PRACTICE VERSUS GRAPHICAL REPRESENTATION FOR MAINTENANCE OF BASIC ARITHMETIC COMPETENCIES: FIRST YEAR PRIMARY by DOROTHY FORREST JOHNSON B . A . , McGlll University, 1938 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Faculty of Education Department of Mathematics Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada Date UsuJLi <j M 7/ ABSTRACT Educators such as Edith Biggs i n Britain and Vincent Glennon and the Cambridge Conference on School Mathematics i n the United States have suggested that the amount of time children spend on direct practice of newly learned s k i l l s and understandings can be greatly reduced. The Americans propose an Integration of this practice with the presentation and learning of new topics. The B r i t i s h favour an acti v i t y approach, where new learnings are put to immediate use, and the need for acquisition and perfection of mathematical competencies becomes obvious to the children. A few Ameri-can research studies have substantiated the merits of reduced practice, at the Intermediate level. This study explores the place of practice for main-tenance of the basic competencies of F i r s t Year Primary children i n B r i t i s h Columbia at the end of the school year. The competencies chosen for study were 1) Numeration: reading, writing and understanding of base ten numerals ^ ,99» and 2) Computation! addition and subtraction operations with sums and minuend s ^ 1 0 * The new material, chosen to be presented as an al t e r -native to direct practice, was Graphical Representation, a unit developed from the Nuffield Project booklet, P i c t o r i a l Representation T i l » Two schools i n the Vancouver area were? used, i l l the f i r s t with a class of 5 ^ children and the second with 3 ^ . Parallel pre-tests and post-tests In the basic compe-tencies were administered. During a three week Intervening interval, the Investigator taught the children, who were divided Into groups,by random selection, as follows» In the f i r s t school, three groups of 18 children were Instructed respectively i n Graphical Representation, i n review and practice, using familiar materials, and In geo-metry, involving no use of numbers (control group). In the second school, two groups of 17 children were Instructed In Graphical Representation, and In review and practice, respec-t i v e l y . At the end of the experiment, there was no significant difference i n the tested numeration competencies of the two experimental groups In their respective schools. The control group showed a slightly lower achievement. Time did not permit a retention test. In the f i r s t school, where computational efficiency was low, the results slightly favoured the review and prac-tice group, over the other groups. In the second school, there was no significant difference between the two groups, regarding progress In computational s k i l l s . Within Its limitations, this study demonstrates the pos s i b i l i t y of maintaining basic competencies, while intro-ducing new topics, at the f i r s t year level. CHAPTER PAGE I. THE PROBLEM, THE RESEARCH DESIGN, AND DEFINITION OF TERMS 1 Statement of the Problem 1 General View . . . . . . . . . . . 1 Specific Area of Investigation . . . . . . . . 3 Hypotheses • 3 Main Hypotheses . . . . . . . . . 3 Supplementary Hypotheses . . . . . . . . . . 4 Research Design . . . . . . . . . . . 4 Definition of Terms 7 I I . SURVEY OF THE LITERATURE 9 Introduction 9 Research at F i r s t Year Primary Level 9 The Place of Practice—Changing Emphases . . . 10 Three Pertinent Suggestions 13 Glennon Proposals and Related Research . . . 14 Proposals of the Cambridge Conference on School Mathematics . . . . . . . . . . . 18 The Role of Practice In Primary Schools 19 The Place of Graphical Representation . . . . . 22 Conclusion • 23 CHAPTER PAGE III . PROCEDURES 25 Introduction • 25 Materials Used 26 Organization . . . . . . . . . . . . . 27 Evaluation Instruments . . . . . . . . . . . . . 28 Basic Competencies, Tests I and II 28 Graphical Representation, Test III . . . . . . 29 Pre-Testing . . . . . . . . • 31 School #1 32 School #2 32 Teaching A c t i v i t i e s . . . . . . . . 33 Graphing Unit as carried out i n School #2 33 Graphing Unit as carried out In School #1 36 Review and Practice Unit as carried out In School #1 38 Review and Practice Unit as carried out i n School #2 39 Geometry Unit as carried out i n School #1 ; ko Confounding Factors 4-0 Computational Practice . . . kO Exposure to Graphing Unit • *K) CHAPTER PAGE Post-Testing . . . . . . . . . . . . 41 School #1 41 School #2 42 IV. STATISTICAL ANALYSIS OF RESULTS 43 Choice of Analysis of Covariance . 43 Assumptions 45 Main Hypotheses . . . . . . . . . . 46 Test I Numeration • • • • 46 Test II Computation o . . . . . 47 Test III Graphical Representation . . . . . . 47 School #1 48 School #2 48 Analysis of Data 50 Main Hypotheses Regarding Numeration S k i l l s 50 Conclusions t Numeration . . . . . . . . . . . 51 Main Hypotheses Regarding Computational S k i l l s . . . . . 52 Conclusions t Computation . . . . . 55 Main Hypotheses Regarding Graphical Representation . . . . . . . . . . . . . . . 58 Conclusions - School #1 . . . . . . . . . . 58 Conclusions - School #2 59 Supplementary Hypotheses! A b i l i t y Groupings • • 6l CHAPTER PAGE School #2 61 Numeration • 62 Supplementary Hypotheses . . . . . . . . . . 63 Conclusions1 Numeration,Ability Groupings • 64 Computation 67 Conclusions! Computation, A b i l i t y Groupings • 67 Graphical Representation, A b i l i t y Groupings . • 69 Summarized Results, Conclusions and Implications . . . . . . . . . . . . . . . . ^ 72 Numeration 72 Computation . . . . . . . . . . 73 Graphical Representation • 75 Supplementary Hypotheses, A b i l i t y Groupings, School #2 . . . . . . . . . . • 76 V. CONCLUSIONS AND RECOMMENDATIONS 78 Introduction • 78 Summary of Findings and Implications 79 Numeration . . . . . . . . . . 79 Computation 80 Graphical Representation 81 Limitations 81 v l l i CHAPTER PAGE Suggestions for the Future 83 Non-Stat lst lcal Outcomes Regarding Graphical Representation . • 84 Concluding Remarks 86 APPENDICES 88 BIBLIOGRAPHY 159 TABLE PAGE I. Grouping of Students at Start of Experimental Period 28 II . School #1 Achievement and 'Mastery* Scores Pre-Test 38 I I I . Grouping of Students at End of Experimental Period . . . . . . 44 IV. Hypothesis Matrix 46 V. Hypothesis Matrix, School #1, Test III 48 VI. Summary of Test Resultst Numeration . . . . . . 49 VII. Tests for Use of Analysis of Covariance, Test It Numeration 50 VIII. Tests of Main Hypotheses, Test It Numeration . . . . . . . . . 51 IX. Summary of Test Results, Test l i t Computation 53 X. Tests for Use of Analysis of Covariance, Test II t Computation 54 XI. Tests of Main Hypotheses, Test l i t Computation 54 XII. t-Test Matrix for Adjusted Group Means, Test l i t Computation 56 XIII. Summary of Test Results, Test IIIt Graphical Representation . . . . . 57 TABLE PAGE XIV. Analysis of Variance, School #1, Test IIIi Graphical Representation . . . . . . 58 XV. Tests for Use of Analysis of Covariance, Test I I I i Graphical Representation, School #2 . . . 59 XVI. Tabulation of Score Gains, Test #lt Numeration, School #2 60 XVII, Tests for Use of Analysis of Covariance, A b i l i t y Groupings, School #2, Test It Numeration , 62 XVIII. Hypothesis Matrixi A b i l i t y Groupings, School #2 63 XIX. Tests of Supplementary Hypotheses, Test I: Numeration, A b i l i t y Groupings, School #2 . . . 64 XX. Tabulation of Score Gains, Test II t . Computation, School #2 . • 65 XXI, Tests for Use of Analysis of Covariance, Test I I i Computation, A b i l i t y Groupings, School #2 . . 66 XXII. Tests of Supplementary Hypotheses, Test l i t Computation, A b i l i t y Groupings, School #2 . . 66 XXIII. Tabulation of Score Gains, Test l i l t Graphical Representation, School #2 68 XXIV. Adjusted Mean Scores, Test l i l t Graphical Representation, A b i l i t y Groupings, School #2 70 TABLE PAGE XXV. Tests for Use of Analysis of Covariance, Test I I I i Graphical Representation, A b i l i t y Groupings, School #2 • • • . • 71 XXVI. Tests of Supplementary Hypotheses, Test I I I i Graphical Representation, A b i l i t y Groupings, School #2 71 LIST OP FIGURES FIGURE • PAGE 1. Research Design 6 2. Worksheet for Group . . . . . . . . 97 3. Worksheet for Group . . 98 4. Worksheet for Group A 1 99 5. Worksheet for Group A g 101 6. Worksheet for Group • . . . • 102 7. Worksheet for Graphical Representation, Test I I I , School #2 158 LIST OF APPENDICES APPENDIX PAGE A. Plans for Teaching . . . . . . . 89 B. Sample Lesson Plans • . 96 C. Test Manual: Description and Instructions for Administering Tests I and II 103 D. Tests I and I I , Form B (Pre-Test) 136" E . Tests I and I I , Form A (Post-Test) 145 F . Block Chart for Graphical Representation Tests . 154 G. Graphical Representation, Test I I I , Pre-Test Questions • 155 H. Graphical Representation, Test I I I , Post-Test, School #1 156 I . Graphical Representation, Test I I I , Post-Test, School #2 157 ACKNOWLEDGEMENT I wish to thank a l l the members of my committee for their interest and encouragement i n the carrying out of this project. Particular appreciation goes to Mr. Tom Bates for his meticulous proof reading and sure guidance towards clear thinking and writ ing. Most Important, my thanks to my chairman, Dr. Tom Howitz, who has offered u n f a i l i n g encouragement and assistance from the start , and has spent long hours a s s i s t i n g In the organization and consolidation of my material . THE PROBLEM, THE RESEARCH DESIGN, AND DEFINITION OF TERMS STATEMENT OF THE PROBLEM This Is a study of the effects of practice versus the introduction of a graphing unit on the malntainance of basic arithmetic competencies of six and seven year old B r i t i s h Columbia children at the end of F i r s t Year Primary. GENERAL VIEW Today 1s world Is one of change, at an ever accelera-t i n g pace* Educators must prepare children for a world as yet unknown. To t h i s end they strive continuously to improve c u r r i c u l a and methods, i n a l l areas of i n s t r u c t i o n . Mathematics was one of the f i r s t c u r r i c u l a to undergo d r a s t i c r e v i s i o n . The vanguard of reformers had long been l e v e l l i n g the c r i t i c i s m that there was too much rote l e a r n -ing and d r i l l i n the " t r a d l t i o n a l , , programs. More recently the Mnewn mathematics programs have been o r i t i o i z e d for not providing suff icient d r i l l for mastery of basic s k i l l s . There i s J u s t i f i c a t i o n for both cri t ic isms and a problem exists to establish a balance between these two positions. In the United States, Vincent Glennon and Marshall Stone have sought examination of the problem of synthesis of curriculum and method, with due regard to child development and psychology. The latter saldi We need fundamental studies i n psychology and mathe-matics, and ultimately on the combination of these find -ings Into a coherent and ef f i c i e n t program of Instruc-tion. 3 In Brita i n and Canada, Edith Biggs and James MacLean have saldt Our experience with young children has shown that we do not yet know how much mathematics children can learn.* The general problem that this study concerns i t s e l f with can be expressed thusi How and to what extent can we increase both the quality and the quantity of mathematics education while at the same time "Maintaining reasonable com-putational ski118".5 Vincent J . Glennon, "...and now Synthesisi a theoretical model for mathematics education", The Arithmetic Teacher. vol. 12, no. 2, (February, 1965) PP. 134-141. 2 Marshall H. Stone, "What i s Expected of Research", i n Needed Research In Mathematical Education. Howard Eehr, ed., A Summary Report of a Conference held at Greystone, New York, October 30-31i 1 9 6 5 » (New York: Teachers' College Press, Teachers* College, Columbia University). 3 I b l d . p.3. **Edlth E. Biggs and James R. MacLean, Freedom to Learn, (Toronto, Canadai Add!son-Wesley, 19o9), P .4 . 5 I b l d . . p.4 SPECIFIC AREA OF INVESTIGATION I n p a r t i c u l a r an attempt was made t o i n c o r p o r a t e the p r a c t i c e needed f o r maintenance of the b a s i c a r i t h m e t i c s k i l l s and u n d e r s t a n d i n g s of F i r s t Year Primary w i t h the l e a r n i n g of new m a t e r i a l from the T e a c h e r s Guide, P i c t o r i a l R e p r e s e n t a t i o n , £T].^ The r e s u l t a n t maintenance of compe-tency was compared w i t h t h a t of a group of c h i l d r e n who spent an e q u a l a l l o t m e n t of time on t r a d i t i o n a l year-end r e v i e w and p r a c t i c e . A c o n t r o l group was exposed t o no a r i t h m e t i c d u r i n g the same p e r i o d , and competency maintenance a s s e s s e d and compared w i t h the oth e r groups. HYPOTHESES Main Hypotheses I t was h y p o t h e s i z e d t h a t the group i n v o l v e d i n G r a p h i c a l R e p r e s e n t a t i o n would m a i n t a i n a l e v e l of competence comparable t o t h a t of the re v i e w and p r a c t i c e group, d u r i n g a t h r e e week p e r i o d . I t was f u r t h e r h y pothesized t h a t b o t h groups would do b e t t e r than the c o n t r o l group, w i t h r e s p e c t t o maintenance and/or improvement of the b a s i c competencies under examination. The s p e c i f i c hypotheses w i l l be found i n Chapter IV, pages 46 t o 49. N u f f i e l d Mathematics P r o j e c t , P i c t o r i a l R e p r e s e n t a -t l o n m. (Londoni W. & R. Chambers and John Murray, 1967. a v a i l a b l e from Longmans Canada L t d . , Don M i l l s , O n t a r i o ) . I t was hypothesized that children of lesser mathe-matical a b i l i t y might profi t more from the novel program than the more able chi ldren. Sufficient s t a t i s t i c a l data for this analysis was only available for some of the children In the o r i g i n a l experiment. The specific supplementary hypotheses regarding the effect of a b i l i t y w i l l be found In Chapter IV, pages 6 l to 63. RESEARCH DESIGN Two schools were available to the Investigator. School #1 had 56 chi ldren, In a large team teaching s i t u a t i o n . School #2 had a single class of 34 chi ldren. Both schools used a modern meaningful approach to mathe-matics, but teaching methods were very dif ferent. To enable comparisons between groups which had previously been taught the same way, the classes were divided so that both ex-perimental methods were used In both schools. To make the numbers v iable, a control group was set up i n School #1 only. Two areas of basic competence were chosen for study 1 1) Numeration, involving reading, writing and understanding of base ten numerals ^ 1 0 0 . 2) Addition and subtraction operations. Pre-tests and p a r a l l e l post-tests In these areas were ad-ministered to the children In the two schools. An attempt was also made to measure a b i l i t y to obtain Information from a block chart, as developed i n the unit on Graphical Repre-sentation. To minimize Hawthorne effects, a l l teaching and testing was carried out by the Investigator. The three teaching groups were set up as followsi Group A (experimental) was introduced to the unit P i c t o r i a l Representation. \1\, with the Investigator incorporating use of the s k i l l s under study, wherever possible. Group B (experimental) was instructed, as a review u n i t , i n practice of the above s k i l l s , using, as far as possible, the methods and materials to which the children were accustomed. Group C (control) was Instructed In an open ended geometry u n i t , care being taken to avoid counting or a d d i -t i o n and subtraction. Children were assigned to their respective groups with the aid of a computer-generated l i s t of random num-bers. The resultant research design i s shown i n Figure 1. School # 1 School # 2 New Material (Graphical A l Representation) c Review and B 0 Practice 1 2 Control °1 (Geometry) FIGURE 1 RESEARCH DESIGN DEFINITION OF TERMS Algorithm. A computational recipe. Concept. A concept Is an abstraction formed by generalization from p a r t i c u l a r s . Curriculum. Subject matter or course content. D r i l l or practice. Repeated use of a s k i l l or r e -peated r e c a l l of facts with the express purpose of learning or Improving the s k i l l or of re-enforclng retention of facts . Developmental A c t i v i t i e s . Those a c t i v i t i e s of teaoher and class intended to Increase understanding of the number system, processes or operations, and the general use-es fulness of number and quantity i n everyday experiences. Projects, constructions and use of manipulative materials would be Included. Meaning Theory. Teaching understanding of computa-t i o n a l process, before formal presentation of an algorithm. Method. Includes how curriculum Is presented, taught and evaluated. Number ' f a c t s ' . Formerly items of ' t a b l e s ' . Number Sentence. A symbolic expression Involving numbers, the relationships ^ » ^ i °r = and frequently an operator such as + or - . K a r l G. Zahn, "Use of class time In eighth-grade arithmetic," The Arithmetic Teacher, v o l . 1 3 , no.2, (Feb .1966). Equation. A number sentence involving the equivalence r e l a t i o n s h i p . Structure. "How things are related."9 Primary (School) i n North America, Includes Kinder-garten to Grade 3, children of ages 5-9. In Europe, general-l y includes children of ages 5-12. (Francei ages 2-12). Elementary (School). North American term, Includes Kindergarten to Grade 7 (±1), i . e . pre-High School. Intermediate Grades. North American term, Includes Grades 4 - 6 or 4 - 7 , ages 9-12 or 9-13. 9Jerome S, Bruner, The Process of Education. (New Yorki Vintage Books, I 9 6 0 ) , p . 7 . SURVEY OP THE LITERATURE INTRODUCTION Very l i t t l e experimental research on the place of practice has been reported In recent l i t e r a t u r e , i n spite of several specif ic suggestions that this be done. In fact there has been a "dearth of any hard research" at the primary l e v e l . ^ RESEARCH AT FIRST YEAR PRIMARY LEVEL The lack of research at this l e v e l was a strong motivating force for this study. The Importance of early childhood years Is being Increasingly recognized by psycho-l o g i s t s and educators around the world. Modern s c i e n t i f i c Investigations appear to back up the old Jesuit saying "Give me a c h i l d u n t i l he i s seven . . . " At present there i s considerable a c t i v i t y at the kindergarten and e a r l i e r l e v e l s , and many Piagetlan type studies are being carried out with primary school chi ldren. For example, Reidesel f s l i s t of "Research Contributions", a "representative selection" Vincent J . Glennon and Leroy G. Callahan, Elementary School Mathematics. A Guide to Current Research. (Association for Supervision and Curriculum Development, National Education Association, 1968), p.75* i n The A r i t h m e t i c Teacher of March, 1970 i n c l u d e d a t l e a s t t h i r t y P i a g e t o r i e n t e d s t u d i e s . On the other hand, th e r e were only f i v e s t u d i e s d e a l i n g s p e c i f i c a l l y w i t h the f i r s t grade. None had any a p p l i c a t i o n t o the problem of the p l a c e of p r a c t i c e . Any other l i s t or summary y i e l d e d the same 3 g e n e r a l p i c t u r e . A l s o , a t the Northwest Mathematics Con-f e r e n c e I n Tacoma, i n 1968, s e v e r a l speakers were heard t o d e p l o r e t h i s l a c k . THE PLACE OP PRACTICE—CHANGING EMPHASES " P r a c t i c e makes p e r f e c t " r u n s an o l d s a y i n g and f o r c e n t u r i e s , i n numerous f i e l d s of l e a r n i n g , p r a c t i c e has been an i m p o r t a n t and n e c e s s a r y f o l l o w - u p of the p r e s e n t a t i o n of new t o p i c s and s k i l l s . D u r i n g the e a r l y y e a r s of the t w e n t i e t h c e n t u r y , d r i l l was c o n s i d e r e d p a r t of the c o g n i t i v e p r o c e s s . The r e s e a r c h of E . L. Thorndlke and o t h e r s confirmed t h a t d r i l l d i d Improve competence i n computation, a much needed s k i l l i n those days. B u s w e l l and Judd, I n t h e i r Summary of E d u c a t i o n a l I n v e s t i g a - t i o n s R e l a t i n g t o A r i t h m e t i c l i s t e d t h i r t y - t h r e e s t u d i e s C. A l a n R i e d e s e l , "Recent Research C o n t r i b u t i o n s t o Elementary S c h o o l Mathematics", The A r i t h m e t i c Teacher, v o l . 17, no. 3, March, 1970. 3 See B i b l i o g r a p h y f o r other l i s t s examined, e.g. J . P red Weaver and Glenadine Gibb, "Mathematics I n the Elementary S c h o o l " , Review of E d u c a t i o n a l Research, v o l , 34, no. 3» June, 1964. which were unanimous i n reporting the benefits of d r i l l . Later, the "meaning theory" of learning, advocated by Brownell^ and others replaced that of d r i l l . Suydam and Riedesel, In their Interpretive Study of Research and Deve- lopment i n Elementary School Mathematics.^ reported consis-tency of findings that the"meaning method" was superior to " t e l l and d r i l l " , for retention and transfer. When Gestalt theories and methods reached North America, Wheeler suggested, In the Tenth Yearbook of the National Council of Teachers of Mathematics, that practice (the newer name for d r i l l ) could be eliminated altogether 7 from the classroom. This suggestion does not seem to have G. T, Buswell and C, H, Judd, Summary of Educational Investigations Relating to Arithmetic. (Chlcagot University of Chicago Press, 1925). ^cf, study by William A, Brownell and Harold E . Mosher Meaningful vs. Mechanical Learning. A Study i n Grade Three Subtraction. Duke University. Studies In Education, v o l . 8. W9, PP. 1-207. ^Marilyn N. Suydam and C. Alan Riedesel, Interpretive Study of Research and Development i n Elementary School Mathematics. F i n a l Report. Pro.leot No. 8-0586. U.S. Offloe of Education, Department of Health, Education and Welfare, (University Park, Pa.i The Pennsylvania State University, June 30, 1969)• ?NCTM, The Teaching of Arithmetic. Tenth Yearbook, (Washington, D . C . i National Council of Teachers of Mathema-t i c s , 1935) as quoted by T, R. McConnell i n "Recent Trends In Learning Theory« Their Application to the Psychology of Arithmetic", Arithmetic i n General Education. Sixteenth Yearbook, (Washington, D . C . i National Council of Teachers of Mathematics, 19^1), pp. 268-289. found many advocates. Is It that practical experience i n classrooms has not provided any supporting evidence which would indicate the worth of experiment In this direction? The Swiss epistemologlst, Jean Piaget, whose work i s being increasingly recognized i n the United States, speaks s p e c i f i c a l l y of practice, saying that "repetitive behavior" o i s an essential part of the learning process. By mid-century, educators were emphasizing the lm-9 portance of teaching concepts and structures. As children were taught new c u r r i c u l a , with new ap-proaches, i t began to appear that, when tested with old-type computational achievement tests, they did better than children on more "conventional" programs. David Page saldi "It now appears that higher proficiency In computation Is an almost J . H . P l a v e l l , The Developmental Psychology of Jean Piaget. (New Jerseyi D. Van Nostrand Co. , I n c . , 1963), P. 57. Q e.g. Jerome S. Bruner, "The Importance of Structure", The Prooess of Education. (New Yorki Vintage Books, Random House, I960), Chapter I I , pp. 17-32, and Esther J . Swenson, "Arithmetic for Pre-school and Primary Grade Children", The Teaching of Arithmetic. F i f t i e t h Yearbook, Part I I , National Society for the Study of Education, (Chicagot University of Chicago Press, 195D» P. 7^ . automatic adjunct." 1 0 Marguerite Brydegaard 1 1 and Hugh Peok 1 2 reported similar results, the latte r especially among children of lower a b i l i t y . However, the general application of these new programs by teachers not specifically trained i n their use, was f r e -quently much less successful. Demands for new learning theories were madei researchers continued to study how children learn, and educators strove to improve teachers 1 t r a i n i n g , ^ But the questions remaini How should children get practice? When? How much Is necessary? THREE PERTINENT SUGGESTIONS Three closely related proposals as to the place of practice, when trying to answer the basic problem of i n -Davld Page, "General Information Release," Univer-sity of I l l i n o i s Arithmetic Project", May, 1 9 6 2 , as quoted by C, D, Hopkins, "An Experiment on Use of Arithmetic Time i n F i f t h Grade", (Doctoral Dissertation, Indiana University, 1 9 6 5 ) t Dissertation Abstracts, vol, 26t 5 2 9 1 t 1 9 6 6 . Microfilms R 2 6 1 6 A . l^A study by Marguerite Brydegaard referred to by Herbert F. spitzer and Paul C. Burns in"Mathematics In the Elementary School", Chapter II of Review of Educational Research. "The Natural Sciences and Mathematics", vol. 3 1 1 no. 3 1 June, 1 9 6 1 , p. 2 ^ 9 . 1 2Peck, Hugh A., "An Evaluation of Topics i n Modern Mathematics", The Arithmetic Teacher, vol, 1 0 , no.5. May ,1963. l^see for example, "Cambridge Conference on Teacher Training", Goals for Mathematical Education of Elementary School Teacher& (Bostont Houghton M i f f l i n Co.. 1967). creasing quantity and quality of instruct ion, have recently been proposed by Vincent J . Glennon, The Cambridge Conference on School Mathematics, and Edith Biggs* Glennon Proposals and Related Research In 1951i Glennon and Buswell proposed that studies be made to determine the optimum d i v i s i o n of arithmetic time between practice and concept development (developmental 14 a c t i v i t i e s ) . Brownell, i n The Arithmetic Teacher of October, 1956, also called for a balance between teaching for meaning or understanding, and training i n computational skills." 1 "^ Very few researchers seem to have taken up this particular ohallenge. In their Interpretive Study of Re-search and Development In Elementary School Mathematics.^ Suydam and Rledesel report four such studies, a l l at the Intermediate l e v e l , by Shlpp and Deer, Shuster and Plgge, Zahn and Hopkins. Shlpp and Deer, In a large, carefully planned study, In grades four, f ive and six, tested understandings, accuracy * G. T. Buswell, "Needed Research In Arithmetic", The Teaching of Arithmetic. F i f t i e t h Yearbook, Part I I , National Society for the Study of Education (Chicago* The University of Chicago Press, 195Dt P. 291. ^ W i l l i a m A. Brownell, "Meaning and Skl l l -Malntalnlng the Balance", The Arithmetic Teacher, v o l . 3, no. 6, October, 1956. ^ M a r i l y n N. Suydam and C. Alan Rledesel, o p . c i t . with s k i l l s , and problem s o l v i n g . ^ They found a definite trend towards higher achievement when a greater percentage of class time was spent on developmental a c t i v i t i e s . The largest proportion of time thus al lotted was seventy-five percent. Shuster andPlgge obtained similar results with a study of addition and subtraction of fractions at the grade 18 f ive l e v e l . Zahn, In "Use of Class Time In Eighth Grade Arithmetic" concluded that f i f t y percent or more of class time spent on developmental a c t i v i t i e s , resulted In higher 19 achievement. ' Hopkins, In his study of "Experimental Use of Time i n F i f t h Grade", concluded that the amount of d r i l l 20 time should be considerably reduced. Most Interestingly he found that the greatest gains i n computational proficiency, 1 7i>onald Eugene Shlpp, "Experimental Study of Achieve-ment and Time Allotment", Dissertation Abstract, v o l . 19 (1), p. 492, 1958, and D. E . Shlpp and G. M. Deer, "Use of Class Time In Arithmetic", The Arithmetic Teacher, v o l . 7, no. 3 . , March, i 9 6 0 , pp. 117-121. 18 Albert H. Shuster and Fred L . Plgge, "Retention Eff iciency of Meaningful Reaching", The Arithmetic Teacher, v o l . 12, no. 1, January, 1965. PP. 24-31. ^ K a r l George Zahn, "Use of Class Time In Eighth Grade Arithmetic", The Arithmetic Teacher, v o l , 13, no. 2, February, 1966. See also Dissertation Abstracts, v o l . 26, p. 6459• May, 1966. 20 C, D. Hopkins, "An Experiment on the Use of A r i t h -metic Time In F i f t h Grade," Doctoral Dissertation, R2616/4, Dissertation Abstracts, v o l . 26, no. 5291, 1966. when practice time was reduced, were made by the students of lower a b i l i t y . In the brief summary of their report, published in 21 The Arithmetic Teacher. March 1970, Suydam and Riedesel say that " D r i l l and practice are necessary for computational accuracy". This seems an oversimplification of the above reported studies, which were examining proportion of time spent on d r i l l rather than the question »to d r i l l or not to d r i l l 1 . They are possibly referring to a study by Ivor 22 G. Meddleton at the fourth grade level, which, i n their 23 Interpretive Report, they say confirmed that "systematic, Bhort review work i n basic mathematics produces significantly higher levels of achievement". More recently, Glennon and Leroy G. Callahan made a stronger suggestion than the Glennon and Buswell one when they wrote* Because of the sequential development of a sound mathematics education program, muoh of the practice on previously learned s k i l l s can be • b u i l t - i n 1 to subse-quently learned materials. 2^ 21 ^Suydam and Riedesel, op.cit. fc Meddleton, Ivor G. "An Experimental Investigation into the Systematic Teaching of Number Combinations i n Arithmetic", B r i t i s h Journal of Educational Psychology, vol. 26i 117-127, June 1956. 23 Suydam and Riedesel, op.cit. ^Vincent J. Glennon and Leroy G. Callahan, Elementary School Mathematics. A Guide to Current Research. (Association for Supervision and Curriculum Development, National Education Association, 1968), p. 81. Brownell also seems to have had this i n mind when he did his studies on f i f t h grade readiness for long d i v i s i o n . 2 ^ Roberta Chlvers, et a l , have Incorporated ' b u i l t - i n * practice i n their text book series, Number Patterns, for example, with the use of parentheses to review addition and m u l t i -p l i c a t i o n ' f a c t s ' during the development of the ' f a c t s ' for larger numbers. 2^ Glennon and Callahan 2 ' ' reported one study by Lelon Capps, "Division of Fractions", which they say showed the efficiency of one method of presentation over another, In that I t provided extra practice i n multipl ication and hence maintained t h i s s k i l l . Capps himself was more cautious, concluding that there were many facets to his problem of which was the more desirable method to use, facets which cal led for examination before either method was advocated 2B or oondemned. In spite of these studies, a very recent survey by ^Brownell, William A. "Arithmetic Readiness as a P r a c t i c a l Classroom Concept", Elementary School Journal. September, 1951 and"Effects of Practising a Complex Arithmetic S k i l l upon Proficiency In Its Constituent S k i l l s " , Journal of Educational Psychology. February, 1953• 2^Roberta Chlvers, J . E . Smith, E r i c D. MacPherson, Alfred P. Hanwell, Number Patterns. Books I and I I , (Torontoi H o l t , Rinehart and Winston, 1966). 2?Glennon and Callahan, o p . c i t . 2 8 L e l o n R. Capps, "Division of Fractions", The Arithmetic Teacher, v o l . 9, n o . l , January, 1962, pp. 10-16. Mllgram showed that In f o r t y - s i x Intermediate classrooms In Pennsylvania, three quarters of class time was being 29 spent on d r i l l or practice. Proposals of the Cambridge Conference on School Mathematics In 1963» the Cambridge Conference on School Mathema-t i c s recommended a step-up of content to gain three years over the period from kindergarten to the end of secondary school1 i t was proposed that this be accomplished by "a new organization of subject matter • • • replacing the unmotiva-ted d r i l l of c l a s s i c a l arithmetic by problems which l l l u s -30 trate new mathematical concepts." This aspect of replacing repetit ive practice with use of s k i l l s i n a new situation does not seem to have been s p e c i f i c a l l y examined to any great extent i n North America. At Nova School, In Dade Park, F l o r i d a , an attempt i s being made to put the overal l currloulum proposals of the Cambridge conference into practice for bright pupils. Professor Fitzgerald reported on this i n 1965i^1 at that time the 2 9 J o e l Mllgram, "Time U t i l i z a t i o n i n Arithmetic Teaching", The Arithmetic Teaoher. v o l . 1 6 , no . 3 t March, 1 9 6 9 . •^Cambridge Conference on School Mathematics, Goals to: School Mathematics. (Bostoni Houghton M i f f l i n Company, 1963) f>. •^See report of discussion following Professor Wllloughby^ report, "Curriculum Experimentation", i n Howard A. Fehr, Needed Research i n Mathematics Education. A summary report of a conference held at Greystone, New York, Oct. 30 and 31, 1965t (New Yorki Teaohers* College Press, Teachers* College, Columbia University, 1965). experience was exposing problems needing further research, among them problems concerned with concept formation, non-verbal Instruction, Individualized Instruction and diagnosis. At the next meeting, the Cambridge Conference on Teacher Training, In 1966, i t was reported that* Experimental work of an even more drastic nature than the 'Goals 1 proposals already has been started In the United States, and the mathematicians of at least one European country, Denmark, have embarked on formal deve-lopment of a school mathematics program that seems to p a r a l l e l closely much of the 'Goals 1 outline.32 At the time of t h i s writing, there are not yet any published results of these endeavours, In journals directed towards mathematics educators. The Role of Practice In Primary Schools. For an account of results of this type of approach with primary aged chi ldren, we turn to Miss E d i t h Biggs. With James R. Maclean, In Freedom to Learn, she saidi I f we can use the natural experiences of children to develop the basic ideas of mathematics i n the f i e l d of number, measurement and shape, while maintaining reasonable computational s k i l l s , we w i l l avoid much of the o r l t l o l s m being directed at many of the 'modern mathematics' programs.33 And again, under the topic Meaningful Practice 1 32Cambrldge Conference, o p . c i t . , p. 7. 33Edlth E . Biggs and James R. MacLean, Freedom to Learn. (Toronto, Canadai Add!son-Wesley, 1969). P. 4. The practice of computational s k i l l s Is based upon the practical a c t i v i t i e s In which the children are engaged. In this way the pupils understand and ap-preciate, the need for polishing and expanding these s k i l l s . ^ The basis for these statements i s found i n the B r i t i s h publication Mathematics i n Primary Schools, prepared by Miss E. E. Biggs,^ and In the UNESCO report of L. G. W. Sealey,^^ The latt e r describes the "upsurge of reform" which took into account "newer knowledge of child development and the on-going work i n cognitive studies". Miss E. E. Biggs, H.M.I*, and her colleagues led "the f i r s t of the organized national efforts to improve early mathematics le a r n i n g " . ^ Mathematics i n Primary Schools. although definitive In Its 'Position 1, could be oalled an Interim report. In the ohapter "Research i n children's 39 method of learning" Miss Biggs reviews the research of 3 2 | ,Ibld.. p. 1 3 . 3^The Schools Council, Mathematics In Primary Schools. Curriculum Bulletin No. 1, Second edition, (London: Her Majesty's Stationery Office), 1 9 6 7 . -^L. G. W. Sealey, "An Outline of Curricular Changes i n Great Britain", i n J. D. Williams, Mathematics Reform In the Primary School (Hamburg: UNESCO Institute for Education, 1967; pp. 106-114. 3 7 I b l d . . p. 1 0 6 . 3 8 I b l d . . p. 1 0 7 . 39The Schools Council, OP.cit.. pp. 5-9. Piaget, Dienes, and others, and summarizes In parts 4) Practice Is necessary to f i x a concept once It has been understood, therefore practice should f o l -low, and not precede discovery.* 0 Specific researoh work Is not quoted for this principle, probably because of Its place In Piaget's scheme, but further on she says: Those teachers who have already organized the work so that the children learn by their own efforts a l l bear testimony to the decrease i n the amount of com-putational practice children require in order to at-tain and maintain computational efficiency. Many teachers have found that they can safely reduce this practice to two periods (and some to one) each week and that the children's efficiency i n computation has improved and not declined In consequence. 1 One direct outcome of this reform movement i n Great Britain i s the Nuffield Mathematics Project which Is deve-loping a new primary school mathematics curriculum on a national scale. The Project i s being developed In close collaboration with L'Instltut des Sciences de 1'Education 42 In Geneva. The latter has provided a team which Is developing "Individual Check-ups" to replace Individual tests. ^ 0 I b l d . . p. 9. ^ I b l d . . p. 41. ^^Nuffield Mathematics Project, P i c t o r i a l Representa-tion m (Londoni W. & R. Chambers and John Murray, 1967, available from Longmans Canada Ltd., Toronto), p. 1. THE PLACE OP GRAPHICAL REPRESENTATION The Importance of early Introduction to concepts of graphing i s being increasingly recognized i n many countries and i n any curriculum movement that i s working for integra-t ion of science and mathematlos programs. The f i l m "Maths Alive" from the United Kingdom showed this v i v i d l y . J Mme. Plcard, i n ^Currlcular Change i n the French Primary Schools", referred to the use of "representations", that is» sketches, diagrams, tables and graphs. This use i s based on Piaget 1 s studies and on experiments i n classes In Francet It provides for the c h i l d a concrete representation of an abstract men-t a l operation, a representation which "precedes symboliza-t ion and f a c i l i t a t e s i t " . Furthermore, Mme. Plcard saysi Another purely educational but not unimportant advantage i s that young children have great d i f f i c u l t y In expressing their thought i n words but experience r e a l sat isfact ion i n being able to express themselves by means of graphs, diagrams, or schematai often, after Inspecting a representation which they have made, they find i t possible to express their thoughts verbally.*o In the United States, John R. Mayor has recently reiterated proposals for the Integration of science and ^"Maths A l i v e " , F i l m , coloured, 30 min. by B r i t i s h Petroleum Company, obtainable Canadat Learning Materials Servioe Unit , Dept. of Education, 559 Jervls S t . , Toronto 5» Ontario. un Nicole Plcard, "Curricular Change In the French Primary Schools", i n J . D. Williams e d . , Mathematics Reform i n the Primary Schools. (Hamburg! UNESCO Institute for Education, 1967), P. 75-76. ^ O p . o l t . . p. 76. mathematics. One of the two f i r s t steps In this d i r e c -t ion would be the early Introduction of graphing. He mentions the MINNEMAST program In Minnesota as one example where this Is already being done. Prom a looal and p r a c t i c a l point of view It was thought that the Introduction of such a relevant and 'open-ended 1 topic would give Interested practising teachers and their students a new tool for their mathematics and science studies In the coming year. CONCLUSION 47 Since the proposal by Glennon and Buswell In 1951» a number of studies have been reported at the Intermediate l e v e l , designed to determine the optimum d i v i s i o n of a r i t h -metic time between practice and concept development. The conclusion i s unanimous that classroom time spent on practice can be greatly reduced. No controlled experimental research studies appear to have been reported at the primary l e v e l . Miss Biggs states that teachers In primary schools In England 48 have been able to considerably reduce practice. 46 John R. Mayor, "Science and Mathematlcst 1970»s — a Decade of Change", The Arithmetic Teacher, v o l . 17» no. 4, A p r i l , 1970. ^Glennon and Buswell, op., c i t . . p. 291. 48 i The Schools Council , oj>. c i t . , p. 41, The present study attempts to incorporate sufficient practice for the maintenance of certain competencies Into the learning of new material (Graphical Representation), as suggested by the Cambridge Conference on School Mathematics i n 1 9 6 3 , ^ and by Glennon and Callahan In 1967,5° while keeping In mind Miss Biggs' remarks and the above men-tioned results with intermediate children. ^Cambridge Conference on School Mathematics, op_. o l t . P . 7. 50 Glennon and Callahan, op_. c i t . , p. 81. CHAPTER III PROCEDURES INTRODUCTION This study attempts to compare the maintenance of arithmetic competencies of groups of children, previously taught by a common method, who spent a period of time In one of the following three waysi a) an introductory unit on graphical representation b) t r a d i t i o n a l year-end review and practice c) a geometry unit involving no arithmetic s k i l l s Two important areas of the B r i t i s h Columbia curriculum were chosen as *basio s k i l l s 1 to be evaluated. I . Numeration; a b i l i t y to recognize, read and write numerals for numbers ^ 9 9 with understanding of base ten numeration. I I . Computatloni addition and subtraction operations In equation form with sums and minuends ^10, The investigator taught a l l groups, to reduce Haw-thorne effects. MATERIALS USED a) An introduction to graphs was chosen as the new material to be introduced using as teacher's guide P i c t o r i a l Representatlon ( T j ,^ from the series of teacher's guides for the Nuffield Mathematics Project. Examination of t h i s booklet Indicated that the material could be used, with careful planning, In place of the regular year-end review of addition and subtraction operations and facts, and of base ten numeration of numbers ^100, Including concepts of greater than and less than, a l l of which are stressed i n the present curriculum i n B r i t i s h Columbia. See Appendix A for detailed plan of the u n i t . b) As far as possible, methods and materials were those to whloh the children were acoustomed. See Appendix A for detailed plan of the u n i t . c) An open-ended geometry unit was planned, using 2 Elementary School Mathematics. Books I and I I . and _~Nuffield Mathematics Project, P i c t o r i a l Representa- t ion [ l i . (Londont W, & R. Chambers and John Murray, 1967, available from Longmans Canada L t d . , Don M i l l s , Ontario). 2 Robert E . E i c h o l z , et a l , Elementary School Mathematics Second E d i t i o n , Books I and I I , (Don M i l l s , Ontarioi Add!son-Wesley Canada L t d . , 1969). tesselatlng as described i n the Nuffield Project booklet, Two schools were available, the f i r s t with 54 c h i l -dren, and the second with 34 children. In the f i r s t school, designated School #1, the children had been i n a single large class under a team teaching situation, thus providing the common method, up to the time of the investigation. A l l three methods were used i n this school, the children being chosen at random to form the Groups A-^, B l t and C-p corres-ponding to the three treatments, with 18 members to a group. In the second school, designated School #2, two groups, &2 a n d B2» w i t n 17 members each, were set up, using random selection! this provided for comparison between the two experimental treatments. Three weeks of teaching were planned, but, i n School #1, three days were lost to other school a c t i v i t i e s and pre-testing took an extra day. This l e f t only eleven days, and caused considerable loss of continuity. Planned teach-ing periods of twenty-five minutes were seldom that long, --Nuffield Mathematics Project, Shape and Size. \2 Nuffield Mathematics Project. (London: W, & R. Chambers Y and John Murray, 1967, available from Longmans (Canada) Ltd., Toronto). See Appendix A for detailed plan of unit. ORGANIZATION owing to administrative procedures, and physical arrange-ments Involved i n class change-overs. Group A most f r e -quently received i t s f u l l time allotment, the other two groups completing their time with assigned seat work under the supervision of the two regular teachers. TABLE I GROUPING OP STUDENTS AT START OP EXPERIMENTAL PERIOD School #1 School #2 Group A l B l c l A 2 B2 No. students 18 18 18 17 17 No. boys 9 9 9 10 10 No. g i r l s 9 9 9 7 7 In school #2, the f u l l three weeks teaching session was u t i l i z e d , each group receiving approximately the twenty five minutes planned for them each day. The lesson for Group A 2 was alternated d a i l y between f i r s t and second session. EVALUATION INSTRUMENTS Basic Competencies. Test I and II Now instruments were developed and subsequently referred to as Test I and Test I I , to test competency i n Numeration and Computation respectively (marked Part I and Part II respectively In the children's booklets). P a r a l l e l power tests, Forms A and B, were constructed to Include extension of topics to Items not normally en-countered by the children u n t i l their second or third year. I t was thought that this would give a better picture of the achievement l e v e l of individual pupils and of the class as a whole. Reading matter was r e s t r i c t e d toi a) words and symbols for equality and inequality b) words and symbols for tens and ones c) symbols for addition and subtraction. The f i r s t of these required three versions, symbolict <C» ^ » 55» t n e Printed word, and the printed word In i . t . a . s c r i pt. For d e t a i l s of test construction, see Test Manual, Appendix C. The tests w i l l be found In Appendices D and E . Graphical Representation. Test III Achievement i n "Graph Reading" or obtaining Infor-mation from a block chart was the special s k i l l chosen for test ing. The unit being taught to Groups A^ and Ag was based on the Teachers' Guide« P i c t o r i a l Representation [T], from the series developed by the Nuffield Mathematics Project In Great Britain. 'Check-up Guides 1 were not yet commercially available for this topic» no answer was received from the head of the project to the letter re-questing help i n this area. Hence tests were Improvised, using a chart-sized copy of "Block Chart of Pets i n Class I" adapted from P i c t o r i a l Representation IT1 (Appendix P). A pre-test, consisting of eight oral questions (see Appendix G), was administered to the children i n School #2, to a third of the class at a time. The children In this group printed their answers on lined newsprint. This pre-testing was not oarrled out In School #1, owing to limit a -tions of time and of class organization. Informal oral testing carried out In groups revealed that a b i l i t y to read a block chart was almost non-existent. For example, although the children knew that there were more dogs than oats, no one In any group challenged the general concensus that there were 10 dogs and 5 cats, rather than 14 and 7» respectively. The same chart was used for the post-test with additional questions added, making fourteen questions in a l l . The format of Test III as presented In School #1 was unfortunate, psychologically. The questions, which were to be read out by the examiner, were printed on the page. This proved rather overpowering for poorer readers and for those who had not yet made the t r a n s i t i o n from i . t . a . s c r i p t to standard orthography. Consequently the children had rather more trouble than expected f inding the correct place on the paper. "This Is the hardest test I have ever written!" was one comment. The c h i l d was thanked for t h i s information and assured that no other c h i l d would ever have to write i t (see Appendix H). A new format was devised for Sohool #2, with a small copy of the chart ap-pearing on the paper, as well as l i t t l e pictures, and ap-propriate spaces to complete the answers to the oral ques-tions (see Appendix I ) . The test attempted to assess the following s k i l l s t 1. A b i l i t y to obtain information d i r e c t l y from the chart. 2. A b i l i t y to construct number sentences and equa-tions expressing the above information and showing how i t was obtained. 3. A b i l i t y to construct other number sentences and equations suggested by the data set out on the chart, PRE-TESTING Form B of Tests I and II was used as pre-test of basic competencies. (See Appendix D) School #1 The Investigator tested the children i n small groups. A group of slower children was taken f i r s t , followed by the rest of the boys and then the rest of the g i r l s . These chi ldren had never been tested i n any way before, so d i f f i -culty was experienced i n obtaining independent responses. More time was needed than anticipated. Three f u l l sessions and part of the f i r s t teaching day were neoessary. The Individual oral testing was accomplished In two afternoonsi the examiner was i n a corner of the classroom where the remainder could not hear, and while the class proceeded with i t s regular lessons, each c h i l d tested would quietly send the next one for his or her turn. The Graph Reading test (Test III) was presented as a short oral discussion session at the beginning of the f i r s t teaching period for each group. School #2 Tests I and I I , Form B, were administered to the whole class at once, with the assistance of the teacher to assure that a l l the children understood instructions, were placing responses i n correct places and were answering a l l questions that they were able to. The tests were completed i n two and one half sessions. D i f f i c u l t y In obtaining independent responses was experienced i n a few cases, due to seating arrangements. Test I I I was administered to a t h i r d of the olass at a time. The I n d i v i d u a l o r a l t e s t i n g was completed i n one afternoon, with the exception of a few absentees who were tested l a t e r i n the week. Use of the foyer assured privacy and Independent responses. TEACHING ACTIVITIES Graphing u n i t as c a r r i e d out In School #2 The a c t i v i t i e s of Group A 2 w i l l be described f i r s t as t h i s group accomplished more than Group A ^ , 1 . Comparison of Number of Boys and G i r l s In Group a) One block per c h i l d , blocks were l i n e d up and comparisons made. The symbols >^ and < were Introduced and used to p r i n t pertinent number sentences on the chalk-board. Sizes of f a m i l i e s were compared. An attempt to develop f a c t s of nine was not too successful, the c h i l d r e n not being ready to r e l a t e arithmetic s k i l l s to actual s i t u a t i o n s . b) Paper squares, one per c h i l d , pink f o r g i r l s and blue f o r boys, were pasted up i n two rows on blank chart paper. A chart story was e l i c i t e d with word and number sentences. Working with partners, the ohlldren then wrote t h e i r own number s t o r i e s and word sentences about the chart. 2 . Birthday Chart Names of months were entered on chart paper, providing twelve horizontal rows. A starting line was drawn and each child pasted up his self portrait, a l l portraits being drawn on the same sized paper. With partners, the children wrote their own word and number sentences. The next day, portraits from Group B2, drawn during an art period, were added to the chart. This gave larger numbers and a different pattern of distribution for Group A 2 to work with. The stories now written by the children showed a developing ability to obtain information from their chart, 3 . Pets a) Picture Charts. A chart was prepared listing the species of pets the children aotually had at home. This chart had no standard size boxes or starting line. Outline drawings were provided, which the children coloured cut out and pasted up. They observed that the line of dogs was longer than that of the cats but there were more cats. The ensuing discussion resulted In an unpastlng and re-arranging to obtain a one-to-one correspondence. Then It was observed that •tadpoles' was such a long word that the f i r s t bowl was under the fourth dog. At the next session various suggestions were tried. By this time, the other group had added their pets, without discussion, and this further confounded things. However, the' children themselves suggested, at this point, that they start again with a new chart. The number of dogs and cats each being ^ 1 0 , inciden-t a l review of tens and ones, and of counting by fives and tens was carried out, chiefly by an e l l c l t a t l o n method. With coloured lines for the fives and tens, the children had a quick way of determining the exact number of any particular kind of animal. b) Vertical Block Graph. A double length chart was prepared as graph paper with one inch squares} the zero l i n e , the fives and tens lines were distinctively marked. To provide an activity Involving larger numbers, pets were grouped according to the number of legs each had. Coloured squares were provided for the children to paste up. This gave numbers i n the forties and f i f t i e s and offered much opportunity for variety of equations and inequalities, and for application of knowledge of tens and one s. c) New Pet Chart. The children pinned rather than pasted up their pictures, so that they could more easily make desired alterations i n position. d) Contaminat1on. Without prior consultation with the investigator, the teacher assigned at least two work-sheets of computational practice to this group. Graphing Unit as Carried out i n School #1 1 . Comparison of Number of Boys and G i r l s i n Group a) One counter per c h i l d , counters were l ined up and comparisons made. The symbols ^ and <^ were introduced and appropriate number sentences e l i c i t e d and printed on the chalkboard. Working In pairs, and using large round counters or one inch linoleum t i l e s , the children compared the sizes of their f a m i l i e s . b) S e l f - p o r t r a i t s were drawn on common-sized pieces of rectangular paper, and pasted up on chart paper. I n d i -vidual writtern work was started using a worksheet for guidance and the chart for information. The children seemed quite at home combining word and number sentences. c) Paper squares, pink and blue, were used for a new chart, following the permanent departure of one mem-ber of the group. Working i n pairs, the children also produced their own smaller oharts. Questions guided their disoovery of the importance of l i n i n g up the squares to show a one-to-one correspondence. One child proposed and printed a question for the others to answer. 2 . Birthday Chart a) S e l f - p o r t r a i t s were drawn and pasted up as des-cribed for School # 2 . Stories about the results were written and discussed. The chart c learly showed a less mature group, chronologically, than i n the other school. This fact , coupled with the lack of continuity of teaching days, could account for the impression that the majority of chi ldren gained l i t t l e from the discussions. After the children from Group had drawn s e l f - p o r t r a i t s for seat work,and added them to the chart, further work was done by Group A^ with the larger numbers thus provided. b) Graph paper. Individual birthday charts were prepared, and the children coloured a square opposite the appropriate month for each member of the c lass . Stories were written In the space provided. Time did not permit adequate assessment with each chi ld of the o r i g i n a l i t y or comprehension of this written work.- During the colouring, It became apparent that the children did not know the r e -lat ionship between a written numeral and tens and ones. The birthday chart was used as a starting point, but i t was necessary to use concrete objects, and reteach this t o p i c . c) V e r t i c a l Block Graph. A long chart was prepared, similar to that used for Pets In School #2. Coloured squares were pasted up, v e r t i c a l l y , i n two columns, comparing the number of birthdays from January to July with those from August to December. The tens l ines were marked for easier counting, but, i f time had permitted, another week could have been spent exploring and consolidating. Review and Practice Unit as Carried Out In School #1 Considerable re-teachlng was involved here, as work with the children soon substantiated the evidence of the pre-tests that very few of them were competent i n the basics assessed i n Tests I and I I . See Table I I . TABLE II SCHOOL #1 ACHIEVEMENT AND •MASTERY1 SCORES PRE-TEST Test Topic Maximum Grade One Class Number Score •Mastery 1 Average Attaining Score Score 'Mastery* I Numeration 42 35 24.2 3 II Addit ion- 40 32 24.6 3 subtraction Number of pupils tested = 56. a) The addition and subtraction facts of 7, 8, 9, 10 were reviewed, using both equations and v e r t i c a l form, but no equations with more than two addends. The investigator's method involved somewhat more use of concrete objects than the demonstration method to which the children were accustomed. b) Considerable time was devoted to numeration, es-p e c i a l l y to discovery and teaching of the relationship be-tween the printed numeral and the number i t represents. Conorete objects were used. Then some work was done c i r c l i n g pictured objects Into groups of tens and u n i t s . Organizing for and assigning the seat work,as r e -quested by the class teachers, cut down the time available for developmental work. However, this meant that the children carried out their usual practice a c t i v i t i e s , with counters available and their own teachers to supervise. Review and Practice Unit as Carried Out In School #2 a) Inequality signs were introduced. b) Addition, subtraction and mult ipl icat ion facts of 9 and 10 were developed with the use of blocks to which the children were accustomed. Review of e a r l i e r facts was Incorporated Into the a c t i v i t y lessons and d a l l y worksheets. These included the use of parentheses, c) Balancing equations, e.g. 5+3=2+ provided for the more able, but was not pushed with the others. d) Numeration review was carried through with a famil iar type of worksheet, with the exception of one recent transferee who required work with objects to count and group. e) Rote counting by ones, f ives and tens, and counting of pictured money were reviewed. f) Adding and subtracting of one from any number less than one hundred was praotlced b r i e f l y . Geometry Unit as Carried Out In School #1 Prom the plan shown i n Appendix A, the following topics were developed! a) F a m i l i a r i t y with two-dimensional shapes. b) Creating and extending patterns. c) Background for l a t e r work with congruence. d) Review of open and closed curves and location of points In r e l a t i o n thereto. e) Geometric shapes from a c o l l e c t i o n of points. CONFOUNDING FACTORS Computational Practice Group A 2 (Graphical Representation) were assigned at least two worksheets for computational practice. This was done by the teacher without consulting the investigator. Exposure to Graphing Unit In School #1, a l l groups were exposed to the results of the work of the Graphical Representation Group. However, this was on display In a partitioned Arithmetic Corner, and the other groups were kept occupied with their lessons when there. Group B^ (Computation) also added their portraits to the birthday chart, but this was done without comment or discussion. In School # 2 , only the group being taught was i n the classroom during a lesson. However, In the self-contained classroom, the other group was exposed to the results of the a c t i v i t i e s for the rest of the day. This could have had an effect on a l e r t yougsters, regarding their knowledge of Graphical Representation. Group B 2 also contributed, with-out comment or discussion, to the birthday chart, and also to the pet chart. POST-TESTING Form A of Tests I and II were used as post-test of basic competencies. (See Appendix E) School #1 The Individual oral testing was smoothly accomplished with the aid of one of the teachers and a duplicate set of f l a s h cards, following the l a s t teaching session. Several unfortunate aspects marred the atmosphere for the pencil and paper testing which commenced on the f i n a l Monday of the school year. Academic a c t i v i t y had v i r t u a l l y ceased through the school and families began premature withdrawal of their children. Group A^, which had l o s t one member at the end of the f i r s t week, l o s t three more. Arrangements had been made to administer the tests to half the class at a time, with the assistance of one teacher, as described on page 32. The other teacher was to keep the other children occupied. Unfortunately, she took them outside to play, creating considerable diversion of attention among those remaining. On the second day, one c h i l d i n Group A^ printed S T O P across his set of addition and subtraction equations, and stopped. This was not discovered u n t i l i t was too late to r e c t i f y , and his results for Test II had to be discarded. Test III was the l a s t to be administered, after recess on Tuesday morning. Perhaps because of the distress due to the poor format d i s -cussed e a r l i e r , the teachers told the children that i t would not count on their report cards. As other a c t i v i t i e s were planned for the afternoon and the remaining days, the c h i l -dren were aware that t h i s was the l a s t work that would be expected of them for the year. School #2 The pencil and paper post-testing was commenced during the regular arithmetic time on the Monday, and completed during the long pre-recess period on the Wednes-day. The whole class was tested at one time, with the teacher a s s i s t i n g as previously indicated. Desks were arranged to encourage independent work. The i n d i v i d u a l oral testing ran smoothly i n tte foyer, during Monday afternoon. The brighter youngsters had to be reassured that there was no catch to i t , when they were presented with the task of reading numerals and completing equations. STATISTICAL ANALYSIS OP RESULTS CHOICE OP ANALYSIS OP COVARIANCE The i n i t i a l score f o r each t e s t was a measure of s k i l l s a c h i e v e d through a combination of a b i l i t y and of the e f f e c t of e i g h t t o n i n e months l e a r n i n g , whereas the f i n a l s c o r e measured the a d d i t i o n a l e f f e c t of t h r e e and a h a l f weeks of I n s t r u c t i o n , I n c l u d i n g the l e a r n i n g which took p l a c e d u r i n g the w r i t i n g of the p r e - t e s t . Hence I t was r e a s o n a b l e t o expect t h a t the i n i t i a l s c o r e s which marked the e x p e r i m e n t a l s t a r t i n g p o i n t would ishow a h i g h p o s i t i v e c o r r e l a t i o n w i t h the f i n a l s c o r e s . A n a l y s i s of c o v a r i a n c e t e s t s on each p a i r of p r e - t e s t and p o s t - t e s t means would determine whether such a r e l a t i o n s h i p d i d Indeed e x i s t , and, i f so, would p r o v i d e the n e c e s s a r y d a t a f o r removing the e f f e c t s of the p r e - t e s t score by c a l c u l a t i o n of a d j u s t e d means. The unequal c e l l s a t the end of the experiment pre-c l u d e d the use of a nested d e s i g n , a n a l y s i s of v a r i a n c e , on the a d j u s t e d means. (See T a b l e I I I f o r n»s I n each group). Hence, o r t h o g o n a l c o n t r a s t s between a d j u s t e d means u s i n g t - t e s t s were planned. TABLE III GROUPING OP STUDENTS AT END OP EXPERIMENTAL PERIOD Group A l B l c l A 2 B 2 No. of 14 17 15 16 17 students TEST I No. boys 8 8 8 10 10 No. g i r l s 6 9 7 6 7 No. of 13 17 18 16 17 students TEST II No. boys 7 8 9 10 10 No. g i r l s 6 9 9 6 7 No. of 13 17 18 14 16 students TEST III No. boys 7 8 9 9 10 No. g i r l s 6 9 9 5 6 Analysis of covariance can be used when there exists a linear 'regression* relationship between i n i t i a l and f i n a l scores and when there i s a common regression line for a l l groups (populations). The BMDX82 computer program for one-way analysis of covariance from the Department of Bio-Mathematics, School of Medicine, University of California at Los Angeles, was made available through the kindness of Dr. S. S. Lee and was used on the IBM 360, Model 67 computer at the University of B r i t i s h Columbia. This s t a t i s t i c a l program provides tests for slope existence, commonality of regression lines and equality of means after adjustment to remove effects of i n i t i a l differences. I t also provides a t-test matrix for adjusted group means and t-tests for any desired contrasts. A short Fortran program was used to determine the exact pro-b a b i l i t i e s of the F values corresponding to the t»s. For the one-way analysis of variance, and contrasts, required to analyse the data for Graphical Representation i n Sohool #1, a companion program, BMDX64, was used. This gave the F values for the desired orthogonal contrasts between means. Assumptions Underlying the above analyses are the assumptions of normal distribution of the populations sampled and of homo-geneity of variance. These are reasonable assumptions since the whole population of the f i r s t year was involved for each s c h o o l ; the s m a l l e r group was over t h i r t y and the subgroups w i t h i n s c h o o l s were randomly s e l e c t e d . Furthermore, as po i n t e d out by Hays, w h i l e an F - t e s t f o r e q u a l i t y of v a r i a n c e i s q u i t e s e n s i t i v e f o r non-normality, t h i s a p p a r e n t l y makes l i t t l e d i f f e r e n c e I n t e s t s c o n c e r n i n g means. 1 MAIN HYPOTHESES T e s t I Numeration The h y p o t h e s i s m a t r i x shown In Table IV p r o v i d e s t e s t s f o r the f o l l o w i n g n u l l hypotheses, a l l c o n c e r n i n g the a d j u s t e d means of the p o s t - t e s t s c o r e s . TABLE IV HYPOTHESIS MATRIX A1 B1 C x A 2 B. 1 - 1 0 0 0 0 0 0 1 -1 1 1 0 - 1 - 1 -1 -1 4 -1 -1 H(l,l)» There w i l l be no s i g n i f i c a n t d i f f e r e n c e be-tween the mean score f o r T e s t I of Groups A^ and B^. Hays, W i l l i a m L., S t a t i s t i c s f o r P s y c h o l o g i s t s (New Y o r k i H o l t , R i n e h a r t and Winston, 1966), p. 352. H(l,2)» There w i l l be no significant difference between the mean scores for Test I of Groups A g and B 2 . H(l,3)i There w i l l be no significant difference between the mean scores for Test I of the treatment groups of School #1 and of School #2, i . e . A-j+B^ w i l l not d i f f e r s i g n i f i c a n t l y from Ag+Bg. H(l,4)i There w i l l be no significant difference between the mean soores for Test I of a l l treatment groups and the mean score of the control group. Test II Computation Using the same hypothesis matrix as shown In Table IV, the n u l l hypotheses were as followsi H(2 , l ) t There w i l l be no significant dlffersnoe between the mean soores for Test II of Groups A^ and B^. H(2,2)i There w i l l be no significant difference between the mean scores for Test II of Groups A 2 and B 2 . H(2,3)i There w i l l be no significant difference between the mean scores for Test II of the treatment groups of School #1 and of Sohool #2. H(2,4)i There w i l l be no significant difference between the mean soores for Test II of a l l treatment groups and the mean score of the control group. Test III Graphical Representation S t a t i s t i c a l comparison between the schools was not made for Test I I I . They would have been meaningless owing to the previously noted differences i n teaching time, content covered and test format. School #1 There were no pre-test scores for School #1. Hence orthogonal contrasts between post-test means were planned, using the Hypothesis Matrix shown in Table V. TABLE V HYPOTHESIS MATRIX SCHOOL #1 TEST III A l B l c l 1 1 1 -1 -2 0 The n u l l hypotheses were as follows: H(3,l)» There w i l l be no significant difference between the mean scores for Test III of the two treatment groups and that of the control group. H(3,2)« There w i l l be no significant difference between the mean scores for Test III of Groups A.^ and B^. School #2 Both pre-test and post-test scores were available for School #2. There was also reason to believe there was a sufficient difference i n mean a b i l i t y between the two groups to make advisable the use of analysis of covariance for this The n u l l hypothesis was as followsi H ( 3 , 3 ) » There w i l l be no significant difference between the mean scores for Test III of Groups A 2 and B 2 . TABLE VT SUMMARY OP TEST RESULTSi NUMERATION Group A l B i c i A 2 B Maximum possible score 42 42 42 42 42 No. cases per group 14 17 15 16 17 P r e - t e s t i Max. observed 38 35 39 35 37 Mln. observed 15 19 8 21 20 Mean 24.57 2 5 . 7 6 24.0 3 0 . 9 3 8 2 9 . 4 7 1 Standard deviation 6 . 6 5 3 4 . 7 9 0 7 . 9 1 0 3 . 7 1 4 4 . 3 6 1 Post-testt Max. observed 38 40 40 40 40 Mln. observed 14 22 16 30 20 Mean 28.643 30.588 2 7 . 6 0 3 5 . 9 3 7 3 3 . 1 7 6 S. D. 5.719 4 . 5 1 5 6 . 6 8 5 3 . 1 9 3 4 . 7 3 3 Adjusted mean 30.286 31.445 2 9 . 6 2 1 33.382 3 1 . 5 8 8 Std. error 0.942 0.845 0.920 0.909 0 . 8 5 7 ANALYSIS OP DATA Main Hypotheses Regarding Numeration S k i l l s For convenience of reference Table VI summarizes the results of the tests administered. A l l groups showed Im-provement from pre-test to post-test, hence i t Is a matter of examining r e l a t i v e Improvement between groups. Table VII shows the r e s u l t s of testing for use of analysis of oovarlance. TABLE VII TESTS FOR USE OP ANALYSIS OP COVARIANCE TEST I i NUMERATION Source of Variance D.P. Sum of Sq. Mean Sq. P-Value Prob. Equality of adj. c e l l means 4 106.361 26.590 2.219 0.074 Zero slope 1 1013.687 1013.687 84.605 0.000 E r r o r 73 874.646 11.981 Equality of slopes if 35.052 8.763 0 .720 0.584 E r r o r 69 839.594 12.168 An examination of this table shows that the pro-b a b i l i t y of no regression slope Is v i r t u a l l y zero and the probabil ity of equal slopes for the different groups Is greater than $0%. Hence the use of analysis of covariance Is j u s t i f i e d . The probability of 7.4£ that the adjusted c e l l means are equal indicates a 'borderline* probability of f inding, among the contrasts, any Important differences due to the different treatments. Table VIII shows the probabil i t ies that the n u l l hypotheses, H ( l , l ) to H(l,4), are true. TABLE VIII TESTS OP MAIN HYPOTHESES TEST 11 NUMERATION Hypothesis t-value D.P. t2=F Probability H ( l , l ) -0.925 73 0.8556 0.361 H ( l , 2 ) 1.4818 73 2.1957 0.139 H(l,3) -1.7225 73 2.9671 0.085 H(l,4) -1.9986 73 3.9944 0.047 Conclusions i Numeration School #1. H(l,l)» The negative t-value Indicates that the * practice* group, B^, did better than the experi-mental group, A^, but there Is a 36^ probability that this could occur by chance, hence H ( l , l ) Is accepted. School #2. H ( l , 2 ) i Experimental group A 2 did better than * practice* group B, but with a probability as large as \h% this can merely be considered as Indicative of a possible trend towards higher achievement In numera-t ion of the graphical representation students. Across schools. H(l,3)» The treatment groups of School #2, A 0 +B 0 , did better than those of School #1, A +B * 2 1 1 with only 8% probabil i ty, suggesting a trend which corrobo-rates the subjective Judgment of the Investigator, that the mean achievement l e v e l of School #2 was higher than that of School #1. H(l,4)t The control group did not gain as much as the experimental groups, the probability of this difference having ocourred by chance being less than 5%, A r e - t e s t , after an opportunity for review and reteachlng, would have helped to determine whether t h i s apparently lower gain was s i g n i f i c a n t . Main Eypotheses Regarding Computational S k i l l s The results of the tests for computational s k i l l s are summarized In Table IX. A l l groups except A^ showed improvement on the post-test. Table X gives the results of the tests for the used of Analysis of Covariance. The probability of no regression slope Is v i r t u a l l y zero. The probability that the slopes are the same Is a borderline case but was considered sufficient to Justify TABLE IX SUMMARY OP TEST RESULTS TEST II t COMPUTATION Group B l C l A2 B 2 Max. possible so ore 40 40 40 40 40 No. oases per group 13 17 18 16 17 Pre-test i Max. observed 34 31 32 40 32 Mln. observed 17 15 12 20 19 Mean 26.154 25.471 23.944 2 9 . $ 0 0 2 7 . 2 9 S. D. 4.562 4 .200 4.976 4.803 3.584 Post-testi Max. observed 36 33 34 40 40 Mln. observed 17 20 12 28 20 Mean 24.462 26.412 2 5 . 1 6 7 33.750 31.765 S. ,D. 5 .027 3 .607 5.555 3 .531 5.333 Adjusted mean 24.646 27.069 26.881 31.616 31.159 Std. error 0.991 0.870 0.872 0.936 0.870 TESTS FOR USE OF ANALYSIS OF COVARIANCE TEST II t COMPUTATION Source of Variance D . F . Sum of Sq. Mean Sq. F-Value Prob. Equality of adj. c e l l means 4 4 9 9 . 0 7 6 124.769 9 . 7 8 8 0 . 0 0 0 Zero slope 1 7 2 1 . 8 9 3 7 2 1 . 8 9 3 5 6 . 6 3 4 0 . 0 0 0 E r r o r 75 956.004 1 2 . 7 4 7 Equality of slopes 4 128.495 32.124 2 . 7 5 6 0 . 0 3 4 TABLE XI TESTS OF MAIN HYPOTHESES TEST II 1 COMPUTATION Hypothesis t-value D . F . t =F Probability H(2 , l) -1.8403 75 3.3867 0 . 0 6 6 H(2,2) 0.3629 75 0.1317 0.717 H(2,3) -5.9092 75 34.9186 0.000 H(2,4) -1.7433 75 3.0392 0.082 the use of the •common slope 1 factor In the use of analysis of covariance and calculation of the adjusted means. The probabil i ty of 0.000 that the adjusted c e l l means are the same indicates there Is some difference of note, either between schools or between treatments. Table XI shows the probabil i t ies that the n u l l hypotheses, H(2 , l ) to H(2,4) are true. Conclusions t Computation H(2 , l ) t The negative t-value confirms that the • p r a c t i c e 1 group did better than the experimental group A-^ with a probability of 6,6% of this happening by chanoe. This trend would correspond to expectations i n the wake of the actual teaching that was oarried out with group B^. H(2,2)t With the effect of i n i t i a l differences being removed by analysis of covariance, there Is nothing s i g n i f i -cant about the difference betweeen the results for groups A and B2» hence H(2,2) Is accepted. H(2,3)» The students In School #2 showed a s i g n i f i -cantly higher achievement than the oomblned experimental groups of School #1, A^ + B^i hence H(2,3) i s rejected. H(2,4)i A l l the experimental groups oomblned showed a trend toward greater improvement than the control group, the probability being 8,2%, The greater part of this d i f -ference i s attributable to the superior performance of the pupils of School #2 as can be seen by an examination of the t - t e s t matrix f o r adjusted group means, shown In Table XII. TABLE XII t-TEST MATRIX FOR ADJUSTED GROUP MEANS TEST II t COMPUTATION Group *1 B l C l A 2 B 2 0.0 B i -1.840 0.0 c i -1.70 0.155 0.0 A 2 -5.095 -3.504 -3.563 0.0 B 2 -4.936 -3.309 -3.433 0.363 0.0 TABLE XIII SUMMARY OP TEST RESULTS TEST III i GRAPHICAL REPRESENTATION Group A l C l A 2 B 2 No. cases per group 13 17 18 14 16 Pre-test i Max. poss. so ore 9 9 Max. observed 7 7 Mln. observed 0 0 Mean 3.643 2 . 6 2 5 S.D. 2.437 2 . 4 1 9 Post-testj Max. poss. score 16 16 16 16 16 Max. observed 9 8 5 13 13 Mln. observed 1 1 0 5 1 Mean 4 . 5 3 9 4 . 0 5 9 2 . 3 3 3 9 . 4 2 5 7 . 2 5 0 S.D. 2 . 5 6 4 3 . 5 5 0 Adjusted mean 8 . 9 3 0 7 . 6 8 7 Std. error 0 . 6 0 5 0 . 5 6 5 Main Hypotheses Regarding Graphical Representations For reasons Indicated In the section setting out the main hypotheses (pages 4?-48 ), hypothesis testing for graphical representation was carried out separately for the two schools. The test results for both schools are summarized In Table XIII. TABLE XIV ANALYSIS OF VARIANCE SCHOOL #1 TEST III t GRAPHICAL REPRESENTATION Source of Variance D.F. Sum of Sq. Mean Sq. F-Value Prob. Mean 1 6 2 4 . 5 5 3 6 2 4 . 5 5 3 1 7 3 . 3 0 3 0 . 0 0 0 Overall treatment 2 4 3 . 7 4 5 2 1 . 8 7 2 6 . 0 6 9 0 . 0 0 5 Contrasts! H(3,U 1 4 3 . 1 6 0 4 3 . 1 6 0 1 1 . 9 7 6 0 . 0 0 1 H ( 3 . 2 ) 1 1 . 6 9 5 1 . 6 9 5 0 . 4 7 0 0 . 5 0 3 Error 45 1 6 2 . 1 7 2 3 . 6 0 4 Conclusions • - School #1 Table XIV shows the F-Values and the probabilities for the hypotheses proposed. H ( 3 , 1 ) J The combined experimental groups of School #1 showed a significantly higher achievement than the control group, the probability of this happening by ohanoe being 0,5%i H ( 3 . D i s r e j e c t e d . H(3,2)» There was no s i g n i f i c a n t d i f f e r e n c e between mean s c o r e s f o r T e s t I I I of Groups and B^j H(3»2) i s a c c e p t e d . C o n c l u s i o n s - Scho o l #2 The r e s u l t s of the t e s t s f o r the use of a n a l y s i s of c o v a r i a n c e are g i v e n i n Table XV. TABLE XV TESTS FOR USE OF ANALYSIS OF COVARIANCE TEST I I I i GRAPHICAL REPRESENTATION SCHOOL #2 Source of V a r i a n c e D.F. Sum of Sq. Mean Sq. F-Value Prob. E q u a l i t y of a d j . c e l l means 1 11.018 11.018 2.2031 0.146 Zero s l o p e 1 139.397 139.397 27.873 0.000 E r r o r 27 135.031 5.001 E q u a l i t y of s l o p e s 1 9.580 9.580 1.895 0.167 E r r o r 26 125.452 4.825 H ( 3 , 3 ) : The p r o b a b i l i t i e s b e i n g v i r t u a l l y z e r o f o r the z e r o s l o p e t e s t and 17$ f o r equal s l o p e s , the use of a n a l y s i s of c o v a r i a n c e would be j u s t i f i e d but the p r o b a b i -TABLE XVI TABULATION OP SCORE GAINS TEST I i NUMERATION SCHOOL #2 A 2 High A 2 Other B 2 High B Other 7 6 8 5 8 6 6 5 7 6 6 5 7 5 5 4 6 4 5 4 5 1 5 3 3 5 3 -2 1 0 -1 - 3 T o t a l 36 44 28 37 n 8 8 6 11 Mean 4 . 5 5 .5 3 . 4 11ty of 15% that the adjusted o e l l means are equal Indicates that no s t a t i s t i c a l significance should be attached to the higher achievement of group A 2 . The experiment had of course been oontamlnated by the participation of group B 2 i n b u i l d i n g the charts and i n their being exposed to the results of the work of the others. H(3,3) i s rejected. However, the observed trend Indicates that a longer range study with better control and more adequate evaluation could be worthwhile. SUPPLEMENTARY HYPOTHESESi ABILITY GROUPINGS SCHOOL #2 Reliable data on the general mathematical a b i l i t i e s of the students i n School #2 were available from a teacher with long experience who had kept regular records of d a l l y work and periodic test ing. Hence plans were made for s t a t i s t i c a l test ing of the effects of a b i l i t y on progress In the three areas of Numeration, Computation and Graphical Re-presentation. O r i g i n a l l y , the children were c l a s s i f i e d roughly as ' h i g h 1 , 'above average', 'average', and'below average'. For s t a t i s t i c a l purposes each treatment group was subdivided Into 'high' and ' o t h e r ' . Score gains on the numeration tests for these groupings are shown In Table XVI, I t was hypothesized that the slower children of Group A 2 had made greater progress than those of Group B 9 or those of higher a b i l i t y In Group A 9 , Numeration Analysis of covariance was carried out on the test r e s u l t s for the four ' a b i l i t y 1 groupings as shown i n Table XVII. Orthogonal contrasts were planned according to Table XVIII, leading to Hypotheses H(4 , l ) , H(4,2) and H(4,3). The hypothesis testing i s shown i n Table XIX. TABLE XVII TESTS FOR USE OF ANALYSIS OF COVARIANCE ABILITY GROUPINGS SCHOOL #2 TEST I t NUMERATION Source of Variance D . F . Sum of Sq. Mean Sq. F-Value Prob. Equality of adj. c e l l means 3 39.822 13.274 1.792 0.170 Zero slope 1 177,421 177.421 23.957 0.000 Error 28 207.364 7.406 Equality of slopes 3 17.330 5.777 0.760 0.530 E r r o r 25 190.034 7.601 An examination of Table XVII shows that the probability of no regression slope Is v i r t u a l l y zero and the probability of equal slopes for the different groups i s greater than 50%• Hence the use of analysis of covariance i s J u s t i f i e d . HYPOTHESIS MATRIX : ABILITY GROUPINGS SCHOOL #2 A 2 High A 0 Other B 2 High B 0 Other 1 1 1 -3 1 -1 0 0 1 1 -2 0 Supplementary Hypotheses U s i n g the h y p o t h e s i s m a t r i x shown i n Tab l e X V I I I , the f o l l o w i n g n u l l hypotheses were formulated t H(4,l)» There w i l l be no s i g n i f i c a n t d i f f e r e n c e between the combined mean s c o r e s of a l l c h i l d r e n I n Group A 2 and of the more a b l e c h i l d r e n I n Group B^ and the other c h i l d r e n I n Group B 2 » H(4,2)i There w i l l be no s i g n i f i c a n t d i f f e r e n c e between the mean score of the more a b l e c h i l d r e n i n Group A 2 and the other c h i l d r e n i n Group Ag. H(4,3): There w i l l be no s i g n i f i c a n t d i f f e r e n c e between the mean score of a l l c h i l d r e n I n Group A 2 and t h a t of the more a b l e c h i l d r e n i n Group B 2 . TABLE XIX TESTS OP SUPPLEMENTARY HYPOTHESES TEST I « NUMERATION ABILITY GROUPINGS SCHOOL #2 Hypothesis t-value D.P. t 2=P Pr o b a b i l i t y H(4,l) 2.273 28 5.164 P=0.029 H(4,2) 0.447 28 0.1998 0.662 H<4,3) 0.281 28 0.079 0.772 Conclusions i Numeration. A b i l i t y Groupings H(4,l ) t There was a s i g n i f i c a n t d i f f e r e n c e between the combined mean scores of the more able c h i l d r e n In Group B 2 and a l l c h i l d r e n In Group A 2 as opposed to the other c h i l d r e n In Group B 2. The p r o b a b i l i t y of chance occurrence being 2,9% t H(4,l) i s rej e c t e d . H(4,2)i There was no s i g n i f i c a n t d i f f e r e n c e between the mean score of the more able c h i l d r e n In Group A 2 and the other c h i l d r e n i n Group A 2 i H(4,2) i s accepted. H(4,3)i There was no s i g n i f i c a n t d i f f e r e n c e between the mean score of a l l c h i l d r e n i n Group A 2 and that of the more able c h i l d r e n i n Group B 2 i H(4,3) i s accepted. The above acceptances and r e j e c t i o n point to the conclusion that a l l c h i l d r e n i n Group A 2 made approximately equal progress and progress equal to the more able c h i l d r e n In Group B«, whereas the other c h i l d r e n In Group B9 made TABLE XX TABULATION OP SCORE GAINS TEST II i COMPUTATION SCHOOL #2 A 2 High A 2 Other B 2 High B 2 Other 12 8 9 9 8 7 7 9 5 6 8 4 4 4 8 4 4 3 4 3 3 1 2 1 3 - 3 0 2 1 0 - 3 Total 34 32 30 4 4 n 8 8 6 11 Mean 4 . 2 5 4 . 0 5 . 0 4 . 0 TESTS FOR USE OF ANALYSIS OF COVARIANCE TEST II i COMPUTATION « ABILITY GROUPINGS SCHOOL #2 Souroe of D . F . Sum of Sq. Mean Sq. F-Value Prob. Equality of 3 88.532 29.511 2.651 0.062 adj. c e l l means Zero slope 1 68.264 68.264 6.132 0.019 E r r o r 28 311.704 11.132 Equality of 3 26.737 8.912 0.781 0.518 slopes E r r o r 25 284.968 11.399 TABLE XXII TESTS OF SUPPLEMENTARY HYPOTHESES TEST II i COMPUTATION t ABILITY GROUPINGS SCHOOL #2 2 Hypothesis t-value D . F . t =F Probability HC5.D 2.238 28 5.008 0 .032 H(5,2) 2.039 28 4.156 0.049 H(5t3) -0.784 28 0.614 0.446 Inferior progress. This indicates strengthening of s k i l l s , hopefully due to new insights gained from application of s k i l l s to the task of obtaining information from a graph. Also i t should be noted that individual children i n Group A 2 made unexpectedly spectacular gains. These results would appear to merit further Investigation with larger and less homogeneous groups. Computation A similar analysis using the same hypotheses, r e -numbered as H(5»U. H(5,2) and H(5,3) was carried out r e -garding computational progress In School #2. The gain adores are tabulated i n Table XX and Table XXI summarizes the tests for analysis of covariance. An examination of Table XXI shows that the probability of no regression slope i s only 6,2% and the probability of equal slopes for the different groups Is greater than 50%, Hence the use of analysis of covariance Is J u s t i f i e d , Conoluslonst Computation. A b i l i t y Groupings, H(5»D« There was a significant difference between the combined mean scores of a l l the children of high a b i l i t y i n Group B 2 and a l l children i n Group A 2 as opposed to the other children i n Group B 2 , the probability of chance occurrence being 3,2%t E(5»D i s rejected. H(5.2)« There was a significant difference between TABULATION OP SCORE GAINS TEST I I I i GRAPHICAL REPRESENTATION SCHOOL #2 A 2 High A 2 Other B 2 High B 2 Other 10 9 6 9 7 7 6 7 7 7 5 7 7 6 4 6 5 4 3 5 5 2 2 5 5 4 2 3 3 1 1 T o t a l 48 35 31 51 n 8 6 6 11 Mean 6.0 5.82 5.18 4.64 the mean score of the more able children In Group A 2 and the other children In Group A 2 , the probability of chance occurrence being 4.9#j H (5»2) i s rejected. H ( 5 » 3 ) * There was no significant difference between the mean score of a l l children In Group A 2 and that of the more able children In Group Bgi H (5»3) Is accepted. There i s l i t t l e significance In the acceptance of H(5,3) (as compared with H(4,3)) since the combined results of H(5»l) and H(5,2) are indicative of the higher achievement of the more able children In both groups. Also this experi-ment was contaminated by practice work given to A 2 . Hence these results can i n no way be considered conclusive. Graphloal Representation - A b i l i t y Groupings I t was suspected that the more able youngsters i n Group B 2 would have made gains i n s k i l l s of graphical representation by virtue of exposure to a greater extent than the other members of their group. The tabulation of soore gains shown i n Table XXIII shows them i n the following rank order» A 2 High, A 2 Other, B 2 High, B 2 Other. However, the standard error of adjusted mean scores i s approximately 1.0 and the difference between A 2 High and B 2 Other adjusted means i s less than 2.0. (See Table XXIV). Hence we can ex-pect ho s t a t i s t i c a l l y s ignif icant results from this data. The s t a t i s t i c a l results of Analysis of Covariance and Hypothesis Testing similar to H (4 , l ) , H(4,2), H(4,3) are shown In Tables XXV and XXVI. TABLE XXIV ADJUSTED MEAN SCORES TEST III $ GRAPHICAL REPRESENTATION ABILITY GROUPINGS SCHOOL #2 Group NO. of Pupils Group Mean Adj. Group Mean Standard E r r o r A 2 High 8 1 1 . 1 2 5 9 . 4 9 7 1 . 1 0 0 5 A 2 Other 6 7 . 1 6 7 8 . 3 1 9 1 . 0 2 7 B 2 High 6 9 . 5 0 0 7 . 8 3 8 1 . 116 B Other 2 10 5 . 9 0 0 7 . 5 0 8 0 . 9 3 2 TESTS FOR USE OF ANALYSIS OF COVARIANCE TEST III» GRAPHICAL REPRESENTATION ABILITY GROUPINGS SCHOOL #2 Source of Variance D . F . Sum of Sq. Mean Sq. F-Value Prob. Equality of adj. c e l l means 3 13.993 4.664 0.883 0.465 Zero slope 1 40.052 40.052 7.582 0.010 E r r o r 25 132.057 5.282 Equality of slope 3 41.518 13.839 3.363 0.037 E r r o r 22 90.539 4.115 TABLE XXVI TESTS OF SUPPLEMENTARY HYPOTHESES TEST III : GRAPHICAL REPRESENTATION ABILITY GROUPINGS SCHOOL #2 Hypothesis t-value D.F. t 2=F Probability a l l A 2 +B 2 High v s . B 2 Other 0.849 25 0.723 0.408 A 9 High vs. c A 2 Other 0.864 25 0.747 0.400 a l l A ? vs. c B 2 High 0.736 25 0.543 0,474 SUMMARIZED RESULTS, CONCLUSIONS AND IMPLICATIONS Numeration Summary of s t a t i s t i c a l results. No difference of s t a t i s t i c a l significance was found between the two experi-mental groups i n School #1. The graphical representation group i n School #2 showed a somewhat higher achievement than the d r i l l and practice group. S t a t i s t i c a l l y this can be considered a trend only, as probability of chance occurrence was 14#. School #2 showed a trend toward higher achievement than the combined experimental groups of School #1, the probability of chance occurrence being 8%, The oontrol group i n School #1 showed less gain In achievement than the combined results of a l l the experi-mental groups. This was s t a t i s t i c a l l y significant at the 0.047 l e v e l . Conclusionsi Working with numbers on the new material for the time of the experiment did not lower the basic s k i l l s tested as compared to s k i l l maintenance on a re-teaching and review program. The difference between schools Is not unusual and could i n part be responsible for the lower achievement of the oontrol group, members of whloh were a l l drawn from School #1. This experiment gives no guidance as to whether the poorer achievement of the control group was educationally signif icant* Implications. Graphical representation as used In this study served as an adequate vehicle for review and maintenance of basic numeration s k i l l s . At the same time the children gained some f a m i l i a r i t y with a new topic which In addition to Its i n t r i n s i c value offered new insights and applications of their s k i l l s . To determine whether the lower achievement of the control group was educationally detrimental, further experi-mental work would be necessary. For example, a r e - t e s t after a review and practice period would have supplied some Infor-mation regarding t h i s particular experiment. More desirable would be fresh experimentation to determine the optimum time to leave numerical s k i l l s unpractlced, for the purpose of pursuing other mathematic topics such as geometry. Computation Summary of s t a t i s t i c a l r e s u l t s . The review and practice group i n School #1 showed somewhat better achievement than the graphical representation group, the probability of chance occurrence being 6.6#. No difference of s t a t i s t i c a l significance was found between the two experimental groups In School #2. The experimental groups In School #2 did very much better than their counterparts i n School #1, the probability of chance occurrence being v i r t u a l l y zero. The control group i n School #1 showed somewhat less gain i n achievement than the combined results of a l l the ex-perimental groups, the probability of chance occurrence being 8,2%, This appears to be mainly accountable for by the difference i n the schools. Examination of the t - t e s t matrix indicates this group was on a par with the review and prac-t ice group, School #1, when compared with School #2. Conclusions. Prom the results of School #1, I t would appear that reteaching of concepts of basic operations, when these have not been adequately mastered, i s a more effective method of improving computational s k i l l s than incidental use, as i n the graphical representation u n i t . In School #2, where the ohlldren had good concepts of numeration and basic operations, and well developed s k i l l s , the group which spent most of i t s time on graphical repre-sentation progressed just as well as that which devoted con-siderable time to development of higher'facts*, those of 9 and 1 0 , i n this oase. The small amounts of practice which were given to the former group could have had a bearing on these r e s u l t s . The difference between the schools Justifies the different treatments resorted to during the course of the experiment. The achievement of the control group was oomparable to that of the experimental groups In the same school. Implications. The experience In School #1 i s an indication of the Importance of spending adequate time on •developmental a c t i v i t i e s * . Prom the results i n Sohool #2 i t would appear that small amounts of direct practice were Just as beneficial for maintalnenoe of competencies as the spending of a major proportion of the time on such a c t i v i t i e s . The achievement of the oontrol group i s a further indication that the amount of practice can be reduced. Graphical Representation Summary of s t a t i s t i c a l results. Sohool #1. Combined experimental groups showed a higher achievement than the control group. This was sta-t i s t i c a l l y significant at the 0.05 l e v e l . There was no significant difference between the achievement of the graphical representation group and the review and practice group. Sohool #2. There was no significant difference between the achievement of the two experimental groups. Conclusions. No conclusions can be drawn regarding the achievement of the children with respect to graphloal representation. The only s t a t i s t i c a l l y significant r e s u l t showed a higher achievement for the oomblned experimental groups of School # 1 over the control groups, but the ex-periment was contaminated by the exposure of the review and practice group to the construction of graphs. Supplementary Hypotheses A b i l i t y Groupings. School #2 Numeration. A l l children of the graphical repre-sentation group combined with the more able children of the review and practice group showed a higher achievement than the other children i n the seoond group, s t a t i s t i c a l l y s ignif icant at the 0.029 l e v e l . The more able and the less able children of the graphical representation group made similar progress. The whole of the graphical representation group showed similar progress to that of the more able children i n the second group. Individual children of lesser a b i l i t y made spectacu-l a r gains i n numeration s k i l l s after working on the graphi-c a l representation u n i t . Conclusions. The fresh approach of graphical repre-sentation appears to have helped the numeration s k i l l s and understandings of less able children to a greater extent than a routine review. Implications. With one class divided Into four groupings the numbers are small for adequate s t a t i s t i c a l analysis . The above results point to the need for further Investigation with larger numbers and less homogeneous groups than this school provided. Computation. Higher achievement appeared to go hand In hand with higher a b i l i t y . As the experiment had been contaminated, In addition to the small numbers Involved, judgment was withheld i n this case. Graphical representation. The groupings showed the following rank order of achievementi A^ more able, A 2 less able, B 2 more able, B 2 less able, but there were no results of s t a t i s t i c a l significance. The experiment was contaminated, the numbers small and the evaluation Instrument inadequate. Hence Judgment was withheld In this case. CONCLUSIONS AND RECOMMENDATIONS INTRODUCTION This study has confirmed i n a l imited way for two particular groups of primary children the results reported by Miss Biggs i n Mathematics In Primary S c h o o l s a s stated e a r l i e r , she claims that with a planned a c t i v i t y approach where children explore mathematical aspects and p o s s i b i l i -t i e s of their world, computational s k i l l s can be maintained with a considerable reduction In time spent on routine practice. I t has indicated the p o s s i b i l i t y that, with an ade-quately controlled approach, primary children would show the same Increased gains In computational efficiency, when more time Is spent on developmental a c t i v i t i e s , as did their Intermediate counterparts, discussed In Chapter I I . In l i n e with the suggestions of the Cambridge Con-Schools Council for the Curriculum and Examinations, Mathematics In Primary Schools. Curriculum B u l l e t i n #1, second e d i t i o n , (Londoni Her Majesty's Stationery Office, 1967) p. xv, p. 49. f e r e n c e on S c h o o l Mathematics 2 and Glennon and C a l l a h a n , 3 new m a t e r i a l , namely G r a p h i c a l R e p r e s e n t a t i o n , was Introduced t o two groups of F i r s t Year c h i l d r e n , without any s i g n i f i c a n t l o s s i n b a s i c a r i t h m e t i c competencies. SUMMARY OF FINDINGS AND IMPLICATIONS At the s t a r t of t h i s study, i t was h y p o t h e s i z e d t h a t a r i t h m e t i c competencies of primary c h i l d r e n a t the end of f i r s t y e a r would be e q u a l l y w e l l - m a i n t a i n e d d u r i n g a t h r e e week p e r i o d spent on a u n i t of G r a p h i c a l R e p r e s e n t a -t i o n , and d u r i n g an e q u i v a l e n t time spent on r o u t i n e r e v i e w and p r a c t i c e . F u r t h e r , i t was hypothesized t h a t m a i n t e n a n c e would be s u p e r i o r t o t h a t of c h i l d r e n who were exposed t o no a r i t h m e t i c a c t i v i t i e s d u r i n g the same p e r i o d . The a r i t h m e t i c competencies t e s t e d were Numeration ( r e a d i n g , w r i t i n g and u n d e r s t a n d i n g of base t e n numerals ^ 9 9 ) and Computation ( a d d i t i o n and s u b t r a c t i o n w i t h sums and minuends ^ 1 0 ) . Numeration S t a t i s t i c a l a n a l y s i s of the d a t a on p r e - t e s t l n g and p o s t - t e s t i n g (see Chapter IV) showed t h a t working w i t h Cambridge Conference on S c h o o l Mathematics, Goals f o r S c h o o l Mathematics, (Boston: Houghton M i f f l i n Company, 1963), P. 7. ^ V i n c e n t J . Glennon and Leroy G. C a l l a h a n , Elementary S c h o o l Mathematics. A Guide t o C u r r e n t Research. ( A s s o c i a t i o n f o r S u p e r v i s i o n and C u r r i c u l u m Development, N a t i o n a l Educa-t i o n A s s o c i a t i o n , 1968), p. 81. numbers on the new material for the time of the experiment did not lower the tested numeration competencies for the time of the experiment. The control group showed somewhat poorer achievement, but the data from this experiment was Insufficient to assess educational significance of t h i s . The implication drawn from these results Is that graphical representation, as used i n this study, served as an adequate vehicle for review and malntalnanoe of basic numeration competencies. In addition, the children gained some f a m i l i a r i t y with a new topic which offered new Insights and application of their s k i l l s , besides i t s own i n t r i n s i c merits. Computation S t a t i s t i c a l analysis of the data on pre-testing and post-testing (see Chapter IV) showed that the review and practice group i n School #1 showed somewhat better achieve-ment than the graphical representation group. The children i n this school had not, for the most part, mastered the concepts of the basic operations being tested. Hence, i t was concluded that the reteaching of these concepts, which was carried out with the review and practice group, was more effective than the Incidental use, as i n the graphical representation group, for i n i t i a l mastery. In School # 2 , where the children had good concepts and well-developed s k i l l s , the two groups progressed equally well. The graphical representation group had done a small amount of practice, hence the implication here is that small amounts of direct practice were Just as beneficial as a major proportion of the time so occupied. In addition, the graphical representation group had the advantage, mentioned above, of the new material. Graphical Representation The attempted assessment of new learning by the graphical representation groups produced no results of statistical significance. Limltatlons The chief limitations of this study can be classified under Time, Place, School Differences, Contamination and Evaluation Instruments. Timet It is impossible from this short experiment to draw any conclusions regarding long range effects. The teaching period was only three and one half weeks and there was no follow-up retention test at a suitable later interval. Placet While there was considerable range of ability in the classrooms used, the children were drawn from a fairly homogenous socio-economic background. School dlfferenoest As indicated, the teaching methods in the two schools had been so different that a divergence of methods during the experiment was inevitable. In addition, the different atmosphere and attitude at the end of the term undoubtedly affected the f i n a l t e s t r e s u l t s . When I t came to the graphical representation t e s t , the ch i l d r e n i n School #1, lacking an adequate understanding of basic operations, were un w i l l i n g to attempt unknown ad d i t i o n or subtraction equations. This resulted i n extremely low mean scores. Contaminationi In both sohools, a l l c h i l d r e n were exposed to the r e s u l t s of the work done by the c h i l d r e n i n the Graphical Representation group. The D r i l l and Practice groups i n p a r t i c u l a r contributed to the graphs that were b u i l t . This e f f e c t would have been more severe i n Sohool #2, where the r e s u l t s were on display i n the main olassroom. I n School #1, the charts were i n the 'Arithmetic Corner' and not continuously within sight of the ch i l d r e n . In addition, i n School #2, tht Graphical Representa-t i o n group was given two or three practice sheets In compu-t a t i o n by t h e i r own teacher. I t was o r i g i n a l l y intended that there be no practice f o r them. The D r i l l and Practice group i n Sohool #2 were In-troduced to the ' f a c t s ' f o r nine and ten, which were new to them. However, t h i s can be considered more as an ex-tension of work already mastered since no new concepts or methods were used. Evaluation Instruments» The evaluation Instruments f o r graphical representation were untried and probably Inadequate. SUGGESTIONS FOR THE FUTURE In retrospect, this study has many of the earmarks of a p i l o t or f e a s i b i l i t y study. Useful conclusions can even be drawn from unforseen occurrences. For example, i t would appear that small amounts of practice are better than none at a l l . Any conclusions from this study have to be confined to the population involved. However, the faot that neither Groups A or C showed any serious losses in numeration or computational s k i l l s over the period of time involved would Indicate that the topic i s worth pursuing further. Both the graphing material and the geometry would lend themselves to a long-term projects work i n either can be interspaced with periods of instruction and development i n the basic s k i l l . One possible follow-up study would eliminate the contamination experienced in the present investigation by having pairs of classes, each pair being taught by one teacher or investigator, over a prolonged period. One class would follow a basic program (as had been done i n School #2 before the a r r i v a l of the investigator) while the other class of each pair would have practice time cut drastically i n order to include other topics. Graphical representation i s one such topic with the added attraction that much review and re-enforcement (practice) can be woven Before any follow-up research i s carried out, though, It Is Imperative that contact be made with B r i t a i n and i n -vestigators be brought up to date with research there, r e l a -tive p a r t i c u l a r l y to Miss Biggs' statement, previously r e -ferred to, that teachers have been able to considerably reduce practice time. NON-STATISTICAL OUTCOMES REGARDING GRAPHICAL REPRESENTATION Graphical representation Is an Important topic i n i t s own r i g h t i n our society today. Children are obviously able to learn a good deal regarding this at a considerably younger age than they do In our current B r i t i s h Columbia curriculum. (Grades 4 and 5)# I f Introduced In the primary grades, It Is essential that every chi ld be actively involved i n the learning process according to Piagetian p r i n c i p l e s . With such Involvement, and careful planning, an a l e r t teacher can bring i n a considerable amount of review of basic s k i l l s . Often new Insights Into meanings of concepts w i l l be gained and new uses for basic s k i l l s w i l l be discovered by the ohlldren. An example from this project was the p r a c t i c a l application of knowledge of 'tens and ones' to interpreta-t i o n of data presented on a v e r t i c a l bar graph. During the development of the unit i n School #2, there was a noticeable redirection of the children's thinking from •blocks and numbers games' to r e a l l i f e situations, a hoped-for outcome of most mathematics programs today. Graphical representation also lends i t s e l f well to correlat ion of mathematics with subjects such as science and language. I t i s a good vehicle for team work and s o c i a l i z i n g , the importance of which i s being increasingly recognized i n the learning processes of young children. The spurts of achievement noted among individual children of average or lower a b i l i t y would merit further investigation. The numbers were too small for adequate s t a t i s t i c a l analysis, but this phenomenon has been noted i n studies of older children. (See Chapter I I , p. 15-16). The unit had advantages for the teacher too. She got to know the children more quickly, found she had less work to do, and that the variety and challenge were much more interesting than 'review*. A note of caution. In the course of the work i n School #1, i t appeared that Graphical Representation was not suitable for teaching the basic concepts of our number system. However, no claim to the contrary was made at any time, nor i s implied i n the Nuffield Guide used. The i n -vestigator would have been well advised, had time permitted, to have brought these children up to a certain standard of p r o f i c i e n c y b e f o r e embarking on the new m a t e r i a l . CONCLUDING REMARKS L. G. W. Sea l e y , i n h i s d i s c u s s i o n of "Some Problems I n v o l v e d i n the I n d u c t i o n ( s i c . ) of New Approaches i n t o Primary S c h o o l s , " p o i n t s t o the d i f f i c u l t i e s of e v a l u a t i n g a new approach, but says, " i t i s o f t e n p o s s i b l e t o p o i n t t o cas e s i n which c h i l d r e n taught by such-and-such a new ap-proach have not s u f f e r e d i n r e s p e c t of attainment on t e s t s of a t r a d i t i o n a l nature."-* I t I s claimed t h a t t h i s has indeed been done i n t h i s study. When d i s c u s s i n g the tremendous d i f f i c u l t i e s I n v o l v e d I n any attempt a t s c i e n t i f i c a l l y c o n t r o l l e d e x p e r i m e n t a l r e s e a r c h , Mr. Seal e y s a y s i • . . w h i l e we may not be a b l e t o o b t a i n evidence of the r e l a t i v e v a l u e of a new approach, we should ap-p r e c i a t e , where the approach c o n t i n u e s t o s u r v i v e , the Importance of i t s s u r v i v a l as evidence of i t s v i a b i l i t y . T h i s seems t o be happening, a t the time of w r i t i n g , w i t h the approach i n primary e d u c a t i o n , of more a c t i v i t y and ^ S e a l e y , L. G. W., "Some Problems I n v o l v e d i n the I n d u c t i o n of New Approaches i n t o Primary S c h o o l s , " i n J . D. W i l l i a m s , ed., Mathematics Reform i n the Primary S c h o o l . A r e p o r t of a meeting of e x p e r t s h e l d i n Hamburg d u r i n g January, 1966, (Hamburg! UNESCO I n s t i t u t e f o r E d u c a t i o n , 1967) P. 54. e x p l o r a t i o n , and l e s s p r a c t i c e , not only s u r v i v i n g but spreading.? The r e s u l t s of t h i s study are a d m i t t e d l y l i m i t e d , but they do show t h a t i t i s p o s s i b l e t o teach more t o p i c s than a t pr e s e n t , and a t the same time m a i n t a i n b a s i c competencies. 'Two Canadian examples of t h i s trend are W. W. Ba t e s and D. I n g l l s , Mathaotlon ( T o r o n t o i Copp C l a r k and Co. L t d . , 1970), and Rob e r t a C h l v e r s e t a l . , P r o j e c t Mathematics. (Toronto: H o l t and R i n e h a r t and Winston of Canada L t d . , 1970). APPENDICES PLANS FOR TEACHING A general plan was drawn up for each group that was to be taught. Observation and pre-test results revealed that groups and B 2 would require quite different t r e a t -ment over and above the difference between t r a d i t i o n a l and i . t . a . orthography. The same plan was used to start off with, for both A groups, but It evolved quite di f ferently i n the two classrooms. A suggested outline was kept prepared for about a week at a time, but the detailed d a i l y planning was done every day, taking into account what had developed during the class sessions. This was part icularly true of the graphing sessions which lent themselves well to a group version of Piaget's ' c l i n i c a l method'. 1 General Plan—A Groups—Graphical Representation Almt To develop a b i l i t i e s and understandings In the areas of (a) construction and (b) Interpretation of p i c t o r i a l and block charts. Evaluation of Uniti (a) Construction to be carried out as group a c t i v i t y See F l a v e l l , J . H . The Developmental Psychology of Jean Piaget. (New Jerseyt D.Van Nostrand Co. , I n c . , 1963)» P.29. accompanied and followed by discussion. I t i s hoped children w i l l suggest further activity. No formal evaluation w i l l be attempted. (b) Interpretation! Evaluation w i l l be attempted with a short test (Test III) i n which a chart i s displayed and the children are asked to print answers to orally presented questions which ask f o r i (i) Information directly obtainable from and pertaining to the chart. ( i l ) construction of number sentences and equations expressing the above information and how i t i s obtained. ( i l l ) construction of other number sentences and equations suggested by the data set out on the chart. General Outline—as adapted from Pict o r i a l Representation Stage Onet Comparison of 2 rows (columns), one object per child. Toplot Comparison of number of boys to number of g i r l s , (a) Concrete objects (e.g. blocks, milk bottle tops). Compare the number of boys to the number of g i r l s in the group, e.g.i diagram. ] | I | | | | | i i — i Nuffield Mathematics Project, Pictorial Representation {TJ , (Londoni W. & R. Chambers and John Murray, 196?, a v a l l -able from Longmans Canada Ltd., Don Mills, Ontario). Variationsi comparison of boys and g i r l s i n the whole class or i n the families of the group. Discuss results and develop appropriate number sentences on large chart of chalkboard. (b) Transit ion to more permanent form of recordingi one inoh squares, blue for boys/ pink for g i r l s , to be pasted onto a chart. Children, working with partners to be encouraged to develop own number and word sentences. Stage Twot Increase In number of data, from comparison of two rows to comparison of several rows. Topici Birthdays. Prepare charts with a row for each month. Provide uniform pieces of paper on which each c h i l d w i l l draw a self portrait to be pasted i n the appropriate place. Children to be encouraged to develop number and word sentences that t e l l about the chart, p a r t i c u l a r l y as i t pertains to their own birthday> or that of a f r i e n d . E l i c i t problems for the children to pose to each other. Stage Threet Transit ion to block chart. Topici Pets. (a) Pet Chart. Horizontal rows prepared for each type of pet owned by a p u p i l . Pupils to colour, out out and paste their pets In appropriate rows. Hopefully a hap-In troduce the symbols ^ and hazard pasting w i l l reveal that the longest row does not have the most pets and a fresh start w i l l be suggested with each pet i n one-to-one correspondence with one above i t . Chart can then be ruled v e r t i c a l l y and use of special l i n e s for 5 's and 10*s ("to save counting 1 ) can be developed. This w i l l provide incidental review of counting by 5 's and 10's and of how many 10*s and ones are represented by a given number, (b) Block Type Chart—Vertical. (Column) Arrangement. New chart, with unit squares, different colours for each pet, to be 1 p i l e d up 1 In columns. Stage Four* Use of squared paper, squares coloured i n . Topicsi Birthdays, pets. Stage Fivei ( i f time, which i s doubtful). Abstract representation by s t r i p s and by bar l i n e . General Plan—Group B-^—Review and Practice Review and Practice of (a) Addition and subtraction facts of 7, 8, 9, 10, using both equation form and v e r t i c a l form. Addition equations with more than two addends were quite unfamiliar to these ohlldren and therefore omitted. (b) Counting by ones, f ives and tens. (c) Simple word problems, oral and printed, which Involved faots and operations being reviewed. (d) Numeration! units and tens and relationship between printed numeral and number It represents! also relat ionship to counting money, arranged In dimes and pennies. Materials and Methods Flannel board and chalk board demonstrations, Involving a few chi ldren, while others watched. Various types of counters available for individual manipulation and s o l u -tlon^and checking of printed exercises (dittoed work-sheets) • Coloured sticks and pegs, especially for numeration. Evaluation! Test I , II and III as described In Chapter I I I . General Plan—Group B 2—Review and Praotioe Teachi symbols >^ , • Having been Introduced, unexpectedly as *the lady from the U n i v e r s i t y 1 who was going to do •experiments* and teach new things, i t was deemed essential for the es-tablishment of good rapport to introduce something newj Also t h i s was requested by the teacher. Review and Praotioe of (a) Addition, subtraction and mult ipl icat ion facts up to 8. <b) Equations with more than 2 addends, sums ^ 8. (c) *Balanclng the equation* e.g. 5+3=2+_. (d) Use of parenthesis as i n Number Patterns I 3 . (e) Numeration! units and tens and relationship between printed numeral and the number i n represents. (f) Counting by ones, 5 's and 10*s. (g) Counting moneyi dimes, nickels and pennies. (h) 'Extensions 1 such as 57+l=_, 43-l*_, I .e. adding and subtracting one to any number <^ 100. Materials and Methods Children to work singly, using one inch ooloured cubes. In oral lessons equations w i l l be e l i c i t e d and printed on the chalkboard. Children to be encouraged to make their own discoveries. Practice work sheets to be done with teacher assistance available. Expectation of amount to be completed to be adjusted to i n d i v i d u a l a b i l i t i e s . Evaluatlont Test I , II and II as described i n Chapter I I I . General Plan—Group Cj,—Geometry General Topict Two dimensional geometric shapes. Proposed sub-topics, materials and methodst (a) Develop f a m i l i a r i t y with o i r c l e s , ovals, triangles and rectangles. Manipulation and sorting of cardboard shapes of various colours and sizes. Tracing and colouring, cutting, sorting and pasting, extending ^Chivers, et a l , Number Patterns Book I (Torontot Holt, Rlnehart and Winston, 19^6). given patterns, and creating new ones. (b) Develop background for later work with con-gruence. Comparison of sizes and colours of similar shapes. (c) Review open and closed curves and related topics suoh as point, l i n e , Inside, outside, on, between. Group work at chalkboard and flannel board. Individual work with dittoed work sheets. (d) Geometric shapes from a collection of points, or (polygonal) with pogboards and elastics. (e) Folding regular shapes Into equal parts (preparation for fractions). (f) Tesselation. Cutting and pasting of regular geo-metric shapes, seeking to discover which w i l l cover an area without leaving gaps. Source Bookst (a) to (e) adapted from Elementary Sohool Mathematics. Second Edition. Book 1. Teacher's Edition, (f) adapted from Shape and Size. ^ Nuffield Mathematics Project.^ Evaluationi No evaluation of geometry was planned as these children were to be subjected to TestsI.II and I I I . Robert E. Eicholtz, et a l , Elementary School Mathematics. Seoond Edition. Book 1. Teacher's Edition. (Addlson-Wessley (Canada) Ltd., 1969, Don Mills, Ontario). ^Nuffield Mathematics Project, Shape and Size W Nuffield Mathematics Project (Londont WAR Chambers and" John Murray, 1967, available from Longmans (Canada) Ltd., Toronto). SAMPLE LESSON PLANS School #1 , Thursday, June 4 . Group B-^ i (15 minutes) Review of addition and subtrac-t ion facts of 8 i Have counters and worksheets ready, i n a c i r c l e on the f l o o r , for each c h i l d . Brief review and assign seat work; allow a start to be sure pupils understand what i s expected. Children take counters and worksheets to their desks, to complete. (See Figure 2 , page 97) Group C^i ( 2 5 minutes) Patterns from geometric shapes. On f lannel board, start a pattern, using cardboard c i r c l e s , triangles and rectangles. Have children complete. Repeat with a new pattern. Have children develop a pattern from scratch. Follow-up of yesterday's lessont Did anyone find an oval shape at home? Seat work assignmentJ cut and paste shapes to complete a pattern begun on a worksheet. (See Figure 3 , P. 98) Group Aj_i ( 2 5 minutes) Paste pictures of selves, drawn yesterday, on a common size of paper, onto chart paper, the g i r l s i n one row r the boys i n another. Develop equations, word sentences and i n e q u a l i t i e s . Start the group off on a worksheet to follow up this development. (See Figure 4 , p.99) Group B.^ (10 minutes) Children return to arithmetic corner for checking of work, and of corrections on yesterday's work. FIGURE 2 SAMPLE WORKSHEET FOR GROUP B± Redrawn from h a l f of an 8|"xl4" worksheet. FIGURE 3 SAMPLE WORKSHEET FOR GROUP Redrawn from h a l f of an 8 | " x l 4 " worksheet. £ a . 9 CT"* -4 5~> 8* c (0 _J3 *0 o - 3 , it £ 3 a . o o 5- £ C TS o s ° S £ £ i ! * FIGURE 4 SAMPLE WORKSHEET FOR GROUP A x Redrawn from h a l f of an 8^"xl4" worksheet. School #2 Wednesday. June 3» Group A^i ( 2 5 minutes) a) Materials 1 each c h i l d to get enough one inch cubes for every member of his family. Reteach ^ and <^ by comparing number of boys to number of g i r l s i n family. Have three or four children write these Inequalities on the chalkboard. b) Materials1 blue paper squares for boys, pink for g i r l s . Paste i n two rows on lined chart paper. Print one or two sentences and equations—words and symbols. c) Seat worki (see Figure 5 t P* 101) Arrange partners, must be from a different reading group. Assign seat work, to be carried out i n foyer, while other group has lesson i n room. This proved to be too ambitious for one day. Hencet Thursday, June 4« Re-do part of worksheet as class pro-ject . "We are learning to write explanations that need sentences and equations." This lesson to follow Group Bg. Group B^i ( 2 5 minutes) Materials! nine one inch cubes for each c h i l d . Introduce symbols ^ and <^ . Review addition and subtraction facts of 9 . Children use* blocks to demonstrate their discoveries, and dictate equations for p r i n t i n g on the chalkboard. Start worksheet with >^ and <^ . (See Figure 6 , p. 102) Thursday, June 4» This group comes f i r s t . Work on 9 and symbols ^ and <^ using worksheet that was Just started yesterday. Red G r o u p Name. Partners name. A . The Bos/s Q^d Qvrls in our Group T h e r e ore bovjs. T t a r e o r e qirb _>ij _ < _ There are more, than . ere ore tewer (less) ft ere art . ren attocjettier. B. Out* families + -V-, = ^ have S cW\\c\ren attoa^etW. C. Other \N0\j5 1b moVe q: -V = a, "D . Mv, family and my partner s family . ar,d I have , people in oar -families . I U e ^ ^ + W , m V pari ner « less 4- • 4-Turn over To prvnT more scouts and e^uotibas. FIGURE 5 SAMPLE WORKSHEET FOR GROUP A g Redrawn from an o % x l l " worksheet. o ro 4-<D, , II -9 O O O O c r -c r r s j ^ ro •+-II 4-• cO it N -t-co 4- 4-OO 00 II II • • 4- 4-• II CM it H II CNl + + CO + 3S 4-t o . Il o u CM o cr» you c o s_ o TXT r - -cr> c "1 ^> o - 1 o - 1 -* 1 1 o U i FIGURE 6 SAMPLE WORKSHEET FOR GROUP B, Redrawn from h a l f of an 84"xl4" worksheet. APPENDIX G MANUAL POWER TEST OF BASIC ARITHMETIC SKILLS -FIRST TEAR PRIMARY-Forrest Johnson University of British Columbia November, 1970 Directions for Administering Test 1 General Description of Test 1 Detailed Instructions • 3 Individual Oral Testing Forms A and B 3 Pencil and Paper Sessions Form A 4 Part I Numeration. 4 Part II Addition and Subtraction 12 Amendments to Form A. 14 Pencil and Paper Sessions Form B 14 Part I Numeration.. 14 Part II Addition and Subtraction.... 21 Scoring. Pencil and Paper Sessions. 24' Suggested Amendments to Form B • 25 Test Construction.... 26 Bibliography. 31 c POWER TEST OF BASIC ARITHMETIC SKILLS -FIRST YEAR PRIMARY-DIRECTIONS FOR ADMINISTERING TEST GENERAL DESCRIPTION OF TEST This test i s composed of two parts, intended to evaluate skills and understandings in the areas of . I. Numeration II. Addition and Subtraction Operations and Facts The skills are separated so that a weakness in one will not be masked by-strength in the other as in traditional tests. An attempt was made during preliminary tryouts to subdivide Numeration intot a) printing of numerals, b) recognition and reading of numerals, c) understanding of base ten numeration system. The resultant rather lengthy test showed such a high degree of internal reliability (KR-20 = 0.9503) that this subdivision was abandoned. The three sub-skills appeared, on the basis of test items selected, to be closely interrelated. The test was specifically designed for use in British Columbia schools, towards the end of the f i r s t year of formal schooling. Part I i s based chiefly on numbers <40 with a smaller percentage of items using numbers between 40 and 100. A few selected items using numbers greater than 100 are included to provide some degree of assessment for advanced; pupils. Estimated time requirement for Part I i s 30-^0 minutes, which should be divided into two sessions. Part II employs chiefly examples with sums or minuends £8; a small percentage of items have sums or minuends of 9 or 10. Some assessment of the advanced pupils is again attempted with selected items using larger numbers (<100), and for the final items, vertical in place of equation form. The estimated time require-ment for Part II i s 15-20 minutes. A short individual oral testing session requires about one or two C minutes with each child. The information on the cover is optional. It may be completed and/or the picture coloured, i f the home-room teacher so desires,during the individual oral testing by the examiner. If the home-room teacher i s carrying out the testing i t i s advisable to do so in small groups. The assistance of a second persons i s important i f a l l members of an average sized class are to be tested at one time; this is to ensure that a l l children are working in the right place, understand the instructions and are doing as much as they are comfortably able to do,. Throughout the testing the examiner should use terms familiar to the class being tested: e.g. 'numeral' or 'number'. Children will need to be encouraged and reassured that they are not expected to be able to do a l l the examples. Some of them are 'about interesting things you will be learning next year'. Teachers, too, need to be reminded that this test covers considerably more content than that to which they are expected to have exposed their classes in the first year at school. Two parallel forms have been developed. Form A and Form B. Page four, which tests the concepts of 'greater than', 'less than', 'equal to', has two alternate versions: k* is designed for those children who have been instructed with the symbols rather than the words. 'i.t.a.4' i s provided for children who have been receiving instruction in the Pitman Initial Teaching Alphabet. Poor readers may, of course, be given help with the reading. An 'i.t.a.' form has also been provided for page five. 'Official' STOP signs are provided periodically, for formal breaks between sittings. However, the test is designed so that a break may be taken at the end of any page. Double lines, part way down a page, are a signal to wait for new instructions. In general the material on any one page is related, so that a 0 break in mid-page is inadvisable. Repetition of essential instructions i s of course necessary at this age level. DETAILED INSTRUCTIONS 1 INDIVIDUAL ORAL TESTING FORHS A AND B The children will come, individually, to the examiner to read aloud the following numerals (Part I) and equations (Part II) prepared on flash cards. Scoring space is provided on the cover. A tick in the fi r s t column indicates a correct response, a tick in the second column indicates an error\ scoring i s done later in the third column. FORM A sample) 7 1) 21 2) 68 3) 12 4) 159 38) 2+4=6 39) 8-3=5 40) 9+0=9 FORM B sample) 9 1) 21 2) 76 3) 12 4) 138 38) 3+2=5 39) 8-2=6 2 40) 7+0=7 Most children are readily satisfied that i t i s the examiner's turn to make a pattern, and are happy to see a l l the ticks appear. The examiner's remarks are of course adjusted to suit the child. More able students have at times had to be reassured that there was *no trick, just some easy numbers to read aloud.* 1 Need not precede pencil and paper testing. Equations of form 8=3+5 were dropped as many children read from right to left. (Mathematically correct.) Others need encouragement. On the other hand, a hesitant child can be moved past the large number, for example, i f he really doesn't want to try i t , by saying 'All right, let's try this equation!'(number sentence). Responses of 'thank you'.'that's fine', etc. can keep the test moving whatever the child answers. Scoring When scoring, an item i s considered correct(one mark) i f a child has corrected himself spontaneously. 'Is* may be substituted for 'equals' i f that is the familiar form. Item kt Reading of digits only i s not acceptable. Item 38» 'Plus', 'add', or 'and* are acceptable. Item 39K yfO* may be read as 'nothing* or 'zero* but 'oh' is not acceptable. Item kOr 'Take away' or 'minus' are acceptable, PENCIL AND PAPER SESSIONS FORM A The area(s) being tested by each group of items i s indicated within square brackets before the oral instructions are given. PART I NUMERATION N. B. REPEAT ALL ESSENTIAL DIRECTIONSt THIS IS NOT A TEST OF LISTENING SKILLS! Page 2 Items 5-6 ^Recognizing printed numerals (aural to reading) with understanding.^] 5. See the baseball bat in the first box and the row of numerals that come after i t . Put your finger on the baseball bat. Put a big X on the greatest (highest) number in that box. 6 . See Snoopy*s house and the row of numerals that come after i t . Put 3 your finger on Snoopy's house . Now put a big X on the numeral one hundred thirty-six. Use the technique 'put your finger on ' whenever advisable to ascertain that the pupils are working in the correct place. Items 7-11 [Aural to printing of numerals with understanding of place value 7. See the box with the l i t t l e Christmas tree i n the corner. Print the numeral 28 i n the box with the Christmas tree. 8. See the box with the l i t t l e stickman i n the corner. Print the numeral 17 i n the box with the stickman. 9. See the box with the l i t t l e f i s h i n the corner. Print the numeral that i s one less than 30 (that comes before 30) i n the box with the l i t t l e f i s h . 10. See the box with the l i t t l e button i n the corner. Print the numeral that comes between 33 and 35 i n the box with the l i t t l e button. 11. See the box with the l i t t l e apple i n the corner. Print the numeral that means four tens and six ones. Print the numeral that means 4 tens and 6 ones i n the box with the apple. Items 12-15 [Matching numeral to picture, involving counting of pictured objects 13. Put your finger on the box with the leaves i n i t . How many leaves are there? Yes, that's right. Now put the point of your pencil inside the box with the leaves. Do you see this long column of numerals? (Demonstrate). Draw a line from the box of leaves to the numeral that t e l l s exactly how many leaves there are. (Check that this item i s corredtly done.) 12-15. Now there are some more boxes, of objects (pictures) for you to match with the correct numeral, BUT BEWARE! SOMEONE HAS PUT IN TOO MANY NUMERALS! Here i s a l i t t l e hint: I f there are lots of things for you to count, you w i l l find that they are i n groups of fives or tens, to help you. (Do not labour this point.) and recognition (Use item 13 as a 'sample') . See note, page 14 rex revision of Form A. 12. See a l l the seagulls f l y i n g . Can you find out how many there are altogether? Then draw a line to the right numeral. BE SURE YOUR LINE STARTS INSIDE THE BOX. 14. Now you can match the cherries to their numeral. Be sure your pencil starts INSIDE the box. 15. Now you can match the stars with the right numeral. (If a child claims he cannot find the correct numeral i t can be suggested that he print the numeral he i s looking for beside the picture). Item 16 1 Reading numeral and printing i t s name.! sample; (Put the numeral 6 on the chalkboard and print the word 'six'. Have the children do the same i n the space provided). 16. Now do you see this big humeral at the bottom of the page? I f you know what i t i s called don't say a word! Just print i t s name the way i t sounds to you. (Spelling w i l l not be marked but the a b i l i t y to put down sound symbols which indicate that the child knows the correct name. Scoring hint given later. See page 24 ). Items 17-18 \ Choosing correct picture for printed numeral. Multiple choice. 17. See the ti n y baseball i n the margin. Put your finger on the baseball. Now look at the numeral next to the baseball. After i t there i s a row of boxes with baseballs i n them. One box has just as many baseballs as the numeral says. Put a big X on the box that has as many baseballs as.the numeral says. Don't forget to look and see i f there i s a quick way of counting by groups. 18. See the tiny seed i n the margin. Put your finger on the seed. Now look at the numeral next to the seed. After i t there i s a row of boxes with seeds i n them. One box had so many seeds that they had to be put into envelopes. Page 3 There are ten seeds i n each envelope. Put a big X on the box that has as many seeds as the numeral says. Don't forget to look and see i f there i s a quick way of counting by groups. Items 19-22 [Number sequences. Counting without picturesJ sample. Now i t i s your turn to count without any pictures and find out what numerals are missing. BEWARE! EACH ROW IS DIFFERENT and some rows are quite tricky! I think everyone can do the row that comes after the tea cup. (sample.) Put your finger on the l i t t l e tea cup. What kind of counting i s i t ? That's right, counting by ones. Let's read togetherj 3# 4, something, 6, 7, some-thing, something, 10, period. Now, please print the correct numerals on the lines where we read 'something'. (Demonstrate on the chalkboard i f necessary and make sure the children complete this linecorrectly). 19. Now put your finger on the l i t t l e feather. Very quietly, to yourself, read the numbers that come after i t . Figure out what kind of counting i t i s . Then f i l l i n the missing numbers l i k e we did before. 20. Now look at the row with the star. It might be a different kind of counting. See i f you can figure i t out and put i n the missing numbers. 21. The row with the balloon i s started but not finished. Please f i n i s h i t carefully. 22. The row with the wizard's hat needs to be finished too. Please f i n i s h i t carefully. Items 23-24 [Making pictures to match the numeral , ]| Now i t i s your turn to make pictures! 23• See the chair with the numeral beside i t , i n a l i t t l e box. Shh, don't t e l l what i t says! In the long empty box beside the chair I want you to make just as many chairs as the numeral says. Make the chairs i n the long empty box. 5 8 C 24. See the l i t t l e star (baseball) with the numeral beside i t . Please draw as many stars (baseballs) as the numeral says. Try to f i t your stars into the long box. (Overflow may of course be placed below). (A few children can pr o f i t from a suggestion of grouping i n fives or tens as they draw). TIME FOR A BREAK Page 4 Items 25-28 {^Concepts of greater than, less than, equals, one greater than, one less than ,J (Three versions are providedj 1) reading the words, as i n Seeing Through 6 Arithmetic, 2) reading the words i n i.t.a.printing, 3) symbols. (Page 4*). (Samples are provided to be done on the chalkboard by the examiner, and by the children i n their booklets) . 25-26 p r i n t i n g correct phrase or symbol sample 1. Let us read what i t says i n the boxes (what the symbols say) at the top of t h i s page. After the moon i t says 8, and then there i s a long line (dotted c i r c l e ) and then 6. Now what can we print on the line (in c i r c l e ) to make a sentence that t e l l s the truth? That's right! Eight i s greater than six. Please print 'is greater than' on the li n e (in the c i r c l e ) . sample 2. Now look at the sun. Which box (symbol) are we going to use to make this a true iehtence? That's right! 48 i s less than 84. 25 and 26. Now there are two sentences for you to do, as quietly as you can. There i s the mushroom sentence and the sentence that starts with a l i t t l e flower. You w i l l have to use one of the boxes (symbols) again. One of the boxes (symbols) has to be used twice. Now see i f you can make both rows t e l l the truth. 5 See note, page \k rei revision of Form A. 6 Maurice L.Hartung,et a l . , Seeing Through Arithmetic, Book I, Teacher's Edition (Toronto: W.J.Gage Ltd., 1965) DON'T FORGET TO REPEAT IMPORTANT INSTRUCTIONS! 27-28. [Choosing correct phra se or symbol. Multiple choice. sample 3« See the l i t t l e pear? Beside i t we can read 21+1. How much does that make? A l l right, remember i n your head how much 21+1 i s . Now we have to find out which phrase (symbol) i n the middle f i t s with 21+1 and with the numeral at the end of the row to make a true sentence. Let's try '21+1 i s greater than 22'. No? (Continue i n this vein u n t i l pupils are satisfied with 'is equal to'). Good, now put a ring around 'is equal to'and read the sentence to yourself to make sure i t i s true. sample 4. Let's try the row that starts with a lemon and do i t just the same way. ( E l i c i t as before u n t i l item i s completed). 27 and 28. Now here are two more jobs for you to do just the same way, a l l by yourself. Look at the row that starts with the racing car. Read quietly and carefully and think what numeral the f i r s t expression stands for. Now read each phrase (symbol) i n the middle and the numeral at the end. Put a ring around the phrase (symbol) i n the middle which makes the line t e l l the truth. Now t r y the row that starts with the s a i l boat. Read quietly and carefully. Think what numeral the f i r s t expression stands for. Now read each phrase (symbol) i n the middle and the numeral at the end. Put a ring around the phrase (symbol) i n the middle which makes the line t e l l the truth. 29« Here are some pens, tied up i n bundles. There are ten pens i n each bundle. Some pens were l e f t over. In the box at the end of the row print the numeral that t e l l s how many pens there are altogether. Pictures, hundreds, tens and ones. value . ] ( i . t . a . 5 has 'wuns' for 'ones'). Numerals to print. 30, 31) 32. Hare are some logs. They were too heavy to put into bundles so someone piled them up. There are ten logs i n each p i l e . Some logs were l e f t over. Please print how many piles of ten there are and how many ones l e f t over. Then printhow many logs there are altogether. 33. HERE IS A REALLY TRICKY ONE! (Very few f i r s t year children are ready for this one). I'D LIKE YOU TO TRY IT. There are l o t s and lots of boxes of erasers. Every box has ten erasers i n i t . There are so many boxes that they had to be piled up. DO YOU SEE THE LITTLE ERASER THAT WAS LEFT OVER? Now see i f you can figure out how many erasers there are altogether. I f you can, don't say a word! Just print the right numeral i n the box at the end of the row. Items 34-37. _tens ones = • Completion, from words to numeral and vice versa . ]| sample. See the box with the l i t t l e b e l l i n the corner. Beside the b e l l i t says, (pause), Three tens and nine ones equals ?' Yes, that's right, t h i r t y -nine. Print 39 on the l i n e . 34-35. Now see the l i t t l e pussy cat. He has two more jobs for you to do a l l by yourself. Read them carefully and print the numeral that the tens and ones stand for, each time. sample. See the box with the butterfly i n the corner. This time there i s a numeral, and you are to print how many tens and how many ones that numeral stands for. Seventy-three i s ? Yes, that's right, seven tens and ? yes, three ones. 36-37. Now the butterfly has two more jobs l i k e that for you to do a l l by yourself. Items 38-39. [understanding of place value, including hundreds, with numbers <10,000 . ] 38. Put your finger on Charlie Brown's baseball cap. Now look at the row of numerals that comes after i t . One of those numerals has a four i n the hundreds place and a four i n the ones place. Put a big X on the numeral that has a four i n the hundreds place and a four i n the ones place. Be sure that you make 39* Put your finger on the baseball. See the big numeral i n the box beside i t . Now look carefully at the row of numerals that come after the box. One of the numerals i n the row t e l l s how many ones the numeral (digit) eight i n the box stands for. Put a big X on the numeral i n the row which t e l l s how many ones the numeral (digit) 8 i n the box stands for. Just make one X please. Items 40-42. [Equations requiring understanding of base ten numeration . 3 40-42. Put your finger on the f l a g . Between the flag and the stop sign are three equations. The numbers look pretty big but i f you READ them carefully, one at a time, you should be able to figure out the right numeral to put i n the box, to make each equation t e l l the truth. You shouldn't have to do any counting. Remember to do some thinking before you f i l l i n a box. I f you are not quite sure, t r y again. (Do not allow excessive time for the children who make rows of dots or otherwise t r y to count). only one X please. - BREAK -- END OF SECOND SITTING -PART II ADDITION AND SUBTRACTION Page 6. Items 1-4. j^Action pictures. Corresponding equations to be completed , J 1. See the box with the l i t t l e chickens i n i t . Something i s happening in the picture. Underneath i s a number story that t e l l s what i s happening. But i t isn't finished. Look carefully at the picture and then print a numeral in the box, to f i n i s h the equation so that i t t e l l s what happens i n the picture. 2. Now look at the bunny rabbits. Can you see what i s happening?. Shh! Don't t e l l ! Just read the equation to yourself and then f i n i s h i t to match the picture. 3. See the ladies with the feathers i n their hats? Those extra lines mean that some of them are walking away. There are two numerals missing i n the equation for this picture. Please print the right numerals i n both boxes so that your equation t e l l s what happens. 4. There are some balls s i t t i n g on this table. There is a ledge around i t so that they won't f a l l off. Some more balls are f a l l i n g down and going to land on the table. Please print a numeral i n the box and a numeral i n the triangle so that the equation t e l l s what w i l l happen. Items 5-9» [choice of + or - sign i n simple equations . 3 sample. Put your finger on the floppy hat i n the next box. This sign says ? (Print + on board) Yes, plus, or add. This sign says ? (Print - on board) Yes, minus or take away. Now here i s an equation with a hole i n the middle. (Put sample on board). The dotty ring means that a sign i s missing. 13 c Maybe i t i s a plus sign; maybe i t i s a minus sign. Let's try reading the equation with a plus sign and see i f i t works. No? Well, how about five minus four equals one? Good! Now you f i n i s h the equation right under the hat on your paper. 5-9. Now there are five more equations to f i n i s h just the same way. Read each one carefully and see i f you need to put a plus or a minus sign i n the hole to make the equation t e l l the truth. Please stop when you come to the double l i n e . Items 10-21. [Addition and subtraction facts ^ 1 0 , 3 The rest of this page i s work you know pretty well. See how many of these equations you can f i n i s h correctly. Some are addition and some are subtraction (take away) so BE CAREFUL! Please don't use your counters (blocks, rods,etc.) just for today. I want to see how many of these equations you are ready to do i n your head. There are three columns of equations to do. Don't stop u n t i l you have finished the whole page. Page_7 Items 22-25. ^Equations with three addends. Items 24 and 25 have = i n beginning position: 9=5+A + A •} See the box with the tree i n the corner. It has some equations that are a l i t t l e longer. There i s one with two triangles to be f i l l e d i n . The triangles are the same shape so they want the same numeral i n them. (If this i s t o t a l l y unfamiliar, demonstrate with a different example, e.g. A+1+A=7)» There are four equations i n this box. Please see i f you can make each one of them t e l l the truth. Items 26-37. [second and third year work . ~Jj (Children should be encouraged to do what they know how to, but counting or the c drawing of pictures for more than one or two examples should be discouraged. Alternatively, picture making can be used to interpret a child's understanding of concepts and operations but i n that case he should not be credited with mastery.) Put your finger on the giant slide. It marks the start of the kind of work that you might be going to do next year. Maybe you already know how to do some of them. If you do, please do them quietly. Do as many as you can. But don't worry, you w i l l be learning how to do a l l of them some day soon. (Papers may be collected as children reach their l i m i t , or they may colour the cover while their classmates work a l i t t l e longer). Items 38-40. Reading equations orally from flash cards. See section on Individual Oral Testing, page 3 . THAT'S ALL! THANK YOU VERY MUCH! AMENDMENTS TO FORM A In the l i g h t of experience during administration and item analysis of Form A,the following amendments were subsequently made to Part I. On page two, item thirteen, which was actually used as a sample during testing, was placed i n the position of item twelve and re-labeled 'sample'. This necessitated re-numbering a l l subsequent items of Part I, there being now only forty-one items i n a l l . Former item twenty-four was also changed to read 34 0 ' s (baseballs). PENCIL AND PAPER SESSIONS, FORM B The area(s) being tested by each group of items i s indicated, within square brackets, before the oral instructions are given. PART I NUMERATION N. B. REPEAT ALL ESSENTIAL DIRECTIONS! THIS IS NOT A TEST OF LISTENING SKILLS! Page 2. 15 Items 5-9» ^>ural to printing of numerals with understanding of place value » 5. See the box with the pussy cat i n the corner. Put your finger on the pussy cat. Print the numeral 27 i n the box with the pussy cat. 6. See the box with the ice-cream cone i n the corner. Put your finger 7 on the ice-cream cone . Print the numeral 16 i n the box with the ice-cream cone. 7. See the box with the l i t t l e flower i n the corner. Print the numeral that i s one less than 40 (comes before 40) i n the box with the l i t t l e flower. 8. See the box with the tennis b a l l i n the corner. Print the numeral that comes between 63 and 65 i n that box. 9. See the box with the party hat i n the corner. Print the numeral that means 3 tens and 5 ones i n that box. Items 10-11. Recognizing printed numerals (aural to reading) with understanding . 10. Put your finger on Charlie Brown's baseball cap. See the row of raomerals that come after i t i n the same box. Put a big X on the greatest(highest) numeral i n that box. 11. Put your finger:on the football. See the row of numerals that come after the football. Put a big X on the numeral one hundred twenty-seven. Items 12-14. {^Matching numeral to picture, involving counting of pictured objects and recognition of numerals ^100 . J sample. Put your finger on the box with the tea cups i n i t . How many tea cups are there? Yes, that's right. Now put the point of your pencil inside the tea cup box. Do you see this long column of numerals? (Demonstrate). Now draw a line from the tea cup box to the numeral that t e l l s exactly how many tea cups there are, (Check that this item i s correctly completed). 7 Use the technique 'Put your finger on ' whenever advisable, to ascertain that the pupils are working i n the correct place. 12-14. Now there are some more boxes of objects (pictures) for you to ^ match with the correct numeral, BUT BEWARES SOMEONE HAS PUT IN TOO MANY NUMERALS! Here i s a l i t t l e hint» If there are lots of things for you to count you will find that they are in groups of fives or tens, to help you. (Do not labour this point). 12. See a l l the l i t t l e stars. Can you find out how many stars there are altogether? Fine, now draw a line to the right numeral. BE SURE YOUR LINE STARTS INSIDE THE BOX. 13. Now you can match the l i t t l e seeds to their numeral. Be sure your pencil starts INSIDE the box. 14. Now you can match the marbles with the right numeral. (If a child claims he cannot find the correct numeral i t can be suggested that he print the numeral he i s looking for beside the picture). Item 15» pleading a numeral and printing i t s name » sample. Put the numeral 4 on the chalkboard and print the word 'four'. Have the children do the same in the space provided. 15» Now do you see this big numeral at the bottom of the page? If you know what i t i s called, don't say a word! Just print i t s name, the way i t sounds to you. (Spelling will not be marked but the ability to put down sound symbols which indicate that the child knows the correct name. Scoring hint given later. See page 24 ), Items 16-17. |_Choosing correct pictures for printed numeral. Multiple choice . 16. See the seagull flying in the margin. Put your finger on the seagull. Now look at the numeral in the box beside the seagull. After i t there i s a row of boxes with seagulls in them. One box has just as many seagulls as the numeral says. Put a big X on the box that has as many seagulls as the numeral says. Don't forget to look and see i f thereiS:a quick way of counting by groups. 17. See the l i t t l e cherry i n the margin. Put your finger on the cherry. Now look at the numeral next to the cherry. After i t there i s a row of boxes with cherries i n them. There were so many cherries i n one box that they had to be put into baskets. There are ten cherries i n each l i t t l e basket. Put a big X on the box that has as many cherries as the numeral says. Don't forget to look and see i f there i s a quick way of counting by groups. Items 18-21. [^ Number sequences. Counting without pictures. Completion . "} sample. Now i t i s your turn to count without any pictures and find out what numerals are missing. BEWARE} EACH ROW IS DIFFERENT! and some rows are quite tricky! I think everyone can do the row that comes after the balloon, (sample) Put your finger on the balloon. What kind of counting i s i t ? That's right! Counting by ones. Let's read together: 3» 4, something, 6, 7, something, 10, period. Now please print the correct numerals on the lines where we read 'something'. (Demonstrate on the chalkboard i f necessary, and make sure the children complete this l i n e correctly). 18. Now put your finger on the l i t t l e Christmas tree. Very quietly, to yourself, read the numbers that come after i t . Figure out what kind of counting i t i s . Then f i l l i n the missing numbers l i k e we did before. 19. Now look at the row with the button. It might be a different kind of counting. See i f you can figure i t out and; put i n the missing numbers. 20. The apple row i s started but not finished. Please f i n i s h i t carefully, 21. The row with the l i t t l e stick man needs to be finished too. Please f i n i s h i t carefully. Items 22-23. match the numeral , 22. Now i t i s your turn to make pictures! See the baseball bat with the numeral beside i t . Shhf Don't t e l l what i t says! In the long empty box C baside the bat I want you to make just as many bats as the numeral says. Make the bats i n the long empty box. Please don't put any tape or other trimming on the handles, u n t i l you have made a l l the bats. 23. See the l i t t l e X with a numeral beside i t ? Please draw as many X's as the numeral says. Try to get your X's to f i t into the long box. (Overflow may of course be placed below). (A few children can profit from a suggestion of grouping i n fives and/or tens as they draw). - TIME FOR A BREAK Page 4. Items 24-27. ^Concepts of greater than, less than, equals, one greater than, one less than . "J (Three versions are provided: 1) reading the words, as i n Seeing Through Arithmetic, 2) reading the words i n i.t.a.printing, 3) symbols. (page 4 *) Samples are provided to be done on the chalkboard by the examiner, and by the children i n their booklets). 24-25. £Printing correct phrase or symbol , ^ sample 1. Let us read what i t says i n the boxes (what the symbols say.) at the top of this page. After the feather i t says 9, and then there i s a long line (dotted c i r c l e ) and then 7. Now what can we print on the li n e (in the c i r c l e ) to make a sentence that t e l l s the truth? That's right! Nine i s greater than seven. Please print 'is greater than' on the line (in the c i r c l e ) . sample 2. Now look at the tea cup. Which box (symbol) are we going to use to make this a true sentence? That's right! 57 i s less than 75• 25 and 26, Now there are two sentences for you to do, as quietly as you can. There i s the star sentence, and the sentence that starts with the wizard's Maurice L.Hartung, et a l . , Seeing Through Arithmetic, Book I, Teacher's Edition, (Toronto: W.J.Gage Ltd., I965) 19 c hat. You will have to use one of the boxes (symbols) again. One of the boxes (symbols) had to be used twice. Now see i f you can make both rows t e l l the truth. DON'T FORGET TO REPEAT IMPORTANT INSTRUCTIONS! 26-27. [choosing correct phrase or symbol. Multiple choice . } sample 3» See the l i t t l e moon. Beside i t we can read 31+1* How much does that make? A l l right, remember in your head how much 31+1 i s . Now we have to find out which phrase (symbol) in the middle f i t s with 31+1 and with the numeral at the end of the row to make a true sentence. Let's try '31+1 i s greater than 32'. No? (Continue in this vein until pupils are satisfied with 'is equal to'). Good, now put a ring around 'is equal to' and read the sentence to yourself to make sure i t i s true. sample 4. Let's try the row that starts with a sun and do i t just the same way. (Elicit as before until item i s completed). 26 and 27. Now here are two more jobs for you to do just the same way, a l l by yourself. Look at the row that starts with a sailboat. Read quietly and carefully, and think what numeral the fi r s t expression stands for. Now read each phrase (symbol) in the middle and the numeral at the end. Put a ring around the phrase (symbol) in the middle which makes the line t e l l the truth. Now try the row that starts with the mushroom. Read quietly and carefully. Think what numeral the f i r s t expression stands for. Now read eech phrase(symbol) in the middle and the numeral at the end. Put a ring around the phrase (symbol) in the middle which makes the line t e l l the truth. Page 5. [place value • (i.t.a .5 has 'wuns' for 'ones'). Items 28-32. ^Pictures! hundreds, tens and ones. Numerals to print » J 28. Here are some pencils tied up in bundles. There are ten pencils in each bundle. Some pencils were left over. In the box at the end of the row, print the numeral that t e l l s how many pencils there are altogether. 29, 30, 31•• Here are some logs. They were too heavy to put into bundles so someone piled them up. There are ten logs i n each p i l e . Some logs were l e f t over. Please print how many pi l e s of ten there are, and how many ones l e f t over. Then print how many logs there are altogether. 32. HERE IS A REALLY TRICKY ONE! (Very few f i r s t year children are ready for this one). I'D LIKE YOU TO TRY IT. There are lots and lots of chalkboard brushes. Every box has ten brushes i n i t . There are so many boxes that they had to be piled up. DO YOU SEE THE BRUSHES THAT WERE LEFT OVER? Now see i f you can figure out how many chalkboard brushes there are altogether. I f you can, don't say a word! Just print the correct numeral i n the box at the end of the row. and vice versa . j sample. See the box with the pear i n the corner. Beside the pear i t says (pause) 'Three tens and nine ones equals ?* Yes, that's right, thirty-nine. Print 39 on the l i n e . 33-34. Now underneath are two more jobs for you to do a l l by yourself. Read them carefully and print the numeral that the tens and ones stand for, each time. sample. See the box with the grapefruit i n i t . This time there i s a numeral and you are to print how many tens and how many ones that numeral stands for. Thirty-five i s ? Yes, that's right, three tens and five ones. 35-36. Now underneath the grapefruit are two more jobs for you to do, just l i k e that one, a l l by yourself. Items 38-39* ["understanding of place value, including hundreds, with numbers 37. Put your finger on Snoopy's night cap. Now look at the row of numerals that comes after i t . One of those numerals has a three i n the hundreds place and a Completion, from words to numeral < 10,000 . 1 three i n the ones place. Put a big X on the numeral that has a three i n the C hundreds place and a three i n the ones place. Be sure that you make ONLY ONE X please. 38. Put your finger on the jet plane. See the big numeral i n the box beside i t . Now look carefully at the row of numerals that comes after the box. One of the numerals i n that row t e l l s how many ones the numeral (digit) seven i n the box stands for. Put a big X on the numeral i n the row whih t e l l s how many ones the numeral (digit) seven i n the box stands for. JUST MAKE ONE X please. Items 39-41. ^Equations requiring understanding of base ten numeration . ^ 39-41. Put your finger on Snoopy's dog house. Between Snoopy's dog house and the stop sign are three equations. The numbers look pretty big but i f you READ them carefully, one at a time, you should be able to figure out the right numeral to put i n the box to make each equation t e l l the truth. You shouldn't have to "do any counting. Remember to do some thinking before you f i l l i n a box. I f you are not quite,sure, t r y again. (Do nbt allow excessive time for the children who make rows of dots or otherwise try to count). - BREAK -- END OF SECOND SITTING -PART II ADDITION AM) SUBTRACTION Page 6. Items 1-4. [Action pictures. Corresponding equations to be completed . ~\ 1. See the box with the bunny rabbits i n i t . Something i s happening i n the picture. Underneath i s a number story that t e l l s what i s happening. But i t isn't finished. Look carefully at the picture and then print a numeral i n the box to f i n i s h the equation so that i t t e l l s what happens i n the picture. c 2. Now look at the picture of the birds. Can you see what i s happening? w Shh! Don't t e l l ! Just read the equation to yourself. Then f i n i s h i t to match the picture. 3. Now look at the funny l i t t l e boys. Can you see which way some of them a re running? Good! Now be careful! There are two numerals missing i n the equation for this picture. Please print the right numerals i n both boxes so that your equation t e l l s what happens. 4. There are some balls s i t t i n g on this table. But some of them have fa l l e n off. That i s what those lines behind them mean. Please print a numeral i n the box and a numeral i n the triangle so that the equation t e l l s what has happened. Items 5-9. [choice of + or - sign i n simple equation's . J sample. Put your finger on the spinning top i n the next box. This sign says ? (Print + on board). Yes, plus, or add. This sign says ? (Print -on board). Yes, minus, or take away. Now here i s an equation with a hole i n the middle. (Put sample on board). The dotty ring means that a sign i s missing. Maybe i t i s a plus sign: maybe i t i s a minus sign. Let's t r y reading the equation with a plus and see i f i t works. No? Well, how about five minus three equals two? Good! Now you f i n i s h the equation right under the top on your paper. 5-9* Now there are five more equations to f i n i s h just the same way. Read each one carefully and see i f you need to put a plus or a minus sign i n the hole to make the equation t e l l the truth. Please stop when you come to the double l i n e . Items 10-21. [Addition and subtraction facts ^ 10 . J The rest of this page i s work you know pretty well. See how many of these equations you can f i n i s h correctly. Some are addition and some are subtraction (take away), so BE CAREFUL! Please don't use your counters (blocks, rods, etc.) just for today, I want to see how many of these equations you are ready to do i n your head. There are three columns of equations to do. Don't stop u n t i l you have finished the whole page. Page 7. Items 22-25, ^Equations with three addends. Items 24 and 25 have = i n beginning position* 7=3+A+ A •] • See the box with the butterfly i n the corner. It has some equations that are a l i t t l e longer. There i s one with two triangles to be f i l l e d i n . The triangles are the same shape so they want the same numeral i n them. (If this i s t o t a l l y four equations i n this box. Please see i f you can make each one of them t e l l the truth. (Children should be encouraged to do what they know how to, but counting, or drawing of pictures for more than one or two examples should be discouraged. Alternatively, picture making can be used to interpret a child's understanding of concepts and operations, but i n that case, he should not be credited with mastery). Put your fingercon the beach b a l l . I t marks the start of some kind of work that you might be going to do next year. Maybe you already know how to do some of them. I f you do, please do them quietly. Do as many as you can, but don't worry, you w i l l be learning how to do a l l of them some day soon. (Papers may be collected as children reach their l i m i t , or they may colour the cover while their classmates work a l i t t l e longer). Items 38-^0. [^Reading equations or a l l y from flash cards . [J (See section on Individual Oral Testing, pages 3-4). There are third year work . THAT'S ALL! THANK YOU VERY MUCH! SCORING, PENCIL AND PAPER SESSIONS General Remarks 24 c One point i s given for each item correct throughout the test. No part marks are awarded. I f a d i g i t i s reversed, e.g. VG' for '2' that i s not considered an error. However, i f two d i g i t s are i n reverse order, e.g. '21' for '12', that i s incorrect. An utterly ambiguous answer should be marked wrong. (It i s a disservice to a child to give him credit for such a response). I f more than one response i s marked, i n the multiple choice items, that i s incorrect, unless an obvious attempt has been made to eliminate a l l but one. Specific Items, Part I Printing names of numbers, item 15 or 16 Correct spelling i s not necessary but the child must be able to print sound symbols which indicate he knows the correct number name and can commit i t to paper i n word form. Example3i Acceptable: hundrtanin, hunderdsivn, wun hudrdsen. ^ Not Acceptablei one oh nin, wun hunderdsevnty. Number sequences, items 19-22 or 18-21 Reversal of a single d i g i t i s acceptable, but two d i g i t s i n reverse order i s an error. Complete row must be correct. Making pictures to match the numeral, items 23-24 or 22-23 The child's intent i s important. The boxes are provided as a convenient guide not a restriction, and no penalty should be attached to overflow. One quick way of marking the large number of stars (balls or X's) i s to run a marking pencil underor around five at a time, u n t i l the answer i s obviously correct or not. Greater than, etc. page 4, items 25-26 or 24-25 Intent i s what should be marked, not correct spelling. Place value Corresponding item numbers and responses are as follows! Form A Form B item 30 2 tens item 31 5 ones item 32 25 logs item 29 2 tens item 30 4 ones item 31 24 logs SUGGESTED AMENDMENTS TO FORM B In the l i g h t of experience during administration of Form B, the following amendments to Part I are considered advisable, to get the children off to a good start. On page two, place items ten and eleven f i r s t , re-numbering them as five and six. Items five to nine would then follow and be re-numbered as items seven to eleven. TEST CONSTRUCTION The P r e l i m i n a r y Form o f the t e s t , c o n t a i n i n g 75 i t e m s i n P a r t I (numerat ion s k i l l s ) and 56 i t e m s i n P a r t I I (Computation S k i l l s ) was a d m i n i s t e r e d t o f o u r c l a s s e s c o n t a i n i n g c h i l d r e n w i t h a wide range o f a b i l i t y , achievement and s o c i o -economic background from two s c h o o l d i s t r i c t s i n the Lower Ma in land o f B r i t i s h Co lumbia . P a r t I ( P r . ) was completed by 86 c h i l d r e n . P a r t I I ( P r . ) w a s completed by 113 c h i l d r e n . Time t a b l e problems prevented the c o m p l e t i o n o f P a r t I by a l l o f the c h i l d r e n i n one of t h e s c h o o l s . Computer a s s i s t e d i t e m a n a l y s i s was c a r r i e d out on the r e s u l t s o f t h i s t e s t i n g on the IBM 360, Model 67, a t t h e U n i v e r s i t y o f B r i t i s h Columbia (Program T I A ) . Two p a r a l l e l forms , Form A and Form B, were developed w i t h the a i d o f t h i s a n a l y s i s . Form B f o l l o w s v e r y c l o s e l y a c t u a l i t e m s used i n the P r e l i m i n a r y Form. Form A which c o n t a i n s many a l t e r n a t e ( p a r a l l e l ) i t e m s was c r o s s - v a l i d a t e d i n another s c h o o l d i s t r i c t i n the Lower M a i n l a n d on two c l a s s e s i n d i f f e r e n t s c h o o l s , c o n t a i n i n g c h i l d r e n w i t h a wide range o f a b i l i t y , achievement and soc io-economic background . Te s t 1(A) was completed by 47 c h i l d r e n . T e s t 11(A) was completed by 49r ; chi ldren. Computer a s s i s t e d i t e m a n a l y s i s was c a r r i e d out as b e f o r e . R e l i a b i l i t y Computer a s s i s t e d i t e m a n a l y s i s gave t h e f o l l o w i n g KR-20 v a l u e s f o r the P r e l i m i n a r y Formi T e s t I ( P r . ) » KR-20=0.9530 T e s t I I ( P r . ) s KR-20±0.9390 Computer assisted item analysis of Form A results gave the following KR-20 values: Test 1(A) : KR-20=0.9435 Test 11(A): KR-20=0.9291 The :;slight reduction i n KR-20 values was to be expected since the tests had been considerably shortened i n order to obtain a reasonably practical length for f i r s t year students. Content V a l i d i t y During test construction the following texts and teacher's guides were 1, 2 consulted: Seeing Through Arithmetic , Elementary School Mathematics , Number 3 5 Patterns , a-iid the Greater Cleveland Mathematics Program . The arithmetic section 5 of the Metropolitan Achievement Tests, Primary I Battery suggested a format pleasing to young children. A few items similar to those i n t h i s test were included i n Part I, thus: two multiple choice items on recognition of printed numerals, and knowledge of "greatest" number (aural to printed) and four items involving writing of numerals from oral instructions. The f i n a l forms also contain two items on place value, similar to items i n Ashlock's Test of Understanding 6 of Selected Properties of a Number. System: Primary Form which was not available i Maurice L.Hartung,et al.,Seeing Through Arithmetic,Book I,Teacher's Edition, (Toronto: W.J.Gage Ltd., 1965). 2 Robert E.Eicholz,et al.,Elementary School Mathematics,Second Edition, Teacher's Edition,Books I and II, (Don Mills,Ontario :Addison-Wesley(Canada) Ltd., 1969). 3 Roberta Chivers,et al.,Number Patterns,Books I and II,Teacher's Edition, (Toronto: Holt Rinehart & Winston, 1966). 4 Educational Research Council<of Greater Cleveland, Greater Cleveland Mathematics Program,Teacher's Guide for F i r s t Grade (Chicago:Science Research Associates,Inc.,1961,1962). 5 Metropolitan Achievement Tests.Primary I Battery,Form B.(New York: Harcourt,Brace and World,Inc.,1959). Robert B.Ashlock and Ronald C.Welch,"A Test of Understandings of Selected Properties of a Number System: Primary Form", Bulletin of the School of Education. Indiana University,Vol.42,No.2,MarcFI^651 at the time of constructinn of the Preliminary Form. Teachers were very helpful during the t r i a l s i n pointing out items which might be ambiguous to the children. They commented favourably on the comprehensiveness of the Preliminary Form but agreed with the administrator that i t was too long to be practical. Concurrent V a l i d i t y Concurrent v a l i d i t y of the Preliminary Form was established by comparison of results of one class with those provided by an able and experienced teacher. Her class l i s t showed the children grouped and approximately ranked according to her testing and observation throughout the year. Correlation was very high between the two sets of results, the only discrepancies being i n the 'middle zone'} those were the children who vary the most from day to day and cannot be too firmly assessed at this age. (See Scatter Diagram, Figure I) PART I - NUMERATION Test Scor X X X *x X X* X X X * *x y -*—' X <s> X X r» • ' 8 T . Q V — X PART II - COMPUTATION Test Score 5b 30 Tfccich 20 to D tr X V X \y X X x •fe—— 1 v XX 2) X JU :— r XX * j — X* 1 • . 27 - S > Assessment/ rx. ^< >- > £ V) > ^ Assess^ ^ .5.51 | _S v- vvitn Sironqers. Figure I Concurrent Validity - Preliminary Form ® Students with personality problems inhibiting rapport Concurrent v a l i d i t y of Form A was affirmed by very high correlation with the year-end results already established i n both schools where corss-validation was carried outt "To a ch i l d " said one teacher. Concurrent v a l i d i t y was similarly established for Form B with a class of 34 children i n another school d i s t r i c t . Summary of Areas Tested Revised Forms A and B PART It NUMERATION Items 1-4. Oral reading of printed numerals. Item 5. Recognition of printed numerals (aural to reading). Item 6 . As item 5 but includes concept of greatest number. Items 7-11. Aural to printing of numerals. Item 9, 'one less than'. Item 10. 'between'. Item 11. Place value, 'tens' and 'ones'. Items 12-14. Matching numeral to pictures, involving counting of pictured object and recognition of numerals 100. Item 15 . Reading a numeral between 100 and 110 and printing i t s name. Items 16-17. Choosing correct pictures for printed numeral, multiple choice. A b i l i t y to count by fives and tens can be used. Items 18-21. Number sequences. Counting without pictures, completion. Items 22-23. Making pictures to match the numeral. Items 24-27. Concepts of greater than, less than, equals, one greater than, one less than, using either the printed words or the symbols. Items 28-32. PicturesJ hundreds, tens^and ones. Numerals to print. Items 33-36. Completion from words to numerals and vice cersa of tens ones= c Items 37-38. Understanding of place value, including hundreds with numbers <^ 10,000. Multiple choice from oral instructions. Items 39-41. Equations requiring understanding of base ten numeration. PART II COMPUTATION Items 1-4. Action pictures. Corresponding equations to be completed. Items 5-9* Choice of + or - sign i n simple equations. Items 10-21. Addition and subtraction facts ^. 10, i n equation form. Items 22-25. Equations with three addends. Items 24 and 25 have = i n beginning position, e.g. 7=3+A+A« Items 26-31. Addition and subtraction facts 11-18, i n equation form. Items 32-35. Extensions of addition and subtraction facts O.0, i n equation form. Items 36-37. Addition and subtraction of two d i g i t numerals i n ver t i c a l computational form, no carrying or borrowing. Items 38-40. Oral reading of complete printed equations. ADDENDA When Form A was administered, the items now numbered 10 and 11 preceded the present items 5 through 9, This gave the children a much better start, psychologically. I t i s recommended that, If these tests are used again, Items 10 and 11 be placed f i r s t , on page 2 of either form, followed by the present Items 5 through 9. C BIBLIOGRAPHY Anastasi, Anne. Psychological Testing. Second and Third Editions (New Yorki MacMillan Co., 1961, 1968). Ashlock, Robert B. "A Test of Understandings of Selected Properties of a and Number System! Primary Form", Bulletin of the School Ronald C.Welch of Education, Indiana University, Vol. 42 No.2., March 1966. Buros, Oscar Krisen, Editorj The Sixth Mental Measurements Yearbook, (Highland Park, New Jersey: 1965). Chlvers,Roberta^et a l , } Number Patterns, Books I and II, Teacher's Edition. (Toronto: Holt Rinehart and Winston.1966). Educational Research Council of Greater Cleveland, Greater Cleveland Mathematics Program, Teacher's Guide for F i r s t Grade, (Chicago:Science Research Associates,Inc.,1961,1962). Eicholz, Robert E. et a l . , Elementary School Mathematics, Second Edition, Teacher's Edition, Books I and II, (Don Mills,Ontario: Addison-Wesley (Canada) Ltd., 1969). Hartung, Maurice L., et a l . , Seeing Through Arithmetic, Book I, Teacher's Editionj(Toronto: W.J.Gage,Ltd., 1965). Magnusson, David } Test Theory, (Reading,Mass.: Addison-Wesley, 1966). Metropolitan Achievement Tests, Primary I Battery, Form B. (New York: Harcourt,Brace and World,Inc., 1959). APPENDIX D I" irst Year - Primary Power Test of Certain Arithmetic Sk i l l s Divis ion H -Forres't Johnson-"or m P a r i 1 D 2. j 10. 13 lo l ( o \5 s a m p l e 12. 13. * ¥ * * * *- * * ^ * * * DO000 00000 0000/1 000 O 0 o o o o o ooo o ooo o o o OO o O O OO o o o 0 0 S a m p l e 4-: 27 127 //3 /3/ IS 81 > H 32. 23 f i Z5-D 3 . ' i 25 V V V V V V V V w y v V V v y v V v V v v - v v V V V V V V y v r \ r / y r r r r V v w v y r r r » \r r r r r r ^ r f r v v r » r v v * ^ v v v v - r - y» y r r v v y- v v v v v v v v v V V V y y V V i ,7* 9 0 0 0 0 • /7 66 a-6 6 %r cr i (fill {[§!!! (SS 1© o ' S a m p l e . 0 2 . 3 5 . 4 9. IS. 20. ^ 3 5 , _ , 3 7 , 3 ? , 4-/. » 2 4 , 2 3 , 2 2 , — , 2 0 , 0 97, q*, qq/_ A 3 0 , 4 - 0 , 5 0 , /7 2Z 23 7/5 2 / f x s S T O P Form Sample I. sampler. f=p $ 7 ^ 3 8 . 25 4-1 7 75 38 14-Sample 3 . J) 31+ I is area ter Hi an \S less than is equal to 32_ s a m p l e 4. i s a r e a an is less than is equal to 28 2U 15-IS g rea te r than is Uss i han i s equal to qual 5 21. iter \\w IS qrea Than is less than is equal to 47 Form & Dlf* > o r < or = /*; } 7 SompU 57/')75 ... ; s a m p l e 3. j) 31+1 S a m p ft / f . S2-/ > > 32 28 or rn B D)itd M-is eekwau S a m p l I. s a m p l z. ^ 3 2 4-1 25*. 7 I S 3 8 sampl 3, j > 3 1 + 1 i s graeter jtian is less jiian i s eekwaul \ u » 3 Z sampl if. is- graeter jhan is less jian is eekwaul tc# 2 3 2 ^ / 5 - I is graeter j)nan is less jhan is eekwaul tc&> \5 Z7. is greet er jtan is \ess jhan i s eekwaul tte> 1+7 Form E> [ F o r tt a : "ones' 'wuns] 05" EE'" 1/ V ens o n e s = lo 5" 32 . n o i to_ J IP I 1Q \ to sample ^ S^ens g ones " 3 3 . 34. | t I Jen 1 ones Z. lens io ones = 35". 3fe. S a m p l e . _ fens 35= ones 83 = _ _ tens — _ ones 41 • - . t ens ones 37. 334 33 343 433> l o 5 7 S 7 70 700 7000 31. * 0 4-1. 4 6 = 4 0 + 32= + 2 STOP 2 0 4 1 0 + 1 0 4 3 = o r m B i . Part E 4 + I =• 5-2-3. s + O L + or - ? d a m pie 5; 3 = 2 S". t, n i • 5 7. 3( ) 4 • 1 5 C) 3 - 2 1. 7C 10. II. a. I f 3 = 6 + 0 -5+5-/5. 17. 5-l-_ 7 - 4 - . 4-0 = 6-5 = 2+7= 4+6 ; 2 / . Form 1"3 D 7 22. 3 + 4 + 1 = 23. 2 + 2+2= 2S '7-3+A+A 2 ( * . 7+ 5-z q . /z -s-3 + 3 = 3 0 . q+7 a _ 1 5- 6 « 3 / . 3 Z . 23+3 = 3 4 . 4-7-•5~-"34+2-. -2= — 2Z + 3 1 3 7 . 36 1 4 (STOP APPENDIX E F i r s t Year - R nmary , 4 . 5 Power Test o{ Certa in Ari thmetic S k i l l s - F o r m A - (revised) M ly name is T e a c h e r S c h o o l D i v i s i o n - Forrest Johnson June m i o 1 Sample 1 2. 3 Fo^m A . - rev ised Par i I a 13 7 i i l b few/3& //3 26> tZi \_L1 S a m p l e i t . '3. 14 0 0? ? 0 ™/0 V V V V V V V Y " V V v r v v v v v v v v -v* v v v v- v' V V v v V V V V V * * * * * *- * * S a m p l e 6 25 5Z 7 32 23 fl IS". /oq Form A C Q revised *- '» *• 24 o o o oo 0 OO O o o OOOOO o O OO O O O Ooo 0 0 o 0 0 OO OO O o OO oo 0 o oo oo 0 o oo oo o O OO OO OOOOO OOOOO o oo ° O o ° o o ° o o o o OOOOO ooooo OOOOO , *7. * e 16 I § H 0 ^ 00 0 0 n o o o 0 DMDO *• sample. © 3,^ (o,l . . 1 0 . • Y 73, ,75, ,77,78, , ,21. - 34,33, 32,__, 30, , ,2 7, 8h5 34 <Zfe STOP - r e v i s e d E 4-IS greater than is equal io 1 tc samp samp 8 4 5 7 13 sample 6 3. 2 1 + 1 is greater than is less tUan is ecjual to 2 2 sample 0 4 6 2 - 1 is greater tUn is less tnan is equal to 1 8 - 1 is greater than is less than ts equal to \Z Z7 A 34+1 is greater than is less than is equal to 3 2 revised IS 9 raster % an i s e e k w a u l tca> sampl i. h sampl i . ^ : 4.3. 5 7 . 2H, 2 5. 31 6 8 L f 5 7 1 3 3. 2 1 + 1 i s graeter ]han i ^ less fttan is eekwaul ]<& 2 2 . S d m p l 0 4 . i s graete^r jkan is less j h a n ' is eekwajul tco Zfc, i s graetex j ^ a n i s less j^kah i s e e k w a u l t o y • IS 34-H Uss ^kan i s eekwajul tce> 3 2 1$. i tdf05 / 1 Slct if v t ens wuns = I 10 I 10 l io I 1 » o l l o l i t > l ^ ^ ^ lo Sftm pi ^ 3 t * 3 ^ f l t e n 3 w u r \ 5 * 4 tens wuns & 3>4 3 5 73* 31 * tens . tens . -tens wuns 3 7 > 4-4- 4-4-3 34-4- 4-34-3 ? , o 54-87 8 SO 3 0 0 8000 3 « . «h0. 3 0 + 1 0 + 1 0 + + = '=+-01 *bi o r S-0/. " - I - H 7 i f •1» "•<H0. •ft • o r -sue- 5-1 ( 39, ;9 91 cliuos 2 - - +• ^ f i f 9 3 'or m A E 7. 2+2+2 '4- + 3+I q-5+A + '7=2 + + I 8 + 3 = Zl. 7+-5 = q + 7 = 30. 15-t« 12-8- 31. IT -S = 3*. 4-3+3 = 34. 37--5" = 33. 2++2 = 35*. k+--3-3b 22. f 31 - 1 + Block Chart of Pet in C ass 1 US (D a." E — 3 I 11 I I 0 0 o (j BLOCK CHART FOR GRAPHICAL REPRESENTATION TESTS Redrawn from a 22"x28" c h a r t . PRE-TEST QUESTIONS With the block chart shown i n Appendix P on display, the following questions were put orally to the children. They printed their answers on lined newsprint paper. 1. How many dogs did the children i n Class 1 have altogether? 2. How many cats did they have? 3. Which were there more of, dogs or cats? 4. How many more were there? 5. How many pets were there altogether? 6. Write a number sentence to prove what you said i n question 5« 7. Do you think they had any chickens? (This particular school had chickens at that time) ("Don't know" was considered a possible correct response.) 8. Why do you say that? POST-TEST SCHOOL #2 With the block chart shown In Appendix F on display, an 8i x 14" worksheet with the following questions on It was distributed to the children. Each question was read orally, and repeated as necessary by the examiner. Adequate lined spaces were provided for answers. K a K b 2(a 2(b 3(a 3(b Ha M b 5(a 5(b 6(a 6(b How many dogs did the children in this class have? Print an equation to show how you could find this out from the chart, without counting them a l l . How many cats did the children i n this class have? Print and equation to show how you could find this out from the chart without counting them a l l . Which were there more of? Dogs or cats? Print a number sentence that t e l l s this* How many more were there? Print an equation to show how to get this result. How many pets were there altogether? Print an equation to show how to get this result. Do you think this class had any white rats for pets? Why do you say that? 7. Print any other number sentences or equations that this chart t e l l s you about, (that you can t e l l from this chart). POST-TEST SCHOOL #2 An 8i x Ik" worksheet similar to Figure 7, page 158, was distributed to the children. The block chart, as shown In Appendix F, was also on display. The following questions were asked orally by the examiner, and repeated as necessary. 1. How many dogs did the children in this class have? 2. Print an equation to show how you could find this out from the chart, without counting them a l l . (use of tens and ones) 3. How many cats were there? k. Print an equation to show how you could find this out without counting them a l l . 5. Which were there more of? Dogs or cats? Put a ring around the right picture. 6. Print a number sentence that tells this. 7. Print another number sentence that t e l l s this too. 8. How many more were there? 9. Print a number sentence to show how you could find this out. 10. How many pets were there altogether? 11. Print an equation to show how to get this. 12. Do you think this class had any white rats for pets? 13. Why do you say that? 15 ./Print any other equations that you can about the pets on this chart. i 6 . ; Pets in Class 1 N qme 10 4* i. u £ 3 i l b. 7. > < S 10. and I I . I5L. Yes or No ? 13-Why? 14. l b , + IS. FIGURE 7 WORKSHEET FOR GRAPHICAL REPRESENTATION TEST I I I SCHOOL #2 Redrawn from an 8i?"xl4" worksheet. BIBLIOGRAPHY A. BOOKS Biggs, E d i t h E . and James R. MacLean. Freedom to Learn. Torontoi Addison-Wesley (Canada) L t d . , 1969. Bruner, Jerome S. The Process of Education. New Yorki Vintage Books, Random House, I960, Buswell, G. T. and C. H. Judd. Summary of Eduoational Investigations Relating to Arithmetic. Chlcagoi University of Chicago Press, 1925. The Cambridge Conference on the Correlation of Science and Mathematics i n the Schools, Goals for the Correlation of Elementary Science and Mathematics. Bostoni Houghton M i f f l i n Company, 1969. Published for Education Develop-ment Center, I n c . , Newton, Massachusetts. The Cambridge Conference on School Mathematics, Goals for School Mathematics. Boston: Houghton M i f f l i n Company, 1963. The Cambridge Conference on Teacher Training. Goals for Mathematical Education of Elementary School Teachers. Boston: Houghton M i f f l i n Company, 196?. F l a v e l l , J . H, The Developmental Psychology of Jean Piaget. New Jersey: D. Van Nostrand Co., Inc. , 1963. Hays, William L . S t a t i s t i c s for Psychologists. New York: Holt, Rinehart and Winston, 1966. Piaget, Jean. The Child *s Conception of Number. London: Routledge and Kegan Paul, L t d . , 19&±» Published i n Switzerland, 19^1. Translated by C. Gattegno and F. M, Hodgson, 1952. B. PUBLICATIONS OF GOVERNMENTS, LEARNED SOCIETIES AND OTHER ORGANIZATIONS Brownell, William A, and Harold E . Mosher. Meaningful vs. Mechanical Learning: A Study i n Grade III Subtraction. Duke University: Studies In Education, Vol VIII, W 9 t pp. 1-207. N a t i o n a l C o u n c i l of Teachers of Mathematics. A r i t h m e t i c i n G e n e r a l E d u c a t i o n . S i x t e e n t h Yearbook. Washington, D.C . i N a t i o n a l C o u n c i l of Teachers of Mathematics, 1941. . E v a l u a t i o n i n Mathematics. Twenty-sixth Yearbook. Washington, D.C.i N a t i o n a l C o u n c i l of Teachers of Mathema-t i c s , 1961. . I n s t r u c t i o n s i n A r i t h m e t i c . T w e n t y - f i f t h Year-book. Washington, D.C . i N a t i o n a l C o u n c i l of Teachers of Mathematics, i 9 6 0 . . Learning; of Mathematics. I t s Theory and P r a c t i c e . T w e n t y - f i r s t Yearbook. Washington, D.C . i N a t i o n a l C o u n c i l of Teachers of Mathematics, 1953. N a t i o n a l S o c i e t y f o r the Study of E d u c a t i o n . The Teaching; of A r i t h m e t i c . F i f t i e t h Yearbook, P a r t I I . Chicago: U n i v e r s i t y of Chicago P r e s s , 1951. S c h o o l s C o u n c i l f o r the C u r r i c u l u m and Examinations. Mathe- mat i c s i n Primary S c h o o l s . C u r r i c u l u m B u l l e t i n No. 1, second e d i t i o n . Londoni Her Majesty's S t a t i o n e r y O f f i c e , 1967. Weber, E v e l y n , A s s o c i a t e E d i t o r and S y l v i a S u n d e r l i n , A s s i s t a n t E d i t o r . Primary E d u c a t i o n : Changing Dimen-s i o n s . Washington, D.C.: A s s o c i a t i o n f o r C h i l d h o o d E d u c a t i o n I n t e r n a t i o n a l , 1966, W i l l i a m s , J . D., ed. Mathematics Reform In the Primary S c h o o l . A r e p o r t of a meeting of e x p e r t s held i n Ham-burg i n January, 1966. Hamburg: UNESCO I n s t i t u t e f o r E d u c a t i o n , I967. C. PERIODICALS Br o w n e l l , W i l l i a m A. " A r i t h m e t i c Readiness as a P r a c t i c a l Classroom Concept," Elementary Sohool J o u r n a l . September, 1951. , " E f f e c t s of P r a c t i s i n g a Complex A r i t h m e t i c S k i l l upon P r o f i c i e n c y i n i t s C o n s t i t u e n t S k i l l s , " J o u r n a l of E d u c a t i o n a l Psychology. February, 1953. , "Meaning and S k i l l — M a i n t a i n i n g the Balance," The A r i t h m e t i c Teacher. I l l , no. 6, October, 1956. Capps, Lelon R. "Division of Fractions," The Arithmetic Teacher. IX, no. 1, January, 1962. Coleman, Josephine K. "Just Plain D r i l l , " The Arithmetic Teacher. December, 1 9 6 l . Davis, Robert B. "Report from the States," Mathematics Teaching. L (Spring, 1970), 6-12. Fehr, Howard A. "Modern Mathematics and Good Pedagogy," The Arithmetic Teacher, X, no. 9, November, 1963. Glennon, Vincent J . ", . , and now Synthesis* a theoretical model for mathematics education," The Arithmetic Teacher, February, 1965. . "Method — a function of a modern program as complement to the content," The Arithmetic Teacher. XII, no. 31 March, 1965. Mayor, John R. "Science and Mathematics 1 1970*s — a Decade of Change," The Arithmetic Teacher. XVII, no. k, A p r i l , 1970. Meddleton, Ivor G. "An Experimental Investigation into the Systematic Teaching of Number Combinations i n Arithmetic, B r i t i s h Journal of Educational Psychology, no. 26 (June, 1956), pp. 117-127t as l i s t e d by Marilyn Suydam and C. Alan Riedesel, o p . c i t . on page 166. Mllgram, J o e l . "Time U t i l i z a t i o n i n Arithmetic Teaching," The Arithmetic Teacher, XVI, no. 3 , March, 1969. Pace, Angela. "The Effect of Instruction upon Development of Concept of Number," Journal of Educational Research, LXII, December, 1968. Peck, Hugh I . "An Evaluation of Topics i n Modern Mathematics The Arithmetic Teacher, X, no. 5, May, 1963. Rappaport, David. "Mathematics — l o g i c a l , psychological, pedagogical," The Arithmetic Teacher. IX, no, 2, Februrary, 1962, Shlpp, D. H. and G. H. Deer. "The Use of Class Time In Arithmetic," The Arithmetic Teacher. VII, no. 3, March, i 9 6 0 . Shuster, Albert H. and Fred L. Pigge, "Retention Efficiency of Meaningful Teaching," The Arithmetic Teacher. XII, no. 1, January, 1965. Unkel, Esther, "Arithmetic Is a Joyous Experience for Elementary Sohool Children In Great B r i t a i n , " The Arithmetic Teacher. XV, no. 2 (February, 1968) 133-137. Weaver, J . Fred. "Using Theories of Learning and Instruc-t ion i n Elementary Sohool Mathematics Research," The Arithmetic Teacher. May, 1969. Wheeler, David, "The Role of the Teacher," Mathematics Teaching, no. 50 (Spring, 1970) , 23-29. Zahn, K a r l George. "The Use of Class Time i n Eighth Grade Arithmetic," The Arithmetic Teacher. XIII, no. 2 , February, 1966. D. TEXT-BOOKS AND TEACHERS' GUIDEBOOKS Bates, W. W. and D, I n g l l s . Mathactlon. TorontoJ Copp Clark and Co. , L t d . , 1970. Chlvers, Roberta, J . E . Smith, E r i c D. MacPherson, Alfred P. Hanwell. Number Patterns. Books I and I I , Teacher's E d i t i o n . Torontot Holt, Rinehart and Winston, 1966. . et a l . Pro.leot Mathematics. Toronto 1 Holt, Rinehart and Winston of Canada, L t d . , 1970. Educational Research Council of Greater Cleveland. Greater Cleveland Mathematics Program. Teacher's Guide for F i r s t Grade. Chicagoi Science Research Associates, I n c . , 1961, 1962. E l c h o l z , Robert E . , et a l . Elementary School Mathematics. Second E d i t i o n . Teacher's E d i t i o n . Books I and I I . Don M i l l s , Ontarioi Addison-Wesley (Canada) L t d . , 1969. Hartung, Maurice L . , et a l . Seeing Through Arithmetic. Book I . Teacher's E d i t i o n . Toronto1 W. J . Gage, L t d . , 1965. Nuffield Mathematics Project. P i c t o r i a l Representation fl"|. London* W. & R. Chambers and John Murray, 1967. Available from Longmans Canada, L t d . , Don M i l l s , Ontario. . Shape and Size \2/ . Londont W. & R. Chambers and John Murray, l^o"7. V Tanyzer, Harold J, and Albert J. Mazurklewicz. mle alfabet bca>k. New Yorki I n i t i a l Teaching Alphabet Publications, Inc., 1965. E. RESEARCH LISTINGS Burns, Paul C. and Arnold R. Davis. "Early Research Contri-butions to Elementary School Mathematics," The Arithmetic Teacher. XVII, no. 1, January, 1970. Buswell, G. T. and C. H. Judd. Summary of Educational In-vestigations Relative to Arithmetic, Chlcagoi University of Chicago Press, 1925. Deans, Edwlna. Elementary School Mathematics New Directions. Washington, D.C.i United States Department of Health, Education and Welfare,Office of Education, 1963. Pehr, Howard A, ed. "Needed Research i n Mathematical Educa-tion." A Summary Report of a Conference held at Greystone, New York, October 30, 31, 1965. New Yorki Teachers 1 College Press, Teachers College, Columbia University. Glennon, Vincent J. "Research Needs in Elementary School Mathematics Education," The Arithmetic Teacher. XIII, no. 5, May, 1966. , and Leroy G. Callahan. Elementary Sohool Mathema-t i c s . A Guide to Current Research. Association for Supervision and Curriculum Development, National Education Association, 1968, Hooten, Joseph R., Junior Advisory Editor. Journal of Research and Development In Education. Vol I, no. 1, P a l l , I967. Proceedings of National Conference on Needed Research i n Mathematics Education. Athens, Georgiat University of Georgia, 1967. Mangrum, Charles and William Morris. "Doctoral Dissertation Research in Science and Mathematics," School Science and Mathematics, LXIX, no. 5. May, 1969. National Council of Teachers of Mathematics. An Analysis of New Mathematics Programs. Washington, D.C.i National Council of Teachers of Mathematics, 1963. N a t i o n a l C o u n c i l of Teachers of Mathematics. J o u r n a l f o r Research In Mathematics E d u c a t i o n , I , no, 1 (January, 1970), I , no. 2 (March, 1970), I , no. 3 (May, 1970). Washington, D.C.i N a t i o n a l C o u n c i l of Teachers of Mathematics. P i k a a r t , Len, e t a l . B i b l i o g r a p h y of Research S t u d i e s . Elementary and Pre-School Mathematics. January, 1954 t o F e bruary, 196~5T Athens, Georgia 1 U n i v e r s i t y of Ge o r g i a , A p r i l , 1967. R i e d e s e l , C. A l a n . "Recent Research C o n t r i b u t i o n s t o Elementary S c h o o l Mathematics," The A r i t h m e t i c Teacher. XVII, no. 3 . March, 1970. . "Researching Research Q u e s t i o n s , " The A r i t h m e t i c Teacher. XVII, no. 5 , May, 1970. . " T o p i c s f o r Research S t u d i e s i n Elementary S c h o o l Mathematics," The A r i t h m e t i c Teacher. XIV, no. 8, December, 1967. . and Len P i k a a r t . "Focus on Research," The A r i t h m e t i c Teacher. XVI, nos. 1-8, January-December, 1969. XVII, nos. 1-5, January-May, 1970. , and M a r i l y n N. Suydam, "Research on Mathematics E d u c a t i o n , Grades K - 8 , f o r 1968," The A r i t h m e t i c Teacher. XVI, no. 6, October, 1969. , M a r i l y n N. Suydam and Len P i k a a r t . 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Practice versus graphical representation for maintainance of basic arithmetic competencies : first year… Johnson, Dorothy Forrest 1971
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Title | Practice versus graphical representation for maintainance of basic arithmetic competencies : first year primary |
Creator |
Johnson, Dorothy Forrest |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | Educators such as Edith Biggs in Britain and Vincent Glennon and the Cambridge Conference on School Mathematics in the United States have suggested that the amount of time children spend on direct practice of newly learned skills and understandings can be greatly reduced. The Americans propose an Integration of this practice with the presentation and learning of new topics. The British favour an activity approach, where new learnings are put to immediate use, and the need for acquisition and perfection of mathematical competencies becomes obvious to the children. A few American research studies have substantiated the merits of reduced practice, at the intermediate level. This study explores the place of practice for maintenance of the basic competencies of First Year Primary children in British Columbia at the end of the school year. The competencies chosen for study were 1) Numeration: reading, writing and understanding of base ten numerals ≤99, and 2) Computation: addition and subtraction operations with sums and minuend ≤10. The new material, chosen to be presented as an alternative to direct practice, was Graphical Representation, a unit developed from the Nuffield Project booklet, Pictorial Representation 1. Two schools in the Vancouver area were used, the first with a class of 54 children and the second with 34. Parallel pre-tests and post-tests in the basic competencies were administered. During a three week Intervening interval, the Investigator taught the children, who were divided into groups, by random selection, as follows: In the first school, three groups of 18 children were instructed respectively in Graphical Representation, in review and practice, using familiar materials, and in geometry, involving no use of numbers (control group). In the second school, two groups of 17 children were Instructed in Graphical Representation, and in review and practice, respectively. At the end of the experiment, there was no significant difference in the tested numeration competencies of the two experimental groups in their respective schools. The control group showed a slightly lower achievement. Time did not permit a retention test. In the first school, where computational efficiency was low, the results slightly favoured the review and practice group, over the other groups. In the second school, there was no significant difference between the two groups, regarding progress in computational skills. Within Its limitations, this study demonstrates the possibility of maintaining basic competencies, while introducing new topics, at the first year level. |
Subject |
Arithmetic -- Study and teaching (Elementary) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-05-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0102020 |
URI | http://hdl.handle.net/2429/34389 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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