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The performance of second year primary children on missing addend sentences Kelleher, Heather 1977

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THE PERFORMANCE OF SECOND YEAR PRIMARY CHILDREN ON MISSING ADDEND SENTENCES by Heather Jane K e l l e h e r B. Ed., U n i v e r s i t y of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF ARTS i n THE FACULTY OF GRADUATE STUDIES Mathematics Department F a c u l t y of Ed u c a t i o n U n i v e r s i t y of B r i t i s h Columbia We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 19 77 Heather Jane Kelleher, 1977 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb i a , I a g ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s , f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thou t my w r i t t e n p e r m i s s i o n . Department o f ^xkvusjoki^v^ The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date CkJh. -*> XW71 A b s t r a c t Research S u p e r v i s o r : Dr. Douglas T. Owens T h i s study examined the accuracy, s o l u t i o n s t r a t e g y use, and l e v e l of response use o f second year c h i l d r e n when s o l v i n g m i s s i n g addend problems at f o u r l e v e l s of d i f f i c u l t y and i n two p l a c e h o l d e r p o s i t i o n s . The r e l a t i o n s h i p between these aspects of M i s s i n g Addend performance and performance on a measure of C l a s s I n c l u s i o n a b i l i t y , was then examined. Subject s of the study were 40 year two students from an urban community i n B r i t i s h Columbia, Canada. A M i s s i n g Addend T e s t and a C l a s s I n c l u s i o n T e s t were adm i n i s t e r e d i n d i v i d u a l l y to a l l s u b j e c t s . The l e v e l of d i f f i c u l t y of the M i s s i n g Addend T e s t items (as d e f i n e d by the magnitude of the constants) a f f e c t e d accuracy. Process e r r o r s were more common than c o n c e p t u a l e r r o r s as the d i f f i c u l t y of the item increased. The l e v e l of d i f f i c u l t y of the item a l s o a f f e c t e d the c h i l d ' s l e v e l of response. C h i l d r e n tended to use more e x t e r n a l i z e d processes and concrete a i d s as the d i f f i c u l t y i n c r e a s e d . The l e v e l of d i f f i c u l t y d i d not, however, appear to a f f e c t s t r a t e g y c h o i c e to the same degree. P l a c e h o l d e r p o s i t i o n was found to have l i t t l e or no e f f e c t on c h i l d r e n ' s accuracy, s t r a t e g y c h o i c e , o.r l e v e l of response use. C h i l d r e n i n t e r p r e t e d the m i s s i n g addend as a i i s i t u a t i o n r e q u i r i n g an incrementing, or a d d i t i v e p rocess i n 68% of the cases. In 9% of the cases they used d e c r e -menting or s u b t r a c t i v e p r o c e s s e s . C h i l d r e n used r e c a l l of b a s i c f a c t combinations to s o l v e 9% of the items. Conceptual m i s i n t e r p r e t a t i o n s of the number sentence, as i n d i c a t e d by the use of an i n c o r r e c t sentence t r a n s f o r m a t i o n , o c c u r r e d i n 8% of the examples. C h i l d r e n omitted items or used u n i d e n t i f i a b l e processes i n 6% of the examples. Of the 248 items where an a d d i t i v e or s u b t r a c t i v e process was used, by f a r the p r e f e r r e d process was a counting procedure. Two c o u n t i n g procedures were particularly popular: C o u n t i n g - A l l and Counting-On. Other i d e n t i f i e d s t r a t e g i e s were Semi-Guesses, S u b s t i t u t i o n procedures, and procedures i n v o l v i n g A s s o c i a t i v e r e a s o n i n g . Concrete m a t e r i a l s were used i n 45% of the examples, and u s u a l l y i n a s s o c i a t i o n w i t h Semi-Guess, S u b s t i t u t i o n , and C o u n t i n g - A l l s t r a t e g i e s . I n t e r n a l i z e d reasoning procedures were used i n 37% of the examples, and u s u a l l y i n a s s o c i a t i o n w i t h the R e c a l l and A s s o c i a t i v e s t r a t e g i e s . F i n g e r s were used as a i d s f o r 17% of the examples, and were used almost e x c l u s i v e l y w i t h the Counting-On s t r a t e g y . I t was concluded t h a t the use of f i n g e r s p r o v i d e d a v a l u a b l e t r a n s i t i o n between e x t e r n a l and i n t e r n a l i z e d s o l u t i o n procedures. I t was a l s o concluded t h a t the a b i l i t y t o count-on was key to the development of more s o p h i s t i c a t e d s o l u t i o n p r o c e s s e s . C l a s s I n c l u s i o n performance was found to be p o s i t i v e l y r e l a t e d (p^.OS) to M i s s i n g Addend performance. C l a s s I n c l u s i o n performance was s i g n i f i c a n t l y r e l a t e d to the use of two s o l u t i o n s t r a t e g i e s . The use of R e c a l l was p o s i t i v e l y r e l a t e d (p <• .01) and the use of an I n c o r r e c t T r a nsformation was n e g a t i v e l y r e l a t e d (p ^ . 0 1 ) . C l a s s I n c l u s i o n performance was not r e l a t e d t o the use of any ofo the i d e n t i f i e d l e v e l s of response. i v CONTENTS Page ABSTRACT * 1 1 CONTENTS V LIST OF TABLES X 1 LIST OF FIGURES X 1 1 1 ACKNOWLEDGEMENT X 1 V Chapter 1. THE PROBLEM 1 D e f i n i t i o n o f Terms 4 Statement of the Problem 5 Purpose of the Study 6 Assumptions of the Study 7 L i m i t a t i o n s o f the Study 7 2. REVIEW OF RELATED LITERATURE 8 Performance on Open Sentences 8 Accuracy on M i s s i n g Addend Open Sentences 10 P l a c e h o l d e r p o s i t i o n 10 L e v e l o f d i f f i c u l t y o f number. combinations 11 Summary of Research on Open Sentence Performance D i f f e r e n c e s 15 M i s s i n g Addend S o l u t i o n S t r a t e g i e s . . . . 16 St u d i e s U t i l i z i n g P u p i l O b servation . . . 17 D e s c r i p t i v e .studies 17 v Chapter Page Studies concerning the sequence of development of solution strategies 23 A P r i o r i Models of Solution Strategies . . 29 Summary of Solution Strategy Literature . . 3 2 Sensory Involvement i n Solution Strategies 34 Class Inclusion A b i l i t y and Missing Addend Performance 38 Studies Investigating the Relationship Between Class Inclusion and Missing Addend Performance 41 Class Inclusion Measures 43 Summary of the Review of Related Literature 45 3. METHOD 4 8 Questions of the Study 4 9 Questions Related to Performance on Missing Addend Sentences ., 49 Questions Related to Strategy Choice . . . 49 Questions Related to Level of Response Usage 50 Questions Related to the Interaction of i Strategy and Level of Response Use . . . 50 Questions Related to Class Inclusion Performance 51 Sample 51 Description of the Sample 51 Selection Technique 51 Missing Addend Background of the Sample 52 v i Chapter Page M i s s i n g Addend Te s t 56 P i l o t T e s t 56 M i s s i n g Addend Te s t Items 59 T e s t M a t e r i a l s 60 T e s t b o o k l e t 60 P u p i l m a t e r i a l s 61 I n t e r v i e w coding form 61 S t r a t e g y C l a s s i f i c a t i o n 62 Semi-Guess procedures . . . . 64 S u b s t i t u t i o n procedures 6 5 C o u n t i n g - A l l procedures . . . . . . . . 65 Counting-On procedures . 66 A s s o c i a t i v e procedures . . . . . . . . 66 R e c a l l or automatic response. . . . . . 67 I n c o r r e c t T r a n s f o r m a t i o n 67 Indeterminate 67 No Attempt 67 L e v e l s of Response 6 8 C l a s s I n c l u s i o n T e s t 70 Items . 70 M a t e r i a l s 73 S c o r i n g Procedure 73 Task P r e s e n t a t i o n and Response Recording Procedure 75 Procedure 75 T e s t A d m i n i s t r a t i o n . . . 76 v i i Chapter Page Data C o l l e c t e d 78 M i s s i n g Addend Te s t data 78 C l a s s I n c l u s i o n Test data 78 R e l i a b i l i t y Study 7 8 Data A n a l y s i s 79 4. RESULTS 80 M i s s i n g Addend Te s t R e s u l t s 80 Item A n a l y s i s R e s u l t s 80 L e v e l o f D i f f i c u l t y 82 P l a c e h o l d e r P o s i t i o n 83 Summary of A n a l y s i s o f V a r i a n c e 84 Summary of M i s s i n g Addend Te s t R e s u l t s . . 86 M i s s i n g Addend S o l u t i o n S t r a t e g i e s 87 Frequency of S t r a t e g y Use 8 8 O v e r a l l 88 By l e v e l of d i f f i c u l t y 90 By p l a c e h o l d e r p o s i t i o n . . . . . . . . 92 I n d i v i d u a l C h i l d r e n ' s S t r a t e g y Choice P a t t e r n s 94 Accuracy of S t r a t e g y Use 96 L e v e l s of Response 100 Frequency of L e v e l o f Response 101 I n d i v i d u a l L e v e l of Response Usage - \ , , P a t t e r n s 10 3 Accuracy o f L e v e l s o f Response 105 I n t e r a c t i o n of L e v e l o f Response and St r a t e g y Choice 107 C l a s s I n c l u s i o n T e s t R e s u l t s 109 v i i i Chapter Page The R e l a t i o n s h i p Between C l a s s I n c l u s i o n Performance and M i s s i n g Addend Te s t V a r i a b l e s I l l M i s s i n g Addend Te s t Accuracy I l l M i s s i n g Addend S t r a t e g y Choice 113 M i s s i n g Addend L e v e l o f Response Usage. . 115 5. SUMMARY, CONCLUSIONS, AND IMPLICATIONS . . . 117 Summary and Conclus i o n s 117 Accuracy on M i s s i n g Addend Sentences . . 117 M i s s i n g Addend S o l u t i o n S t r a t e g i e s . . . 118 A d d i t i v e and s u b t r a c t i v e approaches . . 118 Summary of r e s u l t s f o r each s t r a t e g y . . 121 L e v e l of Response Use 128 E x t e r n a l i z e d l e v e l o f response 128 T r a n s i t i o n a l l e v e l of response 129 I n t e r n a l i z e d l e v e l o f response 129 I n t e r a c t i o n of L e v e l of Response and S t r a t e g y Use 130 C l a s s I n c l u s i o n A b i l i t y and M i s s i n g Addend Performance 131 I m p l i c a t i o n s f o r Classrooms 132 Suggestions f o r F u r t h e r Research 136 REFERENCES . • • • • • • • 139 APPENDIX A. P u p i l Data 1 4 3 B. Sample of the M i s s i n g Addend T e s t Booklet 1 4 4 C. Interview Coding Form i 4 9 D. C l a s s I n c l u s i o n Items 150 " i x Page E. C l a s s I n c l u s i o n Task P r e s e n t a t i o n Card Sample ,152 F. T e s t Scores 153 G. S o l u t i o n S t r a t e g y Usage P a t t e r n s 154 H. L e v e l of Response Usage P a t t e r n s 155 I. R e l i a b i l i t y of S t r a t e g y C l a s s i f i c a t i o n . . 156 J . R e l i a b i l i t y of C l a s s I n c l u s i o n S c o r i n g . . 157 x LIST OF TABLES Table Page 2.1 Age P a t t e r n s f o r A d d i t i o n and S u b t r a c t i o n Processes 27 3.1 M i s s i n g Addend P i l o t T e s t Items 58 3.2 M i s s i n g Addend Te s t Items 60 3.3 C l a s s I n c l u s i o n Task S p e c i f i c a t i o n s . . . . 72 3.4 Order of Mentioning the Greater Subset . . . 72 3.5 C l a s s I n c l u s i o n Task S c o r i n g Procedure . . . 74 4.1 M i s s i n g Addend T e s t Item R e s u l t s 81 4.2 Summary of Data on the L e v e l o f D i f f i c u l t y F a c t o r 82 4.3 Summary of Data on P l a c e h o l d e r P o s i t i o n . . 83 4.4 Summary of ANOVA w i t h Repeated Measures f o r L e v e l o f D i f f i c u l t y and P l a c e h o l d e r P o s i t i o n 84 4.5 Frequency of Use of S o l u t i o n S t r a t e g i e s O v e r a l l 89 4.6 S t r a t e g y Frequency f o r Each L e v e l o f D i f f i c u l t y . . . . . 91 4.7 S t r a t e g y Frequency f o r Each P l a c e h o l d e r P o s i t i o n 93 4.8 S t r a t e g y Choice P a t t e r n Frequencies . . . . 94 4.9 S o l u t i o n S t r a t e g y Accuracy Rates 98 4.10 Frequency of L e v e l of Response O v e r a l l and by L e v e l o f D i f f i c u l t y 101 4.11 P a t t e r n s o f L e v e l o f Response f o r C h i l d r e n Using One S t r a t e g y Seven or E i g h t Times 104 x i Table Page 4.12 Accuracy Rates f o r L e v e l s of Response . . . 105 4.13 C l a s s I n c l u s i o n Task Performance Means . . . 110 4.14 Concrete Versus O r a l Item Performance Means 110 4.15 I n t e r c o r r e l a t i o n s Among Major V a r i a b l e s . . I l l 4.16 C o r r e l a t i o n s Between C l a s s I n c l u s i o n Performance and Frequency of S t r a t e g y Choice 113 4.17 R e l a t i o n s h i p Between L e v e l o f Response Usage and C l a s s I n c l u s i o n Performance . . 116 x i i LIST OF FIGURES F i g u r e Page 3.1 M i s s i n g Addend Experience of the Sample 55 4.1 I n t e r a c t i o n of L e v e l of D i f f i c u l t y and P l a c e h o l d e r P o s i t i o n 85 4.2 Frequency of L e v e l of Response by L e v e l of D i f f i c u l t y 102 4.3 I n t e r a c t i o n of L e v e l o f Response Usage and S t r a t e g y Choice 108 x i i i ACKNOWLEDGEMENT I wish t o thank the members of my t h e s i s committee, Dr, David R o b i t a i l l e and Dr. G a i l S p i t l e r , f o r t h e i r guidance and support. In p a r t i c u l a r , I wish to thank my chairman, Dr. Douglas Owens, f o r h i s p a t i e n c e , i n t e r e s t , and a s s i s t a n c e d u r i n g the completion of t h i s t h e s i s . I would a l s o l i k e to thank s e v e r a l people who have c o n t r i b u t e d i n v a r i o u s ways to t h i s t h e s i s : Lee Herberts, Tom O'Shea, and John T a y l o r f o r t h e i r help i n the s t a t i s t i c a l a n a l y s i s ; the students and te a c h e r s of New Westminster School D i s t r i c t f o r t h e i r c o o p e r a t i o n ; M e r i l e e MacDonald and the Hobart group f o r t h e i r c o n t r i b u t i o n s to the f i n a l copy; Koko C a r l s o n f o r her i n s p i r a t i o n and co u n s e l ; and Steve K e l l e h e r f o r h i s encouragement and support. x i v Chapter 1 THE PROBLEM During the 1960's, programme developers i n t r o d u c e d a v a r i e t y of open sentence forms i n t o primary mathematics t e x t s . U n t i l t h a t time, open sentences of the form a ± b =[~1 were used almost e x c l u s i v e l y , e i t h e r i n h o r i z o n t a l or v e r t i c a l format. Most primary mathematics workbooks and t e x t s p u b l i s h e d i n the 1960's i n t r o d u c e d m i s s i n g addend sentences, t h a t i s , a - b and a = b, w h i l e some a l s o i n t r o d u c e d s u b t r a c t i o n sentences w i t h v a r y i n g placeholder p o s i t i o n s , t h a t i s , a - Q = b and Q- a = b. Less f r e q u e n t l y , the symmetric e q u i v a l e n t of these sentences, t h a t i s • = a ± b, b = [ ] ! a and b = a t [j , was emphasized. Often these forms were presented i n v e r t i c a l as w e l l as h o r i z o n t a l format. The value of d e v e l o p i n g a c h i l d ' s a b i l i t y t o s o l v e open sentences i n v a r i o u s forms has been proposed by s e v e r a l authors. Grouws and Good (1976) suggested t h a t v a r i o u s open sentence forms p r o v i d e computational p r a c t i c e as w e l l as mathematical models f o r many p e r c e p t u a l l y d i f f e r e n t s i t u a t i o n s . Grouws (1971) s t a t e d t h a t open sentences are e s s e n t i a l i n f o r m u l a t i n g c l e a r and p r e c i s e 2 statements of important mathematical and p h y s i c a l r e l a t i o n s h i p s . Flournoy (1964) expressed the o p i n i o n t h a t experience w i t h open sentences p r o v i d e d an important and i n t e r e s t i n g a p p l i c a t i o n of o p e r a t i o n a l r e l a t i o n s h i p s . In p a r t i c u l a r , F l o u r n o y i d e n t i f i e d the m i s s i n g addend sentence as a u s e f u l v e h i c l e f o r such a p p l i c a t i o n s , and recommended the use of m i s s i n g addend sentences s t a r t i n g i n grade one. Groen and P o l l (1973) suggested t h a t m i s s i n g addend sentences are used as a stepping stone from a d d i t i o n t o s u b t r a c t i o n . They proposed t h a t t h i s p r a c t i c e was based i n p a r t on the i n t u i t i v e l y a p p e a l i n g r a t i o n a l e t h a t m i s s i n g addends should develop an understanding of s u b t r a c t i o n as the i n v e r s e of a d d i t i o n . S t e f f e , Spikes and H i r s t e i n (1976) suggested t h a t m i s s i n g addends were i n t r o d u c e d as a means of connecting a d d i t i o n and s u b t r a c t i o n w i t h the purpose of t e a c h i n g c h i l d r e n t o use a d d i t i o n f a c t knowledge to "unlock" s u b t r a c t i o n f a c t s . Reasons f o r i n c l u d i n g m i s s i n g addends a t the f i r s t year l e v e l u n f o r t u n a t e l y were more l o g i c a l l y than e m p i r i c a l l y based (Flournoy, 1964; Groen & P o l l , 1973; S t e f f e e t a l . , 1976), and by the 1970's educators were e x p r e s s i n g concern about the a p p r o p r i a t e n e s s of i n t r o d u c i n g m i s s i n g addend sentences t o c h i l d r e n a t t h i s l e v e l . T h i s concern stemmed from a number of sources. One source was from primary 3 t e a c h e r s who had experienced f i r s t h a n d the d i f f i c u l t y many young c h i l d r e n have wi t h i n t e r p r e t i n g and completing these open sentences. Another source was from r e s e a r c h t h a t examined performance on open sentences and found performance d i f f e r e n c e s f o r sentences of v a r i o u s forms. In an attempt to e x p l a i n these performance d i f f e r e n c e s , s t u d i e s have examined a number of performance v a r i a b l e s i n c l u d i n g the s o l u t i o n s t r a t e g i e s c h i l d r e n use t o s o l v e m i s s i n g addends. St u d i e s t h a t have examined the l o g i c a l s t r u c t u r e s of young c h i l d r e n i n an attempt to determine what types of t h i n k i n g are a v a i l a b l e to them and t h e r e f o r e what asp e c t s of the c u r r i c u l u m are a p p r o p r i a t e to them, have a l s o been a source concerned with the s u i t a b i l i t y of the m i s s i n g addend problem. F i n d i n g s from a l l of these sources of concern i n d i c a t e t h a t the l o g i c a l l y d e r i v e d e x p e c t a t i o n s of the programme developers who i n t r o d u c e d the m i s s i n g addend, may not be being met. A number of s t u d i e s (Howlett, 1973; Peck & Jencks, 1976; S t e f f e e t a l . , 1976; Weaver, 1976) i n d i c a t e t h a t c h i l d r e n do not n e c e s s a r i l y u t i l i z e the u n i f y i n g ideas t h a t supposedly the m i s s i n g addend problem develops, but i n s t e a d use s o l u t i o n methods t h a t v a r y i n s o p h i s t i c a t i o n and r e f l e c t v a r y i n g l e v e l s of m a t u r i t y and understanding of the u n d e r l y i n g concept. These s t u d i e s have i n v e s t i g a t e d the s o l u t i o n methods of year one c h i l d r e n and year t h r e e c h i l d r e n , and have found t h a t these groups process m i s s i n g 4 addends q u i t e d i f f e r e n t l y . Because no s t u d i e s were found t h a t examined the performance of year two c h i l d r e n on m i s s i n g addend sentences, second year c h i l d r e n were chosen f o r t h i s study. I t was hoped t h a t the s t r a t e g i e s used by these c h i l d r e n would i n d i c a t e a p o s s i b l e sequence of t r a n s i t i o n between the l e s s and more mature s t r a t e g i e s i d e n t i f i e d i n the p r e v i o u s s t u d i e s . A f u r t h e r reason f o r u s i n g second year c h i l d r e n was to ensure t h a t the c h i l d r e n had had experience w i t h m i s s i n g addends and with counting and computation w i t h t w o - d i g i t numbers. D e f i n i t i o n of Terms The f o l l o w i n g terms are used throughout the study and are d e f i n e d here f o r convenience. M i s s i n g addend sentences r e f e r t o open a d d i t i o n sentences of the form a + O = b and O + a = b. P l a c e h o l d e r p o s i t i o n r e f e r s t o the l o c a t i o n of the p l a c e h o l d e r (Q) i n the open sentence, namely l e f t p l a c e h o l d e r p o s i t i o n ( • + a = b) and r i g h t p l a c e h o l d e r p o s i t i o n (a + • = b ) . L e v e l of d i f f i c u l t y r e f e r s to the magnitude of the numbers used i n the open sentence. In p a r t i c u l a r , L e v e l One i n v o l v e s b a s i c f a c t s with sums l e s s than t e n , L e v e l Two i n v o l v e s b a s i c f a c t s with sums l e s s than 18, L e v e l Three i n v o l v e s a t w o - d i g i t number p l u s a o n e - d i g i t number, and L e v e l Four i n v o l v e s a t w o - d i g i t number p l u s a t w o - d i g i t number where no regrouping i s r e q u i r e d f o r 5 the u s u a l a d d i t i o n a l g o r i t h m . The B a s i c F a c t Domain r e f e r s t o the group of a l l p o s s i b l e b a s i c f a c t combinations. The Two-Digit Domain r e f e r s t o the group of a l l p o s s i b l e combinations w i t h sums l e s s than 1 0 0 where a t l e a s t one addend i s a t w o - d i g i t number. L e v e l of response r e f e r s t o the observed degree of sensory involvement used i n a s o l u t i o n p r o c e s s . E x t e r n a l i z e d l e v e l of response r e f e r s to s o l u t i o n processes t h a t r e l y on e x t e r n a l c o n c r e t e a i d s and m u l t i s e n s o r y involvement. I n t e r n a l i z e d l e v e l of response r e f e r s t o s o l u t i o n processes where no observable use of a i d s (concrete or m u l t i s e n s o r y ) i s encountered, and i n s t e a d i n t e r n a l i z e d processes are used. The T r a n s i t i o n a l l e v e l of response r e f e r s t o responses t h a t r e l y on v a r i o u s combinations of m u l t i s e n s o r y involvement without the use of c o n c r e t e m a t e r i a l s ; f i n g e r use i s the most common example of t h i s l e v e l . S o l u t i o n s t r a t e g y r e f e r s t o the a c t u a l reasoning process used by the c h i l d t o determine the m i s s i n g addend. For example, counting processes are f r e q u e n t l y used as s o l u t i o n s t r a t e g i e s . Most s t r a t e g i e s can be used a t v a r i o u s l e v e l s of response, f o r example, c o u n t i n g w i t h the help of b l o c k s , f i n g e r s , or an imagined number l i n e . Statement of the Problem T h i s study was designed t o examine the accuracy 6 and t h e s o l u t i o n methods o f y e a r two c h i l d r e n when s o l v i n g m i s s i n g addend s e n t e n c e s . The q u e s t i o n s o f t h e s t u d y c o n c e r n f o u r a s p e c t s o f t h i s p e r f o r m a n c e : (1) t h e e f f e c t o f t h e s e n t e n c e v a r i a b l e s , l e v e l o f d i f f i c u l t y a n d p l a c e h o l d e r p o s i t i o n , o n p e r f o r m a n c e ; (2) t h e e f f e c t o f s o l u t i o n s t r a t e g y on p e r f o r m a n c e ; (.3) t h e e f f e c t o f l e v e l o f r e s p o n s e u s e on p e r f o r m a n c e ; and (.4) t h e r e l a t i o n s h i p o f C l a s s I n c l u s i o n a b i l i t y t o M i s s i n g Addend p e r f o r m a n c e . The s p e c i f i c q u e s t i o n s i n e a c h o f t h e s e f o u r a r e a s grow o u t o f t h e l i t e r a t u r e r e v i e w and a r e l i s t e d a t t h e b e g i n n i n g o f C h a p t e r 3. P u r p o s e o f t h e S t u d y T h i s s t u d y was d e s i g n e d f o r t h e p u r p o s e o f : (1) d e t e r m i n i n g t h e s u i t a b i l i t y o f e m p h a s i z i n g m i s s i n g addend s e n t e n c e s a t t h e y e a r one and two l e v e l s ; (2) d e t e r m i n i n g a p p r o p r i a t e s o l u t i o n s t r a t e g i e s f o r t h i s age g r o u p ; (3) i d e n t i f y i n g a p o s s i b l e s e q u e n c e o f d e v e l o p m e n t o f i n c r e a s i n g l y s o p h i s t i c a t e d s o l u t i o n s t r a t e g i e s t h a t c o u l d i n d i c a t e a t e a c h i n g s e q u e n c e ; (4) d e t e r m i n i n g p r e f e r r e d l e v e l s o f r e s p o n s e f o r t h e i d e n t i f i e d s t r a t e g i e s ; and (5) d e t e r m i n i n g i f a 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 d e v e l o p m e n t a l a b i l i t i e s a n d p e r f o r m a n c e on m i s s i n g addend s e n t e n c e s . 7 Assumptions of the Study I t was assumed t h a t s o l u t i o n s t r a t e g i e s c o u l d be i d e n t i f i e d u s i n g i n d i v i d u a l i n t e r v i e w t e c hniques. I t was assumed t h a t the c h i l d r e n would use s o l u t i o n s t r a t e g i e s t h a t r e f l e c t e d t h e i r c o g n i t i v e a b i l i t i e s . I t was assumed t h a t second year students would be capable of e x p l a i n i n g t h e i r s o l u t i o n s t r a t e g i e s and would re p r e s e n t a v a r i e t y of P i a g e t i a n o p e r a t i o n a l l e v e l s . L i m i t a t i o n s of the Study The f o l l o w i n g l i m i t a t i o n s apply t o t h i s study: (1) The sample was drawn from o n l y one s c h o o l d i s t r i c t . (2) The performance of the c h i l d r e n , i n the i n d i v i d u a l i n t e r v i e w s i t u a t i o n was not n e c e s s a r i l y t y p i c a l of classroom performance. (.3) The study was c r o s s - s e c t i o n a l r a t h e r than l o n g i t u d i n a l in. d e s i g n and t h e r e f o r e p r o v i d e d o n l y l i m i t e d support f o r the proposed sequences of development of both s o l u t i o n s t r a t e g i e s and l e v e l s of response. (4) The number and nature of the items used t o measure performance was l i m i t e d . Chapter 2 REVIEW OF RELATED LITERATURE T h i s chapter d e a l s f i r s t w i t h the l i t e r a t u r e comparing c h i l d r e n ' s performance on open sentences of v a r i o u s types. P a r t i c u l a r a t t e n t i o n i s p a i d to m i s s i n g addend accuracy d i f f e r e n c e s due to p l a c e h o l d e r p o s i t i o n and l e v e l of d i f f i c u l t y of the number combinations; reasons f o r these d i f f e r e n c e s are d i s c u s s e d . Secondly, the l i t e r a t u r e r e l a t e d t o s o l u t i o n s t r a t e g i e s used by primary c h i l d r e n t o s o l v e open sentences i s d i s c u s s e d . F u r t h e r , the r o l e of l e v e l of response as i t r e l a t e s t o these s o l u t i o n s i s i n c l u d e d . T h i r d l y , l i t e r a t u r e r e l a t e d t o C l a s s I n c l u s i o n a b i l i t y as i t r e l a t e s t o performance on m i s s i n g addends i s presented. Performance on Open Sentences A number of s t u d i e s have suggested t h a t performance d i f f e r e n c e s on v a r i o u s forms of open sentences are due to v a r i a t i o n s i n experience with c e r t a i n forms. For i n s t a n c e , the a 1 b =Q i s the most f r e q u e n t l y used open sentence i n most f i r s t year books, while Q= a l b sentences are seen l e s s o f t e n . I t i s not unexpected t h a t c h i l d r e n ' s a b i l i t y t o s o l v e these sentences should v a r y as a f u n c t i o n of t h i s 9 o p p o r t u n i t y t o l e a r n . Weaver (1971) found t h i s t o be the case i n h i s i n v e s t i g a t i o n of the f a c t o r s a s s o c i a t e d with p u p i l s ' performance on simple open a d d i t i o n and s u b t r a c t i o n sentences. He found t h a t accuracy on a 1 b =Q w a s g r e a t e r than a iQ="b which.was-greater than Q t a = b f o r c h i l d r e n i n years one and two. Weaver suggested however t h a t some of the observed d i f f e r e n c e s i n accuracy might have been a t t r i b u t a b l e t o the f a i l u r e o r i n a b i l i t y of c h i l d r e n t o use c e r t a i n mathematical i d e a s or p r o p e r t i e s . Grouws (1971) found t h a t t h i r d grade c h i l d r e n ' s accuracy on a + Q = b was g r e a t e r than Q+ a = b which was g r e a t e r than a -Q = b which, i n t u r n , was g r e a t e r than - a = b. He suggested more i n s t r u c t i o n a l time be devoted t o [ ] - a = b forms t o attempt t o e q u a l i z e performance l e v e l s . Grouws a l s o suggested t h a t c e r t a i n sentence forms may be i n h e r e n t l y more d i f f i c u l t than o t h e r s and i n c r e a s e d experience may not be necessary or s u f f i c i e n t t o improve performance. The i n h e r e n t d i f f i c u l t y of c e r t a i n problem types was examined by S t e f f e and Johnson (1971) u s i n g o r a l problem s i t u a t i o n s . For problems i n v o l v i n g d e s c r i b e d a c t i o n , they found t h a t accuracy on problems of the form a + b =Q was g r e a t e r than a + Q = b which was g r e a t e r than a - b =Q which, i n t u r n was g r e a t e r than Q- a = b. T h i s f i n d i n g , based on o r a l problem s i t u a t i o n s , suggests t h a t something more than experience with open sentence forms c o n t r i b u t e s t o the d i f f i c u l t y of c e r t a i n open sentences. 10 Suppes (1967) has a l s o suggested t h a t c e r t a i n open sentence forms are i n h e r e n t l y more d i f f i c u l t , and a t t r i b u t e d p a r t of t h i s d i f f i c u l t y t o the t r a n s f o r m a t i o n r e q u i r e d f o r s o l u t i o n . For i n s t a n c e , a l b = • i s i n c a n o n i c a l form and can be s o l v e d without t r a n s f o r m a t i o n whereas a t O = b must be transformed to c a n o n i c a l form before s o l u t i o n i s p o s s i b l e (a + D = b >0= b - a > b - a = • ) . Accuracy on M i s s i n g Addend Open Sentences P l a c e h o l d e r p o s i t i o n . S e v e r a l s t u d i e s have examined the comparative d i f f i c u l t y of d i f f e r e n t p l a c e h o l d e r p o s i t i o n s i n m i s s i n g addend sentences. One of the most comprehensive of these s t u d i e s was by Grouws (1971). Grouws i n v e s t i g a t e d the performance of 32 grade t h r e e students on open sentences of the g e n e r a l form a + Q = b, Q + a = b, a - [~] = b, and []] - a = b, where a + b are whole numbers. Sentences i n v o l v e d two number domains: the b a s i c f a c t domain, ^ (a, b, c) such t h a t a, b, and c are c o u n t i n g numbers, a + b = c, a ^ b , l l < c < 1 8 , and 1 < a, b -^ 9^; and the t w o - d i g i t domain,^(a, b, c) such t h a t a, b, and c are counting numbers, a + b = c, a ^ b, 42 c 99, 21 < a, b < 78^. Grouws found t h a t open sentence accuracy was a f f e c t e d by open sentence form. Accuracy on a + [] = b was g r e a t e r than accuracy on a = b, however d i f f e r e n c e s were not s i g n i f i c a n t . Accuracy on [ ] + a = b was not 11 s t a t i s t i c a l l y d i f f e r e n t from accuracy on any of the o t h e r forms. However, accuracy on a +[] = b was g r e a t e r than a - Q = b which was g r e a t e r than Q - a = b, and these d i f f e r e n c e s were s i g n i f i c a n t a t the .05 l e v e l . Weaver (1973) examined the performance of f i r s t , second, and t h i r d year p u p i l s on open a d d i t i o n and s u b t r a c t i o n sentences generated from b a s i c f a c t s having sums between t e n and e i g h t e e n . He i d e n t i f i e d s e v e r a l f a c t o r s t h a t appeared to a f f e c t accuracy on these sentences. Weaver's d e s c r i p t i v e data seemed to i n d i c a t e t h a t d i f f e r e n t i achievement e f f e c t s e x i s t e d a t the t h r e e grade l e v e l s . These e f f e c t s were: o p e r a t i o n ( a d d i t i o n , s u b t r a c t i o n ) , p l a c e h o l d e r p o s i t i o n , symmetric form (a + b = c' , c = a + b) and s o l v a b i l i t y i n the set of whole numbers. Weaver's r e s u l t s f o r m i s s i n g addend p l a c e h o l d e r p o s i t i o n i n d i c a t e g r e a t e r accuracy on a + [ ] = c than Q + a - c items w i t h s o l u t i o n s w i t h i n the s e t of whole numbers a t a l l t h r e e grade l e v e l s . S i g n i f i c a n c e l e v e l s were not r e p o r t e d . Weaver suggested t h a t these f i n d i n g s r e f l e c t i n no s m a l l degree the r e l a t i v e a t t e n t i o n g i v e n to v a r i o u s open sentence forms i n primary mathematics programmes. Suppes, Jerman and B r i a n (1968) concluded t h a t open sentences of the type Q + x = y are more d i f f i c u l t than open sentences of the type x + Q = y. T h i s c o n c l u s i o n was based on the r e s u l t s of an i n v e s t i g a t i o n of t h i r d and f o u r t h grade c h i l d r e n ' s performance on s o l v i n g open 12 sentences i n v o l v i n g a d d i t i o n . However, t h i s performance d i f f e r e n c e may have been confounded by the f a c t t h a t the number s i z e f o r the m i s s i n g addend was not c o n s t a n t f o r the two p l a c e h o l d e r p o s i t i o n s . The format x + = y g e n e r a l l y had o n e - d i g i t numbers f o r the m i s s i n g addend while i n the Q + x = y format, the m i s s i n g addends were t w o - d i g i t numbers. Groen and P o l l (1973), i n a study designed to i n v e s t i g a t e the a l g o r i t h m s used by c h i l d r e n i n s o l v i n g m i s s i n g addend sentences, found no d i f f e r e n c e i n e r r o r r a t e s and o n l y a s l i g h t d i f f e r e n c e i n mean r e a c t i o n times f o r sentences of the form x + Q = y and Q+ x = y. Howlett, (.1973) examined the performance of 115 f i r s t year students on a + [ ] = b and Q+ a = b m i s s i n g addends w i t h s o l u t i o n s w i t h i n the s e t of whole;numbers. Howlett found t h a t performance on []+ a = b was b e t t e r than performance on a +0 = b. Howlett's f i n d i n g as f a r as comparative d i f f i c u l t y of p l a c e h o l d e r p o s i t i o n i s concerned, c o n t r a d i c t s f i n d i n g s from the other s t u d i e s c i t e d . Suppes (1967) proposed s e v e r a l f a c t o r s t h a t c o n t r i b u t e d t o the observed d i f f i c u l t y of an open sentence. Three s t r u c t u r a l v a r i a b l e s were i d e n t i f i e d as c o n t r i b u t o r s t o t h i s d i f f i c u l t y . Suppes i d e n t i f i e d the v a r i a b l e . NSTEPS as the most important and p s y c h o l o g i c a l l y s i g n i f i c a n t v a r i a b l e as f a r as response l a t e n c y was concerned. NSTEPS r e f e r s to the sum of the number of steps r e q u i r e d t o t r a n s f o r m the problem i n t o c a n o n i c a l 13 form (a + b = Q ) , the number of o p e r a t i o n s performed, and the number of d i g i t s t h a t must be h e l d i n memory. The t r a n s f o r m a t i o n step r e f e r s t o t r a n s f o r m i n g the sentence so t h a t the blank or unknown stands by i t s e l f as the o n l y term to the r i g h t of the equal s i g n . For i n s t a n c e , m 1 n =Q i s a l r e a d y i n c a n o n i c a l form, but m +Q = p r e q u i r e s two steps t o t r a n s f o r m i t , f i r s t t o | j = p - m, then t o p - m = £ ] . S i m i l a r l y , Q + m = p r e q u i r e s two st e p s : Q = p - m, then p - m =Q. An a p p l i c a t i o n of the Suppes (1967) process model t o the two sentence forms i n d i c a t e s t h a t they should be of equal d i f f i c u l t y , s i n c e both sentence forms r e q u i r e two steps t o tra n s f o r m them i n t o c a n o n i c a l form ( a l l o t h e r f a c t o r s being e q u a l ) . L e v e l of d i f f i c u l t y of number combinations. Another f a c t o r t h a t has been examined as i t r e l a t e s t o performance on open sentences, i s the s i z e of the number con s t a n t s used i n the open sentence. Many s t u d i e s have examined the r e l a t i v e d i f f i c u l t y of a d d i t i o n and s u b t r a c t i o n combinations i n c a n o n i c a l format (a - b =Q) . Suydam and Weaver (1975) r e p o r t e d t h a t r e s e a r c h g e n e r a l l y • i n d i c a t e s t h a t the s i z e of the addends i s the p r i n c i p a l i n d i c a t o r of d i f f i c u l t y . S t u d i e s such as those by Washburne and Vogel (1928) and Clapp (1924) i l l u s t r a t e the e f f e c t of number s i z e on the r e l a t i v e d i f f i c u l t y of number combinations. With a few exc e p t i o n s such as doubles (for example, 7 + 7 = 14), sums g r e a t e r than t en are g e n e r a l l y c o n s i d e r e d t o be more 14 d i f f i c u l t to l e a r n than sums l e s s than t e n . R e c e n t l y a l i m i t e d number of s t u d i e s have examined the r o l e of number s i z e on the d i f f i c u l t y of open sentences of formats other than a 1 b =Q]. Grouws (1971) found t h a t number s i z e a f f e c t e d performance. His f i n d i n g s i n d i c a t e t h a t performance on items from the t w o - d i g i t domain was not as good as performance on items from the b a s i c f a c t domain. Grouws found t h a t e r r o r s on the t w o - d i g i t items were not p r o c e s s i n g e r r o r s and suggested t h a t the d i f f i c u l t y of the t w o - d i g i t items i n v o l v e d more than the computational complexity of p r o c e s s i n g l a r g e r numbers. A s i g n i f i c a n t i n t e r a c t i o n of open sentence type and number s i z e was found and was a t t r i b u t e d t o the extreme d i f f i c u l t y of the Q- a = b type, i n t h a t t h i s d i f f i c u l t y obscured the i n f l u e n c e of number s i z e . F i n d i n g s by Groen (1967), Parkman (1971), and Suppes and Groen (1967) i n d i c a t e d a r e l a t i o n s h i p between magnitude of constants and open sentence d i f f i c u l t y . These s t u d i e s proposed the use of counting models of v a r i o u s types i n the s o l u t i o n of open sentences, and used response l a t e n c i e s as a measure of sentence d i f f i c u l t y . Parkman found t h a t the minimum addend i n sentences of the form a + b =0 c o n t r i b u t e d more v a r i a n c e than the s i z e of the sum or the s i z e of the g r e a t e r addend. Groen (1967) and Suppes and Groen (1967) supported these f i n d i n g s . 15 Groen and P o l l (1973) have i d e n t i f i e d the s i z e of the m i s s i n g addend t o be a major determiner of; response l a t e n c i e s f o r m i s s i n g addend sentences. T h i s f i n d i n g i s s i m i l a r t o the f i n d i n g s of Groen (1967), Parkman (1971), and Suppes and Groen (1967), f o r a d d i t i o n sentences i n c a n o n i c a l form, i n t h a t the s i z e of the addend t o be added i s the major determiner of response l a t e n c y . In t h e i r overview of the r e l a t i v e d i f f i c u l t y of number combinations, Suydam and Weaver (1975) r e p o r t e d t h a t combinations w i t h a common addend appeared t o be of s i m i l a r but not equal d i f f i c u l t y . They f u r t h e r r e p o r t e d t h a t the doubles appeared to be e a s i e s t along w i t h combinations t h a t i n v o l v e d one or two as an addend. Suppes and Groen (1967) and Groen (1967) a l s o r e c o g n i z e d t h a t doubles had lower l a t e n c i e s and suggested t h a t doubles may be s o l v e d by d i r e c t memory access r a t h e r than the counting models they proposed. Summary of Research on Open Sentence  Performance D i f f e r e n c e s S t u d i e s concerning performance on open sentences of v a r i o u s types i n d i c a t e d t h a t c h i l d r e n i n years one and two have more d i f f i c u l t y w ith m i s s i n g addends than w i t h a + b =d p l a c e h o l d e r p o s i t i o n . Reasons f o r t h i s d i f f e r e n c e i n c l u d e d amount of p r a c t i c e and i n h e r e n t d i f f i c u l t y of sentence types. Stud i e s which have examined d i f f e r e n c e s i n performance on m i s s i n g addend p l a c e h o l d e r p o s i t i o n g e n e r a l l y i n d i c a t e s l i g h t l y b e t t e r performance on a +Q = b than \^}+ a = b examples, or no d i f f e r e n c e i n performance. The p r e s e n t study i n c l u d e d both a +Q = b and Q + a = b p l a c e h o l d e r s i n an attempt to examine r e l a t i v e d i f f i c u l t y f o r year two c h i l d r e n . The magnitude of the c o n s t a n t s was found t o be r e l a t e d t o performance. F u r t h e r , the magnitude of the m i s s i n g addend was i d e n t i f i e d as the major c o n t r i b u t o r t o the d i f f i c u l t y f o r an item. In the p r e s e n t study, f o u r l e v e l s of d i f f i c u l t y (as d e f i n e d by the magnitude, of the numbers) were i n c l u d e d , and a t each l e v e l , the magnitude of the m i s s i n g addend was h e l d c o n s t a n t f o r both p l a c e h o l d e r p o s i t i o n s . Combinations i n v o l v i n g doubles or one or two as an addend were found to be e a s i e r than other combinations. For t h i s reason, these combinations were not used i n t h i s study. M i s s i n g Addend S o l u t i o n S t r a t e g i e s Studies concerning performance d i f f e r e n c e s on m i s s i n g addend sentences suggest t h a t f a c t o r s such as p l a c e h o l d e r p o s i t i o n and number s i z e a f f e c t performance. A number of s t u d i e s have looked a t another a s p e c t of m i s s i n g addend performance i n an attempt to determine p o s s i b l e causes f o r d i f f e r e n c e s i n performance on open sentence forms. These s t u d i e s have focused on the s o l u t i o n s t r a t e g i e s c h i l d r e n u t i l i z e t o s o l v e open sentences, p a r t i c u l a r l y m i s s i n g addend sentences. These s t r a t e g i e s are thought t o i n d i c a t e a p p r o p r i a t e procedures f o r t e a c h e r s to develop w i t h c h i l d r e n i n order to have them m e a n i n g f u l l y complete m i s s i n g addend sentences. Stud i e s concerning s o l u t i o n s t r a t e g i e s are of two types. The f i r s t type i n v o l v e d p u p i l o b s e r v a t i o n as a means of determining s t r a t e g i e s . The second type of study i n v o l v e d a p r i o r i i d e n t i f i c a t i o n of p o s s i b l e s t r a t e g i e s . Response l a t e n c y data was f i t t e d t o l i n e a r r e g r e s s i o n models of the p o s s i b l e s t r a t e g i e s , as a means of determining which s t r a t e g i e s were a c t u a l l y used. Studi e s U t i l i z i n g P u p i l O b s e r v a t i o n Some of these s t u d i e s were of a d e s c r i p t i v e nature. These i n c l u d e d c r o s s - s e c t i o n a l s t u d i e s and experimental s t u d i e s which i n v o l v e d a treatment i n a d d i t i o n t o d e s c r i p t i o n s of s t r a t e g y use. Other s t u d i e s not o n l y d e s c r i b e d s t r a t e g y use but a l s o attempted t o i d e n t i f y a sequence of development of p r o g r e s s i v e l y more mature s o l u t i o n s t r a t e g i e s . The l o n g i t u d i n a l s t u d i e s and the c r o s s - s e c t i o n a l s t u d i e s u t i l i z i n g c h i l d r e n of s e v e r a l age l e v e l s , p r o v i d e d i n f o r m a t i o n concerning the sequence of development of s o l u t i o n s t r a t e g i e s . D e s c r i p t i v e s t u d i e s . S e v e r a l r e c e n t s t u d i e s have u t i l i z e d i n d i v i d u a l i n t e r v i e w techniques to t r y t o i d e n t i f y the s o l u t i o n s t r a t e g i e s used by c h i l d r e n t o s o l v e open sentences of v a r i o u s types. One of these i s the study by Peck and Jencks (1976). Peck and Jencks i n v e s t i g a t e d 18 the a b i l i t y of 142 f i r s t year c h i l d r e n to s o l v e m i s s i n g addend sentences of v a r i o u s forms. I n d i v i d u a l i n t e r v i e w s were used i n an attempt t o answer ques t i o n s c o n c e r n i n g s u c c e s s f u l and u n s u c c e s s f u l s o l u t i o n s t r a t e g i e s and b e h a v i o r s . An i n s t r u c t i o n a l p e r i o d was a l s o used to examine the s u i t a b i l i t y of t e a c h i n g m i s s i n g addends to f i r s t year c h i l d r e n . R e s u l t s concerning s o l u t i o n s t r a t e g i e s showed t h a t of the 103 c h i l d r e n who were s u c c e s s f u l w i t h m i s s i n g addends such as 5 + CU = 7 and C]+ 4 = 9 , 80 used observable c o u n t i n g s t r a t e g i e s such as counting /on f i n g e r s or t a p p i n g . Six c h i l d r e n appeared to count or reason m e n t a l l y , w h i l e 17 used r e c a l l of b a s i c f a c t s . Peck and Jencks found t h a t the c h i l d r e n who used counting s t r a t e g i e s were a b l e to apply t h e i r methods t o c o n c r e t e s i t u a t i o n s and were q u i c k l y a b l e to extend t h e i r a b i l i t y t o s o l v e problems such as 3 Q + 5 = 14, while the c h i l d r e n who u t i l i z e d r e c a l l were unable to g e n e r a l i z e t h e i r procedures t o other problem s i t u a t i o n s . Peck and Jencks suggested t h a t c h i l d r e n c o u l d v e r y w e l l become b e t t e r a t s o l v i n g m i s s i n g number problems i f they were c o n s c i o u s l y taught p h y s i c a l counting and s t r a t e g i e s f o r c o u n t i n g . S t e f f e , Spikes, and H i r s t e i n (1976) i n v e s t i g a t e d performance on o r a l l y presented m i s s i n g addend items as an aspect of an i n v e s t i g a t i o n of two P i a g e t i a n v a r i a b l e s as r e a d i n e s s v a r i a b l e s f o r f i r s t grade mathematics achievement. They a l s o attempted to teach the f i r s t 19 year c h i l d r e n a l o g i c a l method of s o l v i n g the m i s s i n g addend q u e s t i o n but found t h a t the method seemed to be c o n c e p t u a l l y beyond the group. As a r e s u l t of t h e i r f i n d i n g s , S t e f f e e t a l . suggested t h a t c ounting a b i l i t i e s be the determiner of whether or not m i s s i n g addend sentences are presented. They found t h a t c h i l d r e n who were a b l e t o count-on p r e f e r r e d t o use a counting-on w i t h t a l l y method to s o l v e m i s s i n g addend examples r a t h e r than the c o u n t - a l l ( s u b t r a c t i v e ) method they were i n i t i a l l y taught. T h i s s u b t r a c t i v e procedure i n v o l v e d counting back from the sum, the number of the g i v e n addend. T h i s procedure i s l a b e l l e d " s u b t r a c t i v e " f o r the purposes of the present study. C h i l d r e n who were not a b l e t o count-on had d i f f i c u l t y c o n c e p t u a l i z i n g t h i s procedure, however, and tended t o do what i n t h i s study i s r e f e r r e d t o as an " I n c o r r e c t T r a n s f o r m a t i o n " , t h a t i s , i n t e r p r e t i n g 3 + Q= 8 as 3 + 8 = • . S t e f f e et a l . s t a t e d t h a t the c o u n t i n g - a l l ( s u b t r a c t i v e ) procedure was used i n a r o t e f a s h i o n and t h a t the counting-on procedure seemed to be a more meaningful process f o r the c h i l d r e n . S t e f f e et a l . found t h a t procedures where, f o r example, g i v e n 3 + 4 = • the c h i l d counts "1, 2, 3,...4, 5, 6, 7", preceded counting-on procedures where the c h i l d counts "4, 5, 6, 7", and t h a t c h i l d r e n had d i f f i c u l t y a c q u i r i n g the a b i l i t y t o count-on i f i t 20 was not w i t h i n t h e i r c o g n i t i v e competence. Based on these two f i n d i n g s they hypothesized t h a t counting-on procedures were e s s e n t i a l t o meaningful s o l u t i o n of m i s s i n g addends and t h a t a c q u i s i t i o n of counting-on a b i l i t i e s was developmental i n nature. Attempts to t e a c h counting-on s k i l l s o n l y r e s u l t e d i n s i t u a t i o n - s p e c i f i c a l g o r i t h m s and d i d not g e n e r a l i z e t o c o n c e p t u a l l y s i m i l a r problem s i t u a t i o n s . S t e f f e et a l . are i n agreement with Grouws (1971) and Weaver (1976) i n proposing the v a l u e of encouraging c h i l d r e n to use t h e i r own meaningful methods such as counting-on. S t e f f e e t a l . suggest t h a t i n t h e i r own time, c h i l d r e n w i l l develop more e f f i c i e n t s o l u t i o n procedures. Howlett (1973) found t h a t the m i s s i n g addend s o l u t i o n methods used by f i r s t grade c h i l d r e n c l a s s i f i e d as unable to s o l v e the c l a s s i n c l u s i o n problem (Stage 1) were d i f f e r e n t from methods used by c h i l d r e n who were s u c c e s s f u l w i t h the c l a s s i n c l u s i o n problem (Stage 3 ) . The a b i l i t y t o s o l v e P i a g e t i a n c l a s s i n c l u s i o n problems has been shown t o be developmental i n nature (Inhelder & P i a g e t , 1964). Howlett's f i n d i n g s support the p r o p o s a l t h a t the s o p h i s t i c a t i o n of s o l u t i o n s t r a t e g i e s i s developmental i n nature and r e f l e c t s i n c r e a s i n g l e v e l s of m a t u r i t y . Howlett found t h a t the Stage 1 group used f i n g e r s more o f t e n , counted i n t e r n a l l y l e s s o f t e n , and took longer to s o l v e the items than Stage 3 c h i l d r e n . 21 Stage 1 c h i l d r e n i n c o r r e c t l y transformed the number sentence more o f t e n , and were wrong more o f t e n than Stage 3 c h i l d r e n . Although H o w l e t t 1 s r e s u l t s do not s p e c i f i c a l l y i d e n t i f y c o unting s t r a t e g i e s as such, from the data i t appears t h a t a d d i t i v e c ounting p a t t e r n s were used f r e q u e n t l y by both groups. Judging from response l a t e n c i e s i t appears t h a t a number of Stage 3 c h i l d r e n a l s o used r e c a l l of b a s i c f a c t s t o determine the m i s s i n g addend. Grouws (1971) i n v e s t i g a t e d the s o l u t i o n s t r a t e g i e s used by t h i r d grade c h i l d r e n t o s o l v e open sentences of the form a + Q = b, Q+ a = b, a - Q = b, and Q- a = b. He i d e n t i f i e d t h i r t e e n s t r a t e g i e s , f i v e of which had two s u b c l a s s i f i c a t i o n s . Grouws found the most f r e q u e n t l y used s t r a t e g y was use of a formal a d d i t i o n or s u b t r a c t i o n a l g o r i t h m by r e w r i t i n g the h o r i z o n t a l m i s s i n g addend sentence i n computational form and a p p l y i n g an a l g o r i t h m . The next most f r e q u e n t l y used s t r a t e g y was r e c a l l of b a s i c f a c t combinations. R e c a l l was used f o r 13% of the items, but o n l y w i t h items from the b a s i c f a c t domain. Counting on or back was used f o r 11% of the items, w h i l e a s u b s t i t u t i o n procedure ( s u b s t i t u t i n g answers u s i n g a s e r i e s of approximations to determine the m i s s i n g addend) was used i n 7% of the examples. Other s t r a t e g i e s were each used l e s s than 5% of the time. 22 The methods i d e n t i f i e d by Grouws ranged from i n f o r m a l i n t u i t i v e methods to a b s t r a c t formal methods. He proposed t h a t as c h i l d r e n develop i n t e l l e c t u a l l y and g a i n p r a c t i c e i n the use of formal methods, they s o l v e open sentences q u i t e e f f i c i e n t l y , but t h a t i n i t i a l i n s t r u c t i o n should hot impose these formal methods. He suggested t h a t encouraging c h i l d r e n t o f i n d t h e i r own methods may have important c o g n i t i v e and a f f e c t i v e outcomes. Since the use of formal methods was found t o be more accurate and e f f i c i e n t , Grouws suggested t h a t at some p o i n t c h i l d r e n should be encouraged to begin u s i n g more e f f i c i e n t means, but when t h i s t r a n s i t i o n p o i n t should occur depends on the i n d i v i d u a l c h i l d . Weaver (197 6) observed s o l u t i o n s t r a t e g y use as p a r t of an experimental study which taught t h i r d grade c h i l d r e n two s o l u t i o n s t r a t e g i e s f o r s o l v i n g open sentences of the form a + 0 = b, Q + a = b, a - Q = b, and O- a = b, wi t h a and b r e p l a c e d w i t h t h r e e - d i g i t numbers. The f i r s t s o l u t i o n s t r a t e g y taught was a "Guess-and-Test" procedure which i n v o l v e d making s u c c e s s i v e approximations and c o r r e c t i o n s . T h i s s t r a t e g y was then r e p l a c e d by a more d i r e c t and e f f i c i e n t approach i n v o l v i n g the t r a n s f o r m a t i o n of the open sentence t o i t s c a n o n i c a l sentence e q u i v a l e n t ( f o r example, a +Q = b t o b - a =L_P« Although these c h i l d r e n were f a m i l i a r w i t h the i n v e r s e r e l a t i o n s h i p of a d d i t i o n and s u b t r a c t i o n , they had not had experience e x p l i c i t l y changing open sentences and r e w r i t i n g them i n e q u i v a l e n t forms. Weaver found t h a t the e f f e c t i v e n e s s of s t r e s s i n g the t r a n s f o r m a t i o n approach left-much to be d e s i r e d i n t h a t r e l a t i v e l y few p u p i l s a p p a r e n t l y comprehended and used the u n i f y i n g i d e a t h a t was s t r e s s e d . Weaver proposed t h a t the s t r a t e g y of t r a n s f o r m i n g open sentences was too formal and l o g i c a l f o r t h i s age group, and t h a t i n f o r m a l , i n t u i t i v e methods were more a p p r o p r i a t e . He suggested t h a t educators should attempt t o modify approaches to make them "mathematically honest" a t a l e s s s o p h i s t i c a t e d or r i g o r o u s l e v e l (p. 21). S t u d i e s concerning the sequence of development  of s o l u t i o n s s t r a t e g i e s . S t u d i e s by Peck and Jencks (1976), S t e f f e e t a l . (1976), and Howlett (1973) have i n v e s t i g a t e d the s o l u t i o n s t r a t e g i e s of f i r s t year c h i l d r e n w h i le s t u d i e s by Grouws (1971) and Weaver (1976) have examined the s t r a t e g i e s of t h i r d year students. I t appears t h a t the two groups process m i s s i n g addends q u i t e d i f f e r e n t l y . S e v e r a l f a c t o r s , not the l e a s t of which i s mathematical experience, a f f e c t s o l u t i o n s t r a t e g i e s . How m i s s i n g addend s o l u t i o n s t r a t e g i e s change 24 from l e s s to more mature has been d e s c r i b e d by O'Brien and R i c h a r d (1971). O'Brien and Richard i n t e r v i e w e d c h i l d r e n aged s i x years three months to seven years t h r e e months a t the end of f i r s t grade. The m i s s i n g addend problem was presented c o n c r e t e l y r a t h e r than s y m b o l i c a l l y i n the f o l l o w i n g manner: Task 3: M i s s i n g addend - sum 10 or l e s s The t e a c h e r p l a c e s nine c h i p s on the t a b l e i n f r o n t of the c h i l d and asks: "How many c h i p s are i n t h i s s e t ? " . . . "I'm going t o hide some of the c h i p s with my hand. Would you ple a s e c l o s e your eyes?" The teacher covers f o u r of the c h i p s w i t h her hand and says: "Open your eyes now. How many c h i p s are under my hand?" "How do you know?" (p. 3 24) What O'Brien and Ri c h a r d c o n s i d e r e d as s u c c e s s i v e l y more s o p h i s t i c a t e d s o l u t i o n processes are as f o l l o w s : 1. The c h i l d sees no r e l a t i o n s h i p between the c h i p s t h a t are showing and those t h a t are hidden. The c h i l d s t a r e s at the i n t e r v i e w e r ' s hand and does not look a t the v i s i b l e c h i p s . 2. The c h i l d makes a g l o b a l comparison. I f a r e l a t i v e l y l a r g e number of c h i p s are showing, he c i t e s a s m a l l number, and v i c e v e r s a . The c h i l d observes the c h i p s t h a t are showing and may even count them, but he does not e x h i b i t any systematic means of determining the unknown. 3. The c h i l d demonstrates knowledge of a s p e c i f i c r e l a t i o n between the two subordinate c l a s s e s and the s u p e r o r d i n a t e c l a s s and uses t h i s knowledge t o determine the unknown. a) The c h i l d counts on. 25 (.1) I f nine c h i p s were shown o r i g i n a l l y and o n l y f i v e remain when the c h i l d opens h i s eyes, the c h i l d counts to f i v e ( o f t e n u s i n g h i s f i n g e r s ) , then says " S i x , seven, e i g h t , n i ne," and r e p o r t s t h a t f o u r c h i p s are hidden. (2) Some c h i l d r e n s t a r t out w i t h nine ( o f t e n on f i n g e r s ) and separate the nine i n t o two subordinate c l a s s e s , one of which has f i v e members. Then the c h i l d counts the members of the o t h e r s u b o r d i n a t e c l a s s , "One, two, t h r e e , f o u r . " b) The c h i l d c i t e s a number f a c t t h a t i s known and r e l a t e s the p r e s e n t s i t u a t i o n t o t h a t f a c t . For example, with nine o r i g i n a l c h i p s and f i v e showing, the c h i l d might say, "Four, because f i v e and f i v e i s t e n and you o n l y have ni n e . " c) The c h i l d r e c a l l s a number f a c t and r e l a t e s i t t o the p r e s e n t p h y s i c a l s i t u a t i o n . He c i t e s a s i n g l e number as the unknown and, when c a l l e d upon to j u s t i f y h i s response, says, f o r example, "Because 4 + 5 = 9" or "Because 9 take away 4 i s 5." (p. 325) Two s t u d i e s were found t h a t examined the s u c c e s s i v e processes c h i l d r e n use t o s o l v e a d d i t i o n and s u b t r a c t i o n sentences i n c a n o n i c a l form. These sequences may p a r a l l e l to some degree the development of s t r a t e g i e s used f o r m i s s i n g addend sentences, and f o r t h i s reason are i n c l u d e d i n t h i s s e c t i o n . I l g and Ames (1951) s t u d i e d the developmental nature of mathematical behaviours. T h i r t y s u b j e c t s from the ages of f i v e through nine were observed y e a r l y at each age l e v e l . T h i s made p o s s i b l e a l o n g i t u d i n a l as w e l l as a c r o s s - s e c t i o n a l examination of data. Two of the i d e n t i f i e d p a t t e r n s d e a l t with the emergence of a d d i t i o n and s u b t r a c t i o n p r o c e s s e s . Table 2.1 i l l u s t r a t e s age p a t t e r n s f o r these p r o c e s s e s . I l g and Ames i d e n t i f i e d c o u n t i n g - a l l s t r a t e g i e s as the f i r s t t o emerge f o r a d d i t i o n . T h i s r e f e r s t o a c h i l d enumerating both of the addends i n order t o reach the sum. Next, the a b i l i t y t o count-on emerged, a l l o w i n g the c h i l d to m e n t a l l y r e p r e s e n t the f i r s t addend and count-on the second addend. A more e f f i c i e n t counting-on s t r a t e g y i n v o l v e d the process of determining which addend was g r e a t e r and incrementing the s i z e of the s m a l l e r addend. T h i s s t r a t e g y r e q u i r e d an i n t u i t i v e understanding of the commutative p r i n c i p l e . A s h o r t p e r i o d where guessing was f r e q u e n t l y used appeared between i n i t i a l c o u n t i n g - a l l methods and the beginning of the use of r e c a l l . Doubles such as 4 + 4 = 8 were i d e n t i f i e d as the f i r s t t o be memorized. The doubles were then used t o help c h i l d r e n work out other sums by an i n t u i t i v e use of the a s s o c i a t i v e p r i n c i p l e . The s t r a t e g y l a b e l l e d " f i g u r i n g i t out i n head" r e f e r r e d t o such processes as s o l v i n g 7 + 9 =f~] by t h i n k i n g 7 + 7 = 14, 14 + 2 = 16, so 7 + 9 = 16 (or (7 + 7) + 2 = 7 + (7 + 2) = 16). T h i s type of procedure was used to f u r t h e r develop b a s i c f a c t mastery. Table 2.1* Age P a t t e r n s f o r A d d i t i o n and S u b t r a c t i o n Processes Age When Behavior i s P r e v a l e n t A d d i t i o n Behavior 4 Ah 5 5h 6 6h 7 lh 8 Counting a l l , w i t h o b j e c t s Counting a l l , without o b j e c t s Counting on from s m a l l e r addend Counting on from l a r g e r addend Guessing R e c a l l of b a s i c f a c t s (doubles) F i g u r i n g i t out i n head Age When Behavior i s P r e v a l e n t S u b t r a c t i o n Behavior 4 4% 5 5h 6 6h 7 lh 8 Counts a l l , from 1 t o sum, then counts back (with o b j e c t s ) Counts a l l , from 1 t o sum, then counts back (without o b j e c t s ) Guesses Counts back from sum Inverse R e l a t i o n s h i p 6 - 4 = ? 4 + 2 = 6.*. 6 - 4 = 2 — B a s i c f a c t r e c a l l ^ *T h i s t a b l e was org a n i z e d from I l g and Ames, 1951, p. 8. 28 S u b t r a c t i o n p a t t e r n s showed a s i m i l a r sequence of development from c o u n t i n g - a l l t o m e n t a l l y r e p r e s e n t i n g the sum and counting-back. R e l y i n g on a d d i t i o n f a c t s and r e c o g n i z i n g the i n v e r s e r e l a t i o n s h i p i n order t o apply t h i s f a c t knowledge appeared next. Again, r e c a l l of b a s i c f a c t s f o l l o w e d counting processes and developed with the help of grouping s t r a t e g i e s i n v o l v i n g the a s s o c i a t i v e p r i n c i p l e . Brownell (1928) u t i l i z e d i n d i v i d u a l i n t e r v i e w s w i t h 58 c h i l d r e n i n the f i r s t f o u r grades to examine the development of mature methods of d e a l i n g w i t h c o n c r e t e numbers. He s p e c i f i e d f o u r stages i n the development of the a b i l i t y to d e a l with the a d d i t i v e combinations. L e v e l One, counting, i s d e s c r i b e d as the stage where a c h i l d counts a l l , f o r example, g i v e n 4 + 3 = Q the c h i l d counts "1, 2, 3, 4...5, 6, 7". L e v e l Two, p a r t i a l c ounting, i s a counting-on technique where the c h i l d , g i v e n 4 + 3 = D counts "5, 6, 7". L e v e l Three, grouping, i s a procedure u t i l i z i n g the a s s o c i a t i v e p r o p e r t y by r e l y i n g on known combinations, and grouping them to reach the sum. For example, g i v e n 4 + 3 =Q, the c h i l d f i r s t adds 2, then 1 more, or ,(.4 + 2) + 1 = 7. L e v e l Four, m u l t i p l i c a t i o n and c o n v e r s i o n , r e l i e s again on groupings but a t a more s o p h i s t i c a t e d l e v e l . For example, 4 + 3 = ( 4 + 4 ) - 1 . Beyond these f o u r l e v e l s , meaningful h a b i t u a t i o n or r e c a l l develops. 29 The sequence of the l e v e l s i d e n t i f i e d by Brownell corresponds to the sequence i d e n t i f i e d by O'Brien and Richard (1971) and I l g and Ames (.1951) i n t h a t counting s t r a t e g i e s are f i r s t to appear. C o u n t i n g - a l l precedes counting-on i n these sequences. T h i s holds w i t h the f i n d i n g of S t e f f e e t a l . (1976), and serves to i l l u s t r a t e the v a l u e of counting s t r a t e g i e s to the development of more s o p h i s t i c a t e d procedures. A P r i o r i Models of S o l u t i o n S t r a t e g i e s A number of s t u d i e s designed to i n v e s t i g a t e the a l g o r i t h m s used by c h i l d r e n and a d u l t s i n s o l v i n g simple open sentences, have proposed counting models of v a r i o u s types (Groen & Parkman, 1972; Groen & P o l l , 1973; Parkman & Groen, 1971). These s t u d i e s are a l l based on the assumption t h a t r e a c t i o n time to any g i v e n sentence w i l l be a l i n e a r f u n c t i o n of the number of steps r e q u i r e d t o perform the t a s k , and t h a t s e t t i n g time and incrementing or decrementing time are c o n s t a n t . Counting models are e v a l u a t e d by f i t t i n g observed r e a c t i o n times to l i n e a r r e g r e s s i o n l i n e s of the form RT = a + bz where a i s the c o n s t a n t r e p r e s e n t i n g the mean time r e q u i r e d t o set the counter, b i s the c o n s t a n t r e p r e s e n t i n g the mean time r e q u i r e d t o increment or decrement by one, and z i s the s t r u c t u r a l v a r i a b l e d e f i n e d by the counting model under c o n s i d e r a t i o n . Groen and P o l l CI973) proposed three models to account f o r l a t e n c y data on m i s s i n g addend sentences: 1. s e t t i n g the counter t o the g i v e n addend and incrementing to the sum, the s o l u t i o n being the t a l l y of the increment; 2. s e t t i n g the counter t o the sum and decrementing the number of times i n d i c a t e d by the g i v e n addend, the s o l u t i o n being the f i n a l v a l u e of the counter; and 3. e i t h e r of the f i r s t two models depending on which i s f a s t e r , such t h a t the c h i l d judges whether the g i v e n addend i s g r e a t e r or l e s s than the m i s s i n g addend and chooses the process t h a t r e q u i r e s the minimum count. T h e i r f i n d i n g s support the use of model 3 f o r m i s s i n g addends of the form x + Q = y, but no model gave adequate account of the l a t e n c y data f or Q + x = y sentences,. S i m i l a r s t u d i e s have proposed counting models f o r open sentences of the form a 1 b = [ ] . Groen (1967) and Suppes and Groen (1967) proposed f i v e models f o r p o s s i b l e s o l u t i o n s t r a t e g i e s to a + b =Q forms: 1. s e t t i n g the counter a t one and incrementing twice, once f o r the v a l u e of each addend ( c o u n t i n g - a l l ) ; 2. s e t t i n g the counter t o the value of the f i r s t addend and incrementing the value of the second addend; 3. s e t t i n g the counter a t the v a l u e of the second addend and incrementing the v a l u e of the f i r s t addend; 31 4. s e t t i n g the counter a t the va l u e of the minimum addend and incrementing the v a l u e of the maximum addend; and 5. s e t t i n g the counter t o the va l u e of the maximum addend and incrementing f o r the v a l u e of the minimum addend. Suppes and Groen found a good f i t f o r a l i n e a r model based on 5. P r e v i o u s l y c i t e d s t u d i e s have i l l u s t r a t e d the developmental nature of s o l u t i o n s t r a t e g i e s . S t u d i e s by Brownell (1928), I l g and Ames (1951) and S t e f f e et a l . (1976) show t h a t c h i l d r e n f i r s t use a c o u n t i n g - a l l process, then p r o g r e s s t o the more e f f i c i e n t counting-on process. I l g and Ames show t h a t counting-on progresses from counting-on from the f i r s t addend t o counting-on from the maximum addend. T h i s suggests t h a t models one, two, and f i v e are a l l v i a b l e a l t e r n a t i v e s and c o u l d a pply depending on the sample used. A b a s i c assumption of any attempt t o p r e d i c t c o unting s t r a t e g i e s on the b a s i s of the l a t e n c y data of a group of c h i l d r e n i s t h a t a l l c h i l d r e n i n the group use the same counting s t r a t e g y . R e s u l t s of s t u d i e s by Brownell (1928), I l g and Ames (1951), O'Brien and Ri c h a r d (195.7), and S t e f f e e t a l . (1976), p r o v i d e support f o r the hypothesis t h a t c ounting s t r a t e g i e s are developmental i n nature and not u n i v e r s a l t o c h i l d r e n of a g i v e n grade. The f e a s i b i l i t y of u s i n g i n d i v i d u a l i n t e r v i e w procedures f o r d etermining counting s t r a t e g i e s has been i l l u s t r a t e d . For the purposes of t e s t i n g l i n e a r r e g r e s s i o n models, i n i t i a l s c r e e n i n g through i n t e r v i e w s would probably be a v a l u a b l e means of grouping s u b j e c t s b e f o r e a p p l y i n g t h e i r l a t e n c y scores to r e g r e s s i o n a n a l y s e s , thus e n s u r i n g a more homogeneous group upon which t o base p r e d i c t i o n s . Summary of S o l u t i o n S t r a t e g y L i t e r a t u r e S t u d i e s concerning f i r s t year c h i l d r e n ' s s o l u t i o n s t r a t e g i e s (Peck & Jencks, 1976; S t e f f e e t a l . , 1976; Howlett, 1973) i n d i c a t e d t h a t counting s t r a t e g i e s were used most f r e q u e n t l y . C o u n t i n g - a l l was shown t o be one s t r a t e g y , w h ile counting-on was another. R e c a l l was used by some c h i l d r e n , but o n l y f o r f a m i l i a r b a s i c f a c t combinations. T h i r d year c h i l d r e n were found most f r e -q u e n t l y to use formal a l g o r i t h m s i n v o l v i n g sentence t r a n s f o r m a t i o n s (Grouws, 1971). The next most f r e q u e n t l y used s t r a t e g y was r e c a l l , but again t h i s o n l y a p p l i e d to items from the b a s i c f a c t domain. How c h i l d r e n change from u t i l i z i n g c o u n t i n g - a l l s t r a t e g i e s to a p p l y i n g formal a l g o r i t h m s . was answered i n p a r t by t h r e e developmental s t u d i e s of s o l u t i o n processes (O'Brien & Richard, 1971; Brownell, 1928; I l g and- M e s , 1951). These s t u d i e s showed c o u n t i n g - a l l to precede counting-on. Next, v a r i o u s types of grouping procedures were i d e n t i f i e d . These procedures i n v o l v e d 33 incrementing i n groups of u n i t s r a t h e r than by ones, and .usually i n v o l v e d the a s s o c i a t i v e p r i n c i p l e i n some way. R e c a l l of b a s i c f a c t s s t a r t s to appear at t h i s p o i n t . T h i s sequence i s a p p l i c a b l e o n l y t o the b a s i c f a c t domain i n t h a t r e c a l l i s c o n s i d e r e d t o be the most s o p h i s t i c a t e d s t r a t e g y and i t i s o n l y a p p r o p r i a t e to b a s i c f a c t combinations. Presumably the next step f o r c h i l d r e n who can use r e c a l l may be to p r o v i d e them with a method t h a t can apply t o number combinations of any magnitude. For the purposes of e f f i c i e n c y an a l g o r i t h m such as t h a t i d e n t i f i e d by Grouws (1971) and Weaver (1976) may be a p p r o p r i a t e and undoubtedly necessary to f u t u r e work. L a s t l y , s e v e r a l s t u d i e s were c i t e d t h a t proposed counting s t r a t e g i e s as s o l u t i o n procedures. Latency data was then a p p l i e d t o r e g r e s s i o n models of these proposed s t r a t e g i e s as a t e s t of t h e i r use. R e s u l t s of the study d i r e c t l y r e l a t e d to m i s s i n g addend s o l u t i o n s t r a t e g i e s (Groen & P o l l , 1973) suggested f i r s t grade c h i l d r e n choose e i t h e r an incrementing or a decrementing procedure f o r x + LZl = y form depending on which i s most e f f i c i e n t . However, r e s u l t s from i n t e r v i e w s with f i r s t grade c h i l d r e n (Howlett, 1973; Peck & Jencks, 1976; S t e f f e e t a l . , 1976) do not support the Groen and P o l l c o n c l u s i o n . 34 Sensory Involvement i n S o l u t i o n . S t r a t e g i e s S t u d i e s which have i n v o l v e d the o b s e r v a t i o n of young children,' s m i s s i n g addend s o l u t i o n processes (O'Brien and Richard, 1971; Howlett, 1973; S t e f f e e t a l . , 1976) have r e p o r t e d the use of a wide v a r i e t y of a i d s t o these s o l u t i o n processes. These a i d s i n v o l v e sensory input of t h r e e main types: a u d i t o r y - o r a l involvement such as counting or t h i n k i n g aloud, h a p t i c involvement which r e f e r s t o t a c t i l e and k i n e s t h e t i c sensory i n p u t such as m a n i p u l a t i n g counters or u s i n g f i n g e r s ; and v i s u a l involvement such as l o o k i n g a t f i n g e r s , b l o c k s , or p i c t u r e s as an a i d to t h i n k i n g p r o c e s s e s . The degree of use of these a i d s i s c o n s i d e r e d to r e f l e c t to some extent the l e v e l of m a t u r i t y of the c h i l d ' s s o l u t i o n s p r ocesses, as i s i l l u s t r a t e d by the O'Brien and Richard (1971) sequence o f counting s o p h i s t i c a t i o n ; 1. The c h i l d touches the c h i p s , moves them i n t o two c l a s s e s - those counted and those not yet counted - and counts aloud, "One, two t h r e e , . . . . " 2. The c h i l d touches the c h i p s and counts aloud but does not p h y s i c a l l y move the c h i p s . 3. The c h i l d e s t a b l i s h e s a 1 t o 1 correspondence between h i s f i n g e r s and the set of s h i p s . He moves h i s f i n g e r s as he says, "One, two, t h r e e , II 4. The c h i l d e s t a b l i s h e s a 1 to 1 correspondence between c h i p s and p h y s i c a l movements such as head'nods, f i n g e r taps, or f o o t taps ^ as h i s eyes scan the c h i p s and he says aloud, "One, two, t h r e e , . . . . " 5. The c h i l d says the words ".One, two, t h r e e , . . . " as h i s eyes scan the c h i p s . 6. The c h i l d mouths the words "One, two, t h r e e , . . . " as h i s eyes scan the c h i p s . 7. The c h i l d scans the c h i p s with h i s eyes, then says aloud a s i n g l e number, (p. 323) T h i s sequence shows the dependence on a u d i t o r y , v i s u a l and h a p t i c sensory i n p u t as an accompaniment to l e s s mature co u n t i n g p r o c e s s e s . Each stage of the sequence shows a more i n t e r n a l i z e d procedure, due t o the g r a d u a l abandonment of e x t e r n a l sensory a i d s . T h i s sequence a p p l i e s t o enumeration processes but may apply as w e l l t o s o l u t i o n processes e s p e c i a l l y c ounting s t r a t e g i e s . Syer i n "Sensory Learning A p p l i e d to Mathematics" (1953) o u t l i n e s the r o l e of sensory i n p u t as i t r e l a t e s to the development of a b s t r a c t thought. He s t a t e s t h a t i t i s i m p o s s i b l e t o form c o r r e c t concepts of the meanings of p h y s i c a l u n i t s without some concrete experiences w i t h o b j e c t s measured i n these u n i t s . T h i n k i n g i s d e f i n e d by Syer as the m a n i p u l a t i o n of mental images. Sensory l e a r n i n g c r e a t e s these images. For mathematics the most important types of images are the memory image and the i m a g i n a t i o n image. The memory image i s e s t a b l i s h e d through the experience of a c t u a l sensory p e r c e p t i o n , w h i l e the i m a g i n a t i o n image i s c r e a t e d from the combination of p a r t s of memory images. A c t u a l sensory p e r c e p t i o n i s u s u a l l y p r o v i d e d through the u t i l i z a t i o n of c o n c r e t e m a t e r i a l s to r e p r e s e n t mathematical concepts such as the o p e r a t i o n of a d d i t i o n . G r a d u a l l y , c h i l d r e n make the t r a n s i t i o n from c o n c r e t e experience to a b s t r a c t thought. The nature of the t r a n s i t i o n from use of e x t e r n a l a i d s to the use of i n t e r n a l methods i s not e s t a b l i s h e d . A commonly h e l d b e l i e f i s t h a t p i c t o r i a l r e p r e s e n t a t i o n s t h a t r e l y almost e x c l u s i v e l y on the v i s u a l m o d a l i t y b r i d g e t h i s t r a n s i t i o n . T h i s emphasis on sensory i n p u t of a v i s u a l nature does not take advantage of c r i t i c a l sensory input of a t a c t i l e - k i n e s t h e t i c or a u d i t o r y nature. I t may w e l l be t h a t the t r a n s i t i o n from concrete m a n i p u l a t i o n s to a b s t r a c t thought r e l i e s more h e a v i l y on m u l t i - s e n s o r y i n p u t s than some mathematics educators p r e s e n t l y seem to r e a l i z e . Brownell (1928, p. 220) r e c o g n i z e d the need t o b r i d g e the gap between c o n c r e t e and a b s t r a c t numbers through r e c o u r s e to some form of immediate t a c t i l e - k i n e s t h e t i c experience. He s p e c i f i e d the a b i l i t y t o p e r c e i v e groups of o b j e c t s (grouping) as key t o developing the imagery necessary f o r a b s t r a c t number f a c i l i t y . He recommended experience w i t h grouped v i s u a l o b j e c t s ( f o r example, dot p a t t e r n s such as dominoes) as w e l l as experience w i t h t a c t u a l and a u d i t o r y groups (fo r example, tapping p a t t e r n s , both f e l t and heard). T a c t i l e - k i n e s t h e t i c a i d s t o counting and t a l l y i n g are f r e q u e n t l y u t i l i z e d by young c h i l d r e n . A c t i o n s such as f i n g e r c o u n t i n g , p r e s s i n g or tapp i n g the body, f o o t tapping, and head nodding are o f t e n observed as c h i l d r e n count out sums and d i f f e r e n c e s . Often the use of such a i d s i s disc o u r a g e d by parents and educators, presumably based on the argument t h a t these procedures are immature and w i l l slow down mental c a l c u l a t i o n s and b u i l d dependency on f i n g e r s . S t e f f e et a l . (1976) found t h a t the c h i l d r e n they s t u d i e d soon abandoned the use of concrete m a t e r i a l s f o r completing sums and d i f f e r e n c e s and r e l i e d on f i n g e r s . S t e f f e et a l . (p. 175) s t a t e : "The hy p o t h e s i s i s s trong t h a t the use of f i n g e r s i n performing a r i t h m e t i c c a l c u l a t i o n s i s c r i t i c a l i n the fo r m a t i o n of mental o p e r a t i o n s a s s o c i a t e d w i t h c o u n t i n g , and f i n d i n g sums or d i f f e r e n c e s . P h y s i c a l o b j e c t s are e x t e r n a l t o the c h i l d and o f f e r l i t t l e i n the way of sensory impressions". I f , as S t e f f e e t a l . hypothesize, the use of f i n g e r s i s an e s s e n t i a l stage i n the i n t e r n a l i z a t i o n process f o r coun t i n g and o p e r a t i n g w i t h numbers, perhaps educators would be w e l l a d v i s e d t o encourage the use of f i n g e r s and body taps, as a means of i n c r e a s i n g the sensory i n p u t t o a i d the i n t e r n a l i z a t i o n p r o c e s s . T h i s would serve t o broaden the t r a n s i t i o n p rocess from the concre t e to the a b s t r a c t i n t o the m u l t i - s e n s o r y process, r a t h e r than r e s t r i c t i t t o v i s u a l sensory i n p u t . 38 In the present study, the i n t e r v i e w e r r e c o r d e d d e t a i l s of two aspects of p u p i l responses. One aspect concerned the a c t u a l s t r a t e g y used to s o l v e the m i s s i n g addend. The other aspect concerned the a i d s the c h i l d used as a support to t h i s s t r a t e g y . T h i s second aspect was l a b e l l e d " l e v e l of response" f o r the purposes of the study. I t was f e l t t h a t s o l u t i o n s t r a t e g i e s (with the e x c e p t i o n of R e c a l l ) c o u l d c o n c e i v a b l y be handled at almost any l e v e l of response (that i s , c o u l d be used w i t h c o n c r e t e a i d s , f i n g e r s , p e n c i l t a l l i e s , or i n t e r n a l i z e d p r o c e s s e s ) , but t h a t c e r t a i n l e v e l s would be used more f r e q u e n t l y w i t h c e r t a i n s t r a t e g i e s . These l e v e l s d i d not however.constitute c o n c e p t u a l l y d i f f e r e n t s t r a t e g i e s . L e v e l of response was a l s o i n c l u d e d i n the data c o l l e c t i o n as a means of attempting t o i d e n t i f y p r o g r e s s i v e l y more s o p h i s t i c a t e d s o l u t i o n s t r a t e g i e s through t h e i r a s s o c i a t i o n w i t h what have been i d e n t i f i e d as p r o g r e s s i v e l y more s o p h i s t i c a t e d l e v e l s of response. C l a s s I n c l u s i o n A b i l i t y and  M i s s i n g Addend Performance M i s s i n g addend sentences l o g i c a l l y i n v o l v e a comparison of a g i v e n addend t o the sum, or a p a r t to the whole. A c h i l d ' s understanding of t h i s p a r t -whole comparison i s o f t e n r e f l e c t e d by h i s or her method of completing the m i s s i n g addend sentence. C h i l d r e n w i t h no understanding of the m i s s i n g addend q u e s t i o n o f t e n c o n s i d e r the addend and sum as two p a r t s , and add them t o determine the sum. T h i s e r r o r i s c o n c e p t u a l i n nature and may be r e l a t e d t o an i n a b i l i t y t o r e c o g n i z e the part-whole r e l a t i o n s h i p expressed. On the other hand, the a b i l i t y t o s u c c e s s f u l l y s o l v e m i s s i n g addend sentences should r e f l e c t an understanding of t h a t r e l a t i o n s h i p . Furthermore, i t i s suggested t h a t the method c h i l d r e n use t o s o l v e m i s s i n g addends may be r e l a t e d to t h e i r understanding of part-whole r e l a t i o n s h i p s . S t e f f e , Spikes and H i r s t e i n (1976) found t h a t f i r s t grade c h i l d r e n p r e f e r r e d to use an a d d i t i v e approach r a t h e r than a s u b t r a c t i o n sentence t r a n s f o r m a t i o n approach. These f i n d i n g s may i n d i c a t e t h a t u s i n g a s u b t r a c t i v e approach may r e q u i r e a higher l e v e l of c o g n i t i v e a b i l i t i e s than use of a d d i t i v e approaches. For both a d d i t i v e and s u b t r a c t i v e approaches the c h i l d must be a b l e t o s i m u l t a n e o u s l y r e c o g n i z e the addend as a p a r t of the sum. However, to m e a n i n g f u l l y use a t r a n s f o r m a t i o n to a s u b t r a c t i o n sentence (5 +O = 13 to 13 - 5 =0)f t n e c h i l d must a l s o r e c o g n i z e and u t i l i z e an understanding of the i n v e r s e o p e r a t i o n i n a d d i t i o n t o the part-whole r e l a t i o n s h i p . Inhelder and P i a g e t (1964) i d e n t i f i e d the a b i l i t y to c o n s i d e r the whole and i t s p a r t s s i m u l t a n e o u s l y as a developmental c o g n i t i v e a c q u i s i t i o n , and used c l a s s i n c l u s i o n problems as a measure of t h i s c a p a c i t y . They s t a t e d : I t i s one t h i n g to c a r r y out the union expressed by A + A' = B and q u i t e another to understand t h a t i t i s l o g i c a l l y e q u i v a l e n t t o i t s inverseA = B - A' which means t h a t the whole, B, r e t a i n s i t s i d e n t i t y and t h a t the e n t i r e r e l a t i o n can be q u a n t i t a t i v e l y expressed i n the form A<B. The c o n s e r v a t i o n of the whole and p a r t are the two e s s e n t i a l c h a r a c t e r i s t i c s of genuine c l a s s - i n c l u s i o n , (p. 117) Inhelder & P i a g e t i d e n t i f i e d t h r e e stages i n the development of the a b i l i t y t o si m u l t a n e o u s l y r e c o g n i z e the whole and i t s p a r t s . At the f i r s t stage the c h i l d does not yet regard c l a s s B as r e s u l t i n g from the a d d i t i o n of A and A', and c l a s s A as r e s u l t i n g from the s u b t r a c t i o n of A' from B. During the second stage, the c h i l d g r a d u a l l y comes to i n t u i t i v e l y understand t h a t the B c l a s s e s c o n t a i n more elements than the A c l a s s e s . F i n a l l y , the c h i l d immediately grasps t h a t c l a s s B i s l a r g e r than c l a s s A, because he approaches the problem from the p o i n t o f view of a d d i t i v e composition. The c h i l d i s capable of r e c o g n i z i n g A as a subset of B and can determine A 1 by a comparison of A and B. Since m i s s i n g addends l o g i c a l l y i n v o l v e a comparison of a g i v e n p a r t o r addend to the whole or sum, i t appears t h a t C l a s s I n c l u s i o n a b i l i t y may i n d i c a t e a co n c e p t u a l a b i l i t y t o grasp t h i s comparison. Two r e c e n t s t u d i e s have i n v e s t i g a t e d the r o l e of C l a s s I n c l u s i o n a b i l i t y as i t r e l a t e s to m i s s i n g addend performance. The f i n d i n g s are i n c o n s i s t e n t and r e f l e c t t o some degree the i s s u e of how to measure C l a s s I n c l u s i o n a b i l i t y as d e f i n e d by P i a g e t . T h i s i s s u e w i l l be d i s c u s s e d f o l l o w i n g the review of the two s t u d i e s . S t u d i e s I n v e s t i g a t i n g the R e l a t i o n s h i p Between C l a s s  I n c l u s i o n and M i s s i n g Addend Performance Howlett (1973) s t a t e d t h a t f o r a c h i l d t o understand the m i s s i n g addend concept he or she must re c o g n i z e the whole and i t s p a r t s and the r e l a t i o n between them. Howlett used the c l a s s i n c l u s i o n task as d e s c r i b e d by P i a g e t as a measure to determine whether or not the c h i l d understood the whole-part concept and whether or not the c h i l d had developed the c o g n i t i v e a b i l i t i e s necessary to s o l v e m i s s i n g addends. Performance on c l a s s i n c l u s i o n t a s k s was used to c l a s s i f y c h i l d r e n as Stage 3 (capable of s o l v i n g the c l a s s i n c l u s i o n problem) or Stage 1 (incapable of s o l v i n g the c l a s s i n c l u s i o n problem). Stage 2 or t r a n s i t i o n a l s u b j e c t s were excluded from the study. I t was expected t h a t c h i l d r e n c l a s s i f i e d as Stage 3 would perform a t a higher l e v e l of competency on m i s s i n g addend computation than State 1 c h i l d r e n . I t was found t h a t performance on m i s s i n g addends was s i g n i f i c a n t l y r e l a t e d (p<.05) to performance on the c l a s s i n c l u s i o n problem. Based on t h i s f i n d i n g , Howlett recommended t h a t m i s s i n g addend computation be delayed u n t i l the c h i l d i s a b l e t o s o l v e c l a s s i n c l u s i o n t a s k s . S t e f f e , Spikes, and H i r s t e i n (197 6) i n v e s t i g a t e d the r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and v a r i o u s achievement v a r i a b l e s , one of which was M i s s i n g Addend performance. S t e f f e e t a l . s t a t e d t h a t the a b i l i t y to engage i n r e v e r s i b l e t h i n k i n g i s e s s e n t i a l t o understanding the c l a s s i n c l u s i o n problem. T h i s r e v e r s i b l e thought a l l o w s c h i l d r e n t o con c e i v e o f the t o t a l c l a s s as being made up of two sub c l a s s e s a t one and the same time, which i n t u r n a l l o w s a c h i l d t o make comparisons between a su b c l a s s and the t o t a l c l a s s . S t e f f e et a l . used a p r e - t e s t and a p o s t - t e s t measure of C l a s s I n c l u s i o n a b i l i t y . No c h i l d r e n were s u c c e s s f u l w i t h the c l a s s i n c l u s i o n p r e - t e s t w h i l e 3 2% were s u c c e s s f u l w i t h the p o s t - t e s t t h r e e months l a t e r . Because of the p r e - t e s t r e s u l t s which i n d i c a t e d t h a t the c h i l d r e n c o u l d not s o l v e the c l a s s i n c l u s i o n problem, i t was expected t h a t judgments c o u l d be made on the n e c e s s i t y of C l a s s I n c l u s i o n a b i l i t y f o r l e a r n i n g f i r s t grade a r i t h m e t i c content. R e s u l t s of the study i n d i c a t e d t h a t C l a s s I n c l u s i o n performance was not s i g n i f i c a n t l y r e l a t e d to M i s s i n g Addend performance or t o any f i r s t grade achievement v a r i a b l e measured and was t h e r e f o r e of no va l u e as a measure of r e a d i n e s s f o r f i r s t grade mathematics content. 43 However, S t e f f e e t a l . expressed concern over t h e i r p a r t i c u l a r measure of C l a s s I n c l u s i o n a b i l i t y and suggested t h a t many c h i l d r e n who were capable of s o l v i n g the c l a s s i n c l u s i o n q u e s t i o n f a i l e d t o do so because of d i s t r a c t i n g f e a t u r e s of the t a s k s they used as a measure. They suggested t h a t a c r i t i c a l t e s t would e n t a i l a c l a s s i n c l u s i o n measure t h a t d i d not depend on p i c t o r i a l items c o n t a i n i n g d i s t r a c t i n g f e a t u r e s , s i n c e t h e i r study c o n c l u s i v e l y showed t h a t such d i s t r a c t i n g f e a t u r e s had nothing to do wit h pure numerical r e a s o n i n g . The problem of f i n d i n g v a l i d measures of C l a s s I n c l u s i o n a b i l i t y i s one t h a t s e v e r a l r e c e n t s t u d i e s have addressed. C l a s s I n c l u s i o n Measures Current r e s e a r c h on proposed reasons f o r e r r o r s on c l a s s i n c l u s i o n t a s k s have focused on a number of i s s u e s , one of which concerns the e f f e c t of d i s t r a c t i n g f e a t u r e s as mentioned by S t e f f e e t a l . W o h l w i l l (1968) and Winer and Kronberg (1974) found t h a t o r a l p r e s e n t a t i o n of c l a s s i n c l u s i o n items f a c i l i t a t e d s o l u t i o n . W o h l w i l l l a b e l e d t h i s the " v e r b a l f a c i l i t a t i o n e f f e c t " . Schwartz (1970) found the op p o s i t e e f f e c t , and Jennings (1970) found no s i g n i f i c a n t d i f f e r e n c e s . Winer (1974) questioned the v e r b a l f a c i l i t a t i o n e f f e c t . He argued t h a t v a r i a t i o n s i n v e r b a l cues made the d i f f e r e n c e , s i n c e i n the v e r b a l c o n d i t i o n both subsets were mentioned as w e l l as the su p e r o r d i n a t e s e t . K l a h r and Wallace (1972) a l s o 44 a t t r i b u t e d d i f f e r e n c e s i n v e r b a l as compared t o p i c t o r i a l item r e s u l t s t o v e r b a l cues. They noted t h a t subsets were enumerated i n the v e r b a l c o n d i t i o n and not i n the p i c t o r i a l c o n d i t i o n . Ahr and Youniss (1970) and Klahr and Wallace (1972) suggested a q u e s t i o n m i s i n t e r p r e t a t i o n h y p othesis as an e x p l a n a t i o n of c l a s s i n c l u s i o n e r r o r s . These authors suggested t h a t c h i l d r e n confuse and s u b s t i t u t e the l a b e l of a s e t w i t h t h a t of a subset because they c o n s i d e r a subset comparison more l o g i c a l or s e n s i b l e , and assume t h a t to be the q u e s t i o n . Recency e f f e c t s , or the tendency t o respond w i t h the l a s t word heard, were examined by Youssef and Guardo (1972) . T h e i r f i n d i n g s i n d i c a t e d t h a t recency e f f e c t s o btained when no o p p o r t u n i t y f o r s u b s t i t u t i o n (that i s , q u e s t i o n m i s i n t e r p r e t a t i o n ) was p r o v i d e d by v i s u a l d i s t r a c t i o n such as c o n c r e t e m a t e r i a l s or p i c t u r e s . However, when v i s u a l d i s t r a c t i o n was present, s u b s t i t u t i o n or q u e s t i o n m i s i n t e r p r e t a t i o n dominated and overrode the recency e f f e c t s of the a u d i t o r y s t i m u l i . The e f f e c t s of the r e l a t i v e s i z e of s u b c l a s s e s has been examined by Ahr and Youniss (1970) and B r a i n e r d and Kaszor (1973). Ahr and Youniss found t h a t the l e s s the d i s p a r i t y between subsets, the b e t t e r the performance, w h i l e B r a i n e r d and Kaszor found t h a t the g r e a t e r the d i s c r e p a n c y the b e t t e r the performance. 45 These i s s u e s r a i s e q u e s t i o n s concerning the p a r t p l a y e d by l o g i c a l reasoning i n c h i l d r e n ' s answers to c l a s s i n c l u s i o n q u e s t i o n s . Since agreement has by no means been reached on the i s s u e s , an attempt was made to c o n s t r u c t a c l a s s i n c l u s i o n measure f o r t h i s study, t h a t accommodated and c o n t r o l l e d f o r the v a r i e t y of p o i n t s of view. A d e s c r i p t i o n of the c l a s s i n c l u s i o n items i s g i v e n i n Chapter 3 and Appendix D. Summary of the Review of R e l a t e d L i t e r a t u r e I t appears t h a t c h i l d r e n f i n d c e r t a i n types of open sentences more d i f f i c u l t .than others;., S e v e r a l reasons are proposed to e x p l a i n t h i s d i f f e r e n c e . One i s the f a c t t h a t c h i l d r e n have more experience w i t h c e r t a i n types of sentences than o t h e r s . Another i s t h a t c e r t a i n open sentences are i n h e r e n t l y more d i f f i c u l t than o t h e r s . A t h i r d reason i s t h a t c h i l d r e n may not have developed the conceptual l e v e l r e q u i r e d to d e a l w i t h the concepts u n d e r l y i n g the p a r t i c u l a r sentence form. The f a c t o r s of p l a c e h o l d e r p o s i t i o n and number s i z e were d i s c u s s e d as they r e l a t e d t o performance on m i s s i n g addends. T h i s d i s c u s s i o n was i n c l u d e d to p r o v i d e a r a t i o n a l e f o r the d e s i g n of the m i s s i n g addend t e s t . The q u e s t i o n of how c h i l d r e n process m i s s i n g addend sentences i s thought to be of v a l u e i n determining the a p p r o p r i a t e n e s s of i n c l u d i n g m i s s i n g addend computation i n the mathematics c u r r i c u l u m f o r years one and two. I t i s suggested t h a t the methods c h i l d r e n of a p a r t i c u l a r age choose to use t o d e a l w i t h m i s s i n g addends r e f l e c t methods t h a t are meaningful and w i t h i n the c o n c e p t u a l grasp of these c h i l d r e n . S t u d i e s t h a t have used i n d i v i d u a l i n t e r v i e w techniques to determine m i s s i n g addend s o l u t i o n s t r a t e g i e s were reviewed. These s t u d i e s have focused on f i r s t and t h i r d year c h i l d r e n and'have r e v e a l e d t h a t the two age groups process m i s s i n g addends d i f f e r e n t l y . I t i s suggested t h a t s o l u t i o n s t r a t e g i e s g r a d u a l l y develop i n s o p h i s t i c a t i o n , and t h a t the s o l u t i o n s t r a t e g i e s used by a c r o s s - s e c t i o n of second year students may r e f l e c t stages i n the t r a n s i t i o n from l e s s t o more mature s t r a t e g i e s . The m a j o r i t y of f i r s t grade students were found t o use counting s t r a t e g i e s , w h ile t h i r d year students p r e f e r r e d more e f f i c i e n t methods such as the use of a d d i t i o n and s u b t r a c t i o n a l g o r i t h m s . Stu d i e s t h a t have i d e n t i f i e d developmental sequences f o r p r o c e s s i n g open sentences were presented. The r o l e of c o u n t i n g s t r a t e g i e s i n these sequences was emphasized. S i m i l a r l y , the r o l e of l e v e l of response as i t i n t e r a c t s with s o l u t i o n s t r a t e g i e s was examined. I t i s suggested t h a t the s t r a t e g i e s c h i l d r e n use to s o l v e m i s s i n g addends and how a c c u r a t e l y they use these s t r a t e g i e s , may be r e l a t e d t o c o n c e p t u a l l e v e l . C l a s s I n c l u s i o n a b i l i t y as d e f i n e d by P i a g e t has been suggested t o be a l o g i c a l p r e r e q u i s i t e to understanding the m i s s i n g addend problem, and has been used i n other s t u d i e s as a measure of c o n c e p t u a l l e v e l . F i n d i n g s by Howlett (1973) and S t e f f e e t a l . (1976) are i n c o n c l u s i v e as f a r as the r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and M i s s i n g Addend performance i s concerned. F i n a l l y , s t u d i e s were c i t e d i n which an attempt was made t o examine i s s u e s r e l a t e d to the v a l i d i t y of c l a s s i n c l u s i o n measures. Chapter 3 METHOD The purpose of the study was to answer a number of q u e s t i o n s r e l a t e d to c h i l d r e n ' s performance on m i s s i n g addend sentences. These q u e s t i o n s are presented f i r s t . Next, i n f o r m a t i o n i s presented concerning the sample used f o r the study, the sample s e l e c t i o n techniques, and the nature and extent of the c h i l d r e n ' s experience w i t h m i s s i n g addends. Two t e s t s were admi n i s t e r e d i n d i v i d u a l l y t o each s u b j e c t : a M i s s i n g Addend Test and a C l a s s I n c l u s i o n Test. In the second s e c t i o n of the chapter, d e t a i l s of the M i s s i n g Addend Test are presented. The p i l o t t e s t i n g which l e d t o the development of the t e s t items and m a t e r i a l s , i s d i s c u s s e d i n d e t a i l , along with a d e s c r i p t i o n of the s t r a t e g y c l a s s i f i c a t i o n scheme and the l e v e l of response r e c o r d i n g procedure. The d e t a i l s of the C l a s s I n c l u s i o n T e s t items, m a t e r i a l s , and s c o r i n g procedures are then d i s c u s s e d . F i n a l l y , the procedures f o l l o w e d i n the a d m i n i s t r a t i o n of the i n t e r v i e w s , the r e l i a b i l i t y study d e t a i l s , and the data a n a l y s i s techniques are presented. Questions of the Study The g e n e r a l q u e s t i o n s f o r the study a r i s e from concerns expressed by educators r e g a r d i n g the a p p r o p r i a t e n e s s of i n t r o d u c i n g m i s s i n g addends, and i n p a r t i c u l a r , c e r t a i n m i s s i n g addend s o l u t i o n s t r a t e g i e s i n the primary grades. The s p e c i f i c q u e s t i o n s a r i s e from the review of l i t e r a t u r e r e l a t e d to M i s s i n g Addend performance. These q u e s t i o n s are as f o l l o w s : Questions R e l a t e d t o Performance  on M i s s i n g Addend Sentences 1. Does the l e v e l of d i f f i c u l t y of number combinations a f f e c t performance on m i s s i n g addend sentences? 2. Does p l a c e h o l d e r p o s i t i o n a f f e c t performance on m i s s i n g addend sentences? 3. Is t h e r e an i n t e r a c t i o n of l e v e l of d i f f i c u l t y and p l a c e h o l d e r p o s i t i o n ? Questions R e l a t e d to S t r a t e g y Choice 4. What s o l u t i o n s t r a t e g i e s do year two c h i l d r e n use to s o l v e m i s s i n g addend, sentences of the formats a + 0 = h> and Q+ a = b? 5. Which s t r a t e g i e s are used most f r e q u e n t l y ? 6. Does frequency of s t r a t e g y c h o i c e v a r y as a f u n c t i o n of l e v e l of d i f f i c u l t y ? 50 7. Does f r e q u e n c y o f s t r a t e g y c h o i c e v a r y as a f u n c t i o n o f p l a c e h o l d e r p o s i t i o n ? 8. Which s t r a t e g i e s a r e most a c c u r a t e ? 9. Does a c c u r a c y o f a s t r a t e g y v a r y as a f u n c t i o n o f l e v e l o f d i f f i c u l t y ? 10. Does a c c u r a c y o f a s t r a t e g y v a r y as a f u n c t i o n o f p l a c e h o l d e r p o s i t i o n ? Q u e s t i o n s R e l a t e d t o L e v e l  o f R e s p o n s e Usage 11. What l e v e l s o f r e s p o n s e do c h i l d r e n u s e t o s o l v e m i s s i n g addend s e n t e n c e s ? 12. Which l e v e l s a r e u s e d most f r e q u e n t l y ? 13. Does f r e q u e n c y o f l e v e l o f r e s p o n s e u s e v a r y as a f u n c t i o n o f l e v e l o f d i f f i c u l t y ? 14. Does f r e q u e n c y o f l e v e l o f r e s p o n s e u s e v a r y a s a f u n c t i o n o f p l a c e h o l d e r p o s i t i o n ? 15. Which l e v e l o f r e s p o n s e i s most a c c u r a t e ? 16. Does l e v e l o f r e s p o n s e a c c u r a c y v a r y as a f u n c t i o n o f l e v e l o f d i f f i c u l t y ? 17. Does l e v e l o f r e s p o n s e a c c u r a c y v a r y a s a f u n c t i o n o f p l a c e h o l d e r p o s i t i o n ? Q u e s t i o n s R e l a t e d t o t h e I n t e r a c t i o n o f  S t r a t e g y a n d L e v e l o f Response Use 18. I s t h e r e an i n t e r a c t i o n o f s t r a t e g y u s e and l e v e l o f r e s p o n s e u s e , and i f s o , what i s t h e n a t u r e o f t h i s i n t e r a c t i o n ? Questions R e l a t e d t o C l a s s  I n c l u s i o n Performance 19. Is t h e r e a r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and M i s s i n g Addend performance? 20. ,1s there a r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and s t r a t e g y choice? 21. Is t h e r e a r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and l e v e l of response usage? Sample D e s c r i p t i o n of the Sample The sample used i n the study was randomly s e l e c t e d from the second grade students of s c h o o l d i s t r i c t #40, New Westminster, B r i t i s h Columbia. The d i s t r i c t p o p u l a t i o n c o n s i s t e d of 330 year two students from 16 d i f f e r e n t c l a s s e s i n nine d i f f e r e n t s c h o o l s . A wide range of socioeconomic l e v e l s was r e p r e s e n t e d . S e l e c t i o n Technique F i f t y c h i l d r e n were randomly s e l e c t e d (a sample of f o r t y p l u s t e n a l t e r n a t e s ) and the order of s e l e c t i o n was recorded. Three a l t e r n a t e s were used f o r the sample due t o the f a c t t h a t t h r e e c h i l d r e n had moved between the s e l e c t i o n and the i n t e r v i e w i n g p e r i o d s The sample was composed of n i n e t e e n g i r l s and twenty-one boys from t h i r t e e n of the s i x t e e n p o s s i b l e c l a s s e s . The c h i l d r e n ranged i n age from seven years 52 t h r e e months to e i g h t years nine months, wi t h a mean age of seven years nine months. Background i n f o r m a t i o n f o r the sample i s presented i n Appendix A. M i s s i n g Addend Background of the Sample Teacher i n t e r v i e w s were used t o p r o v i d e i n f o r m a t i o n on a number of f a c t o r s r e l a t e d t o m i s s i n g addend experience: (1) the approach used by the teacher to t e a c h m i s s i n g addend s o l u t i o n s t r a t e g i e s ; (2) the t e x t used by the p u p i l s and the degree t o which t h i s t e x t was used; (3) m i s s i n g addend experience p r o v i d e d by the t e a c h e r i n a d d i t i o n to any t e x t e xperience. The teacher i n t e r v i e w s a l s o ensured t h a t the students had worked wi t h both p l a c e h o l d e r p o s i t i o n s (though not n e c e s s a r i l y t o the same degree), and had experience c o u n t i n g , adding, and s u b t r a c t i n g t w o - d i g i t numbers. Teacher i n t e r v i e w s r e v e a l e d a v a r i e t y of approaches to t e a c h i n g m i s s i n g addend s o l u t i o n s t r a t e g i e s . Most teac h e r s were aware of and u t i l i z e d s e v e r a l approaches t o m i s s i n g addends, but a l l emphasized one more than the o t h e r s . The approach emphasized by seven of the t h i r t e e n t e a c h e r s was a counting-on or a d d i t i v e model, where the c h i l d i s encouraged t o 53 s t a r t w i t h the g i v e n addend and increment t o the sum. Four t e a c h e r s emphasized the i n v e r s e r e l a t i o n s h i p of a d d i t i o n and s u b t r a c t i o n , and encouraged c h i l d r e n to t ransform the m i s s i n g addend sentence to i t s s u b t r a c t i o n sentence e q u i v a l e n t . One t eacher used the counting-on model f o r a + f~] = b formats, and the s u b t r a c t i o n model f o r • + a = b. Another method t h a t was mentioned as an a l t e r n a t i v e was a hidden number approach, where c h i l d r e n t r y t o guess the covered number by r e l y i n g on memory of b a s i c f a c t s . When asked, most t e a c h e r s responded t h a t they had more success'with the a d d i t i v e model f o r most c h i l d r e n , and t h a t o n l y some c h i l d r e n seemed a b l e to grasp the s u b t r a c t i v e method. Although s e v e r a l mathematics programmes were mentioned by t e a c h e r s as r e s o u r c e s , o n l y one was named as a core programme. T h i s programme was I n v e s t i g a t i n g  School Mathematics: Book Two, by E i c h o l z e t a l . (1973). E i g h t of the t h i r t e e n t e a c h e r s used the teacher guide and p u p i l t e x t of t h i s programme to v a r y i n g degrees, w h i l e f i v e t e a c h e r s r e l i e d on a v a r i e t y of sources, r a t h e r than one source i n p a r t i c u l a r . An examination of t h i s p u p i l t e x t and t e a c h e r guide p r i o r t o the data c o l l e c t i o n r e v e a l e d t h a t the programme r e l i e s h e a v i l y on mastery of b a s i c a d d i t i o n f a c t s as a p r e r e q u i s i t e to h a n d l i n g m i s s i n g addends. I t i n t r o d u c e s m i s s i n g addends i n the hidden number context by p r e s e n t i n g number f a c t s with one of the numbers covered. C h i l d r e n are not taught a procedure f o r c a l c u l a t i n g the m i s s i n g addend but i n s t e a d are encouraged t o r e c a l l which number i s m i s s i n g from the b a s i c f a c t combination. A f t e r f o u r pages u s i n g the hidden number approach, emphasis s h i f t s t o the i n v e r s e r e l a t i o n s h i p and the corresponding s u b t r a c t i o n sentence. T h i s emphasis i s c o n t i n u e d throughout the book through the p a i r i n g of most m i s s i n g addend equations w i t h t h e i r e q u i v a l e n t s u b t r a c t i o n sentence. For example, • + 3 = 7 > 7 - 3 = • ( E i c h o l z e t a l . , Book 2, p. 57) • + 2 = 7 7 - 2 = • ( E i c h o l z et a l . , Book 2, p. 58) The Year Two t e x t f o r t h i s programme p r o v i d e s experience w i t h m i s s i n g addends i n the l e f t p l a c e h o l d e r p o s i t i o n s o n l y (Q+ a = b ) . The Year One t e x t p r o v i d e s experience o n l y w i t h the a + \3 = b format. Information from the teacher i n t e r v i e w s and the p u p i l t e x t i n d i c a t e d t h a t two major i n t e r p r e t a t i o n s of m i s s i n g addend sentences were used i n the classrooms from which the sample was drawn. One was an a d d i t i v e approach and the o t h e r was a s u b t r a c t i v e approach. Of the c h i l d r e n i n t h i s study, nine were from c l a s s e s where the a d d i t i v e approach may have been the o n l y approach t o which they had been exposed. Twelve c h i l d r e n used a t e x t which emphasized the s u b t r a c t i v e approach, f o u r c h i l d r e n were from c l a s s e s where the s u b t r a c t i v e approach was emphasized by the teacher, and f i f t e e n c h i l d r e n were from c l a s s e s where both t e a c h e r s and t e x t emphasized the s u b t r a c t i v e i n t e r p r e t a t i o n . F i g u r e 3.1 i l l u s t r a t e s t h i s aspect of the mathematical background of the sample. Appendix A p r e s e n t s i n d i v i d u a l p u p i l data concerning m i s s i n g addend experience. A S u b t r a c t i v e approach emphasized by t e a c h e r B S u b t r a c t i v e approach emphasized i n t e x t used * One c h i l d r e c e i v e d i n s t r u c t i o n i n both a d d i t i v e and s u b t r a c t i v e approaches depending on p l a c e -h o l d e r p o s i t i o n . F i g u r e 3.1 M i s s i n g Addend Experience of the Sample A l l t e a c h e r s r e p o r t e d t h a t they p r o v i d e d experience wi t h both p l a c e h o l d e r p o s i t i o n s through the use of e x t r a worksheets and a c t i v i t i e s . Only t h r e e t e a c h e r s , however, 56 p r o v i d e d experience w i t h m i s s i n g addends u s i n g number combinations from beyond the b a s i c f a c t domain. For these reasons, the o p p o r t u n i t y f o r the c h i l d r e n t o l e a r n how t o s o l v e m i s s i n g addends v a r i e d f o r p l a c e h o l d e r p o s i t i o n , number s i z e , and s o l u t i o n s t r a t e g y taught. However, f a m i l i a r i t y w i t h m i s s i n g addends was assured. M i s s i n g Addend Te s t The d e s c r i p t i o n of the development of the M i s s i n g Addend Te s t i n c l u d e s the d e t a i l s of the M i s s i n g Addend p i l o t t e s t , along with the changes t h a t r e s u l t e d from t h i s p i l o t t e s t i n g . Next, the d e f i n i t i o n s of t e s t v a r i a b l e s , a c t u a l t e s t items, and d e s c r i p t i o n s of t e s t m a t e r i a l s are presented. The s o l u t i o n s t r a t e g y scheme i s then d e s c r i b e d along w i t h a d e s c r i p t i o n of the l e v e l of response r e c o r d i n g procedures. P i l o t T e s t P i l o t t e s t i n g was conducted u s i n g f i f t e e n second year students from an ad j a c e n t s c h o o l d i s t r i c t . T h i s was done i n order t o t r y out the proposed items and m a t e r i a l s , t o develop experience i n q u e s t i o n i n g and c l a s s i f y i n g s o l u t i o n s t r a t e g i e s , and t o develop the s t r a t e g y c l a s s i f i c a t i o n scheme. In order to p r o v i d e maximum p u p i l f a m i l i a r i t y w ith the format of the items, the m i s s i n g addend sentences 57 were presented h o r i z o n t a l l y , an empty box was used as the p l a c e h o l d e r , and the e q u a l i t y symbol was to the r i g h t of the a d d i t i o n symbol. These c h a r a c t e r i s t i c s are r e p r e s e n t a t i v e of the m i s s i n g addend sentences most f r e q u e n t l y used i n the classrooms of the s c h o o l d i s t r i c t . Only a d d i t i o n m i s s i n g addend sentences were used. Two p l a c e h o l d e r p o s i t i o n s were used: l e f t p l a c e h o l d e r Q+ a = b, and r i g h t p l a c e h o l d e r a + Q = b. The m i s s i n g addend items were chosen from f o u r l e v e l s of d i f f i c u l t y . The range of l e v e l s of d i f f i c u l t y was used to t e s t f o r comprehension of m i s s i n g addend computation by working beyond the f a m i l i a r b a s i c f a c t domain, and t o t r y t o i d e n t i f y s t r a t e g i e s t h a t can be e f f e c t i v e l y u t i l i z e d beyond the b a s i c f a c t domain. L e v e l One was d e f i n e d as the domain of b a s i c a d d i t i o n f a c t s with sums l e s s than or equal t o t e n . L e v e l Two was d e f i n e d as the domain of the b a s i c a d d i t i o n f a c t s with sums g r e a t e r than ten and l e s s than or equal to e i g h t e e n . L e v e l Three was d e f i n e d as a two d i g i t number p l u s a one d i g i t number, where no regrouping was r e q u i r e d i n the u s u a l a d d i t i o n a l g o r i t h m . L e v e l Four was d e f i n e d as a two d i g i t number p l u s a two d i g i t number where no regrouping was r e q u i r e d i n the u s u a l a d d i t i o n a l g o r i t h m . 58 The t e s t v a r i a b l e s r e s u l t e d i n a 4 x 2 t e s t d e s ign, f o u r l e v e l s of d i f f i c u l t y by two p l a c e h o l d e r p o s i t i o n s , r e s u l t i n g i n e i g h t c e l l s . In order t o ensure a reasonable l e n g t h of time f o r the p u p i l i n t e r v i e w , o n l y one item was used f o r each of the e i g h t c e l l s . The items s e l e c t e d f o r the p i l o t t e s t appear i n Table 3.1. Table 3.1 M i s s i n g Addend P i l o t T e s t Items L e v e l of D i f f i c u l t y L e v e l One L e v e l Two L e v e l Three L e v e l Four P l a c e h o l d e r P o s i t i o n a + • = b • + a = b 5 + • = 8 • + 4 = 7 9 + • = 16 • + 8 = 15 31 + • = 37 • + 32 = 38 34 + • = 76 • + 35 = 77 From the p o s s i b l e combinations f o r each l e v e l , two combinations were s e l e c t e d so t h a t the d i f f e r e n c e i n the sums of any p a i r was one, and one addend was the same i n each p a i r . T h i s addend became the m i s s i n g addend f o r the two p l a c e h o l d e r p o s i t i o n s . T h i s was done t o attempt to make the two items as s i m i l a r i n d i f f i c u l t y as p o s s i b l e without using the same numbers, so t h a t an o v e r a l l comparison of the e f f e c t of p l a c e h o l d e r p o s i t i o n would be p o s s i b l e . These p a i r s of items were chosen t o p r o v i d e a v a r i e t y of b a s i c f a c t combinations. Zero', one, and two were excluded as addends and combinations i n v o l v i n g doubles ( f o r example, 4 + 4 = 8 ) were excluded. T h i s was done t o s t a n d a r d i z e the d i f f i c u l t y of the items, and to make them r e p r e s e n t a t i v e of the proposed l e v e l of d i f f i c u l t y . F o l l o w i n g the p i l o t t e s t , s e v e r a l r e v i s i o n s were made to the t e s t items. The d e f i n i t i o n of L e v e l Two was a l t e r e d so t h a t nine would not be i n c l u d e d as a p o s s i b l e addend, and doubles p l u s one would be excluded. T h i s was done t o s t a n d a r d i z e the item d i f f i c u l t y . The d e f i n i t i o n s of L e v e l s Three and Four were a l t e r e d t o exclude the numeral one i n the g i v e n addend, due t o the number of times c h i l d r e n misread 31 as 13. Sums f o r L e v e l s Three and Four were l i m i t e d t o 5 0 t o reduce the amount of time r e q u i r e d f o r c h i l d r e n who used cou n t i n g t o o b t a i n answers. M i s s i n g Addend T e s t Items P i l o t t e s t r e v i s i o n s r e s u l t e d i n the f o l l o w i n g d e f i n i t i o n s of L e v e l of D i f f i c u l t y . L e v e l One i n c l u d e d b a s i c a d d i t i o n f a c t combinations wi t h sums of t e n or l e s s , e x c l u d i n g combinations i n v o l v i n g addends of 0, 1, or 2 and e x c l u d i n g doubles. L e v e l Two i n c l u d e d b a s i c a d d i t i o n f a c t combinations with sums l e s s than or equal t o e i g h t e e n and g r e a t e r than ten, e x c l u d i n g combinations i n v o l v i n g addends of 0, 1, or 2, and e x c l u d i n g doubles or doubles p l u s one (for example 7 + 8 = 15) . 60 L e v e l Three i n v o l v e d combinations of a t w o - d i g i t number p l u s a o n e - d i g i t number where the m i s s i n g addend was the o n e - d i g i t number and regrouping was not r e q u i r e d . The d i g i t s 0 and 1 were not used and sums were l e s s than 5 0 . L e v e l Four i n v o l v e d combinations of a t w o - d i g i t number p l u s a t w o - d i g i t number where no regrouping was r e q u i r e d . The d i g i t s 0 and 1 were not used and sums were l e s s than 5 0 . Table 3.2 p r e s e n t s the t e s t items used f o r the study. Table 3.2 M i s s i n g Addend Test Items P l a c e h o l d e r P o s i t i o n L e v e l of D i f f i c u l t y a + D = b • + a = b L e v e l One 5 + • = 8 • + 4 = 7 L e v e l Two 5 + • = 13 • + 6 = 14 L e v e l Three 42 + • = 48 • + 4 3 = 4 9 L e v e l Four 23 + • = 38 • + 24 = 39 T e s t M a t e r i a l s T e s t b o o k l e t . Each item was p r i n t e d on a separate sheet 9.5 cm x 2 1 . 5 cm. A sample of the t e s t b o o k l e t i s presented i n Appendix B. With the e x c e p t i o n of the f i r s t item, a l l items were randomly ordered t o attempt t o c o n t r o l f o r the e f f e c t s of order of p r e s e n t a t i o n of items. The item 5 + Q = 8 was always presented f i r s t to attempt t o ensure t h a t c h i l d r e n would experience success on the f i r s t item. 61 The b o o k l e t cover p r o v i d e d a space f o r the c h i l d ' s name and the f o l l o w i n g number sentences: 4 + 2 = D 3 + 6 = • 5 - 3 = • 7 - 2 = • These were p r o v i d e d to ensure t h a t the c h i l d knew how to i n t e r p r e t the symbols r e q u i r e d , knew to read the sentences aloud before completing each one, knew where to r e c o r d the response, and was r e l a x e d enough to begin. Each set of pages was o r g a n i z e d i n a d i f f e r e n t random order, then was s t a p l e d i n t o b o o k l e t s . P u p i l m a t e r i a l s . Approximately s e v e n t y - f i v e 2-cm c o l o u r e d wooden cubes were a v a i l a b l e a t a l l times i f the c h i l d chose t o use them as cou n t e r s . A l s o , p o p s i c l e s t i c k s , some bundled i n tens, were a v a i l a b l e f o r c h i l d r e n who p r e f e r r e d them. Paper was a v a i l a b l e f o r t a l l y i n g or w r i t i n g number sentences, and p e n c i l s were p r o v i d e d . Teacher i n t e r v i e w s r e v e a l e d t h a t these were a p p r o p r i a t e m a t e r i a l s . Interview coding form. Each i n t e r v i e w was recorded on audio tape, and on an i n t e r v i e w coding form (see Appendix C ) . The form p r o v i d e d space f o r r e c o r d i n g the order i n which the items were presented and the c h i l d ' s response. Columns were p r o v i d e d f o r each of the s t r a t e g y c l a s s i f i c a t i o n s . The c h i l d ' s a c t i o n s and comments were recorded under the a p p r o p r i a t e s t r a t e g y column i n the row f o r t h a t p a r t i c u l a r item. The e n t r y a l s o i n c l u d e d a d e s c r i p t i o n of the c h i l d ' s l e v e l of response as w e l l as the s o l u t i o n s t r a t e g y d e t a i l s : . S t r a t e g y C l a s s i f i c a t i o n Through a review of r e l a t e d l i t e r a t u r e , l o g i c a l a n a l y s i s of primary mathematics programmes, and o b s e r v a t i o n dur i n g p i l o t t e s t i n g , the f o l l o w i n g m i s s i n g addend s o l u t i o n s t r a t e g i e s were i d e n t i f i e d : (1) Incrementing procedures where the c h i l d s t a r t e d w i t h the g i v e n addend and added on to reach the sum, while t a l l y i n g the increment; (2) Decrementing procedures where the c h i l d s t a r t e d w i t h the sum and s u b t r a c t e d the g i v e n addend t o be l e f t with the m i s s i n g addend, or decremented to reach the g i v e n addend while t a l l y i n g the decrement; (3) R e c a l l procedures where the c h i l d responded a u t o m a t i c a l l y u s i n g memory o f a b a s i c f a c t combination; (4) S u b s t i t u t i o n procedures where the c h i l d made a response, c o n s i d e r e d the number sentence, and changed i t i n favour of another response; (.5) A s s o c i a t i v e or grouping procedures where the c h i l d used the a s s o c i a t i v e p r i n c i p l e t o s i m p l i f y or " f i g u r e out" the problem by adding or s u b t r a c t i n g groups of u n i t s ; (6) I n c o r r e c t T r a n s f o r m a t i o n procedures where the c h i l d transformed the m i s s i n g addend i n t o a c a n o n i c a l 63 a d d i t i o n sentence ( f o r example, g i v e n 5 +Q = 8, 13 = 8) . ap p a r e n t l y t h i n k s 5 + 8 = 1 3 , r e c o r d s 5 + During the p u p i l i n t e r v i e w s t h i s c l a s s i f i c a t i o n was used. However, f o l l o w i n g the data c o l l e c t i o n the c l a s s i f i c a t i o n scheme was r e o r g a n i z e d t o accommodate the f u l l range of procedures encountered. These procedures f e l l i n t o t hree g e n e r a l c a t e g o r i e s : (a) i n c o r r e c t or u n i d e n t i f i a b l e procedures; (b) r e c a l l or automatic responses; and (c) c o n c e p t u a l l y c o r r e c t s o l u t i o n procedures. The category of i n c o r r e c t of u n i d e n t i f i a b l e procedures i n c l u d e d I n c o r r e c t Transformations, No Attempts, and Indeterminate procedures. The second category was t h a t of R e c a l l or automatic responses. These responses were of a s p e c i a l nature i n t h a t they d i d not i n v o l v e an apparent p r o c e s s f o r s o l u t i o n , but i n s t e a d i n v o l v e d a memory scan technique. Most c h i l d r e n who used R e c a l l c o u l d not e x p l a i n how they found the m i s s i n g addend, they simply s a i d : "I know t h a t one", or " I remembered". The remaining s o l u t i o n procedures f e l l i n t o a t h i r d category, t h a t of c o n c e p t u a l l y c o r r e c t s o l u t i o n procedures. These procedures ranged from haphazard immature s t r a t e g i e s t o s o p h i s t i c a t e d reasoning p r o c e s s e s . F i v e such procedures were i d e n t i f i e d d u r i n g the c o l l e c t i o n of data. 64 A l l f i v e procedures had the common c h a r a c t e r i s t i c t h a t the procedures c o u l d be a p p l i e d i n e i t h e r an a d d i t i v e or a s u b t r a c t i v e context. C h i l d r e n , when fa c e d with a m i s s i n g addend sentence, seemed t o i n t e r p r e t i t as e i t h e r an a d d i t i o n sentence r e q u i r i n g incrementing from the g i v e n addend, or a s u b t r a c t i o n sentence r e q u i r i n g decrementing from the sum. Once t h i s i n i t i a l i n t e r p r e t a t i o n was made c h i l d r e n s o l v e d the problem using one of the f i v e procedures. A second c h a r a c t e r i s t i c of the f i v e c o n c e p t u a l l y c o r r e c t procedures was t h a t they seemed to r e f l e c t l e v e l s of m a t u r i t y i n s o l v i n g m i s s i n g addend equations. From l e a s t to most s o p h i s t i c a t e d , these s t r a t e g i e s have been l a b e l e d Semi-Guess procedures, S u b s t i t u t i o n procedures, C o u n t i n g - A l l procedures, Counting-On procedures, and A s s o c i a t i v e procedures. A d e t a i l e d d e s c r i p t i o n of the f i v e c o n c e p t u a l l y c o r r e c t s o l u t i o n s t r a t e g i e s i s fo l l o w e d by d e s c r i p t i o n s of R e c a l l responses, and i n c o r r e c t or u n i d e n t i f i a b l e procedures. Semi-Guess procedures. The c h i l d c o r r e c t l y i n t e r p r e t e d the m i s s i n g addend sentence i n an a d d i t i v e or s u b t r a c t i v e c ontext. The c h i l d then accounted f o r ( u s u a l l y w i t h blocks) the g i v e n addend or the sum. The c h i l d then added or s u b t r a c t e d a group without c o u n t i n g or checking, a s s e r t e d t h a t t h i s group r e p r e s e n t e d the m i s s i n g addend, and recorded the number of t h i s group as the m i s s i n g addend. Although these procedures were o n l y i d e n t i f i e d when co n c r e t e a i d s were used, they may w e l l have been c l a s s i f i e d simply as a guess i f a c h i l d used them a t an I n t e r n a l i z e d l e v e l of response and was not a b l e t o v e r b a l i z e a p r o c e s s . S u b s t i t u t i o n procedures. The c h i l d c o r r e c t l y i n t e r p r e t e d the m i s s i n g addend sentence i n an a d d i t i v e or s u b t r a c t i v e c o n t e x t . The c h i l d then made an estimate of the m i s s i n g addend, c o n s i d e r e d the estimate i n r e l a t i o n t o the number of sentence, and a l t e r e d i t i n o r d e r to c o r r e c t l y complete the sentence. U s u a l l y t h i s c o r r e c t i o n procedure was systematic i n nature, t h a t i s , the c h i l d r e c o g n i z e d the o r i g i n a l estimate t o be too g r e a t or too s m a l l , and a l t e r e d i t a c c o r d i n g l y . C o u n t i n g - A l l procedures. These procedures i n v o l v e d f u l l y a c counting f o r both addends and the sum. T h i s was u s u a l l y done w i t h c o n c r e t e a i d s although some c h i l d r e n used f i n g e r s or o r a l c ounting sequences. In an a d d i t i v e c o n t e x t the c h i l d u t i l i z e d c o u nting techniques t o r e p r e s e n t the g i v e n addend s t a r t i n g at one, then continued c o u n t i n g to reach the sum. The s i z e of the increment was then t a l l i e d and recorded as the m i s s i n g addend. ^ / / / / For example, 3 + • = 8 1 2 3 4 5 6 7 8 In a s u b t r a c t i v e context, C o u n t i n g - A l l i n v o l v e d a f u l l r e p r e s e n t a t i o n of the sum by counting from one to the sum with or without the use of blocks or fingers. The given addend was then subtracted, and the remaining group was counted and recorded ;as the missing addend. For example, 3 + • = '8 1 2 3 4 5 6 7 8 1 2 3 Counting-On procedures. The c h i l d u t i l i z e d counting techniques but mentally represented either the given addend and counted-on, or the sum and counted-back. Concrete materials or aids may or may not have been used to t a l l y the increment or the decrement. In an additive context, the c h i l d may have dealt with 3 + O = 8 by saying or thinking 3 and counting 4, 5, 6, 7, 8 while t a l l y i n g the increment. For a subtractive context, the c h i l d required the a b i l i t y to count backwards. For example, given 3 + f~l = 8 the c h i l d would have said "8...7, 6, 5, 4, 3" while t a l l y i n g the decrement, or "8...7, 6, 5" while counting-back the number of the given addend. Associative procedures. The c h i l d interpreted the sentence as an addition or subtraction s i t u a t i o n and incremented or decremented not by ones, but by groups. Often basic facts and place value concepts were u t i l i z e d as part of the solution process. An example of use of the additive Associative strategy i s solving 23 + • = 38 by saying "23 + 7 = 30, 3 0 + 8 38, 7 + 8 = 15, so 23 + 15 = 38"(or 23 +(7 + 8)= 38). 67 An example of the s u b t r a c t i v e A s s o c i a t i v e s t r a t e g y i s s o l v i n g 23 + • = 38 by saying "38 10 28, 28 - 5 = 23, 10 + 5 = 15, so 23 + 15 = 38" (or 38 -(10 + 5)= 23). These A s s o c i a t i v e s t r a t e g i e s r e l i e d on mastery of some number f a c t s t o use as t o o l s i n the re a s o n i n g process, and seemed t o be the most f l e x i b l e and s o p h i s t i c a t e d of a l l s t r a t e g i e s encountered. R e c a l l or automatic response. The c h i l d responded immediately with no evidence o f a counting or reasoning procedure. J u s t i f i c a t i o n was u s u a l l y a s s e r t i v e , such as "I j u s t know t h a t one," or "I remembered". I n c o r r e c t T r a n s f o r m a t i o n . The c h i l d i n c o r r e c t l y transformed the m i s s i n g addend equation to a c a n o n i c a l a d d i t i o n sentence. For example, g i v e n 3 + [U = 8, the c h i l d presumably thought 3 + 8 =0, and recorded 3 + [HI = 8 . Indeterminate. The c h i l d gave a response and the i n t e r v i e w e r was unable t o e l i c i t the procedure used, or was unsure of the procedure and was unable to get c l a r i f i c a t i o n from the c h i l d . Guesses were i n c l u d e d i n t h i s category. No Attempt. The c h i l d d i d not attempt the qu e s t i o n . 68 L e v e l s of Response The r e s e a r c h e r h y p o t h e s i z e d t h a t most s t r a t e g i e s c o u l d be used on s e v e r a l d i f f e r e n t l e v e l s of responding, but t h a t these l e v e l s d i d not c o n s t i t u t e c o n c e p t u a l l y d i f f e r e n t s t r a t e g i e s . For example, i f a c h i l d uses an a d d i t i v e c ounting procedure, he or she may use b l o c k s , f i n g e r s , body t a p s , p e n c i l t a l l i e s , an imaginary number l i n e , or o r a l c ounting p a t t e r n s . A l l would e s s e n t i a l l y be examples of the same s t r a t e g y but w i t h i n d i v i d u a l methods of implementation. For t h i s reason, a v a r i e t y of m a t e r i a l s was p r o v i d e d to a l l o w f o r i n d i v i d u a l p r e f e r e n c e s , and to a l l o w the c h i l d ' s c o n c e p t u a l understanding t o be expressed a t any l e v e l of response. No a p r i o r i c l a s s i f i c a t i o n scheme was proposed f o r l e v e l of response. Instead, the r e s e a r c h e r recorded what methods were used, how they were used, and o t h e r p e r t i n e n t aspects of the c h i l d ' s l e v e l of response. F o l l o w i n g the data c o l l e c t i o n , a post hoc scheme f o r l e v e l of response was used to c l a s s i f y the d e s c r i p t i v e data. Over the 32 0 examples a wide v a r i e t y of response l e v e l s was recorded. These responses d i d not f a l l i n t o d i s c r e t e c a t e g o r i e s such as c o n c r e t e or p i c t o r i a l , but i n s t e a d seemed to f a l l onto a continuum of sensory involvement. The lower or l e a s t s o h p i s t i c a t e d e x t r e m i t y of t h i s continuum was c h a r a c t e r i z e d by maximum e x t e r n a l sensory involvement of an a u d i t o r y , v i s u a l , and 69 t a c t i l e - k i n e s t h e t i c nature, i n v o l v i n g l o o k i n g at and ma n i p u l a t i n g concrete m a t e r i a l s , o f t e n while t h i n k i n g or counting aloud. The o t h e r extreme of the continuum was c h a r a c t e r i z e d by i n t e r n a l i z e d c o u n t e r p a r t s t o t h i s sensory involvement: a u d i t o r i z a t i o n , v i s u a l i z a t i o n , o r r e p r e s e n t a t i o n a l thought coupled with i n t e r n a l r e a s o n i n g . Given the many v a r i a t i o n s i n examples of sensory involvement, one means employed to d i s t i n g u i s h i n t e r n a l from e x t e r n a l methods was to separate examples on the b a s i s of use or non-use of co n c r e t e a i d s . In the case of use of b l o c k s o r s t i c k s , c h i l d r e n u s u a l l y b u i l t a c o n c r e t e model of the open sentence, manipulated the d e v i c e s i n some way to determine the m i s s i n g addend, then recorded t h e i r response by re a d i n g the c o n c r e t e i l l u s t r a t i o n . These examples w i l l from now on be r e f e r r e d to as E x t e r n a l i z e d l e v e l of response. These cases of o b v i o u s l y e x t e r n a l r e p r e s e n t a t i o n are compared t o examples of i n t e r n a l reasoning o r r e c a l l where no e x t e r n a l involvement was observed by the i n t e r v i e w e r or mentioned by the c h i l d . Included i n these examples were cases of r e c a l l of a b a s i c c f a c t as w e l l as i n t e r n a l i z e d s t r a t e g i e s o r pr o c e s s e s . These examples w i l l from now on be r e f e r r e d t o as I n t e r n a l i z e d l e v e l of response. The remaining examples (excluding No Attempts) were combinations of i n t e r n a l and e x t e r n a l processes i n v o l v i n g the use of f i n g e r s or body movements i n some 70 way. Most of these examples o c c u r r e d i n c o n j u n c t i o n w i t h c o u n t i n g s t r a t e g i e s . These examples seemed t o i l l u s t r a t e a process of gr a d u a l i n t e r n a l i z a t i o n of counting procedures and mathematical reasoning through r e c e d i n g e x t e r n a l sensory involvement, v i a the hands and body. The use of f i n g e r s v a r i e d from examples of gra s p i n g each f i n g e r as i t was counted o r a l l y , t o l o o k i n g a t , but not moving f i n g e r s w h i l e r e a s o n i n g , t o l i g h t l y p r e s s i n g the body without l o o k i n g w h i l e counting m e n t a l l y . The category T r a n s i t i o n a l l e v e l of  response was designed t o encompass a l l such examples. C l a s s I n c l u s i o n T e s t The C l a s s I n c l u s i o n T e s t was made up of fou r items. The development of these items, the m a t e r i a l s used, the s c o r i n g procedures, and the response r e c o r d i n g procedures are d e s c r i b e d i n d e t a i l . Items The C l a s s I n c l u s i o n T e s t items were chosen t o prov i d e maximum f a m i l i a r i t y w i t h both the v o c a b u l a r y and the c l a s s r e l a t i o n s i n v o l v e d . The g e n e r a l c o n t e x t s of the tasks were borrowed from s e v e r a l sources (Ahr & Youniss, 1970; E l k i n d , 1961; Inhelder & P i a g e t , 1964; Winer & Kronberg, 1974) but the a c t u a l t a s k procedures and q u e s t i o n s were a d a p t a t i o n s of those used i n these s t u d i e s . These a d a p t a t i o n s were the r e s u l t of p i l o t t e s t i n g p o s s i b l e items. Each c l a s s i n c l u s i o n task used i n the study i s d e s c r i b e d i n d e t a i l i n Appendix D. Of the fou r items, Items One and Two were o r a l l y presented while Items Three and Four were accompanied by c o n c r e t e m a t e r i a l s . T h i s was done as a means of c o n t r o l l i n g f o r v e r b a l f a c i l i t a t i o n e f f e c t s on the o v e r a l l c l a s s i n c l u s i o n measure (Winer & Kronberg, 1974; Wo h l w i l l , 1968). One o r a l l y presented and one c o n c r e t e l y presented item i n v o l v e d subset s i z e s s m a l l enough t h a t they encouraged enumeration (.Items Two and Three) , w h i l e Items One and Four i n v o l v e d s i z e s t h a t d i d not encourage enumeration t o the same ex t e n t . T h i s was done as a means of c o n t r o l l i n g f o r the e f f e c t s of enumeration of subsets (Klahr & Wallace, 1972). The r e l a t i v e s i z e of the subsets i n c l u d e d e q u i v a l e n t s e t s (Item Four) and o b v i o u s l y d i s p a r a t e s e t s (Items Two and Three). In Item One, s e t d i s p a r i t y changed depending on the b o y / g i r l r a t i o i n each s u b j e c t ' s c l a s s . T h i s v a r i e t y was i n c l u d e d t o attempt t o c o n t r o l f o r the e f f e c t s of the r e l a t i v e s i z e of s u b c l a s s e s . e f f e c t (Ahr & Youniss, 1970; B r a i n e r d & Kaszor, 1973). 72 Item s p e c i f i c a t i o n s a r e . presented i n Table 3.3. Table 3.3 C l a s s I n c l u s i o n Task S p e c i f i c a t i o n s Item Context P r e s e n t a -t i o n C a r d i n -a l i t y R e l a t i v e Set D i s p a r i t y 1 b o y s / g i r l s , c h i l d r e n o r a l 20< n ^ 3 0 v a r i e d 2 cats/dogs, pets o r a l 5, 2, 7 obvious 3 red/yellow, b l o c k s c o n c r e t e 5, 3, 7 obvious 4 c i r c l e s / t r i a n g l e s , wooden shapes concre t e 12, 12, 24 not obvious Recency e f f e c t s (Wohlwill, 1968; Youssef & Guardo, 1972) were c o n t r o l l e d by v a r y i n g the order of mentioning the g r e a t e r subset both i n i t s i n i t i a l p r e s e n t a t i o n , and i n the c l a s s i n c l u s i o n q u e s t i o n . Table 3.4 p r e s e n t s the f o u r o r d e r s f o r the p r e s e n t a t i o n of the g r e a t e r subset i n each item. Table 3.4 Order of Mentioning the G r e a t e r Subset Subset P r e s e n t a t i o n Orders I n i t i a l P r e s e n t a t i o n C l a s s I n c l u s i o n Question Order 1 ( f , f ) f i r s t f i r s t Order 2 ( f , l ) f i r s t l a s t Order 3 ( l , f ) l a s t f i r s t Order 4 (1,1) l a s t l a s t For example, order one ( f , f ) when used on item.twowould have been worded: " I f you had 5 c a t s and 2 dogs would you have more c a t s or more p e t s ? " . One-fourth of the c h i l d r e n would have had Item twoworded i n t h i s way. To a l l o w f o r v a r i a b l e language f a c i l i t y or vocabulary, the format of p r e s e n t a t i o n of the two concrete items i n v o l v e d having the c h i l d r e n name each subset and the s u p e r o r d i n a t e set i n order t o p r o v i d e the i n t e r v i e w e r w i t h l a b e l s t h a t were meaningful f o r the c h i l d . T h i s a l s o served t o focus the c h i l d ' s a t t e n t i o n on both the subsets and the su p e r o r d i n a t e s e t . I f the c h i l d was not ab l e t o p r o v i d e an a p p r o p r i a t e l a b e l , the i n t e r v i e w e r suggested one. For the o r a l l y presented items, the i n t e r v i e w e r used the c h i l d ' s own c l a s s and the c h i l d ' s own pets ( i f a p p l i c a b l e ) i n the context of the q u e s t i o n . M a t e r i a l s Items One and Two d i d not i n v o l v e c o n c r e t e m a t e r i a l s . Item Three i n v o l v e d the use of f i v e r e d and three y e l l o w 2 cm wooden cubes. Item Four i n v o l v e d the use of c o l o u r e d wooden d i s c s with a r a d i u s of 3.75 cm by 1 cm t h i c k , and c o l o u r e d wooden t r i a n g u l a r b l o c k s w i t h s i d e s of 4 cm. by 1 cm.thick. These b l o c k s are commonly known as A t t r i b u t e B l o c k s . S c o r i n g Procedure Each c h i l d was g i v e n two attempts t o answer a c l a s s i n c l u s i o n q u e s t i o n c o r r e c t l y . T h i s was done to p r o v i d e f o r s i t u a t i o n s where, a c h i l d confused the l a b e l of the s u p e r o r d i n a t e s e t w i t h t h a t of a subset(Youssef & Guardo, 1972), when i n f a c t the c h i l d was capable of r e c o g n i z i n g the part-whole r e l a t i o n s h i p i n v o l v e d . A d e s c r i p t i o n of the s c o r i n g procedure i s presented i n Table 3.5. Table 3.5 C l a s s I n c l u s i o n Task S c o r i n g Procedure Score Response 2 c o r r e c t answer, w i t h a c o r r e c t r a t i o n a l e , on the f i r s t attempt 1 c o r r e c t answer, wi t h no r a t i o n a l e , on the f i r s t attempt 1 c o r r e c t answer on second attempt, w i t h a c o r r e c t r a t i o n a l e 0 c o r r e c t answer on second attempt, w i t h no r a t i o n a l e or an i n c o r r e c t r a t i o n a l e 0 i n c o r r e c t answer on both attempts I f , on the f i r s t attempt, the child answered correctly w i t h a r a t i o n a l e , a score of two was assigned. I f the c h i l d d i d not p r o v i d e a r a t i o n a l e and the i n t e r v i e w e r was unable to e l i c i t one through q u e s t i o n i n g , a score of one was a s s i g n e d . I f the c h i l d ' s answer on the f i r s t attempt was i n c o r r e c t , the i n t e r v i e w e r accepted the response, then p r e s e n t e d the e n t i r e t a s k a g a i n , making sure the c h i l d c o u l d i d e n t i f y the subsets and the s u p e r o r d i n a t e s e t . I f on the second attempt the c h i l d not o n l y answered c o r r e c t l y , but a l s o was a b l e to p r o v i d e a l o g i c a l , c o r r e c t r a t i o n a l e , a score of one was a s s i g n e d . A l l o ther s i t u a t i o n s r e s u l t e d i n a score of zero. Task P r e s e n t a t i o n and Response Recording Procedure A t a b l e of random numbers was used to generate f i f t y o r d e r i n g s f o r Items One to Four. Each ,of the f i f t y item orders was recorded on i n d i v i d u a l c a r d s . The i n t e r v i e w e r used these cards as a guide to each c h i l d ' s c l a s s i n c l u s i o n t a s k s , and recorded responses d i r e c t l y on the cards. A w r i t t e n r e c o r d of each c h i l d ' s responses was l a t e r made from the tape r e c o r d i n g s of the t e s t s e s s i o n s . For each item, the order of mentioning the g r e a t e r subset was then recorded on each c a r d . Using the f o u r p o s s i b l e o r d e r s of mentioning the g r e a t e r subset (f,f;. f , l ; l , f ; 1,1) o r d e r i n g s were generated u s i n g a s e r i e s of 4 x 4 L a t i n Squares. These o r d e r i n g s were then a s s i g n e d to each c l a s s i n c l u s i o n task c a r d . Each c h i l d , then, had a l l four p o s s i b l e o r d e r s of mentioning the g r e a t e r subset but the sequence v a r i e d and the item t o which i t was a p p l i e d v a r i e d f o r each c h i l d . A sample task p r e s e n t a t i o n c a r d i s presented i n Appendix E. Procedure The d e t a i l s of the a d m i n i s t r a t i o n of the M i s s i n g Addend Test and the C l a s s I n c l u s i o n T e s t are presented f i r s t . Next, a d e s c r i p t i o n of the data t h a t were c o l l e c t e d , the r e l i a b i l i t y s t u d i e s f o r t h i s data, and data a n a l y s i s techniques are d e s c r i b e d . T e s t A d m i n i s t r a t i o n Both t e s t s were i n d i v i d u a l l y a d m i n i s t e r e d d u r i n g the same s i t t i n g i n a room other than the classroom. The M i s s i n g Addend T e s t was always a d m i n i s t e r e d before the C l a s s I n c l u s i o n T e s t . Each i n t e r v i e w took between 15 and 25 minutes t o complete. Interviews were conducted between A p r i l 4 and A p r i l 15, 1977, and were a l l conducted by the i n v e s t i g a t o r . A f t e r accompanying the c h i l d t o the t e s t i n g room, the i n t e r v i e w e r then f a m i l i a r i z e d the c h i l d with the tape r e c o r d e r , the m a t e r i a l s t h a t were a v a i l a b l e t o use, and the g e n e r a l purpose of the i n t e r v i e w . The c h i l d was g i v e n a b o o k l e t , and asked to r e c o r d h i s or her f i r s t name i n the space p r o v i d e d . D i r e c t i o n s were then g i v e n f o r the c h i l d to read each p r a c t i c e equation aloud, then t o f i l l i n the answer. Emphasis was p l a c e d on re a d i n g the complete open sentence. Most c h i l d r e n used the word "box" f o r the p l a c e h o l d e r , w h i l e some used the word "something". I f a c h i l d seemedviunsure about how to read the p l a c e h o l d e r , the word "box" was suggested. F o l l o w i n g each p r a c t i c e item, the i n t e r v i e w e r questioned the c h i l d on'how the response had been determined. A f t e r the examples were completed, the i n t e r v i e w e r reminded the c h i l d about the m a t e r i a l s t h a t were a v a i l a b l e , assured the c h i l d t h a t whatever he or she d i d to f i n d an answer was j u s t f i n e , and turned t o the f i r s t item. These reminders were repeated as o f t e n as the i n t e r v i e w e r f e l t was necessary. As each item was presented, the c h i l d read the item aloud and recorded the response. The i n t e r v i e w e r recorded d e t a i l s of the l e v e l of response used and attempted t o determine the s t r a t e g y used by q u e s t i o n i n g the student on h i s or her procedure. The que s t i o n . "How d i d you get your answer?" was used a f t e r each item; f u r t h e r q u e s t i o n i n g v a r i e d depending upon the c l a r i t y of the c h i l d ' s i n i t i a l response. R e l i a n c e by the c h i l d on some e x t e r n a l a i d such as the b l o c k s , p r o v i d e d the i n t e r v i e w e r w i t h c l u e s f o r f u r t h e r q u e s t i o n s . In cases where no observable behaviours were e v i d e n t , the qu e s t i o n s . "What d i d you t h i n k t o y o u r s e l f while you d i d i t ? " , and "What number d i d you s t a r t with?", were asked by the i n t e r v i e w e r . F o l l o w i n g the m i s s i n g addend t e s t , the c h i l d was engaged i n an a c t i v i t y such as p u t t i n g the b l o c k s and s t i c k s a s i d e , then was g i v e n the f o u r c l a s s i n c l u s i o n t a s k s . The i n t e r v i e w e r used the c l a s s i n c l u s i o n t a s k p r e s e n t a t i o n c a r d as a guide f o r the order of items and the order of p r e s e n t a t i o n of subsets, and t o r e c o r d responses a f t e r each q u e s t i o n . Upon completion, the c h i l d was thanked and sent back to c l a s s . 78 Data C o l l e c t e d The f o l l o w i n g data were c o l l e c t e d d u r i n g the i n t e r v i e w s : M i s s i n g Addend T e s t data, 1. p u p i l responses to the e i g h t m i s s i n g addend items, as recorded i n p u p i l b o o k l e t s ; 2. a w r i t t e n d e s c r i p t i o n and an i n i t i a l c l a s s i f i c a t i o n o f the s o l u t i o n procedure used f o r each item, recorded on the i n t e r v i e w coding form; 3. a w r i t t e n d e s c r i p t i o n of the l e v e l of response used f o r each item, recorded on the i n t e r v i e w coding form; 4. tape r e c o r d i n g s of the m i s s i n g addend t e s t a d m i n i s t r a t i o n . C l a s s I n c l u s i o n Test data. 1. a score f o r each c l a s s i n c l u s i o n task w i t h a b r i e f w r i t t e n d e s c r i p t i o n of the response; 2. a tape r e c o r d i n g of the c l a s s i n c l u s i o n task a d m i n i s t r a t i o n (a w r i t t e n t r a n s c r i p t of these r e c o r d i n g s was made subsequent to the c o l l e c t i o n of d a t a ) . R e l i a b i l i t y Study Two observers each observed two d i f f e r e n t M i s s i n g Addend Te s t i n t e r v i e w s and r a t e d m i s s i n g addend s o l u t i o n s t r a t e g i e s , r e s u l t i n g i n 10% of the items (four c h i l d r e n x e i g h t items) being independently r a t e d . S o l u t i o n s t r a t e g y c l a s s i f i c a t i o n s were then compared. R e s u l t s of t h i s comparison are presented i n Chapter 4. 79 The r e l i a b i l i t y of the c l a s s i n c l u s i o n task s c o r i n g was examined f o l l o w i n g the c o l l e c t i o n of data. Using the ta s k cards and the t r a n s c r i p t s from the taped i n t e r v i e w s , an independent r a t e r scored 40% of the items. R e s u l t s of these scores and the i n t e r v i e w e r ' s scores were then compared. R e s u l t s are presented i n Chapter 4. Data A n a l y s i s The data on accuracy of response were computer analysed. The computer programme LERTAP 2.0 was used f o r the Item A n a l y s i s . ANOVA procedures f o r repeated measures were completed u s i n g computer programme BMDP2V. The c l a s s i f i c a t i o n data on the s o l u t i o n methods and l e v e l s of response used, were t r e a t e d u s i n g d e s c r i p t i v e s t a t i s t i c s . The computer programme SPSS.v6.02 was used t o generate contingency t a b l e s used i n t h i s a n a l y s i s . The r e l a t i o n s h i p between M i s s i n g Addend T e s t v a r i a b l e s and C l a s s I n c l u s i o n T e s t v a r i a b l e s was examined u s i n g the c o r r e l a t i o n matrix generated by computer programme BMD02R. A l l computer programmes are s t o r e d a t the U n i v e r s i t y of B r i t i s h Columbia Computing Centre. Chapter 4 RESULTS The r e s u l t s of the study are r e p o r t e d i n the f o l l o w i n g order. F i r s t , performance r e s u l t s f o r the M i s s i n g Addend Te s t are gi v e n . Next, the frequency and accuracy of s t r a t e g y usage and l e v e l of response usage, and the i n t e r a c t i o n of the two v a r i a b l e s , are d e s c r i b e d . F i n a l l y , C l a s s I n c l u s i o n T e s t r e s u l t s and c o r r e l a t i o n s of these r e s u l t s w i t h M i s s i n g Addend T e s t r e s u l t s are presented. M i s s i n g Addend T e s t R e s u l t s Item A n a l y s i s R e s u l t s The M i s s i n g Addend T e s t i n c l u d e d e i g h t items. One item was used t o measure each c e l l of the f o u r by two t e s t d e s i g n (four l e v e l s of d i f f i c u l t y and two p l a c e h o l d e r p o s i t i o n s ) . Table 4.1 l i s t s the a c t u a l items, the number of c o r r e c t responses on each item, the percent c o r r e c t , and the p o i n t b i s e r i a l c o r r e l a t i o n c o e f f i c i e n t of each item w i t h the o v e r a l l t e s t . C h i l d r e n ' s raw scores are l i s t e d i n Appendix F. Table 4.1 M i s s i n g Addend Test Item R e s u l t s L e v e l of P l a c e h o l d e r Item Number of C o r r e c t Percent P o i n t B i s e r i a l D i f f i c u l t y P o s i t i o n Item No. Responses C o r r e c t C o r r e l ; L e v e l One a + • 5 + • = 8 1 35 87.5% 0.58 • + a • + 4 = 7 2 34 85 % 0.60 L e v e l Two a + • 5 + • = 13 3 30 75 % 0.67 • + a • + 6 = 14 4 24 60 % 0.65 L e v e l Three a + • 42 + • = 48 5 21 5 2 • 5 % 0.67 • + a • + 43 = 49 6 21 52.5% 0.75 L e v e l Four a + • 23 + • = 38 7 12 30 % 0. 62 • + a • + 24 = 39 8 19 47.5% 0.77 82 The t o t a l number of c o r r e c t responses was 196 out of a p o s s i b l e 3 2 0 (40 c h i l d r e n x 8 i t e m s ) . The mean score was 4.9 out of e i g h t , with a standard d e v i a t i o n of 2 . 4 3 . O v e r a l l r e l i a b i l i t y of the M i s s i n g Addend T e s t as measured by the Hoyt Estimate of R e l i a b i l i t y was 0.82 (Hoyt, 1 9 4 1 ) . L e v e l of D i f f i c u l t y The e i g h t items on the t e s t r e p r e s e n t e d f o u r l e v e l s of d i f f i c u l t y . The f i r s t two l e v e l s r e p r e s e n t e d the B a s i c F a c t Domain, L e v e l One i n v o l v i n g sums t o ten, and L e v e l Two i n v o l v i n g sums to ei g h t e e n . The oth e r two l e v e l s r e p r e s e n t e d the Two-Digit Domain, L e v e l Three i n v o l v i n g a t w o - d i g i t number p l u s a o n e - d i g i t number, and L e v e l Four i n v o l v i n g a t w o - d i g i t number p l u s a t w o - d i g i t number. Table 4.2 summarizes the data on the l e v e l of d i f f i c u l t y f a c t o r . Table 4.2 Summary of Data on the L e v e l of D i f f i c u l t y F a c t o r T o t a l Number of Mean L e v e l of Number Number of C o r r e c t Percent D i f f i c u l t y of Items Student Items Responses C o r r e c t L e v e l One 2 80 69 8 6 % L e v e l Two 2 80 54 6 8% L e v e l Three 2 80 42 5 3 % L e v e l Four 2 8 0 31 3 9 % T o t a l 8 320 1 9 6 6 1 % 83 The t o t a l number of c o r r e c t responses on items i n L e v e l One was 6 9 , or 86 percent of the p o s s i b l e t o t a l of 8 0 items (two items x f o r t y c h i l d r e n ) . The t o t a l number of c o r r e c t responses on items i n L e v e l Two was 54, or 68 percent. The t o t a l number of c o r r e c t responses on items i n L e v e l Three was 4 2 , or 53 per c e n t ; and L e v e l Four, 3 1 , or 39 percent of the p o s s i b l e t o t a l . P l a c e h o l d e r P o s i t i o n The e i g h t items on the t e s t r e p r e s e n t e d two m i s s i n g addend p l a c e h o l d e r p o s i t i o n s . Of the two items a t each l e v e l of d i f f i c u l t y , one used the format a + O = b and the o t h e r was of the format Q + a = b. Data r e l a t e d t o p l a c e h o l d e r p o s i t i o n are summarized i n Table 4 . 3 . Table 4.3 Summary of Data on P l a c e h o l d e r P o s i t i o n P l a c e - T o t a l Number of Mean ho l d e r Number Number of C o r r e c t Percent P o s i t i o n of Items Student Items Responses C o r r e c t a + • = b 4 1 6 0 98 6 1 % • + a = b 4 1 6 0 98 6 1 % T o t a l 8 320 196 6 1 % The t o t a l number of c o r r e c t responses was 98 f o r both p l a c e h o l d e r p o s i t i o n s , i n d i c a t i n g no main e f f e c t due to p l a c e h o l d e r p o s i t i o n . 84 Summary of Analysis of Variance Computer program BMDP2V ANOVA for repeated measures was used to analyse Missing Addend Test Results, ANOVA re s u l t s are presented i n Table 4.4 Table 4.4 Summary of ANOVA with Repeated Measures for Level of D i f f i c u l t y and Placeholder Position Source Sum of Degrees of Mean Squares Freedom Square F Pro b a b i l i t y F Exceeded Level of D i f f i c u l t y ( L ) Error(L) Placeholder Position(P) Error(P) L X P Interaction 9. 9749 17.2749 0.0 8. 7480 1.0750 3..3250 22.5193 0.000 Error(LXP) 10.1750 117 1 39 3 117 0.1477 0.0 0.0 1.000 0.2244 0.3583 4.1204 0.008 0. 0970 An examination of the table showed that the variable l e v e l of d i f f i c u l t y made a s i g n i f i c a n t contribution to the t o t a l variance (p<.001), and the int e r a c t i o n between l e v e l of d i f f i c u l t y and placeholder p o s i t i o n was s i g n i f i c a n t (p<.01). Performance at each l e v e l of d i f f i c u l t y was compared to performance at a l l other lev e l s using Scheffe's test (Marascuilo & Levin, 1974, p. 132). Levels One and Four d i f f e r e d s i g n i f i c a n t l y (p< .05). 85 However, when L e v e l s One and Two were c o l l a p s e d to r e p r e s e n t the b a s i c f a c t domain, and L e v e l s Three and Four were c o l l a p s e d t o r e p r e s e n t the two d i g i t domain, means f o r the two domains were s i g n i f i c a n t l y d i f f e r e n t (p<..05) as determined by S h e f f e ' s t e s t . o CD u u o o -p c <D O i4 QJ PL) 100% 80% 60% 40% 20% 0% LI L2 L3 L4 L e v e l of D i f f i c u l t y P l a c e h o l d e r P o s i t i o n a + • = b — • + a = b F i g u r e 4.1 I n t e r a c t i o n of L e v e l of D i f f i c u l t y and P l a c e h o l d e r P o s i t i o n F i g u r e 4.1 i s a g r a p h i c d i s p l a y i l l u s t r a t i n g the i n t e r a c t i o n of l e v e l of d i f f i c u l t y x p l a c e h o l d e r p o s i t i o n . Performance on a + Q = b was b e t t e r than performance on [ ] + a = b f o r L e v e l s One and Two. Performance on both p l a c e h o l d e r p o s i t i o n s was the same a t L e v e l Three, while a t L e v e l Four, performance on Q+ a = b was b e t t e r . I t appears t h a t performance on the a + Q = b format was b e t t e r than [ ] + a = b a t L e v e l Two but c o n s i d e r a b l y worse a t L e v e l Four. T h i s d i f f e r e n c e may 86 n o t wholly be due to the i n t e r a c t i o n of the two v a r i a b l e s , but may i n s t e a d be a t t r i b u t e d to d i f f e r e n c e s i n the p a r t i c u l a r items used to measure each l e v e l x p l a c e h o l d e r . Only one item was used t o measure each c e l l o f the 4 x 2 t e s t d e s i g n . The two items used a t any l e v e l of d i f f i c u l t y i n v o l v e d number combinations where the sums d i f f e r e d by one. For example, the two items a t L e v e l One were 5 + Q = 8 and [[] + 4 = 7. Since the number combinations used at any l e v e l were not i d e n t i c a l , d i f f e r e n c e s i n performance a t s p e c i f i c l e v e l s may be a t t r i b u t e d t o the p a r t i c u l a r number combinations r a t h e r than t o p l a c e h o l d e r p o s i t i o n . For t h i s reason, the i n t e r a c t i o n of the two v a r i a b l e s was not pursued f u r t h e r . Summary of• M i s s i n g Addend T e s t R e s u l t s The study addressed t h r e e q u e s t i o n s r e l a t e d t o performance on the M i s s i n g Addend T e s t . One q u e s t i o n concerned whether or not performance was a f f e c t e d by the l e v e l of d i f f i c u l t y of the items. I t appears t h a t performance was s i g n i f i c a n t l y b e t t e r on items from the Basi c F a c t Domain (Levels One and Two) than items from the Two-Digit Domain (Levels Three and F o u r ) , however d i f f e r e n c e s i n performance among the fo u r d i f f i c u l t y l e v e l s were not s i g n i f i c a n t (with the e x c e p t i o n of L e v e l s One and F o u r ) . 87 Another q u e s t i o n concerned whether or not performance was a f f e c t e d by p l a c e h o l d e r p o s i t i o n . R e s u l t s showed t h a t o v e r a l l there was no d i f f e r e n c e a t a l l between performance on a +Q = b items and O + a = b items. The t h i r d q u e s t i o n concerned the p o s s i b l e i n t e r a c t i o n of l e v e l of d i f f i c u l t y and p l a c e h o l d e r p o s i t i o n . A s i g n i f i c a n t i n t e r a c t i o n was i d e n t i f i e d and was a t t r i b u t e d t o the s p e c i f i c number combinations chosen f o r each c e l l of the desi g n . M i s s i n g Addend S o l u t i o n S t r a t e g i e s During the i n d i v i d u a l i n t e r v i e w s , the i n v e s t i g a t o r attempted to determine s o l u t i o n s t r a t e g i e s ' by o b s e r v i n g and q u e s t i o n i n g the c h i l d r e n . A predetermined c l a s s i f i c a t i o n scheme was used t o c l a s s i f y these s t r a t e g i e s d u r i n g the i n t e r v i e w . Upon completion of the data c o l l e c t i o n t h i s scheme was r e o r g a n i z e d , as d i s c u s s e d i n Chapter 3 under " S t r a t e g y C l a s s i f i c a t i o n " . During the i n t e r v i e w s , the Indeterminate category was used f o r s e v e r a l examples where c h i l d r e n used combinations of s t r a t e g i e s , or used as y e t u n i d e n t i f i e d s t r a t e g i e s . There were 17 cases where the r e o r g a n i z a t i o n of the s t r a t e g y c l a s s i f i c a t i o n scheme accomodated what p r e v i o u s l y had been c l a s s i f i e d as Indeterminate. T h i s r e c l a s s i f i c a t i o n was done wit h the help of the i n t e r v i e w e r ' s w r i t t e n d e s c r i p t i o n of the s t r a t e g y and the tape r e c o r d i n g of the c h i l d ' s 88 e x p l a n a t i o n . As a means of examining the r e l i a b i l i t y of the i n t e r v i e w e r ' s c l a s s i f i c a t i o n of s t r a t e g i e s , two independent observers each r a t e d two d i f f e r e n t student i n t e r v i e w s . Approximately 30 minutes were spent ahead of time e x p l a i n i n g the c l a s s i f i c a t i o n scheme and procedure. Ratings by the i n t e r v i e w e r and the independent r a t e r s were then compared, r e s u l t i n g i n agreement on 29 of the 32 items, or 91% of the time. These comparisons are presented i n Appendix I. The r e s u l t s of the s o l u t i o n s t r a t e g y usage d e a l f i r s t w i t h the frequency of usage of each s t r a t e g y o v e r a l l , by l e v e l of d i f f i c u l t y , and by p l a c e h o l d e r p o s i t i o n . These r e s u l t s answer ques t i o n s 5, 6, and 7 which are concerned with the frequency of use o f each s o l u t i o n s t r a t e g y . Secondly, the accuracy o f each s t r a t e g y i s examined o v e r a l l and by l e v e l of d i f f i c u l t y , as a means of answering q u e s t i o n s 8, 9, and 10 concerning accuracy of s t r a t e g i e s . The r e s u l t s of both frequency and accuracy of s t r a t e g y use w i l l be examined f o r a d d i t i v e as opposed to s u b t r a c t i v e i n t e r p r e t a t i o n s , and f o r i n d i v i d u a l s t r a t e g y use. Frequency of St r a t e g y Use O v e r a l l . Table 4.5 p r e s e n t s the frequency of usage of each of the s o l u t i o n s t r a t e g i e s both i n 89 number of examples over the 320 items, and percents o v e r a l l . T able 4.5 Frequency of Use of S o l u t i o n S t r a t e g i e s O v e r a l l Usage  St r a t e g y Frequency Percent Semi-Guess A d d i t i v e 9 3% S u b t r a c t i v e 4 1% S u b s t i t u t i o n A d d i t i v e 7 2% S u b t r a c t i v e 0 0% C o u n t i n g - A l l A d d i t i v e 76 24% S u b t r a c t i v e 20 6% Counting-On A d d i t i v e 97 30% S u b t r a c t i v e 2 1% A s s o c i a t i v e A d d i t i v e 29 9% S u b t r a c t i v e 4 1% R e c a l l 30 9% I n c o r r e c t T r a n s f o r m a t i o n 24 8% Indeterminate 12 4% No Attempt 6 2% T o t a l 320 100% A d d i t i v e s t r a t e g i e s g e n e r a l l y were used much more f r e q u e n t l y than s u b t r a c t i v e approaches. In 68% of the cases c h i l d r e n i n t e r p r e t e d m i s s i n g addend sentences as s i t u a t i o n s r e q u i r i n g an incrementing procedure while i n 9% of the cases c h i l d r e n decremented or s t a r t e d w i t h the sum and s u b t r a c t e d . I n d i v i d u a l p u p i l r e s u l t s i n d i c a t e t h a t the method of i n s t r u c t i o n does not n e c e s s a r i l y d i c t a t e what 90 method children w i l l use to solve missing addends. Of the 19 c h i l d r e n from classes where the teacher emphasized a subtractive approach only f i v e children chose to use a subtractive interpretation; these f i v e account for 18 of the 3 0 subtractive examples. Of the 21 children from classes where the teacher emphasized an additive approach, four of them used a subtraction interpretation a t o t a l of 12 times. These re s u l t s support the assumption that children use strategies that are meaningful to them in preference to methods that they are encouraged to use, but that may be less meaningful. Counting strategies, i n p a r t i c u l a r additive Counting-All and Counting-On, were most frequently used. Semi-Guess and Substitution strategies were l e a s t frequently used. The a v a i l a b i l i t y of concrete materials to aid i n solving the missing addend examples l i k e l y minimized the frequency of children refusing to try the items, since only six examples of t h i s occurred. Likewise, the use of concrete aids provided the interviewer with clues to the solution processes used, and gave r i s e to questions that helped to i d e n t i f y strategies, r e s u l t i n g in only 12 indeterminate examples. By l e v e l of d i f f i c u l t y . The l e v e l of d i f f i c u l t y of the item appeared to a f f e c t the frequency of use of cer t a i n strategies to various degrees. Table 4.6 91 r e p r e s e n t s frequency of s t r a t e g y use broken down by l e v e l of d i f f i c u l t y . The Semi-Guess s t r a t e g y was used o n l y with t w o - d i g i t examples, suggesting the s i z e of the numbers i n v o l v e d may have obscured or confused a method t h a t was s u c c e s s f u l i n the B a s i c F a c t Domain. Usage of S u b s t i t u t i o n and counting s t r a t e g i e s v a r i e d o n l y s l i g h t l y across l e v e l s of d i f f i c u l t y . A s s o c i a t i v e s t r a t e g i e s , which i n v o l v e the use of b a s i c f a c t s , o c c u r r e d i n L e v e l s Two, Three, and Four on l y , presumably because r e c a l l of the b a s i c f a c t s was adequate t o handle L e v e l One. Table 4.6 S t r a t e g y Frequency f o r Each L e v e l of D i f f i c u l t y L e v e l of D i f f i c u l t y S t r a t e g y L e v e l 1 L e v e l 2 L e v e l 3 L e v e l 4 Semi-Guess A d d i t i v e 0 0 2 7 S u b t r a c t i v e 0 0 0 4 S u b s t i t u t i o n A d d i t i v e 3 2 1 1 S u b t r a c t i v e 0 0 0 0 C o u n t i n g - A l l A d d i t i v e 16 25 18 17 S u b t r a c t i v e 6 7 2 5 Counting-On A d d i t i v e 23 23 27 24 S u b t r a c t i v e 0 1 1 0 A s s o c i a t i v e A d d i t i v e 0 10 14 5 S u b t r a c t i v e 0 0 3 1 R e c a l l 26 .4 0 0 I n c o r r e c t Transformation 5' 5 6 8 Indeterminate 1 3 5 3 No Attempt 0 0 1 5 T o t a l 80 80 80 80 The R e c a l l s t r a t e g y a p p l i e d o n l y to the B a s i c F a c t Domain and was used almost completely w i t h the b a s i c f a c t s combinations l e s s than t e n . Since these second year c h i l d r e n were o n l y i n the process of working towards mastery of b a s i c f a c t s t o eighteen, t h i s i s not s u r p r i s i n g . I n c o r r e c t Transformations o c c u r r e d f a i r l y c o n s i s t e n t l y a c r o s s l e v e l s of d i f f i c u l t y . C h i l d r e n who used t h i s T r a n s f o r m a t i o n s t r a t e g y , u s u a l l y d i d so c o n s i s t e n t l y , suggesting t h a t conceptual nature of the e r r o r . F i v e c h i l d r e n accounted f o r the 24 examples of I n c o r r e c t T r a n s f o r m a t i o n . A l l s i x examples of No Attempt o c c u r r e d i n the Two-Digit Domain. T h i s i s not s u r p r i s i n g s i n c e second year c h i l d r e n work with m i s s i n g addends i n v o l v i n g b a s i c f a c t combinations but r a r e l y encounter t w o - d i g i t examples. By p l a c e h o l d e r p o s i t i o n . The p o s i t i o n of the p l a c e h o l d e r i n the m i s s i n g addend sentence appeared t o have almost no e f f e c t on the frequency of s t r a t e g y c h o i c e . Table 4.7 prese n t s the o v e r a l l frequency of s t r a t e g y use f o r each p l a c e h o l d e r p o s i t i o n . Frequencies f o r s t r a t e g y use are r e l a t i v e l y constant a c r o s s the two p l a c e h o l d e r p o s i t i o n s . Only two s t r a t e g y f r e q u e n c i e s seemed to be s l i g h t l y a f f e c t e d by p l a c e h o l d e r p o s i t i o n . S u b s t i t u t i o n s were more fre q u e n t f o r 0+ a = b examples, and a d d i t i v e c o u n t i n g - a l l was more frequent f o r a + Q = b items. Seventy-four p e r c e n t of the t ime, c h i l d r e n used the same s t r a t e g y f o r both items a t a given . l e v e l of d i f f i c u l t y . To some extent t h i s i l l u s t r a t e s an i n t u i t i v e understanding of the commutative p r i n c i p l e e s p e c i a l l y i n cases where a d d i t i v e s t r a t e g i e s were used, i n t h a t c h i l d r e n began t h e i r s o l u t i o n process w i t h whichever addend was presented. Table 4 . 7 St r a t e g y Frequency f o r each . P l a c e h o l d e r P o s i t i o n P l a c e h o l d e r P o s i t i o n S t r a t e g y a + • = b • + a = b Semi-Guess A d d i t i v e 5 4 S u b t r a c t i v e 3 1 S u b s t i t u t i o n A d d i t i v e 1 6 S u b t r a c t i v e 0 0 C o u n t i n g - A l l A d d i t i v e 42 34 S u b t r a c t i v e 11 9 Counting-On A d d i t i v e 48 49 S u b t r a c t i v e 0 2 A s s o c i a t i v e A d d i t i v e 14 15 S u b t r a c t i v e 2 2 R e c a l l 15 15 I n c o r r e c t T r a n s f o r m a t i o n 11 13 Indeterminate 5 7 No Attempt 3 3 T o t a l 160 160 94 I n d i v i d u a l C h i l d r e n ' s S t r a t e g y Choice P a t t e r n s Each c h i l d i n the study was presented w i t h e i g h t m i s s i n g addend sentences i n random o r d e r (with the e x c e p t i o n of the f i r s t item which was always d i f f i c u l t y and two p l a c e h o l d e r p o s i t i o n . Questions 6 and 7 of the study asked whether frequency of s t r a t e g y c h o i c e v a r i e d as a f u n c t i o n of the d i f f i c u l t y of the p l a c e h o l d e r p o s i t i o n of the items. O v e r a l l frequency data has been r e p o r t e d . In a d d i t i o n to t h i s i n f o r m a t i o n , t e s t r e s u l t s p r o v i d e another a n a l y s i s of these q u e s t i o n s , but from the p o i n t o f view of s t r a t e g y c h o i c e p a t t e r n s f o r i n d i v i d u a l c h i l d r e n . Some c h i l d r e n appeared to make s t r a t e g y c h o i c e s independently of the item v a r i a b l e s (level o f d i f f i c u l t y and p l a c e h o l d e r p o s i t i o n ) / while other c h i l d r e n used s t r a t e g i e s i n an i d e n t i f i a b l e p a t t e r n a c r o s s the e i g h t items. Appendix G p r e s e n t s i n d i v i d u a l s t r a t e g y c h o i c e p a t t e r n s f o r the sample. The frequency of occurrence of these p a t t e r n s i s presented i n Table 4.8. 5 +. tZl = 8) . The e i g h t items were at f o u r l e v e l s of Table 4.8 St r a t e g y Choice P a t t e r n Frequencies S t r a t e g y Choice A f f e c t e d by Item L e v e l of D i f f i c u l t y S t r a t e g y Choice A f f e c t e d by P l a c e h o l d e r P o s i t i o n S t r a t e g y Choice Not A f f e c t e d by Tes t V a r i a b l e s U n i d e n t i f i a b l e P a t t e r n t o St r a t e g y Choice 14 13 11 2 T o t a l 40 95 The l e v e l of d i f f i c u l t y of the item appeared to a f f e c t the s t r a t e g y c h o i c e s of 14 c h i l d r e n . The s t r a t e g y c h o i c e p a t t e r n s of 12 of the 14 c h i l d r e n i n d i c a t e t h a t they used what appear to be more s o p h i s t i c a t e d s o l u t i o n s t r a t e g i e s f o r the e a s i e r items, but changed to l e s s s o p h i s t i c a t e d s t r a t e g i e s as the d i f f i c u l t y i n c r e a s e d . F i v e c h i l d r e n used A s s o c i a t i v e or R e c a l l procedures and then changed t o counting s t r a t e g i e s f o r more d i f f i c u l t items. Seven c h i l d r e n changed from counting s t r a t e g i e s t o Semi-Guesses; S u b s t i t u t i o n s , o r No Attempts. There were two cases where c h i l d r e n appeared to be i n f l u e n c e d by l e v e l of d i f f i c u l t y , but changed to what are proposed t o be more s o p h i s t i c a t e d s t r a t e g i e s a t higher d i f f i c u l t y l e v e l s . One c h i l d changed from C o u n t i n g - A l l t o Counting-On, and the ot h e r changed from C o u n t i n g - A l l t o A s s o c i a t i v e f o r more d i f f i c u l t items. The s t r a t e g y c h o i c e s of on l y two c h i l d r e n were a p p a r e n t l y a f f e c t e d by p l a c e h o l d e r p o s i t i o n . One c h i l d I n c o r r e c t l y Transformed a l l Q + a = b examples while C o u n t i n g - A l l f o r a + [~~] = b items. The o t h e r S u b s t i t u t e d f o r Q + a = b and Cou n t e d - A l l f o r a + • = b. These r e s u l t s seem to i n d i c a t e t h a t 3 8 of the 40 c h i l d r e n seemed u n a f f e c t e d by p l a c e h o l d e r p o s i t i o n as f a r as s t r a t e g y c h o i c e was concerned. The s t r a t e g y c h o i c e s of t h i r t e e n c h i l d r e n were c l e a r l y not a f f e c t e d by item v a r i a b l e s . These c h i l d r e n used one s t r a t e g y f o r seven or e i g h t items. There were nine cases of c o n s i s t e n t use of c o u n t i n g procedures, two cases of c o n s i s t e n t a p p l i c a t i o n of A s s o c i a t i v e s t r a t e g i e s and R e c a l l , and two cases of c o n s i s t e n t I n c o r r e c t T r a n s f o r m a t i o n use. In e l e v e n cases no p a t t e r n was i d e n t i f i e d . These c h i l d r e n had more than one e x c e p t i o n to the p a t t e r n s l i s t e d above. I t i s p o s s i b l e t h a t the random p r e s e n t a t i o n a f f e c t e d s t r a t e g y c h o i c e s f o r these c h i l d r e n although t h i s d i d not seem to a f f e c t the p a t t e r n s of the o t h e r 29 c h i l d r e n . Accuracy of S t r a t e g y Use Two c o n s i d e r a t i o n s a f f e c t the a p p r o p r i a t e n e s s of comparing the accuracy r a t i n g s f o r s t r a t e g i e s . One c o n s i d e r a t i o n i s t h a t s t r a t e g i e s are o n l y as a c c u r a t e as the c h i l d r e n who choose t o use them. Since i t i s proposed t h a t the range of s t r a t e g i e s r e f l e c t an h i e r a r c h y of s o p h i s t i c a t i o n , the c h i l d r e n u s i n g the l e a s t s o p h i s t i c a t e d s t r a t e g i e s may w e l l be those w i t h the l e a s t understanding of mathematics g e n e r a l l y , and consequently the l e a s t a c c u r a t e . On the other hand, as Brownell (1928, p. 205) p o i n t e d out, when c h i l d r e n begin to employ a more mature method, they seem to l o s e e f f i c i e n c y as measured by accuracy, so accuracy may depend on the l e n g t h of time c h i l d r e n have been u s i n g the p a r t i c u l a r s t r a t e g y . 97 Secondly, the l e v e l of response used o f t e n a f f e c t e d the accuracy of the s t r a t e g y . For example, o f t e n c h i l d r e n who used b l o c k s v e r y e f f i c i e n t l y f o r some examples t r i e d to use the same s t r a t e g i e s without the a i d of the blocks and made an e r r o r . With these c o n s i d e r a t i o n s i n mind, the s t r a t e g i e s can be examined as o u t l i n e d i n Questions 8, 9, and 10, f o r t h e i r g e n e r a l s u i t a b i l i t y to second year c h i l d r e n . No Attempts, Indeterminate responses, and I n c o r r e c t T ransformations were f o r the most p a r t i n c o r r e c t , so w i l l not be co n s i d e r e d . Table 4.9 p r e s e n t s accuracy r a t e s f o r the other s o l u t i o n s t r a t e g i e s both o v e r a l l , and by l e v e l s of d i f f i c u l t y c o l l a p s e d t o r e p r e s e n t the B a s i c F a c t Domain (Levels One and Two) and the Two D i g i t Domain (Levels Three and F o u r ) . Accuracy r a t e s are based on the r a t i o of c o r r e c t usage to o v e r a l l usage of a s t r a t e g y . Since some s t r a t e g i e s were not f r e q u e n t l y used, t h e i r accuracy r a t e s are not as r e l i a b l e as s t r a t e g i e s t h a t were used more o f t e n . For t h i s reason, the frequency of the usage of the s t r a t e g y i s i n d i c a t e d i n parentheses beside each accuracy r a t e . I f the frequency i s zero no accuracy r a t e i s p o s s i b l e . Semi-Guess procedures always r e s u l t e d i n i n c o r r e c t responses, while S u b s t i t u t i o n procedures were Table 4.9 S o l u t i o n S t r a t e g y Accuracy Rates Accuracy Rate S t r a t e g y O v e r a l l B a s i c Fact Domain Two D i g i t Domain Semi-Guess A d d i t i v e S u b t r a c t i v e 0 0 (9) * (4) (0) (0) 0 0 (9) (4) S u b s t i t u t i o n A d d i t i v e S u b t r a c t i v e .29 (7) (0) .40 (5) (0) 0 (2) (0) C o u n t i n g - A l l A d d i t i v e S u b t r a c t i v e .58 .95 (76) (20) .68 . 92 (41) (13) .46 1.00 (35) (7) Counting-On A d d i t i v e S u b t r a c t i v e .78 1. 00 (97) (2) .93 1.00 (46) (1) .65 1. 00 (51) (1) A s s o c i a t i v e A d d i t i v e S u b t r a c t i v e .76 .25 (29) (4) .80 (10) (0) .74 .25 (19) (4) R e c a l l .97 (30) .97 (30) (0) The frequency of use of each s t r a t e g y i s presented i n parentheses. 99 low i n accuracy. Both procedures are r e l a t i v e l y u n s o p h i s t i c a t e d and o b v i o u s l y u n r e l i a b l e as f a r as accuracy i s concerned. C o u n t i n g - A l l procedures were not used as a c c u r a t e l y as Counting-On procedures. S u b t r a c t i v e c o u n t i n g procedures were used with a hi g h degree of accuracy, though were used much l e s s f r e q u e n t l y than a d d i t i v e countingprocedures. The v a l u e of developing Counting-On a b i l i t y i s i l l u s t r a t e d by the accuracy r a t e s f o r the two counting procedures. Counting-On was c o r r e c t l y used 78% of the time, and was p a r t i c u l a r l y a c c u r a t e f o r the B a s i c F a c t Domain. T h i s i s compared to.58% o v e r a l l accuracy f o r C o u n t i n g - A l l procedures. T h i s d i f f e r e n c e i n favour of the accuracy of Counting-On procedures i s p a r t l y a r e f l e c t i o n of the g e n e r a l a b i l i t y of the c h i l d r e n who use the more s o p h i s t i c a t e d Counting-On s t r a t e g y , but i t a l s o r e f l e c t s the o p p o r t u n i t y f o r e r r o r i n t h a t w i t h Counting-On o n l y one subset has to be enumerated thus l i m i t i n g the o p p o r t u n i t y t o miscount. A d d i t i v e A s s o c i a t i v e s t r a t e g i e s were used q u i t e a c c u r a t e l y o v e r a l l (76% c o r r e c t ) and had the hi g h e s t r a t e of accuracy f o r t w o - d i g i t items when compared t o other a d d i t i v e s t r a t e g i e s . A d d i t i v e A s s o c i a t i v e s t r a t e g i e s showed the l e a s t drop i n accuracy a c r o s s c o l l a p s e d l e v e l s of d i f f i c u l t y , p o s s i b l y due 100 t o the f a c t t h a t the s t r a t e g y o f t e n u t i l i z e d p l a c e value p r i n c i p l e s t o s i m p l i f y the problem, thus lowering the expected d i f f i c u l t y of the t w o - d i g i t examples. The R e c a l l procedure was a s p e c i a l case, i n t h a t i t d i d not i n v o l v e a reasoning process, and was o n l y a p p l i c a b l e t o the B a s i c F a c t Domain. However, c h i l d r e n who used R e c a l l to complete the m i s s i n g addend examples d i d so w i t h a h i g h degree of accuracy, w i t h 29 c o r r e c t responses f o r the 30 R e c a l l uses. P l a c e h o l d e r p o s i t i o n had no e f f e c t on the accuracy of s t r a t e g i e s L e v e l s of Response During the M i s s i n g Addend T e s t i n t e r v i e w s c h i l d r e n were encouraged t o use any a i d s they wished t o help them s o l v e the items. A wide v a r i e t y of a i d s to s o l u t i o n was observed. T h i s v a r i e t y was c l a s s i f i e d a c c o r d i n g t o the scheme d e s c r i b e d i n Chapter 3 under " L e v e l of Response C l a s s i f i c a t i o n " . Three types of responses were i d e n t i f i e d and l a b e l l e d E x t e r n a l , T r a n s i t i o n a l , and I n t e r n a l . E x t e r n a l l e v e l of response r e f e r r e d t o examples i n v o l v i n g the use of c o n c r e t e a i d s such as b l o c k s and/or s t i c k s . T r a n s i t i o n a l l e v e l of response r e f e r r e d t o the use of body movements, i n p a r t i c u l a r f i n g e r movements as a i d s to s o l u t i o n . I n t e r n a l l e v e l of 101 response referred-: t o examples where no e x t e r n a l involvement was observed by the i n t e r v i e w e r or mentioned.by the c h i l d . Frequency of L e v e l of Response Questions 12, 13 and 14 were concerned w i t h the frequency of use of l e v e l s of response. As a means of answering these q u e s t i o n s , frequency data f o r the use of these l e v e l s of response o v e r a l l and by l e v e l of d i f f i c u l t y are r e p o r t e d i n T a b l e 4.10. P l a c e h o l d e r p o s i t i o n had no e f f e c t on l e v e l of response use. Table 4.10 Frequency of L e v e l of Response O v e r a l l and by L e v e l of D i f f i c u l t y Frequency L e v e l of Response O v e r a l l L e v e l 1 L e v e l 2 L e v e l 3 L e v e l 4 E x t e r n a l 144 20 40 36 48 T r a n s i t i o n a l 53 10 11 18 14 I n t e r n a l 117 50 29 25 13 No Attempt 6 0 0 1 5 T o t a l 320 80 80 80 80 O v e r a l l , the E x t e r n a l l e v e l was used most f r e q u e n t l y , f o l l o w e d by the I n t e r n a l l e v e l . The T r a n s i t i o n a l l e v e l was used the l e a s t which may i n p a r t have been caused by the a v a i l a b i l i t y of c o n c r e t e m a t e r i a l s . L e v e l of response i s not independent of s t r a t e g y c h o i c e , however, and t h i s a l s o i n f l u e n c e d the frequency of use of each l e v e l . The T r a n s i t i o n a l 102 l e v e l was used almost e x c l u s i v e l y w i t h counting s t r a t e g i e s , as w i l l be d e s c r i b e d i n some d e t a i l i n the s e c t i o n " I n t e r a c t i o n of S t r a t e g y Choice and L e v e l of Response". I t was expected t h a t l e v e l of response use would a l s o be i n f l u e n c e d by the l e v e l of d i f f i c u l t y of the item. F i g u r e 4.2 d e p i c t s the i n t e r a c t i o n of the two v a r i a b l e s . Across the f o u r l e v e l s of d i f f i c u l t y , two not unexpected trends appear. In the e a s i e s t l e v e l s most c h i l d r e n were a b l e t o use I n t e r n a l i z e d c o u n t i n g , reasoning, or r e c a l l t o determine responses, but as the number sentences became more d i f f i c u l t , i n t e r n a l methods were abandoned. The o p p o s i t e t r e n d i s t r u e i n the case of the E x t e r n a l l e v e l of response. C h i l d r e n r e l i e d more h e a v i l y on m a t e r i a l s as the d i f f i c u l t y l e v e l i n c r e a s e d , although L e v e l Three d i d not encourage the use of m a t e r i a l s to the extent of L e v e l Two. L e v e l of Response E x t e r n a l T r a n s i t i o n a l . . . I n t e r n a l . LI L2 L3 L4 L e v e l of D i f f i c u l t y F i g u r e 4.2 Frequency of L e v e l of Response by L e v e l of D i f f i c u l t y 103 I n t e r e s t i n g l y , the frequency of use of f i n g e r s , or the T r a n s i t i o n a l l e v e l of response, was f a i r l y c o nstant a c r o s s d i f f i c u l t y l e v e l s , but t h i s was not due to c e r t a i n c h i l d r e n who used f i n g e r s c o n s i s t e n t l y . Nineteen c h i l d r e n used f i n g e r s an average of 2.8 times each and o n l y t h r e e c h i l d r e n used f i n g e r s more than f o u r times. I n d i v i d u a l L e v e l of Response Usage P a t t e r n s Question 13 of t h i s study d e a l t w i t h the e f f e c t of l e v e l of d i f f i c u l t y of items on the l e v e l of response used by c h i l d r e n . O v e r a l l frequency data a t each l e v e l of d i f f i c u l t y f o r each l e v e l of response has been presented. Another a n a l y s i s of t h i s same q u e s t i o n i s p r o v i d e d by data on l e v e l of response p a t t e r n s f o r i n d i v i d u a l c h i l d r e n a c r o s s the f o u r l e v e l s of d i f f i c u l t y . Appendix H p r e s e n t s i n d i v i d u a l l e v e l of response p r o f i l e s o r g a n i z e d by the l e v e l of d i f f i c u l t y of the items. I t was expected t h a t as the d i f f i c u l t y of the items i n c r e a s e d , the response l e v e l s used by c h i l d r e n to support s t r a t e g i e s would change. The expected d i r e c t i o n of t h i s change was towards more c o n c r e t e a i d s . Since s t r a t e g y c h o i c e a f f e c t e d l e v e l of response to some degree, o n l y the p a t t e r n s of the t h i r t e e n c h i l d r e n who used one s t r a t e g y f o r seven or e i g h t items w i l l be examined. 104 Table 4.11 p r e s e n t s the l e v e l of response p r o f i l e s f o r these c h i l d r e n . Table 4.11 P a t t e r n s f o r L e v e l of Response f o r C h i l d r e n Using One S t r a t e g y 7 or 8 Times Item Student 1 2 3 4 5 6 7 8 P a t t e r n 01 1 1 1 1 1 0 1 1 06 3 3 3 3 3 3 3 3 11 3 3 3 3 3 3 3 3 18 3 3 3 3 1 1 1 1 19 1 1 1 1 2 1 1 1 21 3 3 2 2 2 2 2 1 22 2 2 2 3 2 2 3 2 27 1 1 1 1 1 1 1 1 30 3 3 3 3 1 2 3 2 36 3 1 1 1 1 1 1 1 38 1 1 1 1 1 1 1 1 39 1 3 3 1 1 1 1 1 40 3 3. 3 1 1 1 1 1 s L e v e l of Response Key s 3 = I n t e r n a l i z e d s 2 = T r a n s i t i o n a l c 1 = E x t e r n a l i z e d s c 0 = No Attempt n s P a t t e r n of Response Key n s = same l e v e l a c r o s s items s s c = change t o more e x t e r n a l i z e d l e v e l c n = no p a t t e r n c Of the 13 cases where c h i l d r e n used one s t r a t e g y seven or e i g h t times, t h e r e were seven cases (s) of c h i l d r e n r e l y i n g on the same l e v e l of response r e g a r d l e s s of the d i f f i c u l t y of the item. Four c h i l d r e n (c) changed to more co n c r e t e l e v e l s as the items i n c r e a s e d i n d i f f i c u l t y , and two c h i l d r e n (n) d i d not show a p a t t e r n i n l e v e l of response c h o i c e . When p a t t e r n s f o r a l l 4 0 c h i l d r e n were l i s t e d r e g a r d l e s s of s t r a t e g y c h o i c e (see Appendix H), 14 105 c h i l d r e n used one l e v e l of response ( s ) , 17 c h i l d r e n worked from one l e v e l t o a more e x t e r n a l i z e d l e v e l (c) as d i f f i c u l t y i n c r e a s e d , and nine c h i l d r e n ' s c h o i c e s f o l l o w e d no obvious p a t t e r n (n). For a p a t t e r n t o be i d e n t i f i a b l e , o n l y one item was allowed t o d e v i a t e from the tr e n d , f o r i n s t a n c e , the l e v e l of response p a t t e r n of a c h i l d who used b l o c k s f o r a l l items except one, was c o n s i d e r e d t o be c o n s i s t e n t and to r e p r e s e n t an E x t e r n a l l e v e l of response p a t t e r n . A c c u r a c y o f L e v e l s of Response As a means of answering Questions 15, 16 and 17 concerned with the accuracy of l e v e l s of response, Table 4.12 p r e s e n t s accuracy r a t e s f o r each l e v e l of response a t each l e v e l of d i f f i c u l t y and o v e r a l l . P l a c e h o l d e r p o s i t i o n had no e f f e c t on the accuracy of l e v e l s of response. The f r e q u e n c i e s upon which the accuracy r a t e s a re based are g i v e n i n b r a c k e t s . Table 4.12 Accuracy Rates f o r L e v e l s of Response L e v e l of D i f f i c u l t y L e v e l of Response O v e r a l l L e v e l 1 L e v e l 2 L e v e l 3 L e v e l 4 E x t e r n a l .55 (144)* .72. (20) .63 (40) .49 (36) .46 (48) T r a n s i t i o n a l .68 (53) 1.0 (10) .91 (11) .56 (18) .43 (14) I n t e r n a l .71 (117) .90 (50) .66 (29) .64 (25) .23 (13) * The frequency of use of each l e v e l i s g i v e n i n parentheses. 106 As mentioned i n the d i s c u s s i o n of accuracy r a t i n g s f o r s t r a t e g i e s , l e v e l of response i s o n l y as a c c u r a t e as the c h i l d r e n who choose to use i t . However, t o some extent these accuracy r a t e s i l l u s t r a t e the v a l u e of c e r t a i n a i d s f o r t h i s group of c h i l d r e n . O v e r a l l accuracy r a t i n g s suggest t h a t the use of c o n c r e t e m a t e r i a l s i s the l e a s t a c c u r a t e l e v e l of response. However, t h i s low accuracy r a t i n g s may be more a r e f l e c t i o n of the a b i l i t y of the c h i l d r e n r e l y i n g on c o n c r e t e m a t e r i a l s than a r e f l e c t i o n of the v a l u e of c o n c r e t e m a t e r i a l s as a i d s to s o l u t i o n . T r a n s i t i o n a l and I n t e r n a l i z e d l e v e l s of response were almost equal i n accuracy o v e r a l l . I t appeared d u r i n g the i n t e r v i e w s t h a t more i n t e r n a l i z e d p rocesses seemed to make process e r r o r s more s a l i e n t so t h a t d e t e c t i o n and c o r r e c t i o n were p o s s i b l e . For i n s t a n c e , miscounting b l o c k s was a f r e q u e n t occurrence, w h i l e miscounting f i n g e r s r a r e l y went undetected. For L e v e l One items the T r a n s i t i o n a l l e v e l of response was most ac c u r a t e while use of c o n c r e t e a i d s was l e a s t a c c u r a t e . For L e v e l Two items the T r a n s i t i o n a l l e v e l was s t i l l most a c c u r a t e while the E x t e r n a l and I n t e r n a l l e v e l s were comparable i n accuracy. For L e v e l Three items I n t e r n a l i z e d processes were most a c c u r a t e , though a l l three l e v e l s were r e l a t i v e l y s i m i l a r i n accuracy. 107 For L e v e l Four items E x t e r n a l processes were of about the same accuracy as f i n g e r s , w h ile I n t e r n a l i z e d p r ocesses dropped c o n s i d e r a b l y i n accuracy. I t i s i n t e r e s t i n g t o note the s i m i l a r i t y i n accuracy f o r E x t e r n a l and T r a n s i t i o n a l l e v e l s c o n s i d e r i n g the f a c t t h a t a t a l l y of 15 was r e q u i r e d t o s o l v e both items, and t h e r e i s an obvious l i m i t a t i o n to using f i n g e r s f o r such a t a l l y . L e v e l of response c h o i c e and accuracy v a r y as a f u n c t i o n of s t r a t e g y use and the c h i l d ' s f a m i l i a r i t y w i t h a g i v e n s t r a t e g y . T h i s i n t e r a c t i o n i s d i s c u s s e d i n the f o l l o w i n g s e c t i o n . I n t e r a c t i o n of L e v e l of Response  and S t r a t e g y Choice Question 18 of the study addressed'the p o s s i b i l i t y and nature of an i n t e r a c t i o n of s t r a t e g y c h o i c e and l e v e l o f response. As a means of i l l u s t r a t i n g t h i s i n t e r a c t i o n , F i g u r e 4.3 p r e s e n t s percentages of l e v e l of response use f o r each s t r a t e g y . I n c o r r e c t s t r a t e g i e s were not i n c l u d e d ( I n c o r r e c t T r a n s f o r m a t i ons, No Attempts, and Indeterminate p r o c e d u r e s ) . R e c a l l procedures r e l y , by d e f i n i t i o n , on i n t e r n a l i z e d p r o c e s s e s , and f o r t h i s reason were a l s o not i n c l u d e d i n t h i s d i s c u s s i o n . S t r a t e g i e s were ordered from l e f t t o r i g h t a c c o r d i n g to what appeared t o be i n c r e a s i n g l y more s o p h i s t i c a t e d s t r a t e g i e s t o the proposed h i e r a r c h y of 108 s t r a t e g y s o p h i s t i c a t i o n . L e a s t s o p h i s t i c a t e d s t r a t e g i e s are on the: l e f t of the h o r i z o n t a l a x i s . L e v e l of Response E x t e r n a l T r a n s i t i o n a l . . . I n t e r n a l Guess t u t i o n A l l On i a t i v e S t r a t e g y F i g u r e 4.3 I n t e r a c t i o n of L e v e l of Response Usage and S t r a t e g y Choice I t appears t h a t a l l s t r a t e g i e s l i s t e d can be handled a t any of the t h r e e l e v e l s of response, however, some s t r a t e g i e s are used more f r e q u e n t l y w i t h c e r t a i n response l e v e l s than o t h e r s . What have been proposed as u n s o p h i s t i c a t e d or immature S t r a t e g i e s (Semi-Guess, S u b s t i t u t i o n , and C o u n t i n g - A l l ) are used most o f t e n along w i t h c o n c r e t e m a t e r i a l s . A s s o c i a t i v e reasoning procedures, the proposed 109 most s o p h i s t i c a t e d s t r a t e g i e s , a re almost always handled a t an I n t e r n a l i z e d l e v e l of response. Since a l l examples of R e c a l l were handled at an I n t e r n a l i z e d l e v e l , and s i n c e r e c a l l of some b a s i c f a c t combinations was r e q u i r e d t o use as t o o l s i n the reasoning process of the A s s o c i a t i v e s t r a t e g y , i t appears t h a t both A s s o c i a t i v e and R e c a l l s t r a t e g i e s r e p r e s e n t more s o p h i s t i c a t e d procedures than the other i d e n t i f i e d s t r a t e g i e s . The Counting-On s t r a t e g y seems to r e p r e s e n t a t r a n s i t i o n p o i n t between the l e s s and more s o p h i s t i c a t e d s t r a t e g i e s due i n p a r t t o i t s a p p l i c a b i l i t y to any of the i d e n t i f i e d l e v e l s of response. C l a s s I n c l u s i o n T e s t R e s u l t s Each c h i l d i n the sample answered f o u r c l a s s i n c l u s i o n problems. Each problem was scored two, one, or zero r e s u l t i n g i n a maximum p o s s i b l e score of e i g h t . P u p i l scores f o r the C l a s s I n c l u s i o n T e s t are presented i n Appendix F. Subsequent t o the c o l l e c t i o n of data, an independent r a t e r scored 40% of the items, u s i n g t r a n s c r i p t s taken from the tape r e c o r d i n g s of the C l a s s I n c l u s i o n Test a d m i n i s t r a t i o n . These scores were then compared t o the i n t e r v i e w e r ' s s c o r e s , r e s u l t i n g i n 85% agreement. T h i s comparison i s presented i n Appendix J . Table 4.13 prese n t s means f o r each of the f o u r 110 tas k s used as the c l a s s i n c l u s i o n measure. C h i l d r e n were most s u c c e s s f u l on Item One and l e a s t s u c c e s s f u l on Item Four. The o v e r a l l mean was 5,6 out of 8. Table 4.13 C l a s s I n c l u s i o n Task Performance Means Items Range Mean S.D. Item 1 - b o y s / g i r l s , c h i l d r e n 0-2 1.7 .61 Item 2 - cats/dogs, pets 0-2 1.375 .87 Item 3 - red/yellow, b l o c k s 0-2 1.275 .95 Item 4 - c i r c l e s / t r i a n g l e s , shapes 0-2 1.25 .84 T o t a l Score 0-8 5.6 2.71 Two items i n v o l v e d the use of co n c r e t e m a t e r i a l s w i t h the o r a l p r e s e n t a t i o n , and two d i d not. Tabl e 4.14 pre s e n t s mean scores f o r the two types of items. Table 4.14 Concrete Versus O r a l Item Performance Means Items Range Mean S.D. O r a l Items (1 and 2) 0-4 3.075 1.585 Concrete Items (3 and 4) 0-4 2.525 1.269 Performance means f o r the two types of items were s t a t i s t i c a l l y d i f f e r e n t (p<.05) as determined by Student's t T e s t f o r dependent measures. I l l The R e l a t i o n s h i p Between C l a s s I n c l u s i o n .Performance  and M i s s i n g Addend Te s t V a r i a b l e s M i s s i n g Addend T e s t Accuracy The r e l a t i o n s h i p between performance on m i s s i n g addend items and c l a s s i n c l u s i o n items was examined as a means of answering Question 1 9 . M i s s i n g Addend performance was c o n s i d e r e d by c o l l a p s e d l e v e l of d i f f i c u l t y , p l a c e h o l d e r p o s i t i o n , and o v e r a l l . C l a s s I n c l u s i o n performance was c o n s i d e r e d by c o n t e x t (concrete v e r s u s o r a l p r e s e n t a t i o n ) and o v e r a l l . The r e l a t i o n s h i p between M i s s i n g Addend Te s t accuracy v a r i a b l e s and C l a s s I n c l u s i o n performance v a r i a b l e s i s presented i n Table 4 . 1 5 , Table 4 . 1 5 I n t e r c o r r e l a t i o n s Among Major V a r i a b l e s M i s s i n g C l a s s I n c l u s i o n V a r i a b l e s Addend V a r i a b l e s T o t a l Score O r a l Concrete T o t a l Score 0 . 3 9 * * 0 . 3 0 0 . 4 3 ** B a s i c F a c t s 0 . 3 2 * 0 . 2 7 0 . 3 3 * Two D i g i t 0 . 3 8 * 0 . 2 7 0 . 4 2 * * a + • . 0 . 4 5 * * 0 . 3 8 * 0 . 4 6 ** • + a 0 . 2 5 0 . 1 6 0 . 3 0 * p <C . 05 ** p < . 0 1 112 Missing Addend variables include overall accuracy, accuracy on items one t o f o u r from the B a s i c F a c t Domain and on items f i v e t o e i g h t from the Two-Digit Domain, and accuracy on the items f o r each p l a c e h o l d e r p o s i t i o n . G l a s s I n c l u s i o n v a r i a b l e s i n c l u d e the o v e r a l l score on the class, i n c l u s i o n measure, score on the o r a l l y presented items (items 1 and 2) and score on the c o n c r e t e l y presented items (item 3 and 4). T o t a l scores f o r the C l a s s I n c l u s i o n T e s t and the M i s s i n g Addend Test, were p o s i t i v e l y and s i g n i f i c a n t l y c o r r e l a t e d (p<.01). Performance on both c o l l a p s e d l e v e l s of d i f f i c u l t y (basic f a c t s score and t w o - d i g i t score) c o r r e l a t e d s i g n i f i c a n t l y (p< .05) with C l a s s I n c l u s i o n performance. However, when the r e l a t i o n s h i p of performance on items of d i f f e r e n t p l a c e h o l d e r p o s i t i o n s and C l a s s I n c l u s i o n performance was examined, o n l y the performance on a + Q = b p l a c e h o l d e r p o s i t i o n was s i g n i f i c a n t l y c o r r e l a t e d (p<C.01) . C l a s s I n c l u s i o n performance on o r a l l y presented items was r e l a t e d s i g n i f i c a n t l y t o o n l y one aspect of M i s s i n g Addend performance, a + Q = b item performance. However, performance on c o n c r e t e l y presented items was s i g n i f i c a n t l y c o r r e l a t e d w i t h a s p e c t s of m i s s i n g addend performance wi t h the e x c e p t i o n of [ ] + a = b item performance. T h i s suggests t h a t c o n c r e t e l y presented items may be a more s t r i n g e n t and consequently more v a l i d measure of c l a s s i n c l u s i o n a b i l i t y . 113 M i s s i n g Addend S t r a t e g y Choice In response t o Question 20, the r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and frequency of s t r a t e g y c h o i c e was examined. Table 4.16 pres e n t s c o r r e l a t i o n s between the two v a r i a b l e s . Table 4.16 C o r r e l a t i o n s Between C l a s s Inclusion.Performance and Frequency of S t r a t e g y Choice V . • ' C l a s s I n c l u s i o n V a r i a b l e s  Score on Score Score on Concrete S t r a t e g y O v e r a l l O r a l Items Items Semi-Guess A d d i t i v e S u b t r a c t i v e -.22 .07 -;12 .14 -.28 -.00 S u b s t i t u t i o n A d d i t i v e S u b t r a c t i v e -.07 .01 -.13 C o u n t - A l l A d d i t i v e S u b t r a c t i v e -.01 .00 .06 -.06 -.07 . 05 Count-On A d d i t i v e S u b t r a c t i v e . 05 -.05 -.03 .11 A s s o c i a t i v e A d d i t i v e S u b t r a c t i v e .26 -.29 .25 -.34* .24 -.23 R e c a l l .39* .38* .36* I n c o r r e c t T r a n s f o r m a t i o n -.37* -.36* -.34* Indeterminate .15 .15 .13 No Attempt -.03 .02 -.07 * p < . 05 114 The use of three s t r a t e g i e s c o r r e l a t e d s i g n i f i c a n t l y w i t h C l a s s I n c l u s i o n performance. The use of an I n c o r r e c t T r a n s f o r m a t i o n procedure (a c o n c e p t u a l v e r s u s process e r r o r ) was n e g a t i v e l y c o r r e l a t e d (p ^.05) w i t h C l a s s I n c l u s i o n performance. The use of the I n c o r r e c t T r a n s f o r m a t i o n i n d i c a t e s v i r t u a l l y no understanding of what i s r e q u i r e d by the number sentence. I t i n d i c a t e d t h a t the c h i l d sees both the addend and the sum as p a r t s of a whole; t h i s i n t e r p r e t a t i o n i s s i m i l a r t o the i n t e r p r e t a t i o n of the c l a s s i n c l u s i o n problem t h a t compares the two subsets r a t h e r than the subset and the -s u p e r o r d i n a t e s e t . The use of the R e c a l l procedure f o r determining the m i s s i n g addend was p o s i t i v e l y c o r r e l a t e d (p<.05) with C l a s s I n c l u s i o n performance. The use of R e c a l l i n t h i s study c o r r e l a t e d s i g n i f i c a n t l y w i t h both o v e r a l l M i s s i n g Addend Te s t performance and the use of the most s o p h i s t i c a t e d of the i d e n t i f i e d procedures, the A s s o c i a t i v e s t r a t e g i e s . The use of one other s t r a t e g y was s i g n i f i c a n t l y c o r r e l a t e d w i t h C l a s s I n c l u s i o n performance. The use of the s u b t r a c t i v e A s s o c i a t i v e s t r a t e g y c o r r e l a t e d n e g a t i v e l y (p<.05) with performance on the o r a l c l a s s i n c l u s i o n items, however t h i s s t r a t e g y was used o n l y f o u r times, so the c o n f i d e n c e to be p l a c e d i n t h i s r e l a t i o n s h i p i s l i m i t e d . 115 M i s s i n g Addend L e v e l of Response Usage As a means of answering Question 21, the r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and frequency of l e v e l o f response use was a l s o examined. I t was proposed t h a t l e v e l s of response ranged from E x t e r n a l i z e d t o I n t e r n a l i z e d methods. To some extent these l e v e l s were d i c t a t e d by the s t r a t e g y used. For i n s t a n c e , the nature of the C o u n t i n g - A l l p rocess made i t d i f f i c u l t t o use without e x t e r n a l support. For the purposes of the c o r r e l a t i o n m a t r i x , the T r a n s i t i o n a l l e v e l o f response was broken down i n t o two p a r t s , r e s u l t i n g i n f o u r l e v e l s : 1. Concrete l e v e l of response which i n v o l v e d the use of b l o c k s or s t i c k s i n the s o l u t i o n ; 2. E x t e r n a l t r a n s i t i o n a l which i n v o l v e d motor a c t i v i t y t h a t appeared t o be e s s e n t i a l t o the s o l u t i o n p r o c e s s , and i n v o l v e d v i s u a l a t t e n t i o n t o t h i s a c t i v i t y ; 3. I n t e r n a l t r a n s i t i o n a l which i n v o l v e d motor a i d s of some s o r t t h a t appeared t o be secondary to the re a s o n i n g p r o c e s s , and d i d not i n v o l v e the v i s u a l m o d a l i t y ; and 4. I n t e r n a l i z e d l e v e l of response where the c h i l d reasoned without any e x t e r n a l support. Table 4.17 pr e s e n t s c o r r e l a t i o n s between these l e v e l s of response and C l a s s I n c l u s i o n performance. No frequency of l e v e l of response use c o r r e l a t e d s i g n i f i c a n t l y w i t h C l a s s I n c l u s i o n performance. R e s u l t s 116 on l e v e l of response usage i n d i c a t e d .-that c h i l d r e n ' s l e v e l of response c h o i c e was a f f e c t e d by both the d i f f i c u l t y of the item and the s t r a t e g y used, so the l a c k of r e l a t i o n s h i p i n t h i s case i s not s u r p r i s i n g . Table 4.17 R e l a t i o n s h i p Between L e v e l of Response Usage and C l a s s I n c l u s i o n Performance C l a s s I n c l u s i o n Performance L e v e l of Response O v e r a l l O r a l Concrete Concrete -.03 -.01 -.04 E x t e r n a l T r a n s i t i o n a l -.01 -.01 -.00 I n t e r n a l T r a n s i t i o n a l -.08 -.13 -.03 I n t e r n a l i z e d r . 07 -.03 .09 Chapter 5 SUMMARY, CONCLUSIONS, AND IMPLICATIONS T h i s study d e a l t w i t h q u e s t i o n s which addressed f i v e g e n e r a l areas. Questions 1, 2, and 3 d e a l t w i t h performance on m i s s i n g addend sentences. Questions 4 t o 10 d e a l t w i t h the frequency and accuracy of s o l u t i o n s t r a t e g y use. Questions 11 to 17 d e a l t w i t h the frequency and accuracy of l e v e l of response use. Question 18 concerned the r e l a t i o n s h i p between s t r a t e g y use and l e v e l of response. Questions 19 to 21 were concerned w i t h the r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and M i s s i n g Addend T e s t performance v a r i a b l e s . The f o l l o w i n g s e c t i o n d e a l s w i t h these areas i n the order presented above. Secondly, i m p l i c a t i o n s f o r classrooms are d i s c u s s e d , and f i n a l l y suggestions f o r f u r t h e r r e s e a r c h are presented. Summary and C o n c l u s i o n s Accuracy on M i s s i n g Addend Sentences Performance r e s u l t s i n d i c a t e d t h a t most of these year two students understood the m i s s i n g addend problem and had v a r y i n g degrees of success w i t h the items depending on the magnitude of the constants i n v o l v e d . T h i s l e v e l of d i f f i c u l t y d i d not serve t o obscure the meaning of the number sentence. What were i d e n t i f i e d as c o n c e p t u a l l y 118 c o r r e c t s t r a t e g i e s were used f o r 91% of the b a s i c f a c t items as compared to 82% of the two d i g i t items. However, the l e v e l of d i f f i c u l t y of the item more d r a s t i c a l l y a f f e c t e d the number of process e r r o r s . Of the 146 c o n c e p t u a l l y c o r r e c t s t r a t e g y uses i n the B a s i c F a c t Domain, 123 or 84% were c o r r e c t . On the o t h e r hand, o n l y 73 or 45% of the 132 c o n c e p t u a l l y c o r r e c t s t r a t e g y uses on t w o - d i g i t items were c o r r e c t . T h i s i l l u s t r a t e s t h a t f o r the l e s s f a m i l i a r t w o - d i g i t items, 18% of the e r r o r s r e s u l t e d from a l a c k of understanding of the problem, while 45% r e s u l t e d from an i n a b i l i t y t o c o r r e c t l y process the problem. The f a c t t h a t the e r r o r s c h i l d r e n made were r a r e l y c o n c e p t u a l i n nature, but i n s t e a d were process e r r o r s , suggests t h a t i f the c h i l d r e n have a r e l i a b l e s o l u t i o n s t r a t e g y , m i s s i n g addend sentences are not i n a p p r o p r i a t e f o r year two c h i l d r e n . M i s s i n g Addend S o l u t i o n S t r a t e g i e s The s o l u t i o n s t r a t e g i e s used by year two c h i l d r e n i n t h i s study to s o l v e m i s s i n g addend items, are summarized i n two ways. F i r s t l y , an a n a l y s i s of the use of a d d i t i v e as opposed t o s u b t r a c t i v e i n t e r p r e t a t i o n s of m i s s i n g addend sentences i s d i s c u s s e d . Secondly, a summary i s g i v e n of each s t r a t e g y used to determine the m i s s i n g addend. A d d i t i v e and s u b t r a c t i v e approaches. M i s s i n g addend sentences were i n i t i a l l y i n t r o d u c e d i n primary t e x t s as a means of r e l a t i n g the o p e r a t i o n s of a d d i t i o n 119 and s u b t r a c t i o n . Whether i n f a c t c h i l d r e n r e c o g n i z e and u t i l i z e the i n v e r s e r e l a t i o n s h i p i n t h e i r s o l u t i o n methods has not been e s t a b l i s h e d . Whether they are developmentally capable of seeing t h i s r e l a t i o n s h i p has a l s o not been e s t a b l i s h e d . T h i s study attempted to determine how c h i l d r e n s o l v e d m i s s i n g addend items. No attempt was made to determine why these s t r a t e g i e s were used. I t was assumed t h a t the methods c h i l d r e n chose to use were ones t h a t were meaningful t o them. The f a c t t h a t c h i l d r e n r a r e l y used a s u b t r a c t i v e approach was i n t e r p r e t e d as an i n d i c a t i o n of a grasp of and a p r e f e r e n c e f o r a d d i t i v e i n t e r p r e t a t i o n s . E i g h t y - e i g h t percent of the examples t h a t c o u l d have been i n t e r p r e t e d e i t h e r as a d d i t i v e or s u b t r a c t i v e s i t u a t i o n s were i n t e r p r e t e d a d d i t i v e l y . Thirty^-nine of the f o r t y c h i l d r e n used a d d i t i v e approaches at l e a s t once, w h i l e nine c h i l d r e n used s u b t r a c t i v e s t r a t e g i e s a t l e a s t once. Two f i n d i n g s suggest the i n a p p r o p r i a t e n e s s of expecting year two c h i l d r e n t o - i n t e r p r e t m i s s i n g addend sentences i n terms of t h e i r s u b t r a c t i o n sentence e q u i v a l e n t s . One i s t h a t these c h i l d r e n , d e s p i t e classroom p r a c t i c e s , p r e f e r t o use a d d i t i v e approaches. They seem to have access t o s e v e r a l r e l i a b l e a d d i t i v e procedures t h a t they m e a n i n g f u l l y apply i n s i t u a t i o n s i n v o l v i n g a v a r i e t y of number s i z e s . 120 The second f i n d i n g i s t h a t r e s u l t s f o r t h i s group o f year two c h i l d r e n i n d i c a t e d a wide range of a b i l i t y t o r e c o g n i z e part-whole r e l a t i o n s h i p s as measured by c l a s s i n c l u s i o n t a s k s . A s i g n i f i c a n t r e l a t i o n s h i p between C l a s s ' I n c l u s i o n performance and M i s s i n g Addend performance was found to e x i s t . L o g i c a l l y i t appears t h a t t h i s r e c o g n i t i o n of part-whole r e l a t i o n s h i p s i s r e q u i r e d f o r an understanding of the m i s s i n g addend problem, and the r e v e r s i b i l i t y of t h i s r e l a t i o n s h i p i s r e q u i r e d f o r c h i l d r e n to m e a n i n g f u l l y use s u b t r a c t i v e approaches. When and how the p r e f e r r e d a d d i t i v e approach i s r e c o g n i z e d as the e q u i v a l e n t of s u b t r a c t i n g the g i v e n addend from the sum has not been e s t a b l i s h e d . S i m i l a r l y , the c h i l d r e n who i n t e r p r e t e d the sentence as a s u b t r a c t i o n sentence may a l s o have l e a r n e d t h a t procedure f o r s o l u t i o n without n e c e s s a r i l y r e c o g n i z i n g the a d d i t i v e a l t e r n a t i v e s and the i n v e r s e r e l a t i o n s h i p . S t e f f e , Spikes, and H i r s t e i n (1976) r e c o g n i z e d t h a t the s u b t r a c t i v e approach (or what they c a l l e d the C o u n t i n g - A l l approach) seemed to be beyond the c o n c e p t u a l a b i l i t i e s of year one c h i l d r e n . S t e f f e e t a l , found t h a t year one c h i l d r e n were ab l e t o l e a r n t o s u b t r a c t but d i d so a l g o r i t h m i c a l l y and d i d not seem to c o n c e p t u a l i z e the problem. Grouws (1971) found t h a t year three c h i l d r e n p r e f e r r e d formal a l g o r i t h m s t h a t i n v o l v e d sentence t r a n s f o r m a t i o n s , but whether or not c h i l d r e n m e a n i n g f u l l y a p p l i e d the a l g o r i t h m s i s not known. Weaver (1976) suggested the i n a p p r o p r i a t e n e s s of emphasizing sentence t r a n s f o r m a t i o n s with year t h r e e c h i l d r e n and suggested they l a c k e d "the mathematical m a t u r i t y t o p r o f i t from an approach to t r a n s f o r m a t i o n s t h a t emphasized c e r t a i n p r o p e r t i e s and combinations t h e r e o f " , (p. 21) The s t u d i e s by S t e f f e e t a l . (1976) and Weaver (.1976) i n d i c a t e d t h a t young c h i l d r e n p r e f e r a d d i t i v e i n t e r p r e t a t i o n s . R e s u l t s from t h i s study i n d i c a t e a s i m i l a r p r e f e r e n c e . One p o s s i b i l i t y i s t h a t t h i s p r e f e r e n c e f o r a d d i t i v e i n t e r p r e t a t i o n s i n d i c a t e s an i n a b i l i t y f o r primary age c h i l d r e n t o l o g i c a l l y grasp sentence t r a n s f o r m a t i o n approaches t o m i s s i n g addend sentences. T h i s suggests t h a t developing the a b i l i t y of year two c h i l d r e n to transform m i s s i n g addend sentences to t h e i r s u b t r a c t i o n sentence e q u i v a l e n t s may be i n a p p r o p r i a t e , t h a t i s , beyond the co n c e p t u a l competence of t h i s age group. Summary of r e s u l t s f o r each s t r a t e g y . For each s t r a t e g y the f o l l o w i n g aspects w i l l be d i s c u s s e d : frequency and accuracy of use; l e v e l s o f response used; d e s c r i p t i o n s of the response p a t t e r n s of the c h i l d r e n who used the s t r a t e g y ; and c o r r e l a t i o n s w i t h v a r i a b l e s of i n t e r e s t . 122 Semi-Guess strategies were used 13 times, always i n the Two-Digit Domain, and 11 times i n Level 4 of that domain. Nine of the 13 examples were additive Semi-Guesses. Of the eight children who used t h i s category, seven had used a counting strategy successfully i n one of the f i r s t three levels of d i f f i c u l t y . This suggests that these children had access to a r e l i a b l e method but were unable to cope with the size of the numbers involved, or lacked the patience to pursue the r e l i a b l e method and instead guessed. The Semi-Guess responses were a l l incorrect. Nine of the 13 examples were handled at an External l e v e l of response. Use of t h i s category did not correlate s i g n i f i c a n t l y with any a t t r i b u t e or achievement v a r i a b l e . The use of t h i s strategy may r e f l e c t p u p i l confidence and attitude more than actual competence. The Substitution strategy was used seven times by three d i f f e r e n t children. Usually they b u i l t the f i r s t addend, added some blocks, counted a l l the blocks, then altered the added group accordingly and recounted. They altered the added group an average of three times for each example, but t h i s procedure resulted i n only one correct example. For a l l seven examples childre n interpreted the missing addend sentence i n an additive context. 123 A d d i t i v e C o u n t i n g - A l l was the second most f r e q u e n t l y used s t r a t e g y ( a f t e r a d d i t i v e Counting-On) and was used f o r 76 of the 320 examples. I t was used most o f t e n f o r L e v e l Two items. I t was used 42 times f o r a = b p l a c e h o l d e r p o s i t i o n as compared t o 34 times f o r Q + a = b. A d d i t i v e C o u n t i n g - A l l was a c c u r a t e 58% of the time, and was s l i g h t l y more a c c u r a t e i n the B a s i c F a c t Domain than i n the Two-Digit Domain. I t was used i n c o n j u n c t i o n w i t h b l o c k s or s t i c k s 8 3% of the time. F i n g e r s were used 14% of the time, and I n t e r n a l i z e d methods 3% of the time. Des p i t e the f a c t t h a t as the l e v e l of d i f f i c u l t y i n c r e a s e d and the g i v e n addend took longer t o b u i l d or r e p r e s e n t , o n l y one c h i l d changed from the time-consuming C o u n t i n g - A l l to the more e f f i c i e n t Counting-On. T h i s may i n d i c a t e t h a t the c o g n i t i v e c a p a c i t y t o m e n t a l l y r e p r e s e n t q u a n t i t y was not a v a i l a b l e to the o t h e r s and they remained c o n f i n e d to the enumeration of both s e t s . There were 20 examples of s u b t r a c t i v e C o u n t i n g - A l l , 15 of which i n v o l v e d the use of b l o c k s or s t i c k s . C h i l d r e n using t h i s s t r a t e g y were h i g h l y a c c u r a t e ; 19 of the 20 examples were c o r r e c t . Three c h i l d r e n used t h i s s t r a t e g y more than once; these t h r e e accounted f o r 16 of the 2 0 examples. A d d i t i v e Counting-On was the most f r e q u e n t l y used s t r a t e g y f o r s o l v i n g the m i s s i n g addend examples. 124 T h i r t y percent of the items i n v o l v e d Counting-On. L e v e l of d i f f i c u l t y d i d not a f f e c t frequency of use; c h i l d r e n a p p l i e d the s t r a t e g y a c r o s s l e v e l s i n d i c a t i n g the s u i t a b i l i t y and a p p l i c a b i l i t y of t h i s s t r a t e g y f o r year two c h i l d r e n . A d d i t i v e Counting-On was a ccurate 78% of the time, 93% of the time f o r B a s i c F a c t Domain items and 65% of the time f o r Two-Digit Domain. A l l t hree l e v e l s of response were used w i t h Counting-On: 27% E x t e r n a l ; 40% T r a n s i t i o n a l ; and 33% I n t e r n a l . I t was the o n l y s t r a t e g y t h a t had a f a i r l y even d i s t r i b u t i o n of the t h r e e l e v e l s of response and had the g r e a t e s t use of the T r a n s i t i o n a l l e v e l . Counting-On i s proposed to be a key a b i l i t y i n the i n t e r n a l i z a t i o n of l e v e l of response and the development of more s o p h i s t i c a t e d s t r a t e g i e s , as i s f u r t h e r e x p l a i n e d i n the s e c t i o n " I n t e r a c t i o n of L e v e l of Response and S t r a t e g y Use". S u b t r a c t i v e Counting-On, or counting-back was used on o n l y two of the 320 items which suggests t h a t p o s s i b l y these c h i l d r e n c o u l d not count backwards w e l l enough t o use the p a t t e r n i n a problem s o l v i n g s i t u a t i o n . A s s o c i a t i v e s t r a t e g i e s were i d e n t i f i e d 33 times, 29 as a d d i t i v e and f o u r as s u b t r a c t i v e . The A s s o c i a t i v e s t r a t e g y when compared to a l l o t h e r s t r a t e g i e s was l e a s t a c c u r a t e f o r the B a s i c F a c t Domain but most 125 a c c u r a t e f o r the Two-Digit Domain. I t was handled a t an I n t e r n a l i z e d l e v e l f o r 3 0 of the 33 examples. Of the 11 c h i l d r e n who used the A s s o c i a t i v e s t r a t e g y , 9 of them used R e c a l l t o determine the m i s s i n g addend on a t l e a s t one item i n the Ba s i c F a c t Domain. Since r e c a l l of some b a s i c f a c t s i s p r e r e q u i s i t e to reasoning with the b a s i c f a c t s , i t i s not s u r p r i s i n g t h a t L e v e l One items would be s o l v e d u s i n g r e c a l l . In L e v e l Two t h i s s t r a t e g y was used t en times, and these uses were c h a r a c t e r i z e d by r e c a l l of a b a s i c f a c t combination with the same sum or a sum c l o s e t o the g i v e n sum. The combination was then a l t e r e d t o make the addend s u i t the g i v e n addend. For example, g i v e n 5 + • = 13, the c h i l d thought 5 + 5 = 10, 10 + 3 = 13, 5 + 3 = 8, so 5 + 8 = 13 or ( 5 + 5 ) + 3 = 5 + (5 + 3) = 13. In L e v e l Three the A s s o c i a t i v e s t r a t e g y was used 17 times, and i n L e v e l Four, 6 times. Use of the A s s o c i a t i v e s t r a t e g y was p o s i t i v e l y c o r r e l a t e d (p<..01) with the use of the R e c a l l s t r a t e g y and wit h the use of I n t e r n a l i z e d processes (p<:.01). R e c a l l was used 3 0 times, and o n l y i n the Bas i c F a c t Domain. Of the 30 examples, 26 o c c u r r e d w i t h combinations l e s s than t e n . C h i l d r e n who r e l i e d on R e c a l l t o determine the m i s s i n g addend were h i g h l y a c c u r a t e . Only one example out of 3 0 examples was i n c o r r e c t . 126 Seventeen c h i l d r e n used R e c a l l a t l e a s t once. The frequency of use of R e c a l l c o r r e l a t e d s i g n i f i c a n t l y w i t h o v e r a l l performance on the M i s s i n g Addend T e s t (p<.01). Since R e c a l l by d e f i n i t i o n r e q u i r e d automatic response and no process, a l l examples were handled without e x t e r n a l involvement. Use of R e c a l l was c o r r e l a t e d (p<.01) w i t h use of the A s s o c i a t i v e s t r a t e g i e s , l e n d i n g support t o the h ypothesis t h a t mastery of some b a s i c f a c t s i s p r e r e q u i s i t e to reasoning i n v o l v i n g the a s s o c i a t i v e p r i n c i p l e . R e c a l l and A s s o c i a t i v e s t r a t e g i e s were proposed to be the most s o p h i s t i c a t e d procedures. R e c a l l a p p l i e d o n l y to the B a s i c F a c t Domain, but was an e s s e n t i a l element i n the use of A s s o c i a t i v e procedures. I t i s suggested t h a t the use of R e c a l l and A s s o c i a t i v e s t r a t e g i e s may i n d i c a t e r e a d i n e s s t o d e a l w i t h number combinations of any magnitude. For these combinations, a more e f f i c i e n t method of s o l u t i o n , such as the a p p l i c a t i o n of a formal s u b t r a c t i o n a l g o r i t h m , would be warranted. R e c a l l was not i d e n t i f i e d as a s t r a t e g y i n the sense of the other s t r a t e g i e s but was i n s t e a d c o n s i d e r e d to be a memory scan technique of some s o r t . The c h i l d r e n who used R e c a l l i n t h i s study a p p a r e n t l y d i d so w i t h meaning, however, i t would be p o s s i b l e to determine the m i s s i n g addend u s i n g v i s u a l memory a b i l i t y and have no idea of the meaning of the sentence. 127 The f a c t t h a t R e c a l l was s i g n i f i c a n t l y c o r r e l a t e d w i t h u s e o f A s s o c i a t i v e s t r a t e g i e s , may s u g g e s t t h a t i t r e p r e s e n t s t h e n e x t most s o p h i s t i c a t e d s o l u t i o n p r o c e d u r e a s B r o w n e l l (.1928) and I l g & Ames (1951) d e s c r i b e d f o r a + b =Q f o r m s . The u s e o f R e c a l l was a l s o p o s i t i v e l y c o r r e l a t e d w i t h C l a s s I n c l u s i o n p e r f o r m a n c e s u g g e s t i n g i t s u s e i s r e l a t e d t o h i g h e r l e v e l s o f c o g n i t i v e a b i l i t i e However, a s a s o l u t i o n p r o c e d u r e f o r number c o n s t a n t s o f g r e a t e r m a g n i t u d e , R e c a l l was o b v i o u s l i m i t a t i o n s . I n c o r r e c t T r a n s f o r m a t i o n s o c c u r r e d 24 t i m e s , and t h e s e r e s p o n s e s were o f c o u r s e i n c o r r e c t . Two c h i l d r e n T r a n s f o r m e d s e v e n o r e i g h t i t e m s , and two c h i l d r e n T r a n s f o r m e d f o u r o f t h e e i g h t i t e m s . Of t h e 24 examples o f I n c o r r e c t T r a n s f o r m a t i o n s 19 were h a n d l e d i n t e r n a l l y . Use o f t h i s s t r a t e g y c o r r e l a t e d n e g a t i v e l y and s i g n i f i c a n t l y w i t h C l a s s I n c l u s i o n p e r f o r m a n c e ( p - ^ . 0 5 ) . No .attempt was made t o c o m p l e t e t h e i t e m i n s i x c a s e s . T h e s e a l l o c c u r r e d w i t h t h e t w o - d i g i t i t e m s and were c o u n t e d a s i n c o r r e c t r e s p o n s e s , and were o m i t t e d f r o m t h e a n a l y s i s ' o f l e v e l o f r e s p o n s e . The I n d e t e r m i n a t e c a t e g o r y was u s e d i n 12 c a s e s and i n o n l y one o f t h e s e c a s e s was t h e c h i l d c o r r e c t . The 12 c a s e s o c c u r r e d where t h e c h i l d w orked a t an I n t e r n a l i z e d l e v e l and was u n a b l e t o v e r b a l i z e t h e p r o c e s s , o r where t h e c h i l d ' s e x p l a n a t i o n was i n c o h e r e n t o r c o n t r a d i c t o r y . 128 The presence of concrete materials and the encouragement to work out loud with any method, served to f a c i l i t a t e the i d e n t i f i c a t i o n of strategies, i n that the children's actions provided the basis for s p e c i f i c questioning on strategies employed. Level of Response Use Three levels of response were i d e n t i f i e d i n the range of responses observed. The le v e l s represented degrees of sensory involvement from less to more int e r n a l i z e d procedures. These three l e v e l s w i l l be discussed separately. Externalized l e v e l of response. This l e v e l was used most frequently, possibly due to the u n f a m i l i a r i t y of the testing s i t u a t i o n and the fac t that blocks and sticks were r e a d i l y available for use. Forty-five percent of the items were solved with the help of concrete aids, 38% of the basic fact items, and 50% of the two d i g i t items. Across the four le v e l s of d i f f i c u l t y the reliance on aids increased, with the exception of Level Three. Twelve children switched to the use of blocks or s t i c k s as the d i f f i c u l t y increased. Externalized responses were least accurate o v e r a l l and for d i f f i c u l t y Levels One, Two and Three. However, the use of blocks and s t i c k s was the most accurate l e v e l of response for the most d i f f i c u l t items, Level Four. 129 Use of the E x t e r n a l i z e d l e v e l of response was p o s i t i v e l y r e l a t e d (p<.01) to use of the G o u n t i n g - A l l s t r a t e g y and n e g a t i v e l y r e l a t e d (p<;01) to use of A s s o c i a t i v e s t r a t e g i e s , which supports the proposed h i e r a r c h y of s t r a t e g y s o p h i s t i c a t i o n . T r a n s i t i o n a l l e v e l of response. T h i s l e v e l was used i n o n l y 53 of the 320 examples, p o s s i b l y due to the a v a i l a b i l i t y of concr e t e m a t e r i a l s . I t was used f a i r l y c o n s i s t e n t l y a c r o s s l e v e l s of d i f f i c u l t y . The T r a n s i t i o n a l l e v e l of response was the most a c c u r a t e l e v e l f o r items from the B a s i c F a c t Domain, and second i n accuracy f o r the Two D i g i t Domain. The T r a n s i t i o n a l l e v e l of response was r a r e l y used w i t h any s t r a t e g y o t h e r than the Counting-On s t r a t e g y , where i t was used more f r e q u e n t l y than e i t h e r of the other l e v e l s . T h i s suggests t h a t the T r a n s i t i o n a l l e v e l i s an important stage i n the i n t e r n a l i z a t i o n process of the Counting-On procedure as d i s c u s s e d i n the s e c t i o n " I n t e r a c t i o n of L e v e l of Response and St r a t e g y Use". I n t e r n a l i z e d l e v e l of response. T h i s l e v e l was used i n 37% of the examples, 49% of the b a s i c f a c t items and 24% of the two d i g i t items. Across the f o u r l e v e l s the use of i n t e r n a l i z e d procedures decreased. I n t e r n a l i z e d procedures were most ac c u r a t e o v e r a l l and l e a s t a c c u r a t e f o r L e v e l Four items. Use 130 of t h i s l e v e l was p o s i t i v e l y r e l a t e d (p<.01) to use of R e c a l l and A s s o c i a t i v e s t r a t e g i e s , and ^negatively r e l a t e d (p-<.01) to use of C o u n t i n g - A l l . I n t e r a c t i o n of L e v e l of Response  and S t r a t e g y Use What were proposed as the l e a s t s o p h i s t i c a t e d s t r a t e g i e s (Semi-Guess, S u b s t i t u t i o n , and C o u n t i n g - A l l ) , were c l o s e l y a s s o c i a t e d w i t h use of concre t e a i d s f o r s o l u t i o n , w h ile what were proposed as the most s o p h i s t i c a t e d s t r a t e g i e s were c l o s e l y a s s o c i a t e d with i n t e r n a l i z e d r e a s o n i n g processes. Research g e n e r a l l y supports the c o n t e n t i o n t h a t r e l i a n c e on concr e t e m a t e r i a l s d i m i n i s h e s as understanding and competence develop. R e p r e s e n t a t i o n a l thought r e p l a c e s a c t u a l experience, and t h i s m a n i p u l a t i o n of images all o w s more s o p h i s t i c a t e d t h i n k i n g processes to develop. An i n t e r e s t i n g t r a n s i t i o n p o i n t o c c u r r e d with the Counting-On procedure. T h i s procedure seems to be key to the i n t e r n a l i z a t i o n process, i n t h a t i t seems to be e q u a l l y s u i t e d t o a l l l e v e l s of response. The nature of counting-on i n d i c a t e s an a c q u i r e d a b i l i t y of a con c e p t u a l nature, t h a t of m e n t a l l y r e p r e s e n t i n g a number and adding on t o or s u b t r a c t i n g from t h i s mental r e p r e s e n t a t i o n . The adding or s u b t r a c t i n g process can be done wi t h the help of 131 c o n c r e t e e x t e r n a l support, f i n g e r s or other semi-i n t e r n a l i z e d a i d s , or i n t e r n a l l y . I t i s proposed t h a t the a c q u i s i t i o n of counting-on a b i l i t i e s p r o v i d e s c h i l d r e n w i t h the o p p o r t u n i t y to move from r e l i a n c e on c o n c r e t e m a t e r i a l s to i n t e r n a l i z e d c ounting methods, and f o r t h i s reason i s an e s s e n t i a l p r e r e q u i s i t e to developing i n t e r n a l i z e d reasoning procedures and meaningful r e c a l l . The r o l e of f i n g e r s i n t h i s t r a n s i t i o n p rocess may be important, as i s suggested by the frequent usage of f i n g e r s f o r the Counting-On pr o c e s s . F i n g e r s were r a r e l y u t i l i z e d except f o r t h i s p a r t i c u l a r s t r a t e g y and t h i s suggests they may c o n t r i b u t e e s s e n t i a l sensory input t o the i n t e r n a l i z a t i o n p r o c e s s . T h i s concurs w i t h S t e f f e , Spikes, and H i r s t e i n (.1976) who proposed t h a t the use of f i n g e r s i s c r i t i c a l i n the f o r m a t i o n of mental o p e r a t i o n s a s s o c i a t e d with c o u n t i n g , f i n d i n g sums, and f i n d i n g d i f f e r e n c e s . C l a s s I n c l u s i o n A b i l i t y and  M i s s i n g Addend Performance C l a s s I n c l u s i o n performance was found to be p o s i t i v e l y r e l a t e d (p.^.01) to M i s s i n g Addend performance. Performance on the two c l a s s i n c l u s i o n t a s k s i n v o l v i n g c oncrete m a t e r i a l s was more c l o s e l y r e l a t e d t o M i s s i n g Addend performance than performance on the o r a l l y presented t a s k s , s u g g e s t i n g t h a t contexts u t i l i z i n g c o n c r e t e 132 m a t e r i a l s p r o v i d e a more r e l i a b l e , t e s t . o f C l a s s I n c l u s i o n a b i l i t y . The observed r e l a t i o n s h i p between C l a s s I n c l u s i o n and M i s s i n g Addend performance concurs w i t h the f i n d i n g s of Howlett (.1973) but i s i n c o n s i s t e n t with f i n d i n g s by S t e f f e , Spikes, and H i r s t e i n (197 6). O v e r a l l C l a s s I n c l u s i o n performance was found to c o r r e l a t e s i g n i f i c a n t l y with two s o l u t i o n s t r a t e g i e s . The use of R e c a l l was p o s i t i v e l y c o r r e l a t e d (p<£.05) and the use of I n c o r r e c t T r a nsformations was n e g a t i v e l y c o r r e l a t e d ( p < . 05) with C l a s s I n c l u s i o n performance. These f i n d i n g s support the proposed r e l a t i o n s h i p of C l a s s I n c l u s i o n a b i l i t y t o M i s s i n g Addend performance and the proposed h i e r a r c h y of solution'.-strategies s o p h i s t i c a t i o n . No l e v e l of response use was s i g n i f i c a n t l y r e l a t e d t o C l a s s I n c l u s i o n performance, i n d i c a t i n g t h a t l e v e l of response was more c l o s e l y r e l a t e d t o other v a r i a b l e s , i n p a r t i c u l a r s t r a t e g y c h o i c e and item d i f f i c u l t y . I m p l i c a t i o n s f o r Classrooms The f i n d i n g s of t h i s study have i m p l i c a t i o n s f o r t e a c h i n g i n i t i a l m i s s i n g addend s o l u t i o n s t r a t e g i e s , f o r d e v eloping a sequence of p r o g r e s s i v e l y more s o p h i s t i c a t e d s t r a t e g i e s , f o r encouraging p a r t i c u l a r l e v e l s of response a t c e r t a i n stages i n t h i s sequence, and f o r i d e n t i f y i n g l e v e l s of r e a d i n e s s f o r the i n t r o d u c t i o n of the m i s s i n g 133 addend problem and p a r t i c u l a r s o l u t i o n s t r a t e g i e s . The i n a b i l i t y t o s o l v e the c l a s s i n c l u s i o n problem may i n d i c a t e an i n a b i l i t y to r e c o g n i z e the part-whole r e l a t i o n s h i p expressed by the number sentence. I n c o r r e c t T ransformations (.5 + 13"-= 8) may i n d i c a t e t h i s i n a b i l i t y . These c h i l d r e n would appear to l a c k the a b i l i t y t o handle the problem and should not be expected to do so. Instead, a + b = Q sentence modeling and counting a c t i v i t i e s would be recommended u n t i l an understanding of the problem i s demonstrated i n work wit h c o n c r e t e m a t e r i a l s . Counting s t r a t e g i e s appear to be the f i r s t r e l i a b l e s o l u t i o n s t r a t e g i e s f o r s o l v i n g m i s s i n g addend sentences. They were the p r e f e r r e d processes f o r t h i s age group and appeared to be a necessary process i n the development of more e f f i c i e n t p r o c e s s e s . As an i n i t i a l t e a c h i n g s t r a t e g y , C o u n t i n g - A l l p r o v i d e s the c h i l d w i t h a r e p r e s e n t a t i o n of the p a r t s and the whole and the r e l a t i o n s h i p between them. I t i s a p p l i c a b l e to both an a d d i t i v e or s u b t r a c t i v e i n t e r p r e t a t i o n and may h elp t o r e i n f o r c e the i n v e r s e r e l a t i o n s h i p . T h i s s t r a t e g y i s recommended f o r i n i t i a l t e a c h i n g of s o l u t i o n s t r a t e g i e s f o r m i s s i n g addends because of i t s c l a r i t y as an i l l u s t r a t i o n of the number sentence and of the part-whole r e l a t i o n s h i p expressed by the sentence. Furthermore, i t appears t h a t C o u n t i n g - A l l processes are p r e r e q u i s i t e 134 to the more e f f i c i e n t Counting-On. I t i s suggested t h a t c h i l d r e n ' t a u g h t C o u n t i n g - A l l would make the t r a n s i t i o n to Counting-On when they were c o n c e p t u a l l y a b l e to do so. Counting-On seems to be of p a r t i c u l a r importance f o r s e v e r a l reasons: 1. i t i s more a c c u r a t e and p r o v i d e s l e s s o p p o r t u n i t y f o r e r r o r ; 2. i t i s more e f f i c i e n t i n t h a t i t e f f e c t i v e l y makes use of a l l the i n f o r m a t i o n ; and 3. i t can be used at a v a r i e t y of l e v e l s of response. Counting-On a b i l i t y seems to i n d i c a t e a c h i l d ' s r e a d i n e s s to begin i n t e r n a l i z i n g s o l u t i o n p r o c e s s e s . I t i s suggested t h a t t e a c h e r s ensure t h a t c h i l d r e n have co n c r e t e m a t e r i a l s to work w i t h u n t i l Counting-On i s a c q u i r e d . However, a t t h a t p o i n t , c h i l d r e n can be encouraged t o i n t e r n a l i z e counting processes f i r s t u s i n g m u l t i s e n s o r y a i d s such as f i n g e r counts and body ta p s , and then g r a d u a l l y m i n i m i z i n g t h i s m u l t i - s e n s o r y i n p u t . The use of f i n g e r s , f i r s t l o o k i n g at f i n g e r s , and then without l o o k i n g , i s s t r o n g l y recommended as a means of i n t e r n a l i z i n g counting processes i n r e a d i n e s s f o r more s o p h i s t i c a t e d procedures. R e s u l t s i n d i c a t e d t h a t v e r y few year two c h i l d r e n t h i n k of a m i s s i n g addend sentence i n terms of i t s s u b t r a c t i o n sentence e q u i v a l e n t . Encouraging c h i l d r e n t o transform the m i s s i n g addend sentence may not be an a p p r o p r i a t e procedure f o r t h i s age group s i n c e they seem to p r e f e r a d d i t i v e methods of s o l u t i o n . I f the s u b t r a c t i o n i n t e r p r e t a t i o n i s used, i t i s suggested t h a t i t be taught a f t e r the development of the a d d i t i v e C o u n t i n g - A l l s t r a t e g y and with the a i d of c o n c r e t e m a t e r i a l s t h a t i l l u s t r a t e the part-whole r e l a t i o n s h i p . Experience"in.backwards counting p a t t e r n s i s a l s o r e q u i r e d f o r c h i l d r e n to proceed to counting-back procedures t h a t a l l o w c h i l d r e n to work without the a i d of c o n c r e t e m a t e r i a l s . R e s u l t s suggest t h a t t e a c h i n g c h i l d r e n to use a R e c a l l procedure t o determine the m i s s i n g addend i s of l i m i t e d v a l u e i n d e veloping u s e f u l s o l u t i o n s t r a t e g i e s f o r m i s s i n g addend sentences. The R e c a l l s t r a t e g y a p p l i e s t o o n l y the B a s i c F a c t Domain and can be used o n l y by c h i l d r e n who have mastered the b a s i c f a c t combinations. I t appears t h a t the more competent c h i l d r e n choose, i n t h e i r own time, to r e l y on R e c a l l r e g a r d l e s s of the s t r a t e g y emphasized i n c l a s s . Sequences of s o p h i s t i c a t i o n of s o l u t i o n processes (Brownell, 19 28; I l g & Ames, 19 51; O'Brien & R i c h a r d , 1971) i n d i c a t e R e c a l l f o l l o w s Counting procedures. The use of R e c a l l and A s s o c i a t i v e processes may i n d i c a t e a r e a d i n e s s f o r more formal s o l u t i o n procedures such as sentence t r a n s f o r m a t i o n s . I t i s 136 recommended t h a t teachers not attempt to each formal s u b t r a c t i v e sentence t r a n s f o r m a t i o n a l g o r i t h m s u n t i l c h i l d r e n have a c q u i r e d the a b i l i t y t o use i n t e r n a l i z e d A s s o c i a t i v e s t r a t e g i e s . The l e v e l s of d i f f i c u l t y used i n t h i s study were not beyond the a b i l i t y of the year two c h i l d r e n who had access to a r e l i a b l e s t r a t e g y . I t i s suggested t h a t experience w i t h open sentences with number co n s t a n t s from other than the B a s i c F a c t Domain be used to develop Counting-On a b i l i t i e s , and to ensure t h a t a c h i l d ' s a b i l i t y t o s o l v e open sentences does not r e l y e n t i r e l y on b a s i c f a c t memory. Both m i s s i n g addend p l a c e h o l d e r p o s i t i o n s appear t o be a p p r o p r i a t e f o r year two c h i l d r e n and may g i v e v a l u a b l e i n t u i t i v e experience w i t h the commutative p r o p e r t y . Suggestions f o r F u r t h e r Research R e s u l t s from t h i s study i n d i c a t e t h a t a r e l a t i o n s h i p between C l a s s I n c l u s i o n performance and M i s s i n g Addend performance e x i s t s , however, a cause and e f f e c t r e l a t i o n s h i p cannot be e s t a b l i s h e d . A l o n g i t u d i n a l study of the development of these a b i l i t i e s i n c h i l d r e n i n years one and two may serve to i l l u s t r a t e whether the development of one a b i l i t y precedes', f a c i l i t a t e s , and/or a l l o w s the development of the o t h e r . S i m i l a r l y 137 a l o n g i t u d i n a l study concerning the development of c h i l d r e n ' s understanding of the i n v e r s e r e l a t i o n s h i p of a d d i t i o n and s u b t r a c t i o n , and i t s r e l a t i o n s h i p t o m i s s i n g addend s o l u t i o n s t r a t e g y use would help t o determine the a p p r o p r i a t e time t o develop sentence t r a n s f o r m a t i o n s . Data of a l o n g i t u d i n a l nature would a l s o be of v a l u e i n determining the r o l e of f i n g e r and body movements i n the i n t e r n a l i z a t i o n of counting p r o c e s s e s . An experimental study t h a t attempted t o t e a c h counting-on a b i l i t y may serve t o i l l u s t r a t e the f e a s i b i l i t y of i n d u c i n g what has been proposed t o be a developmental c o g n i t i v e a c q u i s i t i o n . The a b i l i t y t o count-on may be r e l a t e d to the a b i l i t y t o a c q u i r e a number of mathematical concepts, and would t h e r e f o r e be of v a l u e as a r e a d i n e s s i n d i c a t o r f o r i n i t i a l i n s t r u c t i o n i n these concepts. The concept of p l a c e v a l u e may be r e l a t e d t o counting-on; from the i n t e r v i e w s i t appeared t h a t an i n a b i l i t y or r e l u c t a n c e t o use s t i c k s grouped i n tens was a s s o c i a t e d w i t h C o u n t i n g - A l l s t r a t e g i e s . C h i l d r e n who C o u n t e d - A l l u s u a l l y i n s i s t e d on unbundling the s t i c k s or simply used the s i n g l e s t i c k s f o r items such as 42 + Q = 48. I t i s suggested t h a t the a b i l i t y to accept a group as r e p r e s e n t i n g t en f o r p l a c e v a l u e work i s r e l a t e d to the a b i l i t y t o count-on. The C o u n t - A l l u s e r s had to 138 p e r s o n a l l y account f o r each and every number, whereas the Count-On u s e r s accepted the r e p r e s e n t a t i o n a t f a c e v a l u e . T h i s suggests t h a t a p r e r e q u i s i t e to understanding p l a c e v a l u e may be the a b i l i t y t o Count-On i n t h a t groups of t e n are thought of i n terms of ones u n t i l the c h i l d can m e n t a l l y r e p r e s e n t q u a n t i t i e s and add on to such groupings. A study t o v e r i f y the r e l a t i o n s h i p s and to determine i f counting-on i s p r e r e q u i s i t e t o p l a c e v a l u e would be of c o n s i d e r a b l e b e n e f i t . During the data c o l l e c t i o n , i t became apparent to the r e s e a r c h e r t h a t a h i e r a r c h y of s o p h i s t i c a t i o n i n s o l u t i o n s t r a t e g y was r e p r e s e n t e d . Since t h i s study i n v o l v e d data of a c r o s s - s e c t i o n a l nature, no v e r i f i c a t i o n of t h i s h i e r a r c h y or of the sequence of development of s t r a t e g i e s was p o s s i b l e . A l o n g i t u d i n a l study of how c h i l d r e n ' s s o l u t i o n s t r a t e g i e s change with m a t u r i t y and experience would help to confirm the hypothesis of a sequence of development of p r o g r e s s i v e l y more s o p h i s t i c a t e d m i s s i n g addend s o l u t i o n s t r a t e g i e s . 139 REFERENCES Ahr, P.R. & Youniss, J . Reasons f o r f a i l u r e on the c l a s s i n c l u s i o n problem. C h i l d Development, 1970, 4_1_, 131-143. B r a i n e r d , C.J., Kaszor, P. An a n a l y s i s of two proposed sources of c h i l d r e n ' s c l a s s i n c l u s i o n e r r o r s . Developmental Psychology, 1974, 10, 633-643 Brownell, W.A. The development of c h i l d r e n ' s number ideas i n the primary grades. 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P u p i l E d i t i o n Book 2. Addison-Wesley (Canada) L t d . , 1973. E l k i n d , D. The development of the a d d i t i v e composition of c l a s s e s i n the c h i l d : P i a g e t r e p l i c a t i o n study I I I . J o u r n a l of G e n e t i c Psychology, 1961, 99, 51-57. Flournoy, F. Understanding r e l a t i o n s h i p : an e s s e n t i a l f o r s o l v i n g e quations. Elementary School J o u r n a l , 1964, 64 , 214-217. 1 4 0 Groen, G. An i n v e s t i g a t i o n o f some counting a l g o r i t h m s f o r simple a d d i t i o n problems. S t a n f o r d , C a l i f o r n i a : S t a n f o r d U n i v e r s i t y , I n s t i t u t e of Mathematical S t u d i e s i n the S o c i a l S c i e n c e s , 1 9 6 7 . (Tech. Rep. No. 1 1 8 ) . Groen, G.J., & Parkman, J.M. A chronometric a n a l y s i s o f simple a d d i t i o n . P s y c h o l o g i c a l Review, 1 9 7 2 , 7 9 , 3 2 9 - 3 4 3 . Groen, G.J., & P o l l , M. S u b t r a c t i o n and the s o l u t i o n of open sentence problems. J o u r n a l o f Experimental  C h i l d Psychology, 1 9 7 3 , 1 6 , 2 9 2 - 3 0 2 . Grouws, D.A. D i f f e r e n t i a l performance of t h i r d grade c h i l d r e n i n s o l v i n g open sentences o f f o u r types. Unpublished d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y o f Wisconsin, 1 9 7 1 . Grouws, D.A., & Good, T.L. F a c t o r s a s s o c i a t e d w i t h t h i r d and f o u r t h grade c h i l d r e n ' s performance i n s o l v i n g m u l t i p l i c a t i o n and d i v i s i o n sentences. J o u r n a l f o r Research i n Mathematics Education, 1 9 7 6 , 7 (3) , 1 5 5 - 1 7 1 . Howlett, K.D. A study o f the r e l a t i o n s h i p between p i a g e t i a n c l a s s i n c l u s i o n t a s k s and the a b i l i t y of f i r s t grade c h i l d r e n t o do m i s s i n g addend computation and v e r b a l problems. Unpublished d o c t o r a l d i s s e r t a t i o n , S t a t e U n i v e r s i t y o f New York, B u f f a l o , 1 9 7 3 . Hoyt, C.L. T e s t r e l i a b i l i t y e s t i mated by a n a l y s i s of v a r i a n c e . Psychometrika, 1 9 4 1 , 6>, 1 5 3 - 1 6 0 . I l g , F., & Ames, L.B. Developmental t r e n d s i n a r i t h m e t i c . J o u r n a l of Gen e t i c Psychology, 1 9 5 1 , 79, 3 - 2 8 . I n h e l d e r , B., & Pi a g e t , J . E a r l y growth of l o g i c i n  the c h i l d : c l a s s i f i c a t i o n and s e r i a t i o n . New York: Harper & Row, 19 6 4 . Jennings, J.R. The e f f e c t of v e r b a l and p i c t o r i a l p r e s e n t a t i o n on c l a s s - i n c l u s i o n competence and performance. Psychonomic Scie n c e , 1 9 7 0 , 2 0 , 3 5 7 - 3 5 8 . K l a h r , D., & Wallace, J.G. C l a s s i n c l u s i o n p r o c e s s e s . In S. Farnham-Diggory (Ed.), Information P r o c e s s i n g  i n C h i l d r e n . New York: Academic Press, 1 9 7 2 . 141 M a r a s c u i l o , L.A., & L e v i n , J.R. A p p r o p r i a t e p o s t hoc comparisons f o r i n t e r a c t i o n and nested hypotheses i n a n a l y s i s , of v a r i a n c e d e s i g n s : The e l i m i n a t i o n of type IV e r r o r s . American E d u c a t i o n a l Research  J o u r n a l , 1970, 1_(3) , 397-421. O'Brien, T.C., & Rich a r d , J.V. Interviews t o assess number knowledge. A r i t h m e t i c Teacher, 1971, 18(5) 322-326. Parkman, J.M. Temporal as p e c t s of d i g i t comparisons. Paper presented a t the meeting o f the Midwestern P s y c h o l o g i c a l A s s o c i a t i o n , D e t r o i t , May 1971. Parkman, J.M., & Groen, G.J. Temporal a s p e c t s of simple a d d i t i o n and comparison. J o u r n a l of Experimental  Psychology, 1971, 89, 335-342. Peck, D.M., & Jencks, S.M. M i s s i n g addend problems. School S c i e n c e and Mathematics, 1976, 8_, 647-661. Schwartz, C.R. Developmental aspects of c l a s s i n c l u s i o n . Unpublished d o c t o r a l d i s s e r t a t i o n , U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1970. S t e f f e , L.P., & Johnson, D.C. Problem s o l v i n g performances of f i r s t grade c h i l d r e n . J o u r n a l f o r Research i n  Mathematics Education, 1971, 2{1), 50-64. S t e f f e , L.P., Spikes, W.C., & H i r s t e i n , J . J . Q u a n t i t a t i v e  comparisons and c l a s s i n c l u s i o n as r e a d i n e s s v a r i a b l e s  f o r l e a r n i n g f i r s t grade a r i t h m e t i c a l content. The Georgia, Center .for the Study, of-.Learning. and Teaching Mathematics, U n i v e r s i t y o f Geor g i a , 1976. Suppes, P a t r i c k Some t h e o r e t i c a l models f o r mathematics l e a r n i n g . J o u r n a l of Research and Development i n  Edu c a t i o n , 1967, 1(1), 5-22. Suppes, P., & Groen, G. Some cou n t i n g models f o r f i r s t grade performance data on simple a d d i t i o n f a c t s . In J.M. Scandura (Ed.), Research i n Mathematics  E d u c a t i o n , Washington, D . C : N a t i o n a l C o u n c i l of Teachers o f Mathematics, 1967. Suppes, P., Jerman, M., & B r i a n , D. Computer-assisted  i n s t r u c t i o n : S t a n f o r d ' s 1965-66 a r i t h m e t i c program. New York: Academic Press Inc., 1968. Suydam, M.N., & Weaver, J.F. A d d i t i o n & s u b t r a c t i o n w i t h whole numbers, s e t B. In Using Research: A  Key to Elementary School Mathematics. ERIC Information A n a l y s i s Center, December, 1975. 142 Syer, H.W, Sensory l e a r n i n g a p p l i e d to mathematics. In The Learning of Mathematics. T w e n t y - f i r s t Yearbook. N a t i o n a l C o u n c i l of Teachers of Mathematics. New York: Columbia U n i v e r i s t y of Teachers' C o l l e g e , 1953. Washburne, C., & Vogel, M. Are any number combinations i n h e r e n t l y d i f f i c u l t ? J o u r n a l of E d u c a t i o n a l Research, 1928, 17, 235-255. Weaver, J.F. Some f a c t o r s a s s o c i a t e d with p u p i l s , performance l e v e l s on simple open a d d i t i o n and s u b t r a c t i o n sentences. A r i t h m e t i c Teacher, 1971, 18, 513-519. Weaver, J . The symmetric p r o p e r t y of the e q u a l i t y r e l a t i o n and young c h i l d r e n s ' a b i l i t y t o s o l v e open a d d i t i o n and s u b t r a c t i o n sentences. J o u r n a l  f o r Research i n Mathematics Education., 1973, 4_(1) , 45-56. Weaver, J.F. C a l c u l a t o r i n f l u e n c e d e x p l o r a t i o n s i n school mathematics: A f u r t h e r i n v e s t i g a t i o n of 3rd grade p u p i l s ' performance on open a d d i t i o n and s u b t r a c t i o n sentences. P r o j e c t Paper 7 6-3, Wisconsin Research and Development Center f o r C o g n i t i v e L e a r n i n g , U n i v e r s i t y of Wisconsin, A p r i l 1976. (ERIC Document Reproduction S e r v i c e No. ED 123 089) Winer, G.A. A n a l y s i s of v e r b a l - f a c i l i t a t i o n of c l a s s i n c l u s i o n r easoning. C h i l d Development, 19 74, 45, 224-227. Winer, G.A., & Kronberg, D.D. C h i l d r e n ' s responses to v e r b a l l y and p i c t o r i a l l y p r esented c l a s s - i n c l u s i o n items and to a t a b l e of number c o n s e r v a t i o n . J o u r n a l of Genetic Psychology, 1974, 125, 141-152. W o h l w i l l , J.F. Responses t o c l a s s - i n c l u s i o n q u e s t i o n s f o r v e r b a l l y and p i c t o r i a l l y p r esented items. C h i l d  Development, 1968, 3_9, 449-465. Youssef, Z.I., & Guardo, C.J. The a d d i t i v e composition of c l a s s e s : The r o l e of p e r c e p t u a l cues. J o u r n a l  of G enetic Psychology, 1972, 121, 197-205. 1 4 3 APPENDIX A. Pupil Data I D Sex Age i n Class Text Method Months 1 F 92 1 ISM sub 2 F 87 1 ISM sub 3 M 90 1 ISM sub 4 F 94 1 ISM sub. 5 M 93 2 ISM both 6 M 94 3 ISM sub 7 M 87 3 ISM sub 8 F 96 3 ISM sub 9 F 87 3 ISM sub 10 M 93 4 ISM* add 11 F 91 4 ISM* add 12 M 87 4 ISM* add 13 M 97 4 ISM* add 14 M 91 5 ISM* add 15 M 92 5 ISM* add 16 M 89 5 ISM* add 17 F 100 5 ISM* add 18 M 92 6 ISM sub 19 M 88 7 ISM sub 20 M 91 7 ISM sub 21 F 96 7 ISM sub 22 F 95 7 ISM sub 23 F 98 7 ISM sub 24 M 87 8 ISM* add 25 F 91 8 ISM* add 26 M 89 8 ISM* add 27 F 89 8 ISM* add 28 M 88 9 none add 29 F 96 9 none add 30 F 95 10 none add 31 M 105 11 none sub 32 M 94 11 none sub 33 M 92 11 none sub 34 M 94 11 none sub 35 F 100 12 none add 36 F 97 12 none add 37 F 89 12 none add 38 F 97 13 none add 39 F 96 13 none add 40 M 87 13 none add ISM refers to use of Investigating School Mathematics, Book Two (Eicholz et a l . , 1973) ISM* refers to limited use of the same text none refers to no consistent text use sub refers to classroom emphasis on a subtractive i n t e r p r e t a t i o n of missing addends add refers to classroom emphasis on an additive i n t e r p r e t a t i o n APPENDIX B. Sample of the M i s s i n g Addend T e s t Booklet 144 II II •+- 1 co II i i O N ] I 145 5+ n = 8 + 4 = 7 146 5+n=i3 147 42 + 48 + 43 = 49 148 23+U=38 +24=39 Name Date B i r t h d a t e School Time Cla s s Tape Set # Additive Subtractive Recall Substitution , Associative Transform(I) , Miscellaneous 5 + D = 8 • + 4=7 — 5 + D = 1 3 -• +6=14 4 2 + D =48 • + 43=49 23+D=38 •+24=39 APPENDIX C. Interview Coding Form 150 APPENDIX D. C l a s s I n c l u s i o n Items Item One M a t e r i a l s : none Procedure: I n t e r v i e w e r asked the f o l l o w i n g q u e s t i o n s : 1. You have a l o t of c h i l d r e n i n your c l a s s . Are t h e r e more boys or more g i r l s ? (If the c h i l d was a* boy, boys were mentioned f i r s t . ) 2. Are t h e r e more c h i l d r e n or more boys? (The order of p r e s e n t a t i o n of the s e t and subset was predetermined. The subset boys or g i r l s was used depending on the answer to the preceding question.) 3. Why? Item Two M a t e r i a l s : none Procedure: I n t e r v i e w e r asked the f o l l o w i n g q u e s t i o n s : 1. Do you have any pets? 2. W e l l let;';s p retend you have some more p e t s . L e t ' s pretend you have f i v e c a t s and two dogs ( the c h i l d ' s own pets were used here, i f t h e r e were any). You would.have a l o t of p e t s wouldn't you? Would you have more c a t s or more pets? (The order of mentioning the u n d e r l i n e d words was predetermined.) 3. Why? Item Three M a t e r i a l s : 5. red cubes and 3 y e l l o w cubes Procedure: I n t e r v i e w e r covered the b l o c k s w i t h both hands and asked the f o l l o w i n g q u e s t i o n : 1. What have I got here? 2. That's r i g h t , they are b l o c k s ( or whatever the c h i l d r e f e r r e d t o them a s ) . Some of these b l o c k s are red and some of them are y_ellqw. "(The i n t e r -viewer showed the c h i l d t h e blocks- and 'the c h i l d • t o l d the i n t e r v i e w e r the colours- when asked.) A l t o g e t h e r , "I have some b l o c k s . Have I got more blo c k s or more red b l o c k s ? ( The order of mentioning the set and subset was predetermined. ) 151 3. Why? Item Four M a t e r i a l s : 12 c i r c u l a r and 12 t r i a n g u l a r shaped A t t r i b u t e Blocks Procedure: The i n t e r v i e w e r put out the Blocks and asked the f o l l o w i n g q u e s t i o n s : 1. What have I got here? (The i n t e r v i e w e r i n d i c a t e d the e n t i r e group 2. That's r i g h t , these are wooden shapes (or whatever the c h i l d r e f e r r e d to them a s ) . Some of these wooden shapes are round and some of them are ££i522!ii§E• (The c h i l d p r o v i d e d the l a b e l s f o r the subsets. The i n t e r v i e w e r suggested l a b e l s i f the c h i l d c o u l d not name them and ensured t h a t the c h i l d c o u l d use these l a b e l s . ) A l t o g e t h e r we have a bunch of wooden shapes. Are t h e r e more round shapes or more wooden shapes? (The or d e r of p r e s e n t i n g the s e t and subset was predetermined). 3. Why? r e f e r s t o words where the order of p r e s e n t a t i o n v a r i e d r e f e r s t o words which the c h i l d was encouraged to p r o v i d e Each c h i l d had two o p p o r t u n i t i e s t o answer these c l a s s i n c l u s i o n t a s k s . I f the answer to q u e s t i o n 2 or qu e s t i o n 3 was i n c o r r e c t , the i n t e r v i e w e r posed the tas k again, v a r y i n g the wording s l i g h t l y , and ensuring t h a t the c h i l d c o u l d name the subsets and the su p e r o r d i n a t e se t i n v o l v e d . The s c o r i n g procedure i s o u t l i n e d i n Chapter 3. 152 APPENDIX E. C l a s s I n c l u s i o n Task P r e s e n t a t i o n Card Sample* Name School Item Order P r e s e n t a t i o n Order Score and R a t i o n a l e 3 y e l l o w / r e d , r e d / b l o c k s (1, f) 2 cats/dogs, p e t s / c a t s ( f , 1) 1 / c h i l d r e n ( f , f ) 4 i t r i a n g l e s / c i r c l e s , s h a p e s / c i r c l e s (1, 1) *Item p r e s e n t a t i o n order v a r i e d f o r each c h i l d . APPENDIX F. T e s t Scores 153 Missing Addend Test Class Inclusion Test Items Total Items Total 12345678 Student I D 1234 00000000 0 1 2111 5 10100000 2 2 2202 6 01000000 1 3 2000 2 11100110 5 4 2222 8 11000000 2 5 2222 8 11111111 8 6 2222 8 11010000 3 7 2122 7 11111111 8 8 2222 8 11111111 8 9 2222 8 11101000 4 10 2221 7 11110000 4 11 2211 6 11010101 5 12 0000 0 11111101 7 13 2212 7 11010100 4 14 1001 2 11100010 4 15 0000 0 11111101 7 16 2222 8 11010000 3 17 2222 8 00000000 0 18 0000 0 11111111 !\ 19 2022 6 11110000 20 2000 2 11110000 4 x 21 2122 7 11101000 4 22 2000 2 11101000 4 23 2220 6 11111011 7 24 2222 8 11111101 7 25 2121 6 11111111 8 26 2212 7 11101111 7 27 2222 8 11110111 7 28 1000 1 10101100 4 29 2010 3 10101000 3 30 2222 8 11110001 5 31 1100 2 11111100 6 32 2222 8 11111111 8 33 2222 8 11111101 7 34 2221 7 11101100 5 35 1211 5 01110101 5 36 1201 4 11111101 7 37 2222 8 00000000 0 38 1211 5 11111111 8 39 2222 8 11000001 3 40 2221 7 1= correct 0= i n c o r r e c t 154 APPENDIX G. S o l u t i o n S t r a t e g y Usage P a t t e r n s Student Items I D 12345678 P a t t e r n 1 77777877 C 2 33399411 N 3 22337777 D 4 46435439 N 5 66439493 N 6 66555555 C 7 46534454 N 8 66344444 D 9 43333433 D 10 32323232 P 11 66555515 C 12 44353333 D 13 66655584 D 14 44549411 N 15 33335535 D 16 44333333 D 17 44333788 D 18 77777777 C 19 34444444 C 20 33341188 N 21 44444444 C 22 44442444 C 23 36333311 D 24 36443344 D 25 66434414 N. 26 46444544 N 27 33333333 C 28 63333344 D 29 47474477 N 30 37373737 P 31 44444444 C 32 66655533 D 33 66665553 D 34 33335513 N 35 63499411 N 36 63333333 C 37 66455444 D 38 33333333 C 39 34444444 C 40 44444444 C St r a t e g y I d e n t i f i c a t i o n Key 1 = Semi-Guess 2 = S u b s t i t u t i o n 3 = C o u n t - A l l 4 = Count-On 5 = A s s o c i a t i v e 6 = R e c a l l 7 = I n c o r r e c t T r ansformation 8 = No Attempt 9 = M i s c e l l a n e o u s i n d i c a t e s s u b t r a c t i v e i n t e r p r e t a t i o n o f the s t r a t e g y no l i n e i n d i c a t e s an a d d i t i v e i n t e r p r e t a t i o n of the s t r a t e g y S t r a t e g y Choice P a t t e r n Key N = no i d e n t i f i a b l e p a t t e r n D = s t r a t e g y c h o i c e a f f e c t e d by the d i f f i c u l t y , o f the item P = s t r a t e g y c h o i c e a f f e c t e d by the p l a c e h o l d e r p o s i t i o n C = c o n s i s t e n t choice o f one s t r a t e g y 155 APPENDIX H. L e v e l of Response Usage P a t t e r n s Student Items P a t t e r n Student Items P a t t e r n I D 12345678 , I D 12345678 1 11111011 s 31 33331232 n 2 21231211 n 32 33333322 c 3 11111111 s 33 33333331 s 4 33313313 n 34 11111311 s 5 33313331 n 35 31333212 n 6 33333333 s 36 31111111 s 7 33312222 c 37 33233222 c 8 33111311 c 38 11111111 s 9 32111111 c 39 13311111 c 10 11111011 s 40 33311111 c 11 33333333 s 12 33111111 c 13 33333302 c L e v e l o f Response Key 14 33333333 s 15 33223313 n 16 22111111 c 3= I n t e r n a l i z e d L e v e l 17 33111100 c 2= T r a n s i t i o n a l L e v e l 18 33331111 c 1= E x t e r n a l i z e d L e v e l 19 11112111 s 0= No Attempt 20 22111100 c 21 33222221 c 22 22232232 n L e v e l o f Response P a t t e r n s 23 13111111 s 24 23222222 s 25 33212212 n c= L e v e l o f Response changes to 26 33332311 c more e x t e r n a l i z e d L e v e l 27 11111111 s s= L e v e l o f Response does not 28 32113111 c change a c r o s s items 29 33212211 c n= no p a t t e r n t o L e v e l o f Response 30 13131313 n ( One item i s allowed to d e v i a t e from the p a t t e r n , but 7 or 8 items must i l l u s t r a t e a t r e n d , f o r i t t o be named as such.) 156 APPENDIX I . R e l i a b i l i t y of S t r a t e g y C l a s s i f i c a t i o n Item I n t e r v i e w e r ' s Rater's Comparison C l a s s i f i c a t i o n C l a s s i f i c a t i o n 1 R e c a l l R e c a l l same 2 C o u n t - a l l C o u n t - a l l same 3 Count-on Count-on same 4 Guess Guess same 5 Guess Guess same 6 Count-on Count-on same 7 Semi-guess Indeterminate d i f f e r e n t 8 Semi-guess Indeterminate d i f f e r e n t 1 R e c a l l R e c a l l same 2 R e c a l l R e c a l l same 3 Count-on Count-on same 4 A s s o c i a t i v e A s s o c i a t i v e same 5 A s s o c i a t i v e A s s o c i a t i v e same 6 Count-on Count-on same 7 Count-on Count-on same 8 Count-on Count-on same 1 R e c a l l R e c a l l same 2 R e c a l l R e c a l l same 3... R e c a l l R e c a l l same 4 R e c a l l R e c a l l same 5 A s s o c i a t i v e A s s o c i a t i v e same 6 A s s o c i a t i v e A s s o c i a t i v e same 7 A s s o c i a t i v e A s s o c i a t i v e same 8 C o u n t - a l l Indeterminate d i f f e r e n t 1 R e c a l l R e c a l l same 2 R e c a l l R e c a l l same 3 R e c a l l .i •. R e c a l l same 4 A s s o c i a t i v e A s s o c i a t i v e same 5 A s s o c i a t i v e A s s o c i a t i v e same 6 A s s o c i a t i v e A s s o c i a t i v e same 7 C o u n t - a l l C o u n t - a l l same 8 C o u n t - a l l C o u n t - a l l same 29. out of 32 c l a s s i f i c a t i o n s were i n agreement, r e s u l t i n g i n a r e l i a b i l i t y s c o r e , q f 91%. APPENDIX J . R e l i a b i l i t y o f C l a s s I n c l u s i o n S c o r i n g Item I n t e r v i e w e r ' s Rater's Comparison Score Score 1 2 1 d i f f e r e n t 1 1 0 d i f f e r e n t 1 0 0 same 1 0 0 same 1 2 2 same 1 2 2 same 1 0 0 same 1 0 0 same 1 0 0 same 1 0 0 same 1 1 1 same 1 2 2 same 1 1 1 same 1 2 2 same 1 1 1 same 1 0 0 same 1 0 0 same 2 1 1 same 2 2 2 same 2 0 0 same 2 0 0 same 2 1 1 same 2 2 2 same 2 0 0 same 2 0 0 same 2 1 1 same 2 0 0 same 2 1 2 d i f f e r e n t 2 2 2 same 2 0 1 d i f f e r e n t 2 2 2 same 2 1 0 d i f f e r e n t 2 2 1 d i f f e r e n t 3 2 2 same 3 2 3 same 3 0 0 same 3 2 2 same 3 2 2 same 3 1 1 same 3 1 1 same 3 2 2 same 3 0 0 same 3 1 1 same 3 1 1 same 3 2 2 same 3 1 1 same 3 2 1 d i f f e r e n t 3 2 1 d i f f e r e n t 158 APPENDIX 3. (Continued) Item I n t e r v i e w e r ' s Rater's Comparison Score Score 4 2 2 same 4 1 0 d i f f e r e n t 4 0 0 same 4 0 0 same 4 2 2 same 4 2 2 same 4 1 1 same 4 0 0 same 4 0 0 same 4 0 0 same 4 2 2 same 4 2 2 same 4 2 2 same 4 2 1 d i f f e r e n t 4 2 0 d i f f e r e n t 4 2 2 same 55 out of 6 4 item r a t i n g s were i n agreement r e s u l t i n g i n a r e l i a b i l i t y score of 86%. 

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