A TEST VALIDITY OF ABSTRACT ADDITION OF THE A CONCRETE- HIERARCHY FACTS ON OF KINDERGARTEN CHILDREN by B R I A N WARD Ed., University A THESIS Columbia, SUBMITTED IN FULFILLMENT FOR of B r i t i s h THE OF THE REQUIREMENTS D E G R E E OF OF PARTIAL MASTER ARTS ^ in THE FACULTY OF GRADUATE STUDIES (Department We a c c e p t conforming THE to UNIVERSITY of this Brian thesis the required OF June, © Education) as standard B R I T I S H COLUMBIA 1979 Ward 1979 1972 In p r e s e n t i n g t h i s thesis in partial an a d v a n c e d d e g r e e a t the L i b r a r y I further for shall the U n i v e r s i t y make i t agree that freely this thesis for It Department n f f i n a n c i a l gain shall OTfotJ 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 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1W5 Date BP 7 5-5 1 1 E I agree r e f e r e n c e and copying of this that not copying or for that study. thesis by t h e Head o f my D e p a r t m e n t i s understood permission. the requirements B r i t i s h Columbia, extensive 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 written DE-6 of available for permission for by h i s r e p r e s e n t a t i v e s . of f u l f i l m e n t of or publication be a l l o w e d w i t h o u t my i i ABSTRACT The number study facts of continuum. chies in Literature specify chically Levels is target This both and entire children are 11 to 49 these c h i l d r e n seven test to investigate test designed, subjects Twenty-one study and employed the of the to was and other also with to the balanced sets so the answer testing which level 14 6 and study. items. the with study: between test in be four the the 5 Columbia. 125 Entering concept of boys The years of for the a and questions 14 used numbers. pilot instructions. incorporated of of original subjects most of British the 9 were item Conceptual and ages A Greco-L'atin Each hierar- hierarchy. pretest chosen as the theory behaviours can four-stage with task possibilities s c r e e n e d and would main pilot that a hierar- Concrete-Abstract arranged concerned were the of which school d i s t r i c t in conform standardize balanced seven to facts A learning Concrete-Abstract and the Analytic indicate numbers consider was subjects number 5 in one nature for to c h i l d r e n between subjects chosen 26 in order whether design a hierarchy facts the which for associated with analytic Levels hierarchy a developmental were Task population From of Task is attempts Semi-Concrete, months of eight seven posed in task hierarchy Concrete-Abstract areas: discussed. kindergarten kindergarten reduced and a which of a concept a seven-stage 5 years four theory theory; of 9 along in interaction with Behaviours the reviewed Concrete, was a task-analytic 6 and r e s e a r c h e d and presented population 0 months the Questions validating is a definition designated Abstract. between learning arranged, hierarchy validate the; l e a r n i n g hierarchies; developmental which to addition arithmetic; analytic and attempts Twentygirls. in the Square main was the seven levels of three number set difficulty of on these sets of was, according to established c r i t e r i a , approximately equal. P r i o r t o the t e s t i n g s e s s i o n each s u b j e c t was taught t h r e e items a t each the h y p o t h e s i z e d h i e r a r c h y . Task L e v e l o f T h i s t e a c h i n g was g i v e n t o seven groups o f c h i l d r e n , each group composed of 2 boys and 2 g i r l s . In o r d e r t o d e c i d e the items taught a t each l e v e l o f t h e h i e r a r c h y , a row o f a L a t i n Square was chosen a t random and matched with a balanced s e t o f items. s u b j e c t e d t o a Guttman Scalogram a four-level hierarchy. When A n a l y s i s data supported t h e e x i s t e n c e o f Using a c u t - o f f p o i n t of 2 out o f 3 items c o r r e c t a c o e f f i c i e n t o f r e p r o d u c i b i l i t y o f 0.95 was a c h i e v e d . Ceiling e f f e c t s d i d o c c u r , however, which made i t d i f f i c u l t t o s u b s t a n t i a t e the h y p o t h e s i z e d d i f f e r e n c e s between the Concrete and Semi-Concrete Levels. A p o s s i b l e cause advanced, i s the p r e t e s t which e l i m i n a t e d 76 c h i l d r e n from of the p o p u l a t i o n p r i o r t o b e g i n n i n g the s t u d y , thus p r o d u c i n g a group s u b j e c t s who were most l i k e l y related behaviours. high i n i n t e l l i g e n c e and i n t e l l i g e n c e - iv TABLE OF CONTENTS Page Abstract Table of i i Contents iv List of Tables v i i List of Figures v i i i CHAPTER I. INTRODUCTION TO THE RELATED RESEARCH Task Analytic A Cumulative P R O B L E M AND A R E V I E W Hierarchies Developmental . . . of Knowledge Learning Theory . . as THE of the Related . . . Concrete level Semi-Concrete Abstract . . in . . Research . Task . . . Final level . tasks tasks . 17 . . . . . 19 19 20 20 20 of Terms Task Criterion 13 19 tasks Levels Definition 11 . Continuum level level 3 18 tasks Concrete-Abstract 1 . RESEARCH PROBLEM Concrete-Abstract i\ZZ». . Contrasted with Cumulative Learning Theory . The D e v e l o p m e n t o f Number C o n c e p t s Pre-School Children . . . . . II. THE 1 Theory Implications OF 21 21 f o r Performance Instructional Objective 21 21 Entering Behaviours 21 Learning Hierarchy 21 Task Analysis 21 Page Research Target III. Objectives Population 25 Instrument 26 Pilot 27 Test Subjects 27 Item 27 Difficulty Sequence Teaching 29 Procedures Statistical Analysis 31 Analysis 34 of Variance 34 Descriptive Analysis 35 Guttman R E S U L T S OF Pretest Pilot Scalogram THE PRETEST Analysis 35 AND P I L O T TEST . . . 36 Test R E S U L T S OF 36 Test Implications for the Main of the Pretest Study and P i l o t 41 Test 43 45 of Variance 45 Descriptive Analysis 52 Guttman THE . STUDY Analysis VI. 25 Pretest Testing V. 23 METHODOLOGY Measuring IV. 23 Scalogram Analysis DISCUSSION, LIMITATIONS 53 AND R E C O M M E N D A T I O N S . Discussion Limitations 57 of Recommendations REFERENCES 57 the Study for Future 60 Research . . . 62 64 vi Page APPENDICES . • 67 Appendix A 67 Appendix B 68 Appendix C 70 Appendix D 72 vii LIST OF TABLES TABLES 1 2 Page Items Items Sets 3 4 5 Arranged Grouped of by L e v e l to of Equalize Difficulty . Difficulty . 28 of Three 29 T e s t i n g Sequence as Determined G r e c o - L a t i n Square by a Item Set and i t s Matched Hierarchy i n the Number . of Subjects Level 30 32 Participating i n the Pretest 36 6 Pretest 7 Distribution Times 8 Results of 10 11 Subjects Who F a i l e d the Pretest to Pretest Summary o f Difficulty Criteria 13 and Testing Criteria . Illustrated Category Test 39 40 According to 42 of . the . . . 46 t h e A n a l y s i s of Variance of Item According t o t h e Four E s t a b l i s h e d 49 . 50 A Summary o f t h e A n a l y s i s o f V a r i a n c e o f D i f f i c u l t y f o r Balanced Groups of Items Item . . 51 A Summary o f Hypothesized the Performance Results Seven 16 Schools Summary o f t h e A n a l y s i s o f V a r i a n c e Sequence of Items W i t h i n t h e Test Mean 15 Sex, Who M e t S c r e e n i n g Results of the P i l o t Hierarchy Level 12 14 over Subjects According 9 38 the Analysis Hierarchy Guttman of Scalogram Variance of 52 Analysis - Levels Results Four of by Sex and Item D i f f i c u l t y of Levels Guttman 54 Scalogram Analysis 56 viii LIST OF FIGURES FIGURES 1 Page H y p o t h e s i z e d H i e r a r c h y Showing the Four C o n c e p t u a l L e v e l s and the Seven Task L e v e l s . . 2 3 Hypothesized Hierarchy for A d d i t i o n Facts Showing the C o n c e p t u a l L e v e l s 1 t o 4 and Task L e v e l s 1 t o 7 Mean Responses to Item S e t s at Each L e v e l o f the H i e r a r c h y 22 , 48 33 1 Chapter One I n t r o d u c t i o n to the Problem and a Review o f t h e R e l a t e d Research The f e a s i b i l i t y o f a r r a n g i n g c o n c e p t u a l i n f o r m a t i o n i n a s e q u e n t i a l o r d e r such t h a t a h i e r a r c h y o f s k i l l s i s c o n s i d e r e d t o be has been o f i n t e r e s t f o r some t i m e . established The l e a r n i n g o f one s e t o f behaviours has been p u r p o r t e d t o l e a d a u t o m a t i c a l l y t o , and s u p p o r t the l e a r n i n g o f , a r e l a t e d more d i f f i c u l t set of behaviours. T h i s premise has been b a s i s f o r the d e s i g n o f programmed l e a r n i n g and l e a r n i n g the experiments which have sought to d i m i n i s h or remove f a i l u r e i n m a s t e r i n g a new b e h a v i o u r or s e t o f b e h a v i o u r s . T h i s r e v i e w p r e s e n t s some o f the r e s e a r c h t h a t r e l a t e s t o h i e r a r c h i c a l l e a r n i n g i n f o u r major a r e a s : g e n e r a l r e s e a r c h and r e s e a r c h that affect considerations an u n d e r s t a n d i n g o f t a s k h i e r a r c h i e s i n a r i t h m e t i c ; the t h e o r y o f knowledge t h a t emerges from such a d i s c u s s i o n (the c u m u l a t i v e l e a r n i n g t h e o r y ) ; the r e l a t i o n s h i p o f developmental l e a r n i n g t h e o r y to c u m u l a t i v e l e a r n i n g t h e o r y ; and r e s e a r c h on p r e - s c h o o l c h i l d r e n t h a t bears most d i r e c t l y upon the r e s e a r c h problem and the r e s e a r c h q u e s t i o n s are posed i n t h i s that study. Task A n a l y t i c H i e r a r c h i e s Gagne (1961a) h y p o t h e s i z e d t h a t as a l e a r n e r p r o g r e s s e d up a h i e r a r c h y h i s r a t e o f l e a r n i n g s h o u l d depend i n c r e a s i n g l y on the a t t a i n m e n t attainment or non- o f r e l e v a n t l e a r n i n g s e t s (performance o f a group o f f a c t s s p e c i f i e d l e v e l o f the h i e r a r c h y ) , and d e c r e a s i n g l y on r e l e v a n t the i n d i v i d u a l b r i n g s t o t h e s i t u a t i o n . at a abilities From an experiment designed to 2 a n a l y z e the c l a s s o f t a s k s f o r s o l v i n g l i n e a r e q u a t i o n s , Gagne c o n c l u d e d : (a) r e l e v a n t b a s i c a b i l i t i e s are c l e a r l y of more importance i n the m a s t e r i n g o f a h i e r a r c h y than i r r e l e v a n t b a s i c a b i l i t i e s ; (b) i n s t a n c e s of p o s i t i v e t r a n s f e r t o each l e a r n i n g s e t o r d i n a t e r e l e v a n t l e a r n i n g s e t s are found to occur throughout from s u b the hierarchy; (c) c o r r e l a t i o n s o f r e l e v a n t b a s i c a b i l i t i e s w i t h r a t e s of attainment of l e a r n i n g s e t s a t p r o g r e s s i v e l y h i g h e r l e v e l s o f the h i e r a r c h y show a steeply progressive decrease; and (d) c o r r e l a t i o n s o f r a t e o f a t t a i n m e n t of l e a r n i n g sets with achieve- ment o f r e l e v a n t s u b o r d i n a t e l e a r n i n g s e t s a r e found t o be s y s t e m a t i c a l l y h i g h e r than w i t h the achievement of i r r e l e v a n t s e t s p a r t i c u l a r l y i n t h e upper l e v e l s o f the hierarchy. Gagne' (1962b) then undertook a s t u d y on r e c a l l a b i l i t y and i n t e g r a t i o n presenting subjects w i t h a programme which had t w e l v e s u b o r d i n a t e s u p p o r t i n g two r e l a t e d f i n a l t a s k s . tasks Low and h i g h a b i l i t y s t u d e n t s were t e s t e d t o f i n d out how much r e p e t i t i o n of l e a r n i n g s e t s was r e q u i r e d , and how much d e t a i l was t o be p r o v i d e d , i n o r d e r f o r s t u d e n t s t o i n t e g r a t e information. The s u b j e c t s the were 136 s e v e n t h grade c h i l d r e n who were a s s i g n e d t o groups by achievement ( t e a c h e r grades f o r a r i t h m e t i c because the h y p o t h e s i z e d h i e r a r c h i e s were i n a r i t h m e t i c ) . The experiment showed a h i g h c o r r e l a t i o n between the achievement o f f i n a l t a s k s and the number of s u b o r d i n a t e l e a r n i n g s e t s which were a c q u i r e d . There was a l s o evidence t o suggest t h a t s u b o r d i n a t e l e a r n i n g s e t s i n a h i e r a r c h y positive transfer to a higher l e v e l l e a r n i n g mediate set. Three years l a t e r Gagne'(1965) examined r e t e n t i o n o f i n i t i a l l y knowledge o f n o n - m e t r i c geometry, m a n i p u l a t i n g the i n s t r u c t i o n s learned variable and to g a i n i n f o r m a t i o n t h a t would p e r m i t i n f e r e n c e s t o be made about memory. A f t e r a n i n e week i n t e r v a l measures were t a k e n on: (a) achievement on items t h a t were matched f o r c o n t e n t w i t h the o r i g i n a l achievement t e s t , and (b) performance on each o f the s u b o r d i n a t e t a s k s t h a t were c o n t r i b u t i n g to the f i n a l task. F i v e groups o f s i x t h grade c h i l d r e n , s i x t e e n per g r o u p , worked through a programme w i t h i d e n t i c a l c o n t e n t but composed o f f i v e experimental treatments. different P r a c t i c a l examples of each s u b o r d i n a t e task had: (a) m i n i m a l v a r i e t y i n c o n t e n t , (b) i n t e r m e d i a t e v a r i e t y , (c) great v a r i e t y , (d) no examples, or (e) u n r e l a t e d tasks. The c o r r e l a t i o n between the i n i t i a l achievement t e s t s was +0.64 and the c o r r e l a t i o n between achievement s c o r e s i m m e d i a t e l y a f t e r those o b t a i n e d a f t e r l e a r n i n g and a n i n e week r e t e n t i o n i n t e r v a l was +0.62. C o r r e l a t i o n s between s c o r e s on s u b o r d i n a t e l e a r n i n g h i e r a r c h i e s n i n e weeks, however, were o n l y +0.41 and +0.46. after The s u b o r d i n a t e tasks n e c e s s a r y t o l e a r n the f i n a l t a s k were not w e l l m a i n t a i n e d ; the q u e s t i o n t h a t a r o s e was whether t h e i r importance d i m i n i s h e d once the f i n a l was l e a r n e d . task The l e a r n e r d i d not f o r g e t p a r t i c u l a r p a r t s o f the h i e r a r c h y ; f o r g e t t i n g was e n t i r e l y random. The f i n a l tasks themselves, however, were w e l l m a i n t a i n e d . A C u m u l a t i v e Theory of Knowledge Gagneaand Brown (1961b) undertook a p i e c e o f r e s e a r c h which i n f l u e n c e d Gagn^'s t h i n k i n g about both the a c q u i s i t i o n o f knowledge and 4 the importance o f i n s t r u c t i o n s i n a c q u i r i n g t h a t knowledge. The purpose of t h e study was t o o b t a i n a measure o f t h e l e a r n i n g e f f e c t i v e n e s s of programmed m a t e r i a l i n terms which would permit a r e a s o n a b l e t h a t understanding had been accomplished. and 10 were randomly assigned i n d i v i d u a l l y tested. (the teacher T h i r t y - t h r e e boys i n grades 9 e x p e r i m e n t a l groups and then The programmes were termed ' r u l e and example presented t h e r u l e t h a t 2 + 3 = 3 + 2 s e v e r a l examples); 'guided d i s c o v e r y ' i n the process of discovery acquired to three inference and then t h e c h i l d d i d (the teacher and helped t h e c h i l d was a c t i v e l y i n v o l v e d re-state previously concepts t h a t were r e l e v a n t t o t h e problem); and ' d i s c o v e r y ' ( t r i a l and e r r o r l e a r n i n g where t h e teacher a c t i v e l y involved i n the discovery provided process). s t r u c t u r e , but was not Performance was measured i n terms o f time taken t o s o l v e a problem and t h e number o f h i n t s Learning 1 occurred i n a l l three most l e a r n i n g f o l l o w e d groups but 'guided d i s c o v e r y ' by ' d i s c o v e r y ' and f i n a l l y i n t e r e s t i s t h e f a c t t h a t t h e guided d i s c o v e r y required. produced the ' r u l e and example'. programme not o n l y Of used 'small s t e p s ' , as i s common i n programmed l e a r n i n g , but a l s o r e q u i r e d t h e subject to r e - s t a t e c e r t a i n previously acquired concepts which were r e l e v a n t t o t h e problems i n t h e e x p e r i m e n t a l c o n d i t i o n . of c e r t a i n concepts by t h e s t u d e n t s a t t h e request the p o s i t i v e r e s u l t s o b t a i n e d , focused The r e - s t a t i n g of t h e r e s e a r c h e r , and a t t e n t i o n upon t h e nature of i n s t r u c t i o n s , and a l s o upon t h e n e c e s s i t y o f r e c a l l i n g r e l e v a n t l e a r n i n g s e t s i n order t o o b t a i n a s o l u t i o n t o t h e task subordinate presented. In a t h e o r e t i c a l paper on Knowledge and I n s t r u c t i o n s , Gagne (1962a) / stated that i f l e a r n i n g sets are e s s e n t i a l f o r p o s i t i v e t r a n s f e r the f o l l o w i n g consequences should ensue: 5 (a) i f a h i g h e r l e v e l l e a r n i n g s e t l e v e l s e t s must have been i s passed, a l l related lower passed; (b) i f one or more of the lower l e v e l s e t s have been f a i l e d , the r e l a t e d h i g h e r l e v e l t a s k s must be f a i l e d ; and (c) i f a higher l e v e l set may have been p a s s e d . attributable has been f a i l e d , r e l a t e d lower l e v e l The absence of p o s i t i v e t r a n s f e r sets would be to a d e f i c i e n c y i n i n s t r u c t i o n s . A t h e o r y was o u t l i n e d which d e f i n e d knowledge as an i n f e r r e d c a p a b i l i t y t h a t makes p o s s i b l e the s u c c e s s f u l performance o f a c l a s s o f t a s k s t h a t c o u l d not be performed b e f o r e l e a r n i n g was undertaken. P r o d u c t i v e l e a r n i n g was seen as i n v o l v i n g two v a r i a b l e s : (a) knowledge, and (b) instructions. I t i s i m p o r t a n t t o understand e x a c t l y what Gagne'meant by and ' s a t i s f a c t o r i l y c o m p l e t i n g a f i n a l not n e c e s s a r i l y t r a d i t i o n a l ones. 'instructions' t a s k ' , because h i s d e f i n i t i o n s In o r d e r to have a framework d i s c u s s i n g the l e a r n i n g c o n d i t i o n s necessary for enquiry, that were for is a c t i v i t i e s c h a r a c t e r i z e d by a p r o b l e m - s o l v i n g approach - Gagne'' (1963) delineated the f i n a l t a s k as the i n s t r u c t i o n s , more b r o a d l y , as ' t e r m i n a l c a p a b i l i t y o f the l e a r n e r ' , and 'instructional conditions'. p r a c t i c e and a s u f f i c i e n t background o f e x p e r i e n c e ; o n l y be s o l v e d by the a c q u i s i t i o n o f knowledge. both components o f i n s t r u c t i o n s ; the same. E n q u i r y needed new problems c o u l d D r i l l and d i s c o v e r y were however, d i s c o v e r y and e n q u i r y were not The c o n s t r u c t i o n o f a response by a l e a r n e r was termed c o v e r y ' , whereas ' e n q u i r y ' was the t e r m i n a l t h i n k i n g p r o c e s s . 'dis- A series of r e q u i r e d s t e p s were h y p o t h e s i z e d i n o r d e r f o r a s t u d e n t t o a t t a i n the practice of enquiry: (a) competent performer ( l e a r n i n g of skills), (b) a c q u i s i t i o n o f knowledge, (c) s c i e n t i f i c enquirer (genuine e n q u i r y ; a b i l i t y t o d i s t i n g u i s h good and bad i d e a s ; a b i l i t y t o s o l v e problems by means o f unrestrained i n d u c t i v e t h i n k i n g ) , and (d) independent investigation (e.g. scientist). E x t e n d i n g h i s t h i n k i n g about i n s t r u c t i o n s f u r t h e r , Gagne^ (1964) noted t h a t the p r o b l e m - s o l v e r came to a g i v e n s i t u a t i o n w i t h p r e v i o u s l y l e a r n e d c a p a b i l i t i e s ; the experimenter then communicated w i t h the subject by way o f i n s t r u c t i o n s i n f o u r ways: (a) by i d e n t i f y i n g new s t i m u l i , (b) by i d e n t i f y i n g the expected form of the t e r m i n a l (c) by h e l p i n g t h e s u b j e c t performance, r e c a l l previously acquired c a p a b i l i t i e s , (d) by c h a n n e l i n g t h i n k i n g i n a r e l e v a n t direction. G a g n e ^ c a l l e d f o r r e s e a r c h which would m a n i p u l a t e the v a r i a b l e instructions based upon the h y p o t h e s i s t h a t b e t t e r i n s t r u c t i o n s would f a c i l i t a t e r e c a l l of p r e v i o u s l y acquired relevant concepts. reduced.to (a) s u b j e c t (b) Problem s o l v i n g c o u l d be an e x p e r i m e n t a l s i t u a t i o n where the independent i n t o t h e broad c a t e g o r i e s the variables fell of: c a p a b i l i t y , and instructions. Gagne''and B r o w n ' s (1961b) s t u d y opened the way f o r a s e r i e s of experi ments which c u l m i n a t e d i n the model o f l e a r n i n g p r e s e n t e d by Gagne''in 1968 These experiments emphasized the n e c e s s i t y of exact communication between t e a c h e r and s t u d e n t . The i n s t r u c t i o n s , which are q u i t e c l e a r l y i n a l l forms o f t e a c h e r - p u p i l important l e a r n i n g , t a k e on a new d i m e n s i o n when 7 l o o k e d a t from the v i e w p o i n t o f a t a s k h i e r a r c h y . Each new and h i g h e r x s t a g e can o n l y be l e a r n e d i f the s t u d e n t understands the instructions. A l s o , i n any h y p o t h e s i z e d h i e r a r c h y a h i g h e r l e a r n i n g s e t can o n l y be l e a r n e d when the r e l e v a n t lower s e t s i n the h i e r a r c h y are mastered. Gagne''defined communication i n terms o f s t i m u l u s i d e n t i f i c a t i o n , i d e n t i f i c a t i o n o f the form of t e r m i n a l r e s p o n s e , r e c a l l of p r e v i o u s l y a c q u i r e d c a p a b i l i t i e s , and c h a n n e l i n g t h i n k i n g i n a r e l e v a n t direction. Q u i t e c l e a r l y , however, the problem i s not t h a t s i m p l e ; P e r s o n a l i t y , m o t i v a t i o n a l , and a d d i t i o n a l c o g n i t i v e v a r i a b l e s must be c o n s i d e r e d . One v a r i a b l e which G a g n e ' d i d i n v e s t i g a t e was memory. Because many s u b o r d i n a t e t a s k s w i t h i n h i e r a r c h i e s were f o r g o t t e n , he q u e s t i o n e d whether t h e s e t a s k s were n e c e s s a r y achieved. A further once the t e r m i n a l o b j e c t i v e was problem was posed by the l i m i t e d amount o f i n f o r m a t i o n a v a i l a b l e about how the human memory s t o r e d and coded i n f o r m a t i o n . apparent problem o f f o r g e t f u l n e s s The might be more one o f r e t r i e v a l o f i n f o r m a t i o n from t h e l o n g - t e r m memory s t o r e . Gagne^s model o f l e a r n i n g (based on a t h e o r y o f c u m u l a t i v e l e a r n i n g ) contrasted i n a number o f r e s p e c t s w i t h developmental t h e o r i e s c e n t r a l themes were m a t u r a t i o n , r e a d i n e s s , and c o g n i t i v e He i d e n t i f i e d t h e s t a g e a t which a person c o u l d be as .(a) relevant c a p a b i l i t i e s possessed, whose adaptation. involving and (b) a number o f h i e r a r c h i e s o f c a p a b i l i t i e s t o be a c q u i r e d , such t h a t i t i s p o s s i b l e t o combine s u b o r d i n a t e e n t i t i e s terminal the task. The model took i n t o account t r a n s f e r o f l e a r n i n g : what was l e a r n e d c o u l d be combined w i t h o t h e r l e a r n e d e n t i t i e s transfer. to h e l p a c h i e v e v i a a mechanism o f l e a r n i n g 8 In the second e d i t i o n o f h i s book Gagne'' (1970) d i s c u s s e d the r e l a t i o n s h i p o f v a r i o u s forms o f l e a r n i n g a n d , i n p a r t i c u l a r , noted t h a t the e x i s t e n c e o f p r i o r c a p a b i l i t i e s - i m p o r t a n t i n drawing d i s t i n c t i o n s among c o n d i t i o n s r e q u i r e d f o r l e a r n i n g - was s l i g h t e d or i g n o r e d by most t r a d i t i o n a l learning prototypes. He went on to s t a t e t h a t when b u i l d i n g a h i e r a r c h y the t a s k s s h o u l d be a n a l y z e d such t h a t the l o w e s t ' b o x e s ' i n a hierarchy represented the k i n d of performance a l l s t u d e n t s i n a group could already successfully accomplish. The s t a t e m e n t s i n the were i n t e n d e d to d e s c r i b e a s i n g l e c a p a b i l i t y to be l e a r n e d , 'boxes' representing what the l e a r n e r was a b l e to:';do when l e a r n i n g had been a c c o m p l i s h e d . They were, t h e r e f o r e , s t a t e d i n performance t e r m s . The s u p e r o r d i n a t e c a p a b i l i t y was more r e a d i l y l e a r n e d i f the s u b o r d i n a t e c a p a b i l i t i e s had been p r e v i o u s l y a c q u i r e d and.were r e a d i l y a v a i l a b l e f o r r e c a l l . Each s u b o r d i n a t e c a p a b i l i t y had been i d e n t i f i e d as such because i t was known, o r h y p o t h e s i z e d , to c o n t r i b u t e p o s i t i v e t r a n s f e r superordinate c a p a b i l i t y . to the l e a r n i n g o f a Students who had l e a r n e d s u b o r d i n a t e skills s h o u l d l e a r n s u p e r o r d i n a t e s k i l l s more e a s i l y than t h o s e who had not l e a r n e d them. Engelmarnwas concerned w i t h t h e m a n i p u l a t i o n o f t e a c h i n g v a r i a b l e s ; l i k e Gagne he was i n t e r e s t e d i n t a k i n g i d e a s , c o n c e p t s , and i n f o r m a t i o n , and a r r a n g i n g them i n a h i e r a r c h i c a l f a s h i o n . He argued (1969a) t h a t a concept was a s e t o f c h a r a c t e r i s t i c s shared by a l l ' i n s t a n c e s ' p a r t i c u l a r s e t and o n l y by t h o s e in a ' i n s t a n c e s ' , and t h a t concepts were c l a s s e s and thus amenable t o h i e r a r c h i c a l arrangement. He c o n s i d e r e d i t n e c e s s a r y t o r e a l i z e t h a t c o n c e p t s were always dependent on the c o n t e x t o r u n i v e r s e i n which they were p r e s e n t e d , and t h a t t o a n a l y z e a concept so t h a t i t c o u l d be taught was to d e s c r i b e the concept i n terms o f the 9 minimum set recognize e s s e n t i a l d i s c r i m i n a t i o n s the that an 'instance' For Engelmann, (a) the demonstrations of the concepts, (b) the test istics standing He of of the commented stration, or that least for potential critical when was from to the in what because he a was the concept (b) the rules (c) the rule the therefore, that concept two order to set. components: performs to c o u l d be taught child to the of and had show the demonstrate been both character- his under- a single demon- 'instances' should concentrate make. Engelmann translated a concept, into and to s p e c i f i c a t i o n s , and i t was taught: task not for he had one noted that the range specific ways. unless the the a might learn with load; on from tests move of were determine fail responding. to and tasks, p o s s i b l e to a child convention was a minimum memory behaviour, of in presentation with to were a problem Basically, three when things: taught, responding, responding important and is factor responding acknowledged best specific tasks being that present teacher testing had for one act to The the a particular the find of concept necessary exact know (a) must a member to understanding child taught added in performs understanding of not Engelmann make child concept. yielded terms did teacher misunderstanding test analysis exactly is, no was concept ways specified child that possible behaviour possible Concept the a presentation of not, c o n s i s t e d of the d i s c r i m i n a t i o n s the tests was to and task i t of developing or had concept. non-'instances' in teaching further and was, child and worthwhile. not previously worthwhile. i t is also The mentioned: concept apparent that of a student motivation Engelmann is 10 s u g g e s t i n g a r o l e f o r p e r s o n a l i t y v a r i a b l e s t h a t have some e f f e c t student's upon.a self-esteem. U p r i c h a r d and P h i l l i p s (1977) reviewed t h e r e s e a r c h l i t e r a t u r e o f the 1 9 6 0 ' s and c o n c l u d e d t h a t e p i s t e m o l o g i c a l c o n s i d e r a t i o n s were i n the d e s i g n o f l e a r n i n g h i e r a r c h i e s . They c i t e d Gagne's s t a t e m e n t t h a t the " d e t e r m i n a t i o n o f a h i e r a r c h i c a l sequence o f s u b t a s k s from s i m p l e s t most complex was not e a s i l y a c h i e v e d , " and c l a i m e d t h a t l o g i c a l based upon a t h e o r y o f knowledge, was d i f f i c u l t ignored p s y c h o l o g i c a l f a c t o r s . paramount to analysis, t o v a l i d a t e because i t Engelmann (1969a) however, had c o n s i d e r e d p s y c h o l o g i c a l v a r i a b l e s i n emphasizing the r o l e o f m o t i v a t i o n . The o r d e r i n g o f a h i e r a r c h y would c l e a r l y be i n f l u e n c e d by p r i o r l e a r n i n g , the i n s t r u c t i o n a l s i t u a t i o n , memory and many o t h e r f a c t o r s . the w i l l i n g n e s s of a student to i n t e r a c t w i t h the l e a r n i n g environment a l o n g many dimensions and would no doubt a f f e c t ments s e e k i n g the 'perfect' Additionally, varies the outcome o f any e x p e r i - hierarchy. C u m u l a t i v e L e a r n i n g t h e o r y has enabled e d u c a t i o n a l r e s e a r c h e r s focus on i n s t r u c t i o n a l o b j e c t i v e s . to W h i l e both Gagne''and Engelmann broached t h e i s s u e , B a r b a r a Bateman (1971) f u r t h e r e d its development. L i k e Gagne''"and Engelmann, she b e l i e v e d i n s t r u c t i o n a l o b j e c t i v e s t o be n e c e s s a r y i n d e t e r m i n i n g the g o a l s i n a l e a r n i n g t a s k . She suggested the f o l l o w i n g c r i t e r i a f o r the s e l e c t i o n of o b j e c t i v e s : (a) do the o b j e c t i v e s i n d i c a t e l e a r n i n g outcomes t h a t are appropriate t o the i n s t r u c t i o n a l a r e a ? (b) do the o b j e c t i v e s r e p r e s e n t a l l l o g i c a l l e a r n i n g outcomes of instructional area? ( c ) are the o b j e c t i v e s o b t a i n a b l e by t h e s e s t u d e n t s ? the 11 (d) a r e the o b j e c t i v e s i n harmony w i t h the p h i l o s o p h y o f the s c h o o l i n which the i n s t r u c t i o n i s to be g i v e n ? (e) are the o b j e c t i v e s i n harmony w i t h the t e a c h e r ' s p e r c e i v e d needs f o r the f u t u r e o f the c h i l d and s o c i e t y ? Task a n a l y s i s i s c l e a r l y one way o f s p e c i f y i n g o b j e c t i v e s a c c o r d i n g t o the f o r e g o i n g c r i t e r i a ; i t i s a t f i r s t l o g i c a l , then e m p i r i c a l . s t u d y d e s c r i b e d i n the f o l l o w i n g c h a p t e r s h y p o t h e s i z e s t h a t t a s k i s e s s e n t i a l to successful l e a r n i n g . The analysis W h i l e the b a s i s i s a n a l y t i c a l and h i e r a r c h i c a l , developmental c o n s i d e r a t i o n s w i l l not be i g n o r e d . espoused i s t h a t i n o r d e r t o t e a c h w i t h purpose, The view c l e a r g o a l s need to be o u t l i n e d i n o r d e r f o r t e a c h i n g t o be s u c c e s s f u l w i t h most c h i l d r e n . Developmental L e a r n i n g Theory as C o n t r a s t e d w i t h C u m u l a t i v e L e a r n i n g Theory Engelmann (1967) was p a r t i c u l a r l y c r i t i c a l of the developmental t h e o r i e s o f l e a r n i n g , i n d i c a t i n g t h a t developmental i n t e r p r e t a t i o n s , as those o f P i a g e t , are b a s i c a l l y i r r e l e v a n t to t e a c h e r s , such and t h a t when an i r r e l e v a n t e x p l a n a t i o n i s a c c e p t e d by a t e a c h e r she ceases to f u n c t i o n i n a teaching capacity. Irrelevant explanations ' l o c k the d o o r ' on a p r o b l e m , whereas a t e a c h e r seeks to develop and change a c h i l d of t h i s e x p l a n a t i o n . To irrespective Engelmann, a developmental e x p l a n a t i o n does not i m p l y a remedy t h a t can be a c h i e v e d through the m a n i p u l a t i o n o f e n v i r o n m e n t a l v a r i a b l e s ; as a consequence i t does not d e a l w i t h under the t e a c h e r ' s c o n t r o l . He added t h a t o f t e n the e x p l a n a t i o n cannot be t r a n s l a t e d transfer variables developmental i n t o concepts whereas the t e a c h e r must t h e c h i l d ' s r e l a t i v e d e f i c i e n c y i n t o terms of concepts because these imply a manipulation of v a r i a b l e s , noting that Piaget considered 'conservation' a concept, yet p l a c e d a host o f developmental on the c o n c e p t ' s a c q u i s i t i o n . restrictions He f e e l s t h a t i f i t i s a concept then it 12 can be t a u g h t ; i f i t i n v o l v e s s p e c i f i c c o n t e n t , i t i s c e r t a i n l y r . n o t the p r o d u c t o f c o g n i t i v e s t r u c t u r i n g but r a t h e r the p r o d u c t of e n v i r o n m e n t a l consistency. Many o t h e r a u t h o r s have agreed w i t h Engelmann's a d m i t t e d l y extreme point of view. Kilpatrick (1970) i n d i c a t e d t h a t the c r e a t o r s o f the S c h o o l Mathematics Study Group programmes were guided by l o g i c a l consid- e r a t i o n s r a t h e r than p s y c h o l o g i c a l , c l a i m i n g the e m p i r i c a l t o be the o n l y d e f e n s i b l e approach. P i a g e t , they a r g u e d , was an o b s e r v e r , not a teacher. They c o n s i d e r e d the p r e d i c t i o n s which emerged from h i s d e v e l o p m e n t a l model u n j u s t i f i e d as they were based on i n v e s t i g a t i o n s of the i n g o f c h i l d r e n who had been taught by c o n v e n t i o n a l methods. understandKarplus (1970) s u p p o r t e d t h i s v i e w , n o t i n g t h a t P i a g e t found t h a t f o r m a l operations developed w i t h o u t s p e c i f i c i n s t r u c t i o n , and t h a t t h i s was not an adequate premise t o encompass the r e s u l t s , t h i n k i n g or a t t i t u d e s o f modern science. T h i s d i s c u s s i o n i s i m p o r t a n t f o r many r e a s o n s . Buell (1970) stated t h a t the c o n t r o v e r s i a l i s s u e as t o whether the l e a r n i n g o f c o n c e p t s may be a c c e l e r a t e d i s o f c o n s i d e r a b l e s i g n i f i c a n c e i n view o f the tendency i n r e c e n t y e a r s t o advance the p r e s e n t a t i o n o f s c i e n t i f i c and m a t h e m a t i c a l m a t e r i a l t o e a r l i e r l e v e l s o f the c u r r i c u l u m . Miller (1976) , u s i n g 52 K i n d e r g a r t e n c h i l d r e n from a l o w e r s o c i o - economic background, attempted t o f i n d out whether c o n s e r v a t i o n as d e f i n e d by P i a g e t emerged e a r l i e r i n a c h i l d ' s development i f the t a s k s were p r e s e n t e d n o n - v e r b a l l y and m o t i v a t i o n was maximized. The c h i l d could p i c k the c a n d i e s t h a t he wanted t o eat i n a c o n s e r v a t i o n o f number task, o r the j u i c e he wanted t o d r i n k i n a c o n s e r v a t i o n o f l i q u i d t a s k . Miller commented t h a t language might w e l l be an i m p o r t a n t f a c t o r because the c h i l d r e n seemed t o judge numbers i n terms of l e n g t h , and concept seemed t o be a c t i v a t e d by the word ever, t h a t t h e r e was 'more'. that t h i s He c o n c l u d e d , how- no evidence from h i s study t o suggest that c o n s e r v a t i o n emerged a t an e a r l i e r c h r o n o l o g i c a l age than t h e o r i z e d by Piaget. I t i s very p o s s i b l e t h a t no s o l u t i o n t o the argument e x i s t s a t present. Bruner (1967) took t h i s p o s i t i o n , a s k i n g whether i t was more v a l u a b l e t o p o s t u l a t e s t a g e s of growth or t o t h i n k i n terms of g r a d u a l p r o c e s s e s of growth. He claimed the argument was fruitless. The Development of Number Concepts i n P r e - S c h o o l C h i l d r e n Saxe (1977) analyzed c h i l d r e n ' s c o u n t i n g s t r a t e g i e s with a r r a y s of more than f i v e o b j e c t s with the express purpose of a s s e s s i n g whether developmental changes i n t h e i r s t r a t e g i e s were p a r t i a l l y independent t h e i r a b i l i t y to count a c c u r a t e l y . He concluded that counting accuracy and s t r a t e g i e s were i n t e r r e l a t e d and t h a t they appeared t o the same c o g n i t i v e p r o c e s s . The main p o i n t of i n t e r e s t i n t h i s study was t h a t a t the P r e - q u a n t i t a t i v e l e v e l s almost t o be of connected no a c c u r a t e count- i n g o c c u r r e d , and a t the Q u a n t i t a t i v e l e v e l s c o u n t i n g a c c u r a c y markedly improved. However, a t the T r a n s i t i o n a l l e v e l s , between the P r e - q u a n t i t a t i v e and miscount the Q u a n t i t a t i v e l e v e l s some c h i l d r e n appeared 'purposely' as a r e s u l t of t r y i n g t o i n t e g r a t e t h e i r judgments of numbers with the products of t h e i r c o u n t i n g . emphasized the d i f f i c u l t y of attempting spatial This a r t i c l e to a r b i t r a r i l y d i s t i n g u i s h a sequence of i n t e r r e l a t e d concepts without c h i l d ' s c o g n i t i v e maturation. to taking into consideration a In some c a s e s , however, evidence i n the r e s e a r c h t o s u b s t a n t i a t e the i d e a of t e a c h i n g one concept appears before another. B r a i n e r d (1974) took a group o f p r e - s c h o o l e r s who e v i d e n c e d no p r o f i c i e n c y w i t h e i t h e r t h e o r d i n a l or c a r d i n a l p r o p e r t i e s o f n a t u r a l numbers and t r a i n e d them t o a c q u i r e t h e s e p r o p e r t i e s . 60 female C a u c a s i a n s u b j e c t s H i s 60 male and ranged from 4 years 0 months to 5 years 5 months w i t h a mean age o f 4 years 7 months. The r e s u l t s o f h i s s t u d y showed t h a t both p r o p e r t i e s were t r a i n a b l e , but t h a t t h e o r d i n a l p r o p e r t i e s of numbers were much e a s i e r t o t r a i n than the c a r d i n a l properties. He a l s o noted t h a t the i n f o r m a t i o n l e a r n e d about o r d i n a l p r o p e r t i e s was more e a s i l y t r a n s f e r r e d to other l e a r n i n g s i t u a t i o n s than the i n f o r m a t i o n l e a r n e d about c a r d i n a l p r o p e r t i e s . The r e s u l t s o f t h i s r e s e a r c h i m p l i e d t h a t the u n d e r s t a n d i n g o f the o r d i n a l p r o p e r t i e s o f numbers would precede t h a t of the c a r d i n a l p r o p e r t i e s i n such a way t h a t i t would be f e a s i b l e t o p l a c e them i n a h i e r a r c h i c a l r e l a t i o n s h i p t o one a n o t h e r . S i e g e l (1971) d e s i g n e d a sequence o f e x p e r i m e n t a l s i t u a t i o n s i n o r d e r to a n a l y z e whether c e r t a i n a r i t h m e t i c b e h a v i o u r s formed a h i e r a r c h i c a l r e l a t i o n s h i p t o one a n o t h e r . One o f t h e s t a g e s o f t h a t h y p o t h e s i z e d h i e r a r c h y was d e s i g n e d t o measure concept o f the o r d i n a l p r o p e r t y o f numbers. It i s important to the note, however, t h a t the t a s k s she d e s i g n e d were n o n - v e r b a l t o remove language as a confounding f a c t o r i n the a n a l y s i s o f t h e r e s u l t s . Her h y p o t h e - s i z e d h i e r a r c h y was: (a) magnitude d i s c r i m i n a t i o n s o f s o l i d c o n t i n u o u s a r e a s , (b) magnitude d i s c r i m i n a t i o n s o f d i s c o n t i n u o u s d i s c r e t e (c) e q u i v a l e n c e o f s e t s ( o n e - t o - o n e areas, correspondence), (d) c o n s e r v a t i o n o f numbers ( P i a g e t i a n concept but tested non-verbally), (e) the O r d i n a l p r o p e r t y of numbers f o r example, 1 s t ; 2nd; 3 r d ; 15 ( f ) s e r i a t i o n ( r e c o g n i z i n g the o r d e r o f a s e r i e s o f o b j e c t s that a r e a r r a n g e d i n a s t e p - w i s e f a s h i o n ) , and (g) a d d i t i o n e q u a t i o n s (presented pictorially). The s u b j e c t s were 77 m i d d l e - c l a s s c h i l d r e n r a n g i n g from 3 years 0 months t o 4 years 11 months i n age, w i t h boys and g i r l s a p p r o x i m a t e l y equally represented. S i e g e l attempted t o v a l i d a t e her r e s u l t s by u s i n g the Guttman Scalogram A n a l y s i s t e c h n i q u e , but f a i l e d to o b t a i n the r e q u i r e d v a l u e o f .95 f o r the c o e f f i c i e n t o f r e p r o d u c i b i l i t y . a l l average c o e f f i c i e n t o f r e p r o d u c i b i l i t y was 0.89 and the s i n g l e c o e f f i c i e n t was 0 . 9 3 . Despite t h i s r e s u l t , h y p o t h e s i z e d h i e r a r c h y was o f some i n t e r e s t . The o v e r - highest the o r d e r i n g o f the There was no d i f f e r e n c e between the two s e r i e s o f t a s k s t h a t t e s t e d magnitude d i s c r i m i n a t i o n s , and s e r i a t i o n and a d d i t i o n were a t a p p r o x i m a t e l y the same l e v e l o f difficulty. O r d i n a t i o n o c c u r r e d i n the m i d d l e o f t h e h i e r a r c h y and was c l e a r l y much e a s i e r than s e r i a t i o n and a d d i t i o n . C o n s e r v a t i o n was h a r d e r than one-to-one correspondence but e a s i e r than s e r i a t i o n and addition. O v e r a l l , the p o s i t i o n o f the t a s k s i n r e l a t i o n to one another formed a b r a n c h i n g h i e r a r c h y r a t h e r than a l i n e a r h i e r a r c h y . C l e a r l y , i t would be advantageous to have some c l e a r s p e c i f i c a t i o n s o f c u r r i c u l a , or r a t i o n a l e f o r c u r r i c u l a r c h o i c e s , i n o r d e r t o p l a n more p r e c i s e l y f o r the e d u c a t i o n of p r e - s c h o o l c h i l d r e n . Resnick (1967, 1970) o u t l i n e d an o p e r a t i o n a l d e f i n i t i o n o f the number concept i n the form o f a s e t o f b e h a v i o u r s w h i c h , t a k e n t o g e t h e r , a l l o w a person t o c o n s i d e r the c h i l d t o have mastered the concept o f numbers: (a) o n e - t o - o n e correspondence t o 5, (b) o n e - t o - o n e correspondence to 10, ( c ) r e c o g n i z i n g the numerals from 0 t o 5, . 16 (d) r e c o g n i z i n g the numerals from 0 t o 1 0 , (e) comparison o f Sets ( c o n s e r v a t i o n o f numbers), ( f ) s e r i a t i o n and O r d i n a l p o s i t i o n ( o r g a n i z i n g m a t e r i a l i n o r d e r o f s i z e and d e s i g n a t i n g i t as 1 s t , 2nd, 3 r d , e t c . ) , and (g) a d d i t i o n and s u b t r a c t i o n E q u a t i o n s . Wong, R e s n i c k and Boozer (1971) examined the sequence i n which young c h i l d r e n a c q u i r e the elementary m a t h e m a t i c a l b e h a v i o u r s o f c o u n t i n g , one-to-one correspondence and r e c o g n i t i o n o f n u m e r a l s , h y p o t h e s i z i n g t h a t : (a) r e c o g n i t i o n o f numerals i s l e a r n e d e a r l y i n the p r o c e s s o f l e a r n i n g to c o u n t , (b) c o u n t i n g and r e c o g n i t i o n o f numerals f o r q u a n t i t i e s up t o five would n o r m a l l y be l e a r n e d b e f o r e e i t h e r c l a s s o f s k i l l s was l e a r n e d f o r q u a n t i t i e s up t o ten, ( c ) c o u n t i n g and r e c o g n i t i o n o f numerals a r e l e a r n e d with n e i t h e r c l a s s of tasks p r e r e q u i s i t e to the other, independently and (d) t h e r e i s a p s y c h o l o g i c a l independence of c o u n t i n g and o n e - t o one s k i l l s . The s u b j e c t s were 42 boys and 36 g i r l s between the ages o f 4 y e a r s 6 months and 6 y e a r s 0 months who were a t t e n d i n g k i n d e r g a r t e n . Sixty-three p e r c e n t of the c h i l d r e n were b l a c k and t h i r t y - s e v e n p e r c e n t were w h i t e ; p a r e n t s ' work s t a t u s ranged from unemployed t o e x e c u t i v e - p r o f e s s i o n a l . R e s u l t s i n d i c a t e d t h a t command over n u m e r a l s , numerals a c q u i r e d i n a r e g u l a r sequence, b e g i n n i n g w i t h p e r c e p t u a l matching o f the numerals and c o n c l u d i n g w i t h t h e a s s o c i a t i o n o f s e t s and n u m e r a l s , i s not o r d i n a r i l y learned u n t i l after c o u n t i n g o p e r a t i o n s f o r s e t s of the s i z e by the numerals a r e w e l l e s t a b l i s h e d . Data a l s o s u p p o r t e d p s y c h o l o g i c a l d i f f e r e n t i a t i o n o f c o u n t i n g and o n e - t o - o n e represented the correspondence. I m p l i c a t i o n s o f t h e R e l a t e d Research C o n s i d e r i n g e x t a n t r e s e a r c h i n f o r m a t i o n , i t i s p o s s i b l e t o argue t h a t a r i t h m e t i c b e h a v i o u r s can be arranged i n a h i e r a r c h i c a l manner. Whether i n f a c t t h e y can be arranged so t h a t i t i s p o s s i b l e to c l a i m t h a t c e r t a i n b e h a v i o u r s a r e p r e r e q u i s i t e t o o t h e r s i n forming a concept of numbers i s another m a t t e r . I t does appear r e a s o n a b l e , however, to h y p o t h e s i z e t h a t a d d i t i o n e q u a t i o n s would be a t the top o f a h i e r a r c h y which i n c l u d e d t h e f a c t o r s taken i n t o c o n s i d e r a t i o n by S i e g e l , Saxe, B r a i n e r d , Wong, R e s n i c k and B o o z e r . Resnick (1967, 1970) p l a c e d a d d i t i o n and s u b t r a c t i o n e q u a t i o n s a t the top o f her h y p o t h e s i z e d h i e r a r c h y and S i e g e l (1971) noted t h a t a d d i t i o n e q u a t i o n s presented p i c t o r i a l l y come a t the top o f a h i e r a r c h y a l o n g w i t h s e r i a t i o n . B r a i n a r d (1974) found t h a t o r d i n a l numbers appear to be grasped before c a r d i n a l numbers, S i e g e l (1971) t h a t c o n s e r v a t i o n appeared e a s i e r than s e r i a t i o n and a d d i t i o n , and Wong, R e s n i c k and Boozer (1971) t h a t counting establishes i t s e l f p r i o r t o the r e c o g n i t i o n o f n u m e r a l s . I t appears t h a t i t i s r e a s o n a b l e t o h y p o t h e s i z e a h i e r a r c h y o f arithmetic behaviours. I t seems u n l i k e l y , t h a t t h e s e b e h a v i o u r s form a l i n e a r r e l a t i o n s h i p to one a n o t h e r ; Resnick and Boozer (1971) comment on the h i e r a r c h i e s formed. both S i e g e l will (1971) and Wong, ' b r a n c h i n g ' n a t u r e o f the The Research Problem posed i n Chapter Two w i l l t a k e i n t o account e x i s t i n g r e s e a r c h i n v o l v i n g the concept of number, the b e h a v i o u r s which appear t o be p r e r e q u i s i t e to t h i s c o n c e p t , and t h e nature of h i e r a r c h i e s . 18 Chapter Two The Research Problem Chapter One has o u t l i n e d s i g n i f i c a n t r e s e a r c h by R e s n i c k (1970) and Wong, R e s n i c k and Boozer ( 1 9 7 1 ) , who attempted to present r a t i o n a l arguments f o r t e a c h i n g a s p e c i f i e d group o f a r i t h m e t i c b e h a v i o u r s as a b a s i s f o r d e v e l o p i n g a concept o f numbers. W h i l e R e s n i c k (1967, 1970) suggested a d e f i n i t i o n o f the number c o n c e p t , Wong, R e s n i c k and Boozer went f u r t h e r and d e t a i l e d a h y p o t h e s i z e d h i e r a r c h y o f a r i t h m e t i c b e h a v i o u r s i n a u s e f u l o r d e r f o r c l a s s r o o m t e a c h e r s at the K i n d e r g a r t e n and F i r s t Grade l e v e l . R e s u l t s of t h e i r study s u p p o r t e d the i d e a t h a t some a r i t h m e t i c b e h a v i o u r s are l e a r n e d i n a c l e a r l y d e f i n e d sequence, a l t h o u g h they were q u i c k t o p o i n t out t h a t some b e h a v i o u r s , such as c o u n t i n g and o n e - t o - o n e c o r r e s p o n d e n c e , independent o f one a n o t h e r . appeared to be p s y c h o l o g i c a l l y I f the i d e a o f a h i e r a r c h y i s tenable then i t i s u n l i k e l y t h a t i t would be s t r i c t l y l i n e a r w i t h one b e h a v i o u r l e a d i n g a u t o m a t i c a l l y t o the next most d i f f i c u l t behaviour i n a s t e p - wise f a s h i o n . R e s n i c k (1970) o u t l i n e d seven b e h a v i o u r s which c o u l d be c o n s i d e r e d as one p o s s i b l e o p e r a t i o n a l d e f i n i t i o n o f a number c o n c e p t : (a) o n e - t o - o n e correspondence t o 5, (b) o n e - t o - o n e correspondence t o 1 0 , ( c ) r e c o g n i z i n g the numerals 0 - 5 , (d) r e c o g n i z i n g the numerals 0-10, (e) comparison o f s e t s , ( f ) s e r i a t i o n and o r d i n a l p o s i t i o n , and (g) a d d i t i o n and s u b t r a c t i o n equations. 19 The focus o f t h i s s t u d y i s the mastery o f e q u a t i o n s . Other a s p e c t s of a number c o n c e p t , i n c l u d i n g f a c t o r s such as c o u n t i n g , c o n s e r v a t i o n o f discrete entities, and numeration a r e c o n s i d e r e d t o be p r e r e q u i s i t e s the b e h a v i o u r s i n v o l v e d i n s o l v i n g a d d i t i o n to equations. The r e s e a r c h problem under i n v e s t i g a t i o n i s whether a h i e r a r c h i c a l r e l a t i o n s h i p e x i s t s between c e r t a i n b a s i c number A t e m a j _ a d d i t i o n . The n a d d i t i o n number (li-emsfehosen were t h o s e between 6 and 9 ( f o r 5+1; 4+3; 6+2), example, r a t h e r than those between 1 and 5 i n o r d e r ensure a s u f f i c i e n t degree o f d i f f i c u l t y f o r the s u b j e c t s . s i z e d h i e r a r c h y t o be c o n s i d e r e d i n c l u d e s f o u r d i f f e r e n t to The h y p o t h e conceptual l e v e l s - C o n c r e t e , S e m i - C o n c r e t e , A b s t r a c t - C o n c r e t e and A b s t r a c t . terms a r e d e s c r i b e d on the f o l l o w i n g These pages. C o n c r e t e - A b s t r a c t Continuum C o n c r e t e - A b s t r a c t can be d e f i n e d a l o n g a r e a l e n t i t y - a b s t r a c t e n t i t y continuum, where r e a l e n t i t i e s are a c t u a l p a r t s of the e n v i r o n ment and a b s t r a c t e n t i t i e s a r e numeric r e p r e s e n t a t i o n s is of i t . (An e n t i t y a n y t h i n g to which a numeral can be a s s i g n e d such t h a t i t can be designated quantitively). Concrete l e v e l . t a s k s . C o n c r e t e t a s k s w i l l use o b j e c t s environment which can both be seen and p i c k e d up. from the As a consequence they w i l l be u n l i k e the o t h e r l e v e l s t o be d e s c r i b e d i n t h a t they do not r e p r e s e n t r e a l i t y ; they are i n f a c t a p a r t o f r e a l i t y i t s e l f . crayons and b l o c k s w i l l be used a t the c o n c r e t e l e v e l o f the Semi-concrete l e v e l t a s k s . Semi-concrete l e v e l tasks h y p o t h e s i z e d t o be more a b s t r a c t than the c o n c r e t e t a s k s Both hierarchy. are because p i c t u r e s of o b j e c t s are used t o r e p r e s e n t the r e a l o b j e c t s . The p i c t u r e s seen are o f s h a p e s , f i s h and m a t c h s t i c k s and r e q u i r e the child t o a b s t r a c t at a d i f f e r e n t Level Concret Tasks. Concrete-Abstract symbols i n combination l e v e l tasks. with to be more a b s t r a c t than any c o g n i t i v e l e v e l to t h a t o u t l i n e d under T a l l y Marks. These t a s k s are hypothesized In the p r e v i o u s l e v e l particular not o b j e c t or o b j e c t s i n the environment. of symbols i s hypothesized use (Semi-Concrete) were used whereas a t t h i s l e v e l the t a l l y marks do r e p r e s e n t any of the Semi-Concrete t a s k s i n t h a t they do not r e p r e s e n t a t i o n of r e a l i t y . matchsticks These t a s k s i n v o l v e the use The use to i n c r e a s e the degree of a b s t r a c t n e s s . However, i n t h i s combination the a b s t r a c t n e s s i s decreased t h a t the t a l l y marks can s t i l l to the extent f u n c t i o n as a c o n c r e t e a i d i n s o l v i n g an equation. Abstract l e v e l tasks. A b s t r a c t t a s k s are those i n which numerical symbols a r e used by themselves to r e p r e s e n t e n t i t i e s w i t h i n the environ- ment. Task L e v e l s Some l e v e l s of t h e . h i e r a r c h y w i l l as mentioned above. distinguish These w i l l the h y p o t h e s i z e d w i t h i n the f o u r c o n c e p t u a l task l e v e l s - one levels. Concrete t h e r e i s o n l y one just At the Concrete a t h i r d matchsticks. l e v e l there are - as a At the Semi- s e t s of t a s k s - one uses shapes, At the C o n c r e t e - A b s t r a c t s e t of t a s k s which uses the t a l l y marks, and A b s t r a c t l e v e l t h e r e i s only one to levels the o t h e r , crayons l e v e l of the h i e r a r c h y . tasks described. h i e r a r c h y with seven task l e v e l t h e r e are t h r e e d i f f e r e n t a second f i s h , and levels which uses b l o c k s , and means t o a s s e s s the c o n c r e t e s e t s of be c a l l e d Task L e v e l s i n o r d e r them from the f o u r c o n c e p t u a l Figure 1 presents two contain d i f f e r e n t at s e t of t a s k s which uses numerals. level the 21 D e f i n i t i o n o f Terms The f o l l o w i n g d e f i n i t i o n s a r e e s s e n t i a l to c l a r i f y the d e s i g n procedures, r e s u l t s and c o n c l u s i o n s o f t h i s F i n a l Task: study: A d d i t i o n o f two one d i g i t numerals whose sum i s no g r e a t e r than n i n e . C r i t e r i o n f o r Performance: C o m p l e t i o n o f the t a s k w i t h i n l i m i t s o f the s t a n d a r d i z e d i n s t r u c t i o n s Appendix D ) . (Instructions I f e r r o r o c c u r s , the s u b j e c t c o r r e c t i o n without the for Teaching, has t o i n i t i a t e and complete assistance. Instructional Objective: S u c c e s s f u l c o m p l e t i o n of the F i n a l Task a t the h i g h e s t l e v e l o f t h e h i e r a r c h y ; t h a t i s , a t the most a b s t r a c t level. Entering Behaviours: Those a r i t h m e t i c b e h a v i o u r s which are c o n s i d e r e d p r e r e q u i s i t e t o the u n d e r s t a n d i n g o f b a s i c number i t e m s , and which a l o n g w i t h a demonstrated i l l u s t r a t i v e o f an u n d e r s t a n d i n g mastery of b a s i c a d d i t i o n i t e m s of the concept o f numerals as i n g e n t i t i e s w i t h i n the environment which can be q u a n t i t a t i v e l y Learning Hierarchy: are representdesignated. The r u l e or problem s o l v i n g t a s k i s a n a l y z e d i n t o s i m p l e c a p a b i l i t i e s to be l e a r n e d as p r e r e q u i s i t e s . When such an a n a l y s i s i s c o n t i n u e d p r o g r e s s i v e l y to the p o i n t of d e l i n e a t i n g an e n t i r e s e t o f c a p a b i l i t i e s h a v i n g an o r d e r e d r e l a t i o n s h i p to each then a L e a r n i n g h i e r a r c h y e x i s t s Task a n a l y s i s : (Gagne 1 9 7 0 ) . The p r o c e s s o f i s o l a t i n g , d e s c r i b i n g and i n g a l l the n e c e s s a r y s u b - t a s k s , other, w h i c h , when they a r e m a s t e r e d , enable s u c c e s s f u l mastery o f the i n s t r u c t i o n a l o b j e c t i v e . sequencwill (Bateman 1 9 7 1 ) . 22 Level 4 Abstract Level 3 Abstract-Concrete < Level 2 Semi-Concrete Level 1 Concrete Figure 1 The Seven H y p o t h e s i z e d a Four-Level Hierarchy Test Levels Arranged in Research O b j e c t i v e s 1. To a s c e r t a i n whether a l i n e a r r e l a t i o n s h i p , or a b r a n c h i n g r e l a t i o n s h i p , e x i s t s between the l e v e l s o f t h e h y p o t h e s i z e d h i e r a r c h y . 2. To a s c e r t a i n whether t a s k s h y p o t h e s i z e d to be a t a g i v e n l e v e l of the h i e r a r c h y a r e o f e q u a l d i f f i c u l t y r e l a t i o n s h i p to one 3. and remain i n the same another. To a s c e r t a i n whether any t a s k s w i t h i n any l e v e l o f the chy are o f any g r e a t e r or l e s s e r d i f f i c u l t y t h a t t h e i r r e l a t i o n s h i p t o one another 4. 5. than any o t h e r t a s k s , o f boys and difference girls. To a s c e r t a i n whether any o f the b a s i c number f a c t between 6 and 9 a r e any more d i f f i c u l t than any o t h e r combinations combinations following t h e i r i n i t i a l balancing, according to established 6. such becomes l i n e a r . To a s c e r t a i n whether t h e r e i s any s i g n i f i c a n t between the performances hierar- criteria. To a s c e r t a i n whether any r e l a t i o n s h i p e x i s t s between l e v e l s o f the h y p o t h e s i z e d h i e r a r c h y and the t a s k l e v e l s o f the the hierar- c h y , such t h a t i t i s p o s s i b l e t o say t h a t a h i e r a r c h y e x i s t s which moves from the C o n c r e t e l e v e l t o the A b s t r a c t l e v e l . Target P o p u l a t i o n I n i t i a l l y u s i n g a Grade One p o p u l a t i o n was c o n s i d e r e d but became readi%yapparent designed, too easy. t h a t t h e s e c h i l d r e n would f i n d the t a s k s , it as A l s o , u n l e s s t h e data c o u l d be c o l l e c t e d immediate l y the s t u d e n t s e n t e r e d Grade One, the r e s u l t s would be confounded by t h f a c t t h a t the s t u d e n t s would have been exposed t o a r i t h m e t i c teaching. Thus, K i n d e r g a r t e n c h i l d r e n were s e l e c t e d as the most s u i t a b l e group use i n t h i s study. to However, g i v e n the f a c t o r s d i s c u s s e d to developmental v a r i a b l e s , i t was i n Chapter One necessary p o p u l a t i o n so as t o i n c l u d e o n l y those with to d e l i m i t the c h i l d r e n who l e v e l of c o g n i t i v e development i n mathematics, and had subject a sufficient to exclude would a l r e a d y be a b l e to answer a l l of the q u e s t i o n s regard those t h a t would arise from the seven l e v e l s of the hypothesized hierarchy. a p r e t e s t and the P r e t e s t to e s t a b l i s h a pilot t e s t were designed; whether the s u b j e c t s had the necessary Test to e s t a b l i s h whether any L e v e l One of the h i e r a r c h y To e f f e c t E n t e r i n g Behaviours and who this the Pilot of the s u b j e c t s would f i n d the t a s k s at ( A b s t r a c t ) too easy. Chapter Three Methodology In o r d e r t o i n v e s t i g a t e t h e r e s e a r c h Chapter Two i t was necessary standardized questions specified i n t o c o n t r o l f a c t o r s such as item difficulty, t e s t i n g and t e a c h i n g i n s t r u c t i o n s , item p r e s e n t a t i o n i n a t e s t i n g s i t u a t i o n , and present b e f o r e attempting knowledge o f a r i t h m e t i c behaviours, t o i n t e r p r e t data from each l e v e l o f t h e h i e r a r c h y . I t i s important t o f i n d out whether the items s e l e c t e d a r e s u i t a b l e f o r t h e s u b j e c t s chosen. Standardizing i n s t r u c t i o n s allows each s u b j e c t an equal chance t o l e a r n i n t h e t e a c h i n g s i t u a t i o n and an equal chance t o respond i n t h e t e s t i n g s i t u a t i o n ; randomizing the t e s t i n g procedures e s t a b l i s h e s a c o n t r o l t o d e a l with unequal o p p o r t u n i t i e s t o l e a r n when a c t u a l l y being it i s a l s o important tested. P r i o r t o t h e commencement o f t e a c h i n g t o a s s e s s whichcschildren a r e not developmentally ready t o b e n e f i t from the t e a c h i n g , and which c h i l d r e n have a l r e a d y a c q u i r e d t h e knowledge t h a t i s p r e s e n t e d . Measuring Instrument As i n d i c a t e d p r e v i o u s l y , t h e items s e l e c t e d were t h e 26 b a s i c a d d i t i o n .items whose sums were no l e s s than s i x , and no g r e a t e r than (Appendix A ) . nine To f a c i l i t a t e c o n t r o l o f item d i f f i c u l t y and t o a l l o w t h e c o n s t r u c t i o n o f a measuring instrument which would c o n t r o l f o r f a c t o r s t h a t would a f f e c t t h e t e s t r e s u l t s , f i v e o f these ftemsxwere randomly e l i m i n a t e d , l e a v i n g 21 f a c t s t o be i n c o r p o r a t e d i n t o t h e t e s t i n g and t e a c h i n g procedures. The f i v e items randomly e l i m i n a t e d were: 26 5 + 1 7 + 1 2 + 7 3 + 6 4+5 These f a c t s were used t o e s t a b l i s h whether any o f the s u b j e c t s a l r e a d y answer the a b s t r a c t could q u e s t i o n s a t l e v e l f o u r of the h i e r a r c h y . Pretest One hundred t w e n t y - f i v e s u b j e c t s were a d m i n i s t e r e d a p r e t e s t i n o r d e r t o e s t a b l i s h whether t h e s u b j e c t s had t h e minimum E n t e r i n g B e h a v i o u r s , and to a s c e r t a i n how many s u b j e c t s had a l r e a d y a c q u i r e d the I n s t r u c t i o n a l O b j e c t i v e o f the h y p o t h e s i z e d h i e r a r c h y . Those c h i l d r e n who d i d not have the s p e c i f i e d E n t e r i n g B e h a v i o u r s , and those c h i l d r e n who reached the I n s t r u c t i o n a l O b j e c t i v e , were then e x c l u d e d from the s u b j e c t p o o l . First, There were t h r e e s p e c i f i e d E n t e r i n g B e h a v i o u r s . the s u b j e c t had t o i d e n t i f y v e r b a l l y numerals on c a r d s presented i n a random o r d e r . Second, each s u b j e c t had t o r e c i t e t h e numerals from one t o n i n e . T h i r d , t h e s u b j e c t was r e q u i r e d to count a group o f s i x and then f o u r randomly a r r a n g e d b l o c k s (see Appendix C ) . The f i v e randomly e l i m i n a t e d i t e m s were used t o a s s e s s whether any c h i l d could a l r e a d y r e a c h the I n s t r u c t i o n a l O b j e c t i v e at l e v e l f o u r o f the h i e r a r c h y . Any c h i l d r e s p o n d i n g t o f o u r or f i v e problems c o r r e c t l y w i t h o u t the use of a i d s such as f i n g e r s , paper and p e n c i l , was e x c l u d e d from the pool. Of the 125 s u b j e c t s a s s e s s e d , subject 49 met the r e s e a r c h c r i t e r i a . (For d e t a i l e d i n f o r m a t i o n , see Chapter F o u r ) . Pilot Test The 3 girls) pilot who met study. This subjects in to Test was the research involved order to would questions see at Chapter be giving a random very Level the of the this study but subjects did not of subjects the test, hierarchy. B) and who that (For (4 boys participate (Appendix instructions sample the seven posttest s u c c e s s f u l on Four to criteria standardize a s c e r t a i n whether criteria administered to and in the these procedures, met the i s , and research complete detailed information,, Four). Subjects Subjects comprising for both Squamish School criteria were take part randomly Item in District. used the 21 in were Seven the Pilot study. items rows cell criteria of of chosen from the subjects Test, From t h i s according the the design in items 1. An each equal difficulty. 2. Size addition of size made to and 1933; 49 leaving group row their thus items 14 public 42 schools who met subjects boys and Classes 14 the in the research eligible girls to were the measuring degree of equalize into difficulty problem Vogel three instrument order difficulty.. 1928; levels in were Wheeler of 1939) difficulty arranged to balance The for with were: combination addend is of sum. the up Washburne; twenty-one seven the which (Machatchy arranging than girls children. Difficulty three each and kindergarten selected. The in boys were the and its 'reverse' principal indicator form of tend to be difficulty, of rather 3. Combinations with a common addend appear but not e q u a l d i f f i c u l t y (4+4; to be of s i m i l a r 3+3). 4. The doubles i n a d d i t i o n and those i n which 1 i s added with a g r e a t e r number appear t o be e a s i e s t i n a d d i t i o n . A c c o r d i n g t o these c r i t e r i a , the items can be c l a s s i f i e d A : Easy B : Moderaly C : Hard^ as f o l l o w s : items Difficult items items These 21 items were then d i v i d e d i n t o seven groups o f t h r e e ; a c c o r d i n g t o the c r i t e r i a o u t l i n e d , a l l the problems i n column A would be the e a s i e s t , those i n column B more d i f f i c u l t , and those i n column C the h a r d e s t . Table 1 p r e s e n t s r e s u l t s of t h i s c l a s s i f i c a t i o n ; a l l items w i t h i n each of the columns a r e c o n s i d e r e d to be o f a p p r o x i m a t e l y the same degree o f difficulty. Table 1 Items Arranged by L e v e l of Easy A Items 1 "hard 1 Difficulty B Difficult Moderately Items . Hard C Items 1 + 5 8 + 1 2 + 6 1 + 6 4 + 3 6 + 2 1 + 7 3 + 4 7 + 2 1 + 8 2 + 5 6 + 3 6 + 1 5 + 2 5 + 4 3 + 3 2 + 4 5 + 3 4 + 4 4 + 2 3 + 5 i s used synonomously with the word V 'difficult'. To c o n t r o l f o r item d i f f i c u l t y , these 21 items were then d i v i d e d i n t o seven groups of t h r e e with one from A, one The item from B, and one item i n each group randomly s e l e c t e d from C u n t i l a l l the items were exhausted. groups y i e l d e d by t h i s procedure are shown i n Table Table 2. 2 Items Grouped to E q u a l i z e D i f f i c u l t y of Sets of Three Name of Item Group A Easy litems g Moderately D i f f i c u l t Items C Hard Items a 1 + 5 3 + 3 2 + 6 b 4 + 4 4 + 3 7+2 c 3 + 3 4+2 d 1 + 7 8 + 1 6+2 e 1 + 8 2 + 4 5 + 3 f 6 + 1 5+2 3 + 5 g 1 + 6 2+5 5 + 4 6 + 3 T e s t i n g Sequence Grouping the items to c o n t r o l f o r t h e i r d i f f i c u l t y made p o s s i b l e the c r e a t i o n of a t e s t i n g sequence f o r a s s e s s i n g responses t o the t e a c h i n g created session. by employing a G r e c o - L a t i n The the subjects' t e s t i n g sequence i t s e l f Square d e s i g n that involved placing the 28 s u b j e c t s a t seven d i f f e r e n t l e v e l s , f o u r s u b j e c t s per L e v e l one, f o r i n s t a n c e , would i n v o l v e the f i r s t answering q u e s t i o n s indicated. Table at the hypothesized was level. four subjects i n h i e r a r c h y l e v e l and item 3 shows the l e v e l s of the h i e r a r c h y combined set with the re-grouped i t e m s . Looking task l e v e l i n the h y p o t h e s i z e d t h r e e items 1+8; 2 + 4; at ( f o r example) 6e, h i e r a r c h y and 5 + 3 (see Table Table "6" would be the 'e' would be the group of 2). 3 T e s t i n g Sequence as Determined by a Greco-Latin T-.ask Level Square Subjects Sequence 1 1 - 4 *6e 4b 2f 7c 5g 3d la 2 5 - 8 7f 5c 3g Id 6a 4e 2b 3 9 - 12 ig 6d 4a 2e 7b 5f 3c 4 13 - 16 2a 7e 5b 3f lc 6g 4d 5 17 - 20 3b If 6c 4g 2d 7a 5c 6 21 - .24 4c 2g 7d 5a 3e lb 6f 7 25 - 5d 3a le 6b 4f 2c 7g 28 * i n each combination the number r e f e r s to the l e v e l of the h i e r a r c h y ( F i g . 1) and the l e t t e r r e f e r s to the item s e t (Table 2) The pilot format f o r q u e s t i o n i n g the subjects^was f o r m a l i z e d when the t e s t was administered. Each s u b j e c t was administered three q u e s t i o n s a t each task l e v e l thus r e q u i r i n g a t o t a l of 21 q u e s t i o n s to be answered by each s u b j e c t . the (For d e t a i l e d i n f o r m a t i o n c o n c e r n i n g T e s t i n g I n s t r u c t i o n s see Appendix B ) . 31 Teaching P r o c e d u r e s P r i o r t o the t e s t i n g s e s s i o n each s u b j e c t was taught t h r e e items at each Task L e v e l o f the h y p o t h e s i z e d h i e r a r c h y ; i n s t r u c t i o n was g i v e n t o seven groups of c h i l d r e n , each group composed o f two boys and two g i r l s . Each group l e a r n e d t h r e e problems i n v o l v i n g b l o c k s , t h r e e i n v o l v i n g pictures of f i s h , three i n v o l v i n g shapes, three i n v o l v i n g matchsticks, t h r e e i n v o l v i n g numerals and t a l l y marks, and t h r e e i n v o l v i n g o n l y numerals. In o r d e r t o d e c i d e which t h r e e items were t o be taught a t each l e v e l of the h y p o t h e s i z e d h i e r a r c h y , the f o l l o w i n g row was chosen a t random from a 7 by 7 L a t i n Square. 3, 1, The r e s u l t i n g sequence was: 5, 6, 4, 7, 2. T h i s row was then matched w i t h t h e items as grouped i n T a b l e 2 i n such a way t h a t the " a " group o f items (1 + 5; 3+3; 2+6) was matched w i t h the number 5, t h e " b " group o f problems w i t h number 6 and so o n . R e s u l t s o f t h i s procedure are p r e s e n t e d i n T a b l e 4 . Table 4 Item Set and i t s Matched Level i n the H i e r a r c h y Item Set Name o f Item Group Easy Items Moderately Difficult Items Hard Items Hierarchy Level 5 a 1 + 5 3 + 3 2 + 6 1 6 b 4 + 4 4 + 3 7 + 2 2 4 c 3 + 3 4 + 2 6 + 3 3 7 d 1 + 7 8 + 1 6 + 2 4 3 e 1 + 8 2 + 4 5 + 3 5 1 f 6 + 1 5 + 2 3 + 5 6 2 g 1 + 6 2 + 5 5 + 4 7 The h y p o t h e s i z e d h i e r a r c h y from l e v e l 7 t o l e v e l 1 i s shown i n F i g u r e 2. NUMERALS L e v e l 4' Abstract NUMERALS AND Level 3 TALLY MARKS Abstract-Concrete X SHAPES FISH Level 2 Semi-Concrete MATCHSTICKS 3 4 5 X = 1 jr BLOCKS CRAYONS Level 1 6 Concrete F i g u r e 2. H y p o t h e s i z e d H i e r a r c h y f o r A d d i t i o n F a c t s Showing the C o n c e p t u a l L e v e l s 1 t o 4 and Task L e v e l s 1 t o 7 F i g u r e 2 shows the seven Task L e v e l s o f the h i e r a r c h y . ing items were p r e s e n t e d t o the s u b j e c t s a t Task L e v e l seven f i r s t , Task L e v e l s i x , t h e n Task L e v e l f i v e and so o n , u n t i l a l l the had The t e a c h - been exposed to twenty-one p r o b l e m s . then subjects A t the C o n c r e t e l e v e l - t a s k l e v e l s s i x and seven - two d i f f e r e n t s e t s o f c o n c r e t e t a s k s were g i v e n and a t the S e m i - C o n c r e t e l e v e l t h r e e d i f f e r e n t s e t s of t a s k s were presented, t o a s s e s s whether a s e t o f t a s k s a t the same l e v e l posed any more d i f f i c u l t y that l e v e l (crayons). one i s , i n f a c t , (blocks) f o r s u b j e c t s than another s e t o f t a s k s at The numbering o f t h e t a s k l e v e l s from seven to d e s i g n e d t o a s c e r t a i n (1) whether t h e h i g h e r numbered task l e v e l s are e a s i e r than the lower numbered task l e v e l s , and (2) whether a l i n e a r r e l a t i o n s h i p e x i s t s between these task l e v e l s as from a branching relationship. I t i s p o s s i b l e to c o n c e i v e , f o r i n s t a n c e , t h a t problems i n v o l v i n g " M a t c h s t i c k s " are more a b s t r a c t than involving "Fish". are hypothesized level. The problems " M a t c h s t i c k s " c l o s e l y resemble " T a l l y Marks" which to be more a b s t r a c t than any t a s k s on the Semi-Concrete same r a t i o n a l e a p p l i e s to comparisons between " F i s h " "Shapes" (shapes may s t i c k s may distinct be more a b s t r a c t ) , M a t c h s t i c k s be more a b s t r a c t ) and " B l o c k s " and and and Shapes (match- "Crayons" ( b l o c k s may be more a b s t r a c t ) . The i n s t r u c t i o n s f o r the t e a c h i n g s e s s i o n were s t a n d a r d i z e d so t h a t no s u b j e c t s gained any advantage over any s u b j e c t s were s e t t l e d , r e l a x e d and other s u b j e c t . Once the a t t e n d i n g , i n s t r u c t i o n s were i d e n t i c a l f o r each of the groups ( f o r d e t a i l s of these i n s t r u c t i o n s see Appendix The o n l y v a r i a n c e to the procedures support and D). as o u t l i n e d i n Appendix D concerned encouragement of the s u b j e c t s , and praise for correct responses. Statistical Analysis An a n a l y s i s of v a r i a n c e was c a r r i e d out i n order to c o n s i d e r : (a) p o s s i b l e d i f f e r e n c e s i n the performance of the items when boys a r e compared with girls, (b) p o s s i b l e v a r i a n c e s i n the d i f f i c u l t y when compared with other (c) criteria, and been balanced items items, p o s s i b l e v a r i a n c e s i n the d i f f i c u l t y a f t e r having l e v e l s of c e r t a i n according l e v e l s of groups of problems to the l e v e l s of difficulty 35 (d) p o s s i b l e v a r i a n c e s of d i f f i c u l t y l e v e l between i n d i v i d u a l problems w i t h i n any group, which a l t e r s t h e balance o f the group o f problems. A d e s c r i p t i v e a n a l y s i s was undertaken t o e s t a b l i s h whether: (a) t h e h i e r a r c h y e x h i b i t s evidence o f a l i n e a r r e l a t i o n s h i p between one L e v e l and another, and one Task L e v e l and another, or a branching r e l a t i o n s h i p between these L e v e l s , (b) the t a s k s h y p o t h e s i z e d remained a t t h a t t o be a t any one L e v e l o f t h e h i e r a r c h y level, (c) t h e degree o f d i f f i c u l t y o f a task a t t h e Concrete or Semi- Concrete L e v e l s i s any harder than any other task a t the same L e v e l , and (d) any r e l a t i o n s h i p e x i s t s between the L e v e l s o f t h e hypothesized h i e r a r c h y , and t h e Task L e v e l s o f t h e h i e r a r c h y , such t h a t i t i s p o s s i b l e t o say t h a t a h i e r a r c h y e x i s t s which moved from t h e Concrete t o the A b s t r a c t Level. A Guttman Scalogram A n a l y s i s was e f f e c t e d t o attempt a s t a t i s t i c a l v a l i d a t i o n of the h i e r a r c h y . to t h e hypothesized T h i s t e c h n i q u e analyzes Levels of the hierarchy, according L e v e l s , and a c c o r d i n g the data according t o the Task t o t h e number o f items c o r r e c t l y answered by s u b j e c t s a t any g i v e n L e v e l or Task L e v e l . 36 Chapter Four R e s u l t s o f the P r e t e s t and P i l o t T h i s c h a p t e r w i l l p r e s e n t the r e s u l t s t e s t and d i s c u s s t h e s e r e s u l t s Test o f the p r e t e s t and t h e . p i l o t and t h e i r p o s s i b l e i m p l i c a t i o n s . The p r e t e s t , as mentioned i n Chapter T h r e e , was designed t o f i n d out i f subjects the had the minimum E n t e r i n g B e h a v i o u r s as w e l l as how many c h i l d r e n had a l r e a d y a c q u i r e d the I n s t r u c t i o n a l O b j e c t i v e of the hierarchy. hypothesized The two t e s t s i d e n t i f i e d a p o o l o f s u b j e c t s who would be a t a p p r o x i m a t e l y the same c o n c e p t u a l l e v e l as f a r as t h e i r o f numbers was c o n c e r n e d . The o r i g i n a l s u b j e c t understanding p o o l encompassed a l l o f the k i n d e r g a r t e n c h i l d r e n i n Squamish, a t o t a l of 125. Table 5 o u t l i n e s the breakdown o f t h e s e c h i l d r e n by s c h o o l and by s e x . Table 5 Number o f S u b j e c t s P a r t i c i p a t i n g i n the School .Boys . . Pretest Girls Total 1 8 12 20 2 11 15 26 3 25 15 40 4 19 20 _39 Total 63 62 125 The was to results a s s e s s e d as the of having pretest either established criteria. Behaviours (Conservation and Counting the Instructional Table the these from are 1-9) of and Objective designated are passed The Number the presented (P) or results with five (highest show Blocks, level Table failed problems 'Additional in of both an the item the were at subject according Entering Recognizing that Problems (F) Each 6. Numerals, the hierarchy). level In of the Table 6 Pretest C o n s e r v a t i o n of Number W i t h Blocks School Boys Recognizing Numerals Girls Boys Results # Additional Problems Counting 1-9 Girls Boys Girls Boys Total Girls Boys F P F P F P F P F P F P F P F P F 1. 1 7 2 10 4 4 5 7 1 7 2 10 8 0 11 1 14 2. 3 8 5 10 5 6 8 7 2 9 4 11 10 1 15 0 3. 8 17 4 11 16 9 4 11 2 23 2 13 22 3 15 4. _5 14 _6 14 15 _4 10 10 3 16 1 19 19 0 Total 17 46 17 45 40 23 27 35 8 55 9 53 59 4 F = Failed Girls F P 18 20 28 20 24 32 28 0 48 52 25 35 19 1 42 34 36 44 60 2 124 128 113 135 P P = Passed CO An a n a l y s i s of the r e s u l t s presented i n T a b l e 6 showed t h a t 49 s u b j e c t s s u c c e s s f u l l y met the r e s e a r c h c r i t e r i a . s u b j e c t s a c c o r d i n g to s c h o o l i s p r e s e n t e d i n Table A summaryffof these 7. Table 7 D i s t r i b u t i o n over Sex, Schools and T e s t i n g Times of S u b j e c t s Who Met S c r e e n i n g C r i t e r i a Boys School 1 Morning Girls Afternoon 4 Morning Afternoon 6 Total 10 2 3 2 3 4 12 3 4 2 9 0 15 4 _4 0 _5 3 12 Total 15 4 23 7 49 40 Of the o r i g i n a l 125 s u b j e c t s , 76 were not s u i t a b l e a c c o r d i n g t o the e s t a b l i s h e d c r i t e r i a . the f i r s t Table 8 shows where s u b j e c t s f a i l e d i n three sub-tests of the P r e t e s t . Table 8 S u b j e c t s Who F a i l e d the P r e t e s t I l l u s t r a t e d According to Pretest Category CATEGORY BOYS Conservation Recognizing Counting o f Number with Numerals Blocks 1 - 9 1-9 Total Most s u b j e c t s who f a i l e d of 17 17 40 27 _8 _7 65 51 t h e p r e t e s t were unable t h e numerals from 1 t o 9 when v i s u a l l y presented t h r e e c h i l d r e n who f a i l e d GIRLS to recognize a l l t o them. the t e s t with t h e b l o c k s passed which demanded t h e r e c o g n i t i o n o f numerals. Counting Only the t e s t from 1 t o 9 proved to be o f no a d d i t i o n a l v a l u e i n the p r e t e s t : a l l the c h i l d r e n who failed the c o u n t i n g s u b - t e s t a l s o f a i l e d e i t h e r the s u b - t e s t on b l o c k s or the s u b - t e s t on numeral r e c o g n i t i o n . I t i s i n t e r e s t i n g t o note t h a t f a r more boys than g i r l s f a i l e d t h e s u b - t e s t on number r e c o g n i t i o n ; on t h e other two s u b - t e s t s t h e r e were no s i g n i f i c a n t d i f f e r e n c e s . Those s u b j e c t s who s u c c e s s f u l l y completed the f i r s t three pretests were a d m i n i s t e r e d a f o u r t h t e s t t o f i n d out i f they c o u l d a l r e a d y solve the problems t h a t were t o be placed a t the t o p o f t h e h y p o t h e s i z e d hierarchy. S i x o f t h e s u b j e c t s were a b l e t o c o r r e c t l y answer s u f f i c i e n t problems - f o u r or more - t o be excluded from t h e r e s e a r c h p o o l . Of these s u b j e c t s f o u r were boys and two were g i r l s . Pilot Study The pilot study was c a r r i e d out by a d m i n i s t e r i n g one row o f t h e f i n a l t e s t t o f o u r g i r l s and t h r e e boys randomly s e l e c t e d from t h e research pool. The major purpose f o r t h e p i l o t t e s t was t o a s c e r t a i n whether the s u b j e c t s c o u l d answer most o f t h e q u e s t i o n s on the P o s t t e s t . s u b j e c t s c o u l d answer a s i g n i f i c a n t number o f q u e s t i o n s i n g , then i t was q u i t e l i k e l y I f the p r i o r to teach- t h a t t h e pool o f s u b j e c t s s e l e c t e d f o r the study would a l s o be a b l e t o answer the q u e s t i o n s . A look a t the s t r u c t u r e o f t h e t e s t shows t h e t o t a l p o s s i b l e a t any g i v e n l e v e l o f t h e t e s t was 3 p o i n t s , and the maximum p o s s i b l e s c o r e f o r any s u b j e c t was 21 p o i n t s . None o f t h e seven c h i l d r e n was s u c c e s s f u l on any o f t h e items a t l e v e l s one and two of the h i e r a r c h y ; f i v e o f the seven c h i l d r e n d i d not s c o r e a t l e v e l t h r e e ; t h r e e c h i l d r e n d i d not s c o r e a t l e v e l s f o u r and f i v e ; and two c h i l d r e n d i d not s c o r e a t l e v e l s s i x and seven. Most success was achieved a t the Concrete l e v e l o f t h e h i e r a r c h y , but even here the s u b j e c t s were a b l e t o s c o r e o n l y 8 and 11 p o i n t s r e s p e c t i v e l y - o n l y a f r a c t i o n o f t h e maximum p o s s i b l e o f 21. I f t h e s c o r e s a t each l e v e l o f t h e h i e r a r c h y a r e averaged t h e r e s u l t s would be 0, 0, 5.66 and 9.5; a s c o r e o f 10.5 would be r e q u i r e d t o a c h i e v e a 50 percent l e v e l o f accuracy. One boy and one g i r l scored w e l l on l e v e l s 3 , 4 , 5 , 6 and 7 . subjects, With the e x c e p t i o n o f t h e s e two the most common s c o r e was 0 and 1 . Table 9 , p r o v i d e s u p p o r t on the independent The r e s u l t s , outlined in f o r t h e b e l i e f t h a t t h e r e i s room f o r variance variables. Table 9 R e s u l t s o f the P i l o t Study A c c o r d i n g t o H i e r a r c h y L e v e l TASK LEVEL OF THE HIERARCHY SUBJECTS 1 7 6 5 4 3 2 1 TOTAL 0 1 0 0 0 0 0 1 2 2 2 1 2 0 0 9 1 0 1 1 0 0 0 3 0 3 1 0 0 0 0 4 2 3 2 3 3 0 0 13 1 0 0 1 0 0 0 2 2 2 0 a 0 0 0 _4 8 11 6 6 5 0 0 36 (girl) 2 (girl) 3 (girl) 4 (boy) 5 (boy) 6 (girl) 7 (boy) Total The maximum t o t a l p o s s i b l e i n each c e l l o f t h e t e s t .design"; was 3 . 0 43! I m p l i c a t i o n s o f the P r e t e s t and P i l o t Test f o r t h e Main Study S e v e n t y - s i x s u b j e c t s were r e j e c t e d because they e i t h e r d i d not have the d e s i r e d E n t e r i n g B e h a v i o u r s or because they c o u l d a l r e a d y answer the q u e s t i o n s a t l e v e l one o f the h i e r a r c h y , r e d u c i n g the s u b j e c t from 125 t o 4 9 . T h i r t y o f the 49 c h i l d r e n who met the c r i t e r i a were g i r l s , and n i n e t e e n were b o y s . pool research The f a c t t h a t more g i r l s met the r e s e a r c h c r i t e r i a may be s i g n i f i c a n t , but was not a f a c t o r i n t h i s study because 14 boys and 14 g i r l s were chosen a t random to form the f i n a l p o o l o f subjects. R e s u l t s o f t h e i g p i l o t t e s t l e n d some s u p p o r t to the i d e a o f a C o n c r e t e - A b s t r a c t continuum u s i n g a r i t h m e t i c f a c t s between 6 and 9. With the e x c e p t i o n o f l e v e l s s i x and seven t h e r e appeared to be a l i n e a r p r o g r e s s i o n from the C o n c r e t e t o the A b s t r a c t l e v e l of the h i e r a r c h y . The s u b j e c t s c l e a r l y found the more c o n c r e t e t a s k s e a s i e r than the most abstract tasks. However, i f t h e r e s u l t s a r e averaged a c c o r d i n g t o the f o u r l e v e l s h y p o t h e s i z e d , the h i e r a r c h y appears even more c l e a r l y (9.5, 5.66, 0.0). T h i s data suggest t h a t no d i f f e r e n c e s a r e likely t o be found between problems p r e s e n t e d a t the v a r i o u s Task L e v e l s . For i n s t a n c e , a t the S e m i - C o n c r e t e l e v e l the number o f c o r r e c t responses f o r " F i s h " , "Shapes" and " M a t c h s t i c k s " was 6, 6, and 5 respectively. Another f a c t o r o f i n t e r e s t , a l t h o u g h i t f a l l s o u t s i d e the range o f the r e s e a r c h q u e s t i o n s , i s the impact o f t e a c h i n g on the s u b j e c t s f o r the s t u d y . chosen I f the r e s u l t s o f the p i l o t t e s t a r e used as a g u i d e - l i n e , and i f i t i s r e a s o n a b l e t o g e n e r a l i z e from t h i s group o f seven to the l a r g e r group o f 28, then i t would be r e a s o n a b l e t o expect t h e average s c o r e o f each s u b j e c t t o r i s e . does i t a f f e c t a l l o f t h e items a t each l e v e l some items a t some l e v e l s ? levels the I f t h e t e a c h i n g does have an impact, teachable? Concrete l e v e l ? Will of the h i e r a r c h y or o n l y Are t h e q u e s t i o n s a t t h e more a b s t r a c t the s u b j e c t s c e i l i n g out on t h e q u e s t i o n s a t These q u e s t i o n s , and those s p e c i f i c a l l y from the d e s i g n , w i l l be addressed i n Chapter S i x . formulated 45 Chapter Five R e s u l t s o f t h e Study The r e s u l t s o f t h e main study w i l l be p r e s e n t e d and a n a l y z e d i n t h i s chapter. Firstly, the analyses of variance w i l l be d i s c u s s e d ; s e c o n d l y , whether a h i e r a r c h y i s t e n a b l e g i v e n the r e s u l t s o b t a i n e d i n the study w i l l be c o n s i d e r e d ; and t h i r d l y , t h e evidence statistical concerning v a l i d a t i o n o f t h e h y p o t h e s i z e d h i e r a r c h y o f s k i l l s w i l l be examined. The first s e r i e s o f a n a l y s e s s e t out t o e s t a b l i s h whether or not c e r t a i n b a s i c assumptions u n d e r l y i n g t h e t e s t were t e n a b l e . As the s u b j e c t s were not nested by sex w i t h i n t h e l e v e l o f t h e t e s t , i t was important t h a t each s e t o f items w i t h i n each l e v e l o f t h e h i e r a r c h y t h a t was attempted difficulty. by each group of c h i l d r e n be o f the same l e v e l o f I f some items proved t o be harder t o answer than some o t h e r s , i t would not be p o s s i b l e t o assess t h e impact variable. I f f o r i n s t a n c e , items i n one row proved items i n t h e other rows, and f o u r g i r l s row, i t would be i m p o s s i b l e t o determine o f sex as a t o be e a s i e r answered t h e q u e s t i o n s on t h a t whether t h e g i r l s found some items e a s i e r than d i d t h e boys, o r whether t h e items themselves a c t u a l l y more d i f f i c u l t . it than were As w e l l as a n a l y z i n g the items w i t h i n each row, was necessary t o a n a l y z e t h e items w i t h i n each column t o e s t a b l i s h whether o r d e r e f f e c t s were p r e s e n t . I f questions are presented ina c e r t a i n o r d e r , i t may make i t e a s i e r t o answer t h e q u e s t i o n s t h a t f o l l o w than i f t h e q u e s t i o n s a r e presented i n another order. 46 R e s u l t s presented i n Table 10 i n d i c a t e t h a t n e i t h e r the items w i t h i n row or columns were s i g n i f i c a n t l y d i f f e r e n t t o one another. However, t h e r e was a s i g n i f i c a n t i n t e r a c t i o n between the items i n the rows and the columns. Table 10 Summary o f the A n a l y s i s of V a r i a n c e o f t h e Sequence o f Items W i t h i n the Test Degrees o f Freedom Sum of Squares Sequence (S) 6 2.946 0.491 1.257 Groups o f Items (G) 6 0.946 0.158 0.780 Persons (P:S) 21 8.202 0.391 - Items (I:G) 14 3.167 0.226 2.771 S. x G. 36 30.030 0.834 3.960 * S. x I:G 126 26.548 0.210 - G. x P:S 84 8.167 0.972 1.190 P:S x I:G 292 24.000 0.816 - Total 585 104.006 Source Mean Square F Sequence = Sequence o f items attempted by s u b j e c t s Item Sets = Sets o f t h r e e items * p < .05 T h i s i n t e r a c t i o n appears most l i k e l y t o have been caused by the low s c o r e s o b t a i n e d a t L e v e l 1 ( A b s t r a c t l e v e l ) o f the s k i l l hierarchy. Both the row and column means were h i g h , but w i t h i n each row and column 47 one s c o r e - i n v a r i a b l y f o r t h e problem below t h a t o f the o t h e r s . r e p r e s e n t i n g L e v e l 1 - dropped F i g u r e 3 i l l u s t r a t e s t h a t even t h e h i g h e s t mean o b t a i n e d (0.667 on column 5) was below t h e lowest mean s c o r e f o r any row o r column. The second a n a l y s i s o f v a r i a n c e was c a r r i e d the out t o a s s e s s whether e s t a b l i s h e d l e v e l o f d i f f i c u l t y c r i t e r i a f o r the items had been validated. The items were p l a c e d i n one o f t h r e e c a t e g o r i e s , Moderately D i f f i c u l t , or Hard, (see Table 1 ) . Results presented i n Table 11 showed t h a t l e v e l o f d i f f i c u l t y was s i g n i f i c a n t . no s i g n i f i c a n t The mean s c o r e s f o r t h e items were a c c u r a c y on t h e Easy Items, moderate d i f f i c u l t y , and 68?o 78?o a c c u r a c y on t h e items o f a c c u r a c y on t h e items c o n s i d e r e d d i f f i c u l t ) . As i n d i c a t e d i n T a b l e 12, t h e s u c c e s s r a t e f o r a l l l e v e l s high. group on the b a s i s o f t h e f o u r e s t a b l i s h e d c r i t e r i a (see Chapter Three, Item D i f f i c u l t y ) . (82% There was d i f f e r e n c e between sexes and the items w i t h i n each were d i f f e r e n t i a t e d similar Easy, o f items was T h i s f a c t c o u l d account f o r t h e s i m i l a r i t y of t h e easy and moderately d i f f i c u l t items. With so few e r r o r s made, those items answered i n c o r r e c t l y have a s i g n i f i c a n t ship of the three sets of items. effect upon t h e r e l a t i v e relation- Another p o s s i b l e reason f o r t h e s i m i l a r i t y noted between these items e x i s t s . Item d i f f i c u l t y was d e f i n e d a c c o r d i n g t o f o u r c r i t e r i a ; having grouped t h e 21 items i n t h r e e rows a c c o r d i n g t o these c r i t e r i a - , i t i s c o n c e i v a b l e t h a t the t h r e e rows might not be a b s o l u t e l y d i s t i n c t from one a n o t h e r . An item c l a s s e d as easy, c o u l d q u i t e w e l l have been put with t h e items o f moderate d i f f i c u l t y . T h i s d i d i n f a c t happen with one problem -8+1. A l s o , i t must be remembered t h a t o n l y a s m a l l sampling o f t h e items o c c u r r e d . Often a 00 •3- Sequence 1 VSequence 2 co CD CO Sequence 3 Q. Sequence 4 c o co CD Sequence 5 t-i CD Sequence 6 c co CD Sequence 7 Item Sets Figure 3. Mean responses to Item Sets at each l e v e l of the h i e r a r c h y ( Task L e v e l One o f the h i e r a r c h y ) 49 small sample results in only slight differences Table Summary Difficulty L e v e l of Item Difficulty (L) Item to the (C) (I) L are of Item of Variance Four Established Sum o f Squares 0.612 0.612' 2 1.962 0.981 26 12.034 0.463 - 18 2.418 0.134 0.768 2 0.133 0.663 0.823 - C x L:S 52 4.190 0.806 S x I:L 18 3.949 0.219 C x I:SL 468 81.918 0.175 587 107.216 = sex of child of F 1 x Sex discernible. Criteria Mean Square S Total not 11 Analysis Degrees of Freedom (S) Child the According Source Sex of that Level = Level Item Child Items = Boy o r G i r l = Three A d d i t i o n 0.132 *12.177 1.253 - * Difficulty Items p < .05 Table 12 Mean Performance by Sex and Item L e v e l of D i f f i c u l t y Female Easiest 0.847 Items Difficulty Male ( 1 ) (by Means difficulty) 0.80D 0.821 Moderately Difficult Items 0.800 0.765 0.781 Hard 0.673 0.694 0.684 0.772 0.752 Items Means (by gender) * '* 0.762 Grand Mean (1) There were 7 items i n each group and t h e s c o r e s under 'Male' and 'Female' r e p r e s e n t t h e mean of these s c o r e s 51 The t h i r d a n a l y s i s o f v a r i a n c e was undertaken t o e s t a b l i s h whether any p a r t i c u l a r s e t o f i t e m s was any more d i f f i c u l t other s e t , once they had been b a l a n c e d . item o f moderate d i f f i c u l t y , t o answer than any Each s e t had one easy i t e m , one and one hard i t e m (see T a b l e 2 ) . The r e s u l t s o f t h i s a n a l y s i s which a r e p r e s e n t e d i n Table .12 i n d i c a t e t h a t no group o f i t e m s proved t o be o f any g r e a t e r d i f f i c u l t y than any o t h e r group. Table 13 A Summary o f the A n a l y s i s of V a r i a n c e o f Item Difficulty Source f o r B a l a n c e d Groups o f Items Degrees of Freedom Sum o f Squares Sex (S) 1 0.184 0.184 0.132 Items ( I ) 6 2.286 0.381 0.355 26 36.102 1.389 6 6.102 1.017 C x I:S 156 167.327 1.073 Total 195 212.001 Child (C) S x I Mean Square F 0.948 52 The f o u r t h a n a l y s i s o f v a r i a n c e was e f f e c t e d t o determine i f t h e r e was s t a t i s t i c a l evidence o f a h i e r a r c h y of any type the h i e r a r c h y as h y p o t h e s i z e d ) . (not n e c e s s a r i l y The r e s u l t s o f t h i s a n a l y s i s a r e d e t a i l e d i n Table 14. Table 14 A Summary of t h e A n a l y s i s of Hypothesized Source Degrees of Freedom V a r i a n c e of t h e Hi e r a r c h y Sum o f Squares Mean Square F Sex (S) 1 0.184 0.184 Level of Hierarchy (li') 6 68.071 11.345 26 36.102 1.389 - 6 2.316 0.386 0. 5718 C x L:S 156 105.327 Total 195 212.000 C h i l d (C) S x L 0. 132 16. 804 * * p<..05 The r e s u l t s o f the a n a l y s i s p r o v i d e d a s t r o n g i n d i c a t i o n o f t h e e x i s t e n c e of a hierarchy of s k i l l s . Further analyses considered the hypotheses as they r e l a t e d t o the s e v e n - l e v e l l i n e a r h i e r a r c h y , and a l s o as they r e l a t e d t o t h e f o u r - l e v e l branching hierarchy. The r e s u l t s expressed as mean responses t o the item a t t h e seven t a s k l e v e l s of the h i e r a r c h y and moving from task l e v e l seven t o t a s k l e v e l one were: 2.07; 0.93. 2.71; 2.61; 2.57; 2.68; With the e x c e p t i o n o f l e v e l 4 t h e r e appeared progression of d i f f i c u l t y t o be a from l e v e l one t o l e v e l s e v e n , where l e v e l was the most a b s t r a c t and l e v e l seven the most c o n c r e t e . difference 2.43; However, this was s t r o n g l y e v i d e n t o n l y when l e v e l s one and two were compared w i t h the o t h e r f i v e l e v e l s . The i t e m s at t h e s e l e v e l s were apparently much harder than the items a t any o f the o t h e r f i v e l e v e l s , and the a t l e v e l one were a p p a r e n t l y much more d i f f i c u l t chy: A b s t r a c t , 0 . 9 3 ; Concrete, 2.66. Abstract-Concrete, 2.07; hierar- S e m i - C o n c r e t e , 2 . 5 5 ; and As w i t h the seven s t a g e l i n e a r h i e r a r c h y to be a p r o g r e s s i o n o f d i f f i c u l t y items than those a t l e v e l two. A second a n a l y s i s compared the f o u r l e v e l s o f t h e b r a n c h i n g appeared one from the l e v e l s there hypothesized t o be A b s t r a c t to those h y p o t h e s i z e d to be C o n c r e t e , w i t h the more A b s t r a c t l e v e l s p r o v i n g t o be t h e more d i f f i c u l t . A b s t r a c t - C o n c r e t e l e v e l s were c l e a r l y d i f f e r e n t different The A b s t r a c t and to one a n o t h e r , and from the S e m i - C o n c r e t e and Concrete l e v e l s . the Semi-Concrete l e v e l appeared to be harder t h e r e was v e r y l i t t l e d i f f e r e n c e i n the r e s u l t s Both the r e s u l t s However, a l t h o u g h than the C o n c r e t e level, of these l e v e l s . of the a n a l y s i s o f v a r i a n c e t h a t c o n s i d e r e d h i e r a r c h y , and the r e s u l t s o f the d e s c r i p t i v e a n a l y s i s supported p o s s i b i l i t y t h a t a h i e r a r c h y was f e a s i b l e . In order to t e s t h y p o t h e s i s s t a t i s t i c a l l y the d a t a was s u b j e c t e d t o a Guttman Scalogram The a n a l y s i s c o n s i d e r e d the number of items c o r r e c t l y answered out o f the p o s s i b l e a t each t e s t l e v e l . In e f f e c t , the the A n a l y s i s which c o n s i d e r e d the p o s s i b i l i t y o f both a s e v e n - l e v e l h i e r a r c h y and a f o u r - l e v e l b r a n c h i n g h i e r a r c h y . the linear also three t h i s meant t h a t t h e c u t - o f f 54 p o i n t f o r p a s s i n g a Task L e v e l o f t h e h i e r a r c h y was a r b i t r a r i l y s e t a t 1 out o f 3, 2 out o f 3, or 3 out o f 3 items. c u t - o f f p o i n t s was c o n s i d e r e d Each o f these P a s s - F a i l f o r t h e s e v e n - l e v e l h i e r a r c h y andtthe f o u r - l e v e l hierarchy respectively. Of t h e s i x a n a l y s e s a viable hierarchy. undertaken only two produced evidence At t h e c u t - o f f l e v e l o f two both the s e v e n - l e v e l h i e r a r c h y and the f o u r - l e v e l branching of r e p r o d u c i b i l i t y h i e r a r c h y had high (0.92 and 0.95 r e s p e c t i v e l y ) . the s e v e n - l e v e l l i n e a r h i e r a r c h y , however, r e v e a l e d l e v e l s as h y p o t h e s i z e d analyzed. t o support coefficients C l o s e r examination o f that the various task d i d not emerge i n t h e same o r d e r when s t a t i s t i c a l l y The r e s u l t s o f t h i s a n a l y s i s a r e presented i n Table 15. Table 15 R e s u l t s o f Guttman Scalogram A n a l y s i s Seven L e v e l s Hypothesized Hierarchy Re-Ordered Hierarchy (seven l e v e l s ) Number o f Subjects Who Passed Symbols Symbols 10 Tallies Tallies 20 Matches Fish Shapes Number o f Subjects Who F a i l e d 18 Percentage Percentage of S u b j e c t s of S u b j e c t s Who Passed Who F a i l e d 36 64 8 71 29 25 3 89 11 Matches 25 3 89 11 Fish Crayons 26 2 93 7 Blocks Blocks 27 1 96 4 Crayons Shapes 27 1 96 4 Number o f S u b j e c t s = 28 C o e f f i c i e n t o f R e p r o d u c i b i l i t y 0.918 Task l e v e l s one and two appeared two i n the h y p o t h e s i z e d o r d e r ; Task l e v e l s t o seven, however, d i d n o t . t a s k s , d i d not remain sized. Examination " F i s h " and "Matches", both Semi-Concrete i n the same r e l a t i o n s h i p t o one another as hypothe- o f t h e data showed t h a t 25 s u b j e c t s passed each level and 3 f a i l e d ; the d i f f e r e n c e between the t a s k s a t t h e s e two l e v e l s , then, was s l i g h t . "Shapes" was h y p o t h e s i z e d t o be a Semi-Concrete a c c o r d i n g t o the t h e o r e t i c a l premises proposed, "Crayons" should have been e a s i e r . task; " F i s h " , " B l o c k s " and The s u b j e c t s d i d , i n f a c t , f i n d t h e t a s k s presented with "Shapes" as easy as those p r e s e n t e d with " B l o c k s " . The s t r o n g e s t evidence t o s u p p o r t the h y p o t h e s i s o f a h i e r a r c h y o c c u r r e d when the c u t - o f f p o i n t o f 2 was used with t h e f o u r - l e v e l branching h i e r a r c h y . Table 16. Examination The r e s u l t s o f t h i s a n a l y s i s a r e p r e s e n t e d i n of t h e data i n d i c a t e t h a t the h i e r a r c h y was ordered i n t h e manner h y p o t h e s i z e d - Concrete t o A b s t r a c t . The d i f f e r e n c e between t h e A b s t r a c t and A b s t r a c t - C o n c r e t e l e v e l s , and t h e A b s t r a c t Concrete and Semi-Concrete percentage l e v e l s was c l e a r - c u t ; 36, 71, and 89 o f s u b j e c t s r e s p e c t i v e l y passed each o f t h e s e t h r e e l e v e l s . However, t h e d i f f e r e n c e between the Semi-Concrete although apparent, was not as s t r i k i n g r e s p e c t i v e l y passed these two l e v e l s . and Concrete - 89 and 96 percentage presented a t these l e v e l s , and t h i s their relationship. of subjects As the r e s u l t s of t h e seven- l e v e l l i n e a r h i e r a r c h y showed, a c l e a r d i s t i n c t i o n between and Concrete t a s k s was not apparent. levels, Few s u b j e c t s f a i l e d Semi-Concrete tasks ' c e i l i n g ' e f f e c t may have a f f e c t e d Table 16 R e s u l t s o f Guttman Scalogram Analysis: Four L e v e l s Hypothesized Hierarchy Guttman Scalogram Hierarchy Abstract Abstract 10 18 36 64 AbstractConcrete AbstractConcrete 20 8 71 29 SemiConcrete SemiConcrete 25 3 89 11 Concrete 27 1 96 4 Concrete Number o f Subjects Who Passed Number o f S u b j e c t s = 28 Coefficient of R e p r o d u c i b i l i t y : 0.95 Number o f Subjects Who F a i l e d Percentage Percentage of Subjects of Subjects Who Passed Who F a i l e d 57 Chapter Six D i s c u s s i o n , L i m i t a t i o n s and Recommendations Research Questions posed i n Chapter Two questions as w e l l as pertinent a r i s i n g from the P i l o t T e s t w i l l be addressed i n t h i s chapter i n the l i g h t of t h e o r e t i c a l assumptions o u t l i n e d i n Chapter Three, r e s u l t s presented of v a r i a n c e i n Chapters Four and Five. Results analyses i n d i c a t e d t h a t f a c t o r s such as the sex o f the c h i l d , d i f f i c u l t y , order item e f f e c t w i t h i n the t e s t , and|the r e l a t i v e d i f f i c u l t y groups of items w i t h i n the t e s t , were not s i g n i f i c a n t ; concerning of the and thus, questions the v a l i d a t i o n of a h i e r a r c h y of s k i l l s supported by n i f i c a n t v a r i a t i o n s i n success a t each l e v e l w i l l be of sig- considered. Discussion A l i n e a r h i e r a r c h y of s k i l l s does not appear d e f e n s i b l e . o f v a r i a n c e . i n d i c a t e d a s i g n i f i c a n t amount of v a r i a n c e i n the Analysis data a t t r i b u t a b l e to the h i e r a r c h y , but gave no i n d i c a t i o n as to the of h i e r a r c h y r e s p o n s i b l e . showed t h a t t h e r e was Examination of the means f o r each indeed "Symbols". v a r i a t i o n was.noted between Task l e v e l t h r e e and the mean.for Task l e v e l f o u r was six. The higher However, very Task l e v e l seven than t h a t of Task l e v e l s Guttman Scalogram A n a l y s i s supported the i d e a , o f a h i e r a r c h y but re-arranged ( T a b l e . 14). level a l i n e a r p r o g r e s s i o n moving from Task l e v e l seven "Crayons" to Task l e v e l one and type The the items from the order f a c t t h a t so few of the to argue f o r the of a l i n e a r r e l a t i o n s h i p between these t a s k s . and five linear hypothesized. s u b j e c t s f a i l e d any from t h r e e to seven makes i t very d i f f i c u l t little levels existence While i t i s p o s s i b l e t h a t the linear pattern occurred in tests not do these hierarchy same five have A second does a of to Taken individually, point in The a is one Task hierarchy on the "Shapes" cut-off of two correct tasks, "Blocks" and data pooled, the are the hypothesized discussion subjects there Task of is levels to three seven to Guttman between the Task is is at the two. of evidence Scalogram 2 out Level Abstract of and one at a items are at they analyzed automatically Guttman on at far out three is employed. In terms of levels, for use The hierarchy, in and the i t branching could of the the tasks far Concrete with be the in the the way as with too levels the few making Certainly that hierarchy, the if exception argued when both however, that same The than difficulty, is the easiest discussion. hierarchy, fact, the harder of the Analysis. by be the statistically. Scalogram to of maintain remain appeared these level be of that a l l . given that effect to factor the the another tasks Semi-Concrete in 'ceiling' appear support with of there Task is levels level. to support Analysis 3 items (Table that they the linear In further be constructed. c o n s i d e r e d one the a hand, confounding and strongest failing other the difficulty not responses linearity The point do of to not when sub-test seven-stage support cut-off levels hierarchy clear evidence another some a or equal there items one whether tasks. effects no the well relationship "Shapes" the failed 'ceiling' from on may according to items "Crayon" i t sufficiently relationship the a r i s e because tests, linear question are not 15). of the concept the Four-level correct There of as a basis was a very Abstract-Concrete, and a hierarchy hierarchy for passing clear between arose using or distinction the Abstract- a 59 Concrete and the Semi-Concrete levels. as c l e a r between the Semi-Concrete of the reasons proposed hierarchy. However, the d i s t i n c t i o n was and Concrete l e v e l s , perhaps i n the d i s c u s s i o n of the seven-stage The C o e f f i c i e n t enough to p r o v i d e s t a t i s t i c a l A review of the p i l o t of R e p r o d u c i b i l i t y of 0.95 not f o r one linear i s marginally high validation. t e s t l e d t o support f o r the i d e a s of both a l i n e a r p r o g r e s s i o n from Cpncrete to A b s t r a c t and the p o s s i b i l i t y of a four-level hierarchy. both of these i d e a s . Data from the main study s u p p o r t s , to some e x t e n t , I t i s important t o note t h a t i t i s not p o s s i b l e t o g e n e r a l i z e from the r e s u l t s of the p i l o t t e s t t o the main study o n l y seven c h i l d r e n were i n v o l v e d i n the p i l o t test. v a l u a b l e i d e a s arose which were worth c o n s i d e r i n g . d i f f e r e n c e s between the r e s u l t s of the p i l o t because However, some One of the major t e s t and the main study was the f a c t t h a t so many c h i l d r e n answered most of the q u e s t i o n s on the main study on items a t Task L e v e l s 3 t o 7. On the p i l o t t e s t no s u b j e c t r e c e i v e d more than 11 out of 21 items c o r r e c t on the " B l o c k s " s u b - t e s t . For a l l the o t h e r s u b - t e s t s the mean s c o r e s were w e l l below 50 p e r c e n t of the t o t a l p o s s i b l e . As the seven s u b j e c t s chosen from the r e s e a r c h pool of 49 c h i l d r e n who f o r the P i l o t qualified Test were f o r the study, i t seems u n l i k e l y t h a t the c h a r a c t e r i s t i c s o f t h i s group would be d i f f e r e n t t o those chosen f o r the main study. The major d i f f e r e n c e i n the two situations i s t h a t the s u b j e c t s i n the main study were t a u g h t , w h i l e those i n the p i l o t t e s t were not. The percentage of s u b j e c t s who l e v e l of the h i e r a r c h y , comparing noteworthy: 96%, 93%, l e v e l one: passed a t each the Main Study and the P i l o t T e s t , main study from task l e v e l seven to t a s k l e v e l one: 89%, 89%, 71%, 36%; 38%, pilot was 96%, t e s t from task l e v e l seven t o t a s k 52.4%, 28.6%, 28.6%, 23,8%, 0%, 0%. 60 I t appears t h a t t e a c h i n g was e f f e c t i v e at a l l l e v e l s of the p a r t i c u l a r l y at task l e v e l two t o 11%. The most important " T a l l i e s " , where the change was task l e v e l s one and two At 100%, and, the change i s i n f i n i t e because the p i l o t One deserves mention: i t i s d i f f i c u l t to a s s e s s f a c t o r other than the extent at test teaching, to which s u b j e c t s understood the i n s t r u c t i o n s on the-.-pilot t e s t . The the pilot test i n p a r t , used as a v e h i c l e f o r s t a n d a r d i z i n g the i n s t r u c t i o n s f o r the main study. was results. f i v e the change i s over 200%; b a s e l i n e s were 0% i n each case. was, task, seven the percentage change i s approximately at task l e v e l s t h r e e , f o u r and 0% from p o i n t i s t h a t the more a b s t r a c t the the g r e a t e r i s the percentage i n c r e a s e i n the main study task l e v e l s s i x and hierarchy, I f comprehension of the i n s t r u c t i o n s on the p i l o t poor, t h i s may from the p i l o t have depressed the s c o r e s . t e s t was Abstract-Concrete Another q u e s t i o n and Results i n d i c a t e d that s u b j e c t s were most l i k e l y ready f o r the type of symbolic i n these emerging whether or not the t a s k s a t the A b s t r a c t L e v e l s were t e a c h a b l e . test teaching the involved tasks. L i m i t a t i o n s of the Study From the d i s c u s s i o n thus f a r , i t i s apparent t h a t the high r a t e of success on the more c o n c r e t e t a s k s , which r e s u l t e d i n many s u b j e c t s f i n d i n g the items a t s e v e r a l of the Task L e v e l s too easy, i s a cant f a c t o r i n t h i s study. The f a c t o r s of t e a c h i n g , and signifi- the l a c k of a c t u a l d i f f e r e n c e i n the t a s k s themselves, have a l r e a d y been mentioned; it i s p e r t i n e n t to c o n s i d e r a few p o o l was 125 additional points. s u b j e c t s , e v e n t u a l l y reduced to 49, f o r both the p i l o t t e s t and the main study p o p u l a t i o n were e l i m i n a t e d from the study., The initial from which the were chosen., The subject subjects Thus 60% of s e l e c t i o n of the 49 the subjects was a r r i v e d a t - with t h e e x c e p t i o n o f s i x who c o u l d already answer f o u r or f i v e q u e s t i o n s a t the most a b s t r a c t l e v e l of the h i e r a r chy - by g i v i n g a p r e t e s t which has many developmental c h a r a c t e r i s t i c s to i t . Many 5 year o l d c h i l d r e n , f o r i n s t a n c e , are unable t o conserve d i s c r e t e e n t i t i e s which they w i l l have no d i f f i c u l t y conserving year l a t e r . It i s conceivable who formed the research had c h a r a c t e r i s t i c s which would d i s t i n g u i s h them' from the pool t h a t the 49 s u b j e c t s one 76 not chosen, o t h e r than the a b i l i t y t o s u c c e s s f u l l y complete the t a s k s p r e s e n t e d on the p r e t e s t . Two v a r i a b l e s which c o u l d would be s o c i o - e c o n o m i c s t a t u s and i n t e l l i g e n c e . be considered It i s possible that the c h i l d r e n who formed the r e s e a r c h p o o l were from a h i g h e r socio- economic c l a s s than those who f a i l e d the p r e t e s t ; i t i s a l s o possible t h a t they were more i n t e l l i g e n t than those who f a i l e d the p r e t e s t . The f a c t o r of i n t e l l i g e n c e i s p a r t i c u l a r l y r e l e v a n t t o t h e i s s u e of ' c e i l i n g o u t ' on t h e t a s k s designed. I t may be, f o r i n s t a n c e , that b r i g h t e r c h i l d r e n do not r e q u i r e t a s k s p r e s e n t e d i n a very c o n c r e t e form such as " B l o c k s " or "Crayons" and t h a t any form o f c o n c r e t e e i t h e r two-dimensional, or t h r e e - d i m e n s i o n a l , w i l l s u f f i c e . presentation Many c h i l d r e n who e x p e r i e n c e d i f f i c u l t y with c a r d i n a l numbers r e s o r t t o t h e use of c o n c r e t e a i d s such as f i n g e r s . from t h i s study by the p r e t e s t . previous research, These c h i l d r e n were As mentioned i n t h e d i s c u s s i o n of i t i s not p o s s i b l e t o d i s c u s s c h i l d r e n without c o n s i d e r i n g eliminated skill hierarchies for developmental c h a r a c t e r i s t i c s . This i s p a r t i c u l a r l y t r u e f o r c h i l d r e n a t 5 years o f age where c o g n i t i v e i n g and s t r a t e g i e s a r e r a p i d l y changing. think- Recommendations f o r Future The Research Guttman Scalogram A n a l y s i s v a l i d a t e d the f o u r l e v e l u s i n g two hierarchy out of t h r e e items as the p a s s - f a i l c u t - o f f p o i n t . c o e f f i c i e n t of r e p r o d u c i b i l i t y was significant. f i n d i n g has The any question relevance 0.95 which was statistically t h a t a r i s e s i s the extent to e d u c a t i o n . The The to which this f o r m u l a t i o n of an opera- t i o n a l d e f i n i t i o n of numbers, as o u t l i n e d by Resnick (1970) suggests t h a t a d d i t i o n equations are at the top of the h i e r a r c h y - t h a t i s , at the h i g h e s t l e v e l of d i f f i c u l t y . present The r e s u l t s of the p r e s e n t the p o s s i b i l i t y of t a k i n g a d d i t i o n equations and study presenting them at t h r e e d i f f e r e n t l e v e l s of d i f f i c u l t y b e f o r e a c t u a l l y p r e s e n t i n g them a b s t r a c t l y . R e s u l t s f o r the h i e r a r c h y as p r e s e n t l y c o n s t i t u t e d would need r e p l i c a t i n g b e f o r e i t c o u l d j u s t i f i a b l y be used i n an educational s e t t i n g . A second i s s u e which a f f e c t s the arrangement of items i n the form of a h i e r a r c h y i s item d i f f i c u l t y . some support f o r the c r i t e r i a of d i f f i c u l t y of a d d i t i o n items as l i n e d i n Chapter Three. 4 l e v e l h i e r a r c h y with Further research t h i s study. R e s u l t s showed I t would be p o s s i b l e , then, out- to c o n c e i v e 3 l e v e l s of item d i f f i c u l t y a t each of a level. i s i n d i c a t e d g i v e n the i n f o r m a t i o n a v a i l a b l e from R e p l i c a t i o n of the r e s u l t s u s i n g a l a r g e r p o p u l a t i o n kindergarten c h i l d r e n would be v a l u a b l e , to determine i f the Semi- Concrete and Concrete l e v e l s c o u l d be d i s t i n g u i s h e d simply of by i n c r e a s i n g the number of s u b j e c t s . A l a r g e r p o p u l a t i o n of s u b j e c t s would a l s o a l l o w the v a r i a b l e of age t o be s t u d i e d i n more d e t a i l . 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Journal Research i n s c i e n c e t e a c h i n g , 1963, 1, 144-153. Gagne*, R. M. Problem s o l v i n g . I n A. W. Melbon ( E d . ) , C a t e g o r i e s o f human l e a r n i n g . Gagne', R. M. New York: Some f a c t o r s i n l e a r n i n g non-metric geometry. Development, 1965, Gagne*, R. M. Academic P r e s s , 1964. 30, Child 42-49. C o n t r i b u t i o n s o f l e a r n i n g t o human development. P s y c h o l o g i c a l Review, 1968, 75, 177-191. Gagne'', R. M. The c o n d i t i o n s o f l e a r n i n g . Holt, Rinehart & Winston, 1970. Gronlund, H. E. C o n s t r u c t i n g an achievement t e s t . Englewood Cliffs, New J e r s e y : P r e n t i c e - H a l l , 1970. Karplus, R. Ashley The s c i e n c e c u r r i c u l u m : Improvement study. In I . J . & D. C. Rubadeau ( E d s . ) , E d u c a t i o n a l i m p l i c a t i o n s o f P i a g e t ' s theory. Kilpatrick, J. programme. Toronto, O n t a r i o : C o g n i t i v e theory In I . J . Ashley Serox, 1970. and t h e s c h o o l mathematics study & D. C. Rubadeau ( E d s i ) , i m p l i c a t i o n s of Piaget's theory. Toronto, O n t a r i o : group Educational Xerox, 1970. Mager, R. F., & C l a r k , C. instruction. M i l l e r , S. A. Explorations i n student-controlled P s y c h o l o g i c a l Reports, 1963, 13, Non-verbal assessment of c o n s e r v a t i o n Development, 1976, Resnick, L. B. 47, of number. Resnick, L. B. Development C e n t r e , Learning U n i v e r s i t y of P i t t s b u r g h , Behaviour a n a l y s i s i n c u r r i c u l u m design: sequenced i n t r o d u c t o r y mathematics c u r r i c u l u m . Research and Child 722-728. Design of an e a r l y l e a r n i n g c u r r i c u l u m . Research and Learning 71-76. Development Centre, 1967. A hierarchically Monograph U n i v e r s i t y of 2, Pittsburgh, 1970. Saxe, G. B. A developmental a n a l y s i s of n o t a t i o n a l c o u n t i n g . Development, 1977, S i e g e l , L. S. The in preschool Uprichard, 48, C, Developmental Psychology 1971, A. E. & P h i l l i p s , E. R. An i n t r a c o n c e p t A v a l i d a t i o n study. Mathematics E d u c a t i o n , Wong, M. 1512-1520. Sequence of development of c e r t a i n number concepts children. number a d d i t i o n : Resnick, L. B., 1977, Q_, 1767-1778. 5_, 357-361. a n a l y s i s of rational J o u r n a l f o r Research i n 7-16. & Boozer, R. F. of some e a r l y mathematics behaviours. 42, Child The sequence of development C h i l d Development, 1971, APPENDIX A MEASURING INSTRUMENT B a s i c a d d i t i o n f a c t s . ( u s i n g t h e numbers one t o n i n e sums a r e no l e s s than s i x and no g r e a t e r than n i n e . (1) 1 + 5 (5) 5 + 1 (2) 1 + 6 (6) 6 + 1 (3) 1 + 7 (7) 7 + 1 (4) 1 + 8 (8) 8 + 1 (9) 2 + 4 (13) 4 + 2 (10) 2 + 5 (14) 5 + 2 (11) 2 + 6 (15) 6 + 2 (12) 2 + 7 (16) 7 + 2 (17) 3 + 3 (21) 4 + 3 (18) 3 + 4 (22) 5 + 3 (19) 3 + 5 (23) 6 + 3 (20) 3 + 6 (24) 4 + 4 (26) 5 •+ 4 (25) 4 + 5 Total: 26 F a c t s inclusively) APPENDIX B INSTRUCTIONS Task L e v e l 7. E t o S: FOR THE POSTTEST (Three items were g i v e n ) . Here a r e two s e t s o f crayons (E p o i n t s ) . J o i n these two s e t s of crayons t o make one s e t . How many crayons a r e t h e r e now? S responds: T h i s w i l l be repeated f o r items 2 and 3. Task L e v e l 6. (Three items w i l l be g i v e n ) . Same as Task L e v e l 7. Task L e v e l 5. E t o S: S u b s t i t u t e blocks f o r crayons. (Three i t e m s ) . Here a r e two s e t s o f f i s h two s e t s o f f i s h (E p o i n t s ) . I f you j o i n e d t o make one new s e t , how many f i s h these will t h e r e be? S responds: Task L e v e l s 4 and 3 w i l l be t h e same. matchsticks f o r f i s h . Task L e v e l 2. E t o S: and then (Three items a t each t a s k l e v e l ) . (Three i t e m s ) . Here a r e two numbers which I want you t o add t o g e t h e r (E p o i n t s ) . you. I f you wish you may use the t a l l y marks t o h e l p When you add these numbers t o g e t h e r , how many a r e t h e r e now? S responds: S u b s t i t u t e shapes 69 Task L e v e l 1. E t o S: (Three i t e m s ) . Here a r e two numbers t h a t I want you t o add t o g e t h e r . How many do they make when you add them? begins t o use h i s f i n g e r s , t h e experimenter t h a t t h i s i s not a l l o w e d ) . S responds: ( I f the subject should indicate APPENDIX PRETEST Each child These 8, Each child it were 2, Each with presented 7, 9, was shown verbally. numbers The was p r e s e n t e d numbers 3, ENTERING 1, 6, the test had t o experimenter If the subject by giving orally said recite to the series. If complete E: Would (No E. hesitated, either could The order 5 4, at inclusively. a time a child and asked had to to identify identify a l l nine correctly. child No one t o n i n e i n a random help error he asked then count placed these Subject counts blocks. of the subject i n order inclusively nine help up whether to again subject numbers the repeat to the the f i r s t in subject the series. please. time) to pass. s i x blocks before blocks. you. nine count could was u n s u r e up t o t h e number was p e r m i t t e d one t o Please or both i s permissible at this experimenter Please second the experimenter you count f o r me. the experimenter the f i r s t , the task t h e numbers subject: I would l i k e you to count t h e number n i n e . the BEHAVIOURS t h e numbers one number To p a s s C Count the out loud subject. so that I can hear 71 E. How many a r e t h e r e ? Subject The responds. examiner then spread the b l o c k s out so t h a t they covered surface area. E. the question again. How many a r e t h e r e now? Subject E. Then the examiner asked a greater responds. Are t h e r e more b l o c k s now? Subject responds. (The task was repeated b l o c k s were used. i f error.occurred. The next time, however, f o u r I f the s u b j e c t got t h e c o r r e c t answer, one more item was g i v e n with seven b l o c k s t o make sure the second response guess.) was not a APPENDIX D INSTRUCTIONS FOR Task L e v e l 7 ( r e a l On was 4+4). i n the o t h e r . E: objects) the t a b l e were two (item one The s e t s of crayons i n f r o n t of each There were f o u r p e n c i l s i n one experimenter s a i d t o the s e t s of p e n c i l s (E Here are two pencils ( p o i n t s ) and Watch me TEACHING while Subjects E: Now (E Now (E I count t h i s s e t of p e n c i l s . side.) Now p o i n t s to the s e t on the s u b j e c t s ' l e f t watch w h i l e (E set. I am going to each and Now T h i s i s one s e t of t h i s i s another s e t of p e n c i l s ( p o i n t s ) . I count the other counts you count this side).' you set. Now you count your p o i n t s to each s u b j e c t ' s r i g h t h a n d to take t h i s s e t upside-down to the s u b j e c t s ) and E: four responded. other E: s e t and subjects: points). demonstration s e t on h i s r i g h t h a n d set child (E's s e t i n turn).. l e f t h a n d s e t because i t j o i n i t to t h i s s e t . (E was points then j o i n s them-.).! take t h i s s e t (subjects' righthand s e t ) and j o i n i t to this set. Subjects E: Ei responded. Now, obtained p e n c i l s and how many crayons are t h e r e altogether? a v e r b a l response from each s u b j e c t . E. gave the c o r r e c t answer to the s u b j e c t s . when applicable,)). then counted h i s ( P r a i s e was used 73 The procedure E: demonstrations was Here as are two sets (points) Now many left)) omitted from items two and three. The one of follows: crayons how were Now of and crayons how crayons (E this another are many in is points). there? this set (E This of crayons points set? (E is to points set to set (E on set points). subjects' on subjects', right.)). E: Now you join this set (on subjects' right) to this set (on subjects' left): Subjects E: responded. How Subjects E: There Level This was Task Level E: are Here were are E: to crayons task (E shown two points First, altogether? altogether. (Praised if appropriate). level except 7 crayons were replaced by the card with of fish (E sets to group points count on left the group this set of item points). of to the on on i t . This is card) and this the right of is the one set another of blocks. fish set card). fish. responded. Now, Subjects there 5 fish E: seven identical subjects Subjects are 6 (E E: crayons responded. Task The many count this set of fish. responded. Now if this how many set fish is are added there to this set altogether? (E points to each set), of Subjects E. responded. counts (Repeat Task by them supplies correct for items two Levels 3 and A were 'shapes' Task and Level and E: We a r e going number on hand can card the 5 except 'fish' were together (E replaced f i r s t want the solve for Task Level E.to Subjects: add to and the lefthand these under the number two points and numbers numbers. two and these two numbers help you get give using the items numbers us add marks marks then to the together, (E each rightwe demonstrates their tally together. an answer to I the want you to question. answers. marks and gives the correct answer. three). 1 Do you we u s e d t a l l y together look to items numbers help two number). you the first tally tally subjects. add. t h e s e To the use you same a s the card, the I (Repeated to count Now the before number). with E- a d d s the 2 placed the Subjects three). 'matchsticks'. E. E: and answer. at add marks these this i t to remember two number this to that help numbers (E us? and points number last (E time This tell to the points we a d d e d time me t h e I the want you answer. lefthand to numbers number) other together to First, and number). add you then How 75 many do they make a l t o g e t h e r ? Subjects responded. E-adds the numbers the second. out l o u d by c o u n t i n g (e.g. 5 + 3 - 5-6,7,8)- (Repeated f o r items two and t h r e e ) . on from the f i r s t answer 8) number t o
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A test of the validity of a concrete-abstract hierarchy of addition facts on kindergarten children Ward, Brian 1979
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