instruction .. 84 Evolution of CAI 85 Pedagogical effectiveness of CAI 87 UNIVERSITY OF IBADAN LIBRARY - XI 1 - PAGE Attitudes and CAI 85 2.4.5 Computers and Mathematics 53 “ ̂ ^ Research in Computer-Mathematics = . So 2.4.7 Concls-7usion . . 98 CHAPTER THREE: RESEARCH DESIGN AND PILOT STUDY 100 3.1 Introduction .. .. . „ . . 100 3.2 Population and Sampling Procedure . . .. 105 3.2.1 Selection of Schools for the study 107 3.2.2 Selection of subjects ....... .. 111 3.3 Research Instruments .. .. .. .. 113 3.3.1 Preparation of Instruments: Attitude Scale .................. . . ,, 114 3.3.2 Validation of Instruments =. .. 120 3.3.3 Modification of mental ability Tests A.C.E.R. Higher tests (ML and ML)) „. 125 3.4 \ Pilot Study ....................... . 128 3.4.1 Objective of the Pilcc Study .. .. 128 3.4.2 Procedures for the Pilot Study .. .. 128 3.4.3 Administration of research instruments 129 3.4.4 !he Scoring of different instruments 132 3.5 Analysis of data of pilot study 133 3.5.1 Hypotheses testing for Pilot Study 134 N -F* .'I* U \NIVERSITY OF IBADAN LIBRARY ■< J. 11 PAGE 3.6 -"Results of the Pilot Study 134 3.7 Discussion of results of Pilot Study 146 3.8 Detection of flaws to be corrected for the main study 147 CHAPTE R FOUR: t'HE MATN S 149 4.1 The Design 149 4.2 Population of the 151 4.3 Comperebility of S 154 4./; fier.ihero rig the 160 4.5 Administration of 165 4.5.1 Guideli-.es on the 167 4.6 Data On1lection 170 4,7 He to (Analysis of * 171 CHAPTER FIVE. RESULTS 01 173 5.10 Hypothesis 1 173 5.20 Hypothesis 2 197 5.3P Hypothesis 3 212 5.40 Hypothesis 4 223 5.50 Hypothesis 5 237 5.60 Hypothesis 6 241 CHAPTER SIX: DISCUSSION 248 6 . 1 Relationship of n Its to hypothesis and previous empirical studies 248 6.2 Performance treatment and achievement 248 UNIVERSITY OF IBADAN LIBRARY - I X C E R T I F I C A T I O N I certify that this work was carried out oy Mr. Alade Abimbade in the Department of Teacher Education, University of Ibadan. 8.Sc., M.Sc. (London), M.Ed. Dip. Ed. Tech. (Birm), P.G.D.E. (iuadan) L. I. Biol. Professor of Education and Head, Department of Teacher Education, University of loadan, Nigeria. June, 1987. UNIVERSITY OF IBADAN LIBRARY xiv PAGE Performance treatment and attitudes .. 261 Relating findings to other medio .. ., 272 Educational inplicet ions of the study and recommendatiois . . . . . , . . . . 274 Suggestions fir further study .. .. 261 Summary and Conclusion .. .. .. .. 283 ENCES .......................................... 290 DICES Description 1 Mathematics pre-uost 301 2 Mathematics prs-test answers • • 303 Difficulty Index and discriminatory power (p) sf the mathematics pre-tests 304 *T « Mat heme, tics achievement post'-test . , 205 5 Mathemat ics achievement test answer .. 303 5 Difficulty index, discriminatory power fpj and reliability coefficient, r of the mathematics .. „ . .. 310 Instructional module in ma na.nv.uics 311 Mental ability test ML verbal .. .„ 343 Mental ability test MQ numerical .. 3 4g Attitude questionnaire .. .. 354 Internal consistency reliability coefficient of the attitude score 3 57 "'2 Correlation between 27% upper score and 27% lower score on attitude scale ■MAS and CAS) .. .. .. . 359 UNIVERSITY OF IBADAN LIBRARY - > V - -"ENTICE b PAGE '2 Significant mean difference in mathematics attitude seere MAS ) and calculator attitude score (GAS) for 27% upcer score 36^ 'A Significant mean difference in mathematics attitude score MAS) and calculator 361 attitude score (CAS) for 27% lower score Internal reliability coefficient of mathematics pre-test scores 362 • - Questionnaire directed to te sobers' of mathematics in the secondary schools 36 3 • 7 Data from the main study . . 356 1 o Data from the Pilot study 376 * "5 List of secondary schools in Ibadan Municipality 364 20 Analyses of variances of the post- attitude sr-res, 385 UNIVERSITY OF IBADAN LIBRARY in m xvi LIST OF FIGURES PAGE Abacus: No. 23 1 Model of modern electronic calculator 3 Sequential bases for learning various outcomes .. .. .. .. .• 62 Mediational model .. .. .. .. 64 Solution Process in mathematical Computation .. .. .. •• 96 Paradigm of 3 x 3 factorial design for Pilot study .. .. .. •• 100 Paradigm of 3 x 3 factorial design for the main study .. .. .. •• 149 UNIVERSITY OF IBADAN LIBRARY - xvii - Description PAGF Detailed WASC results in mathematics for Wiperia: June, 1370 . , .. .. ' Percentage failure in I'AS mathematics for Nij-aria: 1965-1977 . „ .. 7 WAEC C.C.E. result of schools in mathematics .. .. . „ . „ .. 110 Mathematics achievement test-plan for conten^ validity .. .. .. . * 116 Mathematics Pre-test items construction format content validity .. 117 Significant mean difference in mathematics attitude score (MAS) and calculator attitude score (CAS) for 27% upper score .. .. .. 123 Significant mean difference in mathematics -ttituch score (MAS) and calculator attitude score (CAS) for 27% lower score .. .. ., ., 124 Internal reliability coefficient, r, of the Pre-test scores .. .. .. .. 125 Analysis of covariance of Post-test scores of groups UCU, ECU and n Cl. . . . 139 Multiple classification of Post-test * scores of groups UCU, RCU and NCU .« 140 Summary of the mean, standard devia'^on and variance of the groups UCU, RCU and NCU ............. ‘ ...................... 140 Analysis of variance of oost-test scores o “ HMA, AMA and LMA r m u r s . . .. 141 UNIVERSITY OF IBADAN LIBRARY PATE 'ultiole classification analysis of post­ zest scores by groups HMA, AMA and LMA = 142 Analysis of covariance of post-test scores of groups HPA, AHA and LMA. .. . .. 143 Multiola classification analysis of post­ test scores by groups HMA, ANA and LNA .. 143 Analysis of variance of the attitude scores of the groups LCU, RCU and NCU .. .. 145 'ulziple classiTication analysis of the attitude scorss fcr the groups UC'J, RCU CJ ............. ' . . . , . . 146 Ana.ysis of \ariaoce of the attitude score of groups rilli, ANA and LNA .. . . . » 147 Multiple clcssification analysis of the attitude sccres of groups HMA, ANA and LNA 143 Summary of the '"'lysis of variance of the NAS and LAS scores cf groups UCU, RCU, NCU 149 Summary of the analysis of variance of NAS and CAS scires of the HMA, ANA, LMA .. 150 Summary of mental ability tests scores for schools 1, 2, 3 in the main study, .. 158 WAEC resu.'ts in mathematics for schools 1, 2, 3 Pres nt age Passes 1980 - 1984 ... 150 Analysis of ./ariance of mean percentage passes of the srhoo.s 1,2,3. Analysis of variance of mean m.Bntal ability tests scores of schools 1,2,3. .. 161 Teacuers variables and content coverage 163 Calculator usage effectiveness by Teachers 164 UNIVERSITY OF IBADAN LIBRARY X I X --E PAGc 1_ Use of instructional materials by schools 165 12 A comparison of percentage of Teacher/ pupils classroom interactions behaviour 167 Percentage distribution of teachers' questions .. .. .. .. .. .. 159 Mean distribution of quest ions/minute of Teachsr/pupiis Interaction .. .. . . 169 22 Summary of the roans, standard deviations, and variances of the three groups (UCU, R CU a n d w Cl) . . 33 Analysis of covariance of Post-achievement test -inures ct UCU, RCU and NCU groups .. 175 34 Mu 1 l 1 p3 ~ Cicissification analysis of post­ test scores by groups UCU, RCU and NCU with mental ability scores .. .. . .. 176 35 Analysis o' vaii^.ics of mental anility test scores of ;he three croups (UCU, RCU and NCU). .. .i' ................... 177 Analysis of covariance of Post-achievement test secies of UCU, RCU and NCU groups .. 176 37 Multiple classification analysis c r cost- test sccres of groups UCU, RCU and NCU with Pre-tes: covariate .. .. .. .. .. 179 I 32 Analysis of varianceof the Pre-test df'the groups UCJ, RCU and'NCU .. .. 180 35 Multiple range tost of Post-test scores by one-way Scheffe procedure (ANOVA) .. .= 161 Summary of t-tests of the Post-test scores cl the groups UCU, RCU and NCU .. .. 182 41 Multiple regression analysis of Post-'^st scores with mental ability tests an.i Pre­ test of tie groups .. .. .. .. 164 UNIVERSITY OF IBADAN LIBRARY X X PAGE Analysis of covariante of Post-test scores of A,8 ,C groups of UCU. (MAT as covariate) 165 Analysis of covariance of Post-test scores of A.3.C groups of UCU (Pre-test as covariate) 136 "ultiple range test of Post-tes: scores of groups A,B,C by ONE-WAV L.SO Pro,edure-ANOVA 187 Summary of t-tests of the post-cast scores of the groups A,5 and C .. .. .. 183 Analysis of covariance ef Post-test scores of groups 0, E and F. .. .. , . .. 189 Multiple classification analysis cf Post­ test scores of groups, 3,L,F with Pre-test scores .. •• •• .. .. .. 190 Multiple range test o- Post-test scores of groups n, E and F by CNE--WAY ANCVA _SD Procedure .. .. . . .. ., 191 Summary of t-tests o- the Post-test scores of groups D ,E and F .. .. .. 131 Analysis of the covariance o- the post-test scores of groups G, H and I. .. .. 132 Multiple classification analysis of r> ost- test scores of groups G ,H and I . . .. 193 Multiple range test of Post-test scores, by one-way AfiOVA LSD procedure of groups G, H and I .............................. 194 Sun.rr.ory of t-tests of Past-test scores of groups G, H and I. ................... 195 Analysis of covariance of Post-test scores of thu groups HMA, AM A, L M A ............. 196 Analysis of variance of mental ability -.cores of the groups HMA, AMA and LMA 199 UNIVERSITY OF IBADAN LIBRARY -- xxi ~ PAGE Analysis of covariance of Post-test scores of the groups Hi'A, AHA and LMA 200 Analyst of varianceof the Pre-test scores of the groups HMA, AHA, LIMA . . 201 Multiple range test of Post-test scores - one- way Scheffe Procedure (AN/VA) of groups HMA, AHA anc LHA .. 202 Multiple ^egression analysis of post-test scores with pre-teat scores of groups HMA, AMA and LHA ............................... 203 Summary of 1-test of ■‘•he post-test scores of groups H.'A, AMA and LMA 204 Analysis of variance of Post-test scores of groups HivA - One-way scheffe Procedure 206 Summary of t-tests of Post-test scores of groups HMA - A,D,G .. 207 Analysis of variance of Post-test scores of groups AMA - One-way Scheffc Procedure .. 208 Summary of t-tests of Post-test scores of AMA (B,E aid H) ......................... 209 Analysis o; varianceof Post-test F'r'^res of groups (LMA) - One-way Scheffe procedure 210 Summary of t-iosts of Post-test scores of groups LMA: C,F, and I 211 Analysis of ccvariance of the post-att■„ J-ude scores of groups UCU, RCU and NCU .. 213 Multiple classification analysis of post­ attitude scores of groups UCU, RCU and NCU ,.ith Pre-attitude covariate. .. 214 Analysis of variance of Pre-attitude "cores of groups UCU, RCU and NCU 215 UNIVERSITY OF IBADAN LIBRARY xxi i j LE PAGE 70 Multiple range test of Post-attitude scores of groups I’CU, RCU ana NCU by Student - Newman Keuls Procedure (SNA) oneway ANOVA 215 71 Multiple regression analysis of t; e post­ attitude scores with pre-atticuda of grouDS UCU, RCiJ, NL’U ............................... 217 72 Summar; of t-tasts of Post-attitude scores of groips UCli, RCU, NCU .. 219 73 Summary of t-ts' cs of Post-attitude scores of the g re ups U CL); A,B and C 220 74 Summary of t-tests of P^st-attitude scores of th. groups RCU: 0. E ana P, 221 75 Summary of the t-tests of Post-attitude moan score scorrs of the groups G,H and I .. .. 222 76 Analysis of covariance of Post-attitude scores of groups HMA, -MA and LMA .. 224 77 Multiple classification analysis of the pose-attitude scores of groups HMA* AMA and ...MA 225 Analysis of variance of Pre-attitude scores of HMA, AMA and LMA ................... 226 / 3 Multiple range test of Post-attitude scores by one-vay ANOVA LSD procedure in groups HMA, AMA and LMA . ................ 22 7 Z — Multiple regression analysis of Post­ attitude scores with pre-attitude score-; for groups HMA and LMA 229 z • Multiole regression analysis of Post­ attitude scores with pre-attitude scores i or greups HMA and AMA 230 z2 Multiple regression analysis of Post attitucs scores with pre-attitude scores for groups AMA and LMA .. 231 UNIVERSITY OF IBADAN LIBRARY xxiii PAGE Summary of t-tests of Post-attitude scores of groups I-ApA, AHA and LMA . . , . . . 232 Summary of t-tests of Post-attitude scores of groups HMA A, Q, C „ .. .. . 233 Summary of t-tsses of Pcst-attitud'’ scores of groups AM A £, E and H .................. 235 Summary of t-tests of Post-attitude scores of groups LMA C,F and I. .. .. 236 Summary of t-tests of MAS and CAS of the groups LICU, RSU, NCU, HMA, AHA and LMA . . 238 Summary of Pearson correlation coefficients of the groups for MAS end CAS ,. .. 239 Multiple regression analysis of the MAS and CAS of the groups. .. .. .. .„ .. 240 Correlation coe-fficients of Post tests with PCA, MAS and C A S .................. . .. 242 Intsrcorrelatic. coefficients of the seven variables .. . . .. .. .. .. 243 Multiple regress analysis of Post-attitude scores with post-test scores of the groups 245 Multiple regression analysis of Post-test scores with MAS and CAS .. 246 UNIVERSITY OF IBADAN LIBRARY CHAPTER ONE INTRODUCTION 1.1 The background to the study The need for man to facilitate mental calculations, and the problem of arithmetic computation with stones in the sand, strokes on the wall, in all probability, were attempts which later gave rise to more sophisticated items, such as the Abacus. Thus, Abacus became the first attempt of a calculating device used to perform arithmetic opera­ tions such as addition, subtraction, multiplication and division. The Egyptians and Greeks had used Abacus many thousands of years ago. Different forms of Abacus can be found among the Indians and Chinese. According to Herodotus, a Greek scholar, the Abacus of the type in (Fig. 1) was used by the Egyptians and the Greeks. 1000 100 10 Fig. 1: Abacus No. 23 1. The Encylopedia Americana, International Edition, New York: Americana Corporation, Vol. 5, 1974, pp. 161 - 163. UNIVERSITY OF IBADAN LIBRARY - 2 - Educationally Abacus is useful as a teaching devics to give young pupils a better understanding of place-value in the decimal system of numeration. It is generally made up of several parallel wires running across the width of a frame, and the beads are strung on the wires and are used as counters (See Fig. 1). For example in a decimal system of numeration, the separate wires represent units of 1 0's, 100's, 1000's and so on. There are usually 10 beads on each wire, so that each bead stands for a unit of place-rvalue in the decimal system of numeration. The simplest kind of calculating machine is an adding machine which mainly performs addition and subtraction operations. However, machines that can perform other mathematical operations such as multiplication, division etc. are called calculators. Both calculators and adding machines are classified as digital devices because the numerical quantities in the machine are represented by a sequence of digits. The first true mechanical calculator, however, was more of an adding machine. It was designed by Blaise Pascal . The adding machine had a ssries of wheels with the numbers of 0 to 9 engraved on their circumferences and could perform addition with ’carrying'. 1 Leibniz designed a real calculating machine that UNIVERSITY OF IBADAN LIBRARY 3 could add, multiply, by repeating addition. In 1822 (Babbage, Charles) built a small six-decimal place calcu­ lating machine which could perform arithmetic operations like addition, subtraction, multiplication and division. Most of the mechanical calculating machines built in,the nineteenth century had register-full-key board (printing or non printing). Most of the full-key board printing or non-printing machines are now electronically operated instead of manual operations. There is also the mechanical rotary calculator. 1.1.1 The Electronic Calculator: The electronic calculators, the subject of this study performs functions midway between those of a mechanical desk calculator and an electronic computer. It adds, subtracts, multiplies, divides and it also automatically stores inter­ mediate answers for farther calculation (Fig. 2). • 'I l S C R E E N M l_d l°FF_J IgN/cj JjL-Cwi lib-j j M+ ( E O m f 8 |h J CD M U J urn IX l La—[ UU L iJ c u lU JlC Z Jl m ! e g Fig. 2: Model of modern Electronic Calculator. UNIVERSITY OF IBADAN LIBRARY - 4 - Most electronic calculators have a cathode ray tube as the out-put device, the computational results appear on the cathode ray screen, which can display as many decimal places as are provided in the calculator. However, the calculator used for this study, and like most hand-held electronic calcu­ lators, can display up to eight places of decimal. Many of these calculators have storage facilities with memory. Though some electronic calculators are programmable externally, most hand-held calculators are not, and the ones for this study were not programmable. Some other electronic calcu­ lators also have internally stored programmes. These programmes usually are shorter than programmes of the large automatic computers. Hand-held calculators are used in this country in higher institutions, industry, commercial and business houses. It was not too long ago that the uses of calculators were allowed in mathematics and science-based courses in higher institutions of the country. However, the primary and secondary schools' pupils are not allowed to use calculators * at all in the school. Several factors are likely to contribute to the opposition of the use of calculator in schools. First,few calculato s are available in schools and in the country in general. In consequence most pupils do not have access to the use of calculators. UNIVERSITY OF IBADAN LIBRARY - 5 - Also pupils are not allowed to ure calculators in classroom instruction and in tests of mathematics in primary and secon­ dary schools. There is a ban on the use of calculators in the West African School Certificate Examinations particularly the General Certificate of Examinations and the Joint Admission and Matriculation Board Examination into the Universities. Four-figure table and slide rule are available in schools and their uses are taught to pupils in the lower forms of secondary schools; and it would have been expected that electronic calculator could be used in the school system to do the same computations, even faster. Would this have been because of limited availability of the device in the country? Or would it be because of a fear of modern technology and its application that is, by teachers and educators? The design of this research was not meant to pursue the probable reasons for teachers’ and educators’ attitudes toward calculator usage (left to future researches) rather it investigated the impact of calculator use on the instructional process. The study however, did try to find out it therg were other empirical studies relating to the attitudes of teachers and educators vis-a-vis calculator usage in schools. In developed countries and to some extent in N i g e r i a many voic j have been heard debating t h e v i r t u e s UNIVERSITY OF IBADAN LIBRARY and dangers of calnu]atcr u^age. A few have cried out in fear that the use of calculator would result in pupils who cannot remember basic facts or do traditional paper- and-pencil computeticn. Teachers, in particular, are concerned about how calculators will affect students compu- tational skills (Palmer) 2. However, this fear of "rot-the- 3 mind theory” has not been supported by research (Suydam) . Although long-term effects of sustained calculator usage are not yet known, there is ample evidence that frequent use of calculators in elementary schools has no detrimental 3 effect on achievement in mathematics (Suydam) . However, the rapid growth and sales of inexpensive calculators and their consequent widespread availability to pupils and teachers demand that the mathematics curriculum be re-examined and that teachers could use calculators as an instructional tool. Throughout the country, mathematics as a school subject has been made compulsory at both primary, secondary and teacher training levels. All school pupils have to take mathematics examination1 in their final year, and this has led to the general anxiety among most2 2. Palmer, H.B.A. Mini calculators in the classroom - What do Teachers Think? Arithmetic Teacher 25(7) 1 978, 27 - 3. Suydam, N. M . "The use of calculators in Pre-college Education: A state of the Art Review". Columbus, Ohio: Calculator Information Center, Hay, 1 979. UNIVERSITY OF IBADAN LIBRARY 7 pupils particularly those who find mathematics difficult. The results of pupils in mathematics examinations at primary and particularly secondary school levels of education have not been encouraging (See tables 1 S 2). TABLE 1 Detailed WASC Result in Mathematics for Nigeria - June. 1970 Grades F 9 P8 P7 C6 C5 C4 A3 A2 A1 Percentage Performances 52 13 10 11 4 3 6 1 1 Total Cand idates 13330 3319 2 4 e o 2704 915 763 1359 319 189 Source of Informat ion : WAEC Annual Report, 1970. TABLE 2 Percent age Failures in WASC Mathematics for Nigeria, 1 965 - 1 976 Year 1 965 1 966 1 96 7 1960 1 96 9 .1 970 1 971 * 1973 1974 Percentage 33 33 34 37 51 53 39 * 55 49 Year * 1976 Percentage * _^ Failure Source of Information: WAEC Annual Reports, 1 965-1 977. * No data were available for 1 972 and 1975. UNIVERSITY OF IBADAN LIBRARY -a- The Federal government of Nigeria through ministries of Education and other educational agencies had made efforts to improve the teaching-learning of mathematics in our school systems and these have been seen in the different programmes launched by government to bring about favourable results in mathematics performance. For example, there was the "traditional to modern mathematics” episode and yet there had been not much improvement. Learning of mathematics involves computational and problem-solving skills. Pupils with good memory are generally proficient in basic computational skills even though they may fail to grasp what should be done in a problem-solving situation (Etlinger)^. A successful educational programme must try to include effective instructional materials, which would for a subject like mathematics incorporate the use of calculator. Mathematics is taught at all levels of education so that it can be used in-real-liks situations, hence much emphasis must be placed on comprehension, analysis and reasoning than on mere memorization cr computational skill which*could be done with calculator. Researchers in developed countries like Japan, United State of America have shown that calculators have a greate advantage in computational use (Suydam) 3. It would therefore be logical to integrate calculator-use in problem-solving situation; and pupils would best be served of rigorous training is given in this area. It has also been 4. Etlinger, L. The Electronic Calculator: A new trend in School Mathematics. Educational Technology Journal, ___________14_D_ec , , 1 974, 43-45. ________________________ UNIVERSITY OF IBADAN LIBRARY 9 found that ^upils waste a lot of time on computation than the analytical part of problems and in many cases, the time- consuming computations involved in mathematics may invariably block the pupils’ minds from even attempting to find the 2 solution of the problem (Palmer) . This is where the cal­ culator has its greatest use because it would relieve pupils of the tedious computational factor in the problem and allow them to concentrate on how the problem would be attacked and solved. Problem-solving is in the higher hierarchy than the concept learning, (GagneO^ and it would therefore, be expedient to study how to integrate calculator- use into the school mathematics programme so as to find out most efficient and effective mode of learning mathematics concept which would bring about a positive attitude towards the subject. The m a i n focus of this study was on electronic calculator-use and instruction with the primary objective of comparing groups that were allowed to use electronic calculators and those that were not allowed to use them. It should be emphasised that the purpose of this investi­ gation however, was not to show pupils how to operate electronic calculators just for computational benefits alone, but rather to try to show how they could be more 5. Gagne’, R.M. The Conditions of Learning. .New York: Holt, Rindhart & Winston, 1970, pp. 155 - 170. UNIVERSITY OF IBADAN LIBRARY effectively and/or effectively used to solve mathematical problems. One difficulty that has constantly surfaced in recent calculator research is the failure of most researchers to carefully build into their research design the experimental treatment in such a way that it would take advantage of the unique capabilities of electronic calculators (Suydam). 3 Also, in developing attitudinal criteria care would have to be taken about pupils’ immediate reactions concerning their feelings about themselves and the problems they would have completed. The present investigation endeavoured to integrate calculator-use into the instructional process, and to administer Likert-type attitudinal measure to assess pupils attitudes towards mathematics as a school subject and their attitudes towards the use of electronic calculator and mathematics in the secondary schools. 1.2 Statement of Problem What effects would the use of calculators have on our school programmes? Should they be used in elementary and t secondary schools mathematics programme? If so, with what level of pupils? How do pupils feel about using calculators in the mathematics programme? Should calculators be used in test such as General Certificate Examinations, Joint Admission and Matriculation Board Examinations? Should the use of calculators be integrated into mathematics curriculum and textbooks ? UNIVERSITY OF IBADAN LIBRARY - 1 1 - Would the pupils who use calculators in instruction and tests perform better than pupils who do not? Would the use of calculators aid the pupils in the development of basic concepts in mathematics? Accurate answers to these questions are essential in assessing the current status of calculator and more importantly, preparing for calculator usage in the mathe­ matics curriculum and for examinations in our schools systems. This study endeavoured to find the effects of the use of hand-held electronic calculators in pupils' attitude and perfor­ mance in mathematics instruction. Specifically answers were sought to the following questions. 1. Would there be any difference in the mathematics achievement scores of those students who use calculator (the treatment groups) and those who do not use calculators (the control group)? 2. Would there be any difference in the attitude of those students who use calculators (treatment groups) and those who do not use calculators (the control group)? 3. Do students differ in their mathematics achievement on the basis of differences in mental abilities? 4. Do students differ in their attitudes towards mathematics and calculator-usage on the basis of differences in mental abilities? 5. Is there any relationship between students mathematics achievement and attitudes? 6 . Is there any relationship between students’ attitudes towards calculator usage and mathematics? In order to answer these questions the following null hypotheses were tested at .05 alpha level: UNIVERSITY OF IBADAN LIBRARY 12 - 3 The Hypotheses: There will be ro significant differences in the achievement scores of pupils who use: (i) Calculators in instruction and tests (E^) groups, (ii) Calculators in tests only (E0) groups, and (iii) no-calculators at all (Eg) groups, i.e. Ho: ME ̂ = ME2 = ME3 at a = .05 There will be no significant difference in the achievement scores of pupil-s of high (C^), average (C2 ̂ anc! •Low ^ 3 ̂ cental abilities, i.e. Ho: M C ̂ ~ = MC2 = MC3 at a = .05 , There will be no significant difference in the attitude towards mathematics and calculator scores of pupils who use: (i) Calculators in instruction and tests (E^) groups, (ii) Calculators in tests only (E2) groups and (iii) no calculators at all (Eg), groups, i.e. Ho : XE1 = XE2 = XE3 at a =.05 UNIVERSITY OF IBADAN LIBRARY - 1 3 - 4. There will be no significant difference in attitude towards mathematics and calculator scores between pupils of high (C ^ , average (C^) and low (C^) mental abilities, i.e. Ho: XC = XC2 = XC3 at a = .05 5. There will be no significant relationship between the attitudes of pupils to mathematics and calculator-use in mathematics as a = .05. 6 . There will be no significant relationship in pupils’ mathematics achievement scores and post attitudes towards mathematics and calculators scores at a - .05. 1.4. Significance of the study From available records, no empirical studies have been undertaken in Nigeria as to the use of electronic hand-held calculator in our school system. However, comments, as to the use of calculator by elementary and secondary school pupils, by teachers, parents, educators, school administrators and concerned citizens have rather been mere speculations. Since there are no empirical studies in the country to back such speculations hence the need for this study. UNIVERSITY OF IBADAN LIBRARY - 1 4 - According Suydam 3 several stuuies on the use of electronic calculator in mathematics at all levels of education anound in many of the developed countries like the United States of America, Britain, yet few studies had been undertaken in the are of differential effects of the electronic calculator and instruction on concept learning in and attitudes to mathe­ matics. Etlinger4 reported studies carried out with respect to the use of calculators in the elementary school mathematics and some of the studies which bother on concept learning and attitudes had inconclusive results. It is the growing awareness of the usefulness of electronic calculators on the part of Nigerians, and the availability of hand-held calculators in and outside school system in the country which have prompted this study. One would want to find out the impact the use of calculators 'would have on the school system. Some interesting and fundamental questions could be raised in respect of the use of electronic calculators in our school system. Does our sqhool. system need a calculator-use policy? Should calculators be available to pupils at all levels of education, primary, secondary and teacher- training colleges? Should calculators be encouragec for the topmost classes in the primary school? Would the teachers allow the use of c a l c u l a t o r s UNIVERSITY OF IBADAN LIBRARY - 1.5 - in the classroom during instructions and tests? Should special trail.mg be needed to use calculators effectively? Would the use of calculators by school pupils not make them mentally lazy? Should mathematics textbook have activities written for calculator usage? Before attempts are made to answer these questions, possible role(s) of electronic calculator in school mathe­ matics programme should be identified. Etlinger4 has characterized two differing views on the use of the calcu­ lators, a functional view and a pedagogical view. In the purely functional view, the calculator is considered as a device much like slide rule, log-table - a device that can do the chores involved in tedious arithmetic computation, thus saving time and frustration. The other view, a peda­ gogical one, looked at the calculator much like a textbook, flash cards, or manipulative device to facilitate learning. Both uses could be good or bad depending on the task to be performed and the age of the pupil. It was on the basis of these views that issues were raised thus: Ci) would calculator maintain a motivational value over several years of pupils’ use or would it become a mor« common place household object? (ii) Can the calculator be made to help with the learning of facts and algorithms or will its availability hinder mathematics learning? UNIVERSITY OF IBADAN LIBRARY 16 Ciii) Will experimentation with the calculator teach children about numbers and operations or merely about the calculator? (iv) What types of directed activities would be most appropriate for pupils at various ages? (v) Will pupils think less about different methods for solving a problem because it is now more expedient to find the easiest or shortest method? Or will children experiment with many methods of solving a problem because it is easy to try different methods on calculator, and interesting to compare solutions? (vi) Will the idea behind the arithmetic operations be more widely understood because the pupils have immediate feedback from the calculators? Will the patterns be easier to grasp? However, a somewhat philosophical question which has often been asked about any new teaching-learning devices be it programmed text, computers, calculators etc is this: Is modern technology providing a learning facility or a learning crutch? Here, it seems educators must become sensitive to the effects of modern technology on their pupils and in particular to whether pupils are becoming UNIVERSITY OF IBADAN LIBRARY - 1 7 - more dependent on motivational and educational devices for thinking and learning. The issues raised above are pertinent to this study and one hoped that che results of this study would be able to offer some answers to these questions. Some studies which were carried out in developed countries like the United States of America and Japan that electronic calculators have some of those useful applications which are relevant to this study. The hand'-held calculator has been found to motivate and encourage students to be inquisitive and creative as they experiment with mathematics ideas e.g. solving relations-equa- tions (Bell)^. For c.a u, li=. j~ ]2 = 1125(|___j is the unknown) Other useful application on the use of electronic calculator among the general public is that the calculator could assist individual consumer on how to become a wiser consumer in the ability to use calculator to compute percentage discount on article purchased in departmental stores. In the classroom instruction calculators can be used to* reinforce the learning of the basic number facts and properties in addition, subtraction, multiplication and division. It can be used to serve as a flexible answer key to verify the results of computatio ., and as a resource tool, it can promote student6 6. Beil, M. Needed R and D on hand-held calculators. Educational Researcher, 977, 6(5), 7-13. UNIVERSITY OF IBADAN LIBRARY - 1 8 - independence in problem-solving. It can be used to solve problems that previously have been too time-consuming or tedious to be done with paper and pencil e.g. to verify the value of exponential ’e ’ by computing its value in series approximation. e = lim (1 + —n)n n — > oo e 1 J_ J_ + J_ + _L1 f 2! + 3! + 4! Despite the useful application of calculators, yet teachers, school administrators and educators are rather skeptical about any innovations which either find their way into schools or catch pupils’ eyes outside the school. Such skepticism may arise as a result of what these people perceived would be the consequence of such innovations within the school system. Educators have, for instance, argued that innovations like teaching machines would merely reinforce rote learning or memory associations rather than encourage pupils to solve problems creatively. While many have even seen television, at one time or another, as a menace to school learning. Some argued that time spent by pupils watching television probably means time lost from an activity like home-work (Bell)5. UNIVERSITY OF IBADAN LIBRARY - 1 9 - Tiost recently in the United States of America and Europe, the Lee of small electronic calculator in schools has come under similar barrage of skepticism .and sporadic C rejection in schools (Bell) . Rather than planning careful investigations to seek answers to important questions relevant to use of calcu­ lators in our school systems, some educators, examining bodies, school administrators have dismissed this new technological innovation as a mere gadgetry or device that is not good enough for the traditional classroom settings. Some people argued that such devices may not yet belong in the mathematics curriculum of primary and secondary schools in this country. Such attitudes suggest that a careful study on the impact of the use of calculators in mathematics instruction was necessary. One of the major spin-offs of the space programme has been the miniature of electronic components to a level which allowed the small hand-held electronic calculators industry to flourish. In fact, current availability has reached the near-saturation point in Japan, Europe, United States of America, and has become popular in Nigeria. One can find them being hawked along every major road in the country. The number of types in the market varies from programmable ones to the non-programmable types but all equipped with UNIVERSITY OF IBADAN LIBRARY 20 different numerical notations and mathematical functions. While the use of calculator is widespread, the debate continuer* as to what performance and psychological bene­ fits may accrue from calculator utilization. g However, Bell had indicated a number of important areas where research and development efforts are needed for better understanding of the impacts of calculators. The groups of people who may be concerned about the effects of calculator usage in schools are parents, teachers, school administrators, educators and examining bodies like West African Examination Council (WAEC) and Joint Admission and Matriculation Board (JAMB) etc. Opponents to the usage of calculators, argued that calculators might have deterimente1 effects on the development of children’s mathematical abilities. Conversely, proponents did assert that calculators may actually facilitate mathematical learning. However, to throw more light on such a debate, the investigator became interested and carriedfo(jt 'this study. Already research efforts in this area are rather scanty in the country; hence the need, in this study, to carry out - carefully planned empirical investigation on the use of electronic calculators in our school system. The 6 6 . Bell, M. Needed R and D on hand-held calculators. Educational Researcher, 1977, 6(5), 7 - 13. UNIVERSITY OF IBADAN LIBRARY -21 - genre of this study was to provide answers to probable advantages of the calculator to the learner in facilitating the teaching learning process of mathematics in the school system. The focus of this study was on pupils in the secondary schools, their learning of mathematical concepts, attitudes as regards calculator-usage and mathematics learning. This study raised issues on what added advantages the probable caleulator-usage ahad on the instructional strategies such as algorithmic computations and problem-solving. 1.5. Limitations of the study The following constitutes the major limitations of the s t u d y : 1. The experiment was carried out only in Ibadan. The situation as to the implication of this study may be different in the rural areas of the country or in other highly industri1ised cities of the world. 2. The duration of the study may have not produced the needed attitudinal changes of this kind of study. For attitudinal changes may take a longer time than six (6) weeks the study covered. 3. It was difficult to ascertain whether the subjects in the restricted calculator groups (RCU) and Non-Caleulator groups (NCU) were using calculators UNIVERSITY OF IBADAN LIBRARY -22- privately especially to practise. This may possibly affect the pupils' attitudes towards the use of calculator. UNIVERSITY OF IBADAN LIBRARY CH,-P7£n TWC REVIEW Or nC-L/'-KE LITE.-AURF AMO RESEARCH The hand-held calculator is : tool used in society today for calculations. fcucators hitherto who have been resisting the use of electronic calculators at primary and secondary school levels m y bo willing to admit the use of calculators if only, where the calculator adds a “-new dimension" of learning to the experience of the pupils and it allows a child to do something or learn something that could not before. Tne most common argjmsnt against the use of calculators in the sonoois .s that treir use would lead to decay of the understanding of arithmetic and loss of computational skills in student. This aversion of many people to the use of calculators is simpJy the resistance to a change in "instruments' in calculation brought about by advancing techno logy. Technological c ongos ore hard to accept at first » and one can easiLy muster up all sorts of specters of doom when first presented vith them. The aversion and resistance to calculators-use in primary and secondary schools have precedence in history. UNIVERSITY OF IBADAN LIBRARY - 2 4 - In the Phaedrus, Plato had a short discussion on the value of reading, which was at that time just coming into vogue. The tradition before then was oral, that is, one memorized stories and recited them. One of the participants in Plato’s dialogue expressed the fear that the coming of written materials would lead to the decay of the ability to memorize and recite works. He was quite right. The ability to do this had generally been lost. But look at what has been gained. The amount of literature that one can absorb has been increased 10,000 fold. One would think it w Ail prove to be the same with calculators. Children would lose the ability to dc sums and'products on paper with pencil, however, at a very early age they would have gained the ability of doing problems with very large numbers quickly and very accurately, and this could be a very exhilarating experience. One should think of the time that will be saved on drill and simple arithmetic computations. This time could be put to good use doing more complicated word problems. Accoraing to Immerzeel^ students would n o t b e 7 7. Immerzeel, 0. et. al.: Teaching mathematics with the hand-held calculator. In M. Suydam Electronic Hand Calculators: The Implications for the Pre- College Education. Final Report for NSF Grant No. EPP 75-16157, 1976. UNIVERSITY OF IBADAN LIBRARY 25 dependent on the calculator rather, students will undertake much more complex problems and students can solve verbal problems using the calculator at about three times the rate the problem could be solved with pencil and paper a lone.® 2.1 Electronic calculator and the school system Despite the availability of the mini-calculator and its ever increasing public usage, there still exists controversy among educators as to its proper usage in the school or whether it should be allowed in the school setting at all. It’s advocate refers to it as an essential implement in the newest mathematics (Higgins)^, as a motivating device Q (Mastbaum) , as a means toward immediate reinforcement of g results and a significant learning strategy (Lewis) . Opponents of calculator like James Mckinney, Professor of mathematics at California Polytechnic State University in Pomona, U.S.A. states his case in the ’Great Calculator Debate", 1974 as follows: ‘ 8. Roberts, D. M. The impact of Electronic Calculators on Educational Performance. Review of Educational Research 1980, 50 (i), 7UTT), 71-98. 9. Lewis, P. "Minicalculators Have Maxi-Impact". Nations Schools 93: 60, 1974 UNIVERSITY OF IBADAN LIBRARY 26 If what we are talking about is reducing tedious calculations then perhaps mini-caleulators can be an aid, but teaching a student to push buttons won’t help him if what he needs is more instruc­ tion ii. actual addition, substraction, multiplication and division - I can’t think of any reason why a fourth or fifth grader* should even see one, after all that's when we are trying to teach basic arithmetic (p. 12)9. This pungent argument holds against the use of electronic calculator in elementary school, whereas this does not go to say that calculator cannot be used in the secondary or higher education levels where the pupil would have mastered the basic concepts of addition, subtraction, multiplication and division of numbers. Here,differences in opinion on the calculator usage bothers on grade-level at which it should be used. However, studies carried out by Schur and Lang 10 with a group of youngsters of elementary school age found that improvement on computational ability of the youngsters * Equivalent of Nigeria Primary 4 or 5 pupil. 10. Schur, J. 0. and Lang, J. W. Just pushing buttons of learning? A case for mini-calculators Arithmetic Teacher 23(7), 1976, 559 - 563. UNIVERSITY OF IBADAN LIBRARY 27 was traceable to the mini-calculator regardless of the learner's sex, and they went on to suggest that mini- calculator does have a place in the elementary school. Similarly in the U.S.A. the Eoard of Directors of the National Council of Teachers of Mathematics has adopted the following position: With the decrease in cost of the mini-calculator its accessibility to students at all LEVELS OF EDUCATION is increasing rapidly. Mathematics teachers should recognise the potential contributi of this calculator as a valuable INSTRUCTIONAL AID in the classroom, the mini-calculator should be used in imaginative ways to reinforce learning and to motivate the learner as he becomes proficient in mathematics (NCTM, Newsletter, December 1974). Studies so far, reviewed are from U.S.A. and Britain and it would be shown that calculator-usage researches:; have been carried out at all levels of education, elementary (primary), secondary and higher schools in these countries. 2.1.1 Elementary School ‘ Most of the studies carried out at this level of education were of the general pre-test-post-test design. Capital letters by the authors: are used to emphasis the relevance of the portion of this study UNIVERSITY OF IBADAN LIBRARY o * 2 6 Hohlfeid . examined the effect of a calculator programmed to provide immediate feedback on working simple multiplication problems. being fifth grade pupils and -pre­ test -post -test design with experimental and control groups it w a s found that the experimental group that used calculator on the test and during instruction performed significantly bstcer than the control group that did not use calculator at ell. Whitaker explored calculator impacts with first grade children in the elementary school. Pupils were randomly assigned to experimental and control groups while the experimental groups used calculator and the control group die not. The treatment lasted for thirty instructional da^,; and it was found that the experimental group had two gain scores to one gain score of the control group but nc altitudinal differences to mathematics were found oetween the groups. Thus, the majority of the stu 'ago . jmpleted at the elementary school showed computational advantages for ages » 6 - 1 1 years for the introduction of calculator usage into the mathematics instruction while the attitudinal benefits 11- Hohlfeld, J.F, Effectiveness of an immediate feedback device for learning basic multiplication facts (Doctoral Dissertation. Indiana University, 1973). 17-• Whitaker, W.H. A study of change in achievement, interest, and attitudinal variates accompany the use of electronic calculators i a first grade mathematics curriculum. Unpublished Ph.D Thesis, University of Southern California, 1977. UNIVERSITY OF IBADAN LIBRARY 2 9 were limited. There was scarcely any conceptual benefits due to calculator-usage. 2.1.2 Sei-onJary Level Most of the studies at the secondary school level reviewed were of the pretest-post-test design. Quinn 13 used eighth and ninth grade students to observe whether the use of programmable calculator would facilitate algebra achievement and attitude towards mathematics. There were experimental and control groups, using same classes from two schools - one experimental and the other control. The experimental classes used programmable calculator while the control did not use calculator. However, no achievement differences were found between experimental and control groups, but the experimental group shewed more favourable attitudes than the control group. Majority of the secondary level studies found computational benefits due to calculator use. However, as the case in the elementary school studies, very little support was found for the hypothesis that calculator benefits improved development as regards conceptual and attitudes areas. It, there­ fore, remains for future researchers to explore conceptual benefits of calculator at secondary school level of education. 13. Quinn, Q. R. The effect of the usage of a program­ mable calculator upon achievement and attitude of eight and ninth grade algebra students. Doctoral Dissertation, Saint Louis University, 1975. UNIVERSITY OF IBADAN LIBRARY 2„1.3 Higher education level The use of caIculatore at this level of education seems to be an accepted norm today. In Nigeria and other parts cf the world, the use of calculators is permitted during instruction and on tests. However, there has been no empirical studies on calculator-usage at this level of education in Nigeria. In developed counti'ies such as United States of America and Britain, many studies have been carried on impacts of calculator in the college mathematics .(Suydam) 3 . Sosebee and Walsh 14 investigated whether students who had used calculators on in-class chemistry examinations would do better than those not using calculators. Using the mean scores from the examinations no differences were found in their scores. g Roberts, Seaman and Lerner were interested in esti­ mating the discrepancy in performance and attitudes of less calculation condition Chard work) against calculation condition (advanced calculator). Using ANOV/A, results showed large differences between calculation modes in favour l of the advanced calculator. Host of the studies at this level provided support for the computational benefits of the calculator. In addition, affective Cattitudina1) effects were found favouring groups 14. Sosebee, J. F. and Walsh, L. M. Pocket calculators and test scores in introductory chemistry, journal of College Science Teaching, 1975, 4,324. UNIVERSITY OF IBADAN LIBRARY which had nsod calculators, This was especially true when the dependent measure tapped specific task, affective responses Reviewed in this study were some findings on the conceptual impact of calculator use in the studies carried out by Roberts and Glynn Q . According to Stultz 1 5 calculator can be used at any grade level not only to check answers but also to debug a problem. Calculator can serve the same purpose as flash cards with quick oral or written response and immediate reinforcement. One of the more practical uses of a calculator in high school mathematics is in the evaluation of formula. At higher education level calculators are needed in statistical problems. 2.1.4 Learning outcomes an mathematics instruction In this study one endeavoured to highlight the compu­ tation, conceptual and ettitudinal effects of calculator use on learning outcomes of mathematics instruction at the secondary school level. Gagne’ and Briggs 1 6 have identified five categories of learning outcomes which are relevant to this study such as: (i) intellectual skills, (ii) cognitive strategies, 15. Stuitz, l_. Electronic calculators in the classroom. Arithmetic Teacher , 21, 1975. 16. Gagne’, R. M. and Briggs, L. 9. Principles of Instruc­ tional Design, 2nd Ed., New York: Holt, Rinehart and Winston^ 1979. UNIVERSITY OF IBADAN LIBRARY 32 (iii) verbal i n formation , (iv) me bar ckills and (v) attitudes. In trying to study the Intellectual skills one examined the individual competency of the learner by measuring the 3 Rs - reading, writing and arithmetic. These competencies were measured in the study using mental ability test of verbal and numerical nature. Writing on cognitive strategies Bruner, et al 1 7 described it as mathemagenic behaviours while Skinner1r called it seIf-management behaviours. Precisely, it is the capability of individual's learning, remembering and thinking behaviour. Gagne 1 6 defined cogni­ tive strategy as a control process, an internally organized skill which governs the learners' own intellectual processing. It selects and guides the internal processes involved in defining and solving new problems. It is this strategy that is needed in problem-solving. It can however, be measured using achievement tests. A mathematics achievement test was used to measure this area of learning outcomes. 17. Bruner, 3. et al. A studv of Thinking, New York: Wiley, 1955. 18. Skinner, 3. F. The Technology of Teaching, New York: Appleton, 1So8. UNIVERSITY OF IBADAN LIBRARY - 3 3 - On verbal information, the learner usually acquires needed information from formal instruction or on incidental learning and such information would be stored in learner’s memory not by memorization, but by constant practice through repetition-practice. Since this study required the subjects to go through an instructional process designed for them it was possible for the learner to acquire the necessary verbal information on the mathematical concepts. This was measured through the teacher-students classroom interaction incorpo­ rated into t he stucy design. Motor skills involve writing, and handling. In this study, some groups of subjects used calculators which they had to manipulate but such skill was not measured in this study. Attitudes: It is an aspect that was measured in this study. One was interested in studying the attitudes of pupils only towards mathematics as a school subject and their attitudes toward the use of calculators iin mathematics. One did not measure such socially approved attitudes as respect for other people, cooperativeness, personal respon­ sibility, self-esteem, but rather attitudes toward knowledge and learning situations. Likert-scale type of attitude questionnaire vas developed and used. Other attitude questionnaire that was appropriate for this study was also used. UNIVERSITY OF IBADAN LIBRARY 34 Fland ers “ recognised the most common learning outcomes as content achievement, skill performance attitudes. This study had therefore recognised these human capabilities, intellectual skills and attitudes. The other learning out­ comes. verbal information and motor skills were demon­ strated during the instructional process and they involved the computational effects of using calculator in mathematics instruction. 2.1.5 Computational effects of calculator Computational benefits would occur when pupils who had used calculators during a treatment could perform routine computations (not solutions to word problems or equations that would be conceptual development or problem-solving) more accurately and/or rapidly than those not having access to calculators during the treatment. Such benefits might occur when either the pupils were allowed to use calculators on the actual task (on test or/and instruction). Sometimes the judgement as to what criterion tests were more compu­ tational than conceptual would appear to be instructive. 19. Flanders-, N. Analyzing Teaching Behaviour, Reading, M ss: Addison-Wesley Publishing Company, 1970, p . 317 - 319. UNIVERSITY OF IBADAN LIBRARY At elementary level., Spencer 2 0 used fifth and sixth graders to observe the impact of calculators on compu­ tational skills and arithmetic reasoning abilities. The experimental group was allowed to use calculator on tests and on all in-class work but the control group was not. At the fifth grade level the experimental group had gain scores on the problem-solving test whereas in the sixth grade, the experimental group had gain scores on the arithmetic computations. In two studies, done a n d g Allen investigated the effects of using hand-held calcu­ lators on mathematics achievement, attitudes and self- concept . In done's work pre-test gain scores with post-test attitude and self-concept were analysed, whereas in Allen's work metric measurement and decimal test gains were examined. In done’s work experimental group performed better on the post-test, however, there was no difference on attitudes and self-concept. With fourth through seventh-grade summer 0 » school pupils, Nelson investigated impacts of calculators on computational skills and attitudes. The experimental groups which used calculator were superior for both compu- 20. Spencer, N . N. Using the ,-iand-held calculator in intermediate grade arithmetic instruction (Doctoral Dissertation) Lehigh University, 1974. UNIVERSITY OF IBADAN LIBRARY tatior. and attitudes. Sutheriin ussd fifth and sixth grade pupils to investigate the effect of calculators on decimal estimation skills. Fupils in experimental and contra] groups were pre-tested, post-tested and given retention tests. Employing the method of Analysis of Variance (ANOVA) for the data analysis significant gains in estimation skills were found in both experimental and control groups. However, there was no difference between experimental and control groups on both the post-test and retention test, 21 Snurnway et al found no measurable detrimental effects for calculator use. Pupils learned basic facts, and achievement was good despite calculator use at grades 2 to 6 , Eikmier^ investigated the use of calculators with low achieving 4th grade pupils in mathematics achievement test and attitudes. He found no significant difference in attitude or achievement gains between calculator and non- calculator groups at this level. Kasnic and Kobrin found i the same results - no significant difference in achievement between calculator and non-calculator groups. Most other findings at this level of education support this position. 2 /|„ Shumway, R. J. et ai, Initial of calculators in Eler.antary school mathematics. Journal for Research in Mathematics Education^ T2^ 119, 141, 1 98 1. UNIVERSITY OF IBADAN LIBRARY - 3 7 - At the secondary school level, Gaslin 8 compared the achievement and attitudes of nineth grade pupils using either conventional or calculator based algorithms for operations on positive rational numbers. Significant treatment effects were found on both post-test achievement measures with experimental group gaining scores on reten­ tion tests and achievement measures; however, no difference on attitudes were found. Similar results were found by Wajeeh and Hut:on°. Whereas Wajeeh found differences between two levels of experimental groups and control groups on both achievement and attitudes, little difference was found between the experimental groups. In Hutton’s study no differences' were found between any of the experimental and control groups on any of the achievement or attitudinal variables and these 0 findings were corroborated by Jamski'and Andersen in their studies on the impact of hand-held calculators on seventh grade learning of decimal/percent conversion algorithms and effects of restricted versus non-restricted use of calculators on mathematics achievement and attitudes. Both studies, like the others tend to show that pupils perform significantly better on computational skills when using calculator but no diffe­ rences as regards attitudes toward the subject. However, Q in a study inv lving a non-mathematics area, Bolesky UNIVERSITY OF IBADAN LIBRARY investigated .ie influence of calculators on achievement in chemistry. Using 2 x 2 factorial design: E experimental condition, calculators on the m e t -test} condition, no calculators on the post-test., control, C condition, calcu­ lators on the post-test, and condition, no calculators on the post-test. Results showed no significant main effects or interaction. At the higher education level,a series of investiga- Q tiens reporter by Roberts and colleagues found computa­ tional benefits of using calculators. For all five studies carried cut by the group, they found that introductory statistics students worxad numerous statistical problems (mean, standard deviations, correlation co-efficient, etc) under a variety of conditions. Three criteria that were common to all studies included the number of correct answers, the time to work problems, and efficiency (number correct per unit of time). In four of the five studies, a post-test attitudinal measure was also administered. i Results from these studies showed that the amount of pre- practice had no effect on the number correct, time, or Q efficiency dependent measures. Roberts and Glynn found for the calculator mada: the advanced machine group was consistently upericr to the basic machine group. Roberts g and Glynn fc nd t at for the instructional work set; that, is, UNIVERSITY OF IBADAN LIBRARY instructions to work fast or work accurately groups who produced more correct answers, longer working times, and less efficient problem solution than the work fast group. Roberts and Fabrey8 found that the advanced machine condition produced more positive ratings on four of the five attitudinal clusters than non- calculator condition. Thus it anpears that most of that data and studies reviewed would support the hypothesis that using calculators during instruction benefits routine calculation and that the benefit is most pronounced when students continue to use the machine while actually performing test computations. Most of the studies were pretest - post test design and th's may constitute some problems in integrating calculator usage in the experimental and control groups, thereby making comparisons difficult. Majority of the studies reviewed still found the experimental groups performing better than control group which lends support to the compu­ tational benefits of the calculator. However, there is * need to investigate effects of calculator on concept learning . 2.1.E Effects of calculator on concept learning Acccrdin- to Suydanf ' calculators can be used to re­ inforce and e. pand many mathematics concepts that may be introduced. However, no research evidences exists to UNIVERSITY OF IBADAN LIBRARY - 0 - support the claim that concepts must be developed prior to calculator-use. Hence the notion of developing;, under­ standing through examples followed by explanation and dis­ cussion is a common technique in mathematics teaching. Almost all of the studies in elementary school mathematics comparing achievement of groups using or not using calcu­ lators favour the calculator groups or reflect no signifi- 22 cant differences (Suydam) . In certain types of mathematics problems, calculator might facilitate concept formation abilities. Fur example, some mathematical principles which require n u m e r o u s , l a b o u r i o u s calcu­ lations in order to be well understood, then concepts would be acquired faster if calculators can aid the student to leap through the computations. In addition and perhaps more importantly, if calculators can reduce frustrating computational errors, then the quality of the concept 0 attainment may be improved. Roberts states that the proposition that calculator usage can have an impact » on mathematical concept formation seemed reasonable. 22. Suydam, M . N» Researches in Mathematics Education, .lournal for Research in Mathematics Education T0C477 W 9 . ‘ ’ UNIVERSITY OF IBADAN LIBRARY -41- But, it is not yet supported by the empirical data available. This is so since few studies made any real attempt to carefully integrate calculator use into the curriculum that would illustrate how calculator can facilitate concept learning. Some studies have been carried out on concept learning but positive results relating to conceptual benefits of calculator usage would not be expected to occur as often as simple computational benefits because conceptual acquisition is a more complex task. Q Kasnic studied the effect of calculator usage on mathematical problem-solving in relation to three levels of ability of the six-grade students tested and using a 2-fact ANCVA analysis with Pretest and ability as a blocking variable he found there were no differences between experimental and control groups, nor were any differences found for the different ability levels between experimental and control groups. This design of using different ability levels has._ » held to substantiate the need for the use of calculator in concept learning. Though the results were not conclusive a similar research design could help to clarify this important area of mathematics learning. UNIVERSITY OF IBADAN LIBRARY N -42- Fischman, Wajeeh, add Hutton 23 investigated effects of a newly developed programme of meaningful and relevant mathematics on achievement and attitudes at secondary school level. Using both AiMOVA and A!\iC0VA for analysis, results showed superiority for both the two experimental groups over the control groups on both achievement and attitudes but little difference was found between the two experimental groups. These results . support . other findings on achie e- ment and attitudes at this level. However, Hutton 23 used t-tests analysis and no differences were found between any of the experimental and control groups on any of the achieve ment or attitudinal variables. Lenhard 23 worked on pre-test and post-test design- experimental and control groups at secondary school level m a t h e m a t i c s i l n a variety of analysis using t-test and ANOVA procedures, it was shown that the higher ability students made fewer concept and computational errors than did the lower ability students; and they also had more I positive atti.udes. In addi:ion, studies by BGiesky and Boling 24 at secondary school level on whether the use of 23. Fischman, 14.L. et. al. The Impact of Electronic Calcula­ tors, Review of Educational Research, 50(1} 71 - 98, 1980 24. Bolesky, E.M. and Boling, I4.A. The impact of Electronic Calculators in Roberts, O.M. Review of Educational Research, 50(1), 1980, 79. UNIVERSITY OF IBADAN LIBRARY calculators would influence students achievement and attitudes in chemistry and in consumer mathematics respectively. Bolesky used 2 x 2 factorial design and the results showed no significant main effects or interaction between, experimental and control groups while Boling findings on achievement and attitudes also showed no significant differences between the experimental and control groups. It would appear, at this level, that most of the findings on studies in the use of calculator for concept learning in mathematics did not show any difference between the calculator-use and non-use- vis-a-vis achieveme and/or attitudes. Most of the studies reviewed at higher education level seem to lay more emphasis on computation and attitudes rather than concepts. Ayers 25' was interested in the effects of situational problem-solving and calculators on statistics performance. Using several analytical procedures (ANOVA, Mann Whitney U-test etc.), results showed better achievement in the calculator groups and more positive attitudes in the situa- » tionals realistic teaching of heuristic groups. No attitudinal differences were found between the groups using or not using calculator. 25. Ayers, S.W. The effects of situational problem-solving and electronic calculating instruments in a College level introductory statistics course Dessertat.ion Abstracts International, 1977, 37A. — UNIVERSITY OF IBADAN LIBRARY 44 0 Loughlin investigated the. effects of using a programmablu electronic calculator on the achievement of students in a calculus course. Using a 2-factor ANOVA (blocking „.ver three ability levels based on previous metnonetics achievement), differences were found in favour of the experimental group, but no differences favoured the control group. It would appeir that there was s^me support for the conceptual impact of calculator use at higher education level. Problem-solving forms a substantial portion of concept development in mathematics. Tie proolem-solver must compre hend the facts, clearly understand whas is needed, and analyze the problem in older to arrive at how tna problem is to be solved. Pupils with sharp, analytical minds may do superior work when solving verbal problems, although they may sometimes lack the basic computational skills needed for problem-solving. It would therefore, biological to integrate calculator-use in problem-solving situations so that pupils would best be served if rigorous training is giving in this area. Abdelsamod 26 found the!, the calculator was considered effective in mathemotics problem- sol.ing at secondary school. 23. Calculator Information Centre, • Piesear-h on Hand-held Calculators. K-12 Oulleting No. 9 Columbus, Ohio, 1982. UNIVERSITY OF IBADAN LIBRARY A calculator can also be u s d very effectively in concept development since many examples can be explored quickly. Consider for example, she concept of the prime numbers. A number can be tastec for primeness quickly by dividing on a calculator. The concept of decimal can be introduced quite early and dev eloped through calculator activities. Many motivational calculator games c - be used to develop- important cone .pta such as place-value, estimation, integers and functional rules (Guthrie and Wiles) ^ccorcsing to Cede studsrzr- receiving both positive arid negative instances (examples) in teaching mathematical concepts die significantly better on an algebraic concept test tinn those given only positive instances only. It would therefore, be worthwhile to explore the use of calculators in the concept formation techniques - examples an: non-examples. It would appear from, studies carried cut on concept (.earning and calculator- use that the learning se;tings in whicn these studies were I conducted did not genera.ly emphasize concept-formation skills, thereby providin' a partial explanation. 2/»,= C ik, W.C.: Teaching of Concepts* Journal for Research in Mathematics Education ed. Suydam, N .M . Vo 1. 1 3, No. 7, 1382” UNIVERSITY OF IBADAN LIBRARY -46- From studies reviewed it would appear that m o s t findings on calcilator-use when students used calculator on pre-test and were n o t a l l o w e d on post-test shewed no difference. Thus, if the calculator-use were to be successful in facilitating learning, the concept attained should really be at the understanding level. Then utilization of calculator in application to principles- and concepts have had relatively little bearing on obtaining the correct Q Solution on a post-test. According to Roberts , the concept formation benefits of calculators will only be resolved when investigators used calculators as a strategy for solving problems. 2.1.7 Effect of Calculator on Attitude Finding attitudinal impact of calculator in instruction may be problem-ridden because of the difficulty in finding the attitudinal criteria to use. For the vast majoritycof the studies reviewed; three factors may have worked against finding attitudinal impacts. First, the measures used were too oriented toward general traits rather than characteris­ tics which were more task specific and sensitive to calculator influences. Second, the time frame for most studies was too short to expect any real change in attitudes. Third, since many of the UNIVERSITY OF IBADAN LIBRARY 47 investigations did not allow :ne experimental groups to use calculators on the post-tos;., many of the students might hove lost interest which in turn, might have influenced che rating on the attitude scales. !v',cst of the studies did not investigate attitudes toward calculator per so but rather in component with achievement' and, „ in most cases, there was no correlation between the achievement and attitudes. Smith" examinee, achievement end attitudinal impacts of calculator usage in teaching methodology to business and economics stuaents. The instructional period lasted for' 10 Jays with Pro-test-post-tost design-experimental and central groups. The experimental group was allowed to use calculator on pra-test and during instruction but they were not allowed on the post-test. It was found that there were no achievement or attitudinal differences between experimental or control groups £ Roberts, Seaman and Lerner' in their study on sirr,ple calculation condition (hardwork) versus, best calculation condition (advanced cal aulator) and the attitudinal data showed large and significant differences between calculation modes with mere positive taci; and self-perceptions being expressed UNIVERSITY OF IBADAN LIBRARY - 4 8 - by students using the advanced calculator. Standifer and Maples investigated the achievement and attitude of third grade students using two types of calculators and found that the hand-held calculator group scored significantly higher on achievement and attitude than the programmed-feedback calculator group at grade level 3. (Elementary School). Connor 2 Qinvestigated a calculator dependent on tri­ gonometry programme and its effect on achievement and attitude towards mathematics of eleventh and twelveth grade college-bound students, and found there were no significant differences on attitude between calculator and non-ca leu lat'"~ groups at secondary school level. It would appear from most of these studies that there seerns to be evidence that calculators influence immediate and specific perceptions, but there was no evidence to support any significant differences between experimental and contr groups or to support more general and lasting attitudinal changes. 2.1.8 Use of Calculators on Tests t One of the most peculiar aspects of calculator research is the extent to which investigators have allowed/ disallowed students to use calculators on tests. According 298 28. Standifer, C.E. and Maples, E.G. Achievement and attitude of 3rd Grade student ...in School Science and mathematics 81, 17-25, 1981. 29. Connor, P.3. A Calculator department Trigonometry Programme and its effect on achievement in and attitude toward mathematics of Eleventh and Twelveth Grade College Bound Students. Un-published Ph.O. Thesis, Temple University, 1981. UNIVERSITY OF IBADAN LIBRARY 49 to Carpenter at ol 30 reported on the results and implications of 1977 - 1978 mathematics assessment of the United States of America National Assessment of Educational Progress (NAEP) 'on "how does the availability of hand-held calculators influence performance in testing.situation?” In the report, several issues related to calculators were raised, for example: timing on exercise? it was found that students to~'< longer time to do problems with calculators than with paper and pencil. This would be surprising because calculators should presumably make computations easier and less time-consuming. The reason for this may be that students lacked confidence in their results, sc they did the problem more than one... C\ perhaps students d'G not know how to interpret the value shown on the calculator’s display. Perhaps students with calculators sought to do the problem in more than one way. The hehaviour of seeking alternative solutions to problems when calculators are available has been reported in several calculator research studies (C^penter, khsatls *- and Shumway, Cobum, F.ayr and SchoSP, Wheatley and White) 31 2. Problem-Solving;^ On exercises requiring problem- _ solving techniques, the performance of 9 and 13 year 3*1 3C. Carpenter, T.P. et a 1 Calculator in testing situation, Arithmetic "eaoher 2b 65-65, 1981. 31. Carpenter, T.P, at ai. Claculators .n testing situations: Results ind Implications from National Assessment Arithmetic Teacher, 28 34-37 Jan, 1981. UNIVERSITY OF IBADAN LIBRARY students with calculator were generally poorer than that ot students witiiout a calculator. Hence calculators dc not solve problems, people do. Strategics such as trial and error that take special advantage of the calculator need to be introduced, developed and encouraged. Problem-solving requires for more than comrutaticn i\ demands understanding, correct choice of operations, and selection of vr’-'es to operate in a particular order Carpenter et a 1 ̂ found that stubents performed routine computation better with aid of a calculator, cut problem-solving scores were poorer i *th calculator et .ages 9, 13, 17, Sons of the studi as did not allow students to use Q calculators on pnot-te sts •'ccerding to Roberts +-he assumption is that this reflects the philosophy that the real question is whether the use o r calculators will harm students’ performance when they are faced with doing calculations by caper and pencil methods. While such an approach is certainly a legitimate ay to test theory, it appears to be & negative or:notation rather thah a positive one. Instead of examining potential positive impacts, the focus is on demonstrating the laris of negative effects. UNIVERSITY OF IBADAN LIBRARY However, it would seem mere realistic, as it is being done, to assume thet calculators may have more positive benefits than negative. Tiiis assumption could be supported, in cases where computational benefits were more when students were allowed to use calculators on tests. On the other hand, it would have been instructive to test for more conceptual benefits (as it is being carried out in this study) by allowing calculator-use on those types of criterion tests - pre-test and post-test. It would therefore be appropriate to use calculators on the post-tests, assuming, of course, that the treatment propei'ly integrated calculators on instr tion, and s.;.e if chare would still be any significant v difference between the experimental and control groups. 2.2 Integrating Calculator into Mathematics Curricula in Schools Before integrating any educational devices into school instruction there must be agreement on what role(s) such devices would play in the school. But viable school roles of electronic calculator will not be established without finding solutions to many problems: problems of philosophy, problems of curriculum and methodology, problems of design, and school management of the calculator themselves. In the belief that solutions to many of these problems could be worked out in actual classrooms, the design and UNIVERSITY OF IBADAN LIBRARY _ c; o - methodology oT this research would try to explore classroom uses of calculators by integrating calculators in the instruction uni tests. According to Palmer 32 attempts to introduce calculators in the classroom, particularly at the elementary grades had been greeted with cautious responses. Teachers and educators could be worried over such innovations and investigations by Palmer 32 revealed the reasons for their caution as follows: (1 ) erosion of the teacher's roles - mechines taking over teaching functions* (2) impersonalization of the teaching - learning process, reduction or elimination of the humane aspects of teaching and learning, (3) stifling of creativity - emphasis on precision, measurement, and mechani­ zation inhibits creative expression. □esoite the anxiety expressed by teachers and educators researchers in the area of Calculator-integration i.ito curriculum had progressed. Using ninth-grade general q mathematics classes, Vaughn examined effects of calculator useage when they were specifically integrated into the curriculum. The design was pre-test-post-test experimental a n d control groups with the s u b j e c t s in the 32 32. Palmer, H.B.A.: Mini Calculators in the Classroom - What do Teachers Think? Arithmetic Teacher 25(7) 1S76, 27-28. UNIVERSITY OF IBADAN LIBRARY 53 experimental group using calculators. Using a stepwise regression analysis , results indicated that the experimental group performed better than the control group on achievement at th^ post-test but there was no difference on attitudes Q of the two groups at the post-tests. .Casterlow who studied the effects - df calculator instruction on the knowledge, skills and attitudes of prospective elementary teachers found that the treatment in which students received teacher-guided instruction with the calculator was more effective than treatments without teacher guidance for preservice elementary teachers.. The calculator was co;v idered effective with 13 to 60 problems strategies at 8 secondary school levei In studies carried out by Hcpkxns . : ' on the effect of a hand-held calculator curriculum in selected fundamentals of mathematics classes he found that students using calculators for instruction gained equally as wall in computation and significantly higher in problem­ solving as students not using calculators at secondary school level. Similarly studies by Smith 8 on a study of the effectiveness of the use of the electronic calculators in teaching simplex method to business and Economics majors found, . regardless of sex or mathematics ..aptitude level. UNIVERSITY OF IBADAN LIBRARY - 5 4 - little difference in mathematics achievement and attitude between students using or not using calculators in the classroom at college level. Most of the studies so far, reviewed have not established any differential effects as regards calculator instruction vis-a-vis achievement and attitude. However, studies by Laursen 33 on the use of calculators in High School General Mathematics which compared achievement, attitude and attendance of general mathematics students who used calculators with students who did not, found that students using calcu­ lator had greater achievement and no significant differences in attitude or attendance were found among the groups. 34 Lawson in a study of the calculators and altered calculator’s effect upon student precaution and utilization of an estimation algorithm found that calculator-use did not affect performance in estimation. Students of lowest ability made the most errors when using calculators compared with other ability levels at secondary school level. There could be no appropriate word(s) to describe the problems surrounding the integration of calculator in the3 4 33. Laursen, K .D .: A study of Calculators in High School General Mathematics ... Unpublished Ph.D. Thesis Brigham Young University, 1976. 34. Lawson, T.J.A.:- A study of the calculators and altered calculators effect ... Unpublished Ph.D. Thesis State University of New York at Buffalo, 1977. UNIVERSITY OF IBADAN LIBRARY -[35- curriculum than the supgustion reflected in Recommendation (E) of the Report of the. Conference on Needed Research and Develop- rnent on Hand-held Calculators in school mathemati. cs 3 3 Materials should be developed to exploit the cal­ culator as a teaching tool at every point in the curriculum to test a variety of ideas and possibi­ lities pending emergence of calculator - integrated curricu 1 urn ( P. 7 ) „ From available research reports there seem to be little doubt about the computational value associated with calculator use. Sufficient calculator-pretraining enables one to work problems more accurately, rapidly and efficiently. Also calcu­ lators would allow one to complete more problems per unit of time- thus in effect affording greater amount4 of practice Suydam 22 . However, for conceptual and attitudinal impacts due to calcula­ tor use, there is less concensus as to what facts can be gleaned from the research literature, and it would appear that this is the issue being raised in this study. Also, at a curriculum level, there is less agreement as to what should be done re­ garding the question of whether to incorporate calculators into instruction. For those who believe that calculators should be used in the schools, the question may still remain as to when, class and level of education for calculator to be introduced into the school mathematics programme. 2.3 Definition; Concepts and Concept Learning: Sultz^4 defined a concept as a "bounded region in the cognitive space which is reacted to as an entity".* 34 351 Calculator Information Center: Research on Hand-Held Calculators, K-12. Bulletin No. 9, Columbus, Ohio 1977. 34. Sultz, see copeland, R.W. Mathematics and its elementary teacher Philadelphia: W.B. Saunders Company, 1976, 15-65. UNIVERSITY OF IBADAN LIBRARY 56 Saltz also went further to define concept learning as, "the association and bounding of Lhe set of attributes. The first step in learning is perceiving through the sensory modalities which leads tc both concept formation and concept utilization, in any teaching and learning or problem solving situations. Saltz could mean "bounded" here as referring to those common characteristics which justify the inclusion or exclusion of anything within the frame of reference of the entity, idea, objects, event etc. A concept is not formed in a vacuum. This is why :he instructor should provide the learner with the real object or event to perceive. According to Bolton35 "concept learning is the process whereby one comes to distinguish between those elements which are an essential part of the concept and those which are not”. Concept learning as a process presupposes the learner is active, can justify the characteristics which bound them together and the words to describe them, time is saved as the learner interacts directly with his environment, and t avoids unnecessary interaction, and can recognise and generalise. For example, a student who has a true concept of equation will be able to generalise to all forms of equations be it simple, simultaneous or quadratic.3 5 35 Bolror. N. Concept Formation: U.K.: Pergamon Press Ltd. , Pp. 64' - 139, 1977. UNIVERSITY OF IBADAN LIBRARY - 5 7 - When uiiicept if an entity has been formed it has to be tried out. At this stage, there should be minimum interaction between the learner and the instructor. The need for instructional material would be ssential so as to facilitate the definitive properties of the concept. Expounding on problems of conceptual learning (Eng]eman ) 36 defined concept as "a set of characteristics shared by all instances (examplers) in a particular set and only by these instances" (p.87)., Thus, vh 1 the relationship between teaching and concept analysis is made explicit, the teaching sequence O 7 can be evaluated more precisely. Markle defined defines gpnccp" os a class or category where all the members shaxo a particular ©©mbination of critical proper-ties not sharei by any other class. Gagne’ defined concepts as a capability that makes it possible for an individual to identify a stimulus as a mernbei of a class having some caaracaeristics in common, even though such still may other­ wise differ from each other markedly. There are concrete and abstract concepts. Concrete concepts identifies an object properly or object attribute(s) (Colour, shape e.g. rojna, square, blue, three etc.). A b s t r a c t concept 37. Markle, S. M. Problems of conceptual learning. Journal of Education Technology Vol.1. No. 1.. 11r19 1970. UNIVERSITY OF IBADAN LIBRARY -58- needs to identify the referents of the language used in concept concept definition, to discriminate and generalise. Thereby the subject formulates rules as in problem-solving. For example, the concept of fruit can be concrete, and is shared by all classes of fruit and nothing more. Here, ffo'Sfc referents to concept are noun-like particularly for tangible objects. However, concepts learned before, during and after school years vary widely in ways which make generalisations about them unsafe. Concepts which are abstract like "equation" in algebra can be distinct from other mathematical concepts. Hence "language" plays a key role in achieving concepts especially abstract ones and in using attained concepts to learn related principles and to solving problems. How then do we learn concepts 2.3.1 Criteria For Conceptual Learning A learner who has fully grasped a concept can give two relevant kinds of responses. He can generalise to instances O C (exemplars) and discriminate non-instances (Machner) . Genera 1isation, by definition, involves a new situation, one that the learner has never encountered before. Conceptual learning is therefore easily discriminated from rote learning. UNIVERSITY OF IBADAN LIBRARY - 5 9 - Concept learning should be taught with examples and non­ examples so that the learner can generalise and discriminate either overtly or convertly. Teaching concepts would require a strong focusing strategy on those examples and non-examples and the mode of generalisation. For mathematical concepts the use of instructional materials like electronic calculator could be useful to illustrate examples and non-examples in word-problems. This would allow the learner a greater exposure to the full range of referents. Lack of sufficient discrimi­ nation or undergeneralisation may lead to the concept not being learnt. HeT’e adequacy or otherwise of the instructional strategy would determine whether the concepts have been learnt or not. In designing an effective instructional situation in the school, the learning of concepts is of central concern since school learning is predominantly conceptual in nature. Mode: or concept presentation and acquisition that could be found in research literature a^e comparing "inductive versus deductive mode", "rule-example versus example-rule”, and "discovery versus guided discovery or deductive mode" o c (Clark) ,and the amount of time, a learner is exposed to an example in a critical information processing variable in concept acquisition (Horland and Weiss) 35 , Yudin and Kate In most studies time,has frequently being held UNIVERSITY OF IBADAN LIBRARY constant whilo exploring mace cf presentation. Koran and Freeman" ' 1 found that for sc.iool instruction when biological concepts are being taught a deductive approach is more efficient. However, as concepts become more complex or abstract, and can contribute to higher levels of learning or a wider range of objectives, inductive method should be useful strategies to explore (Gagne)3'7. 2.3.3 Concept Formation and Attainment: Concept Learning Working on the theory of abstraction Locke 33 and Hume37 89 40 stated that concept was formed through a process in which the person recognizes similarities or identical elements in a set of objects, the person thus abstracts these resemblances away fromthe other properties of the set of objects that are not rele­ vant to the concept e.g. concept of man - features of man. According to Turner ' 3 once a concept has been formed or attains the person will be able to do two distinct things: firstly he recognise its relevant attributes and secondly he will know how they are related to one another. There can be three forms of relationship between attributes: a conjunctive relationship when the concept requires all the r e l e v a n t 37. Gagne, R.M. Categories of human learning, N.Y. AcademicPPress, 1366. 38. Locke, 3.: Essay on the Human Understanding. Oxford: Clarendon Press, 1690. 33. Hum. , D.A. Treaties of Human Nature, Oxford: The Clarendon Press, 173S. 40. Turner, 3.: Psychology for the Classroom. London Metheven, 1977, p.48. UNIVERSITY OF IBADAN LIBRARY -61- attributes to be present. Secondly, there can be disjunctive relationship when either one or other attribute or both are present, the concept then exists. The third form of relationship is the relational one when it is the relationship between two attributes which defines the concept. Bruner, Goodnow .41 • . . . and Austin made important distinction in learning concepts, that is, between concept formation and concept attainment: Concept formation is an initial creative action which results in the formation of super-ordinate classes or abstract categories. However, concept attainment is more often of interest to teachers and this means the activity of finding examplars of a concept which is already in the mind; or attempting to reconstruct the concept that is already in someone else’s mind. Concept formation is more of the fundamental process while concept attainment is the more familiar, which teachers examine on tests. for attaining concepts, Bruner et. al. 41 'itemized the followings ft (a) identifying objects vb) reducing the necessity of constant learning, (c) reducing the complexity of the environment (d) providing direction for useful activity, and (e) .rdering and relating different types of events. 41 41. Bruner, J. Et. al.: A Study of Thinking: New York; Wiley, lITST! UNIVERSITY OF IBADAN LIBRARY - e2 Concepts are therefore, building blocks for understand)ing after correct perception has boon made, and using principles which in turn, arc critical in solving problems. Fig. 3: Sequential bases for learning various outcomes (Adapted from Bruner, 3.S. Creative Thinking) L a n g u a g e p l a y s a key role in achieving corcepts and in using attained concepts to learn related principles and to solving problems. From the sequential basis one can define concept as ordered information about the unaracteristics of one or more things - objects. UNIVERSITY OF IBADAN LIBRARY events or processes that enables any particular thing or class of things to be diPfcrentiated from also related to other things or classes. Concepts can therefore, be thought of as both mental constructs of individual and also identifiable public entities (Bruner) 41 Each individual attains mental construct or concepts according to his unique learning experiences and maturation pattern. Once attained concepts play a key role in his thinking about the word in a language, and in the intellectual growth. Concepts are the fundamental agents of intellectual w o r t<, 2.3.3 Associative Versus Hediationa1 Theories of Concept Learning The simplest theory is the associative one which sees concept lenrn ng as a matter of associating positive and negative instances of stimuli, to reward or punishment. For example, in developing the concept of a bird the subject, can associate pictures of birds with a positive responses whereas mediational theory says the subject uses covert cues with which he organizes his behaviour. UNIVERSITY OF IBADAN LIBRARY — ci 4 ~ STIMULUS MEOIATIONAL RESPONSE AND NEW STIMULI RESPONSE O (FIGURE) - R1 Number - 3 - S.1 + 1- YELLOW (COLOUR) - P? Colour - Yellow - S2 + 1- SQUARES (SHAPE 1 - R_ Shape - Square - S + 1- 3 Fig. 4: riediational Model (Adapted from Turner, J . in Psychology for t'.s Classroom. London: Netheuen 1977, p.45) Some theorists, especially those within the S-R. tradition, regard associative learning or conditioning as the major, if not the only form of learning . While some have brought in connectism 'O’ i.e. S — > 0 ---R:S for stimulus, R for response, and :0 is belipved as a set of concepts or mediators in learning others have suggested that there are types of learning, some more advanced than others, and that associative learning occurs at early stage in development but is superseded by structured or cognitive ‘ learning (BoIton)^ . Kendler and Kendler 3E maintained that two theories are necessary to explain concept learning. A single unit S *■ R theory to account for the behaviour of animals and children who cannot make use of symbolic mediation, a n d UNIVERSITY OF IBADAN LIBRARY nediational S — r. theory tor older children and adults. Higher-order coi.cept formation would depend on: 35 (i) organisation of the material to be learned (ii) the motivation for concept learning and (iii) creativity in formation and (iv) individual differences. 2.3.5 Mathematical Concept Formation Psychologists who have investigated mathanatical concept formation have identified the inadequacies of traditional 'stilulus-response” learning theory, and the need for a theory of structured learning. Biggs03 asserted the necessity of constructing theory which will account for the process by which the learner acquires meaning or structure, rather than a response o r action Draws upon Piagetian theory for this purpose. Skemp 35 , distinquishes between primary and secondary concepts: A primary concept is derived from other concepts;for instance, some mathematics 1 concepts, such as "three", are primary, since they are formed through inspection of collection of three objects while others are secondary for they consist of generalizations about the properties of individual members. Thus, 0 x 7 = 56 is understandable on UNIVERSITY OF IBADAN LIBRARY -66- the primary conceptual level, whilst 8 (x + y) = 8x + 8y is a secondary concept. Skemp believed that many of the difficulties experienced in mathematics learning stem from the transition from primary to secondary modes of representa tion and understanding, ana he proposed a schematic theory of learning to account for the way in which successful instruction should progress from structured primary to structured secondary knowledge. However, Dienes 35 argued for the existence of a number of stages in mathematical concept formation. In the first state, the person’s behaviour is playful and haphazard at the second stage it becomes more regular and purposeful, It is confined to oractice in handling situations in which the rule-structure of the subject matter is relevant and results in a more or less unconscious stamping-in of the rule. At the third stage, analytic thinking about the rule becomes possible, and it exists now as an object of thought. Thereby forming the basis for the conceptual attainment. Dienes' apparatus consists of the Multibase Arithmetic Blocks which is structured in geometrical progression and these blocks are used to teach place-value in number system How do we "each concepts to pupils today? UNIVERSITY OF IBADAN LIBRARY - 6 7 - At the present time, there are a number of different methods in use in the classroom for teaching mathematical concepts. There are the traditional methods with their emphasis upon repetition, the early use of symbolism, computational efficiency and extrinsic motivation. In contrast to these methods, is discovery learning in which the aim is to motivate the child to develop understanding on the basis of his own experience, and the emphasis is upon wide experience, the gradual introduction of numerir.al symbols only after concrete experience with numerical relationships, problem-solving rather than computation and intrinsic motivation. There are other specific methods, sjch as Dienes Blocks and Cuisenaire rods, for use in the develop­ ment of certain basic concepts. In Britain, the Nuffield Mathematics Project, following closely the ideas of Piaget, developed techniques of instruction and methods of assessment for a wide range of concepts, for example, tran­ sitivity, conservation etc. However, Biggs 35' hypothesized that the traditional methods of teaching should be mere successful with respect to tests of mechanical arithmeoic,. whereas structural and possibly motivational methods would surpass the traditional methods when conceptual understanding rather than rote learning, was demanded. In fact, he found very little UNIVERSITY OF IBADAN LIBRARY ~ li 6 - difference i.n the performance of children, whether mechanical or conceptual who had been trained by traditional abd uni-mood, structural methods, although there was tendency for the more intelligent children to benefit from- uni-model (i.e. Hienses' method), as against the less intelligent. 2 .3 . 0 Researches in Concept Learning On a project carried out by Harris and Harris 43 on concept attainment abilities they found that achievement in language arts and mathematics was related to three abilities - numerical, word fluency and memory. Piland and Lemka''3 ^' studied the effects of ability grouping or concept learning. They investigated the effect of conceptual training and transfer of ability grouping on intelligence, sex and temporal tests. Results indicated that (a) ability grouping has no significant effect on concept learning under any effect of the variables of the experiment and (b) high ability subjects (students) are better able to obtain mathematic concepts than medium ability or low ability subjects. The non-significant effect of ability grouping is seen as an important finding in the light of its present emphasis in our secondary schools.. Host of our secondary schools today "roup their students into Science, Arts or Commercial4 3 43 Harris and Harris. Concept Attainment Abilities Project. Journal for Research in Hatch Education (JMEE) 9(5). 1378, 334 - 336. UNIVERSITY OF IBADAN LIBRARY - 6 9 - classes based on the students performances in form three. This would raise issues whether such grouping has any significant effect on the results of schools in the West African School Certificates Examinations. The results . cited in this study found the effect as non-significant in a controlled experimental setting, but one wonders whether an effect that is not significant in a controlled setting will work in the school setting. Studies have also been carried out on the teaching of concepts in schools. According to CookZ/ students receiving both positive and negative instances (e molars) in teaching mathematics concepts did significantly otter on an algebra concepts tests than those given only positive instances . Studies in the organisation of content elements in instructions 1 materials has lone been an important issue in educational planning (Ausubel, Bruner, et. al) . From research findings, an emperically based set of instructional design strategies has been developed to organized content elements in concept teaching (Klausmeier, et. al.) 45 These design strategies include (a) the relationship4 5 44. Ausubel, D.P. et. al. Educational Psychology; A Cognitive View, Few York: Holt, Rinehart,"and Winston, 1966. 45. Klausmeier, H.3. et. al.: Analyses ofConcept Learning. New Yrok: Academic Press, T9uB. UNIVERSITY OF IBADAN LIBRARY - 7 0 - between examples (b) the relationship between examples and non-examplesj (c) the ordering of examples and instructional help, (d) developing a procedure for selecting an appropriate number of examples, and (e) the relationship between co-ordinate concepts, Carroll and Methner 45 regarded concept learning as 'the identification of concept attributes w torch can be’ .generalized to newly encountered examples and discriminate examples from non-examples. 2.3.6 Facilitating Concept Learning In the instructional process for concept learning, a definition of the concept should be presented in terms of its critical attributes between the examples and non-examples 46 zi 5 (Tennyspn ) . Johnson and Stratton's study demonstrated the effectiveness of definition in concept learning. The results of this study indicated that students who were given a definition performed significantly better on classification of new examples, definition of the concept, isentence completion, and selection of synonyms. Klansmeier and Feldman45 f o u n d t h a t a d e f i n i t i o n 46 46 Tennyson, R.D., et. al.: The Teaching of Concepts: A Review of Instructional Design Research Literature. Review of Education Research 50 (1) 55-57. 1960. UNIVERSITY OF IBADAN LIBRARY -71* provided about the same amount of learning facilitation as on e r a t i o n a l s e t o f e x a m p l e s an d n o n - e x a m p l e s . M a r k l e a nd 46 T i e m a n n ’ s s t u d y s h o w e d t h a t a c o n c e p t d e f i n i t i o n b e s t f a c i l i t a t e s c o n c e p t a t t a i n m e n t when s t a t e d i n t e r m s o f c r i t i c a l a t t r i b u t e s o f t h e c o n c e p t . i 46 F o r i n s t r u c t i o n a l p r o c e s s T e n n y s o n a n d P a r k p r o p o s e d a f o u r - s t e p p r o c e s s f o r c o n c e p t t e a c h i n g : ( 1 ) The t a x o n o m i c a l s t r u c t u r e o f t h e c o n t e n t s h o u l d be d e t e r m i n e d . The t h r e e l e v e l s o f c o n c e p t s t r u c t u r e - s u p e r - o r d i n a t e , c o - o r d i n a t e , an d s u b o r d i n a t e - s h o u l d be a n a l y s e d w i t h i d e n t i f i c a t i o n o f c r i t i c a l a nd v a r i a b l e a t t r i b u t e s ; ( 2 ) A d e f i n i t i o n o f t h e c o n c e p t s h o u l d be p r e p a r e d i n t e r m s o f t h e c r i t i c a l a t t r i b u t e s a nd a p o o l o f e x a m p l e s s h o u l d be p r e p a r e d on t h e b a s i s o f c r i t i c a l a nd v a r i a b l e a t t r i b u t e s ! ( 3 ) The e x a m p l e s s h o u l d be a r r a n g e d i n r a t i o n a l s e t s by a p p r o p r i a t e m a n i p u l a t i o n o f t h e a t t r i b u t e s . W i t h i n a r a t i o n a l s e t , c o n t a i n i n g o ne e x a m p l e f r o m e a c h c o - o r d i n a t e c o n c e p t , t h e e x a m p l e , s h o u l d h a v e s i m i l a r v a r i a b l e a t t r i b u t e s y ( 4 ) The p r e s e n t a t i o n o r d e r o f t h e r a t i o n a l s e t s s h o u l d be a r r a n g e d a c c o r d i n g t o t h e d i v e r g e n c y a nd d i f f i c u l t y l e v e l among e x a m p l e s o f t h e c o n c e p t , t h e p r e s e n t a t i o n o r d e r o f t h e e x a m p l e s w i t h i n t h e a t i o n a l s e t s s h o u l d be d e c i d e d a c c o r d i n g t o j p d n t e i n f o r m a t i o n a b o u t t h e l e a r n e r ' s k n o w l e d g e s t a t e ( p .65 ) . UNIVERSITY OF IBADAN LIBRARY - 7 2 - Some of the following ractors identified by Klansmeier, Ghatala and Frayer 46 could affect concept planning: (1 ) characteristics of the learner, (2 ) characteristics of the instructional situation ^nd (3) characteristics of the concept. Obviously,the learner's age and to some extent, ability will affect concept learning, but Klansmeier and his associates gave reasons for ability being more important. They quoted Wi. ci. ott’s 46 results that children who scored highly on mathematics tests scored more highly on a test of concept mastery. Klansmeier and f’leirke listed six functions of instruction in concept learning (1 ) to acquaint the subject with the structures material; (2 ) to acquaint the subject with the response desired; (3) to inform the subject of a strategy or method to apply for the solution of the task; (4) to provide substantive information, (5) to provide a set of relevant information examples) and (6) to change the level of motivation of the subject (pupils). I The extent which teachers use instruction which fulfil these purposes will determine, t h e s u c c e s s of their pupils in attaining the concepts, that is, the more UNIVERSITY OF IBADAN LIBRARY relevant dimensions there are, the more difficult it is to attain. Similarly, abstract concepts are more difficult to learn than concrete ones (head and Dick) 47 . It is necessary to consider these factors as well as objectives, instructional materials and methods of assessment in concept learning. 2*3»7 Attitude Towards Mathematics In the last few decades educators have become more and more concerned with the affective outcomes of educational programmes. Many teachers believe that a student's attitude towards a school subject will affect that student’s achievement in the subject (Michaels, Forsyth)^. Teachers generally are interested in pupil's attitudes towards the subject they are teaching? teachers of mathematics are parti­ cularly concerned about pupils feeling about their subject because mathematics has a reputation for being unpopular. Zacharias (in Time, 1975) contends that fear of mathematics is widespread among school children. From the above comments, it would be difficult to over­ emphasize the importance of attitudes in school learning. 47 47. Reed, H.D. and Dick, R.D.: The learning and generali­ zation of abstract and concrete concepts. Journal of Verbal Learning and Verbal Behaviour, >, 46(5 - 490, 1968 UNIVERSITY OF IBADAN LIBRARY 74 in ti 3-Firs'- place, it is so evident to the classroom teacher t h a t the students attitudes toward his subject-matter, toward cooperating with him as the teacher and his classmates, towards attending school, toward giving attention to the communication presented to him and toward the art of learning itself, are all of great importance in determining how readily the pupil would learn. 48 According to Gagne’ the school aims-to inculcate in the pupil some attitudes as a result of teaching-learning experiences. Attitudes of tolerance, honesty, good citizenship ere of tea mentioned as t-cols of L.du~~tion in the schools. Whatever the particular content of an attitude, it functions to affect "approaching" or "avoiding”. In so doing, an attitude i^-Fiuences a large set of specific behaviours of the individual. What therefore is attitude? Camp boll 48 defined r?ttitude as "consistency in response to social objects”. According to Allport:" "an attitude is a mental and neutral state of readiness, organized through experience, exerting a directive or dynamic influence upon the individual's response to all objects and situations with which it is related” (p.56). Since human beings do exhibit attitudes hence attitudes are complex states or predeposition ot human beings which affect their behaviour towards people, ~4£r Gagne’r R .M .1 Principles of Instructional Design 2nd Ed. New York: Holi Rinehart and Winston, Pp. 55-79? 1376. UNIVERSITY OF IBADAN LIBRARY 75 things and events (Gagne1 and Briggs) 48 , Many investigators have studied and emphasised in their writings, the concep- 4 9 tion of ~n attitude as a '-ystsm of beliefs (Fishbein) , or as a state arising from a conflict or disparity in beliefs (Festinger) 3 0 . These views serve to point out the cognitive aspects of attitudes. Other investigators deal with the affective components, the feelings given rise to or which accompany them as in liking and disliking. Attitudes relating to learning outcomes in the ’affec- C A tive domain’ are described by Kruthwohl, Bloom and Masia . C e r t a i n attitudes are related to action as a result of invigorated emotion. Hence, what action does the attitude support? An attitude influences a choice of action on the part of individual, According to Gagne’ and Briggs'4 0 attitude is an internal state which affect an individual choice of action toward some object, person or event. Such action has led to attitude crisis. Attitude crisis may exist in mathematics learning among teachers, pupils and in society at large. Burton 52 suggested that crisis of attitude among mathematics teachers is displayed in a number of ways the most common of which is confide«ce- failure. 49. Fishbein, M.A. Current studies in social Psy., N.Y.: Holt, Rinehart & Winston, 1965. 50 . Festinger, L.A. Theory of cognitive Dissonance: N.Y.: Harper & Low, 1957. 51 . Krawthwohl, D.R, et al. Taxonomy of Educ. Obj. Handbook II: Affective Doman, New York: McKay, 1964. 52. In Morris-̂ T"! Math Anxiety: Teaching to avoid it. ----T— k~ 7t:rc'> UNIVERSITY OF IBADAN LIBRARY - 7 6 - Insecurities which can be traced back to their own experience of mathematics in school. The crisis of attitude amonr children can almost be identified by performance - failure and dislike. The crisis of attitude in society is generally seen in the com­ plaints of, for example, employers, that school leavers are not numerate or students perform badly in external examination or children cannot "add" 52 2.3 Mathematics Learning and Attitudes Learning of mathematics cannot be divorced from the way the subject is caught in schools. Many psychologists have considered how children learn mathematics. One school of thought is that cf behaviourists. Their premise is, "Provide proper conditioning and you can ^et human beings to behave in any way you want". This view is represented by B.F. Skinner, Robert Pagne, et. al. ^ However, another school of thought is that of the developmental psychologists as represented by Jean Piaget. They both differ in their approaches on how mathematics should be taught. Those favouring learning by dis- covery or invention: like Piaget and Bruner 53 advocated maximum opportunity for physical exploration by the student. Hence, solutions for problems and generalizations should result from the student's own action on his environment, and fron his own mental operations. 53 53. Bruner, J. et. al.: The Process of Education, Cambridge, Massachusetts j Harvard University Press, 1962. UNIVERSITY OF IBADAN LIBRARY - 7 7 - Those preferring guided learning as Gagne’ and Skinner,48 emphasized the impjrtance of carefully sequenced instructional experiences (information processing) through maximum guidance by the teacher and/or instructional material. Basic ’associations" of facts are stressed. Here, association refers to the familiar stimulus-response of S--- -̂R machanism. Control the stimulus to obtain the desired response. Gagne’ emphasized task analysis-what do you want the learner to be able to do? The capability must be stated specifically and behaviourally» ' Piaget prefers "assimilation and accommodation" to controlled associations of stimuli to responses. It should be noted that the logical processes involved in mathematics must be based on the psychological structures available to the pupil. These structures would change as the pupil matures physiologically and neurologically, aod as the child has the necessary experiences in the physical world. These experiences must involve actions performed on objects and communication with other people such as the teacher and peer group. UNIVERSITY OF IBADAN LIBRARY - 7 8 - 2.3.3 Research in Attitudes Imnsell, Brooks and Retry 54 studied ability grouping in mathematica achievement and pulils’ attitudes toward mathematica and they found that self-concept appeared to decrease as placement within the ability groups decreased, with the high-ranked pupils in the high level class having the highest scores. In the same study by Brasseli, Brooks and Retry 54 the low-ranked students in the medium-level class appeared to have the lowest self-concepts of all. While anxiety and self-concept do not only inversely correlate to each other, they may be related to the learning context. finally, the group found that mathematics self-concept and mathematics anxiety 45 54. Brasseli, Anne, et. al: Ability Grouping, Mathematics Achievement and Pupils Attitude toward mathematics. Journal for research in mathematics education 11 (4~H 1 380, TT1-119. UNIVERSITY OF IBADAN LIBRARY 73 appear to :ia important correlates of mathematics achievement. Quinrl'1 ' st-ucied the casual relationship between mathematics achievement and. attitude in grades 3 tc 6 and found some significant correlations between attitude and achievement of elementary schoo] pupils 55 Schofield ‘ cn a study cf teacher’s effect on cognitive and affective of pupils’ outcomes in elementary school mathematics found that high achievement and high attitudes of teachers were each significantly related to high achievement in pupils, but related to least favourable pupil’s attitudes toward mathematics in igrade 6 (elementary school). It would appear that most research findings tend to correlate achievement to attitudes: That pupils with hijh achievement tend to have positive attitude toward the subject and vice-vcrra ltd addition, self- concept and anxiety have significant e-ffect on pupils’ mathematics achievement and attitude. This only suggests that teachers must attend tc seIf-cancepc enhancement and anxiety reduction in mathematics. The correlation between attitude and achievement varies not only with grade level* h5. debutield, H.L.: Teacher Effects on Cognitive and Affective Pupil Outcomes in Elementary School Journal of Education Psychology. 73: 462-471, 1961.' UNIVERSITY OF IBADAN LIBRARY - 6 0 - but also with the sex of the student and is generally somewhat higher for girls. Thus, girls’ mathematics marks are more predictable from their attitude than boys’ marks (B e h r ) ^ . 2.3.10 Instructional Design and Attitudes How c a n o n e d e s i g n i n s t r u c t i o n t o f a c i l i t a t e o r e l i c i t a t t i t u d e c h a n g e o r f o r m a t i o n ? C e r t a i n l y t h e met hod o f i n s t r u c t i o n t o be e m p l o y e d i n e s t a b l i s h i n g d e s i r e d a t t i t u d e s d i f f e r c o n s i d e r a b l e f r o m t h o s e a p p l i c a b l e t o t h e l e a r n i n g o f i n t e l l e c t u a l s k i l l s and v e r b a l i n f o r m a t i o n ( G a g n e ’ ) 46 While McGuire 57 identified the use of persuasive communication, Skinne-r 18 s u g g e s t e d the idea of arranging contingencies of reinforcements by some preferred or rewarding activity. Teachers of mathematics should use instructional strategies such as incorporating learning resources like calculators which could tend to reduce cor’putational problems and thus limit mathematics anxiety in pupils. The use of calculators has been found to be enj■ oyable and moti. vating to pupils in the* classroom (Suydam).22 Hence, this and other activities perceived to be enjoyable and motivating may be adopted to reduce pupils anxiety. 576 56. Behr, A . f\l. , Achievement, aptitude and attitude in Mathematics, Two-Year College Mathematics Journal 4, 72-74, 1970. 57. McGuire, M.3. Handbook of Social Psychology. 2nd Ed. Vol. 3, Reading Mass: Addison-Wesley, 1969. UNIVERSITY OF IBADAN LIBRARY 81 In addition pupils' attitude toward the teacher may be important in the formation of mathematics attitudes. Teachers should therefore, be aware of the fact that at this critical period of attitude formation in high school the teachers’ attitudes toward mathematics and their pupils are important and may be the determinants of pupils’ attitudes toward mathematics. I t may not be a new phenomenon t h a t a l a r g e segment o f t h e p a p u l a t i o n f e a r s m a t h e m a t i c s tGaugh, Kogelman and Warren) What can be d o n e? The f o l i o w i i 0 c o n s t i o c t i v e t e c h n i q u e s and s t r a t e g i e s can be helpful: (1 ) creating a positive, supportive classroom atmosphere - for example, a teacher who takes time to listen intently while a pupij. asks a question, and responds with a willingness- to explain, will create an atmosphere in which students feel at ease asking questions. The math - anxious are especially sensitive to criticism and pupils with a low self-concept are reluctant to take risks (Morris) Hence they do not ask questions i (2) There is need to stress t understanding of the thought process. The process and product aspect of mathematics problem-solving are important. Too much emphasis should not be placed on one to the detriment of the other, Pupils should therefore be encouraged to think; (3) The teacher should dispel the * *5B. Morris, J.: Mach Anxiety: Teaching to avoid it. Mathematics Teacher, NCTM, 74(6) 413 - 417, 1381. UNIVERSITY OF IBADAN LIBRARY ’mathphobia' in his pupils. If the teacher is not afraid to say he ’does nut know' when it is the honest answer the pupils would have confidence in the ability of the teacher. The impression that the teacher knows all the answers should be dispelled because it creates defeatism on the part of che pupils especially the weaker ones. (4) Provide new positive mathematics experiences. (5) Use a p p r o p r i a t e i n s t r u c t i o n a l materials to teach content. (6) Make sure each c o n c e p t i s understood before continuing to new one. ( 7 ) Reduce t e n s i o n and pressure in mathematics classes. (6) G i v e p o s i t i v e feedback on written tests, (9) Teachers should not o n l y be sensitive but determined. Apart from above suggestions, anxiety which is a learned response to a negative experience according to Morris J should be prevented. Thus, the above techniques and strategies can be tried by teachers’ of mathematics asc as to reduce mathematics failure in our schools. It would be appropriate to subject each of the above suggested strategies or techniques 'to empirical studies so as to determine the their.efficacy or otherwise on concept learning and attitudes in mathematics. , 2.4 Caleulater and Other Instructional Devices Researches on instructional techniques and materials 39. Morris, 3.: Math. Anxiety: Teaching to avoid it. Mathematics Teacher, NCTM, 74 (6) 413-417, 1981. UNIVERSITY OF IBADAN LIBRARY - 6 3 such as programmed text/instruction, individualised instruction, computer-based instruction etc. have been carried out and their effects on the teaching-learning process do produce interesting results. Tne results of researches on programmed instruction on teaching and learning of mathematics, (for example, Abimbade 60 in a study on relative effectiveness o f programmed i n s t r u c t i o n to t r a d i t i o n a l method o f t e a c h i n g s e c o n d a r y s c h o o l m a t h e m a t i c s ) showed t h a t t he programmed i n s t r u c t i o n gr oup did a c h i e v e s i g n i f i c a n t l y b e t t e r results than the 61 t r a d i t i o n a l g r o u p . K a l e j a i y e c a r r i e d o u t a s t u d y on t h e i n d i v i d u a l d i f f e r e n c e t o programmed m a t e r i a l i n t h e new mathematics and found that programmed text was effective in changing the attitudes of pupils towards the new mathematics. Balogun et. a l . ^ Adewakun^ and Okunrotifa^ have all reported positive results in favour of programmed instruction, ■Jurgemeyer ~ also got positive results with programmed instruction vis-a-vis the advent of new directions in me'dia technology such as Video discs, interactive Video and micro-computers. 0*612 » 60. Abimbade, A.: The relative effectiveness of Programmed Instruction to the traditional teaching of secondary school mathematics. Unpublished PI.Ed. Dissertation, University of Ibadan, 1963. 61. Kalejaiye, A.O.: Individual differences to programmed instruction in t.ie new mathematics, West African journal of Education, 15 (3) Oct. 1971. 62. Jurprmeyer, F.H.: Programmed Instruction: Lessons it can teach us. Educational Technology, 14-49, May, 1982. UN VERSITY OF IBADAN LIBRARY *" G - Crat-ford 63 reported more favourable attitudes towards mathematics as a result of computer-assisted instruction with seventh graders. But working with the same group, Johnson 54 found no differential increases to positive attitudes toward mathematics. It would appear that little studies are available comparing the use of calculators and other instruc­ tional devices. However, interesting results relating to the use of computer's in mathematics instruction -would be discussed so as to identify those areas applicable to calculator-use. 2.4. 1 Computer Based Instruction Computer-based instruction is used here purposely to encompass a broader spectrum of computer applications: computer-assisted instruction (CAI), and computer-managed instruction (CHI). Computer-assisted instruction implies the concept of tutorial instruction, drill and practice, presents instructional material to the learner, a6eepts and judges responses from the learner, provides feedback and alters the flow of subsequent instructional material on the basis of the learner’s responses whereas the computer- managed instruction relies principally on the record-keeping and summarising power of the computer. However, one’s 463 63. Crawford, A.N. and Johnson, F.E. see Aikan, L.R., In "Update on Attitudes and other Affective variables in learning mathematics” Review of Educational Research 46(2), 293 - 311, 1976. — 64. Johnson, R.E.: The effect of activity oriented leason on the achievement and attitudes of 7th grade students in mathematics. Dissertation Abstracts International 1971, 32, 5C5A. UNIVERSITY OF IBADAN LIBRARY - J - emphasis * in tlu? rtviov- shal 1 be the computer-assisted instruction which is more relevant to the objective of the present studv. 2.4.2 Evolution of Get■■puter-Assisted-Instruction (CAP Followin'-' dwindling interest in prof rammed learning and with the emergent of new technology, educators have askew: how far has ecucation/instruction been individualised with the new technolory like computer? Burns and Bozeman 65 proponents of .CAI asserted that computer’s support for the instructional process offered the promise of greater student achievement, more efficient use of human and material resources, improved attitudes towards the learning process, and enh?ncerent of quality education in general. fer-antf rh. first users of C.A.I, were • embers of the Commuter Industry vie, .in the late EO’s user! computer-based instructor to train their own personnel (Suppes and Macken)u° , Educator’s interest focussed on programmed instruction as a means toward individualised instruction. Educationally, C.A.I. was an almost natural combination of emerging conputer technology and the programmed instruction movement *»• 7 (Scheen and Hunt)"'. Among tho early C.A.I, 6* C5. Burns, P.K. and Bezeman, U.C.: Computer-Assisted Instruction and ; ,aths:iatics Achievement: Is there a Relations!*)! Educational Techno logy, 1301 6 6. Suppes, et. } I. Institution for Mathematical Studies in the Social Sciences, California, Standford University, USA, 1307. 07. Schoen and Hunt: In Burns and Bozeman. Educational Technology, 1371, UNIVERSITY OF IBADAN LIBRARY models to emerge was the Standford Project*. It began in 196j and its original aim centred on the achievement of a small tutorial system intended to provide instruction in elementary mathematics and language arts. It went through different phases such as Stan iford Drill-and Practice System. There were other projects like The Individual Communication System (INDICOM) launched in 1S67 in the Waterford, Michigan School Districtj then PLATO (Programmed Logic for Automatic Teaching Operations) System which originated in the co­ ordinated Science Laboratory at the University of Illinois, U.S.A. The PLATO Project has also gone through many phases. For example PLATO IV system (1981) which continues to be operational, supports several hundred terminals at dis­ persed locations. Each terminal site is provided access to a central lesson library. The powerful and relatively facile author Language of PLATO called TUTOR accommodates simul­ taneous system time sharing by students and teacher as lesson materials. The projects have, at the elementary and secon­ dary school levels been concentrated in the areas of mathe­ matics and language arts, There are many other systems all over the world but it appears that .the Nigerian educational systems has not been able to catch up with these new tech­ nological approaches to education. This may probably be due to * Under Patrick Suppes at the Institute for Mathematical Studies in the Social Sciences at Stan 'ford Univer- ______ sity, U .S .A . UNIVERSITY OF IBADAN LIBRARY 87 the cost of computer hardware v/hich is generally very expensive. 2.4,3 Pedagogical effectiveness of CAT Published studies comparing the effectiveness of CAI to traditional instruction report conflicting and incon­ clusive results. Most studies, however, generally conclude that an instructional programme supplemented with CAI is at least as effective or more than a programme utilizing only traditional instructional methods (Magidson) 6 8 Research in the area of CAI effectiveness has typically investigated one or more of five criterion variables? student achievement, student attitudes toward. CAI and towards subject-matter, time-savings relative to unit completion and/or mastery learning, learning retention and cost factors. Results of a review of research lite­ rature on CAI effectiveness on the above criteria as 5 g compiled by Edwards et al are as follows: (i) All studies reviewed have shown normal instruc­ tion supplemented by CAI to be more effective than the normal instruction alone? (ii) then CAI was substituted, in whole or in part, for traditional instruction, 45 percent of the studies 63. Magidson, E.M. Issue overview: Trends in CAI Educational Technology 18(4), 1978, 5-8. 69, Edwards, J. et al. How effective is CAI? Review io . 5. 1975. UNIVERSITY OF IBADAN LIBRARY 08 demonstrated grea t2 r achievement by CAI students, while 40 percent found little or no difference) 15 percent showee mixed results. (iii) Based on available results, it cannot be concludeo that any given CAI mode is mire effective relative to student achievement, than other modes. (iv) CAI has been shown to be equally effective relative to student achievement, when compared with other non-traditional instructional methods. (v) All studies showed that it took less time for students to learn through CAI than through other methods. (vi) There is more evidence that learning retention levels of CAI students may not be as high as those of traditionally taught students. Host of the researches reviewed in this study came from United States., of America which, appear to be the only country that has the largest published work in computer-instruction. There are nc available records on computer inst-notion research in this country except the work of Ohuoha on comput'r’s utilization in Nigerian Universities with guidelines for improved utilization (Columbia University, Teacher College, 1981). With the advent of technological approaches to education and instruction in Europe, U.S.S.R. UNIVERSITY OF IBADAN LIBRARY 5S and America, there was renewed interest in student reactions to instructional technique. Several studies have reported students’ attitudes toward computer-assisted Instruction (CAI) (Brown, Filip and Murphy)^ 2.4 .A Attitudes and CAI Attitudes are probably dependent on the kind of experience individuals have with instructional device, Resenberg, Rezuikoff, Stroebei and Ericson70 1 72* reported that nurses atti­ tudes toward computers became favourable after they had actually worked with computer. Wodtke77 found that those students who performed well when instructed by a com­ puter had more favourable attitudes toward it than those students who performed poorly. He concluded that favourable or unfavourable attituoes of students toward a teaching method could be the result of their experience with the particular method of instruction. Mathis 73 found that College students generally have posi­ tive attitudes towards computers when j»s also found that tho CAI has a major arJv’sntage over other teaching machines or 70. Brown (1967, Filip and Murphy (19671. Communications, Mass without meaning. Educational Technology, April 15, 1967, 4-5, 71. Rosenberg, M.3. et al. Developing effective Instruc­ tional manuals and computers. Educational Technology, Ban. 1967. 72. Wodtl.e, R.C. Computers and Learning strategies. EducationalTechnology, Dec. 1965, 1-5. 73. Math is j A"! College students attitudes toward CAI, Journal of Educational Psychology, 61(1), 1970. UNIVERSITY OF IBADAN LIBRARY 9 0 programmed texts in that students can branch immediately to 66 easier materials if they begin to make many errors (Supps?) Most of the studies showed favourable attitude toward CAI, but, would the use of calculator generate such favourable attitudes? It would have boon possible to infer the result of such investigation but this may not be so until empirical study integrating the use of calculator into instruction is carried out. However, from studies so far carried out there are evidences on the positive note for CAI. For Hansen et'al 3 in "What Teachers think - Every high school graduate should know about Computers" found that teachers supported ~he idea that students should have some minimal understanding aDout computers, but the extent of coverage o* computer topics was minimal Kleinian, G. and otheiT / working on microcomputers and hyperactive children found that children did almost twice as many exercises with the computer as they did with paper and pencil, although no differences were found in proportion to correctness or time 77 at a^es 6 - 1 4 years. De-tblaaaio' studied attitudes toward computers in High School mathematics courses and found 76 Hansen, T.P. et al. What Teachers Think Every High School Graduate should know about Computers. School Science and Mathematics 81: 457-472; October, 1961. 76 KleimaT] G"! et al. Micro-computers and hyperactive children in Creative Computing 7: 93-94; March 1901. 77 De-Blassio, 3.K. and Bell, F.H.: Attitudes Toward Computers in High School Mathematics Courses. International Journal of Mathematics Education in Science and Technology. T2~, 17-56, Feb . 1 §01 . UNIVERSITY OF IBADAN LIBRARY - 9 1 - positive correlations between students' attitudes toward using a computer and attitudes toward mathematics and instructional setting, plus achievement variables for grades 11 and 12. These results are quite significant because it relates effects of compute on mathematics achievement and attitudes which has a good bearing with the present study. However Casner 7 8 in a study of attitudes toward mathematica of eighth grade students receiving computer-assisted Instruction and students receiving conventional classroom instruction found no significant differences in attitudes toward mathematics between girls using or not using CAI, but boys using CAI showed less negative attitudes towards mathematics at secondary school - 79 Ray studies the effects of computer' assisted test construction on achievement in first year algebra and found that students using the computer assisted test with repeat- able testing achieved high score than students taking a traditional course at secondary school level. Not only is the computer used for testing but Sears 80 *c o m p -a r e d 78. Casner, 3.L.: S Study of AtMtudes Toward Mathematics of 8th Grade Students receiving CAI and Students receiving conventional classroom instruction Unpublished Ph.D. Thesis,* University of Kansas 1977. 79 Ray, K.L. The effects of Computer-Assisted Test Construction on achievement in first year algebra. Unpublished Ph.D. Thesis, University of Southern Calfornia, 1977. 80. Sears, L.D. A problem of the Effects of Teaching a Course in aleebra II and Trigonometry via Traditional Method and Other Methods. Unpublished Ph.D. I'nesis, University of Houston, Texas, 1977. UNIVERSITY OF IBADAN LIBRARY 92 seme teaching methods in solving algebraic and trigomomefcrical problems and found tt at there were no significant differences in achievement between the methods including computer- assisted instruction at the Secondary Sc'ool. However, 81 Vincent studied the computer’s c a p a b i l i t i e s a n d effects of supplementary computer assisted instruction on the mathematics achievement and attitude towards ■ , mathematics of high school students.: He found'that students using CAI drill and practice programme had better achievement and attitudes than those not using CAI at grades 9 - 12. Essentially, most of the studies reviewed have demonstrated some relationship between . ■’mputer and mathematics. It would appear f r m research literature that mathematics instruction can be facilitated with the uue of computer. Before any conclusion can be made it would be necessary to establish the basis for computing in mathematics instruction. The process for mathematical computation shall be discussed so as to elucidate or> the place of calculators and computer in mathematical computations. 81. Vincent, A.T.: The effects of supplementary CAI on the mathematics achievement and Attitude in EHR High School. Unpublished Ph.D. Thesis. University of Cincinnati, 1977. UNIVERSITY OF IBADAN LIBRARY 9 3 2.4.5 Computers and mathematics Electronic caiculatoi's like computers have speed, internal memory but requires human direction at each step in a computational routine. The electronic calculators have limited facilities in terms of stored program. The stored progr-a,... characteristically marks the difference between computers and electronic calculators. Computer’s stored program usually includes logical tests to determine which of many possible program steps should be taken at important junctures in the program. Thus, the stored program is net just a sequence of steps to follow in computation of computer programming but typically includes all possible paths that computation might take within the scope of the problem, being programmed. Computers and other calculating machines owed their developments to the works of nineteenth century English Q2 mathematician George Boole and his "Algebra of logic" popularly called Boolean Algebra which represents logic in mathematical symbols and provides rules for* calculating the truth or faisity of statements. Other works were those of Herman Hollerith on machine purchased-card while Howard Aiksr in 1937 designed a machine that could automatically perform a sequence of arithmetic operations. However, the 82. Boole, George: The Laws of thought, 1854. UNIVERSITY OF IBADAN LIBRARY 94 remarkable work by Charles Babbage in 1812^ who devised a machine nailed a "differet f’ial machine” which could auto­ matically perform simple computations needed for trigono­ metric and logarithmic tsb?,es. It was his invention of analytic engine which led to todays switching networks and internal storage for data processing. Electronic calculator like computer is an important technological advance because it extends and expands the capabilities of man. Unaided, man is rather "puny". Variety of• oompufcers/calculators have reduced or eliminated druggery connected with extended computations. Calculator or computer cannot perform any operations which cannot also be performed by human being but the calculator or computer can operate at very high speed, store and retrieve data at high spaed However, man can reason ^euristically and he is best fujiteri to think, reason and discover? whereas a computer is best adapted to calculate, manipulate and compare at very supersonic rosed. To every mathematical problem there is, a solution- process (See Fig. 5). Once a problem has been identified in the reai-.mrld, the objectives of a solution-process are defined. Fc v solution, some abstraction based on human interpretations would produce iriformation/data that are processable b̂ numerical methods to provide solution. The information may be generated in variocs ways such as mathe­ matical •: jrmula or numerical data which would be defined, processed and solved. UNIVERSITY OF IBADAN LIBRARY 95 The numerical method are applied in the. comp-utational phase, which is a decisive portion of the total problem solution. This processing of information with the feedback mechanism Source: Modified from Paul, G. A. Introduction to Scien­ tific Computing, New York; MerecHTR L'orp., 19/1, pg. 9 UNIVERSITY OF IBADAN LIBRARY 96 2 C4.6 Research • in ccmputer-rrathematics Many researches have been reported in computer and mathematics but those ones dealing with concept learning have to he identified. Most of the studies reviwed recorded significant differences between groups using computer for computatotion and those not using. However, for concept learning there are conflicting reports as to any advantage i. n the use of computers. Wright B3 investigated selected decision-making, processes for aspects of a computer- assisted and mastery learning model in basic mathematics and found no significant differences between four treatments varying the type of drill and practice. Similarly Cheshire 84 studied thn eFfect of learning computer programming skills on developing cognitive abilities and found no significant difference in problem-solving scores between computer programming and algebra classes classes at grade nine. However, Cranford 85 in a study of the effects of computer-r.-sisted instruction on achievement in mathematics S3. Wright, E.B. Investigation of Selected Decision-Making Processes for aspects of a computer-assisted and Mastery Learning model in Basic Mathematics. Ph.D. Thesis, The Pennsylvania State University, 1977. 84. Cheshire, F.O. The effect of learning Computer Programming Skills on developing cognitive abili­ ties. Unpublished Ph.D. Thesis, Arizona State University, 1981. 85. Cranford, H.R. A study of the effects of CAI in Mathe­ matics on the achievemant and attitude of pupils Grades 5 and 6 in a rural setting. Unpublished Ph.D. Thesis, University of Southern Mississippi, 1976 . UNIVERSITY OF IBADAN LIBRARY 97 and attitude of pupils in grades five and six in a rural setting found that the groups which used a computer drill-and-practice program achieved at a faster rate on computation and applications tests. Little difference was, however, found in understanding. Some of these studies have ehowr» J no significant differences in learning of concept by computer but some studies where computers have been integrated into the instructions gave positive results. Deloatch d i d a c o m p a r a t i v e study on the use of computer programming activities in an introductory college mathematics course for disadvantaged students and found the computer-augmented instruction to have significant positive .3. feet on the mathematical attitudes of disadvantaged students, but not upon their achievement at College level. Lamb 87 in qra? study on ?uhe co'-ordination of graph theory and computer science at the secondary school reported that students with access to computer terminals scored significantly higher in achievement of graph theory content than students without such access. BE) Oeloatch, S.3.: A comparative study of use of Programming abilities in an introductory college mathematics for disadvantage students. Unpublished Ph.D Thesis, Indiana University, 1977. 92 Lamb, R.L.: A study on the coordination of Graph Theory and Computer Science at the Secondary School. Unpublished Ph.D Thesis, Georgia State University, School of Education, 1976. UNIVERSITY OF IBADAN LIBRARY 98 From studies carried out so far one can gather that the computer could be programmed to serve as a test gene­ rator anc administrator. Computers have been used in educational settings and for instructional processes, individualising instruction, testing or drilling for compe­ tency in basic facts. Computers can be used as teaching- aids that help to achieve the objectives for mathematics learning. When access to computer is available students will be able to use the computer for programming the solu­ tions to prrblemsi for simulating situations in order to V test hypotheses, for gamingi as a study of probability and statistics? as well as for testing, practising, drilling and tutoring. 2.4.5 Con cl us 1 on Most of these researches reviewed have been carried out abroad by different researchers functioning in different environments. Their findings may be affected by environments, to apply their conclusions to the Nigerian situations directly may be inappropriate because the Nigerian environment differs from those of pupils in Europe and United States of America who are more exposed to stimulating and sophisticated environments. It would be appreciated that the input of the environment on the individual could play a significant role on the psychological thinking, perception, reasoning and UNIVERSITY OF IBADAN LIBRARY - 9 9 - learning of the individual. For pupils in an enriching and stimulating environment, they would be motivated and are likely going to learn better and achieve better in schools than pupils not exposed to such environment. It only means that pupils in under-developed countries of the world will of necessity be at a disadvantage. The advanced and industralised countries have always provided a wider range of experiences. This limitation portends itself in the area of technological based-instruction for example the use of calculators or computers in instruction. Besides,, only- -very few studies on concept ••learning in mathematics have been carried out here in Nigeria. Falokun 96 , Oni 09 and Ogunyemi and Beltie qn' have all investigated concept formation in mathematics using different experimental variables. However, none of them has tried to integrate the use of electronic calculator into the instructional process to find out the effect of such a device on the concept learning and attitudes to mathematics in the secondary school. 88. Falokun, C.O.: Concept formation in algebraic equations and problem-solving among form V students in Qyo State of Nigeria. Unpublished M.Ph. Thesis, University of Ibadan, 1983. 89. Oni, E.O.: Conceptual difficulties with ionic equations as function of intellectual development among secondary school students. Unpublished M.A. Thesis, University of Ife, Nigeria, 1982. 90. Ogunyemi,F. and Beltie,J.: An investigation of cognitive preferences in mathematics among high and low achievers in the Nigerian Secondary Schools. African Journal of Educational Research, Vol. 1 ppV 97-1'trS', 'T974'.------------------------- UNIVERSITY OF IBADAN LIBRARY - -ioo - CHAPTER THREE RESEARCH DESIGN AND PILOT STUDY 3 * 1 IntraduetIon The research is basically experimental using a 3 x 3 factorial design. The independent, variables or main effects are mental ability levels and different modes of instructional presentations, while the dependent variables are achievement and attitude measures. MENTAL ABILITY LEVELS a c h i e v e m e n t MEASURES HIGH MENTAL AVERAGE A 01 LITY MENTAL LOW MENTAL ATTITUDE C1 ABI LI TY ABILITY MEASURES C2 C3 USE OF CALCULATOR F1 IN INSTRUCTION AND A B C cn TESTS (UNRESTRICTEt3 (1 2) (12) (1 2) o4 GROUPS)cc CD * USE OF CALCULATOR H- E2 IN TESTS ONLY D E F 2 UJ (RESTRICTED CROUPS (1 2) (12) (1 2) ts—c < NON-USE OF CALCU- LoUr E3 LATOR IN INSTRUC­ G H I i- TION AND TESTS (1 2) (1 2) (12) Fig. G: Paradigm of 3 x 3 factorial design of pilot study. Number of cells, K = 9, Number of subjects in each cell, n = 12. Total Number of subjects, N ® 108 UNIVERSITY OF IBADAN LIBRARY 10 1 - The design is comparable three-group, before-after design proposed by Solomon 91 . If provided a way to avaid possible interactive effects due to the pretest or pre-attitude and also allowed for both to be used as covariates in the data analysis. The main hypotheses were tested on the effects of calculator use and instruction on concept learning and attitudes towards mathematics and calculator. The experimental and control groups were selected for treatments at random from population of form five pupils in the secondary school. The structure of this type of research design, according to Kerlinger 92 is where two or more independent variables are juxtaposed in order to study their independent and interactive effects on dependent variable. In this study there were two indeperm ant variables - mental ability levels and modes of presentation and two dependent variables - achievement and attitude measures. The assignment of the subject of the study to treatment groups was randomly based on their mental ability levels. The rental ability levels of the subject were obtained from the results of their Mental Language/Verbal (ML) and Mental numerical. (MQ) tests. 91. Solomon, R. in Kerlinger, F.N. Foundation *or Behavioural Researc h, 2nd Ed., New York: Ho It Rinehart and Winston Inc., 1973', 339 - 37 5. 92. Kerlinger, F.N. Foundation of Behavioural Research 2nd Ed., New York! Holt, Rinehart and Winston x n c. , 197.'', UNIVERSITY OF IBADAN LIBRARY 1 0 2 The tests were developed and validated by Australian Council for Educational Research (A.C.E.R. ) and they had been used by different researchers in the Faculty of Education, University of Ibadan, Nigeria. However, Campbell and Stanley S3 have noted the possi­ bility that some errors might result from employing the usual statistics appropriate for the random assignment of individual pupils to treatment groups, such as the ANQVA (Analysis of Variance) for intact classes assigned to treatment groups. In this study therefore, the use of the pre-test and pre-attitude mean scopes were used as covariates to serve for adjusting the initial differences within and between groups. I f T^, Tq and represent the pretests, and the treatments and f.4 and T~b the post-test, the design paradigm can be represented in this format: P re-trea tmen t T reatment Post -treatment T i X! T 2 T 3 x 2 T 4 T 5 T 6 93. Campbell, 0. and Stanley, 3. Experimental and Quasi Experimental Designs for Research, Stokiej Illinois: Rand MciTally, 1 963 . UNIVERSITY OF IBADAN LIBRARY 1 0 3 Both the pre-andrpos t treatments incorporated the attitude end achievement measures. Levels of treatments varied depending on the groups. One of the groups received the full treatment: calculator-use in instruction and tests, other experimental group received treatment of calculator- use in tests only and the control group did not receive any treatment on calculator, that is, non-calculator-use. The aim of this research was to find out how instruc­ tional product could facilitate teaching-learning processes, hence the design is an empirical study on the improvement of instruction. The experimental-control group design using equated experimental group subjects and control group subjects tt rough randomization, equal number of subjects In each cell provided an effective comparison in this factorial design. According to Kerlinger 32 research design has two basic purposes: (!) to provide answers to research questions and (2) to control variance. UNIVERSITY OF IBADAN LIBRARY 104 What the research design does is therefore to help the investigator obtain answers to questions of research and also to control the experimental, extraneous, and error variances of the particular research problem under study. The focus of this study arose as a result of educators’ concern on : (a) possible change in pupil’s attitude toward calculator and school mathematics# (b) possible interference with pupils growth in knowledge of basic mathematical facts and paper-pencil computationsj Cc) possible changes in children’s scores on standardized achievement tests in mathematics. (d) pc'.nntial development of additional mathematics uonjwpts related to calculator; (e) possible change in computational power of pupils when using calculators and (f) facilitation of mathematics conceptual learning and problem-solving skills of pupils through the use of calculator. UNIVERSITY OF IBADAN LIBRARY 105 3.2 Population and sampling procodurs The form five pupils, in the secondary school in-Oyo State, Nigeria at the time of this study constituted the population, and subjects used were from the sampled popu­ lation. The subjects wora all enrolled in the sampled schools at the time of study. The choice of this categor of students was considered appropriate because they alrea had the requisite knowledge of basic mathematical operations of addition, subtraction, multiplication and division. Their average age was within the piagetian operational rang;: whe r- symbolic a bo traction :> possible. They were already familiar with mathematics as a school subject, anc hence, they were sufficiently predisposed to the learning of suitably prepared structural materials in equations. The sampling technique adopted far this investigation was the random sampling approach. Kerlinger 92 defined this as the method of drawing a proportion (sample) of a population or universe so that all passible samples of fixed size have same probability of being selected . This approach is regarded by statisticians to be the most practical and free of bias. For instance, Kerlinger aluo observed that a sample drawn at random is unbiased in the sense that no member of the population has any more chance UNIVERSITY OF IBADAN LIBRARY 106 PC of being selected than any other member" . However, because of the nature of this research design it was not practicable to randomly select schools throughout the state. Therefore, n mulei -stage random sampling selection of the secondary schools in Ibadan city for this study was carried out. The selection of subjects into treatment groups were also carried out through random sampling technique. S6. P i s hier, R. The design of Experiments, 5th Ed. , Jew York: hafner, 1 951 , pg. IT. UNIVERSITY OF IBADAN LIBRARY 107 3.2.1 Selection, of Schools for the Study As at the time of this study there were 750 secon­ dary schools in Oyo State and 95 secondary schools in the Ibadan Municipality. Different schools were selected for: (i) validation of research instruments. (ii) pilot study. (iii) main study. The choice of schools in Ibadan city for this study was because of the accessibility of the researcher to the subjects used in this experimental study. (i) For the validation of research instruments, United Secondary School, Ijokodo was chosen. The school was established in 1 980, and it is made up of boys and girls. The school first presented pupils for West African Examination (WAEC) May/June 1 955 General Certificate of Education, Ordinary Level (G.C.E., 0/L) Examinations. The school has both Junior Secondary School and Senior Secondary School (JSS/SSS) in line with the Mew Matiinal Policy UNIVERSITY OF IBADAN LIBRARY - i oa - on Education 6-3-3-4 system. As at the time of this study, thr school had a population of about i500 pupils. The sohool had five arms of form five and the arms were divided into Arts and Science classes. The research instruments for validation were mathematics achievement test and attitudes towards mathe­ matics and calculator use in mathematics questionnaire. The mathematics pre-test had earlier jean validated in previous study^^. (ii) The multi-stage random technique was used for the selection of the school for the pilot study. Gut of the S5 secondary schools in the Ibadan Municipality only 33 of the schools we.w mixed [boys and girls). Only sixteen of the ■'.hoc) s were established more than ten years ago. Most of the 33 schools were esta­ blished in 1 380, and they first presented pupil for West African Examination Council/General Certificate of Education ordinary level exarni,*r nation in 196b. Host of these schools have relatively large number of pupils averaging about 1000 or more pupils in each school. UNIVERSITY OF IBADAN LIBRARY 1 0 9 Those schools established more than ten years ago have larger number of pupils average about 2000 pupils per school and more equipped than 1960 schools. nil the schools have OSS/ SSS. Five of the schools were randomly selected out of the 16 schools. On the basis of their West African Examination Council (WAEC) results in General Certificate of Education, ordinary level (G.C.E., 0/L) mathematics for the last five years (198G-1S84) five of the schools were comparable, (see Table 3). One of the schools Ahmadiyya Grammar School, cleyele, Ibadan was selected by ballot for the pilot study. t ii1J The procedure for the selection of schools for the main study would be discussed in » chapter four of this report. however, the pilot and main study schools were relatively apart. This was to reduce any possible experimental contamination between the subjects in the study. UNIVERSITY OF IBADAN LIBRARY 110 TABLE 3 WAEC/GCE rssu 11s_ of eampled schools — mathematics PERCENTAGE PASSES FDR THE YEAR SCHOOLS % AVE. PASSES 1960 1981 1902 1983 1984 ADEKILE* GOODWILL GRAN. SCH., 37.8 46.3 65.6 53 47.3 50 APE RIN, IBADAN. HOLY TRINITY GRAN. SCH... OLD 43 24.6 45. j 45.5 85 48 IFE ROAD, IBADAN IBADAN CITY A C A DE ivi V , E LE T A , 62 55.6 42.4 41 50 50.2 IBADAN. AH N A D I Y Y A GRAN. SCH . , E LE YE LE , 45 2u o5 42 32 45 IBADAN . ! » IS L A M I C HIGH SCH,., B A S O R U N , 59 52 63 37 31 48 IBADAN. UNIVERSITY OF IBADAN LIBRARY 111 3.2.2 Selection of Subjects The subjects used for the study were mainly secondary school pupils. The pupils were considered as represen­ tative of the country’s secondary school population because they came from different ethnic, cultural, reli­ gious and socio-economic background. The assumption was that the pupils in this group were not different in any way from pupils in other comparable schools in Nigeria. The study was conducted with the form five pupils who were in their first term of their final year in the school. There were 216 pupils (boys and girls) in form five of the pilot study school. The whole of the form five pupils of the school took the A.C.EvR. mental ability tests (ML and m ). Out of the 216 pupils who took the tests those pupils who scored between 32 and 51 were within the high mental ability level and 36 of them were randomly selected into the high mental ability (HMA) groups. Those who scored between 27 and 31 were within average mental ability level and 35 of them were randomly selected into the average mental ability (AMA) groups. Similarly, those who scored UNIVERSITY OF IBADAN LIBRARY 1 1 2 betwesei:i20 and 2E were wiiriin the low mental ability level and 36 of them were randomly selected into the low mental ability (LHA) groups. Twsrve pupils (subjects) per group from the HMA were randomly selected into calculator in instruction and tests group (E,j), calculator in tesiss only group (E25, and non- calculator group (£3 )* The same random selection was done for AHA into E.. , E£ and E^ groups. The same procedure was carried out for LMA groups. A total of 108 pupils took part in the piloc study. The same procedure of subjects selection was carried out in the main study except that the 3 treatment groups '..'ere in 3 schools. The nature of the research design alleged for the use of boys and girls schools for the pilot and main studies. However, to control for sex variable one would have used girls only and boys only schools. Sut this design controlled for sex variable by having nearly equal number (of boys and girls in each cell/group. Kerlinger'9- 2 observed that both girls and boys are used in an experiment, randomi­ zation can be usee in order to balance the individual differences that are concomitant to sex. Then the number of girls and boys in each experimental group will be UNIVERSITY OF IBADAN LIBRARY 113 approximately equal". The girls and boys were assigned randomly in nearly equal numbers to the groups in the pilot and main studies. 3.3 Research Instruments: These are ACER Higher test ML and MQ published by Australian Council for Educational Research (ACER) (See Appendixes 8 & 9) which are standardized tests in verbal and numerical abilities. However, the tests were modi­ fied for the ui.gerian situation. There were also author- prepared instruments such os: (1) Attitude measures towards mathematics and calculator. (2) Mathematics pretest. (3) Mathematics achievsnent/post-test. (4) Instructional Module in Mathematics. In this study, mathematics achievement test and mathe­ matics post-test meant the same test UNIVERSITY OF IBADAN LIBRARY 1 1 4 3.3.1 Preparation of Instrumsnts (1) Attitude Scale Working from the basis of 20-item seals developed by Aiken and Oreger 97 and using the Likert scaling procedures, the attitude scales were two Likert-type A & B with five response-options. One scale was used to measure attitudes toward mathematics (A) and the other to measure attitudes toward the use of calculator in mathematics by the pupils (B). For scale A, there were 12-items and for Scale B there were 14-items,- Pupils responded to each item by choosing one of five Likert alternatives: strongly agree (5), agree C4), undecided (3), disagree (2) and strongly disagree (1). There were equal number of positive and negative items in each scale. The positive items were scores, 5, 4, 3, 2;1 as shown on scale . The negative items were reversed for purposes of scoring. The same response alternatives were used with all items. The instrument was field- tested, and pupils were interviewed to determine how valid the scales were for reflecting the pupils’ atti­ tudes. The pupils’ responses were registered in the ^pace adjacent to an item. (See Appendix 10) . 97. Aiken, L. R. Jr. Personality correlates of attitude toward mathematics. Journal of Educational Research, 1 963, 56 , UNIVERSITY OF IBADAN LIBRARY 115 { 2 ) NathBmatics pre-test: From the learning experiences of the pupils a sample of question-items on linear equations was prepared. The construction of the test items was based on the analysis of the objectives in the cognitive domain, Knowledge, comprehension and application. For the task analysis, we have algebraic operations, expressions, identifying euqations, forming equations and solving equations to comply with the following behavioural objectives: (i) to identify expressions such as: ax ♦ b from linear equations of the form: a x + b = x where a and b are constants. (ii) to form linear equations with one variable: 2£-J-± - 6 - 0 (iii) to solve linear equations of one variable: 3x + x + 2 = 10 The numbers. rfor each of the behavioural objectives and corresponding topics were worked out. On the whole, 15 question test-item was constructed. The test-items were multiple choice objective questions format, and the pupils were expected to complete the test in 30 minutes. It was field-tested for validity and reliability. UNIVERSITY OF IBADAN LIBRARY 1 16 T A S I F 4 Mathematics pre-test items construction f°rmat for content validity BEH AVIO U RA L OBJECTIVES TOPI CS TOTAL KNOWLEDGE COM3 RE TENSION / m i CATION 1 Alge braic operations 1 1 - 2 2 Algebraic Expressions 1 1 1 3 2 Identifying Equations 1 1 - 2 4 Forming Equations 1 1 1 3 2 Solving Equations 2 2 1 5 TOTAL 6 6 3 15 ( 3 ) Mathematics post-test: The test-item content was based on equations: simple, simulatenous, and quadratic. The test was applicable to forms four and five pupils of secondary school wbo had covered these aspects of the seondary school mathematics curriculum. The test-item selection was based on the following objectives: (i) to provide pupils with basic facts on algebraic concepts, (ii) to develop pupils’ computational skills in mathematics, (iii) to identify relations in mathematical concepts. (iv) to solve simple forms of different equations. (v) to translate word problems to equation and solve them. The objectives were translated into the test ^lan relating each objective to the cognitive domain and the appropriate task. UNIVERSITY OF IBADAN LIBRARY - 117 TABLE 5 flathemct. cs post-test plan for content validity CO GNITIVE DOMAIN TOPICS COMP UTAH CNAL to ta l ALGORITHMS CONCEPTS APPLICA­ PROBLEM TIONS SOLVING Sirrple Equation 4 4 3 3 15 Simultaneous Equations 3 3 2 2 10 Quadratic Equations 3 1 1 - 5 TOTAL 10 0 6 6 30 The test items were made up of 30 multiple choice objective questions. Each test-item had five options lettered A, B, C, D and E. One of the options was the correct answer. The pupils ' ore required to answer all the questions in 40 minutes, ^ur validity and reliability of the test, a pilot study for the test validation was carried out. The school used to carry out both validity and reliability tests was United Secondary School, Ijokodc, Ibadan. (4) Instructional rrrJule The module had been prepared in response to the need for appropriate and adequate mode of instructional presen­ tation. From the analysis of some Nigerian textbooks on secondary school mathematics carried out it was observed that they did not meet the need of calculator-use in instruction. It became necessary to prepare both the module and calculator in instruction guide (See Appendix 7). UNIVERSITY OF IBADAN LIBRARY - 1 1 8 O0 According to Boll'' , the main difficulty, of course, is that few existing school mathematics textbooks have really interesting problems that exploit the power given by calculators. Presentation sequence in the teaching of mathematical concepts plays an important role in the learning of the subject. Suppes, like Gagne’̂ , subscribes to the idea of the importance of accounting for content structure in the study of learning and sequencing. Suppes, Hyman and Oerman^ stated that in the cognitive domain mathematics provides one of the clearest examples of complex learning and performance, for the structure of the subject-itself provides n .ne /ous constraints on any adequate theory . A substantial amount of Suppes’ work reflects the attitude contained ia the following statement: For anyone interested in the psychological foundations of mathematical concept formation S0. Bell, Max. S. Calculators in Elementary Schools? Some tentative guidelines and questions based on classroom experience. The Arithmetic Teacher 23(7), No. 1975, pp. 502 - 507. UNIVERSITY OF IBADAN LIBRARY 119 i i s natural to ask what is the sort of connection that holds between the logical structure of mathematical concepts and the psychological processes of acquisition of the concept.8 (p.73). Here the need to determine the hierarchies of learning mathematical concepts which would conform with psychological principles becomes necessary as well as the mediational procedures. To this end, the ideas of Ausubel 99 on Advance Organizers ar.d Gagne's learning hierarchy theory were utilized in the construction of the instructional module used in this study. Since calculators were used as part of treatment in the study the instructional module was developed to tap the intrinsic capabilities of the calcu­ lator. Hence, the instructional module was prepared with 99. Aus ube 1, 0. P. The Psychology of Meaningful Verbal Learning-. New York, Grune and Stralton, T5IT37— 100. Gagne-’, R. M. Learning Hierarchies. Educational Psychologists 6(1), 1968, 3-6. UNIVERSITY OF IBADAN LIBRARY tile; following objectives in view: (i) To introduce the pupils to the concept of equations., simple, simultaneous and quadratic. (ii) To identify different forms of equations: simple, simultaneous and quadratic. (iii) To solve simple forms of different equations: simole, simultaneous and quadratic. (iv) To translate wo rd-p ro blems into equational format: simple and simultaneous. ( v) To -.ulve the word-problem-equations: simple and •.imv.lt&neous. The above objectives were related to the cognitive domain in the learning content. Hence, appropriate instruc­ tional module on simple, simultaaoous and quadratic equations was developed. (See Appendix 7). 3.3.2 Validation of instruments (1) Attitude Questionnaire: The attitude measures-scale for mathematics and calculator, prepared by the author was validated for use at United Secondary School, Ijokodo. The 12-item and 14-item of attitude toward mathematics and attitudes toward calculator UNIVERSITY OF IBADAN LIBRARY 1 2 1 were duvelope J in line with Aiken and Dreger using tho I ikert S . a 1' procedure. The subjects were form five pupil' of the school in which two classes were selected ^randomly (by ballot) as sample for the validation process- pupils responded to each item by choosing one of five Likart alternatives: strongly agree, agree, undecided, disagree and strongly disagree . The sessions were conducted during the free periods of the sampled classes with their mathematics teacher m attendance. The purpose of the questionnaire was explained, and that it would be followed with an achievement test. The questionnaire was first administered before achievement test so that the test might not interfoie with thtio response set. On the whole 80 pupils tv- ponded to the questionnaire. Out of the 80 pupils, 40 pupils were randomly selected for testing the validity and reliability of the scale. For the internal consistency reliability coefficient of the attitude measure. Pearson product moment used to compute the correlation coefficient between odd and even (r = 0.98), using Spearman-3rown coefficient the reliability coefficient was found to be c.99 ffor the calculation see Appendix 11 ). The calculated correlation coefficient of r = 0.98 is significant for N = 20 at a = .05 r = 0.423. UNIVERSITY OF IBADAN LIBRARY - 122 - While another widely used index of item discriminabi1ity was the critical ratio based upon the means and variances of the upper and lower 27% of the sampled distribution. The correlation co-efficients of mathematics-attitude and ca1culator-attitude were computed (See Appendix 12). The mean significant difference of the mathemacical attitude scale and calculator attitude scale was computed (See Appendix 13). Shaw a n d Wright1D1 s t a t e d that Aiken and Hreger,in 1964, reported a test-re-test reliabili­ ty coefficient :. this study, rhe reliability co -efficient obtained would be considered adequate coreiriaring the levels of difficulty and discrimination of the test-item. Blood and Budd^^ furthered opined that a reliability co-efficient for class­ room test should be at least 0.60 (Appendix 6). 3.3.3 Modification of mental ability tests - ACER higher cests'ffL and MT Both A. C.E.R, Higher Test: ML and MQ for verbal and numerical abilities were developed by Australian Council for Educational Research in Australia, which is socially.. and culturally different from Nigeria. Though the tests have been found to be applicable to Nigerian setting ( Egbugara)1 06 , it w o u 1 d ' b e necessary to make some modifications if they would he used effectively and appropriately in N i ge ri a. The A. C.E.R. ML test deals with questions on language and vocabulary of English Language. Both Australia and Nigeria have English Language as their Lingua Franca the test may therefore, not suffer much reliability and validity in terms of structure and lexis. However, some of the test-items contained culturally 105. Blood, D. I. and Budd, W. C. Educational Measurement and Evaluation, New York: Harper and ftow, V577~. 106. Egbugara, U. 0. Effects of Three Levels of Advance Organizers cn Achievement of Some Nigerian Secondary School Physics Students, Ibadan: Unpublished Ph.D. Thesis, University of Ibadan, Nigeria, 1 964. UNIVERSITY OF IBADAN LIBRARY 1 2 7 biased words and expressions e.g. the use of Alastian dog which had to he replaced with a more familiar Nigerian name. Those, test-items which shored propensity for Cultural bias were notified by changing then bo socially and culturally accepted..a words in Nigeria. Whereas in the A.C.E.R. MQ test of numerical ability most of the test [tarns were appropriate except in cases where units; of mone'< had to be changed. The Australian pound and penny had to be changed to Naira and Kobo respectively. These changes did not have structural effect on the test-item or their meaning. The durations of the tests ML and MQ had to be changed. Instead of the 20 minutes allocated for each test it was change to 30 minutes so as to give the pupils enough tirrm to read and answer the questions. Secondly it gave room to correct typographical mistakes or non-clarity. Nonetheless, the tests can be considered to be valid and reliable "or the level of puoils after those modifications had been made. It would be understood that those changes could not possibly have affected the validity or reliability of the tests because nothing structurally in the tests were changed. The validity index and reliability coefficient of the tests were not supplied but from available records the tescs had been ('standardised. UNIVERSITY OF IBADAN LIBRARY 1 2 6 3 . 4 T h e P i l o t S t u d y 3.4.1 Objectives of the Pilot Study (i) To validate and modify instruments. C i i j To simulate experimental conditions. (iii) Trial run for the entire experimental plan. (iv) To detect flaws so as to increase the probability of a good research. 3.4.2 Proceriure for the Pilot Study The pilot study began in early October 1 985 . The Form V mathematics teacher provided adequate support. There were 216 pupils in form V of the Ahmadiyya Grammar School, Eleyele, where the pilot study was conducted. Because of the time­ table arrangement of the school, some of the tests were conducted after school hours. The school was organising evening remedial classes for Form V pupils from 2 p.m. to 3.30 p.m. There was 30 minutes session everyday for each group. The sampled pupils were divided into S treatment groups based initially on their mental ability scores on A.C.E.R. Higher Tests ML and MQ, and randomly selected into those groups. The groups were randomly selected into treat­ ment groups by taken cognizance of their relative perfor­ mances on the tests. The calculator groups were instructed by the author while the mathematics teacher helped to instruct the non-calculator groups. The instructional module UNIVERSITY OF IBADAN LIBRARY was used by all the groups. The pilot study lasted for six weeks. Thare wore breaks in between the days of administration either due to pupils being engaged for a school programme or the author/school mathematics teacher not able to attend. However, records of attendance wars kept and dates for the administration of the instruments. By the end of the pilot study only an average of ten pupils per group totalling ninety (90) pupils completed the study. 3.4.3 Administration of research instruments: The first instrument administered during the pilot study was the A. C.E.R. Higher tests ML and MQ . The mental ability tests were used to divide the pupils into different treatment groups. Most of the pupils in Form V of the school took the tests (185 out of 216). The tests took place after the pupils preparatory classes. The school mathematics teacher and the author administered the tests from 4 p.m. to 5.30 p.m. on 21-10-05. The pupils had been informed of the test by their mathematics teacher. Hence the pupils showed enthusiasm towards the test. They were told the purpose of the test that it was not supposed to grade them but to assist in diagnosing their problems in mathematics. The ML test was first taken and followed by MQ test by all the classes. The UNIVERSITY OF IBADAN LIBRARY -130- time of the day the tests were administered had effects on pupils. Some pupils complained of tiredness. In fairness the pupils had been receiving lessons before the test began. It would appear that the tests could only be taken at that time so as to avoid any contamination, and leakage of test-materials Hence all the pupils had to take the tests at the same time. Each of the tests had a duration of thirty minutes. The pupils were provided with individual question and answer sheets (Appendices 8 and 9). There were thirty-six question- items on each of the tests. There were anough examples on each of the tests and the pupils were required to respond to all the thirty six test-items. After the tests had been completed by the pupils he results of the tests of ML and MQ were used to divide the pupils into treatment groups. The test scores (X) of each pupil, were added together to determine his/fter relative position. Since the test was meant to divide the pupils into three different ability levels those pupils who scored 32 £ x £ 51 were grouped into High Mental ability level, pupils who scored 27 £ x £ 31 were grouped Average Mental ability level, and those pupils who scored 28 < x < were groupec Low Ability level. The c.fferent mental ability levels were then randomly selected (by ballot) into the three different treatment groups) UNIVERSITY OF IBADAN LIBRARY 131 Calculator unrestricted group, calculator restricted groups and Non-Calculat-r-use in i ns truction/test group. The mathematics pre-test and pre-attitude questionnaire were not administered to all the nine groups before, instruction began. There were four instructional sessions of thirty- five minutes per soarsion for each of the group. On the whole, twenty-four instructional sessions were conducted during the pilot study for the six groups by the author. The mathematics teacher had only one group of thirty-six pupils of four sessions because three groups were put into one. He had to do this because there was no need for the use of calculator and the pupils were going to receive the same learning experiences. Immediately the instructional process was coming to an end, the pupils had to take the mathematics post test and respond to the attitude measures. The unrestricted and restricted calculator-use groups used the calculator on the post test while the remaining three groups .of non-calculator- use did not use calculator on the test but they responded to the attitude measures. The post test of thirty-items had a duration of forty minutes. When the mathematics post test had been completed the pupils were supplied with the attitude measures which they freely responded to. UNIVERSITY OF IBADAN LIBRARY 132 3*4.4 The Scoring of different instruments; ML and M3 m-. vi c.ncfics pre-test, post-test, attitudes measures. Both tiL and , Vj rests wu re scored on a scale of 1-35. There rare 36 test-items on each of the tests. For the mathematics or- -test it was scored on scale of (1-15). (here were 15 test-items on the test. Similarly, the mathe­ matics achievement test was ccorsd on the basis of number of question on the test (1-30). The raw scores on the tests were not converted but were directly used in the various analyses. The tests scores (X) in ML and MQ were added together for each pupil so the possible range of score 1 i X < 72 was used to divide the pupils into different ability groupings. Methoc of summaL.d ratings was used for attitude scores. Items wore worded positively and negatively. For positively worded iter 'hey were scored 5, 4, 3, 2, 1 and negatively wGruQfc! items -he scoring was reversed as 1, 2, 3, 4, 5. Items scores were added (i) for both Mathematics Attitude Scale (MAS) and Calculator Attitude Scale ‘(CAS) (ii) for each attitude scales MAS and CAT. The item score was assumed to be the weighted sum of the common factor and a factor specific to the item, The common factor was the general attitude variable that we were trying to measure. For MAS ♦ CAS Scores (X) could range between 26 < X < 130, for MAS alone scores (X) range 12 _< X _< BO and CAS: 14 < X _< 70. UNIVERSITY OF IBADAN LIBRARY 1 3 3 3 . 5 A n a .;, .• ~ c ? d a t a O f p i l o t s t u d y Ariaiyaxs of l j l ., comprised mainly the comparison of achievement test mean scores and attitude measures. The computer library programme LIB y2$P was particularly useful for the one and two way factorial analysis of variance and covariance. This programme enabled the use of the attitude towards mathematics and calculator scores (ATS) to adjust the achievement scores of the groups. Pupils’ scores (X) in the mental ability tests were grouped into different anility range: Low ability ( LA : 20 X _< 25) , A vs ra ge abi li ty ( A. A. : 27 <_ X <_ 31) , and High ability (H.A.: 32 <_ X .5.51) respectively. These score categories were used as the selection basis for examining the effects of cognitive, numerical and verbal aptitude levels on i is post-test and attitude scores. Also multiple 2 correlation coefficients R, R derived from analysis of covariance were computed to determine the relationships between scores of the attitude measures and* post-test. All tests of significance we re carried out at the 5% alpha level, and all computations were aided by the University computer. However, the computer results of analyses were given to the nearest significant levels e.g, 0.001 or 0.01 etc. UNIVERSITY OF IBADAN LIBRARY -134- 3.6 Results of Pilot Study The results were analysed and discussed in relation to the hypotheses earlier stated. That is: Hypothesis One: There will be no significant difference in the mean achievement scores of those groups of pupils who use (i) Calculator in tests and instruction - the unrestricted groups (UCin (ii) Calculators in tests only - the restricted groups (RCU) and (iii) No calculators - use at all (NCU). That is : Ho: fi,- = fir- IV at a = .05 E 1 E2 E3 TABLE 3 Analysis of variance of Post-test Scores of Groups UCU, RCU, and NC U SOURCE df SUM OF MEAN F- P SQUARES SQUARES RATIO SS MS Covariates VAR02 - ATS 1 50.34S 50.349 4.627 0.032* Main Effects 2 7.014 3.507 0.322 0.999 ns GRP Explained 3 57.363 19.121 1.757 0.160 ns Residual 06 335.705 10.061 TOTAL 89 993.140 11.159 * Significant at p < .05 ns: Not significant at p = .05. UNIVERSITY OF IBADAN LIBRARY - 1 3 5 - TABLE 10 Multiple Classification of Post-Test Scores By Croups UCU, FCU and NCU Grand Mean 8.18 Variable Category N Unadjusted BETA Adjusted Beta Groups Dev’n for inde­ pendents Dev ’ n 1 30 -0.26 -0., 1 3 n 30 0.39 0.,39 3 30 -0.11 0.09 -0.,26 MULTIPLE R SQUARED = C.058 MULTIPLE R = 0 . 2 4 0 TABLE 11 Summary of the Mean, Standard Deviation and Variance of the Groups UCU, RCU, NCU Variable N Mean Std. Dev. Variance VAR 01:- MAT 90 26.6555 7.6190 56.06 VAR 02:- ATS 90 77.4556 12.9114 156.70 VAR 03:- ACT 90 6.1778 3.3405 11.16 VAR 04:- MAS 90 42.6555 8.5554 73.195 VAR 05:- e AS 90 34.6333 11.0933 123.06 UNIVERSITY OF IBADAN LIBRARY -136- From table 9, it showed that there was significant difference when attitude scores were used as covariates to post-test scores. However, it would be inconclusive to say that the post-test scores had significant difference at a = .05. It was not sufficient to reject the null hypothesis based on this result. However, it would be necessary to run significant level test for the groups in the main study. Hypothesis two: There will he no significant difference in the mean achievement scorer of those groups of pupils of high iri.M.A.), average (A.M.A.), and Low (L.M.A.) mental abilities. That is: Ho: lv!C1 = MC2 = MC3 at = .05. TABLE 12 Analysis of variance of Post-test Scores of HMA, ANA, and LMA Groups SCJRCE df sun of MEAN F RATIO P SQUARES SQUARES ss MS Plain Effects 2 273.489 136.744 16.531 0 .001* * * GRP Explained 2 273.489 136.744 16.531 0.001 Residual 87 719.659 8.272 TOTAL 89 990.147 11.159 *** Highly significant at p < .001. UNIVERSITY OF IBADAN LIBRARY 13 7 TABLE 13 Multiple C1assificatim Analysis of Post-Test Scores by Groups: HMA, AMA and LMA Grand Mean = 8.16 Variable + Category N Unadjusted ABjusted for Groups Dev' n BETA Independents Dev’n BETA 1 30 1.36 1 .96 2 30 0.32 0.32 3 30 -2.28 -2.28 0,52 0.52 MULTIPLE R SQUARED -- UP i.275 MULTIPLE R = 0.,525 There appeared to be significant difference in the mean scores of the groups of different mental ability levels. This significance difference could only be ascertained whan a multiple - range test - post - hoc analysis is performed in the main study. UNIVERSITY OF IBADAN LIBRARY - t S b - TABLE 1* Analys is of covariance of Post-Test Scores of groups HMA, AMA and LMA SUM OF MEAN SOURCE df SQUARES SQUARES F SS MS RATIO P Covariates VAR02 - ATS 1 50.343 50.348 6.372 0.015* Main Effects GRP 2 263.301 131.650 16.662 0 .0 0 1*** Explained 3 313.649 104.550 13.232 0 .0 0 1*** Residual 56 679.498 7.9C1 TOTAL 69 993.147 11.159 ***. cantr,at p < .0 0 1. * Significant at p < .05 . TABLE 15 Multiple Classification Analysis of Post-Test Scores by Groups, HMA, AMA and LMA. ACT - GROUPS - Grand Mean = 8.18 Variable + Category Group N Unad lusted Adjusted for Dsv’n BETA Independents & Covaricates Dev’n BETA 1 30 1 .96 1.96 2 30 0.32 0.25 3 30 -2.28 -2.21 0.52 0.52 Multiple R Squared 0.316 Multiple R 0.56? UNIVERSITY OF IBADAN LIBRARY 139 On the basis of the sampled data and the analysis carried out the null hypothesis two was rejected. That is: There will be no significant difference in the achievement scores of those groups of pupils in HMA, AMA and LMA at a = .05 could be rejected. There would be need to carry out the post-hoc analysis for the treatment effects particularly in the main study. Hypotheses three: Ho: There will be no significant difference in the mean attitude towards mathematics and calculator scores of those groups of pupils who use (i) calculators in tests and instruction - unrestricted groups (UCU), C ii) Calculators in tests only - the restricted groups (RCU), and (iii) No Calculators use at ail (NCU), i.e. Ho: XE1 = XE2 * XE3 at a = .05 UNIVERSITY OF IBADAN LIBRARY 140 TABLE 1c Analysis of variance of the Attitude Scores (ATS) of the groups UCU, RCU and NCU GUM OF MEAN SOURCE df SQUARES SQUARES F SG MS RATIO P Main Effects Group 2 385.0B6 192.543 1.159 ' 31 9 ' rts Explained 2 385.090 192.545 1.159 . . 319*as Residual 87 14451.105 166.105 TOTAL o, o 14336.135 165.599 ns'"- 4 o t significant at p - .05 UNIVERSITY OF IBADAN LIBRARY T A B L E 1 7 Multiple class1 Flection Analysis of the Attitude Scores for the Croups UCU, RC'J and IMCU GRAND MEAN = 77.,46 Variable + Category UNADJUSTED ADJUSTED FOR Groups N Dev’n BETA INDEPENDENTS Dev'.n BETA 1 30 2.54 2.54 2 30 -0. 02 -0.02 3 30 -2.52 -2.52 0.6 o . 6 MULTIPLE R SQUARED - 0.026 MULTIPLE R = 0.151 Ins analysis showed that the result of the attitude scores was not significant, None of F-ratio was significant and the null hypothesis was therefore not rejected. Hence, * there was no significant difference in the mean attitude scores of the groups. However, this would also be tested in the main study. UNIVERSITY OF IBADAN LIBRARY 142 Hypothesis four There .ill be no significant difference in attitude towards mat mmatics and calculator scores of those groups of pupils of HMA, AHA and LMA. That is: Ho: X = X = Xr at a - .05 L ̂ L'p L ^ TABLE 13 Analysis of Variance of the Attitude Scores of tiraups HMA, AMA and LiMA SUM OF MEAN SOURCE df SQUARES SQUARES F SIGNI FI CANT SS MS RATIO OF F Mai r, Effects 2 96.355 48.178 0.284 .999 ns GRP E xplained 2 96.355 48.173 0.284 .999 ns Residual 97 14739.867 169.424 TOTAL as 1483.229 166.6 99 ns - Not significant p = .05 UNIVERSITY OF IBAD N LIBRARY - 1 4 3 - T ABLE 19 Multiple Classification Analysis of the Attitude Scores for groups HMA, AMA and LMA Grand Mean * 77.46 Variable + Category N L’nad j us ted Adjusted for Independents Group Dev' Beta Dev'n Beta 1 30 0.04 0.04 2 30 -1.29 -1.29 3 30 1.24 1.24 0 . G 6 0.08 MULTIPLE R SQUARED = 0.005 MULTIPLE R = 0.051 The analysis of the result of the attitude scores of the groups HMA, AMA, and LMA seemed to suggest that there was no significant difference in the mean scores, and hence, the null hypothesis was accepted. Hypothesis five: There will be no significant relationship in pupils’ attitudes toward mathematics and their attitudes toward the use of calculators in secondary school mathematics as a = .05 UNIVERSITY OF IBADAN LIBRARY -1 44- TABLE 20 Summary of the Analysis of Variance of the MAS and CAS Scares or Groups U C U , RCU and NCU GROUPS df F-ratio SIGNIF OF F SIGNIF LEVE UCU 2? 0.237 .339 ns MAS RCU 23 0.943 .999 ns NCU 23 0.131 .999 ns - UCU 23 1.654 .209 ns CAS RCU 29 ! .179 .324 ns NCU 29 u «25 .939 ns ns - Not sign ificon t at p = .05 It would appea r from the ana lysis of this result that there was nr signrfic ant difference in the attitude scores of tha three groups. There was no difference in the group n^jans of these three groups, for the MAS and CAS, the groups could have the same attitudes to mathematics and calculator. However, this could have been due to some factors or treatment which this pilot study did not envisage there was no pre-attitude treatment for the groups to ascertain groups attitude before treatment. UNIVERSITY OF IBADAN LIBRARY - 1 4 5 - This '..•ic’d have tc be t aken care of in the main study. TABLE 21 Summary of the Analysis cf Variance of NAS and CAS Scores of the FI HA AHA and LMA groups ARCSPS df F-RATIir. SIGH IF SIGNIF LEVEL U OF F at a = .05 HHA 29 0.Q50 ,999 ns MAS ANA 29 0.022 .999 ns LNA 29 0. S39 .999 ns HNA 29 2.05 i .146 ns CAS ANA 29 3.014 .054 ns LNA 2 j 0.530 .999 ns ns - Not significant at = .05 The group means were no : significant at; a = .05, there was no difference .at all in the mean scores . It would appear as if there was no difference in the attitudes of pupils of » differert mental abilities . t ;-, iy seamed to have the same attitudes toward mathematics and calculators. Finally Pearson correlation was ■. sed to test relation­ ship or the groups achievement scores to attitude scores, r = 0.225, n = 90 at a = .05, This showed that there was no high relationship in the achievement scores UNIVERSITY OF IBADAN LIBRARY 146 and the attitude scores. For the MAS and CAS r = 0.156, N = 30 a = .07 The relationship of the mathematics attitude scores and calculator attitude scores was not significant at a = .05. This was already evident in the analysis of variance of the variance of the groups. Nonetheless, this might have arisen out of uncontrolled variance of pre -attitudes and other extraneous factors. Since the ANOVA was not signi ficant in all the groups, tests of significance for the treatment groups were not carried out. 3.7 Discus si c_m on the result of Pilot Study The number of subjects in the groups could have affected the statistical analysis as related to significance level. For larger number of subjects the result could have possibly been different. The pilot study was able to est a ­ blish that there was significant difference in post-test scores of the groups UCU, RCU, NCL) which led to the rejec­ tion of the null hypothesis. Though, no tests of s i g n i f i ­ cance for the treatment groups were further carried out, there was not enough evidence of equalisation of the groups through pre-test and pre-atti tudes. Hence it would be difficult to ascertain what had contributed to the UNIVERSITY OF IBADAN LIBRARY difference in post scores eiuher it was the treatment or other factors. As regards the groups' atti­ tudes, most of the tests showed no significant difference in the mean attitude scores. This would likely be that the groups had similar attitudes towards mathematics and electronic calculator. Further tests in the main study might reveal more information about these findings. 3 .8 Detection of flaws corrected for the main study The pilot study revealed certain aspects of research procedures which needed to he c o r r e c t e d before the main study: (i) the number of schools (ii) number of subjects in each group (i i i) no pre-test and pre-attitude question­ naire were administered. The pilot study began with 12 subjects per groups but ended with 10 subjects per group. Like in most experimental studies that would go on for weeks, arrangoment should be made to take care of subject mortality. The number of schools increased to three in the main study which could reduce subject contamination of treatment. In the pilot study instructions and tests were carried out after the school hours - because of the problem of school’s time-table, Most of the pupils complained of physical tiredness. The problem of time-table was tackled during the main study when the study took place during the school hours and in the mornings. For the main study. UNIVERSITY OF IBADAN LIBRARY three comparable schools were used, it became necessary to have cooperating teachers from the different schools to assist in the instruction >nd testing. Efforts were made to brier all the pupils in the study and the cooperating teachers about the purpose of the study and its implications to mathematics education in Nigeria. Some of the limitations experienced during the pilot study related mostly to incorporating the instructional time into the normal school-ho urs. Pupils' trepidation as regards the Handling of the calculators and pupils' anxiety in the face of calculator during pilot study was somehow remedied for the main study. The pupils could not take the instructional module and calculators home to practise. For this kind of study, the experimental groups could have been much more motivated by allowing them to take the modules and calculators home for practice. However, the non-ealeu■ator groups was discouraged from the use of calculators. In fact, they would not be exposed or advised to use calculator either in class or at home throughout the study. The main study took cognizance of the pilot study results on the use or non-use of calcu­ lator and some plausible answers to the introduction or otherwise of technological devices such as calculators, computers etc. into instructional systems. UNIVERSITY OF IBADAN LIBRARY - 14 2 CHAPTER FOUR THE MAIN STUDY METHODOLOGY 4.1 Design The experimental design for the main study was slightly modified from the pilot study design so as to take care of the flaws identified in ihe pilot study. One of the corrections carried out was to make use of three compar­ able secondary schools (mixed) in Ibadan instead of one school used for the pilot study. Three groups were used in each of the three randomly selected schools. MENTAL ABILITY LEVELS SCHOOL HIGH MENTAL AVERAGE MEN TAL LOW MENTAL ABILITY ABILITY ABILITY 'H. M. A. CC1) A.M.A. (L) L.M.A. (C3) Lin res triete d SCHOOL Groups (UCU) A B C E 1 Calculator in 1 2 3Inst ruction (n=14) (n -14 (n = 14 and tests Restricted SCHOOL Groups (RCU) D E F Calculator in 4 5 6b2 tests only (n=14) (n=14) (n=14) SCHOOL No Calculator G H IUse at All 7 8 9 E 3 Groups (NCU) (n=14) (n * 1 4) (n=14) Fig. 7: A paradigm of 3 x 3 factorial design for the main study. N = 126 subjects K = 9 UNIVERSITY OF IBADAN LIBRARY - 1 5 0 - Variables used in the main soudy VAR 01 MAT - Mental ability test scores VAL 01 PEA - Pre-attitude questionnaire scores VAR 0 3 PET - Pre-test scores VAR 04 P0A - Post attitude questionnaire scores VAR 05 POT - Post test scores VAR 05 MAS - Mathematics attitude questionnaire scores VAR 07 CAS - Calculator attitude questionnaire scores In the main study, no intact classes were used and therefore, the use of covariates allowed the pre-test mean scores and pre-attitude mean scores to serve in adjusting the initial differences or equalizing factors within and between groups. In addition, the pre-test served as a measure of the level of pupils’ prior familiarity with the selected learning material content on which the test was based. The pre- attitude questionnaire administration was to help to establish prior attitude of subjects towards mathematics and calculators, and if, there would be any attitudinal change as a result of the treatment or otherwise. If T, represents the Pre-test or Pre-attitude, the treatment one - the use of calculator in tests and instruction, restricted calculator use in tests only UNIVERSITY OF IBADAN LIBRARY and T2 the post treatment tests or attitude measure; and R means randomization cf treatments to groups. Then, the design can generally be represented as 1 0 1 x a W'S : R T1 X1 T2 T1 X2 T2 T1 T2 4.2 Population of the main study Secondary schools in Ibadan Municipality constituted the population of the study and ths three mixed secondary schools used for the study were selected by the following method. There were ninety-five (S5) Junior and Senior Secondary Schools in Ibadan Municipality at the time of the study. A multi _-age stratified random sampling technique 107 was used in selecting the schools. First stage, schools in Ibadan were stratified on the basis of th.sp that offered students for the West African School Certificate Examinations of WAEC in for the last ten years and those which did not. There were thirty-three schools in this category. Second stage, all those schools selected in first stage were stratified nn the basis of whether they were mixed 107. Cnein, I. "An Introduction to Sampling". In C. Selltiz at al. Research Methods in Social relations ■ New York. Holt, Rinehart and Winston, 1959, pp. 509 - 545. UNIVERSITY OF IBADAN LIBRARY - 152 - schools or not (See Appendix 19). There were sixtenn (16) schools in this category. The third stage, using a random sample (by ballot) five schools from the mixed schools were selected. Out of the five schools only three of the schools satisfied the condition of comparability and were selected. The'three (3) schools we r e : 1. Holy Trinity Grammar School, Old Ife Road, Ibadan. 2. Islernic High School, 8 asorun, Ibadan. 3. Ibadan City Academy, Eleta, Ibadan. The three schools were then randomly selected (by ballot) into treatme'vb'.; groups with school 1 as the experimental school - the unrestricted calculator groups (UCU) (to use calculator in instruction and tests); experimental school E^ - the restricted calculator groups (RCU) (to use calculator in the tests only) and school 3 the control - the non-calculator groups (NCU) (Eg)* Sjbjects were then randomly selected into these t r e a t ­ ments and control groups. ^■2.1 Subjects of the main study Form five pupils in their first term of their last-year in the secondary school were used as the subjects for the study, In each of the three snhnnic ai l_ UNIVERSITY OF IBADAN L BRARY 153 th* mantel ability tests (verbal and numerical) so as to be able to divxdo them into different ability levels. In school (1) eighty four (G4j pupils took the tests and forty eight pupils were selected into the different abilitv l evels (see Table 22 ) . In school (2) seventy six (76) pupils took the tests and only forty eight were selected into the different ability levels. In school (3) one hundred and fifty nine (159) pupils took the tests and forty-eight (48) pupils were selected. All selections were done randomly for the different ability levels. TABLE 22 Summary of Mental Ability Tests Scores for Schools 1, 2. 3 in the (lain Study SCHOOLS; M ! MEAN SO RANGE OF ABILITY LEVELS SCORES '■ ■ X i H n A ANA ! LMA! 1 I 84 | 28.64 7.3 i|37 - 52 ! 30 - 36 2 5 - 2 9 2 1 76 I 32.78 10.59! 41 - 55 ' 33 - *40 25 - 32 i | i 3 |159 ' 33.025 i 8,97 41 - 52 ■ 34 - 40 25 - 33l j ! An average of sixteen pupils per group started the programme. There were nine (?) groups in all with the total number of subjects that started as one hundred and UNIVERSITY OF IBADAN LIBRARY 154 and forty four (144). However, by the end of the programme some of the pupils had dropped out which left an average of fourteen suLjects per group. Where there were more than 14 subjects per group the extra(s) were randomly dropped on the basis of sex. Like in the pilot stuiy the groups were equalized on sex at all times. 4.3 Comparability of Schools The study had taken care of sex variable by having equal number of girls and boys in each group. For the comparability of the schools the following conditions were considered: i. Results of the schools in the West Afriean Examinations Council (WAEC) examinations and mental ability tests, ii. Age of the schools - this had been taken care of during the selection, iii. Qualifications of teachers, iy. Sequencing of topics in the scheme of work. v. Opinions of teachers and pupils, v i . Training of teachers for the programme. The WAEC results of the three schools in mathematics from 1980-1904 were obtained and analysed. UNIVERSITY OF IBADAN LIBRARY -155- TABLE 23 NAEC Results in Mathanatics for Schools 1, 2, 3 Percentage Pa-ses: 1980 - 19S4 SCHOOLS 1980 1981 1982 1983 1984 AVERAGEPASSES HOLY TRINITY GRAM SCHOOL 43 24.6 45.5 45.5 85 48 .ISLAMIC HIGH SCHOOL 59 52 33 37 31 48 IBADAN CITY 42.4 ACADEMY 62 55.6 41 50 50.2 TABLE 24 Analysis of Variance of Mean Percentage Passes of the Schools 1, 2, 3. F SOURCE df SUM SQUARES MEAN SQUARE SIGNIF.SS ns RATIO LEVEL BETWEEN GROUPS 2 1235,222 116.611 0.185 NS WITHIN GROUPS 12 7556.488 629.715 * TOTAL 14 7789.81 NS: Not Significant at ? = .05 UNIVERSITY OF IBADAN LIBRARY -15 b- This table on the mean percentage passes on the schools showed that there was no significant difference in their mean passes. It cculd then be inferred that the three schools might have performed relatively equally in the last five years 1980 - 1984. Hence, the three schools were possibly comparable on this basis. When the results of all the pupils who took the mental ability tests in the three schools were obtained and analysed it helped to determine if the schools were comparable on the mental ability tests scores. TABLE 25 Analysis of Variance of Kean Mental Ability Test Scores of Schools 1, 2, 3 - SOURCE Df SUN OF NEAN F-RATI0 SIGNIF SQUARES SQUARES LEVEL SS NS BETWEEN GROUPS 2 1359.41 679.705 . 0.873 NS WITHIN GROUPS 316 246034.30 778.59 TOTAL 318 2473'3,71 NS: Not signi •ficarit at p = .05. i UNIVERSITY OF IBADAN LIBRARY - 157 - The table showed that there was no significant difference in the mental ability test scores of the three schools. Though the number cf pupils who Look the tests in the schools were not equal: = 64, ^ 3 75 and = 159 it would appear that the groups .could be compared on the test scores and being relatively equal statistically. To obtain information on the other conditions of comparability, a questionnaire was constructed by the researcher (See Appendix 161. The face and content validity of the questionnaire were carried out by this investigator and some lecturers in the Teacher Education Department, University of Ibadan. Teachers of mathematics in the three schools responded to the items on the questionnaire. UNIVERSITY OF IBADAN LIBRARY -15a- Tsacher's Variables and Content Coverage School School School Average 1 2 3 No. of Mathematics Teachers in Sampled 5 6 5 5.3 Schools Total Years of Experience of the 59 37 24 40 Teachers Syllabus (Covered ) (Partially ) 0.4 0.33 0.4 0.30 (in the year) Syllabus (Covered ) (Fully in j O.b 0.67 0.6 0.62 (the Year ) The teachers in the three schools indicated that they had taught equations to their pupils at different terns of the year for different classes before the pupils reached first tern of Perm V. UNIVERSITY OF IBADAN LIBRARY - 1riS - TA3LE: 27 Calculator ‘Jsage F.ffectiveness by Teachers Not Total Mo. of PEf4:f^eecrt^-i ve Effective Effective Responses SCHCOL 1 2 3 5 SCHOOL 2 3 3 3 SCHOOL 3 3 2 c Nona judged Calculator to be very effective at the secondary school level - this cannot be a conclusive evidence on Calculator effectiveness at secondary .school. UNIVERSITY OF IBADAN LIBRARY -150- TABLE ?8 Use of Instructional Materials by Schools Materials School 1 School 2 School 3 Four figure Table + + + Calcularor + + - Mathematics Set + + + Slide Rule - + - Geoboard - + - Other boards (Graphboard + - + etc.) Objects (Sticks, Shapes etc.) + - + + Used in the school Not used in the school. From the available data, it would appear that the three schools were comparable. » 4.4 Monitoring the Cooperating Teachers The design of the study involved conducting tests and having classroom instructions during the school hours. Because of ’’.he problems of schools’ distances and time-table it became .operative to solicit the assistance a n d UNIVERSITY OF IBADAN LIBRARY -1 6 '\ - cooperation of mathematics teachers in the sampled schools. Through the Principals, the i orm V mathematics teachers in the schools were briefed on the purpose of the study. Fortunately all the t h r e e school teachers agreed to assist in the programme. The teachers were shown the format of the "Teacher - pupil - material Interaction model" developed from Ogunniyi 108 Laboratory Interaction Categories (LIC) - a modified version of Flanders' Interaction Categories. This method was used to bear credence tc Flanders' findings that, teaching behaviour is the most potent, single controllable factur that can alter learning opportunities in the classroom 1 11. In order to determine the teachers’ classroom effectiveness and behavioural characteristics of the pupils the investigator decided tc observe Teacher - pupil - material interaction in the three schools. Each of the teachers was observed for thirty minutes three times in a week and the records of observations were then analysed. 1D8 Ogunniyi, fl.B.: An analysis of laboratory activities in selected Nigerian secondary Schools. European lournaTlc rfo-f- -S--c-i-e-n-c--e- -E--d-u-c-a-t--i-o-n-. 1983 Vol.5, No. 2, UNIVERSI Y OF IBADAN LIBRARY - 162- Permission was sought end granted from the Principals of sampled schools to observe 'he cooperating teachers and pupils during their lessons. Thus, arrangement.was made to observe the teachers rand pupils wheneveb they had mathematics on their school time-table, and was held with I only these pupils in the study. TABLE 29 A Comparison of Percentage of Teacher/ Classroom Interactions Behaviour PERCENTAGE RATING MEAN SO TEACHER’S CATEGORIES SCHOOL SCHOOL SCHOOL X 1 2 3 A - Accepts Feeling 0.3 0.19 0.42 0.30 0.13 G - Gives Verbal reward C.3 0.19 0.22 0.24 0.029 R - Reinforces response 2.02 1.57 2.20 1.93 0.32 0 - Questions 19,35 18.89 13. 96 17.066 7.89 L - Lectures 12.55 15.42 14.59 14.19 1.43 D - Directs 15.54 8.33 8. 61 10.93 4.25 C - Criticises 4. 744 1.55 11.62 5.97 5.10 M - Manipulates Materials 2.61 5.09 0.42 2-7066 4.68* S - Supervises 2.64 4.3 9.07 5.353 3.32 * The sta; dard deviation calculated for the materials do appear to be high compered with others. Table 29 continues next page with the pupils’ categories. UNIVERSITY OF IBADAN LIBRARY 16 3 PUPILS’ CATEGOf IE5 PERCENTAGE RATINGS |V)EAN SCHOOL SCHOOL SCHOOL X 1 2 3 RQ - Responds to questions 13.35 18.49 12.41 14.95 3.16 10 - Initiates questions 1.26 1.56 1 .04 1 .287 0.26 IT - Initiates talk 2.60 3.75 0.82 2.39 1 .48 CA - Calculates Using materials 9.83 6.48 6.17 7.49 2.05 PD - Reads, Writes/and/or Draws 11 .29 7.21 13.78 10.76 3.32 N - Non-productive activities 1 .74 6.91 7.84 5.50 3.28 n = 3 recordings in each school. N = 14 - 16 (No. nf students per group). Variability in the use of materials as shown by the standard deviation seemed to be high, and this would appear to sug­ gest the difference in the teacher - students interaction. The interaction pattern, -from iutlo 2-9 showed that the » three teachers were direct teachers, they did not use materials much. Hence the teachersof schools 1 and 2 were advised to use more of the materials - particularly teacher of school (1) who had to use calculator throughout the study. UNIVERSITY OF IBADAN LIBRARY -164- TABLE 30 Percentage Distribution of Teachers* Qusstions CATEGORY OF PERCENTAGE RATINGS MEAN QUESTIONS SCHOOL 1 SCHOOL 2 SCHOOL 3 X SD FACTUAL 41.76 27.5 41.02 36.76 8.028 RHETORICAL 44.37 57.9 39.66 47.38 9.41 LEADING 7.55 11.0 8.58 9.31 2.22 PROBING 6.36 2.7 5.51 4.86 1 .9 TABLE 31 Mean Distribution of Questions/Minute o i- Teacher/Pupils Interaction TIME IN MINUTES GROUP 1 - 10 11 - 20 21 - 30 * TEACHER’S OF SCHOOLS 1,2,3 1 .2 1.016 1.18X PUPILS OF SCHOOLS 1, 2, 3 1.17 0.7 0.85X UNIVERSITY OF IBADAN LIBRARY 165 Tables 29-3 t showed that the teachers had relatively the same pattern or Teacher-Student and material interaction. Thus, it could be inferred that the three teachers were direct teachers, The only variation is in the use of materials where they have been designed to be structurally different as indicated in the research design. 4.5 Administration of Instruments When the cooperating teachers had been found to be comparable by the Teacher-pupil material interaction model they were advised about how to administer the instruments. The pupils had been separated into different ability levels through the mental ability tests scores, they all had to take the pre-test and respond to the pre-attitude question­ naire. The sessions were held for thirty minutes (7.30 - 8.00 a.m.) on Mondays, Tuesdays and Wednesdays in the three schools. Thursdays and Fridays were used to hold dialogue with the teachers. The pre-test and pre-attitude responses were collected from tne teachers before the treatments started. The teachers were advised to keep record of attendance of pupils. Each of the teacher was supplied with the instructional modflile. They UNIVERSITY OF IBADAN LIBRARY were advised tc follow the instructional format in the module and they were instructed not to use any other tekt for the study. The calculator groups were supplied with hand-held calculators. Each pupil In the two treatment groups (E>i and used calculator on the pre-test. The next treatment was instruction which was carried out by the cooperating teachers. The unrestricted calculator group used calculator throughout the treatment period. UNIVERSITY OF IBADAN LIBRARY -167- 4.5.1 Guidelines on the Use of Calculators Calculators were used by the experimental groups: E^ and E^. E^ : Calculator use^ in instruction and tests. E2 : Calculator use in tests only. The subjects in these groups and the cooperating teachers were instructed by the investigator on how to use the calculator. The instruction on how to use the calculator took place two days prior to the commencement of the six- week duration of the study. E^ groups who were in the same school received the instruction first day and they were followed by groups the second day. There were three sessions per day, and each of the instructional sessions was held with high, average and low mental ability groups respectively. Each session had a duration of thirty minutes and they were held immediately after the school hours (2 - 3.30 p.m,) in each of the schools. Fourteen calculators were made available to each group ( one per pupil). This allowed the pupils to get familiar with the calculators. PROCEDURE: The operational keys of the calculator were shown: addition (+), subtraction (-), multiplication (x), division ( 7 ) , .quare root (/~j percentage (%), memory storage N+ ,lvl~ R.C M)„ fSse Fig. 2). When the pupils could identify and UNIVERSITY OF IBADAN LIBRARY -168- operate them, the following example wqs done with the groups using the calculator! Simplify the expression: 55 x 10 - 7.22 (7. 22 7 10.96) ION: OPERATIONAL KEYS DISPLAY ON SCREEN Punch ON/C 0 . Punch 5 twice 55. Punch x 55. Punch 10 10. Punch = 550. Punch - 550. Punch 7 points(.) and 2 twice 7.22 Punch = 542.78 Punch rV: 542.70 Punch 7,point (.) and 2 twice 7.22M Punch 7 7.2 2|V* Punch 1 M and 0 , point(.) 9 S 6 10.96 Punch = 0.6587591^ Punch M“ 0.65875911'1 Punch R. CM 542.12l25n Ans. = 542.12125 This answer was checked with the paper-and pencil calculation and comparison was made between the answer from calculator and the paper-and-penci1 procedure. The pupils were ^hen asked to practise with rare exercises on calculator. The teachers checked the pupils’ work and made corrections where necessary. UNIVERSITY OF IBADAN LIBRA Y - 1 6 3 - 2. Calculators were used with the instructional module copies of which were supplied to the (Calculator in instruction and tests) groups only. The teacher of E1 groups was specifi­ cally instructed that the pupils in his group should use only the module and no other textbooks should be used by them. After the operational uses of the calculator had been done on the first day, the investigator advised the groups to continue the next day on the use of calculators with their copies of the instructional module. The general instruction on the use of calculator with the instructional module can be found at the end of the module (see Appendix 7). The other two cooperating teachers were given the instructional modules to be used as the teaching and learning material. The pupils in their groups were not supplied with the module. These other groups did all the cumulations with paper - and - pencil as they were used to in their normal mathematics class lesson. During the four weeks of instruction the cooperating teachers were closely monitored. Each of the teachers was observed three times a week and records were kept. This was to make sure that they were carrying out the objectives of the programme. Attempts by the teachers to deviate from the f UNIVERSITY OF IBADAN LIBRARY -1 70- expressed ob.jendives were corrected. Throughout the duration of instruction the teacher candidly cooperated. By the sixth week the post-test and post-attit.ude " questionnaire were administered. 4.6 Data Collection All data used in this study were collected from the sampledschooIs. The mental ability test scores of the nine groups were extracted from total scores cf those form five pupils who took the ML and MQ tests in the three schools (See Table 22.) The pre and post tests scores, and pre and post attitude questionnaire scores were collated by the investigator. The pre-test scores were 15 points and could have been doubled to equal the post-test scores of 30 points. However, this could have statistically made no difference in results because analyses were done with the means of the scores. UNIVERSITY OF IBADAN LIBRARY -171- 4.7 Data Analysis of the Main Study Analysis of data comprised the comparison of post- treatment and post-attitude mean scores (POT, PDA) with the prs-t:st and pre-attitude mean sccrcs (PET,PEA) respectively. The computer library programme LIB020P, was used for: (i) all the analyses of covariance (ii) Analyse^ of variance (iii) the significant mean effects Comparison o-c significance e-f the means the multiple tests using: (a) Student - Newman Keuls (SNK at a = .05) (b) Scheffe alpha is .05 ) i J (c) LSD alpha is .05 ) all ONE WAV (d) T u key alpha is .05 ! AN0VA Civ) Pearson correlation coefficients.. (v) Hu Itiple regression analyses (vi) Frequency distributions. UNIVERSITY OF IBADAN LIBRARY -172- All tests of significance were carried out at P = .05, and all corr.putat.ons/prograrnmin^ were with the aid of the University Computing Center, except for the t - tests comparision of means which were done with the hand-held calculator. UNIVERSITY OF IBADAN LIBRARY -173- CHAPTER t-'IVE RESULTS OF THE NAIM STUDY The results of the study were discussed in relation to the null hypothesis earlier stated. i 5.10 Hypothesis 1 There will be no significant difference in the achievement mean scores of those groups of pupils who used (i) calculators in tests and instruction - the unrestricted group (UCU} (ii) calculators only in tests - the restricted group (RCU) and (iii) No-calculators at all (NCU). That is: H 0: ME1 = ME7 = ME3 at o - .05 TABLE 32 S ummary of the means, standard deviations and variances of the three Groups (UCU, RCU and NCU) ‘Variables N Mean SD wariances NAT 126 36.903 7,6779 58.950 * PEA 126 81.2937 11.4200 130.4164 PET 126 9.1349 3.3331 11.1095 POA 126 84.0397 13.1954 144.1186 CQT 126 14.2391 4.9565 24.5668 NAS 126 47.4683 7.3065 53.3703 CAS 126 36.5873 11.8701 140.8993 UNIVERSITY OF IBADAN LIBRARY -174- Fach of the groups unristricted calculator (UCU) as , restricted nalrulator (RCU) as amd non-calculator (NCU) as E^was made up of three groups of three mental ability levels: high mental ability (HHA) as , average mental ability (AMA) as and low mental ability (LMA) as and this gave the total -umber of groups to be nine (See fig. 6). For the computer programming and analysis of the data each of the three groups merged into one to give three groups in all for treatments and also three groups for mental ability levels. That is V M , l- 3 = 1) : UCU V (4, 5, 6 = 2) : RCU (7, a, 9 = 3) : NCU V M , 4, 7) = 1) r HMA V (2, 5, 0) = 2) : AMA V (3, 6, 9) = 3) : LMA All tests of significance wei'e carried out at p = .05 but the use of computer for the analysis gave the results of the statistical computations to the nearest signi­ ficant levels. For example there were .001, .01, etc p-levels, and for this study, they are highly significant. Other signi­ ficant levels different from p<.05 or p > .05 were used as they were received from the computer-print-out. They did not affect the interpretations of the results. UNIVERSITY OF IBADAN LIBRARY CO LU -175- TABLE 33 Analysis of Covariance of Post Achievement Test Scores of UCU, RCU, and NCU Groups Source df Sum of Mean F P squares squares Ratio SS MS COVARIATE MAT 1 1S5.819 195.819 12.341 .001*** MAIN EFFECTS 2 939.265 469.633 29.598 .001*** EXPLAINED 3 1135.084 378.361 23.846 .001*** RESIDUAL 122 1935.758 15.687 TOTAL 125 3070.842 24.567 *** Highly Significant at p < .001 When the mental ability test scores (MAT) were used as a covariate on the post-test scores therei was significant difference at v. - .001 of the group - means. UNIVERSITY OF IBADAN LIBRARY 17 S TABLE 34 Multiple Clas~ification Analysis of Post-Test Scores 3y Groups UCLJ, RuU end NCU with Mental Ability Scores GRAND MEAN = 14.24 ADJUSTtD INDEPENDENTS VARIABLE ?, CATEGORY UNADJUSTED + COVARIATES GRP N D E V ’N BETA D E V ’N BETA 1 42 3.14 4.01 2 '2 -1 . 1n -1.31 42 -2.05 -2.40 0.46 0.58 MULTIPLE R SCIJARED = 0.370 MULTIPLE R = 0.60S From = .37.- indicated that only 37% of the variance in the cri terion measure of po st-tost scores was associated with the nenta1 ability test scores whereas the remainiog63% of the varian cc might have been due to treatment or some tc error. UNIVERSITY OF IBADAN LIBRARY 177 TABLE 3E Analysis of Variance of Mental Ability Test Scores the three Groups (UCU, RCU and MCU) SUM OF MEAN SOURCE df SQUARES SS SQUARES F- SIGNIF . MS RATI0 OF F MAIN EFFECTS 2 GRP 748.619 374.309 5.935 0.004** EXPLAINED 2 748.621 374.311 5.935 0.004** RESIDUAL 123 1756.938 63.065 TOTAL 125 8505.559 68.044 ** Significant at p < .01 The mental ability scores were used to divide the pupils into different ability levels. Hence, one would expect a significant difference in the means of the mental scores of the groups. When the mental ability scores of the groups were used as the covariate to post-best scores there, were significant differences'in the covariate, main effects, and explained variance of the groups. Hence, there was signi ficant difference in the. post-test scores of the groups. Since aii the groups took the pre-test which was to serve js an equalizing factor, an analysis of covariance of the post-test was carried out using pre-test as the covariate UNIVERSITY OF IBADAN LIBRARY - 178 - TABLE 36 Analysis of Covariance of Post-Achievement Test Scores, of UCU, RCU and NCU groups sun of nEAN SOURCE df SQUARES SQUARES p SIGNIF . SS ns RATIO OF F COVARIATE (PET) 1 33.992 33.992 1.713 0.190ns MAIN EFFECTS (GRP) 2 615.025 307.912 15.516 0.01*** EXPLAINED 3 649.017 216.606 10.915 0.001*** RESIDUAL i 22 2421.025 19.844 TOTAL 125 3070.842 21.567 COVARIATE BETA PET 0.007 *** Highly significant at p < .o01 ns: .iot significant at p < .05 UNIVERSITY OF IBADAN LIBRARY 179 TABLE 37 Multiple Classification Analysis of Post-test Scores of Groups UCU, RCU and NCU with Pre-test covariate GRAND MEAN = 14.24 VARIABLE + CATEGORY UNADJUSTED ADJUSTED FOR GRP N DEV 'N ETA DEV’N BETA 1 42 3.14 3.10 2 42 -1.10 -1.06 O ’2 -2.05 -2.02 0.46 0.45 MULTIPLE R SQUARED = 0.212 MULTIPLE R 0.460 2 From R = 0.21, indicated that only 21% of the variance in the criterion measure of post-test scores was associated with the pre-test scores whereas the remaining 79% of the var* ance might have been due to treatment or some t "• f=rror. UNIVERSITY OF IBADAN LIBRARY 180 TABLE 38 Analys is of variance of the Pre-test of the 2rsy.roups UCU, RCU end NCU. SOURCL df SUN OF MEAN F- SIGNIF. SQUARES SQUARE RATI0 LEVEL MAIN EFFECTS GRP 2 8515.531 4257.766 0.852 .999 ns EXPLAINED 2 8515,553 4257.781 0.852 .999 ns RESIDUAL 123 614953.436 4999.621 TOTAL 125 523469.000 4987.750 ns - Not significant at P = .05 The m e a n •Pro-test scores of the groups did not show any significant difference at a = .05. This would suggest that the three groups were equalized by the pre-test. He ice any difference on the post-test scores would likely be du to the treatment. To further test for the contribution of each treatment groups (UCU, RCU) to the significant difference from the control group (NCU) a post-hoc analysis was carried out to determine any significant differ nee among the groups. If a significant difference did exist, which of the groups was better than cne other was determined by mult-ple range test of one way Scneffe and t-^osts. It UNIVERSITY OF IBADAN LIBRARY would bb noted that LSD or Scheffe could only be applied if only and if there was significant difference at ct = .05. TABLE 39 Multiple Range Test of Fost-test spores by One-Way Scheffe Procedure - ANG7A+ SOURCE df sun OF MEAN F F SIGNIF SQUARES SQUARES RATIO PR0B LEVEL oS ns BETWEEN GROUPS 2 541.332 320.560 16.23 0.001 * * * WITHIN GROUPS 123 2429.5273 19.7522 TOTAL 125 3070.6594 * * * Highly sign! ficant at p < .001 The group s were rearranged into groups of 2 for t-tests as shown in the table of t-test.(Table 40). + Oneway LSD, Scheffe, TUKEY-HSD and SNK were carried out, and they showed the same result. UNIVERSITY OF IBADAN LIBRARY 102 TABLE 40 Summary of t-tests of the Post-test Scores Groups UCU, RCU and NCU GROUPS X ?N SO SO t-rat io SIGNIF LEVEL 1. UCU 42 17.30 4.403 20.10 5.354 .001 *** NCU 42 12.191 4.40 19.304 2. UCU 42 17 38 4.463 20.10 RCU 42 13.143 4.44 19.704 4.35 .001 *** 3. RCU 42 13.43 4.44 19.704 NCU 42 12.191 4.40 19.334 0.90 ns * * * i-< ̂ rhl'V■ 'sign if icant at p! < . U 01 ns Not Signi ficant at p = .05 Based on these two analyses one might suggest that there would be significant difference in the mean post-test scores of the groups. Hence hypothesis one was rejected. Further statistical test (sea Tab?e;.41) of multiple regression analysis where the dependent ■ • variable - post­ test scores and independent variables were mental ability UNIVERSITY OF IBADAN LIBRARY 183 scores and pre-test scores.Gf the three croups showed significant difference (F (2,123) = 7.29 at P < .01). The results of the multiple regression analysis showed that there w a s a linear correlation between the dependent variable^ post-tost scores, and the independent variables (the mental ability and pre-test scores^. This correlation meant a significant relationship between the post-test scores anu the mental ability and the pre-ucst scores. Though, t h e hypothesis was rejected on the basis of the statistical tests one would want to determine what main effects if any, the treatments had on each group of the treatments and control. Considered for analysis was the unrestricted calculator groups (UCU) - those who used calculator on tests and in instruction that is,(A, 3, C1 . UNIVERSITY OF IBADAN LIBRARY 184 TABLE 41 Multiple Repression Analysis of Post-test scores with mental ability and Pre-test scores (ABC, DEF, GHI Groups) ANALYSIS of SUM OF MEAN F SIGNIF “ VARIANCE DF SQUARES SQUARE RATIO LEVEL FEGRESSI ON 2 . 325.33122 162.66561 7.26745 .01** RESIDUAL 123 2745.52592 22.32135 VARIABLES IN THE EQUATION VARIABLE B BETA STD ERROR F -RATIO SIGNF. LEVEL VAR 01 - MAT 0.19728 0.32833 0.05461 13.052 .01** VAR 03 - I'ET 0.01536 0.21891 0.00638 5.802 .05* (CONSTANT) 0.75831 VARIABLE MULTIPLE f P VAP 01 - MAT 0.25252 0.06377 VAR 03 - PET 0.32540 0.10594 * •:< Significant at p < .01 Significant at o < .05 UNIVERSITY OF IBADAN LIBRARY 185 TA3LE & 2 Analysis of Covariance of Post-test Scores or A, 3, C, Groups of UCU (MAT as covariate) Source df Sum of Mean Squares Squares F- Signif. SS MS ratio Level Covariate HAT 1 180.965 100.965 13.277 0.001 *** MAIN EFFECTS GRP 2 •’24.973 62.489 4.585 0.016* • • EXPLAINED 3 305.944 101.951 7.485 0 .001*** RESIDUAL 38 517.957 13.630 TOTAL 41 823.900 20.095 * * * iii-Lly si-gn.iricant at p < .001 ■* 'Significant a«t p < •. 05 UNIVERSITY OF IBADAN LIBRARY 186 TABLE 43 Analysis of Covarianc e of Post-test Scores of A, B, C Groups of UCU . (Pre-test as covariate} Source df Sum of Mean F- Signif. Squares Squares ratio Level SS MS COVARIATE PRE-TEST 1 3.470 3.470 0.255 0.999 ns MAIN EFFECTS GRP 2 302.392 151.196 1 ,.091 0.001 *** EXPLAINED 3 305.862 101.954 7,479 0.001 *** RESIDUAL 38 518.038 13.633 TOTAL 41 823.900 20.095 *** Highly Significant at P < ,001 rS: not significant at p = .05 From the above table it appeared t h a t there was no significant difference in the mean scores of the groups. Since the mean • post-test scores wsr*e significantly different for the three groups, a One-way Multiple range test using LSC procedure was used to determine the level of significance due to treatment. UNIVERSITY OF IBADAN LIBRARY 157 TABLE 44 Multiple rar.tjc test of Post-test Scores of Groups A, B, C. One-Way LSD Procedure - ANOVA Source df Sum of Mean F r Si gnif. Squares Squares Ratio Prob Level SS MS BETWEEN GROUPS 2 302.4727 151.2353 11.312 .001 * * * WITHIN GROUPS 39 521.4336 13.3701 TOTAL 41 923.9063 ***Hi£tiTy Significant at P < .001 The three groups were re-arranged into group of twcs for t-test. UNIVERSITY OF IBADAN LIBRARY 186 TABLE 4 5 Summary of t-test of the Post-test Scores of tne Groups A, B, C. S i gn if. GROUPS N X SD SD^ t-ratio Level A 14 20.7143 4.9835 23.640 4.22 .001*** 1 . C 14 14.1429 3.0091 9.055 A 14 20.7143 4.9635 24.840 2 . B 14 17.2857 2.4940 6.220 3.01 .01 ** B 14 17.2857 2.4940 6.220 3. C 14 14.1429 3.0091 9.055 3.19 .01 ** ♦ ** HighlySig nificant at d <.001 ** Significant at p <' .01 F rom statistical tests, the post-test mean scores were significantly different for the three groups and hence, hypothesis one was again rejected. The post-test scores of the restricted calculator groups were considered for analysis. The groups D, E, F who used calculator in tests only. UNIVERSITY OF IBADAN LIBRARY 189 TABLE 45 Analysis of Covariance o f Post-test Scores cf Groups C, P•— t F . Source df Sum of Mean F - Signif. Squares Squares Ratio of F COVARIATE PRE-TEST 1 317.210 317.210 25.035 .001 - MAIN EFFECTS GRP 2 30.940 15.470 1 .270 .292 ns EXPLAINED 3 348.150 116.050 9.525 .001 *** RESIDUAL 38 452.990 12.184 TOTAL 41 311.140 19.784 ns : Not significant at P = .05 * * * H ghly Signifies nt at p < .001 Whatever w a s * responsib1e for the main effect not being signific ant at ct = .05 could be explained by the multiple classification analysis. So as to determine the level of significance: the multiple range test - LSD procedure was carried out. In addition to further determine the treatment effects the three groups were rearranged into groups of two and t-tests were done. UNIVERSITY OF IBADAN LIBRARY - 190 TABLE 47 Multiple Classification Analysis of Post-test Scores of Groups D, E, F with Pre-test scores GRAND MEAN = 13.14 AD3USTFC FOR VARIAELE + CATEGORY UNADJUSTED INDEPENDENTS GRP N O E V ’N ETA + COVARIATE DE V ’N RETA 1 14 2.86 1 .36 2 14 0.50 -0.C5 3 14 -3.36 0.53 0.22 MULTIPLE R SQUARED = 0.429 MULTIPLE R = 0.655 = 0.429, indicated that only 42,9% of the variance . the criterion measure of Post-test scores v,as associated with the pre-test scores whereas the remaining 57.1% of the variance might have been due to treatment or some to error. In order to determine the treatment effect, Qne-way ANOVA using LSD procedure of the multiple range test was carried out, a n H the t-tests of the groups. UNIVERSITY OF IBADAN 1 O L VJ o IBRARY 131 TABLE 48 Multiple Range Test_of Post-Test Scores of Groups D.E,F, by Dnc Way ANOVA LSD Procedure dOJ RCE df SUM OF MEANSQUARES SQUARES F F SIGNIF SS MS RATIO PROP LEVEL BETWEEN GROUPS 2 275.5703 137.7852 10.033 .001 WITHIN GROUPS 39 535.5742 13.7327 TOTAL 41 811.1445 TABLE 49 Summary of ''.-tests of the Post-test of Groups D,E, F . GROUPS N X SD SD^ i. -r-a.i i. o SIGft,IF L£VEL D 14 16.000 3.4194 11.5922 1 F 14 9.7857 3.1908 10.1812 4.973 .001 *** D 14 16.000 3.4194 11.6922 1 2 E 14 13,543 -4.3959 19.3239 1.58 ns Lr. 14 13.643 4.3959 19.3239 3 F 14 9.7857 3.1908 10.1812 2.39 .05* ns : Not Significant at P - .05 Highly Significant a t P < .001 * Significant at p < .05 UNIVERSITY OF IBADAN LIBRARY 192 For the '•■roups n. F. F the means of the post-test scores could be considered to be significantly Hi-r Parent. Hence t h e hypothesis was again rejected. The post-test scores of the non-calculator groups were also considered for analysis. The groups b, H and I did not use calculator in tests and instruction and were treated as the control group. TABLE 50 Analysis of the Covariance of the Post-test Scores of Groups G, H, I SOURCE df SUN OF MEAN F-RATIO SIGNIF SQUARES SQUARES OF F SS MS COVARIATE PRE-TEST 1 385.968 385.968 38.813 0.001 *** MAIN EFFECTS 2 30.679 15.333 1.543 0.226 ns GRP EXPLAINED O 416.646 13.862 13.968 0.001 *** RESIDUAL 38 377.827 9,54 TOTAL 41 734.473 13.377 ns : Not Significant at p f .05 * * * H i• ciiy Significant at p < ,001 UNIVERSITY OF IBADAN LIBRARY 193 Whatever was responsible for the main effects not to be significant at Ot = .05 could be explained by the multiple classification analysis. TABLE 5. Multiple Classification Analysis of Post-test Scours of Groups G, H, I. GRAND MEANS = 12.19 VARIABLE ♦ CATEGORY UNADJUSTED AQ1USTED FOR INDEPENDENTS ♦ COVARIATE GRP N DEV’N ETA DEV’N BETA 1 14 c . 45 -0.52 2 14 1.^1 1 .20 3 14 -2.26 -0.70 0.33 0.20 MULTIPLE R SQUARED = 0.524 MULTIPLE R = 0.724 R 2 = 0.524, indicated that 52.4% of the variance in the criterion measure of Post-test scores was associated with the 'pro-test scores (quite large). Whereas the rest 47.6% of the variance might have been due to treatment or some to error. UNIVERSITY OF IBADAN LIBRARY 194 In order to determine the level of significant difference, and if this was due to treatment the multiple range test one-way LSD procedure and t-tests of the post-test mean scores of the groups were carried out. TABLE 52 Multiple Range test of Post-test Scores by One-way ANQVA Procedure of Groups G,H, I SOURCE df SUM OF MEAN F F SIGNF SQUARES SQUARES RAilO PROP LEVEL SS MS BETWEEN GROUPS 2 120.3320 60,1660 3.481 .04 * WITHIN GROUFS 39 674.1445 17.2856 TOTAL 4-1 794.4756 < * Significant at P < .06 UNIVERSITY OF IBADAN LIBRARY 19 5 TABLE 53 Summary of t-test of Post-test Scores of Groups G,H,I GROUPS N X SO SD2 t-ratio SIGNIFLEVEL G 14 12.543 5.0931 25.9397 1 I 14 9.9285 4.1224 15.994 1.55 ns G 14 12.645 .5.0931 25.9397 2 H 14 14.0000 2.9872 8.9234 -0.85 ns H 14 14.000 2.9872 8.9234 3 I 14 9.9286 4.1224 16.994 2.992 .01 ** ns; Not significant at P = .05 ** Significant at P .01 The effects of the covariate as represented by thr pre-jest seemed toi have affected the level of significar.ca of post-test scores as evident in the high cuvariates F-ratio value. Though the multiple range test of LSD producedure on the post-test scores showed significance of F, it was n o t high enough to have obliterated the effects of pre­ test scores. In addition, the t-test was significant for H and I alone of the three groups tested. UNIVERSITY OF IBADAN LIBRARY Since F-ratio was significant as shown in the multiple range test, hypothesis one was again rejected. The results of the analyses carried out using multiple regression analysis, one-way AIM OVA, multiple range test and t - tests of post-test scores (Tables 33 - 41) showed that there were significant differences in the mean scores of the groups, hypothesis one was conclusively rejected. It also demonstrated that (i) the unrestricted calculator group (UCU), those pupils who used calculator in tests and instruction were better in performance than restricted group (RCU), those who used calculator only in the tests as indicated by the comparison of the means of post-test scores analysis and t-test, and (ii), the unrestricted calculator groups (UCU) were also significantly better in performance than the non-ca1culator group (IMCU) as indicated in the post-test scores analysis - t-test, and (iii) there was no significant difference in the performance cf the restricted calculator groups (RCU) as indicated in the p o s t ­ test scores analysis (See Tables 46 - 49), This was further collaborated by the results of each of the groups in UCU, RCU and NCU as indicated on Tables 42 - 53, For UCU, Table 45 indicated that the high mental ability UNIVERSITY OF IBADAN LIBRARY 197 group of this unrestricted calculator group performed significantly better than the low mental ability, as could be expected, and also the same group did better than the average mental ability group, while the average mental ability group cxd also better than the low mental aoility group. For RCU, the high mental ability group of tie restricted calculator group did significantly better than the low mental ability group, There was no significant difference L Lween the performance of. high and average groups but there was a difference in the performance c.p the average and low mental ability groups in favour of average mental ability (Table 49). For NCU, ^he control group, that is, non-calculator group there was no difference in the performances of high and low; high and average except average and low mental ability groups Table 53) 5.20 Hypothesis 2 . There will oe no significant difference in the achieve- » menv scores of those groups of pupils of high, average and low mental abilities. That is: Ho: ■ HC^ = f1 C^ at a 3 .05. UNIVERSITY OF IBADAN LIBRARY 198 TABLE 54 Analysis of Covariance of Post-test Scores of the_ _ groups HMA, AMA and LMA SOURCE SUM OF MEANSQUARES SQUARES F SIGN IF ss MS RATIO OF F COVARIATE MAT 1 195,021 195.821 9.915 0 .002 * » ' MAIN EFEE CTS GRP 2 4G5.539 232.600 11.763 0 .001 *** EXPLAINED 3 661.420 220.473 1 i.<64 0 .001 *** RESIDUAL 122 2403.422 19.749 TOTAL 125 3070.842 24.567 ** Significant at P < . .01 4. * * Highly Significant at P a .001 The mental ability scores could not have had confounding affects on the post-test score since the groups had been randomly selected on the basis of their scores in the mental ability tests. This significant difference *• in the mental qhiiity mean sccres-of the groups was expected. This had bean corroborated the AN OVA table of PI AT , ( i. e b 1 s .55), UNIVERSITY OF IBADAN LIBRARY 139 TAE3LF 55 Analys is of Variance of Mental Ability Scores of the Grojps HMA , AMA, LMA SOURCE df SUM DF MEAN F 9IGNIF SQUARES SQUARES RATIO DF F SS MS! MAIN EFFECTS z 5610.996 2805.438 119.215 0.0u1*** GRP EXPLAINED z 5610.996 2805.498 119.215 0.001*** RESIDUAL 123 2034.574 23.533 TOTAL 125 6505.570 58.045 *** Highly Significant at p < .001 There was high 3 ignifica.it difference on the mental ability scores since the groups were not equalized o , th s basis, However, the effects of the pre-test on the treatment w a s able to demonstrate if actually the significance » difference in the post-test scores was due to experimental treatment or other factors. UNIVERSITY OF IBADAN LIBRARY 200 TABLE 56 Analysis of Covariance of Post-test Scores of the Groups HMA, AHA, LMA SOURCE df SUM OF MEAN F SIGNIF SQUARES SQUARES RATIO F SS SS COVARIATE (PRE-TEST) 1 33.992 33.392 1.689 0.193 ns MAIN EFFEC Q GRP 2 561.739 290.853 14.454 0.001 *** EXPLAINED 3 515-731 205.244 10.199 0.001 *** RESIDUAL 122 2455.11 1 20.124 TOTAL 125 3070.842 24.567 !« * t Highly Significant at p <. .001 ns Net Sign! -r i can t at p = .05 Whatever could have acco untad for the covariates no: to b e signifies nt at o< = .05 might have been due to the equalizing f ac tor of the pre- test on th e groups, as it could be date'- cad on the ANOVA table of Pr e-test. UNIVERSITY OF IBADAN LIBRARY 2Q TABLE 57 Analysis of Variance of the Pre-test Scores of cha Groups HMfi, ANA and ENA SOURCE df SUM OF MEAN F SIGNIF OF F SQUARES SQUARES RATIO S3 MS MAIN EFFECTS GROP 2 11415,445 5707.723 1.147 0.321 ns EXPLAINED 2 11415.500 5757.750 1.147 0.321 ns r e s i d u a l 123 612054.375 4976*051 TOTAL 125 623469.675 4937.758 ns : Not Significant atP~ = .05 The mean pre-test scores of the groups did not show any significant difference at a = .05. This would suggest that too three groups were equalized on the basis of the pre-test scores. Hence any differer.ee on the post-test scores would likely be due to the* treatment given to the groups. To further test for the level or significance so as to ascertain which was due iu treatment or other factors,a post hoc analysis was carried out. UNIVERSITY OF IBADAN LIBRARY 202 If a significant difference did exist which of the groups was better than the other. Tc do this, multiple range test one-way Scheffe procedure and t-tests were carried out. TABLE 55 Multiple Range Test of Post-test scores One-Way Scheffe Procedure (ANOVA) of Groups HMA, AMA, ar. LMA SOURCE df SUM OF MEAN F F SQUARES SQUARES RATIO PR0B SS MS SOURCE GROUPS 2 594.3945 237.4*73 14.776 0 .001*** WITHIN GROUPS 123 2475,,9646 20.1293 TOTAL 125 3070,.8594 ***Highly Significant at p < .Cc . Similarly a multiple regression analysis of Post-test scores, table 59, showed significant difference: (F (2,123) - 28.395 at p < .001) UNIVERSITY OF IBADAN LIBRARY 203 TABLE 5 S Multiple Repression Analysis of Post-test scores with M ental ability and Fre-test scores IADG, BEH, CFI Croups) ANALYSIS OF SUM OF MEAN F SIGNIF. VARIANCE DF SQUARES SQUARE RATI 0 LEVEL REGRESSION 2 720.12596 360.06298 28.398 .001*** RESIDUAL 123 1559.53277 12.67913 VARIABLES IN THE EG LI ATI ON VARIABLE SIGNIFB BETA STD ERROP! F -PATIO LEVEL VAR 01 - MAT 0.12145 0.21836 0.0467’ 6.746 .01 ** VAR 03 - PET 0.54684 0.42681 0.1077; I 25.773 .001** (CONSTAN') 3.41246 VARIABLE MULTIPLE R R2 VAR 01 - MAT 0.41533 0.17254 VAR 03 - PET 0.56204 0.31589 *** Highly significant at p < .001 ** Significant at p < .01 UNIVERSITY OF IBADAN LIBRARY 204 TABLE 63 Summary of t-rest of -he Post-test Scores Scores of Groups HMA, ANA, LMA GROUPS N X SO SD2 t SIGNIFRATIO LEVEL HMA 42 16.4524 5.5703 31.0349 4.9 0 .001*** 1 LMA 42 11.2o57 3.9589 15.6723 HMA 42 16.4524 5.5709 31.0349 2 AM A 42 14.9762 3.6990 13.6826 1 .43 ns ANA 42 14.3762 3.6930 13.6826 3 LMA 42 11.2857 3.9589 15.6723 4.4 0.001 *** NS : Not Significant at p = .05 ***' Highly Significant at p < .001 Tables 54 - 60 showed that the ability groups differed significantly from each other. The analysis of » covariance of Post-test mean scores showed significant difference at p < .001. The treatment effect too was significant as shown by the multiple range test using One-w? ' Scheffe Procedure. Based on the significance difference of the mean scores of the high, average and low UNIVERSITY OF IBADAN LIBRARY 205 mental ability groups t h e hypothesis that there would be no significant difference in their mean post-test scores was rejected. Since the groups were significantly different in their post-tost mean scores,the three groups were compared using t-test, (Table 60 ). , It was found that high mental aoility groups pupils Perf0rmed bette than the low mental ability group. The average mental ability group pupils also did significantly better than the low mental ability groups. However, there seemed to be no difference in the performances of high and average menta ability groups. Though h y p o t h e s i s two was rejected on the basis of the above statistical tests .one would want to determine what main effects, if any, the treatments had on each group of different mental abilities. The post-test scores of high mental ability groups (HMA) from each of the treatment groups UCu, RCU, and control groups !\ICU were tested for signififcant difference. UNIVERSITY OF IBADAN LIBRARY 206 TABLE 61 Analysis of Variance of Post-test Scores of Groups (HMA) One -w-iy Scheffe Procedure SOURCE df SUM OF MEAN F F SQUARES SQUARES RATIO PROB SS MS BETWEEN GROUPS 2 460.3359 230.1680 11.054 0.001 * WITHIN GROUPS 39 612.0742 20.6224 TOTAL 41 1272.4102 **** Highly' Significant at P < .001 Table 5-1 of "Multiple range test showed that there was significant difference in the mean post-test scores of the groups. Further test would show the relative performance of each group. UNIVERSITY OF IBADAN LIBRARY 207 TABLE 62 Su mmery of c- tests of Post-test scores of groups (HMA) (A, D, G) GROUPS !M 51 SO soz T-RATI0 SIGNIF LEVEL A 14 20.7143 4.9635 24.3353 1 G 14 12.64^9 4.0931 25.9387 4.24 -> 001*** A 14 20.7143 4.9835 24.8353 2 D 14 16.000 3.4194 11.6923 2.92 0.01 ** D 14 16.000 3.4194 1 1.6923 3 G 14 12.0429 5.0931 25.3387 2.05 0.05 * *** Highly significant at P *.< =001 ** Significant a4- p,,< .01 Significant at' p < .05 The above t-test table shows that hi«jh mental ability group of unrestricted calculator groups (A1 ; performed » significantly better than the high mental ability groups of restricted calculator groups ;(D)' and ncn-c"-lculator groups (GJ. ■ Similarly the high mental ability group of restri ted groups performed better than the control groups - the non-calculator groups in the Post-test. This would UNIVERSITY OF IBADAN LIBRARY 206 suggest that the treatment was effective on the groups and confirmed the rejection of hypothesis two. The post test scores af Average mental ability groups (A H A ) frcm each of the treatment groups UC1', RCU and control group PJCU ware tested for significant difference. TABLE C 3 Analysis of variance of Post-test Scores of Groups AfiA - One-way Scheffe Procedure SOURCE df SUM OF MEAN F F SQUARES SQUARES RATIO PROP SS MS BETWEEN GROUPS 2 112.9063 56.4531 4.914 o.oll* WITHIN GROUPS 33 44B .0742 11.4891 TOTAL 41 560.9605 * Significant at p < .05. Table 63 of Multiple range test showed that there was significant difference in the mean po-t-test scores, at. p < .01. Further test would show the relative performance of each group. UNIVERSITY OF IBADAN LIBRARY TA3LE G4 Summery of t-tests of Fost of Post-test S curias of groups AM A (B, E, H) GROUPS N X SO sc2 t-rat io SIGNIFLEVEL B 14 17.2857 2.434 6.220 1 H 14 14.000 2.9872 8.9234 3.16 w . 01 * * 0 14 17.2857 2.494 6.220 2 F. 14 1 3.642 w/ 4.3959 19.3240 2.7 0.05* E 14 13.6423 4.3953 19.3240 3 H 14 14.00G0 2.9872 8.9234 -0.25 ns ** Significant at P < .01 * Significant at p < .05 ns Net Significant at o = .05 The abevs t-tests table showed that average mental ability group of unrestricted calculator groups '(B) - performed significantly better than the average mental ability groups of restricted calculated group(E)and the Non-calculator group (H). However, there seemed to be no significant difference in the mean post-test scores of average mental ability groups (c) and (H)].4 UNIVERSITY OF IBADAN LIBRARY 210 » In other words .the restricted calculator - average mental ability group did net perform better than the non-calculator - average mental ability group. The result showed that the treatment was effective or the experimental groups and confirmed the rejection hypothesis two. The post-test scores of low mental ability groups CLMA) from each of the treatment groups UCU, RCU arH control group NCU were tested for significant difference. TABLE u5 Analysis of Variance of Post-test Scores of Groups (LMA) - One-way Scheffe Procedure SOURCE df SUN OF MEAN F F SQUARES SQUARES RATIO PROP SS NS BETWEEN 7 GROUPS 171.5703 85.7052 7.103 .002** WITHIN GROUPS so »471.0039 12.0770 TOTAL 41 642.5742 ** Significant at p < .01 UNIVERSITY OF IBADAN LIBRARY Table 65 of multiple range test showed that there was signi-Tionnt diffarcoce in the mean post-test scores at p < .01. Further t-tasts would show the relative performance of each group. TABLE 66 Summary of t-tests of Post-test Scores of Groups (LMA) C, F . I . GR0UFS N 2X SD SD^ t- SIGNIFratio LEVEL C 14 14.1429 3.0091 3.0547 . 1 I 14 99.9286 4.1224 16.3942 3.1 0 .01 ** C 14 14.1429 3.0091 9.0547 2 F 14 9.7857 3.1906 10.1312 3.72 0.001 * C 14 9.7357 3.1308 10.1812 3 I 14 3.9265 4.1224 16.9942 j. 10 ns *** Highly Significant cat p < .00*1 ** Significant at p _< .01 ns Not Significant at P = .05 UNIVERSITY OF IBADAN LIBRARY 212 The abev/e t-tests tabic showed that Low mental ability group of unrestricted calcu.ator groups ' - C performed significantly hotter than (■> ) Low mental ability group from Non-Calculator - Control group I and (ii) Restricted calculator group F. However, there WuS no significant difference in the mean scores 0 " the low mental ability groups from RCU ai.d NCU. It would therefore, appea that there was no difference in the performance of these two groups. ihough trie result showed that the treatment was relatively effective. Again hypothesis two was rejected. 5.30 Hypothesis 3. There will significant difference in the mean attitude scores of those groups of ouoils who use (i) Cal­ culators in tests and instruction (unrestricted Groups) (UCU), (ii) Calculators in tests only (restricted groups) (RCU), and (iii) Non-calculators at all-control groups (NCU). That is : at a .05 UNIVERSITY OF IBADAN LIBRARY 213 TABLE 67 Analysis oF Covaricoce of tha Pcst-Attitude Scoros of Groups UCU, RCU, NCU SOURCE df SUM OF MEAN F-RATIO SIGN IF SQUARES SQUARE LEVEL SS MS COVARIATE PRE-ATTITUDE 1 2 5 0 2 . 7 2 5 2 5 0 2 .728 1 7 . 0 6 4 0.01 1 *** MAIN EFFECTS GRP 1 3 5 7 . 9 7 2 w 8 3 . 9 8 5 4 . 6 6 3 0 . 0 1 1 * * EXPLr I NED 3 3 5 7 0 , 7 0 3 1 2 9 0 . 2 3 4 8 . 7 9 7 0 . 0 0 1 * * * RESIDUAL 122 1 7 8 9 3 . 3 3 1 14 .6 71 TOTAL 125 2 1 7 5 4 , 5 9 4 17 4. 11 * * * H i g h l y " S i g n i f i c a nt at ,p <. .001 * Significant at p.< .05 UNIVERSITY OF IBADAN LIBRARY 214 TABLE 63 Multiple Classification Analysis of Post-Attitude Scores of Groups UCU, RCU and NCU with Pre-Attitude Covariate GRAND MEAN = 84.04 VARIABLE + CATEGORY UNADJUSTED ADJUSTEn FOR INDEPENDENTS + COVARIATES GRP N DEV’N ' ETA DEV’N BETA 1 42 3.53 4.72 2 42 -2.94 -2.54 3 42 -0.59 -2.10 0.20 0.25 MULTIPLE R SOU’.RED = 0.178 MULTIPLE R = 0.422 2 R = 0.178 - , indicated that 17.8% of the variance in the criterion measure of Post-attitude scores* was associated with the Pre-attitude scores whereas the rest 82.2% of the variance might have been due to treatment or to some error. UNIVERSITY OF IBADAN LIBRARY 215 TABLE 59 Analysis of Variance cf Pre-attitude Scores of Groups UCU, RCU and UCU SOURCE df SUM OF MEAN SIGNIF SQUARES SQUARES F F SS MS RATIO OF F MAIM EFFECTS GRP 2 831.048 415.524 0.424 0.999 ns EXPLAINED 2 831.063 514.531 0.424 0.999 ns RESIDUAL 123 120632.0u3 981.155 TOTAL 125 121513.125 972.105 ns : Mot Significant at cx. = .05 Table 69 showed that there was no significant difference in pre-attitude scores cf the three groups. This would mean that the groups were equalized on their attitudes before the treatment. Any other variance that would have accounted for the difference, if any, in the post-attitude score -oof the groups might have been due to the treatment. Since the covariate, main effects, and explained were significant (See Table 67 ) it would be necessary to determine (i) the level of significance of the means of the UNIVERSITY OF IBADAN LIBRARY 216 postattitudes scores and (ii) the level of significance of the attitudinal charge between Pro-attitudes and Post­ attitudes mean scores. The one-way multiple range test using student - Newman Keuls (SNK) procedure was used to determine the level of significance due to treatment or otherwise while multiple regression analysis was used to determine the level of significance of ti.o attitudinal change. TABLE 70 Multiple range Test of Post-attitude Scores of Groups UCU, RCU, NCU by Student - Newman - Keuls Procedure (SNK) ONE-WAY ANOVA SOURCE df SUM IF MEAN F-RATIO F-PROB SQUARES SQUARES SS MS BETWEEN GROUPS 2 47'. 3750 235.6875 1.217 0.299 ns WITHIN GROUPS I 22 2381'.. 000 193.5935 TOTAL 125 2420;.3750 t NS = Net Significant at p = ,05 The table showed that the means of the groups were not significantly different at p = .05. This would mean that there was no difference in the mean attitude snores of the UNIVERSITY OF IBADAN LIBRARY 217 TABLE 71 ilu 11ipIn Regression Analysis of Post-attitude scores vit'h "Pre-attitude scores (ABC, Dt:F, GH I Groups) — ANALYSES OF OF SUN OF MEAN F SIGNIF.VARIANCE SQUARES SQUARE RATIO LEVEL ' REGRESSION 1 2502.77415 2502.77415 15.11170 .001*** RFSIDUAL 124 19262.02744 155.33893 VARIABLES IN THE EQUATION SIGNIG. VARIABLE B BETA ! STD ERROR F-RATIO LEVEL VAR 02 - PEA 0.39182 0.33910 0.09752 16.112 (CONSTANT) 52.18706 . . . . VARIABLE MULTIPLE R R2 VAR 02 - PEA 0.33910 0.11499 * * !*■ Highly significant at p < .DQ1 UNIVERSITY OF IBADAN LIBRARY * |I •* .105). Since F-value was not significant,it implied tha~ the means of MAS and CAS were not different significantly. Hence, there would not be any relationship between the MAS and CAS. From t-tests table, it w a s, possible to ascertain the effects of the treatment on the groups. The t-tests of the groups were all significantly different which would imply that individual groups would have different attitudes to mathematics and to calculators. That is, for the groups there w a s some relationship between the MAS and CAS. i 5.6 Hypothesis 6 There will toe noe significant relationship in pupils' achievement and attitudes toward mathematics ano calculators at ex = .05. UNIVERSITY OF IBADAN LIBRARY 1 TABLE 90 Correlation Coefficients of Post-tests with POA, HAS and CAS P 0 A M A S C A S GROUPS N R SIGNIF SIGNIF r SIGNIFLEVEL r LEVEL LEVEL UCU 42 0.174 ns 0.419 0 .01** -0.14 Ho RCU 42 0.05 ns 0.17 ns -0.06 ns NCU 42 0.26 ns 0.38 0.05 0.05 ns HMA 42 0.14 ns 0.21 ns -0.01 ns AMA 42 0.28 ns 0.24 ns 0.21 ns LMA 42 0.04 ns 0.195 ns -0.11 ns ** Significant at p .01 * Significant at p. < .05 ns Net Significant at p dz .05 The table 91 showed the intercorrel^tion coefficients of the seven variables:fn = 126, r = 0.308, p < 0.001 for P0T/MAS), (n = 126, r = -0.05 not significant at P = .05. for P0T/CAS), (n = 126, r •= 0.194 significant at P < .05 for P0T/P0A). UNIVERSITY OF IBADAN LIBRARY TABLE S1 Correlation Coreffininnts of the variables for all the groups IN = 126) r ~ MAT PEA FET POA POT MAS CAS VAR 01 VAR 02 VAR 03 VAR 04 VAR 05 VAR 06 VAR 07 VAR 01 - -0-06684 0.46162 0.12407 0.41538 0.26162 -0. ,2098 VAR 02 - - -0.05212 0.33910 -0.06909 0.22119 0.24045 VAR 03 - - - 0.14665 0.52761 0.37356 -0.06875 VAR 04 - - - - 0.12756 0.45257 0.83539 VAR 05 - - - - - 0.30747 -0.04551 VAR 06 - - - - - - -0.11176 VAR 07 - - - - - - UNIVERSITY OF IBADAN LIBRARY 244 The two tables showed that most of the groups did not have any significant relationship between achievement scores and attitude scores at p < .05. Though correlations did exist in seme group (MAS) especially UCU ana NLU, t h i s would not be adequate to generalize the relationship. However, the general pattern was that low negative correlation or very nearly zero correlation Occurredin calculator attitude and achievement. The table of correlation coefficients showed that there were significant relationships between the groups’ (i) Pre- test and Post? test (ii) Pre -attitudes and Post-attitudes (i) N * 126 POT: r = 0.528, p 0.001 (ii) N = 126 POA: r = 0.339, p < . 0.001 Any differences that would have occured in the Post­ attitude and Post-test scores could have been due to treatment. The main effects of Post-attitude and Post­ test scores have been found to be significantly different (see Tables 29, 54, 67 and 76). Since both we‘re significantly different, they would appear to have some relationship. To determine the level of significance of the relationship between post-test scores and post-attitude UNIVERSITY OF IBADAN LIBRARY - 245 TABLE 92 Multiple Regression Analysis of Post-attitude scores with Post-test scores for all the groups (.PDA with POT ANALYSIS OF SUM OF MEAN F SI GNIF. VARIANCE DF SQUARES SJ UARE RATIO LEVEI RE GRES SI ON 1 91 1.65634 911.65634 4 . o3r06 . u5* RESIDUAL 124 233" 1.^4525 188.46101 VARIABLES IN THE EQUATION 31 GNIF. VARIABLES B BETA STD ERROR F-RATIO LEVEL VAR 05 - POT 0.54486 0.19376 0.24774 4 .u37 . u5* (CONSTANT) 75.70253 VARIABLE MULTIPLE R VAR 05 - POT 0.19376 0.03754 * Significant at p<,05 UNIVERSITY OF IBADAN LIBRARY 246 TABLE S3 Multiple Regression Analysis of Post-test scores fTET calculator atTTiTEude scores and -'athematics attitude score's for all the groups ANALYSIS of SUM OF MEAN F SIGNIF. VARI AiNICE OF SJuARES SO UARE RATIO UEVEL RE GRES SI ON 2 275.10200 137.55101 6.u515 S .ul** REST DEAL 123 2795.75514 22.7 2 972 — VARIABLES IN THE EOUATION ox lx r. VARIABLE B BETA STD ERROR F -r ATI 0 lE\£L VAR 07 - CAS 0.03025 0.u7246 0.03615 6.700 .u1** VAR 06 - MAS 0 .s02S0 0.2S362 0. _»5374 11 .097 .u01*** (.CONSTANT) 3.j 14 16 VARIABLE MULTIPLE R R VAR 07 - CAS 0.03908 0.00153 VAR 06 - MAS 0.29931 0.06 958 * * * Highly significant at p < .001 * * Significant at p < .01 UNIVERSITY OF IBADAN LIBRARY - 247 - scores of the groups, a multiple regression analysis of Post-attitudes as dependent variable and post-test as independent variable was done. The table showed that there was significant difference in the post-atLitude and post test scores of the groups (See Table 92) (F (1 ,124 = 4.84, P < .05). In order to ascertain that this significant difference was not due to chance or error, a multiple regression analysis of Post-test scores as dependent variable, anc calculator attitude scores and mathematics attitude scores as independent variables were carried out. (Table 93) showed that there was a significant difference in the Post­ test scores, and CAS and HAS; F (2.123} = 6.052, P < .001), The lwo regression analyses showed that there was significant relationship between Post-test scores and post attitude scores of all the groups. On the basis of the analysis, the hypothesis v s rejected. That is, there w a s a significant relationship between the post-test scores and the post-attitude scores. In addition, it was ; found that there was a linear correlation between post-test scores and mathematics attitude scores and calculator attitude scores. UNIVERSITY OF IBADAN LIBRARY 248 CHAPTER SIX DISCUSSION 6.1 Relationship of Results to hypotheses end Previous Empirical Studies The results of this study would be discussed in relation to the hypotheses tested. The purpose of this study was to investigate the effects of the use of hand held electronic calculators on outcomes in mathematics instruction. The learning outcomes investigated were primarily achievement and attitudes. The criterion measures of achievement were pre- post-tests scores, and for attitudes were pre- post-attitude scores. 6.2 Performance Treatments and achievement: The hypotheses which dealt with 'achievement were one and two: I. Hypothesis one; There will be rto> significant difference in the mean achievement scores of pupils who use (i) calculators in instruction and tests (UCU) (ii) calcuators in test only (restricted groups -(RCU) and C i i i) no calculators at all - control groups,(NCU. This hypothesis was rejected because significant difference was found in the mean post-test scores of the groups (F (2, 123) = 16.234, fj < .001). Also the groups UNIVERSITY OF IBADAN LIBRARY - 2 4 9 - that used calculators throughout (UCU) were significantly better than no calculator group (NCU) ( t (02) = 5.35, P < .001) and also UCU groups were significantly better than the calculator on the test only groups (RCU)? (t (82) » 4.35, P < .001). lo w e v e r , no significant difference was found between the groups RCU and NCU, (t (82) = 0.98, P > .05) which showed that ineither of t h e s e two g r o u p s w a s b e t t e r . He nce, Similar finaings had been obtained by Gaslin who compared the achievement and attitudes of high school pupils. Pupils in treatment groups and E ̂ were allowed to use calculators on post-tests and retention tests and no calculator or C control group. Significant treatment effects were found on both post-test achievement measures with E^ > E.j > C and the retention test with E_ > E. = C; however, no significant differences on I attitudes were found. These finaings were also collaborated by Finehman .. However, Quinn 13 and Hutton 3 did not find any significant differences between calculator and non calculator groups on the achiovement variable. Other 5-tudies which support the findings of this study are Q those of Andersen who was interested in the effects of restricted versus unrestricted use of calculators in mathematics UNIVERSITY OF IBADAN LIBRARY -250- achievement and attitudes - the .‘unrestricted groups, - the restricted groups and £ no calculators groups. The pupils were pre-post-tested, using ANCOVA as the principal analysis procedure it was found that E^= E2 ̂ 0 on achievement and attitudes. Recent studies by Hedren found classes (in Sweden) using calculators, whenever they could be of use, were as competent as control classes on mental arthemetic and calculations with simple algorithms, and had better understadding of numbers and problem-solving (ages 10-12). This was further supported by Mellon 111 and Kelly 112 whose studies found calculators to be effective. Kelly 112 found that calculator enhanced the use of deductive reasoning, ability to explain strategies in retropect,in retropect and implementation of strategies to increase understanding of problems. He also found that the specific processes to promote effective solution and ability to evaluate were aided by the use of calculators. 110 Hedren, Rolf: The Hand-held Calculator, at the Intermediate Level. Educational Studies in Mathematics Vol. 16, 11. 163-179, May 1985 111 Mellon, .Joam: Calculator Based Units in Demands and percents for 7th Grade Students. Unpublished Ph.D. Thesis: Columbia University Teachers College: DAI 46A, Sept. 1985. 112 Kelly, M.G. The effect cf the use of the Hand-held calculator on the development of problem­ solving strategies: Utah State University, 1984: DAI, 45A 3571, June. 1985. UNIVERSITY OF IBADAN LIBRARY - 251 - Padberg 113 , using calculators to discjver simple theorem in an example from number theory, concluded that "calculator enables one to generate a sufficient number of examples that one can easily get through to conjectures or theo-ams”. All these pointed out that calculators are not only used as computational device but can also be used in concept formation a problem-solving techniques. >In his work, Cheung 114 d e s c r i b e d how a scientific calculator can be used to introduce the method of successive substitutions in generating approximate solutions to a number of equations chat can be expressed in the form y = f(x) including trigonometric equations such as y = sin (cos (tan x)). So far, .he empirical studies reviewed and the results of this study on achievement shewed that most calculator groups performed better than or equalised other groups; and in no case was the use of calculator had . debilitating effect at secondary school level. 113. Padberg, F.F.: Using calculators to Discover simple Theorems - An examples from number Theory Arithemetic Teacher, NCTN, Vol. 28, No. 8 , 1981, pp 114 Cheung, Y.L.: Using scientific calculators to demonstrate the method of successive substitution. flat hematics Teacher. Vol.79 No. 1, Jan. 198t>7 pg. 15-17. UNIVERSITY OF IBADAN LIBRARY 252 The results of the groups (UCU) unrestricted calculator groups showed that there was significant difference in the mean post-test scores of A » B and C groups: CF (2,39) 11.312, p -n. .001). The t-tests showed that high mental ability group performed significantly better than average and low mental ability groups (Table 45 ) This showed that high mental ability pupils that used calculator performed better' than other calculator groups of the average or low mental ability groups. Here, the use of calculator is an added advantage in the instructional process and tests. The results of groups RCU - the restricted calculator groups showed that there was significant difference in the mean post-test scores of D, E, and F groups. (’• F (2,39) = 10.033, p < .001)* The t-tests also showed that the high mental ability group of the RCU group that used calculators on the tests only performed significantly better than others in the group. Though there was no significant difference in mean scores of high and average mental ability groups, yet the average mental ability group performed better than low mental ability group (Table 40 ) • Since the main effects was not significant (Table 46 and 5C ), it would appear that the effect of using the UNIVERSITY OF IBADAN LIBRARY 253 calculator on the tests alone would not be as effective as incorporating the use of calculators in the instructional process and tests. The non-significant difference of the main effects could be attributed to either the treatment or erro-F and the - R2 = 0.429 ' * indicated that 42.9% of the variance in the post-test scores was associated with the pre-test scores. This might have been quite high to have provided some effects on the post-test scores. The. comparison of the mean post-test scores showed significant difference/' t "h e means were significantly differnt as a result of the treatment. This was confirmed by the t-tescs (see table 49 ). The results of non-calculator NCU groups showed significant difference in the mean post-test scores (see table 50 )• Also the main effect, was not significant. 2 The multiple classification table showed that R - 0.524, indicating that 52.4% of the variance in the criterion measure of post-test scores was associated with pre-test scores. This is quite large, whereas the rest 47.6% of the » variance might have been due to treatment or some to error. The treatment might have not nullified the effects of the pre-test as to make the main effects to be non-significant. UNIVERSITY OF IBADAN LIBRARY -25 4 - The means of the post-test scores we e then compared by multiple range test so as to determine their significance: (F(2,39) = 3.48, P < .05). The means were found to be significantly different. For the control group, m e r e was no difference in means of the high and low menta 1 ability groupo, high and average mental ability groups except for the average and low mental ability groups (See Table 53). This finding is supported by those of Hembree and Dessart 115 that the use of calculators in testing pjradvo&f much higher achievement scores than paper and pencil efforts, both in working exsrcises and in problem-solving. They went further to show that this applied to all grade and ability levels. In particular, it applies to low and high ability pupils in problem solving". The better problem-solving performance is a result of improved computation and process selection. However, some other results were contrary especially the findings of the United States National Assessment of Educa­ tional progress (NAEP) which have discovered some area of mathematics where pupils who did not use calculators 115. Hembree, Ray and Dessart, D.J. Effects of Hand-Held Calculators in Pre-College Mathematics Education: A mental-analysis. "Journal for Research in Mathematics Education, 17, March, 1986:'83-99. UNIVERSITY OF IBADAN LIBRARY -255- fared better that pupils equipped with calculators (Driscoll) lU. Indeed, when pupils lacked the understanding of a concept, the use of calculators offered no advantage (Driscoll) 117 The t-tests (Table 40) showed that no significant difference was found between restricted calculator groups and non-calculator groups which implied that the use of calculators only on tests would not result in higher achievement in mathematics. Many researchers in the calculator field have advanced reasons that such pupxls might i'icrb nave confidence in the results displayed on the calculator screen due to non-continuous practice with calculator (Carpenter et. a l . ) ^ ' ^ 1 . Other related studies that supported the findings on achievement are those of Murphy 118 who round that students with unrestricted use of calculator achieved higher problem-solving scores than students not using calculators for i n s t r u c t i o n * 116. Driscoll, M.3. Research within Reach: Elementary School Mathematics, Raston, .Va: National Council of Teachers of Mathematics^ 1 £)01. pp 117. _________ _____ Research within Reach: Secondary School Mathematics. Reston Va: National Council of Teachers of Mathematics, 1982, ppp 118 Murphy, N.K., The effects of a Calculator Treatment on Achievement and attitude Towards Problem- solving in 7th Grade Mathematics. (Doctoral Dissertation, University of Denver, 1981) DAI 42A, 20Q8-2009, 1981. UNIVERSITY OF IBADAN LIBRARY - 2 5 6 - or tests, However, Rule 119 found no significant difference between groups which used or did not use calculators for a unit on functions. While his findings on computational benefit of calculators was in comformity with the findings if this study since all the groups had equal time to complete instructions and tests. The time was not incorporated as a criterion measure in this study, but other studies have found that using calcu­ lators during instruction and tests produced a significant improvement in less time than without calculators (Stewart)1 19 Hence, further studies in Nigeria night incorporate time variable as a criterion measure. II. Hypothesis ~wo: There w i 11 be no significant difference in the mean post-test scores of high, average and low mental ability groups. This hypothesis was rejected because there was significant difference in the mean post-test scores of the groups: (F (2,123) = 14.776, p < .001). The ability » groups differed significantly on their post-test scores as the high mental ability groups (HMA) performed 119 Stewart, J.T, See Suydam, M.N. In Research in Mathematics Reported in 1981, 3RME, 1981. UNIVERSITY OF IBADAN LIBRARY -257 - significantly better lower mental ability groups: (t (62) = 4.9, P < .001), and the average mental ability groups also did bettwe than the low ability groups: (t (62) = 4.4. P < .001). However, it appeared there was no significant diffehence in the performance df high and average mental ability groups (t (82) = 0.98, P > .05). The 0 findings are consistent with other similar studies. Zepp examined whether there was an interaction between the -use of calculator and different ability levels high, medium and low in secondary school pupils’ solutions to proportion problems and found no differences. This was collaborated by Bolesky , whereas, Fischman0 and Laursen0 found significant differences in achievement scores of students in different ability groups . 1 Brassell, et. al. 120 found positive correlation between mathematics achievement and ability groups. The result of the high mental ability groups showed that there was significant difference in the mean pose-test scores of groups A, D and G. (F(2.3S) = 11.054, p < .001). * 120. Brassell, et. al. Ability Grouping , Mathematics Achievement and Pupils Attitudes toward Mathematics. JRME, Vol.11, No.1. Jan. 1980, UNIVERSITY OF IBADAN LIBRARY - 2 5 8 - Table 62 - 66 showed that the high mental ability group that used calculators on instruction and test per­ formed significantly better than high mental ability group that used calculators in teste only and that restricted calculator groups high mental ability performed significahtly better than the high mental ability group of non-calculator group. The pattern also showed that the unrestricted calculator groups in A, B and C performed better than groups in 0, E and F and d » H, and I. This pattern of performances was irrespective of whether the groups were high, average or low mental ability levels (HMA > AMA > LMA), However, there were few deviations in the pattern. For example, with the ANA groups (B,E,H) there was no significant difference between E (the restricted group) and H (Non-calculator group) performance on their post-test scores: (t (26) = -0.25, P > .05). Nichols 22 found among students using calculators in College basic mathematics that those having higher aptitudes in mathematics showed significantly higher attitude scores than students having lower aptitudes. This supports the finding of this study. Similarly in the low mental ability groups (C,F.I.) there was n. significant difference in theperformance of UNIVERSITY OF IBADAN LIBRARY -25 9 - the restricted calculator gro'ip (T) and the non-calculator group (I) (t ̂ 2 6 ) = -0.10, P > .05). Based on these results, one is inclined to say that tho unrestricted calculator groups whether in the high mental ability levels (HMA), the average mental ability level (AMA) or the low mental ability level (LMA) seemed to have performed better than other treatment groups irrespective of mental ability levels. That is: A performed better than 0 or G B performed better than E or H C performed better than C or I 121 The finding of this study is supported by Lenhard . In a variety of analyses using t-tests and ANOVA he found that the higher ability pupils made fewer concept and computation eri'ors than the lower ability pupils. In addition the higher ability pupils performed significantly better than the low mental ability pupils in mathematics achievement tests. » 121. Lenhard, R.W. Hand-held Calculators in the Mathematics Classroom as Stuart Public School, Stuart Nebraska (Doctoral Dissertation, Montana State University, 1976). Dissertation Abstract International, W 7 7 1 7 A ~ m ----------------- UNIVERSITY OF IBADAN LIBRARY -260 - On the contrary Kasnic *122 using a 2-factor ANOVA with pre-test ability as a blocking variable found that there were no differences between calculator groups andcontrol groups, nor were any differences found for the different ability levels between calculator groups and the control groups. However, Miller 123 who examined whether calculators would be effective instructional aids in developing tne concept and skill of lcng-division obtained similar results. When separate ANOVA analyses for low and high ability groups were done, the results showed that the performances of calculator groups of low mental ability groups were significantly better than the Non-calculator groups of low mental ability. 'The high mental ability of restricted and unrestricted calculator groups performed .equally as the high mental ability of the control groups on achievement and attitudes. The finding in this study was- also supported by that of Lawson 124 who found 122. Kasnic, M.J.: The effect of using hand-held calculators on Mathematical Problem-solving ability anong six grade students. (Doctoral Dissertation, Oklahoma State University, 1977) DAI, 1978, J8A, 5311. 123. Miller, D.F. Effectiveness of Using, Minicalculators as an Instructional Aid in Developing thw Concept and Skill of Long Division... (Doctoral Disserta­ tion, Florida State University 1977), DAI, 1977, 37A, 6327. 124. Laws' n, K.W. Use of Calculators in High School General Mathematics... (Doctoral Dissertation, Brigham Young University, 1978) DAI, 39A. UNIVERSITY OF IBADAN LIBRARY that the use of calculators die not affect performance in estimation but students of lowest ability made the most errors with the calculators compared with other •’hility levels: higner or average. This finding in collaboration with -fche ones in this study implied that ability levels of the students is an important factor as far as mathematics achievement ty calculator is concerned. 6 .3 Performance Tyeatments and Attitudes Hypothesis Th. ee There will be no significant difference in the mean attitude towards mathematics and calculator scores of those groups of pup Is who use (i) ca -̂cu • a cars in tests and instruction - the Unrestricted groups U'OU), (ii) Calculators in tests onLy - restricted groups - RCU. and (iii) : i Non-calculator - control groups NCU. The table of analysis of cr.varia.ice of the post - attitude . scores with pre-attitude scores as covariate showed * that the groups’ covariates were significant: (F(l,24) = 17.OCT, P < .001] . Similarly the main effect was also significant: (F(2,123) * 4.663, P < .01) and the explained variance was also signi­ ficant: (F(3,122) = 6.797, P < .001). But the comparison of the means of the post-attitudes scores using multiple range test, (SNK) student that is, student Newman UNIVERSITY OF IBADAN LIBRARY 262 Kaul test showed the means were not significantly different: 2(2,123) = 1.217 p = 0.299, i.e. P > .j S) This was also confirmed by the statistical test of straight analysis of variance of the post-attitude scores (See Appendix 20). But the pre- and post-attitude scores of the groups were found to be significantly different as was shown by multiple regression analysis. (Table 71 ). The t-1cs15 revealed that t h e hypothesis was not rejected f o r groups UCU and NCU-j RCU and NCUj but significant difference did exist between UCU and RCU. Hence t h e hypothesis was rejected for the two groups. This would imply that there was a significant difference in the means of the post attitude scores of the unrestricted calculator groups and the restricted calculator groups. Over-all, there w a s attitudinal change among tbe groups as revealed by the regression table. Conclusively , hypothesis throe cannot be rejected in its entirety. This hypothesis was not rejected though significant difference was found in the mean pre- and post-attitudes scores cf the groups by regression analysis: (F (1,124) = 16.1117, P < < .001) The significant difference obtained in the pre- and post-attitude scores implied that there was attitudinal UNIVERSITY OF IBADAN LIBRARY 263 change among the groups. Generally the groups had a change of attitude towards mathematics and calculator. However, there was no significant difference in the mean post-atti-- tude scores of the groups as shown by the multiple range test of the treatment groups and control group: (F(2,123) = 1.217, P > .05). The non-significance of the attitude scores at this level might have been caused by treatment effects or va r­ iances due tq other extraneous variables or error. However, to determine the level of treatment effects, the results of t-tests were discussed. It was found that there was no significant difference in the post-attitude scores of the unrestricted calculator groups and non-calculator group (E1 = E 3):(t(82) = 1.33, P > .05) and (t(82) = 0.97, P > .05) (^2 = (restricted calculator group and control group). However, significant difference did exist between the post­ attitude scores of unrestricted calculator and restricted calculator groups: C t (8 2) = 2.22, P < .05) (E-̂ > £2 ). The findings on attitudes is in conformity with, other related studies Hutton8,. Boling8, Whitaker12 and Q Bolesky where no differences were found between calculator groups and non-calculator groups. Similarly (Gaslin and 0 Vaughn) found E = C, (experimental = control), and this was Q contrary to 'findings by Fischman, Zepp and Andersen . UNIVERSITY OF IBADAN LIBRARY - 264 - Koop 12• 5 and Me* llon 111 -Found significant differences between calculator and non-calculator groups. In Bulletin No. 9 of the Calculator Information Center^ , of the seven findings reported on attitudes toward mathematics in calculator studies, six of the findings produced non­ significant differences, (which agreed with the results of this study) whereas one group did produce a significant difference in favour of calculator based instruction. In this study, the results indicated that calculator instructed pupils did at least as well as, if not better than the non-calculator instructed pupils on attitudes. In addition, both the cooperating teacher and the investigator found that teaching with calculators was much less onerous than teaching without calculators. The results on Table 72 showed that there was no significant difference in the mean attitude scores of those pupils who used calculator throughout - the unrestricted groups despite differences in mental ability levels. Similarly there was no significant difference in the mean attitude scores of the restricted calculator t groups despite differences in their mental ability levels. For the groups G, H and I of the non-calculator groups, there were no significant differences between G and I, G and I was found to be significantly different at. p = .05 by one-tailed test. 125 Koop, J.8. Calculator use in the Corrmunity College arithmetic course, 3RME, 13(1) 1982, 50 - 60. UNIVERSITY OF IBADAN LIBRARY 265 t(26) = 1.93 for one-tailed test, t-ratio is significant. This finding is supported by Gaslin 8 , Hutton 8 who found no differences on attitudes of the groups. Contrary to this finding is that of Lenhard 8 , Zepp 8 , Fisch .an 8 who found differences on the attitudes of the different groups. Recent studies by Hembree and Dessart 11 5 found that pupils using calculators had better attitude towards mathematics than pupils not using calculator and their findings applied across all grades and ability levels. Some other studies involving younger populations according to Koop 125 have reported a more positi.ve attitude towards mathematics when students were allowed to use Q calculator. However, Dyce and Gooden found no significant changes in student attitudes. Host of the studies reviewed did not discuss the relationship between the pre- and post­ attitudes they only reported differences or none in the post-attitude scores of groups. However, what is most important is to recognize from the findings of various studies whether there are differences in attitudes scores or not i g when calculator was used. The studies of Anderson and Ayers 25 actually pin-pointed these difference where calculator groups were described as having higher attitude scores then non-calculator groups. A UNIVERSITY OF IBADAN LIBRARY 266 Hypothesis Four There will be no significant difference in the mean post attitude towards mathematics and calculator scores of those groups of pupils of low, average and high mental abilities. The table of analysis of covariance of the p o s t ­ attitude scores ith pre-attitude scores as covariate showed that the groups' covariates were significant: (F(1,124) = 16,810, p <',001). Similarly the main effects was also significant: (F(2,123) = 3.688, p < .05) and :he c x p l m n e u variance was also significant (F(3,122) = 8,062, p c .001). But the comparison of the means of the post -atti tudes icores using multiple-range test-SNK (student Newman Keul test) showed that the groups means were not significantly different: (F(2,123) = 2,417, p =.091, ns). Thic was also confirmed by the statistical test of straight analysis of variance of the post-attitude scores (see Appendix 20). UNIVERSITY OF IBADAN LIBRARY 267 The pre- and post-attitude scores of the groups were found to be significantly different as was shown by multiple regression analysis, (Table 71). This would imply that there had been attitudinal change among the groups as revealed by the regression table. The t-tests also revealed tu>at the null hypothesis was rejected among HMA and LMA because significant difference did exist between the means of the post-attitude scores of the gro.ps. However, the null hypothesis was not rejected between HMA and AMA? AMA and LMA because no significant differences were found in tfv° means of the post-attitude scores of the groups. Conseo* ently hypothesis four cannot be rejected in its entirety. This hypothesis was not rejected though significant difference was found in the means of the post-attitude scores of the groups (Tables 76). It was also found that the high mental ability groups had better attitudes than low mental ability groups: (t (62) = 2.07, P < .05). No significant differences were found between (i) HMA and AMA: (t (82) = 0.2S, P >.05) and (ii) AMA and LMA: (t(82) = 1.72, P > .05). Since the groups HMA and LMA showed some significant differences in their post-attitude scores, the high mental ability groups seamed to have better attitude towaids UNIVERSITY OF IBADAN LIBRARY 268 mathematics and calculators. Attitude toward mathematics has been found directly related to aspired school grades and ability levels ( Spickerman ) . Considering the t-tests for the within groups of high mental ability. A, D and Gj none of the post-attitude scores was significant except for D and G which was t(26) = -1.99 significant at P = .05 (one tailed test). The high mental ability group of the non-calculator group had better attitude than the high mental ability group of the restricted calculator group. In the average mental ability groups of the unrestricted calculator group had better attitudes than the restricted calculator group. There were no significant differences in the mean attitude scores of the other groups. For the low ability groups there were no significant differences in the mean attitude scores of the treatment groups anrl control group. However, the results did indicate that those groups of unrestricted calculator had better attitudes in respect of ability levels. This finding is supported by those of Ayers 25 and- AnderseiT 0 who found that 126 Spickerman, W.R.A.: A study of the relationships between attitudes toward mathematics and some selected pupil characteristics in a Kentucky High School (Doctoral dissertation. University of Kentucky, (1965) DAI, 1970, 30, 2733A. UNIVERSITY OF IBADAN LIBRARY 26 S attitudes improved when calculators were used without g restrictions. 3ut othe^ findings like those of Elliot , found no significant difference tetween groups using calculators or paper and pencil on problem-solving. Connor found that there was attitudinal difference between calculator and non-calculator groups. Futherman 127 findings on attitudes support those of this study which found ability to have played an important causal role in the attitudinal process . Other studies not necess­ arily in the use of calculator but media studies have found similar -results. The results of these research works would be discussed later in this report. Hypothesis -Pive There will be no significant relationship in the groups attitudes toward mathematics (MAS) and attitudes toward calculator (CAS). The t-tests (Tatle 07) showed that there were signi­ ficant differences between mathematics attitude scores and calculator attitudes scores in all the groups. This would imply that some relationship did exist between the MAS 127 Futherman, Robert: A causal analysis of expectancies and values concerning mathematics (the University of Michigan, 1S0O), DAI, 341B, 3620, 1981. UNIVERSITY OF IBADAN LIBRARY 270 and CAS. When this relationship was determined using Pearson correlations approach.it was found that r, correlation coeffir.icrrjts were not significant at p = .05 for all the groups and the F-value: (r(1,124) = 1.57 also was not significant at p = .05). This hypothesis was not rejected, that is, there was no significant relationship between the MAS and CAS. If any relationship did exist it could be said t h a t it was not significant at P = .05. Hence any change in attitude possibly towards mathematics might not necessarily mean change of attitude towards calculator. Calculator was used for six weeks by the treatment groups and some of the pupils had not been exposed to instruction where calculators were used. Q Robert found evidence., of calculators influencing immediate and specific attituainal perceptions, but evidence supporting more general and lasting changes of attitudes Was not available. VI VI Hypothesis 6- » There will be no significant relationship in pupils' achievement and attitudes towards mathematics and calculator. The results showed that:(N = ̂2 6 , r = 0.303, p < .001) for post-test scores against mathematics attitude scores, UNIVERSITY OF IBADAN LIBRARY 271 i\! = 123, r = 0.05. riot significant at P = .05 for post­ test scores against mathematics scores and N = 126, r - 0.194, P < .05 for pose-test scores against post­ attitudes . There was a significant relationship between po^t tests scores and post-attitude scores. When multiple regression analysis of post-test as dependent variable and post-attitudes as independent variable was done, the following result 'as obtained: (F(l,124) = 4,04, P < .05,. Similarly with post-test scores as dependent variable, and MAS and CAS as independent varieties the result of the multiple regression analysis also showed: (F(2,123) = 6.052, P ' .C0i), Botn results showed some, significant differences hence some correlations existed a n d hypothesis 6 was rejected. That is, there was significant relationship between post-test scores and post-attitudes scores. This finding is supported by Quinn’s study who investigated the causal relationship between mathematics achievement and attit»udes, and found some significant correlations between mathematics attitudes and achievement at grades 3 and 5. Similarly Gordon ® 120 Gordon, D.W. A profile of High and Low Achievers in Mathematics (Doctoral dissertation, Duke University 1971) DAI, 4639 - 4640, Feb. 1970. UNIVERSITY OF IBADAN LIBRARY 272 on e profile of high and low achievers in mathematics found that attitudes to mathematics to be related to students' levels of achievement. This was not corroborated by 21 Shimway et al who found that children’s attitudes towards calculators were more positive than their attitudes toward mathematics. 12 Q Other contrary findings were those of Corey “ and Wolf and Blixt"1 30 who suggested that attitudes toward mathematics are causally predominant over mathematics achievement for their common variance. The findings in this study are compar­ able with those of other media when used in mathematics in­ struction . 6 . Relating findings to other media Calculators as an electronic medium could be compared, in capability for mathematics instruction with any other media. UNESCO, according to Balogun 131 in "New methods and Techniques in Education” listed the following media: (11 Radio and Television (2 ) Electro nic Computers and (3) Programmed learning and application. 12S Corey, J.F.O. The relationship between attitude ... and academic achievement, (Doctoral dissertatiop. The University of Rochester, 197B), DAI, 3SA, 2824 - 2625, Nov. 1978. 130 Wolf, F.T1. and Blixt S.L. A cross-sectional and cross- lagged panel analysis of mathematics achievement and attitudes. Educational and Psychological Measurement, 41: 1981, 829 - 834. 131 Balogun, T.A. Programmed Learning and the teaching of Science, West African Journal of Education (WA3E) 15 (2 ), 1971TT0'9"-irS’: UNIVERSITY OF IBADAN LIBRARY 273 Relatively, most of the listed media had been studied with limited results in Nigeria especially radio, televi­ sion, programmed learning (print)? but the investigator had not found any reported studies on electronic computers or calculators in classroom instruction in Nigeria. DeBlassio '7'7 found positive correlations between students’ attitudes toward using a computer and attitudes toward mathematics and instructional settings plus achieve- ment variables. However, Earle 13' 1 on student attitude, toward geometry using computer assisted instruction found that there were no significant differences between treat­ ment groups in attitudes towards mathematics. Some find- ings of the study by Backens 132 in mathematics by Television and Wilson s 13w 3 study by audio tutorial course in mathema­ tics compared with the ones in this study where favourable attitudes were found towards mathematics. But Wilkinson 134 who studied the effect of supplementary materials upon 132 Earle, H.F. Student attitudes toward geometry. (Doctoral dissertation. University of Maryland, 1972) M I / 1372, 34, 1059A. 133 Backens, V.W. The effects of teaching beginning mathematics by Television (Doctoral dissertation? North Texas State University, 1970), DAI, 1970, 31, 5143A. 134 Wilson, P.M. Do students learn from and like an audio-tutorial course in Freshman mathematics? Two year College Mathematics Journal, 1972, 3(2), 32 - 41. 135 Wilkinson, G.G. The effect of Supplementary materials upon academic achievement and attitudes toward mathematics... (Doctoral dissertation, North Texas State University, 1971 DAI, 1971, 32, 1994A. UNIVERSITY OF IBADAN LIBRARY 274 academic achievement in and attitudes towards mathematics showed that there were no signixicant differences in impro­ vement in attitudes towards mathematics. Similarly K o lm os ^5 studied the effects of instructional media in teaching and learning beginning statistics, and found that 'Carrel' groups did not significantly differ in their attitudes toward mathematics. Despite some - favourable ana few conflicting findings on the use of media in mathematics instruction it would remain how much of those favourable findings have been disseminated to the users in the schools and other learning environments. 6o 5 Educational Implications nf the Study and Recommendations The res' Its of this study have shown that the use of calculators by teachers would be an advantage in the teaching and learni ,g of mathematics in the secondary schools. However, what prospects and problems await the use of calculators in mathematics education? There would prob^tly be problems in the calculators entering t mathematics classrooms in Nigeria if the experience of the United States of America’s school systems is anything to go by. 13S Kolmos, A.S. Effects of Instructional Media in Teaching Beginning statistics (Doctoral disser­ tation, University of Illunous, 1970) DAI, 1970, 31, 4600A. UNIVERSITY OF IBADAN LIBRARY 275 Hsmbree 115 observed that in December 1974, the. National Council of Teachers of Mathematics (NCTh) issued a far-reaching statement that urgort the use of hand-held calculators in schools (IMCTM, 1974). The council found that the core of mathematics instruction in the elementary school grades was computation, pleasantly coinciding with the calculator's first purpose, and a host of other intentions of its use were envisioned (Suydam”)0 suph as: to aic algorithmic instruction, facilitate concept attainment, reduce the need for memorization, enlarge the scope of problem-solving, motivate students and encourage discovery, exploration and creativity. With the reduced cost of calculators it appeared accessible to the school systems. Howevei , orare. than a decade later, not only has the calculator failed to redirect the curriculum, it has even 11 failed to enter most L!.S. mathematics classroom. (Hembree J Fewer than 20 percent of elementary school teachers and fewer than 35 percent of secondary school teachers have employed the calculator during, instruction * (Suydam) 136 1 136 Suydam, N. M. The use of calculators in Pre-College Education; Fifth Annual State of the Art Review, Columbus, Ohio: Calculator Information Center, 1982 ('.ERIC Document Reproduction Service No. ED22QI8). UNIVERSITY OF IBADAN LIBRARY 276 Gome have suggested that the use o '7 calculators in school might not produce entirely positive effects, even computers, so far, have not c h an ge che system of teaching of mathema- tics in any fundamental way (Bell) 137 What must have teen the cause(s) of such development in an advanced technological society such as U.S.A? Perhaps several causes have inhibited the services of cal­ culators in schools. First, not all teachers and educators applauded the N C T M ’s poisition: indeed its statement on calculators provoked a barrage of skeptical comments, warn­ ing that such devices would replace skills with paper and pencil. It was tnis kind of reaction that led to "one of the largest bodies of research on any topic or material in mathematics education” (.Suydam) 136 The questions that have often teen raised are (i) Do calculators threaten basic skills? (ii) What benefits would be derived from calculator usage not already offered in the use of pencil and paper?. The answer to the first question had consistently seemed to be no particularly at the secondary school level where *1 137 Bell, F.H. Can Computers Really Improve School Mathematics? Mathematics Teacher Vol. 71, No. 5, 1 S70. UNIVERSITY OF IBADAN LIBRARY basic skills would have teen developed with manipulative materials, paper and pencil (fuydam) , To the second question, the benefits of calculators have remained some­ what in doubt, since many studies presented amtiguous fin ings (Hambree) 115 . While some studies recorded no harmful effects of calculators’ use in upper grades of elementary schools and in high schools some failed to shd-w significant differences in either students’ achievement or their attitudes towards mathematics. Teachers and educators may then ask, why bother w i t h calculators? In addition most of the studies available are in published dissertations where most of the findings have not yet been popularized i.n the education system. Hence, teachers and researchers need to reassess che calculator in what B e | l e ^ ^ called its "crucial implications for mathematics education” . Nonetheless, the findings of this study and others like that of Hembree and □ essart^'*'^ have shown that (a) Calculators greatly benefit student achievement especially for low, average and high ability pupils. » Cb) Positive attitudes about c=lculators might help to reduce students dread to tedious calculations and word problems. 139 Suydam, IM.M. The use of calculators in Pre-College Education: A state of the art review. Columbus, Ohio: Calculator Information Centre, 1979. 140 Begle, G.G. Critical Variables in Mathematics Education, Washington D.C: Mathematical Association of America and NCTM, 1979. UNIVERSITY OF IBADAN LIBRARY Driscoll 116 has affirmed teachers enthusiasm towards the calculator-use and positive attempts at the incor­ poration of calculators into the California State Board of Education mathematics programmes have been reported Yet, the question of how the device could best be used for effective instruction in the classroom demanded research attention (Suydam) 136 , (Hembree and Dessart) 115 152 and by the U.S. National Science Foundation report It is apparent that with the availability cf calculators, it will be impossible for schools to ignore them. Teachers now use them, and some pupils do have wrist-watches equipped with digital calculators. In higher education little restrictions (if any) are being placed on the use of calculators. In the secondary school, at least calculators may s oner or later replace slide-rules and books of tables in the service of existing curricula. The following opinions may be advanced while others may be mere speculations as to the use of calculators in and out of school, for it could be said:_________________________________ t 141 California State Board of Education Mathematics Framework for California Public Schools, Kindergaten through Grade Twelve, Prel. Ed. Sacraments: ’California State Department of Public Instruction, 1985. '42 National Science Foundation: Programme Solicitation: Materials fof elementary School Mathematics Instruction (NSF 55 CD), Washington D.C.: National Science Foundation 1986. UNIVERSITY OF IBADAN LIBRARY - 2 7 9 - (a) that calculators represent the kind of phenomenal change in technology that can substantially change the society* (b) one aspect of the change in society is that people outside of school are already using calculators to accomplish what we now make the principal component of years of schooling* and (c) for schools to ignore this challenge poseo great risks to the proper mathematics education of the youngsters, and td) in fact, that calculators have already had a conside­ rable impact on education in and out of school. Recommendation: This study has revealed that the use of calculators has no harmful effects on the secondary school mathematics, it • i s therefore, recommended that: 1. S c h o d authorities should allow the use of calculators to supplement teachers and pupils instructional aids in the teaching and learning of mathematics. The versatility of hand-held calculators cannot be in doubt. Car* r 143 observed how calculators could be useful in the teaching 143. Carr, Jane, M. Get away from the Table: Make Interest more Interesting. Mathematics Teacher. Vol. 79, N o . 9, D e c ., 1966. UNIVERSITY OF IBADAN LIBRARY - 2 e o - and learning of a mathematics cour.-se for business majors. He opined that several instructors who had to teach the course agreed that hand-held calculators could be used to solve most problems by choosing the oppropr'iartG formula and the solving for many variables. This has been found to be useful particularly in this study for the solution of _b+ ^b2-4ac quadratic equations by general formula (say: * 2a where a, b and c are constants]. Similarly calculators can easily be used to create a series of iterations to approximate value to some desired degree of accuracy (e.g. ^ where m = 0 ,1 ,2 ,3,...) and in other areas of concept learning in mathematics. 2. That examining bodies for secondary education should allow the use of calculators in tests. From this results calculators can easily replace the use of Tables or Slide-rules in schools because one gains not only greater speed and accuracy but also the advantage of computing values that are very large. 11 1 By using the method of successive substitutiqm,Cheung observed ithat secondary school pupils would have little difficulty in understanding equations and their solutions. 3. Where possible,books should be written to incorporate exercises that would involve drills in mathematics. UNIVERSITY OF IBADAN LIBRARY 281 and thereby encourage pupils’ effective participation in algorithmic computations and problem-so1ving. 4. Calculators may be purchased at great expense, it would be advisable for schools to operate a pool of calculators so that each class can contribute and benefit from such an arrangement. One advantage about calculators is that-it is not as fragile as most other media. Most, if not all can operate on battery. Operation^ of calculators are easily learnt, even by elementary school pupils and its durability and versatility are some advantages. 6.6 Suggestions on Further Research As mentioned in the previous sections, this study has raised a number of pertinent and philosophical questions which were probably outside the scone of this investigation. Hence, they coule not be answered by the present study in any great depth. Indeed some of the questions raised would constitute an extension of this undertaking and answers to them would serve to uncover unresolved issues associated with this investigation. The following suggestions for further research studies on the Nigerian educational scene are predicated on these unresolvea issues. Specifically there is need to inquire into the following areas of possible calculator-use: UNIVERSITY OF IBADAN LIBRARY - 262 la Primary School level: In primary school at what class pan the calculator’ be integrated into mathematics primary uohoel curriculum in Nigeria so as not to have h a r m f u l effects on the pupils’ basic computational ski 11sJ. 2a Secondary School level: The findings of this study like mo3+- otners have shown that calculators can effect­ ively be used to enhance higher achievement and positi- ' attitudes in mathematics. It would be appropriate if this study can be replicated with a larger population, longer duration, more criterion measures in attitudes and achieve ment, and possible sex differential in relation to calcu­ lator usage It -would also be appropriate to have further research studies to look at: (a) the effectiveness and efficiency of calculator-use compared with other media in instruction e.g. programmed text on print or disc, computers etc. (bj the development of mathematical concepts, problem­ solving capabilities with calculators, . (c) teachers, parents, educators and school administra­ tors attitude towards calculator-use in mathematics instruction and (d) the calculator cost-effectiveness in relation to other media. UNIVERSITY OF IBADAN LIBRARY 263 3- On higher education; Since the use of calculators in higher education is allowed, it will be appropriate to hove research studies tint would investigate what effects calculator use may have on'»(i} concept development, problem­ solving and attitudes toward remedial mathematics in higher education in Nigeria, (ii) the teaching and learning of the physical sciences. 4. The possible impact of the use of hand-held calculator on other school suujects particularly the physical sciences and statistics. 6 .7 Summary and Conclusion The purpose of the study was to determine the effects of the use of electronic calculator on the learning outcomes of mathematics instruction. The criterion measures of the learning outcomes of the study were achievement in mathematics and attitudes toward mathematics and calculator. The study was limited to secondary school mathematics and the achievement measure was in terms of computetional, conceptual and problem-solving aspects of linear equations while the attitude was limited to attitudes toward mathematics and calculator use in mathematics instruction. The study did not investigate other personality variables of attitudes such as self-concept, anxiety etc. UNIVERSITY OF IBADAN LIBRARY 254 Most of toe literature and empirical researches consulted dealt with the studies that were carried out in the United States of America or reported there. The pilot and main study used a 3 x 3 factorial design with three levels of ability! high, average and low and with two treatment groups and control group. It was a pre-test - post-test control group design. The treatments were calculator in instruction and tests, and calculator in tests only. For the pilot study, all the nine groups were in the same school, However, for the main study, three comparable schools were used. Each school was randomly assigned to treatment groups and control groups. There were three groupg of different mental abilities in each school. For the pilot study, 50 subjects completed the study and for the main study 126 subjects completed the study with an average of 14 subjects per group. There were equal number of boys and girls per g r o u p . The three schools were selected by multi-stage sampling from the population of mixed secondary schools established mere than 10 years ago in the Ibadan Municipality. The instruments used for the study were : (i) Frs-test items in mathematics (ii) Post-test items in mathematics UNIVERSITY OF IBADAN LIBRARY 265 (iii) Modified ACER ML and MQ (verbal and numerical) tests . (iv) Attitude questionnaire (v) Mathematics teachers attitude and school inventory. (vi) Instructional module (vii) Modified Flanders classroom interaction model. The duration of the study was for six weeks. Six hypotht,es were stated and the results of the study were presented and discussed based d n ’these hypotheses tested in the study. The major findings were on the hypotheses teaueo while the subsidiary findings were derived from them. Major findings: 1. The mecn peso-test scores of those groups who used calculators on instruction and tests (the unrestricted groups), was higher chan calculators in tests only (the restricted groups) ar.d the non-caIculator groups precisely, the calculator groups have higher achieve­ ment than non-caleui?'or groups in mathematics. 2, The mean post-test scores of those groups of high, mental ability was higher than those of average and low mental ability groups. UNIVERSITY OF IBADAN LIBRARY - 2 8 6 - This would mean that pupils of high mental ability would have higher achievement in mathematics than pupils of other ability levels. 3. There was no significant difference in the mean post-attitude scores of pupils i n the unrestricted calculator, restricted calculator and nor,-ca lc ulator groups. Since their means were not significantly different, their attitudes was not different towards mathematics and calculator-use But significant difference did exist between the maa of the post-attitude scores of unrestricted calculator and restricted calculator groups. Hence this had an inconclus.ve results. 4. There was no significant difference in the mean p o s t ­ attitude scores of pupils of high,, average, and low mental ability levels. Since their means were not signi­ ficantly different their attitudes was not different towards mathematics and calculator use. For the high mental ability and lew mental ability groups,there was significant i difference in the means of their post-attitude scores. For the other groups there was no significant difference. This implied that high mental ability group pupils had better att tudes towards mathematics and calculator than low mentau ability pupils. UNIVERSITY OF IBADAN LIBRARY - 2 6 7 - 5. There was no significant relationship between the pupils attitudes towards mathematics and attitudes towards calculator. Hence, attitudes towards mathematics was not related to attitudes towards calculator-usage in mathematics. 6 , There was significant relationship in the post-attitude scores of pupils and their post-test scores. Achievement was found to be significantly related to positive attituaes towards mathematics and calculator usage. Subsidiary findings: (i) The restricted groups showed no significant advantage over the non-calculator groups in the mathematics achievement scores. (ii) The high mental ability pupils have better altitude towards mathematics than other ability gr ou ps. (iii) The post-test scores of the pupils were more positively correlated to mathematics attitudes than calculator attitudes. (iv) Although, there were no significant differences in the means of the post-attitude scores of the groups yet, thore were attitudinal changes among the groups. UNIVERSITY OF IBADAN LIBRARY 288 The findings on achievement measure seemed to be conclusive in this study while those findings relating to attitude measure seemed inconclusive. Despite th- posi­ tive cori «latiori between achievement and attitudes of the groups further research studies should be carried out on attitude variables as regards to calculator usage. This study of electronic calculator in mathematics instruction presents opportunities for further research on the influ­ enced) of different electronic media: computers, video­ tapes, television etc. in mathematics instruction on such outcomes as conceptual development, problem-solving, creativity anr< attitudinal variables. On the whoie, others in Nigeria or elsewhere may replicate this study with larger population and longer duration. Tc utilize the findings on the use of calcula­ tors like many other media in instruction would demand appropriate techniques in disseminating related research results to learners, teachers, educators and school administrators . Many issues raised in this study may not be resolved effectively by experimental research alone but rather in conjunction with survey, philosophical and clinical researches. UNIVERSITY OF IBADAN LIBRARY 289 So far, experimental research study in the use of calculators in instruction in Nigeria like most media studies, would entail a lot of expenses in term- of pro­ curements of calculators. Handling and operations of suci. calculators demand care by pupils and teachers. Unlike some other tools and instructional aids the cal­ culator. is rather easily controllable by the learner and may obviously become the successor to slide-rule, brok of tables for pupils in the school systems. It would appear that there is the social acceptability of electro­ nic calculator in the business world as a tool but the reluctance of the school systems (primary and ssoondary) to accept t h 3 device as a teaching and learning aid remain an issue to ce resolved. However, the uesign and possible implementation of the findings of this study can help to provide insight into the integration of calculators into the classroom instruct­ ion and mathematics curriculum. The result of the investi­ gation would obviously be of interest not only to class­ room teachers, but also to parents, educators, school administrators and examining bodies. UNIVERSITY OF IBADAN LIBRARY 290 REFERENCES Abimbade, A. The relative effectiveness of Programmed Instruction to the teaching of Secondary School Mathematics, Unpublished M.Ed. Dissertation, Ibadan: University of Ibadan, 1983. Abo-Elkhair, M. E. An investigation of the Effectiveness of using minicalculators to teach basic concepts of average in the Upper Elementary Grades (Doctoral Dissertation, The Florida State University, 1980) Dissertation Abstracts International (DAI) Jan., 1980, 41 A".2"980 .-------------------------------------- Adewakun, S. 0. Programmed Learning Project in Mathematics, University of I badan, Nigeria, 1968. ”~ Aiken, Howe d, The automatic sequence controlled calculator - Mark I t Masil Harvard University Press, U.S.A. 1937. Aiken, Lewis R. 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Unpublished Ph.D. Thesis, Temple University 1981 . Crawford, A.N. A pilot study of computer assisted drill and practice in 7th grade remedial mathematics. Cali­ fornia Journal of Educational Research 21, 1970. Gambrot, Faye, et. al. Correlates of sex differences in attitudes towards involvement and improvement with computers. Jo7u1r-n8a5l. of Vocational Behaviour 27,August 1986, ~ Day, Roger,R. A problem-solving components for Junior High School Mathematics. Arithmetic Teacher, 34(2> Oct. 1986, 14-17. Deblassio, J.K, and Bell, F.H. Attitudes toward Computer in High School Mathematics Courses. Internationa 1 Journal of Mathematics Education in SFTimce and Technology, T2~, Feb. i 98 f", 47-56 . Deloatch, S. A Comparative study of use of programming abilities in an introductory college mathematics for disadvantage students. Unpublished Ph.D. Thesis, Indiana University, 1977. Driscoll, M. Research within Reach: Elementary Mathematics, Va: National Council of Teachers of Mathematics Report, U.S.A., 1931. Dyce, B.A. The effect of incorporating the mini-calculator into a community Junior College basic mathematics course. (Doctoral dissertation. The University of Florida, 1977) Dissertation Abstracts International, 38A, 1978, 43. Eckmier, J.L. An investigation of the use of calculator with low achieving 4th Grade Students in Mathematics Achievement and Attitude. Doctoral Dissertation, Univers ty of Southern California, 1978). Dissertation Abstracts International, 38A, June 1978, 71D § . I UNIVERSITY OF IBADAN LIBRARY 294 - Egbugara, W, 0. Effects of Three Levels of Advance Organizers on achievement of sene Nigerian Secondary School Physics Students. Unpublished Ph.D. Thesis, I badan, Univers ity of Ibadan, 1984. Ericksen, G. L.j Ryan, 3, 3. A study of the effects of Experimental programs on pupil achievement J.S.A.: State Department of Education Report No, MNL-TR, 66-4 St. Pauls: State Department, 1966. Etlinger, L. The electronic calculators A new trend in school mathematics. Educational Technology 3ournals 14, Dec. 1-74, 43 - AT. Falokun, C. 0. Concept formation in a algebraic equations and problem-solving among form five students in Oyo State of Nigeria. Unpublished M.Ph. Thesis, Ibadan, University of Ibadan, 1983, Flanagan, 3. C. General consideration in the selection of test items. 3ournal of Educational Psychology 30, 1939, 674 - 680. Flanders, N. Analyzing Teaching Behaviour. Reading, ass 1 AJ’Li.' ” r-Wesley Publishing Company, 1970. Fisher, Ronald. The Design of Experiment 6th Ed. New York : Haf uerj 1 951. Gadzella, B . 11, et a x . Mathematics Course Grades and Attitudes in Mathematics for students enrolled in Three University Colleges. Psychological Reports 57 Part I , Dec. 1875, 767 - 772. Gagne, R. M. Learning hierarchies. Educational Psycho­ logists . 6(1) 1968, 3 - 6 . _____________. The Conditions of Learning, NeW York: Holt, Rinehart and Winston 1979, ~i5T5 - 170. Gagne, R. M. and Briggs, L. Principles of Instructional 1 - ’ , New York! Holt, Rinehart and Winston, Gawronski, 3. D. and Coblentz, D . : Calculators and the Mathematics Curriculum. The Arithmetic Teacher (NCTM) 23(7). Nov. 1 976, 510 - T T . UNIVERSITY OF IBADAN LIBRARY - 2 9 5 Gooden, C.L. Some effects of using minicalculators in mathematics. (Doctoral dissertation, Kent State University 1978). Dissertation Abstracts International 3 9 A , Nov. 1978, 28'd'0-260T. ' ' Gordon, B.W. A profile of High and Low Achievers in Mathematic (Doctoral Dissertation, Duke University, 1978). Dissertation Abstracts International, 38A, Feb. 1978. 4TT39 -4640 . — — — Hansen, T.P. et. al. What Teachers Think every High School Graduate Should Know about Computers. School Science and Mathematics, 81, October 1981, 467 - 472. Hector, J.H. and Frandsen, H. Calculator Algorithms for Fractions with Community College Students. Journal for Research in Mathematics Education 12(5) November 1981, 349 - 356. Hedren, Rolf. The hand-held calculator at the intermediate level. Education Studies in Mathematics 16: May 1985 163 - 179. Hembree, Ray and Dessert, D.J. Effects of Hand held Calculators in Pre-College Mathematics,Education: A Meta-analysis. Journal for Research in Mathematics Education (NCTM) l7'(4)7 March 1986, 83 - 99. Hembree, Ray. Research Gives Calculator a Green Light. Arithematic Teacher, (NCTM) 34(1), September 1986. Hohlfield, J.R. The impact of electronic calculators on educational performance. Review of Educational Research 50( 1 ), 1980, 71 - 78. Immerzeel, G. e t » al. Teaching Mathematics with the Hand-held Calculator in N.M. Suydam. Electronic Hand Calculators: The implications for the Pre-College Education Final Report, 1976. * Jurgemeyer, F.H. Programmed Instruction: Lessons it can teach us. Educational Technology, May 1982, 14 - 48. Kalejaiye, >.Q. Individual Differences to Programmed Instructio in tie New Mathematics. West African Journal of Edi ation 15(3) October 1971, 201 - 205. Kasnic, M.J. The Effect of Using hand held calculators on mathematical problem-solving among 6th grade students (Doctoral Dissertation, Oklahoma State University, 1977) DA I, 1978, 38 A , 5311 UNIVERSITY OF IBADAN LIBRARY - 296 - Kelley, T.L. The selection of Upper and Lower groups for the Validation of Test Items. Journal of Educational Psychology 30, 1939, 17 - 24. • Kelley, M.G. The Effect of the use of the Hand held calculator on the Devexopment of Problem solving strategies (Doctoral Dissertation, Utah State University, 1964). Dissertation Abstracts International 45A: Dec. 1985. T3'71. ' ~ ~ Kerlinger, F .N . foundations of Behavioura1 Research, 2nd Ed., New Yo'rk Holt, Rinehart and WTnston Inc., 1983. Klausmeier, H.J. et. al. A,nalyses of Concept Learning, New York: Academic Press, 1966. Kleiman, G. et. al. Micro-computers and hyperactive- children in Creative Thinking 7, March 1981; 93 - 94. Koop, J.B. Calculator use in the Community College Arithematic Course. Journal for Research in Mathematics Education. 13 (1), 19521. 50-60.' " Lamb, R.L. A Study on the Coordination of Graph Theory and Computer Science at the Secondary School. Unpublished Ph.D. Thesis: f'ecrgia State University, 1976. Lawson, T.J., A Study of the Calculators and Altered Calcula­ tors effect. Unpublished Ph.D. Thesis. State University of New York at Buffalo, 1977. Leibniz, G.W. Eine Blographie Ed, Guhrauer, G.E., 2 Vols. 1842. In Encyclopedia Ameri'cana^ International Edition, N.Y. Americana Corporation, Vol. 5, 1974, pp 161-163. Magidson, E.M. Issue Overview: Trends in CAI, Educational Technology, 18 (4) 1978, 5-8. ~ I Mbrkle, S.M. Problems of Conceptual Learning. Journal of Educational Technology, 1(1), 1970. Mathis, A. College Students’ Attitudes Towards CAI. Journal of Educational Psychology _ 6 1(1), 1970 Mellon, J . Calculator based units in decimals and percents fo Seventh Grade Students (Doctoral Dissertation, Columbia University, Teachers College, 1985). Dissertation Abstracts International 4 6 A , Sept. 1985. UNIVERSITY OF IBADAN LIBRARY =297 Meyer, R.A. Concept Attainment Abilities Project. Journal f,or Research in riat_h_emdtic_s_ Education 9(5) 1975, 334-336. Moar, Eli, The Pocket Calculator as a Teaching Aid. The Mathematic,: Teacher (NCTM). 1966, Oct. 1976. Morris, J. Math Anxiety: Teaching to Avoia it. Mathematics Teacher 74(6), 1961, 413 - 417. Murphy, N.K. The Effects of a Calculator Treatment on Achiene- ment and Attitude problem solving in 7th Grade Mathematics (Doctoral Dissertation, University of Denver, 1981). Dissertation Abstracts International 42A. Noiv, 1981, 2008 - 2009. Nichols, W.E. The Use of Electronic Calculators in a Basic Mathematics For College Students. (North Texas State University, 1975) DAI 36A: 7919, June 1976. Ogunniyi, M.B. Educational Measurement and Evaluation. Lagos Longman (Fig.) Ltd., 1984. ________________ An Analysis of Laboratory Activities in Selected Nigerian Secondary Schools, European Journal of, Science Education 5(2), 1983, 195 - 201. Ogunyemi, F. and Bettis, J. An Investigation of Cognitive Preferences in Mathematics Among High and Low Achievers in the Nigerian Secondary Schools. African Journal of Educational Research. 1. 1974, 97-105. Okunrotifa, P.0. Programmed Learning and the Teaching of Geography. West African Journal of Education 14 (3), October 1970, 203 - 207. Oni, E.0. Conceptual Difficulties With Ionic Equations as Function of Intellectual Development Among Secondary School Students. Unpublished M.A. Thesis, Ife: University of Ife, Nigeria 1982. Padberg, F.F. Using Calculators to Discover Simple Theorems - An Example from Number Theory. Arithemetic Teacher. 28(8), April, 1981. Palmer, H .r H., Minicalculators in the Classroom. What do Tee ,ners Think? Arithmetic Teacher 25(7), 1978. 27 - 28. ' ~ ~ UNIVERSITY OF IBADAN LIBRARY -298- Pascal Blaise. L * oeuvrs Scidntif iqu,e de Blaise Pascal ed. Maire Albert', PariV 1912 ' In Encyclopedia Americana, Int. Ed. N.Y. Americana Corporation, Vol. 5, 1974, pp 161 - 163. Pollok, H§nry. 0. Hand Held Calculators and Potential Redesign of the School Mathematics Curriculum. The Mathematics Teacher. 73(4), 1977, 293 - 297. Powers, S. et. al. Generalization of the Mathematics Attribution Scale Norms to Academically Talented High School Students. Psychological Reoorts. 57, Oct. 1985, 475 - 478. Ray, K.L. The Effects of Computer-Assisted Test Construction on Achievement in First Year Algebra. Unpublished Ph.D. Thesis, University of Southern California, 1 977. Reed, H.D. and Dick, R.D. The learning and Generalization of Abstract and Concrete Concepts. Journal of Verbal Learning and Verbal Behaviour, 7, 1968, 485 - 490. Reys, R.E. e t . al. Hand Calculators: What is happening in Schools Today. Arithmetic Teacher 27(6), Feb. 1980 38 - 43. Roberts, D.M. The impact of Electronic Calculators on Educational Performance, Review of Educational Research 50 (1 ), 1980, 71 - 98. Saltz and Copeland, R.W. Mathematics and the Elementary Teacher. Philadelphia: W.B. Saunders Company, 1976, IT - 65. Sears, L.0. A problem of the effects of teaching a course in Algebra II and Trigonometry via Traditional method and other methods. Unpublished Ph.D. Thesife, University of Houston, Texas, 1977. She.-., M. and Wright, 3. Scales of the Measurement of Attitudes. New York: McGraw-Hxl1, 1967. Shu'-way, R. . e t . al. Initial Effects of Calculators in Elementary School Mathematics. Journal ofor Research in othematics Education ,12. 1981, 119 - 141. Si-onson, Michael R. Media and Methods. Educational Communi­ cation and Technology Journal (ECTJ) 28(1), Spring 1980, 47 ' - 61. UNIVERSITY OF IBADAN LIBRARY 299 smit~, d. L. A study of the effectiveness of the use of the electronic calculator (Doct ral Dissertation, North Texas State University, 1977). Dissertation Abstracts International. 38A, Jan. 1971^ 3988"! Spickerman, W. R. A. A study of the relationships between attitudes toward mathematics and some selected pupil characteristics in a Kentucky High School. (Doctoral Dissertation, University of Kentucky, 1965). Disser- -ation Abstracts International, 1970, 2733A. Standifer, C. E. and Mapies, E. G. Achievement and attitude of 3rd Grace students in School Science and Mathematics 61, 1981, 17 - 24. Suppes, Patrick CAI at Stanford. In Man and Computer (Proceedings f International Conference, Bordeaux, 1970) Besel Kerger, 1972. Suydam, N. M. Electronic Hand Calculators. The implications for Pre-College Education Final Report. Washington D.C., National Science Foundation 1976, _ . The use of Calculators in Pre-College F lucr.’-iD,, i A state of the Art Review Columbus, Ohio: Calculates Information Center, May 1979. Suydam, N. M. and Weaver, 3. F. Research on Mathematics Education reported in 1977. Journal of Research in Mathematics Education 9(4), July 1978. ___________ . Research on Mathematics Education reported in Mathematics Education reported in 1978, Journal for Research in Mathematics Education 10(4), 3 uTy 1979"! Suydam, N. M. Research on Mathematics Education reported in 1981. Journal for Research in Mathematics Education 13(4), Tiny^rasT;---------------------------- •----------- . Research on Mathematics Education reported in ------r a r ; ' Journal for Research in Mathematics Education 1 7 (4), T u i y T w : ---------------------------------------- _____ _______. The use of electronic Hand Calculator in 8 re-College Education: Fourth Annual State of the Art Review. Columbus, Ohio: Calculator Information Center, 1981. UNIVERSITY OF IBADAN LIBRARY 300 Suydam, N. M. The use of electronic Calculator in Pre- College Education: Fifth Annua . State of the Art Review, Columbus, Ohio: Calculator Information Center 1982. Szetella, Walter. Hand-Held Calculators and the I.earning of Trigonometric Ratios. Journal for Research in Mathematics Education. (N CfM) 10(2), March 1979, '111 - 118 . Tennyson, R. D. et al. The teaching of concepts: A re­ view of instructional Design Research Literature. Review of Educational Research. 50C1), 1980, 55 - 67. Vincent, A. T. The effects of Supplementary CAI on the Mathematics Achievement and Attitude in EMA High School. Unpublished Ph.D. Thesis, University of Cincinnati, 1977. Wilson, R. L. Computers and Mathematics. West African Journal of Education 12(2), June 1968, 131 - 133. Wright, E. B. Investigation of Selected decision-making process for aspects of a computer-assisted and mastery u. jarrrVig rnu.'jl in basic mathematics. Unpublished Ph.D Thesis, ~Te Pennsylvania State University, 1977. Zepp, R. A. Fatterns and Computation on proportions, problems and thrir interactions with the use of Pocket Calculators in 9th Grade and College ^Doctoral Dissertation, Ohio State University, 1975). Disser­ tation Abstracts International 36A, 1975, 5181. UNIVERSITY OF IBADAN LIBRARY 301 AP PE ND IX 1 M a t h e m a t i c s P r e - t e s t INSTRUCTIONS; TIME: 30 Minutes 1) Answer all the questions 2) All questions carry equal marks 3* Each question is followed by five options lettered A — E find out the correct option to each question and write the correct answer on your answer sheet. Example: If ~ — 3x — 2(1 — x), which of the following is true for x ? A ^ t B = "2 , C ta B ■Jf E = ■§■ Answer; C - ~ 4 ANSWER THE FOLLOWING QUESTIONS: 1. Simplify: -§-(2|- + 6 ■§■) + 3r» ^Simplify: -§-(2f- + % ) «*• 3f) k " f* B " f ’ ° s 3> = 3sr* E = 2 2. Ademola is x years old. Write an expression to represent his age two years ago. A = x years, B «= (x — 2)yrs., C = (x+2) yrs, B = 2yrs, „E s» ^x yrs. 3. Given rod below: 2r 1 1777771/} (3t + 2b) cm From the rod (3t + 2b) centimetres long, a part 2tcm is cut off. What is the length of the reminder ? A = (5* + 2b)on, B = (3t + 2b^cm, C = (t +2b)cm B =3 21 im, E t [5t - 2b)cm. 2 4. If 2p m — 4n • Find P when m = 20, n a 2, A s= B = 6j C = 3, I) s= 2, E » 1. 5. Express P Nairas q Kobos in Kobo? . A = 10pq, B 100P + Q, fl= 100 pq6 Find the value of p if 2(p»4) ~ 2.; A *= 3| B =3 4; C 5t B 3= 6, E a 8 UNIVERSITY OF IBADAN LIBRARY 302 7* Given that x *» — 2 gnd y «= .-jrj find xy 2 — x2 y, A a — 2g-| B e» ««10, C ta 2-g-, D ea 6 , E = 8 , 9* Simplify:. 2x(3x — 4) — 2x($x — 4)* A = 6x2 , B r- 2x“ - 16x , C = 2x2 +- 16x , D e 2x2 * E * 16x If □ - 8 - 2 ( □ - 3), What the value of O A «= 5 » B e - 5 , © - 2 , D = - 2 , E = 6. > If i + | • 7 . 7 1 What7 LS.the value of A ? ? A = 1, B B 7 J C SS 3» D e 4, E - None of the above. \L -J* j ] I • — I----- 4 The above is g balance, what is the vglue of x ? A e 10fr« B *: 1-0-* C IS 4s-* D ss 2Mr. "FI =3 di- £ i h I ? ± \ 3 * * 13, ----------- -- -----— — — -- ***** The aV ^ is a balance,.what is the value of P ? A * - l3» B = 21, C «= 9, D = 7 , E a ~ 21, ^4* If 3b + b — 2b — b 0 — 6 + 6 ,,What is b ? A = 12, B = P , C a «• 12, D = 11 E = > — 1, 15« If 15 subtracted from TO times x, the result is 55, find the value of x, A * 10x, B » 5 5 , C . 1«x - 15, B - 7. E = 4. UNIVERSITY OF IBADAN LIBRARY -303- APPENDIX 2 M at hemat i cs P r e - t e s t Answers NOS ANSWERS VALUES 1 c 7/6 2 B x-zy 3 C t + 2b 4 D 2 5 B 100p + i 6 C 5 7 A -2.5 0 A 9 D o2 x2 1 0 . D -2 11 C 3 12 E 4.5 13 B 21 14 B 0 15 D 7 UNIVERSITY OF IBADAN LIBRARY 0 O *3- “ 304- APPENDIX 3 Difficulty Index and Discriminatory Power (p) of the Mathematics Pre-test TEST-ITEM DIFFICULTY INDICES DISCRIMINATORY POWER % 1 0.45 55 2 0.25 75 3 0.25 75 4 0.25 75 5 0.50 50 6 0.60 40 7 0.75 25 e 0.25 25 9 0.40 50 10 0.40 60 11 0. 20 60 12 0.15 25 13 0.45 55 14 0.50 50 15 0.25 75 AVERAGE DIFFICULTY INDEX OF PRE-TEST = -0.40 From R/T x 100 = DIFFICULTY INDEX R ..... No. Right, T ..... Total Test Items. AVERAGE DISCRIMINATORY POWER OF PRE-TEST = 63.3% From D'.SCRININATORY POWER P = ~ -Rl RU ... No. of Pupils in the Upper 27% of the group who got the item right. R. ... No. of Pupils in the Lower 27% of the group who got the item right. UNIVERSITY OF IBADAN LIBRARY 305 APPENDIX k Mathematics Achievement test NAME* .......... ...... ..................... . AGEs ........................... SEX; ........ . SCHOOL; * ........ ...................... ..... DATE OP TEST This booklet contains an example and 30 multiple-choice objective questions. INSTRUCTIONS; Please answer all the 30 multiple-choice objective questions. You should write out only the correct letter. IMPORTANT; You have Ip? minutes to complete the test. Some questions are easier than others. If you find any question is too hard, leave it out and come back to it later if you have time. Do not spend too much time on any question. Examples Find x in the equation x = -ab - ay if a = 1+ b = 1 and y = - 2 A = -U, B = lq, 0 = 8, D = 12, E = 12. The correct answer is B. You should write only B. I Now Answer the following questions-; 1. Find the value of x if f = 5b - x when y = -2 . 2 and b = 1 . 5 A = -9.70, b = 5.50, c = 9.70, D = -5.50, e = -5.30 2. If •§■ - 5£x = 2 ( 1 - x ) , which of the following is true for x A =1.33, B = 0.75 c = -0.75, D = -1.33, E = 0 .6 7 UNIVERSITY OF IBADAN LIBRARY 306 21. A lorry carrying concrete "blocks weighs 8 6 4 kg when loaded. The blocks weigh three time as much as the empty lorry. Find the weight of the lorry. Hint, Weight of lorry = Lorry + Concrete - weight of concrete. A = 2880kg, B 25920kg, C = 34560kg, I) = 2Io0kg, E A 960kg. 2 2 . "Dairo is buying books for her friends one of the books they want costs if2.00 more than the other. She buys 5 °f the more expensive book and 3 of the cheaper book. Hint, Let x = expensive book,y= cheaper book, Form Simul taneous equations and find the cost of the..cheaper book. -.75, B == S3.50, c = S2, I) = SI. 5 0, E = S3.00. If J©hn earned S48 for 16 hours work, what was his average wage per hour? ■3.00, B =??4«00 $ C = S2, D = S5.0C, 2A. Akin and Ayo were given 20k to share. Akin is younger than Ayo,so he has to get 20k less than Ayo. How much did Ayo get? Let Akin's, share be ' x and Ayo's share be y form simultaneous equation. A = 45kobo, B = 55-kobo, C = 35kobo, D _ 7Ckobo, E = 20 kobo # 2 5 . From question 24, how much did Aki n got? A = 45kcbo, B = 55kobo C = 35kobo, D *= 70kobo, E = 20kobo. * 26. The sum of* three consecutive numbers is 123. Find one of the numb e r s (Hi nt e consocu.ti.v6 numbers are 6 5 5 8 6 ,8 7 ) r A = 21, B = 30, C =41, B= 52, E = 6 6 . 2". A bookshelf holds x .books each 1200mm thick. The same hookah elf can hold * (x-: 3) books each 900mm thick. What is the length of the" shelf? = I®8o;0nm, B = 14600mm, C = 15000mm D - 16800mm, E = 19200mm. 28. S +■ 0.5s +S + 3000 = 18, 000, Find S. A = 15, 000, B = 6000, C = 9000, D = 7500, E = 5000. 29. If .X = 2 + if4+16 Find X. A = I.23>.6, B= -3.236, C= -1.632, B= 3.236, E = 1.632. 30, If "b‘_ - 4&0 = 0 when a = -0.675 C =-75*00 Find b. A = +1422.3, B= -142.3, C =-14.23, D = 202.5, E =14.23. UNIVERSITY 0 00 0 OF IBADAN LIBRARY 0 0 V9O; ii (h 307 2 • 2 5 Cfi’wen that x - 2 . 0 and y - 0 .5 , find the value of xy— x y , A. - 2.50, B - V 10, C - 2.50, I) =-- 6 , E - 8 ,, A= - 2.^0 4 Given the simultaneous equations 2x - 3y + 2 = 0 and 3x + 2y = 23, calculate the value of (x - y ) . A - 21, S = 5, C = 4.2, D = 2- E = 1.0 3 5 5 :Pwo quantities, y and x are connected by a linear relation of the form ' y ” kx + c where k and c are constants. If x = 60 when y =10 and x = 240 . when y = 1 0 0 , find the equation connecting y. and x. y = .■§3c + 2 0 , B; y — -gac — 2 0 , k+c Ds y = 2k - IIO, E; y 2k - 130 2 2 2 If x = - I, y = 2 and Z = 3, evaluate x +y + Z - 3xyz A = -32, B = - 4 , 0 = 4, J* „ 14, E = 32. Evaluate y - 3x - /\x - 5, if x = 2 .1 2 . = 0.0032, B = - 0 .4 8 3 2 , 0 = - 0 .0032, D = 13.4832 E = -13.4832. 9. If x = 2, evaluate x - (3-(x -(2-x))+4) A = -5, B — 3, C = 2, D - 5 j E = 6 * $6 - 3xf 1 16 - 5*J ihe above figure is a balance, what is the value of x? A = -5, B = 1.25, C = 5, D = -1.25, E = II UNIVERSITY OF 0 IB IIADAN LIBRARY 12 o Find the value of 4^ - y if x +- y = - 8 and 2x - y = 7 • A = 3, B = 7, C = 8 , D = 13, E = 17. 13 If m = 4-8- -- pq. find q_ when m = 5 s P = 2 and a = - 4*5 A = 11.5, B = - 6.5, C = 6.5, D = -II . 5, E = II.0 14. Given that x + 2y = 4 and 3x + 4y = 6 find the value of (x + ;$ = 5s B = 3.5s c = 1 . 0 D = 2 . 5 , E = - I .0 T S If E = -gin fv'2 - u 2)V find m, when E = 270, V = 10 and u = 8 A = 154, S = 36, G = 15, d = 7.5, E = 0.13 16. Given triangle ABC where AB = 4^ - I BC = 3 ^ + 2 and CA = 2x + 3. If the perimeter of this triangle is 148cm, find x. £ J/.-+. AB + BC + CA = 148 A\— 40 4 Jr O 9 B == 1 6 , C = 1 6 *?, D = 1 6 2 E = 59. S 'x 4-2 17 If : _ 1 - 'i. “ 3 ), what is the value lx =• — O 9 3 = 2 , C = -5, D = -5s E = 6 18. The formula F = 9G + 32 is a temperature relationship of Fahreheit (f ) and Centigrate (C). Find the temperature when F •= C. A = 40, B = II. 4s CL = 8 , I) = -II4, E = - 40. 19. Given that 3x + 2y = 18, and px - 2y = 1 4 , Find the value of x.. A = 0.5s B = I, C = 2, D = 3, E = 4 20 If 80(2-I) - 60s, Find z A = 4 ~ B = -4.0, C = 4~ f D = 7.0, E = 4.0 h l k UNIVERSITY OF IBADAN LIBRARY f \ K C\J 1! CO t 309- APPENDIX 5 Mathematics achievements test answers 1. c 16. B 2. D 17. B 3. A 18. A 4. E 19. E 5. B 20 . E 6. E 21 . D 7. A 22 . D B. B 23 . A 9 . B 24. B 10 . C 25 . C 1 1 . A 2 6 . C 12 . E 27 . A 13. D 28 . B 14. C 29 . D 15. C 3 3 . E 4. UNIVERSITY OF IBADAN LIBRARY -31C • APPENDIX 6 Difficulty Index and Discriminatory Power (P) of the Mathematics Achievement Test TEST DIFFICULTY DISCRIMINATORY TEST DIFFICULTY DISCRIMINATORY ITEMS INDICES POWER (P) % ITEMS INDICES POWER % 1 0.69 50 16 Q. 53 17 2 0.40 42 17 0.31 17 3 0.51 42 18 0.22 17 4 0.40 50 19 0.78 42 5 0.51 40 20 0.56 58 6 0.51 58 21 0.04 08 7 0.40 58 22 0.42 42 8 0.73 25 23 0.73 42 9 0.16 33 24 0.49 33 10 0.56 58 25 0.49 33 11 0.58 17 26 0.60 33 12 0.62 50 27 0.47 08 13 0.33 42 28 0.40 42 14 0.20 17 29 0.22 33 15 0.17 56 30 0.20 00 AVERAGE DIFFICULTY INDEX = 0.46 AVERAGE DISCRIMINATORY POWER P = 42% r = reliability coefficient of the whole test = 0.54 a = standard deviation of the test scores = 3.95 c2= 15.61 X = the mean of the test scores = 13.87 UNIVERSITY OF IBADAN LIBRARY APPENDIX 7 Instructional Nodule on Equations TOPICS: (i) Simple equation. (ii) Simultaneous equations. (iii) Quadratic equation. OIRECTED TO FORM FIVE PUPILS SECONDARY SCHOOL, NIGERIA This module takes cognizance of the fact that pupils are already familiar with some aspects of equations. It has been prepared to be jsed with calculator. The general instruction on the use of calculator with this module is at the end. Computational algorithms have been incorporated in all aspects of the exercises in the module. OBJECTIVES OF TIE LEARNING CONTENT: (i) To introduce the pupils to the concept of equations simple, simul'aneous and quadratic. (ii) To identify different forms of equations: simple, simultaneous and quadratic. (iii) To solve different equations - simple, simultaneous and quadratic. (iv) To translate word-problems into equational format - simple and simultaneous equations. (v) To solve the word-problem equations simple and s imultaneous. PREAMBLE: 'ihe module is divided into FOUR parts. Each part consti tutej»an instructional content relating to each objective. Th instructional content shall cover a period of 40 minutes. It made up of examples on the concept to be developed and graded UNIVERSITY OF IBADAN LIBRARY 3 12 exercises# The pupils shall he allowed to work at their own pace. Each instructional content shall he followed with exercises ^̂ ĥich the pupil3 shall respond to. The pupils shall he made to use only this module for the project# PART I INTRODUCTION Two types of linear equations shall he considered in detail} while quadratics id discussed with examples# 1. Linear simple equation in one variable: This is an equation having one as the power of its variable# It has a single solution. Consider this linear equation: x sa 9 is the solution# Please note that x is the unknown variable and the solution of x = 9 will satisfy that equation# Give other examples# 20 Simultaneous Linear equations in two variables called the unknowns. These equations consist of two 1incap equations in two unknown-variables# The equations are solved by eliminating one of the two unkown variables and thereby reduce the equations to one linear equation in one unknown variable; which is then solved as simple linear equation. The solution to the other variable is obtained by substituting the solution already obtained into one of the two given equations and then solve the resulting linear equation. Consider the equations: x + y 20 x - y 6 UNIVERSITY OF IBADAN LIBRARY 313 Please note that x and y axe the two unknown variables in these 9 equations: • Solving these equations: X = 13 i the two unknowns y - 7 ) These values of x and y shall satisfy the two equations 13 + 7 = 20 13 ~ 7 = » Other examples shall be given. 3. The third form of equation, quadratics is an equation in one variable, having 2 as the highest power of its variable. It has at most 2 solutions. A general quadratic equation is of the form ^ ax + bx + c es 0 where a, b, and c are constants 2 To solve the quadratic equation ax + bk + c = 0, the factor method is used when the expression ax^ + bx + c can be factorised. However, if it cannot be factorised, the formula x = ~ b + < /b 2 ~ 4ac ----- is used to 2a solve for x which is the unknown variable and will have two values. To give examples e.g, x2 - 3x + 2 == j* PAPlT II SIKPLF EQUATIONS An equation is an open sentence with an equality sign. For example, 4 + s = x + 2 x i s r statement which expresses a true state­ ment if 2 is substituted for x. That is when x = 2 what is on the left hand side (L.H ~ S (4 + x ) ) must be equal to the right hand side (R.H. S s— (x + 2x). The expression 4 + x is equal to 6 when x = 2 and x + 2x also is equal to 6. UNIVERSITY OF IBADAN LIBRARY 31^ Any of the four basic operations of elementary mathematics (addition| subtraction, multiplication and division) car. be performed on both sides of an equation; though the expression will be altered yet the equation will still be true. Examples: — 3x -s. •; 4x~2 = 5j 3x + 4 = 7 etc. Worked Examples i) Consider the equation 22 = 7x ~ 6 Solution: Add 6 to both sides 22 + 6 = 'fx — +*• 2 8 = 7x Divide both sides by ; 267 IS 7 7 4 = x The solution is x = 4 It may also be necessary" for us to check if our solution is correct or not? Thus if x = 4 then 22 = 7(4) — 6 = 2 8 - 8 L.H.S. = R.H.S. : 22 = 2 The solution is correct. Consider the equation x + 2 2 .5 " SOLUTION Multiply through by 22.5 (the L.C.I.) Thus 22.5x + x = 3.5 X 2 2 .5 23.5s 78.75 Divide both sides by 23.5 - 2 3 ^ J M L 23.5 23.5 x 3.35 UNIVERSITY OF IBADAN LIBRARY Check 3.35 + 2^ ’ = 3.50 3.35 + t.15 = 3.50 The so lu tio n i s correct# iii) Consider the equation SOLUTION 2.2 + 3.1 * 4.4 M ultiply through by 31.944 (th e LCM) 14.52 x + 9«68x + 7.26bx = 31.94 1 4*52x + 9 *68z + 7 , 2 6x = 31*944 3 1.463: <= 31.944 31.Jj5x 311 jr.ir - if .. Divide both sides hy x « 1 , * 1 5 Check 0.46 + 0.31 + 0.23 = 1,* 'The so lu tio n i s x = 1.015. Solve the equation x — 3.5 = - s ~ + 1 .5 2T 5 3,5 SOLUTIONS Multiply both sides of the equation by 8.7 5 , the LCM :* 3.5x - 30,625 t= 2.5x + 13.125 ft Adds 30,625 to both sides 3.5x = 2.5x + 43.75 Subtract 2.5x from both sides 3.5x - 2.5x = 2.5x + 43.75 - 2.5x 43.75 2.5 ” = 3.5 17.5 - 3.5 «* 1 2 .5 + 1 .5 14 .0 = 14 ,0 The correct so lu tio n i s x = 43.75 UNIVERSITY OF IBADAN LIBRARY 3 1 6 v) Solve the equation: x - 20 - 3x - 40 - 100 3 ' 4 Multiply both sides of the equation by 12 (the 4 (x - 2 0) - 3(3x - 40) = 1200 Remove brackets; „ 80 - 9X + 120 r= 1200 Collect like terms; 4x - 9x = 1200 + 80 - 120 - 5x » 1160 Divide both sides by ♦ 5 rj6o -5 X e -232 Check -232 - 20 - 3(>»232) - 40 100 3 4 ~ ^ 2 „ =§2§ J i J0 _ 100 3 4 + m m 100 3 T 4 —8 4 + 18 4 = 1*0 The correct solution is x - — 232 PUPIL'S IK CLASS Practise Exercise: Solve the following equ&tions: 1} P - ± _ U ? _ 3( & J ^ = 2 2 .5 10 ii) 4 + 1 ■ 0 ,7 5 t - 1 iii) 3x — 2 (x + 3) « 7 — x iv) x + = 12 v) ~ (y ~ 4) - UNIVERSITY OF IBADAN LIBRARY 317 SOLUTION i) P + 1.2 5 10 = 22,5 Multiply both sides of tl o equation by 10 (the L.C.M.) 2(P + 1.2) - 3(P-1.5) » 225 Remove brackets: 2P + 2.4 - 3P + 4.5 = 225 .Collect the like terms: 2P — 3P = 225 - 2 . 4 - 4 .5 -P « 225 - .,9 1* . -P - 218.1 -P » -218.1 Check -2 1 8 .1 + 1,2 - 3 (1 .2 1 8 .1 - 1 . 5 ) = 2 2 .5 5 “ 10 - 43.38 + 6 5 .8 8 s= 2 2 .5 2 2 .5 = 2 2 .5 The correct solution is F= — 213.1 ii) H r - <>.75 M ultiply both sides by (t — 1 ) t + 1 = 0„75t — 0.75. Remove bracket* Collect the like terns: t - 0.75t = - O .7 5 - 1 0.25t = - 0.25 Divide both sides by 0.25 rj 1aZ5 „ 0 .2 5 • 0 .2 5 -Ml. 0 .2 5 Check jMHj■ = 0 .7 5 UNIVERSITY OF IBADAN LIBRARY 6 0.75 t = 0.75 •.75 = *.75 The correct solution is t b 7 iii) 3x - 2 (x + 3) = 7- x 3x - 2x - b = 7- x Rem*ve bracket 3x — 2x — x = 7 + 6 Gdllect the like terms X + X = 13 Divide both sides by 2, 2x = 13 2x = 13 l x = 6.5 Check: 3x 6 , 5 — 2 (6.5 + 3 ) = 7 - 6 . 5 19.5 - 2(9 5) = 0 .5 19.5 - 19 = 0,5 0.5 = 0.5 The correct solution is 0.5, iv) x + -§x = 12 Multiply both sides of the equation, by 3: ♦ 3x + x = 36 4X = 3 6 Divide both sides by 4: x = 9 Check: 9 + -§-(9) = 12 9 + 3 = 1 2 12 = 12 The correct solution, ig t = Q. UNIVERSITY OF IBADAN LIBRARY 319 v) y (y - 4) » | y Multiply "both sides of the equation by J y - 4 = 3y Collect the like terms y - 2 s y = 4 -2y = 4 Divide both sides of the equation by —2 = & _ 4 -2 “ -2 y - -2 Check j (~2 ~ 4) = 7 1 j ~ 7 ■ 8̂ 7 7 The correct solution is y« ~ 2 . WGitD PROBLEMS LJADINC TO SIMPLE ECb'ATIOHS Statements which bear relationships with one another are often considered equations. For example; If I double a number and add 2 ot it, the result is 8 # What is the number ? By trial ana error with « different number c«rrmnations one may get the answer as 3 , but this waste time. However, the question could easily be solved if put in an algebraic form. Suppose the number is x. 2x is the same as ’twice x f. So ; 2x + 2 = 8 which is an equation SOLUTION: 2x = 8~2 2x = 6 UNIVERSITY OF IBADAN LIBRARY Check 2 x 3 + 2 8 6 + 2 = 8 8 « 8 The correct solution is x = _3 which is the number, WORKED EXAMPLES; Consider the followings The sum of three consecutive odd numbers is 93# What are the numbers ? SOLUTION': Let x - the first odd number Then x + 2 = the next oonsecutive odd number And x + 4 = the third oonsecutive odd number First number + second number + third number = 93 x + (x + 2) + (x + 4) = 93 X + 2 +- x +4 = 93 3x + 6 = 93 3x +6 + (-6) = 9 3 + c ~ 6 ) 3x * 87 3x 3 s ^ X a 29 first number x + 2 ts 31 second number x + 4 =3 33 third number. A father is now six tim 03 S/3 old as his son. In twenty— tw* and half years from now, the father will be three times as old as his son will then be. How old is each of them now ? UNIVERSITY OF IBADAN LIBRARY 321 SOLUTION: Lei: x years be the Son’s present age* Then 6x years is the father’s present age* Also, (x + 22*5) years is the son’s age 22.5 years ftom now* And (6x + 22,5) years is the father’s age 22*5 years fro-: now* The father’s age be three times the son’s age (x + 22*5) years* •X + 2 2 .5 3 (x + 2 2 .5 ) 6x + 2 2 .5 = 3x + 67.5 6x - 3x = 67,5 - 2 2 .5 3x ss 45 X 1 5 , the son 6x = 9 0, the fat! iii) A man drives from Ibadan to Qyo, a distance of 55 km*, in 45 minutes. Where the surface is good, he drives at 90km/hr.| where it is bad, at 3#km/hr. Find the number of 1cm of good surface. SOLUTION: Suppose there are x km of good surface. Then there are (55 — x) km of bad surface. Over the good surface, he drives at 90km/hr. Therefore, he drives xkm at 90km/hr. The time taken is x/?0 hours* Over the bad surface, he drives at 60km/hr. Therefore he drives (55-x)km at 60km/hr, The time taken is hrs* The total time taken is 45 minutes or ■§• hour. 2L. + 5 1 - - * 90 + 60 Multiply both sides of the equation by 360, UNIVERSITY OF IBADAN LIBRARY (the L.G.M*) 3 2 2 Ax +,6 (5 5 - x) - - Ax + 33C — 6x - 2*7# Remove brackets Ax — ix **= 270 - 330 Collect the like terms ~ 2x r» —60 —2n —b® — 2 “ — 2 Divide both sides by —2 x - 30 There are 30km of good surface and 2 5km of bad surface Check; He drives 30km at 90 km/hr, Time taken = vy hr* - 2 5 minutes* Total Time taken - (20 + 25) a 45 minutes* PUPILS * IH—CLASS PRACTISE EXERCISES i) A rectangle has its length four times as long as its width* Its perimeter is 720 an, find the dimension of the width and 1 ength* ii) A boy is paid 50 kobo for each day he works and is fined 25 kobo for each day he fails to work. After 20 days, he is paid Wf .0 0. For how many days has he worked ? iii) A girl is 10 years older than her brother. In 5 year's time, she will be twice as old as the boy. How old is the boy ? How old is the sister ? SOLUTION: Let the width of the rectangle be w Length of the rectangle = 4w Perimeter = 2 (l + b) 1 1= length, b *= width UNIVERSITY OF IBADAN LIBRARY 323 720 2 (4w + w) 720 2(5w) 720 10w Divide both sides of the equation by 10: 120 10w 10 1t * w » 72 The width is 72cm and length is 268 cm. Check: 720 = 2 (7 +b) 7 2 0 = 2 (7 2 + 288) 720 = 2 (360) 72 0 = 7 2 0. Dimensions 72cm 72cm and 288 an. ii) Let y represent the number of days he worked. Number of days he did not work (20 — y) Amount paid for days he worked = 50y kobo Amount fined for days he did not work = (20 — y ) (2 5) Kobo Total amount paid for working = 700 kobo 50y - (20 - y) (2 5 ) = 700 5Qjr - 500 + 25y = J00 7 5y = 7 0 0 + 5 0 0 75y » 1200 Divide both sides by 75 1200 75 y 16 He has worked for 16 days he did not work for 4 days* UNIVERSITY OF IBADAN LIBRARY - 3 2 k - Check: 16 x 50 4 x 25 rs 700 800 100 vs 700 700 700 iii) Let the age of the boy now be x years. •Hie age of the girl now is (x + 1 0) years. In five years times the age of the boy will be (x + 5) years. Age of the girl in five years time is 2 (x + 5 ) years (x + 10) + 5 = 2 (x + 5 ) x + 15 = 2x + 10 1 5 - 1 0 a 2x - X 5 = x Age of the bey is 5 years Age of the sister is 15 years. PART III SIMULTANEOUS EQUATIONS So far, the equations that have been solved involved only one unknown each, but if we need to find two unknowns, we require two equations to involve the unknowns. Consider these equations: j k - y « -13 | called simultaneous equations. 2x + y a 17 ) The equations are solved by calculation either by substitution method or elimination method. It can also be solved graphically. However, for this module, one shall concentrate on the two methods: Method of Substitution: This is by solving one of the equations for one of the variables in terms of the other. One has to select the UNIVERSITY OF IBADAN LIBRARY 325 one variable to be solved more easily by inspection. For equations: 3x - y = -13 i) 2x + y = 17 ii) From equation (i) y a 3x + 13 One will now substitute value of y in the second equation and get the resulting equation in terrss of x alone. 2x + ( 3 x + 1 3 ) = 1 7 2 x + 3 x = 1 7 — 1 3 5x a 4 X « 4"c -■ 5 One now substitutes x*= 4 m. 5 the first, or second, equat..ion as it m a y be appropriate from equation (i) y « 3x + 13 y - 3x ^ + 13 y » J§ + 13 y = 4 +13 y 2= 155 Check: From equation (i) 3 ( f ) - 15§ » - 1 3 12 , r2 32| - 152§ » “13 - 1 3 = - 1 3 This method is used for cases in which one of the coefficients of x and y in one of the two equations is 1. Method of Elimination: The method of elimination by addition or subtraction is very effective where the coefficients of x and y are UNIVERSITY OF IBADAN LIBRARY - 326 - than 1. Consider the equations: 3x - y = -13 ------- (i) 2x + 3y = 17 -------(ii) Multiply equation (i) by 3 and then add to (ii): 9x - 3y = -39 ■I*.* ,fr. 22 Adding: 11 x =2 —22 X B —2 Substitute x = -2 in either equation (i) or (ii) Wow in equation (i) 3(-2) - y = — 13 -6 + 13 - y 7 » y The values x = —2 and y = 7 give the solution of the two equations: Check In equation (ii) 2x + 3y = 17 2 (-2) + 3(7) 17 -4 + 21 17 17 17 PUPILS’ IN CLASS PRACTISE EXERCISES: *■ • • .&■ t Solve the equations (i) ■§* + & = 4 1 Z (ii) 16+ + 3v = 14 —4t U IVERSITY OF < IB l A o D ii AN LIBRARY - 327 - SOLUTION: (i) ■§* + |y = 4 - - (i) 1 iy - = 6 - - (ii) (i) x 6 gives 3x + 2y = 24 - (iii) (ii) x 12 gives - 4x « 2 - - (iv) Multiply (iii) by 3 an.d (iv) by 2 9x + 6y = 72 ~ (v) — 8x + 6y = 4 — (vi) Subtract (vi) from (v) 17x «= 68 x = 4 Substitute for x = 4 in equation (i) i(4 ) + -Jy = 4 2 + $ r = 4 -5y » 2 y - « x = 4 , y = 6 are the solutions to the equations* Check In equation (ii) i(6) - m - 1 64 . 4 1 3 = 3 18 - 16 12 2__ 1 12 S 1 1 7 5 (ii) l6t + 3v = 14 ----------- (i) V - 10 = —4ts v + 4t <= 10 (ii) 3v + l6t 14 ------- (i) UNIVERSITY OF IBADAN LIBRARY - 5 2 8 - V + 4t = 10 ■ ------- (ii) (ii) x 3 gives 3v + 12t = 3 0 (iii) Subtract (ii) from (iii) l6t - 12t = 1 4 - 3 0 4-k = -16 t = -4 Substitute t = —4 in equation (i) V = -4t + 10 V = -4 X' —4 +■ 1# V = 16 + it = 26 V = 26 Check In equation (i) l6t + 3v = 1 4 16 x (—4) + 3(26) = 14 - 6 4 + 78 = 14 14 = 14 t= -4, V = 26 are the solutions of the equations, TORlVfROBLIKS LEADING TO SBIULTAUEOUS EQUATIONS Consider the following statements; The sum of two numbers is 25 and their difference is 15* What are the two numbers ? SOLUTION: Let the numbers be x and y x + y = 2 5 Sum x - y = 1 5 difference The statements lead to two equations with the unknown numbers UNIVERSITY OF IBADAN LIBRARY 329 as x and y solving the equations: X + y = 25 ( i ) x - y = 15 ( i i ) Adding ( i ) and ( i i ) x ** 20 Substitute x = 20 in either equations ( i ) ar ( i i ) In equation ( i ) 20 +y = 25 y = 2 5 - 2 0 y « 5 Check: In equation ('ii) x — y = 15 2* — 5 = 15 15 = 15 The numbers are 20 and 5 , PUPILS*t IN-CLASS PRACTISE EXERCISES: i) 20 spoons and 200 forks cost H 10; 6 spoons and 100 forks cost X (x *-» 2) — 1 (x*“ 2) a 6 (x — 1 ) (x — 2 a 0 Either x - 1 = 0 o r x ~ 2 = 0 x = 1 or 2 . Check: When x = 1 : I2- 3(l) +- 2 = 0 1 - 3 + 2 = 0 2 _ 2 = • When x = 2 ; 22 - 3(2) + 2 = • 4 - 6 + 2 = 0 6 — 6 = • The solution of the equation is x «= 1 or 2 ii) Solve the quadratic equation: 2x — 7 x + - 3 <= 0 *2-------- Using the formula x = - b~ x/b — 4ac 2a Wher e a = 2 ̂ h « —7 j o c 3 x = - (-7) + / 7 - 7 ) 2 - 4 x 2 x 3 2 x 2 x = 7 + V 49 - 24 s = 7 + v/25 " ~ „ _ 7 + 5 12 2 . 7 + - 5 12 * T* °r 4 i-e. ? . 5 2 x = 3 or 0,5 Check when x = 3 , 2(3)^ «• 7 (3) +- 3 = 1;8 — 21 +• 3 « 0 21 - 21 = Jt When x = i , 2 » )2 - T » ) + 3 UNIVERSITY OF IBADAN LIBRARY 333 2 x i - | + 3 » 1 i - | + 3 « v 3g- — 3§- + 3 t= 0 The solution of the equation is x » 3 or -J-, IN— CLASS PUPILS PRACTISE EXERCISES Solve the equations (i) 2x 2— 5x + 2 = 0 (ii) 2y 2 - 3y - 21 = 0 (iii) ' ~ = 5x. SOLUTION; 2 i) 2x — 5x + 2 = 0, Check for factorisation: 2 x 2 = 4 , factors of 4 are - 4 — 1 to give — 5, Hence: 2x2 — 4x — x + 2 = 0 2x(x ~ 2 ) - 1 (x — 2) = ^ 2x — 1 = • or x - 2 = 0 x = ■§• or 2 Check: When x = 2(i ) 2 - ̂ (2) + 2 = 0 2 4* 2 = 0 % “ Sfr «= 0 When x = 2, 2(2)* - 3(2) +- 2 » 8 - 1 0 + 2 0 10 + 10 0 The solution of the equation x = -§■ or 2. „ 2 ii) - 3y - 21 = 0 Check for factorisation: 2 x —21 = — 4 2; far less than — 3» cannot be factorised* Using the general formula: y = -b + jr. 4ac 2 a UNIVERSITY OF IBADAN LIBRARY 23 4 - = 2 J "b « j C r 1 — — 4 x 2 x --21 2 x 2 y * 1 + yT77 y = 3f 13>30 i . e . 3̂ I3r 30 or 3 — 13.30 y » 16,30 4 o r - 1 0 . 3 07 y a 4.075 or ~ 2,575 Check; When x = 4.»75j 2(4 .075)2 - 3 (4.075) - 2 1 a 0 33.20 - 12,15 - 21 - 0 33.20 ~ 33.20 « 0 When x = - 2, 575, 2 ( -2. 575, 2 ( -2. 575) - 3 ( - 2. 575) - 21 =» 0 1 3 .2 6 1 - 5 + 7.725 - 21 21 - 21 [Icnee, the solution of the equation is: x r= 4.075 or -2,575. i i i ) Solve; J3 1 r: - 5x 3 sa 5x (x - 1) 3 E3 5_ 2 x 5r 5, x2 — 5x «» Ji =r- < / ? r ------4---------x = *-«t) + — ao , = 5, t 1 SS —5? 0 UNIVERSITY OF IBADAN LIBRARY * 135 - x <=> -(—5) j, yj~(-5) ̂- 4 j 5 z -3 2 x 5 » 5 + •/ 25 + 60 TO x = 5 £ 9.22 tassasna-aya*'-! 3=- TO X = 14.22 10 or -4.2210 X = 1.422 or - 0.422 Check: When x = 1.422 5(1.422f - 5(1.422) - 3 » 9 10,110 - 7.11 - 3 0 10.110 - TO.110 0 When x a -0*422 5{ ^ c422)2 ~ 5 (-Q.422) - 3 = 9 0.890/]2 + 2 . 1 1 - 3 « § 0,89 + 2.T1 - 3 = 9 3.0 - 3.0 ss 9 Fencej the so lu tion o f th e equation i s x * 1.422 or -C.422. E N D Pages 336-342 contain the instruction on the ~se of calculator with the module UNIVERSITY OF IBADAN LIBRARY *336- GENERAL INSTRUCTION ON THE USE OF CALCULATOR ----------------C7TTH TrTSTT̂ nJCTTiSlTAII RSTJCIlTE------------ Introduction There are different types of calculators in the market ranging from scientific, non-programmable to programmable ones. However, the type used for this instructional module is shown in the figure below. It has capacity for values up to 6 digits or 7 places of decimal, and it is battery operated. S C R E E N - t-r 1 v n i°Ff i [™ji (r . cm-I Li-i 1 m 1 r a J X J L d L ? J M U -1;m I6 ,! 1 * | i i. j U J h i m U J U J , U J T Model of Modern Electronic Calculator This type of calculator is capable of performing the following mathematical operations (the operational keys): 1. Addition (+) 2. Subtraction (-) 3. Multiplication (x) 4. Division (?) 5. Add to the memory (M+) 6. Subtract from the memory (M~) 7. Retrieve and compute memory (R.CM). UNIVERSITY OF IBADAN LIBRARY -3 3 7 - P p , ‘ar from memory (CE) 9. Square root (/"") 10. Compute percentage (%) 11. Decimal point (.3 12. Equality ( = 3 Other operational keys are: DN/C Starts the calculator and clears with zero (0) appealing on the screen. OFF Shut off the calculator and it also has an automatic shut off which turns the power-off when the calcu­ lator is not used for 7 minutes. Numerals: 0,1 - 9. Finally we have the display screen. With this ba^K^round one can proceed to discuss how this calcula 'or c i . L c nd to solve mathematical problems on simple,, simultaneous and quadratic equations. Example 1 of Part II: Using the calculator s°lve the simple equation:- 22 = x - 6 . SOLUTION OPERATIONAL KEYS DISPLAY ON SCREEN Punch ON/C 0 . If 2 twice 2 2 . M if Fl+ 2 2 . if = 2 2 .^ if 7 7." if X 7,M Try x = 4 )„ 4 4 .mor any ) other va- )” N+ 26. fl lue 3 x=1 ,2,3,etc.3 " - and 6 6 .M if = 2 2 . UNIVERSITY OF IBADAN LIBRARY -3 3 8 - which is equal to 22 the value which we have started with. Thus, x = 4 is the solution. You amay wish to try other values x = 1, 2, 3, etc. and check if they will be equal to 22. [Hint: To solve simple equations in fraction forms it is easier to change the fractions into whole numbers.] Example I of Part III Use the calculator to solve the simultaneous equations} 3x - y = -13 H i ) + y = 17 f ii) The solution of these equations implies obtaining values of x and y which will satisfy equations (i) and (ii). SOLUTION: Eliminate one of the values by method of substitution to reduce it into simple linear equation in just one variable. Then use the calculator to so-1ve the linear equation in one variable. By substitution, y = 3x + 13 from equation (i) Put (i) in (ii) 2x + 3x + 13 = 17 5x = 17 - 13 Simple linear equation: 5x = 4 » STEP 1 -- OPERATIONAL KEYS DISPLAY ON SCREEN Punch ON/C U • n 5 5. Try di-Fferent it 5. values o -f x X x = 1, J, 4/5 r 1 5. ft a 5. Screen value not equal to 4 f? 5 5. f 9 X 5. UNIVERSITY OF IBADAN LIBRARY - 339 - ->ch (') and 5 0.5 n _ 2.5 Not equal to 4 ” 5 5. ” X 5. ” point (.) and 8 0.8 ft _ 4 Equal to 4 So x 0.8 is one of the solution. STEP 2: Put x = 0.8 in equation C i 3 y = 3x + 13. y = 3(0.8) + 13: So as to obtain value of y OPERATIONAL KEYS DISPLAY ON SCREEN Punch 0N/C 0. ” 3 3. ” X 3. " point (.) and 8 1 0.8 2.4 2.4 1 a d 3 13 15.4 y = 15.A is a solution The solution of the simultaneous equations are x = 0.8 and y = 15.^ Example of III of Part IV Solve the quadratic equation l 2xZ - 7x + 3 = 0 First check if the equation can be factorised. If not, then apply the general formula. Most importantly, all quadratic equations can be solved by the general formula and it is therefore, easier and faster with the calculator to user the formula. UNIVERSITY OF IBADAN LIBRARY 340 General formula x = '-b± /b2 - 4ac ---------?i------ where a, b and c are constants from the general form of quadratic equation ax2 + bx + c = 0 From this example: a = 2, b = -7, c = 3 x = - ( - 7 ) + /(- 7 )2 - 4 x~T~7 3~" 2 x 2 Pupils njte that - x - = + - -k + = - X = -1 x -7 -+ y/-7 x -7 - 4 x 2 x 3 SOLUTION: St art the operation with the Square root valus from right to left. OPERATIONAL KEYS DISPLAY ON SCREEN Punch ON/C 0 . If 7 7. n X 7. ft 7 7. if = 49. ft N+ 4 9 . M I 4 4.'" I X 4.M I _ M2 2 . I 8n .MX I . N3 3. I = 24.^ I M— 24. I R.CM 25.M UNIVERSITY OF IBADAN LIBRARY CM X CM ■341 Punch r 5. ’ M 5. Positive ” + 5. solution 7 7. ) ” + 1 2 .'M For quadratic equation solution we have ± (the square root values are negative and positive) OPERATIONAL KEYS DISPLAY ON SCREEN Punch - and 5 5. Negative) „ ) 5. - solution) + and 7 7. n 2 .(1 Final solution 12 2x2 2x2 POSITIVE CALCULATION NEGATIVE CALCULATION OPERATIONS SCREEN OPERATION SCREEN Punch 1 and 2 12 Punch 2 2. It r and 2 2 " f and 2 2 . it = 6 " 1 . ft 7 2 2 " f and 2 2 . It - 3 - = 0.5 x = 3 or 0.5 Note that negative value sign appears to the right of the number on the screen unlike the way it is wr’tten down on paper. Check for values of x = 3 or 0.5 in the equation 2x -• 7x + 3 - 0, a^d you determine if values on the right hand side (R.H.S.) will be equal to zero. For x = 3, 2 (3) 2 - 7(3) * 3, is it equal tc zero? UNIVERSITY OF IBADAN LIBRARY -3 4 2 - OPERATIONAL KEYS DISPLAY ON SCREEN Punch O N /C 0 . 2 2 . H X 2 . ft 3 3 . '• X 6 . 3 3 . ft = 1 8 ft M+ 1 8 99 7 7 " 7 "Mft V M It 3 3 f» — 21 M 9? N— 21 n it n nr [VI 3 - it 3 3 M ft M + 3 99 R . c n 0 Here the v a ] e x = 3 satisfies the equation. You use the came steps for x = 0.5 it should also satisf- equation » The few worked exam^les have been carried out using the calculator you will now practise with your teacher some of trie in-class exercises. UNIVERSITY OF IBADAN LIBRARY -3 '-3 - APPENDIX 8 A.C.E.R. ML TEST Name: ................. Age Now .............. Class............. This is a test to see how well you can thing. It contains questions of different kinds. Sore examples and practice questions will be given to show you how to answer the questions. Example A. Four of the following are alike in some way, write the number of the other two in thebrackets at the end of the line. I. tea 2. Cofee 3. Cocoa 5 Pencil 6. Milk (3 8 5) Question I. Four of the following are alike in some way. Write numbers of the other two in the brackets. 1. apple 2. pear 3. potato 4. babana 5. carrot 6. oranges (3 & 4) Question 2. Four of the following are alike in some way. Write the numbers of the other two in the brackets. 1. door 2. window 3. coat 4. wall 5. roof 6. book (3 & 6) EXAMPLE TOWEL IB TO WATER AS BLOTTING PAPER IS TO - I. school 2. ink 3. writing 4. deak 5. pen (2) QUESTION 3. Hand is to finger as foot is to 1. leg 2. arm 3. too 4. man 5. ankle (3) Question 4. Newspaper to tG see as Wireless is to 1. wire 2. hear 3. dial 4, car 5. deaf (2) EXAMPLE: Which two of the following statements mean most nearly the same? 1. Too many cooks spoils the broosh. 2. Make hay while the sun shines. 3. A stitch in time saves nine 4. It’s a long lane that has no turning. (2, 3) 5. Strike whine the iron is hot. Question 5. Which two of the following statements mean most nearly the sane? 1. A careless: master makes a negligent servant. 2. To resist him that is set in authority is evil. 3. Little is done when many command. 4. When the cat is away the nice play. 5. Where there are seven shephards there is no flock. (3,5). UNIVERSITY OF IBADAN LIBRARY -344- Question 6. Our gdog bit the postman yesterday? Which of the following statement are together 1. Our dog is the only German shepard dog in the street. 2. The postman was late yesterday 3. The postman is in bed because a German shepard dog bit him yesterday in our street 4. Dogs seem to dislike postmen. 5. The postman had sore leg last week ( ) You will have 30 minutes to do the test. Some questions are earier than ot others. Try each question as you come to it, but if you find any question is too hard, leave it out and come back to it later if you have time. Do not spend too much time on any one question. Try to get as many right as possible. 1. Of the following are alike in sane way. Write the number of the other two in the brackets. 1. table 2. chair 3. man 4. bed 5. cupboard 6. towel ( ) 2. FILTHY is to DISEASE as CLEAN is to: 1. dirty 2. safety 3. water 4. illness 5. hea1th. 3. Four of the following are alike in some way. Write the numbers of the other two in the brackets. 1. tube 2. artery 3. tunnel 4. string 5. pipe 6. wire ( ) 4. INCH is to SPACE as SECOND is to: 1. hour 2. age 3. time 4. clock 5. time ( ) 5. Four of the following are alike in some way. Write the number of the other two in the. brackets. 1. lagoon 2. pool 3. swamp 4. lake 5. march 6. pond ( ) 6. PIN is to HEAD as NEEDLE is to: 1 . prick 2. sew 3. eye 4. point 5. threa ( ) 7 Four of the following are alike in some way. Write the number of the other two in the brackets. • 1. onlooker 2. spectator 3. critic 4. eye-witness 5. author 6. bystander ( ) 3. HEAT Is to ASHES as CARPENTRY is to: 1. carpenter 2. sawdust 3. chest 4. furniture 5. wood ( ) 9. Four of the following are alike in some way. Write the numbers of the other two in the brackets. 1 1. sponge 2. water 3. map 4. towel 5. blotting paper 6. dirt. UNIVERSITY OF IBADAN LIBRARY 10 Which two -L5EB DOES ROT L! ' Ilf HUMS STREET? 1 All the buildings in Hume Street are modern. 2 All the buildings in Hume Street are flats. 3 Mr. Reed lives in comfort 4. Mr. Reed does not live in a flat. a Mr. Reed lives'five miles from town. ( ) 25 If these words were rearaaged correctly to form a sentense, with what letter would the middle word begin? *26 GATE is to FEHCE as PORT is to- land 2 Coast 3 town 4 sea 5 destination ( ) GO STRAIGHT OH BITE THE HEAT PAGE UNIVERSITY OF IBADAN LIBRARY 27 Which two of the following statements mean most nearly the same? 1 I t ’s' petty expenses that empty tho purse. 2 Snail gains bring riches in. 3 Even the weak are strong when united. 4 Constant dripping wears away the stone. 5 A chain is as strong as its weakest link. 28 Four of the following are alike in some way. Write the numbers of the other two in the brackets. I ruler 2 heat 3 clock 4 thermometer 5 rainguage 6 yard ( ) 29 Which two of the following statements mean most nearly the same0 1 .Repentance is poor consolation. 2 bore haste less speed. 3 Quick decisions often breed regret. 4 H e ’ll have a bucket of tears for a cup of joy. 5 Harry in haste, repent in Leisure. ( ) 30 DRUuVJI93? is to PLAY as COMPOSER is to- i orchestra 2 piano 3 symphony 4 performance 5 concert ( ) 31 Which of the following statements together prove that 'TODAY IS COLDER THih YESTERDAY? I Ivory Friday this month was a cold day. oc. To-morrow is the first day of the month. 3 Last Thursday was a hot day. HA- The last day of each month this year has been the coldest day in the month 5 Summer is nearly over. I fugitive 2 enemy 3 evacuee 4 escapee p prisoner 6 truant ( ) 33 Which two of the following statements mear. most nearly the same? 1 A great fortune is a great slavery 2 Dexter beans and bacon in freedom than cakes and ale in bondage. 3 Put a chain round the n .ck of a slave and the end fastens round your own 4 Lean liberty is bettei chan fat slavery. 3 Stone walls do not a prison make. UNIVERSITY OF IBADAN LIBRARY - 346 34 In certain oode the English word BOARD is written oODVI. Ifihnt the English word PAT •'an in this code? 33 Which two of the following statements mean most nearly the same ? I Forwarned is forearmed. ? The loss which is unknown is no loss at all 3 Do man is happy that noes not think so. 4 Uneasy lies the head that wears a crown. 5 Where ignorance is bliss, 'tis felly to "be wise. 36 BUTTLE is to DUEL as CHORUS IS to- I twins 2 duet 3 selection 4 music 5 Song. ( LOOK BACK OVER YOUR WORK. UNIVERSITY OF IBADAN LIBRARY -34&- APPENDIX 9 A.C.E.R. MQ TEST Name: ............... . . i ............................ ............*..............Age now: Date of Test: ....... . Birthday: School: ........... .............. . Class: .. ... . ■ ..............- ........ . .— .. . .. This is a test to see how well you can think. It contains questions of different kinds, Some exarrples and practice question will be given to show you how to answer the questions. EXAI^LE: (a) Find out how the following numbers go Write the missing numbers in the brackets: 2 5 8 * 1 4 17* 23 .. .. (11 & 20) Question 1: Find out how the fallowing numbers. Write the missing numbers in the brackets. 4 5 6 * 5 * 7 10 * .. .. (8 & 9) Question 2: Find out how the following numbers go. Write the missing numbers in the brackets. 1 3 5 7 * 11 * 15 . . . . (9 & 13) Question 3.: Find out how the following numbers go. Write the missing numbers in the brackets 26 23 20 17 14 * 8 * .. .. (11 & 5) EXAMPLE (b) Find the number which should be in the square with the question mark and write it in the brackets 3 5 7 6 8 10 11 ? Question 4: In this table two numbers are missing, find the number which should be in the square with the question mark and write it in the brackets. 2 5 9 6 • 13 11 14 UNIVERSITY OF IBADAN LIBRARY cn - 3 5 0 - Question 5 . %_ Find the number which should be in the q square with the question mark, and write it in the brackets. 1 — 3 “ 1 ------- • • I* . 7 (19)3 . . jL - 7 Question 6. ’W -** the number which, should be in.the eauare with the question mark, and write it in the orackets. • 13 . 9 I 15 1 1 ? l 9 . • 1 You will have 20 minutes to do the test. Some questions are easier than others. Try each question as you come to it, but if you find any question is too hail, leave it out and come back to it later if you have time. S .f A It T ho not spend too much time on any one question. Try to get as many right as possible. 1. Find out how the following numbers go. Write the missing numbers in the brackets; 1 5 - 13 - 2 21 25 29 .......... (9 & 17) 2. What change should I got from a ricte if I buy two theatre tickets @ 2 5k " . . . . . . (50k) I 3. Find the number which should be in the square with the question mark, and write it in the brackets. 2 1 5 7 5 10 (1 1 ) 12 ? ________ u, Find out how the following numbers go. Write the missing numbers in the brackets. 19 9 18 6 - 7 16 - .......... (17 & 6 ) 5. Oliver is +hree times as old as his sister Pat. Their father who is 35» is seven times as old as Pat. How old is Oliver ? ( 1 5 yrs.) UNIVERSITY OF IBADAN LIBRARY Find the number which should be in the square with the question and write it in the brackets. I & 10 . 17 ... 8.. 19 (23) I 12 16 ? Find out how the following numbers go. Write the missing number in the brackets; 512 256 128 6 4 - T6 - 4 •• (32 & 8) Which of the followirg prices for oranges is the cheapest? (?) 5k each; (?> ?0 for 45k; (3) 5 for 24k; (4) 4 for ?8k (5 ) 3 for 12k (3 for 12k) Find the number which should be in the square with the question and write it in the brackets. | 32 2 ( 6 ) L * 16 4 f 96 -24___ ? Find out how the following numbers go. White the missing numbers in the brackets; 87 78 76 67 65 56 54 - .. (65 & 45) The total cost of ten books bound in leather is 820,00. Each book in an ordinary edition costs one Naira. How much extra do I pay on each book for the leather binding? ( HTO.OO) Find the number which should be in the square with the question mark, and write it in the brackets. .. 2 .....4 _ r 8 6 . 9 . . .. (72)2 4 16 ___ •> John and lary are twine whose ages together are half their mother’s Their father, who is three years’ older than their r her, is 51. Hew old is John ? .. .. ( 12) Find the number which should be in the square with the question mark, and write it in the bracket. 1 3 25 . to9. .2 __ _z__ ? (13) UNIVERSITY OF IBADAN LIBRARY - 352 T5. It took me four times as long to climb a mountain 6000m high as it took me to come down. I decended 3000m in an hour. How many hours did it take to climb u p ? .. .. { 8 *) 16 * Find the number which should be in the square with the question mark, and write it in the brackets. 1 0 9 .. 4 ... 12 36 ? 48 144 17. What are two numbers whose stun is 16, such that the first divided by the second gives three ? ,. .. »* (4» 12) Find out how the following numbers go. Write the missing numbers in the brackets 0 ~ 3 5 6 8 - 11 .. •» (2 . 9 ) 19. Find the number which should be in the square with the question mark and write in the brackets. r! r .... Ts 9 ........ 5 . . ..7 5 ? . . ( 3 ) i z z ? 0 1 20. Find out how the following numbers go. Write the missing numbers in the brackets? 4 8 7 - 13 26 - .59 .. .. .♦ (44 & 25) 21. If nine framed pictures cost H£7«00: and each picture unframed only costs ono-third as much, how many unframed pictures could I buy for the same money ? .« ( 2?) 22. Find the number which should be in the square with the question mark, and write it in the brackets. 4 4 ..... (5 8 • ........ 12 ? 23. Find ov c how the following numbers go. Write the missing number in the brackets. 1 3 - - 81 243 729 ................ (9 &’ 2 7) 24. I bought an equal number of 5k magnazined and 2§k exercise books, which cost me 45k altogether. How many of each did I buy? .. •• •• ( 6 ) 25. Find out how the following numbers go. Write the missing number in the brackets. 41 35 30 26 21 20 • 0 • • 0 • C 23 } —A •00 UNIVERSITY OF IBADAN LIBRARY - 353 - 2 6 # A vegetable farmer finds that by selling his carrots at 40k per kilogram, he rakes exactly the same profit as by selling at 30k per br oh, what is the average weight of each bunch of his carrots ? . . •• •• (it kg,) 27, A furniture dealer bought 12 chairs at M 4 8cOO. In selling them, he received as much for two chairs as he had paid for three chairs# What was the selling price for the twelve . chairs ? •• •• c# •• (H7 2 .0 0) 28* Find the number which should be in the square with the question mark, and write its number in the brackets# 16 3 "— 5----- • ( 3 )2 2 _____ 2____ 3 ? 29# I can buy 5§kg0 of potatoes for S3#30k, How much do I pay for 4§kg, ? .. •• •• O#2*70> 30, In a class of 46 pupils, there are 8 more boys than girls; How many beys are there ? ,• , • (27 boys) 31# Find the number which should be in the square with the question mark, and write it in the brackets. 0 . __ 1.. ft 6) 18 2 27____ • 24 .... 32, Three new books cost 45k, 90k and 81,05 respectively. If I buy ther second-hand, I only pay two thirds of the new price# How much monqv do I save 7 •• •• (8®k) 33# A piece of wood 35cm long is to be cut in three parts, each successive part being twice as long as the previous part# What is the length of the longest ? #. •• (20cm) 34* A Kitten is 3 days old and a puppy is 11 days old. In how many days will the puppy be twice as old as kitten ? ( 5 days) 35# A daix^ serves a mixt'ire of two parts cream and three parts milk# How maiy litres of cream will it take to make 15 litres of the mixture ? «. •# •• •• (6 litres). 36. Find out.how the following numbers go# Write the missing numbers In the brackets, 87 7 4 63 54 47 - 39 • • (42, 38) UNIVERSITY OF IBADAN LIBRARY 3 St- - APPENDIX 10 Attitude questionnaire to be completed by pupil's ■ _ in secondary rchools oft the use of eloctrooic calculator P lNeaamsee , ofc oSmchpoleotl:e . .t.h..i..s.. ...s..e..c...t.i. .o..n...:............... Address of S c h o o l ......... ......................... Male/Pemale:................ Age: C la ss:.................................................................................. ...................... This i s not an examination. P lea se , g ive your honest opinion on each item . You should only th ick under the most appropriate response according to the format. SA - 5 Strongly Agree A - 4 Agree U - 3 No Opinion (Undecided) D - 2 Disagree SD - 1 Strongly Disagree. ATTITUDES TOWARD MATHEMATICS 5 4 3 2 1 SA A u. D SD Mathematics i s one of the subjects I have always enjoyed studying I approach mathematics c la ss with a fe e lin g of h es ita t io n resu ltin g from fear of not being able +o do mathematics. » I l ik e mathematics, and I am happier in a mathematics c la ss than in any other c la s s . Mathematics i s very in te r e s tin g to me because I enjoy working with numbers* Mathematics makes me fe e l secure, and at the same tim ^t i s stim ulating. When I hear the word 'M athematics', I have fe e lin g of d is l ik e . My mind goes blank, and I am unable to think Mclaethacrmlny.+ .io orn r-emember anything when doing UNIVERSITY OF IBADAN LIBRARY 355 c: __4J 3 2 1 SA~ A " U D SD 8. I feel a sense of insecurity when attempting Mathematical problems. 9. I feel at ease with Mathematics and I like it very much. 10, Mathematics makes me feel uncomfortable, restless, irritable, and impatient. $1. I am poor in formulae and they scale me whenever I do Mathematics. 12. I always react positively to Mathematics because it is enjoyable. B. ATTITUDES TOWARD THE USE OP ELECTRONIC CALCULATORS IN MATHEMATICS 1, Calculator increases one's computational skill in mathematics, schools should encourage it’s use. 2, I think four figure table is more useful in the classroom and in examinations, calcu­ lator will not be necessary. 3o Calculators will make one lazy, it should not be used in Mathematics. 4* I will like to see people use more of calculator s« 5. Sometimes I feel that the calculators are • desirable and sometimes I doubts ijt„ 6. The calculator is one of the few tilings I en.ioy using in Mathematics. 7» Pupils who use calculators in the classroom and in examinations should be punished. 8, Many pup .is lack the ability to do simple calcul. eons, so calculators can be useful. 9* The use of calculators by pupils in schools should be decided by teachers alone. 10, Pupils should be permitted to use calculators only in their final examinations. UNIVERSITY OF IBADAN LIBRARY - 35 6 - . _ . . . . V * 5 4 3 2 1 SA A u D SD 11« There would be very little progress in Mathematics without the calculator. 12, The computational advantage, .of the calculator is bound to weaken the mental ability of those who use it. 13. As of now calculators are completely bad for school pupils. 14. Calculator will help me in solving Mathematics ^problems. | UNIVERSITY OF IBADAN LIBRARY 357 AP PE ND IX 11 Internal consistency reliability coefficient of the attitude meaaure N o . X (Odd) ! Y (Even) X2 Y2 | XY 1 . 105 j 101 11025 10201 '10605 ij 2 . 101 99 10201 9001 9999 3. 96 ; 95 9216 9025 ! 9120 S X Y/= 126779 4. 94 92 0036 0464 8648 n £(XY) = 2535^00 5. 09 00 7921 7744 7832 ^ X ^ Y = 2453650 6 . 00 | 06 7921 7396 7568 N|X2 = 2507100 7. 05 04 7225 7056 7140 NjY2 = 240°100 0 . 04 83 7056 6009 b972 ( 2 X)2= 2505009 9 . 6561 6241 81 79 6399 ( £ Y )2 = 2402500 1 0 . 70 6004 5929 600677 11 . 77 77 5929 5929 5929 1 2 . 77 77 5929 5029 5929 13. 76 76 I 5776 5776 5776 ! 14. 76 5776 5776 5776 I 76 15. 72 71 j 5104 5041 5112 I 16. 69 69 | 4761 4751 4761 17. 69 60 ! 4761 4761 4692 1 0 . 6' j 57 ; 4096 4761 4692 19. 67 52 | 3249 2704 2964 2 0 . 45 44 • 2025 | 1936 j 1900 i .. ■? ! UNIVERSITY OF IBADAN LIBRARY 356 Computing: = NC> XY) - (^.X) (^ Y) l/ffifc*2) - (£.X)2(N£Y2) - CL Y) ^ « 81900 83862.219 = 0.98 Y = 0.98 (reliability coefficient of odd and Even) (Split half reliability) * Using Spearman-Brown formula, R = N(r)/(1 + (N-1)r) R is the reliability co-efficient R = Nr/1+(N-1)r) = 19.6/19.62 = 0.99 UNIVERSITY OF IBADAN LIBRARY -359- APPENDIX 12 Correlation between 27% Upper Score and 27% Lower Score on Attitude Scale (MAS And CAS) 27% Upper Score 27% Lower Score 0 D2 S/R X Rank of Y Rank of X-Y X Y 1 105 10 69 9 1.0 1 2 101 8.5 69 9 -0.5 0.25 3 101 8.5 69 9 -0.5 0.25 4 99 7 7 7 0 0 5 96 6 6 6 0 0 6 95 5 4.5 4.5 0.5 0.25 7 94 4 4.5 4.5 -0.5 0.25 8 92 3 3 3 0 0 9 89 2 2 2 0 0 10 88 1 1 1 01 0 Using Spearman - Rank Order Correlation Coefficient r - 1 - t o t s !) n ■ 10 0.98 UNIVERSITY OF IBADAN LIBRARY X -i< '""to - 360 - APPENDIX 13 Significant m a m difference in mathematics attitude cco-re Ci'iAS) and ccicuiat i attitude score UV-.S) -"or' t?0, uop -r score SEX M A S C A SAGE S/N IN YEARS m7F X Y Y2 1 . 17 M 53 2809 52 2704 2 . 18 M 49 2401 52 2704 3. 17 M 44 1936 57 3249 4. 16 F 45 2025 54 2916 5. 1 B F 34 1156 62 3844 6 . 14 F 43 1849 52 2704 7. 17 M 49 2401 45 2025 8. 15 F 51 2601 41 1681 9. 16 F 51 2601 38 1441 1 0 . 17 F 26 676 36 1 2d6 Mean Age = 16 .2 yrs. X = X/N = 445 /10 N = 10 Y = EY/N = 489n r = 48.9 • a 2 X = IX2 N E X 2 20455 - —19S8-0-2-.-5N-1 TF T" ~ 9 = 72.5 0 2 = £ Y^ y N-1 - N E y ’ 24564 2397N-1 = — g— " - --- g— 2. 1 = 72.43 t-ratio -X - X = 48.9 - 44.5 4.4 ~2---- 2 d x - 6 W t1r / 7 2.5 y 14."4 9 3 y / ~ W v 1.16 at X 05 UNIVERSITY OF IBADAN LIBRARY - 3 6 1 - APPENDIX 14 Significant mean Difference in Mathematics attitude Score (MAS) and Calculated Attitude Score (CAS) for ( 27% Lower Score AGE SEX MAS CAS S/N IN YEARS M/F X Xa Y Y2 1 16 M 4 1 1681 28 784 2 16 F 43 1649 26 676 3 16 F 47 2209 22 484 4 18 M 21 4441 47 2009 5 16 F 31 961 33 1089 6 17 M 35 1225 22 484 7 15 F 33 1521 18 324 6 19 W 28 784 24 576 9 17 M ' 21 441 24 576 10 15 M 21 441 23 529 Mean Age - 16. 6 years N * 10, N - 10 307/1G - 30.7, y X = e n /n - * X = 30.7, Y = 26. 7 a 2 e x 2 NEX2 _ 11553 9424.9 X N -1 N-1 9 ~ 9 = 236. 5 a_ 2 .1 Y2 NEY2 7731 7125.9 602,1 y |\u 1 ~ N-1 9 9 T t - ratio = X - Y - 3Q.7 - 26.7 / a 2 + a2 J 236.5 + 66.9 __21 JL N ~ W "TITx N'y t-ratio = — — --- = 0.73 at a 0.05 V/ " T O T UNIVERSITY OF IBADAN LIBRARY u c n CD to 3G2 APPENDIX 1b Internal reliability co -e ff irient of mathematics pr^-t^st scores S/N AGE SEX SCORE OUT OF 15 1 . 14 F 5 2 . 14 M 11 3. 18 M 7 4. 18 M 12 5. 18 M 14 6 . 16 F n3 7. 16 F 7 8. 14 F 6 9. 17 M 9 1 0 . 13 F 10 1 1 . 15 M 3 1 2 . 17 F 4 « No. of test-items = 15 RANGE = 3 - 1 4 MEAN X = 8.00 ST/ .JARD DEVIATION, SD: 3.35 RELIABILITY COEFFICIENCE, r = 0.67 F emale 6 , Male 6 . UNIVERSITY OF IBADAN LIBRARY 363 APPENDIX 16 Questionnaire directed to teactiers of mathematics in the secondary school. ■J 5 V » PLEASE COMPLETE: Name of School _______ ______________ __ The School was established in 19_________________ __________ Your Sex: _______________ Your Age in Year* ______________ Your qualifications to date: _ _______ ___________ _____ Your years of experience in the teaching oi mathematics: Classess taught mathematics to date: Class No. of Years V ______________ IV ______________ III ______________ I & II ______________ One would appreciate your responses to the following items please complete the correct response where applicable: 1 . If you teach mathematics in forms IV or/and V? do you always cover the mathematics syllabus fully or partially in pre -/ring your schemes of work? underline partially or fully. UNIVERSITY OF IBADAN LIBRARY - 364 - Which of the following types of equations have you taught in your class? Equations Class Ci) Simole (ii) S imultaneous (iii) Quadratic 3. List the types of instruments used by you or your students in facilitating quick and reliable numerical calculations. 1 . 4. 2 . 5. 3 . 6 . 4. If you teach in forms IV or/and V you would assess your student's performance in mathematics as (i) above average, C i i D average and (iii) below average: Class arm Assessment £. You would consider the electronic calculator as (i) very (i) effective (ii) effective (iii) not effective teaching learning aid. UNIVERSITY OF IBADAN LIBRARY - 365 - 6 . If you consider the electronic calculator as an effective teaching - learning aidj indicate in which class or classes it can be used.* * Face and content validity of the questionnaire were carried ou by the author and some lecturers in the Teacher Education department, University of Ibadan. UNIVERSITY OF IBADAN LIBRARY - 3 66 - APPENDIX 17 Data from Study GROUP A VARIABLES VAR 01 VAR 02 VAR 03 VAR 04 VAR 05 VAL 06 MAT PEA PET POA POT MAS F 42 79 9 78 25 49 M 37 64 10 86 26 47 41 M 52 72 12 106 22 53 53 M 33 77 11 103 1 3 49 54 F 37 74 9 115 18 55 60 F 38 105 fl 99 12 49 50 M 44 74 14 78 24 59 19 F 37 87 6 90 20 46 44 M 39 d3 11 107 23 54 53 M 49 78 4 109 24 50 59 M 39 30 9 79 19 56 23 F 39 60 13 77 20 ‘ 43 34 F 37 60 6 77 15 27 F 45 93 13 54 23 51 Male (7) Female (7) UNIVERSITY OF IBADAN LIBRARY - 367 VAR 01 : MAT Mental ability TEst Scores (72) VAR 02: PEA Pr8-Attitude questionnaire scores (130) VAR 03: PET Pre-test scores '15) VAR 04 : POA Post-Attitude questionnaire scores (130) VAR 05: POT Post-test scores (30) VAR 06 : MAS Mathematics Attitude scores (60) VAR 07: CAS - Calculator Attitude Scores (70) UNIVERSITY OF IBADAN LIBRARY 366 GROUP B SEX MAT PEA PET POA POT MAS CAS M 34 78 10 86 19 54 32 F i 36 74 7 87 16 54 33 M 34 75 10 113 17 56 57 F 30 56 8 74 15 4r 26 M 35 78 5 74 12 41 33 F 36 63 5 83 19 56 27 F 30 104 9 103 20 49 54 M 35 89 6 111 17 50 61 F 30 85 9 77 22 50 27 F 36 7“7 6 72 19 54 16 M 3? 87 12 98 15 51 47 F 35 83 7 89 18 41 48 ' M 31 87 8 81 16 38 4 3 M 36 78 1 1 123 17 57 66 MALE (7) FEMALE (7) UNIVERSITY OF IBADAN LIBRARY 369 GROUP C Nos. SEX MAT PEA PET POA POT MAS CAS 1 . M 26 79 2 83 11 45 38 2, F 27 80 9 103 11 38 65 3. M 27 78 11 93 17 57 36 4. F 26 75 6 84 11 45 39 5. M 27 68 11 84 13 31 53 6. M 23 72 11 80 16 48 32 7. F 27 27 6 72 12 35 37 e. M 29 90 6 74 14 37 37 9. F 29 67 5 73 12 38 35 10. F 2 - 72 9 100 14 55 45 1 1 . F 28 32 4 89 20 44 45 12. M 27 86 7 77 12 49 28 13. M 28 77 5 59 16 30 29 14. F 26 78 4 78 19 * 42 36 MALE (7) FEMALE (7) UNIVERSITY OF IBADAN LIBRARY r 370 - - GROUP 0 NOS. SEX MAT PEA PET POA POT MAS CAS 1 . F 47 72 11 81 18 47 34 2. M 44 88 12 85 1 6 51 34 3. F 46 79 13 91 17 53 38 4, M 42 58 13 86 9 49 37 5. F 53 69 15 86 20 46 40 6. M 54 S4 12 78 16 44 34 7, F 50 74 11 78 21 51 27 a. F 50 84 14 83 17 50 33 9. F 45 63 12 78 19 53 25 10. F 48 85 9 93 11 47 45 1 1 M 45 85 10 82 17 51 31 12. M 50 71 13 71 17 51 20 13. M 50 73 12 77 14 53 24 14. M 46 76 12 69 12 ' 49 20 MALE (7) FEMALE (7) UNIVERSITY OF IBADAN LIBRARY 371 - GROUP E MAT PEA PET POA POT MAS CAS F 47 72 11 81 18 47 34 M 44 88 12 85 16 51 34 F 46 79 13 91 17 53 38 M 43 58 13 86 9 49 3 7 F 58 69 15 86 20 46 40 M 54 64 12 78 16 44 34 F 50 74 11 78 21 51 27 F 50 84 14 83 17 50 33 F 45 63 12 78 19 53 25 F 48 06 9 92 11 47 45 m 45 85 10 82 17 51 31 M 50 71 13 71 17 51 20 M 50 73 12 77 14 53 24 M 46 76 12 69 12 49 20 MALE (7) FEMALE (7) UN mI XVERSITY OF IBADAN LIBRARY 372 GROUP F NOS. SEX MAT PEA PET POA POT MAS CAS 1 . M 30 99 10 71 12 57 14 2. M 32 08 6 90 13 56 34 3. M 30 82 7 83 17 52 31 4. M 29 81 5 80 5 54 26 5. M 32 80 7 82 8 56 26 6. M 32 112 10 119 10 57 62 7. F 2B 79 7 84 12 52 32 a. F 31 72 6 58 8 32 26 9. F 30 6R 7 68 10 46 22 10. F 30 97 5 75 11 55 20 1 1 . F 20 89 6 82 6 36 46 12. M 27 95 6 82 9 44 38 13. F 25 81 9 77 10 48 29 14. F 32 105 2 86 6 48 38 MALE (7) FEMALE (7) UNIVERSITY OF IBADAN LIBRARY 373 GROUP G SEX MAT PEA PET POA POT MAS M 44 85 9 97 9 52 M 52 90 7 97 9 51 46 M 48 88 6 72 8 51 21 M 45 72 5 80 14 48 32 M 51 78 9 79 12 50 29 M 49 99 5 95 9 49 46 Ml 43 61 5 77 7 32 45 F 43 102 11 96 12 51 45 F 41 93 14 89 9 55 34 F 41 95 15 77 12 41 36 F 51 S3 14 100 14 53 47 F 47 94 15 107 25 47 5 0 rc~ 45 79 13 81 20 52 29 F 52 83 15 79 17 53 26 MALE (7) FEMALE (7) UNIVERSITY OF IBADAN LIBRARY - 374 GROUP H NOS. SEX MAT PEA PET POA POT MAS CAS 1 . M 39 114 11 105 19 55 50 2 . M 37 84 11 00 12 47 33 3. M 40 98 13 91 19 48 43 4. M 40 63 8 79 14 43 36 5. M 37 88 4 83 12 45 36 6. M 35 95 8 82 18 42 40 7. M 40 84 7 74 10 45 29 8. F 39 79 14 78 18 42 36 9. F 37 101 13 113 14 52 61 10. F 36 73 9 72 11 48 24 11 . F 40 78 12 80 14 47 33 12. F 38 02 7 78 11 45 33 13. ci 40 82 12 80 14 49 31 14. C? 34 84 9 91 12 .47 44 MALE (7) FEMALE (71 UNIVERSITY OF IBADAN LIBRARY GROUP I NOS. SEX MAT PEA PET POA POT MAS CAS 1 . M 33 70 13 82 19 50 32 2. M 33 92 2 76 8 31 45 3. n 30 90 3 65 7 35 30 4. M 33 76 4 82 8 48 34 5. M 26 74 5 77 8 46 29 6. M 27 94 13 75 16 51 24 7. M 25 77 12 74 14 54 20 6, F 27 64 5 50 6 20 30 9. F 33 94 4 G5 9 47 18 10. F 33 67 5 104 6 46 56 11 F 31 72 6 78 7 36 42 12. F 33 03 6 79 12 29 50 13. F 23 104 9 105 or- 47 58 14. F 25 69 11 81 14 47 32 MALE (71 FEMALE ( 7 ) UNIVERSITY OF IBADAN LIBRARY 37b APPENDIX 16 Data xrom the Pilot Study GROUP A NOS. SEX VARIABLES VAR 01 VAR 02 VAR 03 VAR 04 VAR 05 MAT ATS ACT MAS CAS 1 . M 400 89 7 52 37 2 . F 38 101 8 13 52 3. F 37 74 10 38 36 A. M 43 76 9 47 29 5. F 34 79 7 45 34 6 . N 43 67 1 6 49 18 7. M 44 60 9 31 29 9. M 44 66 14 39 27 9 . F 33 73 12 44 29 1 0. F 33 76 11 43 33 MALE (5) FEMALE (5) VAR 01 : NAT - Mental abillty scores (72) VAR 02 ! A" - At t i t u d e questionnaire scores (130) VAR 03 : ACT - Post-ach ievement Scores (30) VAR 04 : MAS - Mathematics Attitude Scores (60) VAR 05 : CAS - Calculator Attitude Scores (70) UNIVERSITY OF IBADAN LIBRARY 377 GROUP B NOS. SEX MAT ATS ACT MAS CAS 1 M 29 86 9 36 48 2 . F 30 69 10 41 28 3. M 30 57 7 35 22 4. F 29 87 5 47 40 5. F 29 36 12 35 51 6 . M 28 101 9 44 57 7. F 28 90 5 44 46 8. F 26 31 8 44 37 9. M 26 69 6 43 26 1 0 . m 27 95 7 43 52 MALE (5) FEMALE (5) GROUP C NOS. SEX MAT ATC ACT MAS CAS 1 F 23 52 4 28 24 2 . M 22 84 4 43 41 3. M 22 97 6 48 49 4. F 18 92 9 51 41 5. I 21 77 5 41 36 6 . M 22 98 7 50 48 7. F 18 99 5 45 54 8 . F 16 78 9 58 20 9. F 23 64 4 31 33 1 0 . M 21 77 3 36 41 MALE (5) FEMALE (5) UNIVERSI Y OF IBADAN LIBRARY 376 GROUP 0 SEX MAT ATS ACT MAS CAS M 35 76 9 39 37 M 36 88 11 28 60 M 36 94 14 49 45 M r 7 77 13 41 36 M 42 63 8 30 33 F 32 —/* J 6 4b 30 P 34 58 14 36 22 F 32 31 10 44 37 F 32 89 7 51 38 F 44 96 6 34 62 MALE (5) FEMALE (5) UNIVERSITY OF IBADAN LIBRARY 379 GROUP E NOS. SEX NAT ATS ACT NAS CAS 1 M 28 76 10 46 30 2. E 22 72 9 35 38 3. M 28 72 11 20 52 4. F >7 88 7 47 41 5. E 28 68 8 43 25 6. M 29 79 9 4 8 31 7. F 2/ 58 10 34 24 8. M 3D 73 12 52 21 9 . F 29 75 12 52 23 10. N 27 76 6 42 34 MALE (5) FEMALE (5) l UNIVERSITY OF IBADAN LIBRARY GROUP F SEX MAT ATS ACT MAS CAS M 20 88 26 62 F 23 105 5 53 52 F 23 85 2 45 40 M 22 73 7 45 20 F 19 76 8 56 20 F 20 76 10 39 37 F 19 69 7 47 22 M 20 57 8 39 18 M 21 70 5 52 26 M 19 83 4 50 33 MALE (5) FEMALE (5) UNIVERSITY OF IBADAN LIBRARY 331 - GROUP G SEX MAT ATS ACT MAS CAS M 46 77 6 58 19 F 35 84 8 46 38 M 36 82 7 52 30 F 33 88 13 54 34 M 32 79 6 50 29 F 40 72 11 *'■1 31 p 4^ 71 4 40 31 M 33 66 13 21 47 M 33 68 19 38 30 F 33 77 16 43 34 MALE (5) FEMALE (5) UNIVERSITY OF IBADAN LIBRARY 382 GROUP H SEX MAT ATS •ACT MAS CAS M 27 45 11 21 24 n 29 92 5 55 37 F 27 81 8 54 37 F 26 76 10 49 27 F 26 84 6 44 36 p 26 80 4 aq 43 M 2 7 92 4 49 43 M 28 85 14 48 37 F 29 44 12 21 23 M 30 47 g 32 15 MALE [5) FEMALE (5) UNIVERSITY OF IBADAN LIBRARY - jea - GROUP I SEX MAT ATS ACT MAS CAS M 21 82 5 48 34 M 13 58 7 38 30 F 20 75 7 50 25 F 20 99 6 39 60 M 22 78 3 36 42 F 21 79 4 50 29 F 20 65 9 47 10 M 22 90 5 44 46 M 19 50 4 30 20 F 13 70 6 40 30 MALE (5) FEMALE (5) UNIVERSITY OF IBADAN LIBRARY 364 APPENDIX 10 Li-sN; of-sGCOfl-dary schools rn Ibadan iiurrictpality aainple-G for the stuq,y Name of School YearEstablished 1. Mount Olivet Grammar School, Bodija, Ibadan 1965 2. *-* *Hnly Trinity Grammar School, Old Ife Road, xbadan 1966 3. Bishop Phillips Academy, Iwo Road, Ibadan 1964 3. *Adekile Goodwill Grammar School, Aperin, Ibadan 1964 5. C.A.C. Grammar School, Aperin, Ibadan 1960 6 . Adelagun Memorial Grammar School, Odinjo, Ibadan. 1967 7. ^ * I b a d a n City Academy, Eleta, Ibadan. 1946 6 . *Ahmadiyya Grammar School, Eleyele, Ibadan 1955 9. Baptist Grammar School, Idi-Isin, Ibadan 1966 10. Ibadan Grammar School, Molete, Ibadan 1913 11. Methodist High School, Express Road, Ibadan. 1961 12. Eyinni High School, Lagos Road, Ibadan 1966 13. Renascent High School, Aremo, Ibadan. 1964 14. African Church Grammar School, Apata, Ibadan! 1960 15 .* Islamic High School, Basorun, Ibadan 1957 16. Oke ’Badan High School, Oluyoro, Ibadan. 1964 * F: s schools selected for pilot and main study ** Three schools selected for the main study UNIVERSITY OF IBADAN LIBRARY 385 APPENDIX 20 Analysis of Variance of Post-attitude Scores of H.MA, ANA and LHA Groups *■ SOURCE df SUM OF MEAN F* SIGN IF SQUARES SQUARES RATIO OF F MAIN EFFECTS 2 822.9375 411.4688 2.418 0.091 ns EXPLAINED 2 822.9376 411.4689 2.410 0.091 ns RESIDUAL 123 20942.000 170.2602 TOTALQ 125 21764.9376 174.120 ns : Not Significant at F = .04 Anal ysis cf Variance of Post-Attitude Scores of UCU, RCU and NCU Groups SOURCE DF SUM OF MEAN F- SIGNIF SQUARES SQUARES RATI0 Or F 1 MAIN EFFECTS 2 471.443 235.722 1 .218 0.299 ns EXPLAINED 2 471.445 235.723 1 .218 0.299 ns RESIDUAL 123 23811.625 193.590 TOTAL 125 24283.070 194.265 ns Not Significant at P .05 UNIVERSITY OF IBADAN LIBRARY