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Profoundly deaf students’ performance on arithmetical word problems Lapawa, Marguerite Clauvet 1980

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PROFOUNDLY DEAF STUDENTS* PERFORMANCE ON ARITHMETICAL WORD PROBLEMS by MARGUERITE CHAUVET LAPAWA B.Ed., U n i v e r s i t y of A l b e r t a , 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of V S p e c i a l Education We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA (c) September, 1980 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 t h e r equ i rement 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 ia , I a g r e e t h a t 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 that 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 thout my w r i t t e n p e r m i s s i o n . Department o f ~~DPi?C>\<i\u j£"v>tA.cftTi pA The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 ABSTRACT The i n v e s t i g a t i o n examined the performance of profoundly hearing impaired students on one-step word problems i n a r i t h m e t i c . Students were administered a computation task and those who met the pass r e q u i r e -ments were given a word problem task. These ninety subjects were d i v i d e d i n t o four age groups as f o l l o w s : 8-11 years; 12-13 years; 14-15 years; and 16+ years. S t a t i s t i c a l treatment of the data showed no s i g n i f i c a n t d i f f e r e n c e s when age and gender were examined f o r any of the word problems which i n v o l v e d the four operations of a d d i t i o n , sub-t r a c t i o n , m u l t i p l i c a t i o n or d i v i s i o n . S i g n i f i c a n t d i f f e r e n c e s however, were found f o r type of question f o r a d d i t i o n and s u b t r a c t i o n word prob-lems. There was a s i g n i f i c a n t i n t e r a c t i o n between type of ques t i o n , age, and gender f o r d i v i s i o n problems. When age and gender were c o l l a p s e d , a second a n a l y s i s revealed that operation, type of question, and the i n t e r a c t i o n between operation and type of question were a l l s i g n i f i c a n t . E r r o r a n a l y s i s revealed that profoundly hearing impaired students, when faced w i t h a word problem r e q u i r i n g s u b t r a c t i o n or d i v i s i o n , were l i k e l y to e i t h e r add or m u l t i p l y . Educational i m p l i c a t i o n s are discussed and suggestions made f o r fu t h e r research. i i TABLE OF CONTENTS Page TITLE PAGE i ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENT v i i CHAPTER I . INTRODUCTION 1 DEFINITION OF TERMS . . 4 CHAPTER I I . PROBLEM 6 STATEMENT OF THE PROBLEM . . . . . . 6 REVIEW OF THE LITERATURE 6 STATEMENT OF HYPOTHESES 20 CHAPTER I I I . METHOD 21 SAMPLE POPULATION 21 TEST INSTRUMENTS 22 SCORING 26 PROCEDURE 27 DATA ANALYSIS 28 CHAPTER IV. RESULTS AND DISCUSSION 29 i i i i v Page CHAPTER V. CONCLUSIONS 51 SUMMARY 51 LIMITATIONS 54 IMPLICATIONS 54 FUTURE RESEARCH 56 BIBLIOGRAPHY 58 APPENDIX A 6 4 LIST OF TABLES TABLE Page I Number and Percentage D i s t r i b u t i o n of Gender W i t h i n Age 22 I I Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question (Addition) 30 I I I Performance Means and Standard Deviations Type of Question by Age Group (Addition) 30 IV E r r o r A n a l y s i s f o r A d d i t i o n , 32 V Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question (Subtraction) . . . . 34 VI Performance Means and Standard D e v i a t i o n s Type of Question by Age Group (Subtraction) 34 VI I E r r o r A n a l y s i s f o r S u b t r a c t i o n 36 V I I I Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question ( M u l t i p l i c a t i o n ) . . . 38 IX Performance Means and Standard Deviations Type of Question by Age Group ( M u l t i p l i c a t i o n ) 38 X E r r o r A n a l y s i s f o r M u l t i p l i c a t i o n 40 XI Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question ( D i v i s i o n ) 42 X I I Performance Means and Standard D e v i a t i o n s Type of Question by Age Group ( D i v i s i o n ) 42 X I I I E r r o r A n a l y s i s f o r D i v i s i o n 44 XIV Summary ANOVA ( u n i v a r i a t e ) Mathematical Operation by Type of Question 46 XV Performance Means and Standard D e v i a t i o n s Mathematical Operation by Type of Question 46 XVI T o t a l E r r o r A n a l y s i s 50 v LIST OF FIGURES FIGURE Page 1 I n t e r a c t i o n Graph, Type of Question by Age 35 2 I n t e r a c t i o n Graphs, Gender by Age f o r Both Types of Question 43 3 I n t e r a c t i o n Graph, Type of Question by Operation 48 v i . ACKNOWLEDGEMENTS The w r i t e r wishes to acknowledge the ass i s t a n c e which she rec e i v e d from the a d m i n i s t r a t i o n and s t a f f of J e r i c h o H i l l School f o r the Deaf. I t was through t h e i r cooperation that the work of data c o l l e c t i o n was f a c i l i t a t e d . To the members of my committee Dr. T. Rogers and Dr. P. L e s l i e , go my thanks f o r t h e i r advice and time on t h i s study. A s p e c i a l note of a p p r e c i a t i o n to Dr. B. R. Clarke my a d v i s o r , whose encouragement and p o s i t i v e t h i n k i n g made t h i s study a s p e c i a l e d u c a t i o n a l experience. v i i CHAPTER I INTRODUCTION The word problem has long held an important place i n i n s t r u c t i o n i n elementary school a r i t h m e t i c . Nevertheless, there i s a general l a c k of agreement as to j u s t which of the many v a r i a b l e s s t u d i e d s i g n i f i c a n t l y i n f l u e n c e one's a b i l i t y to s o l v e v e r b a l problems. Researchers ( C o f f i n g 1941, Hansen 1944, Van der Linde 1964, Meyer 1978, and Webb 1979) studying one or more f a c t o r s r e l a t i n g to problem s o l v i n g have i n general concluded the f o l l o w i n g : a) i n t e l l i g e n c e was s i g n i f i c a n t l y r e l a t e d to problem s o l v i n g a b i l i t y , b) problems i n a d d i t i o n were the e a s i e s t to solve followed by s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n i n that order, c) sex d i f f e r e n c e s do not appear to e f f e c t prob-lem s o l v i n g a b i l i t y , d) no s i g n i f i c a n t d i f f e r e n c e s e x i s t i n achievement between problems i n textbooks and problems w r i t t e n by c h i l d r e n using f a m i l i a r s e t t i n g s and people, e) computational d i f f i c u l t i e s appear to be a major d e t e r r e n t to f i n d i n g c o r r e c t answers, f) reading a b i l i t y i s p o s i t i v e l y c o r r e l a t e d w i t h problem s o l v i n g a b i l i t y (see Suydam and Weaver 1977) . C o f f i n g (1941) concluded that a p o s i t i v e r e l a t i o n e x i s t e d between s i l e n t reading and the a b i l i t y to solve reasoning problems. Van der Linde (1964) however, s t a t e d that vocabulary was one of the main f a c t o r s i n the same a b i l i t y . Other i n v e s t i g a t i o n s (Monroe and Engelhart 1933, Faulk and Landry 1 2 1961) demonstrated the advantage of a given method of problem a n a l y s i s as an a i d to problem s o l v i n g and showed that encouraging c h i l d r e n to s o l v e problems i n a v a r i e t y of ways appears to help c h i l d r e n develop problem s o l v i n g s k i l l s . Results showed i n s t r u c t i o n i n systematic approaches, reading o r i e n t e d approaches or s p e c i f i c s k i l l i n s t r u c t i o n were a s s o c i a t e d w i t h s u c c e s s f u l problem s o l v i n g . While some i n v e s t i g a t o r s examined s k i l l s a s s o c i a t e d w i t h the a b i l i t y of problem s o l v i n g or methods of i n s t r u c t i o n , others t r i e d to d i s c o v e r s p e c i f i c causes of f a i l u r e i n s o l v i n g a r i t h m e t i c problems ( R o l i n g , Blume and Morehart 1924, Stevenson 1925, and Lenore 1930) . A composite of t h e i r f i n d i n g s regarding causes of d i f f i c u l t y i n c l u d e s : 1) p h y s i c a l and mental d e f e c t s ; 2) d i f f i c u l t i e s w i t h word r e c o g n i t i o n ; 3) l a c k of knowledge of vocabulary and t e c h n i c a l terms; 4) carelessness i n reading the problem; 5) focus upon numbers r a t h e r than meaning; 6) confusion caused by numbers l a r g e r than those commonly encountered; 7) d i r e c t i o n by v e r b a l cues r a t h e r than the mathematical r e l a t i o n s h i p s w i t h i n the problems; 8) l a c k of a b i l i t y to understand and compare q u a n t i t a t i v e r e l a t i o n s ; 9) l a c k of a b i l i t y i n fundamental operations; 10) lack of knowledge of b a s i c a r i t h m e t i c f a c t s , r u l e s , and formulae; 11) i n a b i l i t y to t h i n k r e f l e c t i v e l y ; 12) l a c k of a b i l i t y to choose the main computa-t i o n a l process i n c l u d i n g i n a b i l i t y to recognize the main elements; 13) inaccurate copying of numerals; 14) lack of v a r i e t y of good problems r e s u l t i n g i n adaptation of the p u p i l to the problems ra t h e r than the problem to the p u p i l ; 15) lack of i n t e r e s t and e f f o r t ; 16) f a i l u r e to r e g u l a r l y v e r i f y r e s u l t s ; and 17) poor teaching, i n c l u d i n g f a i l u r e to help p u p i l s t r a n s l a t e problems i n t o t h e i r own experiences. Teachers of the deaf are quick to acknowledge the problems t h e i r 3 hearing impaired students face when attempting to sol v e word problems i n a r i t h m e t i c . Research i n d i c a t e s that the average deaf student's achieve-ment i n a r i t h m e t i c word problems i s g e n e r a l l y low (Hargis 1969, Suppes 1974) . In h i s review of the l i t e r a t u r e on c o g n i t i o n i n handicapped c h i l d r e n , Suppes reported no s t u d i e s other than h i s own had been found regarding the mathematical a b i l i t i e s of deaf c h i l d r e n beyond the l e v e l of computational a r i t h m e t i c . These st u d i e s which he reported were p r i -m a r i l y assessments of achievements on standardized t e s t s . Suppes con-cluded that o b j e c t i v e features of the curriculum, f o r example, whether a v e r t i c a l a d d i t i o n problem r e q u i r e d c a r r y i n g or not, dominate the ease or d i f f i c u l t y of e x e r c i s e s i n much the same way f o r both deaf and normal hearing c h i l d r e n . The low language l e v e l of deaf c h i l d r e n , however, i s a f a c t o r i n f r u s t r a t i n g reading e f f o r t s i n c l u d i n g the a b i l i t y to solve word problems. Goetzinger and Rousey (1959) using the WAIS and Stanford B i n e t per-formance subtests tested 101 hearing impaired students ages fourteen to twenty-one years. The mean scores i n a r i t h m e t i c reasoning were found to be l e s s than s i x t h grade and means i n paragraph meaning approximately f o u r t h grade. Hine (1970) t e s t e d 104 students 7.8 to 16.5 years of age wi t h an average hearing l o s s of 60 dB. The Schonell's E s s e n t i a l Mechanical Problem A r i t h m e t i c Test was administered and the r e s u l t s showed that hearing impaired students aged ten performed as w e l l as an average hearing student of eigh t and deaf students aged f i f t e e n scored equal to that of a hearing ten year o l d . Messerley and Aram (1980) confirmed previous s t u d i e s w i t h t h e i r ' i n v e s t i g a t i o n on seventeen year o l d students. The Stanford Achievement Test s c a l e scores of these students on mathematics a p p l i c a t i o n s were at the seventh grade - a l e v e l reached by hearing students by the age of twelve. 4 In a recent B r i t i s h Columbia study (Rogers et a l . 1978) 148 hearing impaired students aged f i v e to seventeen years and o l d e r were t e s t e d . The scaled scores on the Stanford Achievement Test, S p e c i a l E d i t i o n f o r Hearing Impaired Students (SAT-HI) f o r mathematics a p p l i c a t i o n s ranged from 147.5 ( f o u r t h grade) f o r the f i v e to ei g h t year o l d group to 172.0 (seventh grade) f o r the seventeen plus age group. Performance increased w i t h i n c r e a s i n g age but no s i g n i f i c a n t d i f f e r e n c e s among hearing l o s s categories were found f o r mathematics a p p l i c a t i o n s . With the mean adjusted f o r age and s e v e r i t y of hearing l o s s the performance of students w i t h at l e a s t one a d d i t i o n a l handicap was s i g n i f i c a n t l y lower than students w i t h no a d d i t i o n a l handicap. I t would appear that language f a c t o r s p l a y an important r o l e i n success of a r i t h m e t i c problem s o l v i n g . Further i t has been found that hearing impaired students are se v e r e l y retarded i n t h e i r language achievement (Cooper and Rosenstein 1966). I t would the r e f o r e f o l l o w that the hearing-impaired student would have d i f f i c u l t y with word prob-lems and the d i f f i c u l t y i s caused mainly by h i s low l e v e l of language achievement (Hargis 1969) . D e f i n i t i o n of Terms Throughout the study v a r i o u s terms w i l l be used as defined below: 1. Hearing Impairment - The term describes the hearing threshold l e v e l (HTL) which may range from m i l d moderate (<59 dB), marked severe (60^-89 dB) to a profound (>90 dB) degree. I t incorporates the c o n d i t i o n s known as hard of hearing and deafness. 2. Degree of Hearing Loss - Hearing l o s s i s discussed i n terms of aver-age hearing threshold i n d e c i b e l s (dB) using the American N a t i o n a l Standards I n s t i t u t e (ANSI) c r i t e r i a and i s equal to the a r i t h m e t i c mean of the pure tone thresholds obtained at 5 0 0 , 1 0 0 0 , and 2000 Hz i n the b e t t e r ear. Profound Hearing Loss - An average hearing threshold l e v e l of 90 dB or greater (>90 dB) i n the b e t t e r ear. Word Problems - Those problems t y p i c a l l y found i n elementary math-ematics textbooks that are presented i n words. To sol v e the word problem, the operations of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and/or d i v i s i o n are used. The term "word problem" may a l s o be st a t e d as a v e r b a l problem or s t o r y problem. One-step Word Problem - A word problem that r e q u i r e s only one math-em a t i c a l operation f o r i t s s o l u t i o n . CHAPTER I I PROBLEM I . Statement of the Problem The s p e c i f i c purpose of t h i s study was to i n v e s t i g a t e the f o l l o w i n g questions: 1) I s there an order of d i f f i c u l t y i n one-step word problems r e q u i r i n g one of the four operations of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , or d i v i s i o n ? 2) Is the a b i l i t y to solve word problems r e l a t e d to age? 3) I s the a b i l i t y to sol v e word problems as s o c i a t e d w i t h gender? 4) Does the frequency of a question type as found i n elementary math-ematics t e x t s , a f f e c t the a b i l i t y to s o l v e word problems? I I . Review of the L i t e r a t u r e Researchers have been e x p l o r i n g the many d i f f i c u l t i e s a s s o c i a t e d w i t h the s o l u t i o n of a r i t h m e t i c word problems f o r many years. As problem s o l v i n g i s viewed as a v i t a l part of doing mathematics, i t t h e r e f o r e seems n a t u r a l to analyze c a r e f u l l y what i s i n v o l v e d i n the process so that appropriate l e a r n i n g environments and i n s t r u c t i o n a l techniques can be developed. Research w i l l be discussed under the c l a s s i f i c a t i o n s of general i n t e l l i g e n c e , computation, reading a b i l i t y , mathematical vocab-u l a r y and syntax,and language i n f l u e n c e s on mathematical development w i t h normally hearing s u b j e c t s . In the f i n a l s e c t i o n p e r t i n e n t research w i t h hearing impaired students w i l l then be examined. 6 7 General I n t e l l i g e n c e - P i t t s (1952), E r i k s o n (1958) and Wrigley (1958) found that there i s a c l o s e connection between mathematical and general a b i l i t y and that high general i n t e l l i g e n c e i s an important f a c -tor f o r success i n mathematics and problem s o l v i n g . E r i k s o n examined f a c t o r s of i n t e l l i g e n c e and reading a b i l i t y and t h e i r e f f e c t s on achieve-ment i n a r i t h m e t i c concepts and problem s o l v i n g f o r s i x t h graders. I t was found that i n t e l l i g e n c e and reading a b i l i t y each c o r r e l a t e d pos-i t i v e l y w i t h both concepts and s o l v i n g of problems. Wrigley (1958) i n h i s study of students aged fourteen years and older concluded that high general i n t e l l i g e n c e can account f o r a large p a r t of the v a r i a b i l i t y shared by v e r b a l and mathematical a b i l i t y , but a s i g n i f i c a n t degree of overlap between the l a s t two v a r i a b l e s remains unexplained. He s t a t e d that "there e x i s t s a c l e a r l y i d e n t i f i a b l e mathematical group f a c t o r . The d i f f e r e n t branches of mathematics are l i n k e d together more c l o s e l y than they would be i f a general a b i l i t y i s e l i m i n a t e d , v e r b a l a b i l i t y has l i t t l e connection w i t h mathematical a b i l i t y (page 77)." Further research i m p l i e d that reading ( l i n g u i s t i c ) a b i l i t y i s the "unexplained" v a r i a b l e s i n c e d i f f e r e n c e s i n reading a b i l i t y may be used to e x p l a i n the p o s i t i v e c o r r e l a t i o n between scores on mathematics t e s t s and i n t e l l i g e n c e t e s t s (Muscio 1962). Computation - Stevens (1932) i n h i s i n v e s t i g a t i o n examined four v a r i a b l e s thought to be a s s o c i a t e d w i t h problem s o l v i n g a b i l i t y : 1) a b i l i t y i n s i l e n t reading; 2) power i n the fundamental operations of a r i t h m e t i c ; 3) power i n s o l v i n g reasonable problems i n a r i t h m e t i c ; and 4) " g e n e r a l " i n t e l l i g e n c e t e s t scores. The r e s u l t s i n d i c a t e d that power i n the fundamental operations of a r i t h m e t i c was most c l o s e l y r e l a t e d to a b i l i t y i n problem s o l v i n g when s t a t i s t i c a l l y the e f f e c t of general reading a b i l i t y was h e l d constant. 8 Engelhart (1932) a l s o s t u d i e d the e f f e c t s of reading a b i l i t y , i n t e l l i g e n c e and computational a b i l i t y on problem s o l v i n g a b i l i t y . Examination of the f i n d i n g s showed that the l a r g e s t amount (42.05%) of variance i n problem s o l v i n g a b i l i t y was accounted f o r by computational a b i l i t y . S i x t h graders were t e s t e d by Hansen (1944) on twenty-eight f a c t o r s r e l a t e d to a r i t h m e t i c reasoning, reading and general reasoning a b i l i t y . The upper and lower twenty-seven per cent on a t e s t of problem s o l v i n g were compared and r e s u l t s i n d i c a t e d that those f a c t o r s associated w i t h numbers and reasoning appeared to be most c l o s e l y r e l a t e d to s u c c e s s f u l problem s o l v i n g . Chase (1960) administered a s e r i e s of standardized t e s t s i n i n t e l -l i g e n c e , a r i t h m e t i c computation, a r i t h m e t i c problem s o l v i n g and reading to s i x t h graders. He wanted to discover the s k i l l s and i n t e l l e c t u a l f a c t o r s p r i m a r i l y r e l a t e d to the a b i l i t y to solve word problems. Of the f i f t e e n v a r i a b l e s t e s t e d only three - computation, reading to note d e t a i l s , and fundamental knowledge i n a r i t h m e t i c — were i d e n t i f i e d as of major importance i n the a b i l i t y of problem s o l v i n g . D i f f e r e n c e s between matched p a i r s of h i g h and low achievers i n seventh grade problem s o l v i n g were i n v e s t i g a t e d by Alexander (1960). In favor of the high achievers d i f f e r e n c e s i n c l u d e d : 1) s p e c i f i c mental a b i l i t i e s , 2) q u a n t i t a t i v e s k i l l s , 3) general reading s k i l l s , 4) problem s o l v i n g reading s k i l l s , and 5) a b i l i t y to i n t e r p r e t q u a n t i t a t i v e f a c t s and r e l a t i o n s h i p s . Results of recent n a t i o n a l assessments (Carpenter et a l . 1975, 1976, 1980) showed that computation s k i l l s of t h i r t e e n year olds were almost as good as a d u l t s on the four operations i n v o l v i n g whole numbers. Although 9. the performance on one-step word problems was s l i g h t l y lower than per-formance on computation e x e r c i s e s r e q u i r i n g the same operations, the d i f f e r e n c e was l e s s than ten per cent. Carpenter et a l . (1980) s t a t e d that i t appeared that i f c h i l d r e n had not mastered a computational a l g o r i t h m they could not generate s o l u t i o n s f o r even simple problems th a t might have been solved i n t u i t i v e l y or by using p h y s i c a l representa-t i v e s . Reading A b i l i t y - Measures of general reading a b i l i t i e s were found to c o r r e l a t e p o s i t i v e l y w i t h scores on a r i t h m e t i c and mathematics t e s t s (Balow 1964, Chase 1960, C o f f i n g 1941, and Muscio 1962). C o f f i n g (1941) administered standardized t e s t s of paragraph meaning and a r i t h m e t i c reasoning to students i n grade f o u r through e i g h t . A p o s i t i v e c o r r e l a t i o n was found between s i l e n t reading and a b i l i t y to solve reasoning problems. Balow (1964) found that f o r any given l e v e l of computation a b i l i t y , problem s o l v i n g increased as reading a b i l i t y increased and that f o r any given l e v e l of reading a b i l i t y , problem s o l v i n g increased as computation a b i l i t y i n c r e a s e d . Muscio (1962) examined the r e l a t i o n s h i p between s i x t h graders quan-t i t a t i v e understanding and c e r t a i n mental a b i l i t i e s , achievements, and a t t i t u d e s . The r e s u l t s i n d i c a t e d that achievement on the measure of q u a n t i t a t i v e understanding was c l o s e l y r e l a t e d to achievement on measures of general reading a b i l i t i e s , computation, reasoning and mathematical vocabulary. Some researchers regard reading not as a ge n e r a l i z e d a b i l i t y but as a composite of s p e c i f i c s k i l l s and ther e f o r e examined the i n f l u e n c e of various s p e c i f i c reading a b i l i t i e s on success i n problem s o l v i n g . Newcomb (1922), Stevenson (1924) and Washburne and Osburne (1926) i n v e s t i g a t e d the worth of systematic and l o g i c a l procedures i n s o l v i n g problems as compared to students' own methods. The three s t u d i e s had experimental groups r e c e i v e i n s t r u c t i o n i n s p e c i f i c s k i l l s of reading and a n a l y z i n g problems, vocabulary, and guides to reasoning. Newcomb and Stevenson found that the s p e c i f i c i n s t r u c t i o n increased seventh and eigh t h grade students' a b i l i t y to solve problems. Washburne and Osburne, however, concluded that students s o l v i n g problems using t h e i r own techniques were more s u c c e s s f u l at problem s o l v i n g than students using a s p e c i a l technique or f i n d i n g s i m i l a r i t i e s between d i f f i c u l t and easy problems. Monroe and Engelhart (1933) s t u d i e d the e f f e c t i v e n e s s of a program of systematic i n s t r u c t i o n i n the reading of v e r b a l problems. The experimental group r e c e i v e d p r a c t i c e i n reading of problems, restatement of problem, diagraming, accurate reading of f i g u r e s , and emphasis on terms i n d i c a t i n g the process to be used. The mean gains i n readingj computation and problem s o l v i n g were found not to be s i g n i f i c a n t - a f i n d i n g not i n accord w i t h the m a j o r i t y of the s t u d i e s . Faulk and Landry (1961) i n a s i m i l a r study t e s t e d twenty-two c l a s s e s of s i x t h grade students. Again the experimental group had i n s t r u c t i o n i n a systematic approach to problem s o l v i n g . Students s t u d i e d vocabulary, meanings of words, diagraming and e s t i m a t i n g . The r e s u l t s showed the gains i n achievement were s i g n i f i c a n t f o r t h i s group. C a l l and Wiggen (1966) compared e f f e c t s of a reading o r i e n t e d approach to a conventional approach to teaching problem s o l v i n g . The reading approach i n c l u d e d vocabulary, use of context to get the meaning and seeing r e l a t i o n s h i p s . Findings i n d i c a t e d that the reading o r i e n t e d approach was more s u c c e s s f u l i n improving the a b i l i t y of problem s o l v i n g . Treacy (1944) stud i e d eighteen f a c t o r s i n the areas of a r i t h m e t i c problem s o l v i n g , mental a b i l i t y , and reading a b i l i t y i n a study of seventh graders. When mental age was h e l d constant, the r e s u l t s showed good achievers i n problem s o l v i n g s u p e r i o r to poor achievers i n four reading s k i l l s a l l a ssociated w i t h vocabulary, q u a n t i t a t i v e r e l a t i o n -s h i p s , vocabulary i n context, vocabulary ( i . e . i s o l a t e d words), and a r i t h m e t i c vocabulary. Corle (1958) t e s t e d seventy-four p u p i l s on e i g h t word problems and concluded that the a b i l i t y of problem s o l v i n g was greater i f students understood the meaning of the problem, used word and number c l u e s , and understood the meaning of the words. Henney (1971) wrote that to s o l v e problems the students must be able to 1) read (recognize, comprehend) the words of the problem, 2) v i s u a l i z e the s i t u a t i o n , 3) recognize the question asked, 4) note impor-ta n t f a c t s given, 5) i n f e r the r e l a t i o n s h i p of the f a c t s , and 6) i n t e r -p ret the s o l u t i o n obtained i n terms of the question asked. Henney s t a t e d that although there i s a l a c k of agreement on which f a c t o r s have the gr e a t e s t i n f l u e n c e , there i s consensus that c h i l d r e n must be able to recognize words and comprehend thought u n i t s i n the problem, l o g i c a l l y i n t e r p r e t the problem s i t u a t i o n , and organize the i n f o r m a t i o n i n such a way as to lead to an answer to the question. I t was found that the s p e c i f i c reading a b i l i t i e s were no more h i g h l y c o r r e l a t e d than general reading a b i l i t y w i t h a r i t h m e t i c problem s o l v i n g . Mathematical Vocabulary and Syntax - Research done by Hansen (1944) and Johnson (1949) i n d i c a t e d that knowledge of vocabulary was important i n s o l v i n g mathematical problems f o r students i n grades s i x and e i g h t . A study which examined seventh grade, students, as p r e v i o u s l y s t a t e d , found that four reading s k i l l s that a f f e c t e d achievement i n problem s o l v i n g were a l l a s s o c i a t e d w i t h vocabulary (Treacy 1944). To elaborate on e a r l i e r s t u d i e s of mathematical vocabulary, Olander and Ehmer (1971) and L i n v i l l e (1976) a l s o examined vocabulary i n math-ematics. Olander and Ehmer tested 1200 students i n grades f o u r , f i v e and six. on one hundred vocabulary items and compared them w i t h students' performances i n 1930. I t was found that grades four and f i v e w i t h means of 49 and 57 performed b e t t e r than the 1930 students of the same grades but the s i x t h graders i n 1930 scored higher than those of t h i s study. L i n v i l l e (1976) tes t e d f o u r t h grade students on four t e s t s i n v o l v i n g l e v e l s of d i f f i c u l t y i n syntax and vocabulary. The study showed there were s i g n i f i c a n t main e f f e c t s i n favor of easy syntax and easy vocabulary t e s t s w i t h vocabulary being more c r u c i a l . I t was found that students scored higher on word problems w i t h easy vocabulary than d i f f i c u l t vocabulary across l e v e l s of syntax and higher on problems w i t h easy syntax than d i f f i c u l t syntax across l e v e l s of vocabulary. Another approach to studying the r e l a t i o n s h i p of knowledge of vocabulary to achievement i n mathematics i s to determine whether s p e c i f i c t r a i n i n g i n vocabulary has an e f f e c t on mathematical performances. Johnson (1944) found that f o r seventh, graders s p e c i f i c i n s t r u c t i o n i n vocabulary l e d to s i g n i f i c a n t growth i n 1) the knowledge of the s p e c i f i c terms and 2) the s o l u t i o n of numerical problems i n v o l v i n g the use of the terms. Van der Linde (1964) tes t e d f i f t h graders a f t e r i n s t r u c t i o n on 242 terms (.8 terms studied each week) . R e s ults showed a s i g n i f i c a n t gain f o r the experimental group on a r i t h m e t i c concepts and problem s o l v i n g . In agreement w i t h previous f i n d i n g s , Lyda and Duncan (1967) i n s t r u c -ted students i n grade two on 178 terms from f i v e c a t e g o r i e s - a r i t h m e t i c , 13 time, measurement, q u a n t i t y , and geometry. I t was found that a d i r e c t study of q u a n t i t a t i v e vocabulary c o n t r i b u t e s s i g n i f i c a n t l y to growth i n problem s o l v i n g . Research that concerns the r e l a t i o n s h i p of vocabulary and syntax to ease of reading has inv o l v e d the a p p l i c a t i o n of r e a d a b i l i t y formulas to mathematical t e x t s and problems. Heddens and Smith (1964) examined f i v e textbooks used f o r elementary mathematics using the Spache formula f o r grades one to three and the D a l e - C h a l l formula f o r grades four to s i x . I t was found that 1) the r e a d a b i l i t y l e v e l of the s e l e c t e d t e x t s seemed to be g e n e r a l l y above the assigned grade l e v e l , 2) there was consid e r a b l e v a r i a t i o n of r e a d a b i l i t y l e v e l s among the textbooks con-sidered and 3) the v a r i a t i o n w i t h i n each book i n d i c a t e d that some por-t i o n s of the t e x t s should be comprehended by most students, w h i l e other p o r t i o n s of the same te x t were w r i t t e n on a r e l a t i v e l y more d i f f i c u l t l e v e l . Smith (1971) a l s o a p p l i e d the D a l e - C h a l l formula to the word prob-lems i n s i x t h grade t e x t s and three mathematical achievement t e s t s . I t was found that the average r e a d a b i l i t y of the problems f e l l w i t h i n bounds f o r the grade l e v e l although the r e s u l t s i n d i c a t e d wide v a r i a t i o n from problem to problem. Smith s t a t e d , "In problem s o l v i n g s i t u a t i o n s that i n v o l v e v e r b a l m a t e r i a l s , a person i s o r d i n a r i l y expected to read a statement, analyze the data, use computational s k i l l s , and a r r i v e at the c o r r e c t answer. I f a c h i l d cannot read the statement, then h i s a b i l i t i e s to t h i n k , analyze and compute are hampered . . . I t i s a l s o p o s s i b l e the achievement t e s t scores i n problem s o l v i n g i n mathematics sometimes r e f l e c t a c h i l d ' s l i m i t a t i o n i n reading r a t h e r than h i s mathematical performance, (p.559)" Language Factors - I n v e s t i g a t o r s have r e f e r r e d to v a r i o u s aspects of the i n t e r a c t i o n between language development and the growth of math-ematical understanding. Kramer (1933) studied four f a c t o r s : 1) sentence form ( d e c l a r a t i v e versus complex i n t e r r o g a t i v e ) ; 2) s t y l e (language d e t a i l versus concise, compressed language); 3) vocabulary ( f a m i l i a r versus u n f a m i l i a r ) ; and 4) problem s i t u a t i o n ( i n t e r e s t i n g versus u n i n t e r -e s t i n g ) . She t e s t e d 237 s i x t h grade students. Regarding sentence form the d i f f e r e n c e s were small but i n favor of the complex i n t e r r o g a t i v e form. Concise compressed language and f a m i l i a r vocabulary r e s u l t e d i n greater success i n problem s o l v i n g . The d i f f e r e n c e s i n success w i t h i n t e r e s t i n g and u n i n t e r e s t i n g m a t e r i a l were n e g l i g i b l e . Kramer stat e d that cue words, r a t h e r than f a c t s and requirements of the problem were a f a c t o r i n determining the s e l e c t i o n of process. The numbers, i n par-t i c u l a r r e l a t i o n s h i p s and p a t t e r n s , appeared to serve as cues to mathematical operations. Expanding on the i n v e s t i g a t i o n of Kramer, with, regards to sentence form and s t y l e are s t u d i e s by S c h e l l and Burns (1965), S t e f f e (1968) and U n d e r h i l l (1977). S c h e l l and Burns examined second graders on three types of s u b t r a c t i o n problems: 1) "take away" - a q u a n t i t y i s removed (transformational) and what remains i s to be i d e n t i f i e d ; 2) "how many more are needed"; 3) "comparison or d i f f e r e n c e " - q u a n t i t i e s matched and excess of one over the other i s i d e n t i f i e d . Twelve questions of each type were administered. R e s u l t s showed that the students performed best on the "take-away" problems, followed by "how many more are needed" and "com-pa r i s o n " i n that order. S t e f f e (.1968) studied the f a c t o r s of conservation on number capa-b i l i t i e s and performances on t r a n s f o r m a t i o n a l and non-transformational a d d i t i o n word problems. R e s u l t s i n d i c a t e d that high conservers performed b e t t e r than low conservers and that t r a n s f o r m a t i o n a l problems were e a s i e r to s o l v e than non-transformational problems. U n d e r h i l l (1977) elaborated on S t e f f e ' s r e s u l t s by examining k i n d e r -garten and f i r s t graders i n a d d i t i o n and s u b t r a c t i o n problems w i t h the use of mani p u l a t i v e s . He found that a d d i t i o n ( t r a n s f o r m a t i o n a l and non-transformational) and t r a n s f o r m a t i o n a l "take away" s u b t r a c t i o n (how many l e f t ? ) were e a s i e r to solve than the non-transformational comparison s u b t r a c t i o n (how many more?) and t r a n s f o r m a t i o n a l a d d i t i v e s u b t r a c t i o n C3 + • = 5) . Rose and Rose (1961) stated that childhood t r a i n i n g i n p r e c i s e language i s e s s e n t i a l f o r performing w e l l i n mathematics. Piaget (1954) and Bruner (1966) a l s o s t r e s s e d the importance to mathematical a b i l i t y of language a b i l i t y . Growth i n l i n g u i s t i c a b i l i t y according to Piaget f o l l o w s the development of concrete o p e r a t i o n a l thought ra t h e r than preceding i t , although language i s important i n the completion of such c o g n i t i v e s t r u c t u r e s . In co n t r a s t Bruner et a l . maintain that the development of adequate terminology i s e s s e n t i a l to c o g n i t i v e growth. • Using P i a g e t ' s c l a s s i f i c a t i o n s of concrete and formal operations, Days, Wheatley and Kulm (1979) examined the processes used by f i f t y -e i ght students i n grade e i g h t . Two types of word problems were used: 1) simple s t r u c t u r e and 2) complex s t r u c t u r e . No d i f f e r e n c e s were found between the concrete and formal o p e r a t i o n a l students i n the use of understanding, r e p r e s e n t a t i o n a l , and r e c a l l processes. Use of pro-d u c t i o n and e v a l u a t i o n processes d i d d i f f e r between the two groups. Problem s t r u c t u r e played a bigger r o l e i n determining process use by formal o p e r a t i o n a l students than by students at the concrete l e v e l . The research on a r i t h m e t i c problem s o l v i n g shows that a knowledge of a r i t h m e t i c vocabulary i s p e r t i n e n t to achievement i n s o l v i n g word problems. Reading comprehension and a r i t h m e t i c achievement tend to be p o s i t i v e l y r e l a t e d . Almost without exception i n s t r u c t i o n i n vocabulary and/or reading s k i l l s i n a r i t h m e t i c paid o f f i n terms of higher achieve-ment . Research, with. Hearing Impaired Students - Although performances on standardized achievement t e s t s of deaf students i n d i c a t e a r e l a t i v e l y greater a b i l i t y i n mathematics than other academic areas, i t i s s t i l l below the norms e s t a b l i s h e d f o r hearing students. Goetzinger and Rousey (1959) tes t e d 101 deaf students between the ages of fourteen and twenty-one on the performance subtests of the WAIS and Stanford B i n e t . R e s u l t s showed a mean grade l e v e l score of 6.5 In computation and 6.0 f o r a r i t h -metic reasoning. These scores are approximately equal to those of an average hearing ten year o l d . Trybus. and Karehmer. (1977) c o l l e c t e d n a t i o n a l data on achievement scores and concluded that the mean growth l i n e f o r hearing students i n mathematics, was equivalent to the n i n e t i e t h p e r c e n t i l e f o r the hearing impaired. In other words only ten per cent of the hearing impaired students o b t a i n scores equal to that of the average hearing student of the same age. R e s u l t s showed that the average twenty year o l d hearing impaired person scored j u s t below the eighth grade l e v e l (equal to that of an average hearing student at t h i r t e e n y e a r s ) . Trybus and Karchmer stated " . . . a group of hearing impaired c h i l d r e n whose reading com-prehension scores at a t h i r d grade l e v e l w i l l t y p i c a l l y o b t a i n scores about one grade l e v e l lower than t h i r d on the vocabulary t e s t , and one to two grades higher on the mathematics computation (p.68)." 17 Academic achievement t e s t data were c o l l e c t e d by Rogers et a l . (1978) on 383 hearing impaired students aged f i v e to seventeen years and ol d e r . The r e s u l t s showed that f o r computation, student performance increased s i g n i f i c a n t l y with, i n c r e a s i n g age. The scaled scores ranged from 131.1 to 178.1 which, corresponds to raw scores obtained by normal hearing students of t h i r d to seventh grade (ages eight to twelve y e a r s ) . Messerly and Aram (1980) examined Stanford Achievement Test scores of seventeen year o l d deaf students- of hearing parents and deaf students of deaf parents. Although the sample was small (n.= 16) and not rep r e -s e n t a t i v e of the n a t i o n a l group, the r e s u l t s showed the mean scores of both, mathematics computation and mathematics a p p l i c a t i o n s were below the eigh t h grade l e v e l . The f i n d i n g s again were i n accord w i t h previous s t u d i e s . Achievement scores on standardized t e s t s i n d i c a t e that the average deaf c h i l d achieves c l o s e to normal l e v e l s i n a r i t h m e t i c computation. Their a r i t h m e t i c reasoning s k i l l s however are g e n e r a l l y w e l l below normal. This i s b e l i e v e d to be a r e s u l t p r i m a r i l y of low l e v e l s of reading and language (Hargis 1969, Suppes. 1974). Cooper and Rosenstein (1966) under-took an extensive survey examining the language s k i l l s of reading and w r i t t e n expression i n deaf c h i l d r e n . The researchers reported that these students were s i g n i f i c a n t l y retarded i n t h e i r achievement t e s t scores i n terms of both, reading and w r i t t e n expression. In a study of deaf students aged 10.5 and 16.5 years, F u r t h (1966a) found reading grade equivalents of 2.7 and 3.5 r e s p e c t i v e l y . He concluded that the measurement of reading d i s a b i l i t y presupposes l i n g u i s t i c com-petence which i s not present i n the deaf. The low reading l e v e l of the deaf does not c o n s t i t u t e a reading d e f i c i e n c y but r a t h e r a l i n g u i s t i c Incompetence. Many other s t u d i e s using r e s u l t s from standardized achievement t e s t s a l s o obtained low reading l e v e l s f o r deaf students (Goetzinger and Rousey 1959, Vernon 1970, 1971, Conrad 1977, Karchmer, Milone and Wolk 1979, and Messerly and Aram 1980). Vernon tes t e d deaf students aged f i f t e e n years on the Stanford Achievement Test and found reading scores l e s s than the f i f t h grade. Conrad administered t e s t s to f i f t e e n and s i x t e e n year o l d students and discovered reading ages approximately equal to the average hearing nine year olds. His r e s u l t s a l s o showed that reading a b i l i t y depended s i g n i f i c a n t l y on degree of hearing l o s s . In agreement w i t h these f i n d i n g s was the i n v e s t i g a t i o n of Karchmer, Milone and Wolk (1979). Seven thousand students were t e s t e d on the Stanford Achievement Test, S p e c i a l E d i t i o n f o r Hearing Impaired Students (SAT-HI). Reading comprehension scores d i f f e r e d s i g n i f i c a n t l y w i t h degree of hearing l o s s . With the i n c r e a s e of s e v e r i t y of hearing l o s s , student performances d e c l i n e d . In a recent n a t i o n a l survey (Trybus and Karchmer 1977) data showed that i n reading, the average hearing-impaired student aged twenty and over scored at the grade equivalent of 4.5 (below or b a r e l y at news-paper l i t e r a c y ) . At best only ten per cent of the eighteen year o l d s were reading at or above the eighth, grade l e v e l . An average hearing student reaches t h i s l e v e l before age fourteen. Rogers et a l . (1978) reported reading comprehension scores f o r hearing impaired students on the SAT-HI t e s t . Grade l e v e l mean per-formances ranged from 1.5 f o r f i v e to eight year olds to 4.9 f o r students seventeen years and o l d e r . Student performance i n reading comprehension Increased w i t h age but appeared to have a d e c l i n e w i t h increase of hearing l o s s . Demographic v a r i a b l e s which s i g n i f i c a n t l y a f f e c t e d reading comprehension, adjusted f o r age and s e v e r i t y of hearing l o s s were found as f o l l o w s : 1) g i r l s d i d b e t t e r than boys; 2) the students u s i n g personal hearing a i d s performed b e t t e r than the group not using a i d s ; 3) the group w i t h a d d i t i o n a l e d ucational handicapping c o n d i t i o n s scored lower than the group without these handicaps; 4) the o r a l communication students outperformed the t o t a l communication group; 5) the students i n s p e c i a l c l a s s e s and schools performed below the students e n r o l l e d i n r e g u l a r school. In t h e i r examination of academic achievement of hearing impaired students Rogers and Clarke (1980) found three personal v a r i a b l e s (age, s e v e r i t y of l o s s , and a d d i t i o n a l educational handicaps) and one manipulable v a r i a b l e (educati o n a l s e t t i n g ) were s i g n i f i c a n t p r e d i c t o r s of reading comprehension. For mathematics computation and a p p l i c a t i o n s only personal v a r i a b l e s (age and a d d i t i o n a l e d ucational handicaps) con-t r i b u t e d s i g n i f i c a n t l y to the r e g r e s s i o n equation. Hamp (1972) i n a study of 367 deaf and p a r t i a l l y hearing c h i l d r e n ages nine to f i f t e e n years stated that reading performances improved only 0.8 of a grade between ages eleven to s i x t e e n years. I t was concluded that the e f f e c t s of age and i n t e l l i g e n c e were of greater s i g n i f i c a n c e than the degree of hearing impairment. In a survey of c o g n i t i o n of deaf c h i l d r e n , Suppes (.1974) wrote that the c o g n i t i v e performances of deaf c h i l d r e n i s as: good as that of normal hearing c h i l d r e n when the c o g n i t i v e task does not d i r e c t l y i n v o l v e v e r b a l s k i l l s . In almost a l l surveys examining achievement scores, r e s u l t s showed the deaf to be s e v e r e l y retarded i n t h e i r language achievement (Cooper and Rosenstein 1966) . Fu r t h (.1966) has concluded that language and words behind the math concepts are not always c l e a r and may run counter to conventional usage i n the mind of a young c h i l d . 2.0 According to Hargis (1969) " I t i s apparent that the deaf c h i l d ' s d e f i c i e n c y i n a r i t h m e t i c reasoning i s caused p r i m a r i l y by h i s low l e v e l of language achievement. He may master the mechanical aspect of a r i t h -metic (computational s k i l l s ) ; but when an a r i t h m e t i c problem r e q u i r e s understanding of the language the deaf c h i l d o f t e n f a i l s (p.766)." Suppes (1974) was i n agreement when he wrote that the competence i h a standard n a t u r a l language i s the outstanding defect and problem of deaf persons. I I I . Statement of Hypotheses 1) One-step word problems r e q u i r i n g the operation of a d d i t i o n w i l l be e a s i e s t f o r deaf students to s o l v e followed by problems r e q u i r i n g the operations of s u b t r a c t i o n , m u l t i p l i c a t i o n or d i v i s i o n i n that order. 2) The a b i l i t y to solve a r i t h m e t i c word problems w i l l i ncrease as age i n c r e a s e s . 3) The a b i l i t y to s o l v e word problems w i l l be s i g n i f i c a n t l y d i f f e r e n t between deaf males and females. 4) The frequency of a question type as found i n elementary math-ematics t e x t s w i l l a f f e c t the a b i l i t y to s o l v e word problems f o r hearing impaired students i n that the more f r e q u e n t l y o c c u r r i n g question type w i l l be more e a s i l y solved. CHAPTER I I I METHOD I. Sample P o p u l a t i o n For the purpose of t h i s study the subjects were s e l e c t e d from a popu l a t i o n of students attending J e r i c h o H i l l School f o r the Deaf and who met the f o l l o w i n g c r i t e r i a : a) had reached t h e i r e ighth but not twentieth b i r t h d a y and were attending a c l a s s f o r the hearing Impaired under the j u r i s d i c t i o n of J e r i c h o H i l l School on May 1 s t , 1980. This included fourteen c l a s s e s on campus and eight s e l f - c o n t a i n e d c l a s s e s s i t u a t e d i n r e g u l a r schools. Throughout the study the f o l l o w i n g age groups were used: 8-11 years 12 & 13 years 14 & 15 years 16+ years b) had a profound hearing l o s s from b i r t h or. p r i o r to the age of two years. c) had been judged by school personnel as having no d i s a b i l i t y which would prevent task completion, and i n the p r o f e s s i o n a l o p i n i o n of the teacher had no reason to be excluded from the study. A l l the students who met the above c r i t e r i a were administered the computation task (see Instruments). Of the 116 students who attempted the f i r s t t e s t , f i f t e e n were deleted from the study f o r not meeting the pass requirement ( I . e . at l e a s t four of the s i x questions c o r r e c t i n each of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n q u e s t i o n s ) . 21 Another eleven students had to be dropped from the study because they did not complete the computation and/or word problem tasks. The f i n a l study sample was comprised of ninety s u b j e c t s . Table 1 shows the cr o s s -t a b u l a t i o n of the four age groups w i t h gender. TABLE 1 NUMBER AND PERCENTAGE DISTRIBUTION OF GENDER WITHIN AGE Gender Age i n Years Male Female T o t a l f % f % f % 8-11 13 68, .4 6 31 .6 19 100 .0 12&13 8 50, .0 8 50 .0 16 100 .0 14&15 13 50, .0 13 50 .0 26 100 .0 16+ 19 65, .5 10 34 .5 29 100 .0 T o t a l 53 58, .9 37 41 .1 90 100 .0 I I . Instruments Computational Task - I n t h i s study, the s o l u t i o n methods used by profoundly hearing impaired students due to language s t r u c t u r e r a t h e r than computation were examined. To ensure that a computation v a r i a b l e would not a f f e c t the s o l v i n g of the word problems a computational task was administered f i r s t . This task c o n s i s t e d of three pages - the f i r s t page r e q u i r e d i n f o r m a t i o n from the students such as name, date of b i r t h , and teacher's name. On the same page four examples, one f o r each type of o p e r a t i o n , were given to introduce the students to the type of questions, the task format, and the method of response. (See Appendix A). The f o l l o w i n g two pages comprised the a c t u a l computation tasks. 23 These c o n s i s t e d of twenty-four questions - s i x on each mathematical operation ( a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n ) . S i x p a i r s of numbers were used i n the twenty-four questions: 6 and 2; 8 and 2; 10 and 2; 3 and 3; 6 and 3; and 9 and 3. These p a r t i c u l a r p a i r s were chosen as these were the only combination of d i g i t s between 1 and 10 which would r e s u l t i n a d i f f e r e n t answer f o r each of the four operations. e.g. 6 and 2 6 + 2 = 8 6 - 2 = 4 6 X 2 = 12 6 * 2 = 3 Each p a i r of numbers was a p p l i e d to each of the four operations, r e s u l t i n g i n the twenty-four questions. These questions were randomly assigned a p o s i t i o n from one to twenty-four. (See Appendix A). The mathematical sentences on the computation task were w r i t t e n h o r i z o n t a l l y , the reason being that although a d d i t i o n , s u b t r a c t i o n , and m u l t i p l i c a t i o n can be represented v e r t i c a l l y , d i v i s i o n i s not w r i t t e n i n t h i s manner i n the textbooks. The l a r g e r number of the number p a i r i n a l l cases, was w r i t t e n f i r s t to ensure that the questions would give no i n d i c a t i o n as to which operation was t o be used. For each question a l l p o s s i b l e answers were l i s t e d h o r i z o n t a l l y below the sentence from smallest to l a r g e s t . The student was asked to c i r c l e the c o r r e c t answer. e.g. A) 6 + 2 = 3 4 8 12 The twenty-four items were l e t t e r e d (A t o X) to ensure that the students would not get confused as to whether a number was p a r t of the question or not. The operations were assigned as f o l l o w s : Page 2 a) a d d i t i o n b) m u l t i p l i c a t i o n c) m u l t i p l i c a t i o n d) d i v i s i o n e) a d d i t i o n f) s u b t r a c t i o n g) d i v i s i o n h) m u l t i p l i c a t i o n Page 3 m) a d d i t i o n o) a d d i t i o n q) a d d i t i o n s) s u b t r a c t i o n n) d i v i s i o n p) s u b t r a c t i o n r) s u b t r a c t i o n t) a d d i t i o n i ) m u l t i p l i c a t i o n j ) m u l t i p l i c a t i o n u) m u l t i p l i c a t i o n v) d i v i s i o n k) d i v i s i o n 1) d i v i s i o n w) s u b t r a c t i o n x) s u b t r a c t i o n Word Problem Task - The second instrument used i n t h i s study was the one-step word problem task (see Appendix A). The " I n v e s t i g a t i n g School Mathematics" t e x t s , grades two through s i x i n c l u s i v e , were examined to determine the most popular forms of questions used f o r each of the four operations. The two most common forms of each operation were then sel e c t e d f o r t h i s study (Type A and Type B, see below). To ensure that a language v a r i a b l e would not a f f e c t the performance of the students, the vocabulary used was r e s t r i c t e d to "John," " B i l l , " "Susan," " t o y s , " "boxes," "had" (as main v e r b ) , "found," " l o s t , " "sawj" and "put" i n the st o r y p a r t of the problems. To c o n t r o l the syntax of the problems, a l l statements p r i o r to the a c t u a l question were w r i t t e n as simple, a c t i v e , d e c l a r a t i v e s t a t e -ments i n the form of "John had 6 boxes." The one exception to t h i s p a t t e r n i s the wording of the Type B m u l t i p l i c a t i o n problems. I t was worded as f o l l o w s : B i l l had _ Susan saw boxes. times as many boxes, How many boxes d i d Susan see? A l l the questions were w r i t t e n as they appeared i n the textbooks w i t h the c o n t r o l l e d vocabulary i n s e r t e d . Type A problems were the most common type of question (found i n the te x t s ) w i t h Type B problems next i n frequency. Examples, f o r both types of problems f o r each operation are: A d d i t i o n John had B i l l had toys. toys, How many toys i n a l l ? S u b t r a c t i o n B Susan had Susan found How many boxes were there? boxes. boxes. Susan had John had toys, boxes. How many more toys than boxes? M u l t i p l i c a t i o n A John had John had boxes. toys i n each box. How many toys i n a l l ? D i v i s i o n John had John l o s t How many toys l e f t ? t oys. toys, boxes. _ times as many B i l l had Susan saw boxes. How many boxes d i d Susan see? B i l l had B i l l had toys. boxes, How many toys i n each box? John had B i l l had How many boxes? toys. toys i n each box. Li k e the computation task, t h i s second task had twenty-four ques-t i o n s . These questions used e x a c t l y the same number p a i r and ope r a t i o n as the corresponding question on the computation task. For example, item A on both the computation task and word problem task was the a d d i -t i o n of s i x and two. Because there were two types (A and B) of question f o r each operation, these were randomly assigned - r e s u l t i n g i n three items f o r each type of question f o r each operation (e.g. three Type A a d d i t i o n problems, three Type B a d d i t i o n problems). The type A and Type B questions were arranged as f o l l o w s : Page 1 A) Type A a d d i t i o n B) Type A m u l t i p l i c a t i o n c ) Type A m u l t i p l i c a t i o n D) Type B d i v i s i o n E) Type B a d d i t i o n F) Type B s u b t r a c t i o n G) Type A d i v i s i o n H) Type B m u l t i p l i c a t i o n Page 2 I) Type A m u l t i p l i c a t i o n J) Type B m u l t i p l i c a t i o n K) Type B d i v i s i o n L) Type B d i v i s i o n M) Type B a d d i t i o n N) Type A d i v i s i o n 0) Type B a d d i t i o n P) Type A s u b t r a c t i o n Page 3 Q) Type A a d d i t i o n R) Type B s u b t r a c t i o n s ) Type A s u b t r a c t i o n T) Type A a d d i t i o n u ) Type B m u l t i p l i c a t i o n V) Type A d i v i s i o n W) Type B s u b t r a c t i o n v. X) Type A s u b t r a c t i o n Again the l a r g e r d i g i t of the number p a i r was given f i r s t to ensure that no i n d i c a t i o n as to type of op e r a t i o n could be derived from the order of the number. Following the p a t t e r n of the computation task, the answers were l i s t e d h o r i z o n t a l l y below the word problems and the student was asked to c i r c l e the c o r r e c t answer: e.g. m) Susan had 9 boxes. Susan found 3 boxes. How many boxes were there? 3 6 12 27 I I I . Scoring Computation Task - There was a t o t a l of twenty-four questions w i t h s i x items f o r each operation of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n . _The students were required to c o r r e c t l y answer at l e a s t four of the s i x questions f o r each operation. Of the n i n e t y students who met t h i s r e q u i r e -ment seventy-four had no e r r o r s on the e n t i r e task. Nine students had one e r r o r and seven students had two or more e r r o r s . The computational e r r o r s made, appeared to be random - that i s no c h i l d made c o n s i s t e n t e r r o r s . Word Problem Task - Again there were twenty-four questions w i t h s i x items f o r each operation. As w e l l there were two types of question f o r each op e r a t i o n . I f an item on the computation task was a d d i t i o n , the item on the word task was a l s o a d d i t i o n (Type A or Type B). As the type of question as w e l l as the operations were being examined, the students r e c e i v e d a score out of three f o r each type of question f o r each operation. For example, on Type A a d d i t i o n questions (A, Q and T) a student could o b t a i n a score of 3, 2, 1, or 0 ( i n d i c a t i n g . t h e number c o r -r e c t ) . Each student r e c e i v e d eight such scores - two f o r each i n a d d i t i o n (Type A and Type B), s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n . There was a t o t a l of 270 items (90 students x 3 questions) f o r each type of question f o r each operation. Each operation had 540 items w i t h a t o t a l of 2,160 items on the word problem task f o r the e n t i r e sample popu-l a t i o n . IV. Procedure The procedure of t h i s study can be d i v i d e d i n t o two p a r t s - the compu-t a t i o n task and the word problem task. Teachers were approached by the i n v e s t i g a t o r to i n d i c a t e which of t h e i r students f i t t e d the study c r i t e r i a and could perform the four operations on number f a c t s up to t h i r t y . These students were then given the f i r s t task -the computation task which was administered by the teachers to t h e i r own p u p i l s . The teachers were.masked to help the students complete the f i r s t 28 .page to ensure the students understood what was r e q u i r e d f o r the task. I f i n the e s t i m a t i o n of the teacher, the students needed more examples, the teachers were asked to provide a d d i t i o n a l questions s i m i l a r to the ones on the f i r s t page. The r e s u l t s were scored by the w r i t e r and v e r i f i e d by d i s -c u s s i o n w i t h the c l a s s teacher. The students who met the c r i t e r i a of c o r r e c t l y answering four of the s i x questions f o r each operation were given the word problem task approx-imately two weeks l a t e r . To meet the school's convenience, t h i s second task was administered on three separate days by the i n v e s t i g a t o r , to three d i f f e r e n t groups. The students were grouped f o r t h i s task according to the l e v e l of t h e i r c l a s s : primary, intermediate, or s e n i o r . The s c o r i n g f o r t h i s task was a l s o done by the w r i t e r . V. Data A n a l y s i s The data on the students along w i t h the r e s u l t s of the word problem task were then keypunched onto computer cards. Hypotheses 1, 2, 3, and 4 were t e s t e d using a 4 x 2 x 2 (age-by-gender-by-type) f i x e d e f f e c t s design, w i t h repeated measures on the t h i r d f a c t o r . The problems (corresponding to the four operations were examined w i t h respect to age, gender, and type of question (A or B). The computer program used was BMD P2V-Analysis of v a r -iance and covariance w i t h repeated measures. As type of questions was the only s i g n i f i c a n t source of v a r i a n c e iden-i t i f i e d i n the above a n a l y s i s (see Chapter 4 ) , age and gender were c o l l a p s e d and an a n a l y s i s of type of question by operation was completed by using a repeated measures a n a l y s i s . As w e l l , a q u a l i t a t i v e e r r o r a n a l y s i s was c a r r i e d out f o r the operations s e p a r a t e l y and combined. That i s , the e r r o r s were examined f o r each operation and then the t o t a l number of e r r o r s were studied . CHAPTER IV RESULTS AND DISCUSSION The analyses were conducted i n three phases as s t a t e d p r e v i o u s l y . F i r s t , the word problems corresponding to the four operations of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n were examined separately w i t h respect to age, gender and type of question (A or B). As w i l l be shown, the only s i g n i f i c a n t source of v a r i a n c e was type. Therefore i t was decided to c o l l a p s e age and gender, thereby a l l o w i n g an a n a l y s i s of type of question by operation (repeated measures ANOVA). F i n a l l y the d i s t r i -b u t i o n of e r r o r scores was examined. The r e s u l t s of the analyses described i n the l a s t s e c t i o n of the previous chapter are presented i n t h i s chapter. F i r s t , data are present s e p a r a t e l y f o r each operation i n three t a b l e s . The f i r s t t a b l e summarizes the s i g n i f i c a n c e t e s t f o r the d i f f e r e n c e s f o r type of question, age and gender. The second t a b l e contains the means and standard d e v i a t i o n s of both types of question f o r each age group and f o r the t o t a l sample. E r r o r a n a l y s i s f o r the operation i s shown i n the t h i r d t a b l e . A d d i t i o n - There were no s i g n i f i c a n t d i f f e r e n c e s f o r age or gender on a d d i t i o n word problems (see Table I I ) . However type of question was s i g n i -f i c a n t (F = 10.54, df = 1, 82; p<.01). The o v e r a l l mean f o r Type A ques-t i o n s i n a d d i t i o n was 2.63 ( p o s s i b l e t o t a l of 3.00) (see Table I I I ) . For Type B questions i n a d d i t i o n the o v e r a l l mean was 2.27 (see Table I I I ) . The o v e r a l l , mean of Type A and B questions f o r the e n t i r e 29 TABLE I I : Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question A d d i t i o n Source of Variance df Mean Square F Between persons age 3 1.33463 1.32 gender 1 2.97090 2.94 age x gender 3 1.13084 1.12 e r r o r 82 1.01222 Wi t h i n persons type 1 4.32867 10.54** type x age 3 0.56130 1.37 type x gender 1 0.01484 0.04 type x age x gender 3 0.54960 1.34 e r r o r 82 0.41059 ** = p<.01 TABLE I I I Performance Means and Standard D e v i a t i o n s Type of Question by Age Group A d d i t i o n  Type of Question Age i n Years A B T o t a l X SD X SD X SD Males 2.00 0.82 1.69 1.03 1.85 0.98 8-11 Females 2.50 0.55 2.67 0.82 2.58 0.69 Both 2.16 0.73 2.00 0.96 2.08 0.85 Males 2.50 1.07 2.50 0.76 2.50 0.42 12 & 13 Females 3.00 0.00 2.63 0.52 2.81 0.26 Both 2.75 0.54 2.57 0.64 2.66 0.59 Males 2.92 0.28 2.31 1.18 2.62 0.73 14 & 15 Females 2.77 0.60 2.38 0.77 2.58 0.69 Both 2.85 0.44 2.34 0.98 2.60 0.71 Males 2.58 0.77 2.26 1.15 2.42 0.96 16+ Females 2.90 0.32 2.10 1.20 2.50 0.76 Both 2.69 0.61 2.21 1.17 2.45 0.94 O v e r a l l 2.63 0.57 2.27 0.98 2.45 0.78 sample was 2.45. Although the means f o r Type B questions i n a d d i t i o n were lower than the means f o r Type A questions f o r each age group, the d i f f e r e n c e was not s i g n i f i c a n t . The d i f f e r e n c e i n performance on the two types of a d d i t i o n questions might be explained by the d i f f e r e n c e i n wording of the two questions. The Type A a d d i t i o n question (How many toys i n a l l ? ) may have suggested to the students that a t o t a l was r e q u i r e d . The phrase " i n a l l " suggests a bigger answer. On the other hand, the Type B a d d i t i o n question (How many boxes were there?) may have given l i t t l e i n d i c a t i o n of the type of ope r a t i o n to be used s i n c e "how many" occurs i n a l l problems i n v o l v i n g a l l four processes. The t o t a l number of items f o r a d d i t i o n Type A was 270 (3 questions by 90 subjects) (see Table I V ) . The t o t a l number of e r r o r s was 33 (.12.2%). These e r r o r s were d i s t r i b u t e d as f o l l o w s : on 20 occasions (60.6%) m u l t i p l i c a t i o n was used in s t e a d of a d d i t i o n ; s u b t r a c t i o n 10 times (30.3%) and d i v i s i o n 3 times (.9.1%). The l a r g e percentage of m u l t i p l i -c a t i o n e r r o r s i s probably due to the f a c t that the question form f o r Type A a d d i t i o n and Type A m u l t i p l i c a t i o n are i d e n t i c a l (How many toys i n a l l ? ) . As sta t e d p r e v i o u s l y , t h i s question form probably i n d i c a t e d that a t o t a l was r e q u i r e d . Type B a d d i t i o n questions, as shown i n Table IV, were answered i n c o r r e c t l y 66 times (.24.4%). Of the t o t a l number of e r r o r s f o r t h i s type of question 38 items (57.6%) were m u l t i p l i e d ; 19 items (28.8%) were subtracted; and 9 items (.13.6%) were d i v i d e d . Although the percentage of e r r o r s made was doubled f o r Type B questions (24.4% compared w i t h 12.2%) the d i s t r i b u t i o n of e r r o r s was s i m i l a r . In both cases the m a j o r i t y of students who made e r r o r s g e n e r a l l y e l e c t e d to m u l t i p l y . TABLE IV Er r o r A n a l y s i s f o r A d d i t i o n Type of Question A B T o t a l s AgeoGroup T o t a l Composite* T o t a l Composite* T o t a l Composite* n (%) n (%) n (%) n (%) n (%) n (%) 6 (37.5)- 9 (47.4)- 15 (42.9)-8-11 16 (28.1) 9 (56.3)x 19 (33.3) 8 (42.1)x 35 (30.7) 17 (48.6)x 1 (6.2)- 2 (10.5)- 3 :(8.5)-0 (0.0)- 1 (14.3)- 1 (9.1)-12&13 4 (8.3) 3 (75.0)x 7 (14.6) 5 (71.4)x 11 (11.5) 8 (72.7)x 1 (25.0)- 1 (14.3)- 2 (18.2)-1 (25.0)- 5 (29.4)- 6 (28.6)-14&15 4 (5.1) 3 (75.0)x 17 (21.8) 10 (58.9)x 21 (13.5) 13 (61.9)x 0 (0.0)- 2 (11.7)- 2 (9.5)-3 (33.3)- 4 (17.4)- 7 (21.9)-16+ 9 (10.3) 5 (55.6)x 23 (26.4) 15 (65.2)x 32 (18.4) 20 (62.5)x 1 (11.1)- 4 (17.4)- 5 (15.6)-f 10 (30.3)- 19 (28.8)- 29 (29.3)-T o t a l 33 (12.2) 20 (60.6)x 66 (24.4) 38 (57.6)x 99 (18.3) 58 (58.6)x 3 (9.1)- 9 (13.6)- 12 (12.1)-* = The composite numbers and percentages add up to the t o t a l e r r o r s (100%) f o r each age group. S u b t r a c t i o n - The differences> f or age and gender again were not s i g n i f i c a n t (see Table V). The d i f f e r e n c e s i n type of question, however, again was s i g n i f i c a n t (F = 88.44; df = 1, 82; p<.01). The o v e r a l l mean performance f o r Type A s u b t r a c t i o n questions was 0.74 and f o r Type B questions was 2.23 (see Table V I ) . The Type A means ranged from 0.53 to 1.44. The means f o r Type B ranged from 1.37 to 2.72. For s u b t r a c t i o n , the i n t e r a c t i o n between type of question and age was a l s o s i g n i f i c a n t (F = 5.08; df = 3, 82; p<.01). Figure 1 shows the i n t e r a c t i o n between the two types of question. The age group means f o r Type A problems were 0.53 (8-11 y e a r s ) ; 1.44 (12 & 13 y e a r s ) ; 0.73 (14 & 15 y e a r s ) ; and 0.52 (16+ years); f o r Type B questions, the-means were 1. (.8-11 y e a r s ) ; 2.19 (12 & 13 y e a r s ) ; 2.35 (14 & 15 y e a r s ) ; and 2.72 (16+ y e a r s ) . Type A s u b t r a c t i o n questions were un i f o r m l y more d i f f i c u l t f o r a l l age groups. As would be expected, the mean performances f o r the Type B s u b t r a c t i o n questions improved as age increased. With the Type A questions although, the means of the two younger groups (8-11 and 12 & 13 years) were much the same as i n the Type B questions. The two ol d e r groups (14 & 15 and.16+ years) showed a decrease i n performance on Type A questions. In f a c t , t h e means f o r these two older groups were s i m i l a r to that of the 8 to 11 year o l d group. The Type A s u b t r a c t i o n questions (How many more toys than boxes?) proved to have the most d i f f i c u l t question form of the e n t i r e task w i t h 202 (.74.8%) of the questions answered i n c o r r e c t l y (see Table V I I ) . The Type A s u b t r a c t i o n question e r r o r s were d i s t r i b u t e d as f o l l o w s : 81 (40.1%) of the t o t a l 270 items were added; 64 (31.7%) were m u l t i p l i e d ; and 57 C28.2%) were d i v i d e d . The two younger age groups added r a t h e r than m u l t i p l i e d or d i v i d e d ; the 14 and 15 year o l d group evenly added or 34 TABLE V Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question S u b t r a c t i o n Source of Variance df Mean Square F Between persons age 3 2.66885 1.85 gender 1 3.10919 2.16 age x gender 3 2.89893 2.01 e r r o r 82 1.43970 W i t h i n persons type 1 81.38360 88.44** type x age 3 4.67552 5.08** type x gender 1 2.26199 2.46 type x age x gender 3 0.63295 0.69 e r r o r 82 0.92022 ** = p<.01 TABLE VI Performance Means and Standard Dev i a t i o n s Type of Question.-by Age Group S u b t r a c t i o n Type of Question Age i n Years .A B T o t a l X SD X SD X SD Males 0.38 0.65 0.84 0.98 0.61 0.82 8-11 Females 0.83 1.17 2.50 1.22 1.67 1.20 Both 0.53 0.81 1.37 1.06 0.95 0.94 Males 1.38 1.41 2.00 1.20 1.69 1.31 12 & 13 Females 1.50 1.60 2.38 1.06 1.94 1.33 Both 1.44 1.51 2.19 1.13 1.82 1.32 Males 0.85 1.28 2.46 1.13 1.65 1.21 14 & 15 Females 0.62 1.19 2.23 1.30 1.42 1.25 Both 0.73 1.23 2.35 1.22 1.54 1.23 Males 0.58 1.12 2.63 0.68 1.61 0.90 16+ Females 0.40 0.97 2.90 0.32 1.65 0.65 Both 0.52 1.07 2.72 0.56 1.62 0.82 O v e r a l l 0.74 1.14 2.23 0.96 1.49 1.05 A +• -A T y p e A -4. Type B Figure 1: I n t e r a c t i o n Graph, Type of Question by Age .TABLE VII -Error A n a l y s i s f o r S u b t r a c t i o n type of Question A B T o t a l s Age Group T o t a l Composite* T o t a l Composite* t o t a l Composite* n (%) n (%) n (%) n (%) n (%) n (%) 27 (57.4)+ 21 (67.7)+ 48 (61.5)+ 8-11 47 (82.5) 14 (29.8)x .. 31 (54.4) 4 (12.9)x 78 (68.4) 18 (23.1)x 6 (12.8)* 6 (19.4)* 12 (15.4)* 15 (60.0)+ 3 (23.1)+ 18 (47.4)+ 12&13 25 (52.1) 2 (8.0)x 13 (27.1) 0 (0.0)x 38 (39.6) 2 (5.2)x 8 (32.0)* 10 (76.9)* 18 (47.4)* 22 (37.3)+ 14 (82.4)+ 36 (47.4)+ 14&15 59 (75.6) 12 (20.3)x 17 (21.7) 2 (11.8)x 76 (48.7) 14 (18.4)x 25 (42.2)* 1 :(5.8)* 26 (34.2)* 2 (25.0)c|> 2 (2.5)<j> 16+ 71 (81.6) " 17 (23.9)+ 8 (9.2) 3 (37.5)+ 79 (45.4) 20 (25.3)+ 36 (50.8)x 1 (12.5)x 37 (46.9)x 18 (25.3)* 2 (25.0)* 20 (25.3)* 2 (2.9)(J) 2 ( 0 . 7 H T o t a l 202 (74.8) 81 (40.1)+ 69 (25.6) 41 (59.4)+ 271 (50.2) 122 (45.0)+ 64 (31.7)x 7 (10.1)x 71 (26.2)x 57 (28.2)* 19 (27.6)* 76 (28.1)* * = Composite numbers and percentages add up to the t o t a l e r r o r s (100%) f o r each age group. m u l t i p l i e d ; whereas the o l d e s t group (16+ years) m u l t i p l i e d . I t would appear that f o r the two youngest groups of students that the "more" i n question Type A suggested a d d i t i o n . For the next older group (14 & 15 years) the same word might have i n d i c a t e d a t o t a l or l a r g e r number was r e q u i r e d r e s u l t i n g i n an even d i s t r i b u t i o n of e r r o r s of a d d i t i o n or mul-t i p l i c a t i o n f o r these s u b t r a c t i o n items. I t might be that f o r the o l d e s t group (16+ years) "more" i n d i c a t e d a l a r g e r number and the operation of m u l t i p l i c a t i o n . The Type B s u b t r a c t i o n questions (How many toys l e f t ? ) were answered i n c o r r e c t l y 69 times (25.6%) (see Table V I I ) . Of these e r r o r s 41 (59.4%) of the items were added; 19 items (27.6%) were d i v i d e d ; 7 items (10.1%) were m u l t i p l i e d and 2 items (2.9%) were no responses. A l l the age groups w i t h the exception of the 12 and 13 year olds made a greater percentage of e r r o r s by adding r a t h e r than m u l t i p l y i n g or d i v i d i n g . This second group made more e r r o r s by d i v i d i n g . U n l i k e Type A s u b t r a c t i o n questions there was no i n d i c a t i o n i n the Type B question of which operation was to be used. A l s o the o v e r a l l e r r o r p a t t e r n shows that students tend to add when no i n d i c a t i o n of operation i s contained i n the question form. The 12 and 13 year olds showed deviant behavior i n t h e i r e r r o r p a t t e r n by d i v i d i n g . This type of e r r o r might be a r e s u l t of work done on d i v i s i o n problems. When the students are asked to f i n d the remainder, o f t e n the question i s "How much i s l e f t ? " The word " l e f t " may suggest d i v i s i o n to these students. M u l t i p l i c a t i o n - Examination of Table V I I I shows that age, gender, and type of question were not s i g n i f i c a n t f o r m u l t i p l i c a t i o n word problems. The o v e r a l l mean performance f o r Types A and B ranged from 0.76 (8-11 years) to 1.64 (16+ years) w i t h a t o t a l mean performance of 1.35 (see Table I X ) . Although the performance on m u l t i p l i c a t i o n questions improved s l i g h t l y w i t h i n c r e a s i n g age these d i f f e r e n c e s were not s t a t i s t i c a l l y s i g n i f i c a n t . 38 TABLE V I I I Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question M u l t i p l i c a t i o n Source of Variance df Mean Square F Between persons age gender age x gender e r r o r 3 1 3 82 3.70168 7.95995 1.31622 2.01221 1.84 3.96 0.65 Wi t h i n persons type 1 type x age 3 type x gender 1 type x age x gender 3 e r r o r 82 0.45002 1.71605 0.01903 1.65317 0.96415 0.47 1.78 0.02 1.71 TABLE IX Performance Means and Standard Deviations Type of Question by Age Group M u l t i p l i c a t i o n Type of Question Age i n Years A B T o t a l X SD X SD X SD Males 0.46 0.52 0.54 0.78 0.50 0.65 8-11 Females 1.83 1.17 0.83 0.98 1.33 1.08 Both 0.89 0.67 0.63 0.82 0.76 0.75 Males 1.13 1.55 0.88 1.36 1.00 1.46 12 & 13 Females 1.25 1.39 2.00 1.20 1.63 1.30 Both 1.19 1.47 1.44 1,28 1.32 1.38 Males 1.00 1.15 1.62 1.45 1.31 1.30 14 & 15 Females 1.38 1.45 1.85 1.41 1.62 1.43 Both 1.19 1.30 1.74 1.43 1.47 1.37 Males 1.68 1.16 1.58 1.26 1.63 1.21 16+ Females 1.50 1.18 1.80 1.23 1.65 1.21 Both 1.62 1.17 1.66 1.25 1.64 1.21 O v e r a l l 1.27 0.97 1.42 1.13 1.35 1.18 Of the Type A m u l t i p l i c a t i o n questions (How many toys i n a l l ? ) 155 of the 270 items (57.4%) were answered i n c o r r e c t l y (see Table X). Of these e r r o r s 124 problems (80%) were added; 17 problems (10.9%) were m u l t i p l i e d ; and 14 problems (9.1%) were d i v i d e d . This question type i s i d e n t i c a l to the Type A a d d i t i o n question and the r e s u l t i s ther e f o r e not unexpected. As w i t h a d d i t i o n , the " i n a l l " phrase may have proved ambiguous because of the suggestion that a greater number was req u i r e d f o r the s o l u t i o n . The f a c t that a d d i t i o n and m u l t i p l i c a t i o n responses were s e l e c t e d most f r e q u e n t l y supports t h i s t h e s i s . Table X a l s o shows 142 (52.6%) of Type B m u l t i p l i c a t i o n questions i n c o r r e c t l y answered. This question type (How many boxes d i d Susan see?) produced the f o l l o w i n g e r r o r s : i n s t e a d of m u l t i p l i c a t i o n being used, on 89 items (62.7%) a d d i t i o n was used; on 32 items (22.5%) s u b t r a c t i o n was used; on 20 items (14.1%) d i v i s i o n was used; and on 1 item (0.7%) there was no response. As the wording i n these problems was d i f f e r e n t to the other questions on the task the e n t i r e word problem was examined to f i n d a suggestion f o r the higher percentage of s u b t r a c t i o n used i n e r r o r com-pared w i t h Type A questions (22.5% f o r B compared w i t h 10.9% f o r A). e.g. B i l l had boxes. Susan saw times as many boxes. How many boxes d i d Susan see? I t appears the term "times" gave l i t t l e i n d i c a t i o n of m u l t i p l i c a t i o n to those students who made the e r r o r s . There i s no apparent reason f o r the i n c r e a s e of the s u b t r a c t i o n o p eration instead of m u l t i p l i c a t i o n f o r Type B questions. TABLE X Er r o r A n a l y s i s f o r M u l t i p l i c a t i o n Type of Question A B T o t a l Age Group To t a l Composite* T o t a l Composite* T o t a l Composite* n (%) n (%) n (%) n (%) n (%) h (%) 30 (75.0)+ 28 (62.2)+ 58 (68.2)+ 8-11 40 (70.1) 7 (17.5)- 45 (78.9) 12 (26.7)- 85 (74.6) 19 (22.4)-3 (7.5)* 5 (11.1)* 8 (9.4)* 1 (4.0)<j> 1 (1.8)* 12&13 29 (60.4) 24 (82.7)+ 25 (52.1) 18 (72.0)+ 54 (56.3) 42 (77.8)+ 4 (13.7)- 4 (16.0)- 8 (14.8)-1 (3.6)* 2 (8.0)* 3 (5.6)* 36 (78.3)+ 18 (54.5)+ 54 (68.4)+ 14&15 46 (58.9) 2 (4.3)- 33 (42.3) 9 (27.3)- 79 (50.6) 11 (13.9)-8 (17.4)* 6 (18.2)* 14 (11.5)* 34 (85.0)+ 25 (64.2)+ 59 (74.6)+ 16+ 40 (45.9) 4 (10.0)- 39 (44.8) 7 (17.9)- 79 (45.4) 11 (13.9)-2 (5.0)* 7 (17.9)* 9 (11.5)* 1 (0.7)<f> 1 (0.4)<J) T o t a l 155 (57.4) 124 (80.0)+ 142 (52.6) 89 (62.7)+ 297 (55) 213 (71.7)+ 17 (10.9)- 32 (22.5)- 49 (16.5)-14 (9.1)* 20 (14.1)* 34 (11.4)* * = Composite numbers and percentages add up to the t o t a l e r r o r s (100%) f o r each , age group. D i v i s i o n - As w i t h the m u l t i p l i c a t i o n problems, no s i g n i f i c a n t d i f f e r e n c e s among age group-, gender or type of question were found f o r d i v i s i o n questions. There was however, at the .05 l e v e l a s i g n i f i c a n t i n t e r a c t i o n between type of question, age and gender (F = 3.66; df = 3, 82; p<.05) (see Table X I ) . The o v e r a l l mean f o r Type A was 1.29; f o r Type B was 1.13 w i t h a t o t a l mean performance f o r both of 1.21 (see Table X I I ) . Figure 2 shows that both females and males i n the three youngest age groups (8-11, 12 & 13, 14 & 15) perform i n much the same manner f o r both type of questions. For both question types the highest score f o r g i r l s was the 12 and 13 year o l d group but the highest score f o r boys was the 14 and 15 year o l d group. The o l d e s t group (16+ years) of students had the most d i s p a r a t e performance. On Type A d i v i s i o n questions the females (X = 1.50) scored higher than the males (X = 1.26), but on Type B ques-t i o n s these 16+ year females scored lower (X = 0.80) than the males (X = 1.53). The g i r l s performance was a l s o e r r a t i c w i t h respect to the t h i r d age group (14 and 15 y e a r s ) . Again on Type A questions the o l d e s t females performed b e t t e r than the younger ones (X = 1.50 compared w i t h 1.00). On Type B questions however the s i t u a t i o n was reversed and the o l d e s t group of g i r l s d i d not perform as w e l l as the younger group (0.80 compared w i t h 1.08). This r e s u l t seems to defy p l a u s i b l e i n t e r p r e t a t i o n and the p r o b a b i l i t y of a Type I e r r o r cannot be overlooked. Table X I I I shows the e r r o r a n a l y s i s f o r both types of d i v i s i o n problems. The Type A questions (How many toys i n each box?) had 154 out of 270 items (57%) answered i n c o r r e c t l y . Of these errors,77 items (50%) were added in s t e a d of d i v i d e d ; 64 items (41.6%) were m u l t i p l i e d ; 12 items (7.8%) were subtracted; and 1 item (0.6%) had no response. This percentage 42 TABLE XI Three Way A n a l y s i s of Variance ( u n i v a r i a t e ) Age by Gender by Type of Question M u l t i p l i c a t i o n Source of Variance df Mean Square F Between persons age 3 5.85997 2.27 gender 1 0.20276 0.08 age x gender 3 3.32126 1.29 e r r o r 82 2.57601 W i t h i n persons type 1 1.41392 3.59 type x age 3 0.02541 0.06 type x gender 1 0.03660 0.09 type x age x gender 3 1.44275 3.66* e r r o r 82 0.39403 * = p<.05 TABLE XII Performance Means and Standard Deviations Type of Question by Age Group D i v i s i o n Type of Question Age i n Years A B t o t a l X SD X SD X SD Males 0.62 0.65 0.38 0.65 0.50 0.65 8-11 Females 0.67 0.82 0.67 1.21 0.67 1.02 Both 0.64 0.70 0.47 0.83 0.56 0.77 Males 1.25 1.49 1.13 1.36 1.19 1.43 12 & 13 Females 1.88 1.55 1.63 1.41 1.75 1.48 Both 1.57 1.52 1.38 1.39 1.48 1.46 Males 2.08 1.26 1.54 1.13 1.81 1.20 14 & 15 Females 1.00 1.35 1.08 1.32 1.04 1.34 Both 1.54 1.31 1.31 1.23 1.43 1.27 Males 1.26 1.28 1.53 1.22 1.39 1.25 16+ Females 1.50 1.43 0.80 1.14 1.15 1.29 Both 1.34 1.33 1.28 1.19 1.31 1.26 O v e r a l l 1.29 1.23 1.13 1.16 1.21 1.20 PI -.0) g CD o c e u o 14-1 M Q) 3.0 f 2.0 4-1.0 t D i v i s i o n Type A H h Females © © Males 3.0 f •4-_| ; 1 5-11 12 & 13 14 & 15 Age i n Years 2.0 + 1.0 + — 1 0 16+ D i v i s i o n Type B A 4- Females © © Males •4- + 8-11 12 & 13 14 &15 Age i n Years 16+ Figure 2: I n t e r a c t i o n Graphs, Gender by Age f o r Both Types of Question 4^  TABLE X I I I E r r o r A n a l y s i s f o r D i v i s i o n Type of Question A B T o t a l Age Group T o t a l Composite* T o t a l Composite* T o t a l Composite* n (%) n (%) n (%) n (%) n (%) n (%) 25 (55.6)+ 22 (45.8)+ 47 (50.5)+ 8-11 45 (78.9) 4 (8.9)- 48 (84.2) 10 (20.8)- 93 (81.6) 14 (15.1)-16 (35.5)x 16 (33.4Jx 32 (34.4)x 1 (4.4)cj) 1 (3.8)cj) 2 (4.1)(f> 12&13 23 (47.9) 13 (56.5)+ 25 (54.2) 13 (50.0)+ 49 (51.0) 26 (53.1)+ 2 (8.7)- 8 (30.8)- 10 (20.4)-7 (30.4)x 4 (15.4)x 11 (22.4)x 21 (55.3)+ 20 (44.4)+ 41 (49.4)+ 14&15 38 (48.9) 2 (5.3)- 45 (57.7) 10 (22.2)- 83 (53.2) 12 (14.5)-15 (39.4)x 15 (33.4)x 30 (36.1)x 18 (37.5)+ 19 (38.0)+ 37 (37.8)+ 16+ 48 (55.2) 4 (8.3)- 50 (57.5) 10 (20.0)- 98 (56.3) 14 (14.3)-26 (54.2)x 21 (42.0)x 47 (47.9)x 1 (0.6)<t> 1 (0. 6)cb 2 (0.6)(f) T o t a l 154 (57.0) 77 (50.0)+ 169 (62.6) 74 (43.8)+ 323 (59.8) 151 (46.7)+ 12 (7.8)- 38 (22.5)- 50 (15.5)-64 (41.6)x 56 (33.1)x 120 (37.2)x * = Composite numbers and percentages add up to the t o t a l e r r o r s (100%) f o r each age group. 45 of e r r o r s was t y p i c a l f o r a l l age groups except the o l d e s t . They chose ( i f they made an er r o r ) to m u l t i p l y (54.2%) r a t h e r than add (37.5%). The Type B questions (How many boxes?) were answered i n c o r r e c t l y 169 times (62.6%). The e r r o r s were then d i s t r i b u t e d as f o l l o w s : 74 questions (43.8%) were added; 56 questions (33.1%) were m u l t i p l i e d ; 38 questions (22.5%) were subtracted; and 1 question (0.6%) had no response. Those d i v i s i o n questions that were added was the most common e r r o r f o r the three younger groups and the o l d e s t group (16+ years) e i t h e r mul-t i p l i e d (42%) or added (38%). For both Type A and B d i v i s i o n questions the o v e r a l l number of students who e l e c t e d to add and not d i v i d e was s i m i l a r . On the other hand,there was a higher percentage of i n c o r r e c t response based on s u b t r a c t i o n f o r Type B problems, but the reason i s not apparent from the question forms. Operation By Type - The r e s u l t s of the second a n a l y s i s are presented i n two t a b l e s and one graph. The f i r s t t a b l e summarizes the a n a l y s i s of var i a n c e conducted on the four operations and type of question, t r e a t i n g each f a c t o r as a repeated f a c t o r . The second t a b l e contains the means and standard d e v i a t i o n s of each c e l l corresponding to the i n t e r a c t i o n between the two f a c t o r s . The graph a l s o d e p i c t s t h i s i n t e r a c t i o n . As shown i n Table XIV the u n i v a r i a t e ANOVA showed that o p e r a t i o n , type, and operation by type of question were a l l s i g n i f i c a n t at the .01 l e v e l of s i g n i f i c a n c e . The o v e r a l l mean performance f o r a d d i t i o n was 2.45; f o r s u b t r a c t i o n - 1.49; f o r m u l t i p l i c a t i o n - 1.35; and f o r d i v i s i o n - 1.21, w i t h the t o t a l mean f o r a l l four operations being 1.62 (see Table XV). Scheffe's t e s t (p<.05) was used to determine s i g n i f i c a n t d i f f e r e n c e s between operation and type of question. The r e s u l t s were as f o l l o w s : 46-TABLE XIV Summary ANOVA ( u n i v a r i a t e ) Mathematical Operation by Type of Question Source of Variance df Mean Square F operation e r r o r 3 267 56.94954 1.12557 50.60** type e r r o r 1 89 14.16806 0.57817 24.51** operation x type e r r o r 3 267 31.27176 0.76801 40.72** ** = p<.01 TABLE XV Performance Means and Standard Deviations Mathematical Operation by Type of Question Type of Question A _ B T o t a l Operation X SD X SD X SD A d d i t i o n 2.63 0.69 2.27 1.00 2.45 0.85 S u b t r a c t i o n 0.74 1.18 2.23 1.14 1.49 1.16 M u l t i p l i c a t i o n 1.27 1.23 1.42 1.29 1.35 1.26 D i v i s i o n 1.29 1.30 1.13 1.21 1.21 1.26 T o t a l 1.48 1.10 1.76 1.16 1.62 1.13 Type A . Type B • T o t a l (A and B) ad d i t i o n > s u b t r a c t i o n a d d i t i o n > m u l t i p l i c a t i o n a d d i t i o n > s u b t r a c t i o n a d d i t i o n > m u l t i p l i c a t i o n . a d d i t i o n > d i v l s i o n a d d l t i o n > m u l t i p l i c a t i o n a d d i t i o n > d i v i s i o n s u b t r a c t i o n > m u l t i p l i c a t i o n a d d i t i o n > d i v i s i o n m u l t l p l i c a t i o n > s u b t r a c t i o n s u b t r a c t i o n > d i v i s i o n d i v i s i o n > s u b t r a c t i o n As might be expected, the performances on a d d i t i o n problems on Type A, Type B, or both types together, were s i g n i f i c a n t l y higher than the performances on word problems i n v o l v i n g the other three operations. For Type A ques-t i o n s the mean f o r s u b t r a c t i o n problems was s i g n i f i c a n t l y lower than the performances on m u l t i p l i c a t i o n or d i v i s i o n problems. For Type B questions however, the reverse was true - the performance on s u b t r a c t i o n problems was s i g n i f i c a n t l y greater than that on word problems r e q u i r i n g m u l t i p l i -c a t i o n or d i v i s i o n . The d i f f e r e n c e s between t o t a l mean performances.on Type A (mean of 1.48) and Type B (mean of 1.76) were not s i g n i f i c a n t . However, the main e f f e c t s noted above need to be i n t e r p r e t e d i n l i g h t of the s i g n i f i c a n t i n t e r a c t i o n between the two f a c t o r s . Scheffe's t e s t (p<.05) was a l s o used to examine s i g n i f i c a n t d i f f e r e n c e s between types of questions. The only s i g n i f i c a n t d i f f e r e n c e found was between Type A and Type B s u b t r a c t i o n problems w i t h the performance on Type B questions s i g -n i f i c a n t l y exceeding that of Type A questions. This i s i l l u s t r a t e d i n Figure 3. In a d d i t i o n the mean performance f o r both types of question f o r each mathematical operation are a l s o shown. The graph of the o v e r a l l means i n d i c a t e s that the a d d i t i o n mean (2.45) was followed i n order by s u b t r a c t i o n (X = 1.49), m u l t i p l i c a t i o n (X = 1.34) and d i v i s i o n (X = 1.21), as discussed e a r l i e r . However, the presence of the d i s o r d i n a l i n t e r a c t i o n i n d i c a t e s that t h i s trend i s not c o n s i s t e n t across type. Add 1 Subt Operation 1 Mult Div gure 3: I n t e r a c t i o n Graph, Type of Question by Operation Total Errors - On the e n t i r e task 990 (45.8%) of the questions were answered i n c o r r e c t l y . Of these errors 486 (40.9%) were due to addition; 249 (30.5%) were due to m u l t i p l i c a t i o n ; 128 (15.3%) were due to subtrac-tion; 122 (12.9%) were due to d i v i s i o n ; and 5 (0.4%) were not answered. The fewest errors were made on the addition questions (18.3% errors) while the most errors were made on d i v i s i o n questions (59.8% errors) (see Table XVI). As a general r u l e i t appears that when confronted with a word prob-lem for which the so l u t i o n i s unclear, students w i l l e l e c t more often to add or multiply rather than subtract or di v i d e . These r e s u l t s must be interpreted with caution. This add or multiply strategy which appears to apply to subtraction and d i v i s i o n problems may also occur with addition and m u l t i p l i c a t i o n . If t h i s i s so, then scores on these l a t t e r operations are perhaps i n f l a t e d . The strategy would r e s u l t i n some correct answers whereas t h i s would not be possible i n subtraction and d i v i s i o n . This aspect merits further i n v e s t i g a t i o n (possibly through an exploration of the reasons given by students for t h e i r s o l u t i o n ) . TABLE XVI T o t a l E r r o r A n a l y s i s Operation n T o t a l (%) E r r o r s Composite* n (%) 29 (29.3)-A d d i t i o n 99 (18.3) 58 (58.6)* 12 (12.1)-2 (0.7)4> S u b t r a c t i o n 271 (50.2) 122 (45.0)+ 71 (26.2) x 76 (28.1)-1 (0.4)* M u l t i p l i c a t i o n 297 (55.0) 213 (71.7)+ 49 (16.5)-34 (11.4)-2 (0.6)<f> D i v i s i o n 323 (59.8) 151 (46.7)+ 50 (15.5)-120 (37.2)* 5 (0.4)<1) 486 (40.9)+ T o t a l 990 (45.8) 128 (15.3)-249 ( 3 0 . 5 ) x 122 (12.9)-* = Composite numbers and percentages add up to the t o t a l e r r o r s (100%) f o r each operation. CHAPTER V CONCLUSIONS I. Summary This study examined the performance of profoundly hearing impaired students on one-step a r i t h m e t i c a l word problems. One hundred s i x t e e n students were administered a computation task and those n i n e t y who met the pass requirement (at l e a s t four of the s i x questions c o r r e c t f o r each operation) were s e l e c t e d f o r the study i n which they were given a word problem task. The subjects were d i v i d e d i n t o four age groups as f o l l o w s : 8-11 years; 12 & 13 years; 14 & 15 years; and 16+ years. S t a t i s t i c a l treatment of the data showed no s i g n i f i c a n t d i f f e r e n c e s when age and gender were examined f o r any of the word problems which inv o l v e d the four operations of a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n . For the word problem r e q u i r i n g the operation of a d d i t i o n however, the type of question was s i g n i f i c a n t . R e s ults revealed that Type A a d d i t i o n problems ( i . e . How many toys i n a l l ? ) were e a s i e r to sol v e than the Type B problems ( i . e . How many boxes were t h e r e ? ) . Although the percentage of e r r o r s f o r Type B questions was twice that of Type A ques-t i o n s , the d i s t r i b u t i o n of e r r o r s was s i m i l a r . For both types of a d d i t i o n problems, the m a j o r i t y of e r r o r s c o n s i s t e d of m u l t i p l i c a t i o n i n l i e u of a d d i t i o n . The d i f f e r e n c e i n the type of question was a l s o s i g n i f i c a n t f o r sub-t r a c t i o n problems w i t h Type A questions ( i . e . How many more toys than boxes?) being u n i f o r m l y more d i f f i c u l t f o r a l l age groups. The i n t e r a c t i o n 51 52 between type of question and age was a l s o found to be s i g n i f i c a n t . The mean performances f o r Type B s u b t r a c t i o n problems ( i . e . How many toys l e f t ? ) improved as age increased whereas the means f o r Type A showed a decrease i n performance f o r the two older groups (14 & 15, 16+ y e a r s ) . Examination of e r r o r s showed that f o r Type A questions younger students added whereas o l d e r students m u l t i p l i e d r a t h e r than subtracted. For Type B problems, a l l the age groups w i t h the exception of the 12 and 13 year old s made a greater percentage of e r r o r s by adding i n s t e a d of sub-t r a c t i n g . This group made more e r r o r s by d i v i d i n g . O v e r a l l the m a j o r i t y of students who made e r r o r s tended to add ra t h e r than m u l t i p l y or d i v i d e . U n l i k e a d d i t i o n and s u b t r a c t i o n problems, type of question was not s i g n i f i c a n t f o r m u l t i p l i c a t i o n word problems. Although the performance on these questions improved s l i g h t l y w i t h i n c r e a s i n g age, the d i f f e r e n c e s were not s t a t i s t i c a l l y s i g n i f i c a n t . E r r o r a n a l y s i s f o r m u l t i p l i c a t i o n problems showed that i n the m a j o r i t y of cases f o r both Type A ( i . e . How many toys i n a l l ? ) and B questions ( I . e . How many boxes d i d Susan see?), the students added r a t h e r than m u l t i p l i e d . Type of question was a l s o found not to be s i g n i f i c a n t f o r d i v i s i o n problems. There was, however, a s i g n i f i c a n t i n t e r a c t i o n between type of question, age, and gender. Both, females and males i n the three younger age groups (8-11, 12 & 13, 14 & 15 years) performed i n a s i m i l a r manner f o r both, types of questions. The o l d e s t group (16+ years) had the most d i s p a r a t e performance. On Type A questions ( i . e How many toys i n each box?) the g i r l s from t h i s group scored higher than the boys but on the Type B. questions ( i . e . How many boxes?) the reverse was t r u e . Also on Type A questions the o l d e s t g i r l s ' mean performance was b e t t e r than the younger g i r l s (14 & 15 years) but on the Type B problems the s i t u a t i o n again was reversed. E r r o r a n a l y s i s f o r d i v i s i o n word problems showed that f o r both Type A and B questions, a s i m i l a r number of students e l e c t e d to add and not d i v i d e . For both types of d i v i s i o n questions, a d d i t i o n was the most common er r o r s t r a t e g y f o r the three younger age groups. The o l d e s t group of students chose to m u l t i p l y on Type A problems and e i t h e r m u l t i p l y or add on Type B questions. When age and gender were c o l l a p s e d , a second a n a l y s i s revealed that o p e r a t i o n , type of question and the i n t e r a c t i o n between operation and type of question were a l l s i g n i f i c a n t . The mean performances f o r the t o t a l sample popu l a t i o n i n d i c a t e d that a d d i t i o n word problems were the e a s i e s t to solve followed by s u b t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n i n that order. Type A word problems f o r the four operations appeared s l i g h t l y more d i f f i c u l t than Type B. questions. Performances on a d d i -t i o n problems were s i g n i f i c a n t l y greater than performances on problems i n v o l v i n g the other three operations. For Type A questions the means f o r s u b t r a c t i o n problems was s i g n i f i c a n t l y lower than the performances on m u l t i p l i c a t i o n or d i v i s i o n problems. For Type B questions however, the reverse was t r u e - the mean on the s u b t r a c t i o n problems was s i g n i -f i c a n t l y greater than that on word problems r e q u i r i n g m u l t i p l i c a t i o n or d i v i s i o n . A n a l y s i s of the t o t a l e r r o r s on the task showed that a d d i t i o n had the lowest percentage of e r r o r s (18.3%). The other three groups of problems had s i m i l a r percentage ranging from 50.2% to 59.8% (move e r r o r s than c o r r e c t responses). I t would appear that i f an e r r o r was made the ma j o r i t y of students e i t h e r added or m u l t i p l i e d r a t h e r than subtracted or d i v i d e d . I I . L i m i t a t i o n s A f a c t o r of concern i s the l a c k of c o n t r o l over many v a r i a b l e s i n a hearing impaired student population. While the study attempted to con-t r o l degree of hearing l o s s and age, other v a r i a b l e s such as e t i o l o g y , e d u c a t i o n a l placement and educational treatment were r e l a t i v e l y uncon-t r o l l e d . Sample s i z e was small because the study was l i m i t e d to subjects attending a c l a s s under the j u r i s d i c t i o n of the p r o v i n c i a l school f o r the deaf and who s a t i s f i e d the c r i t e r i a of the study. G e n e r a l i z a t i o n of the r e s u l t s to the deaf population as a whole i s not warranted i n that there may be an u n c o n t r o l l e d sampling e r r o r . A f u r t h e r l i m i t a t i o n i s the small number of problems tested f o r each, operation which r e s t r i c t e d the r e l i a b i l i t y of the measurement. However In t h i s p i l o t study i n order not to discourage the students w i t h a lengthy task, greater p r e c i s i o n was s a c r i f i c e d f o r b r e v i t y , h o p e f u l l y to o b t a i n sustained e f f o r t from the students. I I I . E d u c ational I m p l i c a t i o n s Each, of a number of f a c t o r s appeared to p l a y a r o l e i n s u c c e s s f u l problem s o l v i n g , but no s i n g l e one seemed to be paramount. Beyond r e c o g n i t i o n of the f a c t that i n t e l l i g e n c e , a r i t h m e t i c a l s k i l l s , and reading s k i l l s are elements i n v o l v e d i n problem s o l v i n g , the conclusions of most s t u d i e s were not i n accord (see Suydam 1967). S t a t i s t i c a l treatment on the performance of profoundly deaf students on word problems i n t h i s study showed low scores on three of the four operations (sub-t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n ) . Even though computational s k i l l s syntax and vocabulary were c o n t r o l l e d , r e s u l t s i n d i c a t e d that these 55 students had great d i f f i c u l t y i n s o l v i n g seemingly simple one-step a r i t h m e t i c a l word problems. As already d i s c u s s e d , previous s t u d i e s have i n d i c a t e d that s p e c i f i c t r a i n i n g i n vocabulary (Lyda and Duncan 1967), and programs of systematic i n s t r u c t i o n i n the reading of word problems (Faulk and Landry 1961) aided student performance. Teachers of the deaf may w e l l consider these approaches and i n c l u d e them i n t h e i r i n s t r u c t i o n of a r i t h m e t i c . Because words may have m u l t i p l e meanings, as was i n d i c a t e d i n t h i s study, vocabulary should be taught i n context. "More" does not n e c e s s a r i l y mean a greater q u a n t i t y . F u r t h (1966b) claims that language and words behind mathematical concepts are not always c l e a r and may run counter to conventional usage i n the mind of a young deaf c h i l d . For example when asked to f i n d the p i l e of blocks which "has" more, the c h i l d tends to s e l e c t the p i l e which "needs" more. Perhaps a c o n t r a s t i n g of "wanting" more and "having" more (Ling 1978) may be h e l p f u l i n developing a f u l l e r understanding of the c o n t e x t u a l dependent nature of more. In a d d i t i o n to vocabulary i n s t r u c t i o n , mathematical punctuation and a b b r e v i a t i o n s should a l s o be s t u d i e d . The two most common question forms f o r each o p e r a t i o n were s e l e c t e d f o r t h i s study from an elementary mathematics s e r i e s ( I n v e s t i g a t i n g School Mathematics) p r e s e n t l y used i n B r i t i s h Columbia. Although the students encounter these questions f r e q u e n t l y , r e s u l t s showed that g e n e r a l l y they do not f u l l y understand what operation was required from the wording of the problem. Teachers should r e a l i z e that reading s k i l l s encountered i n l i t e r a t u r e are not i d e n t i c a l w i t h those i n mathematics and s p e c i a l i z e d work on the reading of mathematics i s r e q u i r e d (Barney 1972). This could be accomplished p a r t i a l l y by the student r e s t a t i n g or 56 dramatizing the word problem to ensure comprehension of the s i t u a t i o n posed i n the problem. Borron (.1975) stat e d that to s o l v e problems s u c c e s s f u l l y three p r e r e q u i s i t e s - reading s k i l l , understanding of mathematical processes and t h e i r a p p l i c a t i o n , and computational s k i l l -must be met so that the students w i l l be attempting only one new s k i l l - choosing the proper operation to s o l v e the problem. In t h i s study computational s k i l l was c o n t r o l l e d t h e r e f o r e i t would.appear that reading s k i l l and understanding of mathematical processes and t h e i r a p p l i c a t i o n should be examined as to how these f a c t o r s a f f e c t the performance of hearing impaired students on word problems. IV. Future Research As t h i s study examined a r e s t r i c t e d sample p o p u l a t i o n , i t would be i n t e r e s t i n g and v a l u a b l e to i n v e s t i g a t e how hearing impaired students from v a r i o u s e d u c a t i o n a l placements and i n s t r u c t i o n would perform on word problem t a s k s . Factors such as v a r y i n g degrees of hearing l o s s and type of communication system used need a l s o to be explored. One of the shortcomings of many s t u d i e s , i s the use of c o r r e c t answers as the c r i t e r i o n and corresponding avoidance of l o o k i n g at the problem s o l v i n g process. In other words most s t u d i e s examined e r r o r type r a t h e r than e r r o r cause. A f r u i t f u l area of research would c o n s i s t of an examination of how hearing impaired students reason when they attempt to solve a word problem. A f u r t h e r c o n s i d e r a t i o n f o r f u t u r e research i n v o l v e s s t u d i e s of techniques f o r teaching hearing impaired students to comprehend the mathematical r e l a t i o n s h i p s expressed by the syntax and numerals of a word problem. 57 In the d i s c u s s i o n . o f educational i m p l i c a t i o n s , i t was f u r t h e r suggested that a need f o r research r e l a t e d to the development of measures of s p e c i f i c mathematical reading s k i l l s . H o p e f u l l y d i a g n o s t i c t o o l s would be developed from t h i s research to a i d teachers of mathematics. BIBLIOGRAPHY Alexander, Vincent E. Seventh graders a b i l i t y to solve problems. School Science and Mathematics, 1960, 60, 603-606. Balow, I r v i n g H. Reading and computation a b i l i t y as determinants of problem s o l v i n g . The A r i t h m e t i c Teacher, 1964, _11, 18-22. Barney, Leroy. Problems a s s o c i a t e d w i t h the reading of a r i t h m e t i c . The A r i t h m e t i c Teacher, 1972, 19, 131-133. Borron, Roberta. Helping deaf c h i l d r e n l e a r n to solv e a d d i t i o n and s u b t r a c t i o n v e r b a l problems. American Annals of the Deaf, 1975, 120, 346-349. Bruner, J.S., Olver, R., and G r e e n f i e l d , P. Studies i n c o g n i t i v e growth. New York: Wiley, 1966. C a l l , Russel J . , and Wiggen, Neal A. Reading and mathematics. The  Mathematics Teacher, 1966, 59, 149-157. Carpenter, T.P., Coburn, T.G., Reys, R.E., and Wilson, J.W. Re s u l t s and i m p l i c a t i o n s of the NAEP mathematics assessment: elementary school. The A r i t h m e t i c Reader, 1975, 12, 438-450. Carpenter, T.P., Coburn, T.G., Reys, R.E., and Wilson, J.W. Su b t r a c t i o n : what do students know? The A r i t h m e t i c Teacher, 1975, 12, 653-657. Carpenter, T.P., Coburn, T.G., Reys, R.E., and Wilson, J.W. Notes from the n a t i o n a l assessment: word problem. The A r i t h m e t i c Teacher, 1976, 23, 389-393. Carpenter, T.P., C o r b i t t , M.K., Kepner, H.S., L i n d q u i s t , M.M., and Reys, R.E. R e s u l t s of the second NAEP mathematics assessment: secondary school. The Mathematics Teacher, 1980, 73, 329-338. Chase, C l i n t o n I . The p o s i t i o n of c e r t a i n v a r i a b l e s i n the p r e d i c t i o n of problem s o l v i n g i n a r i t h m e t i c . J o u r n a l of Educational  Research, 1960, 54, 9-14. C o f f i n g , Esther A. The r e l a t i o n s h i p between s i l e n t reading a b i l i t y and a r i t h m e t i c a l a b i l i t y . School Science and Mathematics, 1941, 41, 10-14. Conrad, R. The reading a b i l i t y of deaf s c h o o l - l e a v e r s . B r i t i s h J o u r n a l  of E d u c a t i o n a l Research, 1977, 47., 138-148. 58 59 Cooper, Robert L., and Rosenstein, Joseph. Language a c q u i s i t i o n of deaf c h i l d r e n . V o l t a Review, 1966, 68, 58-67. C o r l e , Clyde G. Thought processes i n grade s i x problems. The A r i t h m e t i c  Teacher, 1958, 5, 193-203. Days, Harold C , Wheatley, Grayson H., and Kulm, Gerald. Problem s t r u c t u r e , c o g n i t i v e l e v e l , and problem s o l v i n g performance. Jo u r n a l f o r Research i n Mathematics Education, 1979, 10, 135-146. Earp, N. Wesley. Observations on teaching-reading i n mathematics. J o u r n a l of Reading, 1970, 13, 529-532. Earp, N. Wesley. Problems of reading i n mathematics. School Science and  Mathematics, 1971, 71, 129-133. E i c h o l z , Robert E., O'Daffer, Phares G., and Fleenor, Charles R. I n v e s t i g a t i n g school mathematics. Grades 2-6. Don M i l l s , O ntario: Addison-Wesley (Canada) L t d . , 1973. Engelhart, Max D. The r e l a t i v e c o n t r i b u t i o n of c e r t a i n f a c t o r s to i n d i v i d u a l d i f f e r e n c e s i n a r i t h m e t i c a l problem s o l v i n g . J o u r n a l  of Experimental Education, 1932, 1, 19-27. E r i k s o n , Leland H. C e r t a i n a b i l i t y f a c t o r s and t h e i r e f f e c t on a r i t h m e t i c achievement. The A r i t h m e t i c Teacher, 1958, j j , 287-291. Faulk, Charles J . , and Landry, Thomas B. An approach to problem s o l v i n g . The A r i t h m e t i c Teacher, 1961, 8, 157-160. Fu r t h , Hans G. A comparison of reading t e s t norms of deaf and hearing c h i l d r e n . American Annals of the Deaf, 1966a, 111, 461-462. F u r t h , Hans G. Thinking without language. London: C o l l i e r - M a c M i l l a n , 1966b. Goetzinger, C P . , and Rousey, C.L. Educational achievement of deaf c h i l d r e n . American Annals of the Deaf, 1959, 104, 221-231. Hamp, N.W. Reading attainment and some as s o c i a t e d f a c t o r s i n deaf and p a r t i a l l y hearing c h i l d r e n . Teacher of the Deaf, 1972, 70, 203-215. Hansen, C a r l W. F a c t o r s a s s o c i a t e d w i t h s u c c e s s f u l achievement i n problem s o l v i n g i n s i x t h grade a r i t h m e t i c . J o u r n a l of  Educational Research, 1944, 38, 111-118. Ha r g i s , Charles H. The grammar of the noun phrase and a r i t h m e t i c i n s t r u c t i o n f o r deaf c h i l d r e n . American Annals of the Deaf, 1969, 114, 766-769. Heddens, James W., and Smith, Kenneth J . The r e a d a b i l i t y of elementary mathematics books. The A r i t h m e t i c Teacher, 1964, 11, 466-468. 60 Henney, Maribeth. Improving mathematics v e r b a l problem s o l v i n g a b i l i t y through reading i n s t r u c t i o n . The A r i t h m e t i c Teacher, 1971, 18, 223-229. Hernandez, Norma G. Word problem s k i t s f o r the deaf. The A r i t h m e t i c  Teacher, 1979, 27, 14-16. Hine, W.D. The attainments of c h i l d r e n w i t h p a r t i a l hearing. Teacher  of the Deaf, 1970, 68, 129-135. Johnson, Harry C. The e f f e c t of i n s t r u c t o r i n mathematical vocabulary upon problem s o l v i n g In a r i t h m e t i c . J o u r n a l of Educational  Research, 1944, 38, 97-110. Johnson, J.T. On the nature of problem s o l v i n g i n a r i t h m e t i c . J o u r n a l  of E d u cational Research, 1949, 43, 110-115. Johnson, Kerry A. A survey of mathematical programs, m a t e r i a l s , and methods i n schools f o r the deaf. American Annals of the Deaf, 1977, 122, 19-25. Karchmer, Michael A., Milone, Michael N., and Wolk, Steve. Educational s i g n i f i c a n c e of hearing l o s s at three l e v e l s of s e v e r i t y . American Annals of the Deaf, 1979, 124, 97-109. Kramer, Grace A. The e f f e c t of c e r t a i n f a c t o r s i n the v e r b a l a r i t h m e t i c  problem upon c h i l d r e n ' s success i n the s o l u t i o n . Johns Hopkins U n i v e r s i t y Studies i n Education, Baltimore: The Johns Hopkins Press, 1933. Lenore, John. D i f f i c u l t i e s i n s o l v i n g problems i n a r i t h m e t i c . The  Elementary School J o u r n a l , 1930, 31, 202-215. L e s t e r , J r . , Frank K. Ideas about problem s o l v i n g : a look at some p s y c h o l o g i c a l research. The A r i t h m e t i c Teacher, 1977, 25, 12-14. L i n g , Agnes. B a s i c number and mathematical concepts f o r young hearing impaired c h i l d r e n . V o l t a Review, 1978, 8£, 46-50. L i n v i l l e , W i l l i a m J . Syntax, vocabulary, and the v e r b a l a r i t h m e t i c problem. School Science and Mathematics, 1976, 7_6, 152-158. Lyda, W.J. A r i t h m e t i c i n the secondary school curriculum. The Mathematics  Teacher, 1947, 40, 387-388. Lyda, W.J., and Duncan, Frances M. Q u a n t i t a t i v e vocabulary and problem s o l v i n g . The A r i t h m e t i c Teacher, 1967, 14, 289-291. Messerly, C a r o l L., and Aram, Dorothy M. Academic achievement of hearing impaired students of hearing parents and of hearing impaired parents: another look, V o l t a Review, 1980, J32, 28-32. 61 Meyer, Ruth Ann. Mathematical problem s o l v i n g performance and i n t e l l e c t u a l a b i l i t i e s of f o u r t h grade c h i l d r e n . J o u r n a l f o r Research i n Mathematics Education, 1978, 9, 334-348. Monroe, Walter S., and Engelhart, Max D. The e f f e c t i v e n e s s of systematic i n s t r u c t i o n i n reading v e r b a l problems i n a r i t h m e t i c . Elementary  School J o u r n a l , 1933, 33, 377-381. Muscio, Robert. Factors r e l a t i n g to q u a n t i t a t i v e understanding i n the s i x t h grade. The A r i t h m e t i c Teacher, 1962, _9, 259-262. Newcomb, R.S. Teaching p u p i l s how to solve problems i n a r i t h m e t i c . Elementary School J o u r n a l , 1922, 23, 183-189. Olander, Herbert T., and Ehmer, Charles L. What p u p i l s know about vocabulary i n mathematics - 1930 and 1968. Elementary School  J o u r n a l , 1971, 71, 361-367. Otterburn, Margaret K., and Nicholson, A.R. The language of (CSE) mathematics. Mathematics i n School, 1976, _5, 18-20. P i a g e t , J . The c o n s t r u c t i o n of r e a l i t y i n the c h i l d . New York: Basic Books, 1954. P i t t s , Raymond J . R e l a t i o n s h i p s between f u n c t i o n a l competence i n math-ematics and reading grade l e v e l s , mental a b i l i t y and age. Jo u r n a l of Educational Psychology, 1952, 43, 486-492. Richardson, L l o y d I . The r o l e s of s t r a t e g i e s f o r teaching p u p i l s to solve v e r b a l problems. The A r i t h m e t i c Teacher, 1975, 22_, 414-421. Rogers, W. Todd, and C l a r k e , B.R. C o r r e l a t e s of academic achievement of hearing impaired students. Canadian J o u r n a l of Education, 1980, 5, 27-39. Rogers, W.T., L e s l i e , P.T., Cl a r k e , B.R., Booth, J.A., and Horvath, A. Academic achievement of hearing impaired students: comparison among s e l e c t e d subpopulations. B.C. Jo u r n a l of S p e c i a l Education, 1978, 2, 183-213. R o l i n g , P e a r l , Blume, C l a r a L., and Morehart, Mary S. S p e c i f i c causes_of f a i l u r e i n a r i t h m e t i c problems. Educational Research B u l l e t i n , 1924, 3_, 271-272. Rose, A l v i n W., and Rose, Helen Cureton. I n t e l l i g e n c e , s i b l i n g p o s i t i o n , and s o c i o c u l t u r a l background as f a c t o r s i n a r i t h m e t i c performance. The A r i t h m e t i c Teacher, 1961, 8, 50-56. S c h e l l , Leo M., and Burns, Paul C. P u p i l performance w i t h three types of s u b t r a c t i o n s i t u a t i o n s . School Science and Mathematics, 1962, 62, 208-214. Smith, Frank. The r e a d a b i l i t y of s i x t h grade word problems. School Science and Mathematics, 1971, 71, 559-562. 62 S t e f f e , L e s l i e P. The r e l a t i o n s h i p of conservation of numerous to problem s o l v i n g a b i l i t i e s of f i r s t grade c h i l d r e n . The A r i t h m e t i c  Teacher, 1968, 15, 47-52. Stevens, B.A. Problem s o l v i n g i n a r i t h m e t i c . J o u r n a l of Educational  Research, 1932, 25, 253-260. Stevenson, P.R. D i f f i c u l t i e s i n problem s o l v i n g . J o u r n a l of Educational  Research, 1925, 11, 95-103. Suppes, P a t r i c k . A survey of c o g n i t i o n i n handicapped c h i l d r e n . Review  of E d u c a t i o n a l Research, 1974, 44, 145-176. Suydam, M a r i l y n N. The s t a t u s of research on elementary school mathematics. The A r i t h m e t i c Teacher, 1967, 14, 684-689. Suydam, M a r i l y n N., and Weaver, J . Fred. Research on problem s o l v i n g : i m p l i c a t i o n s f o r elementary school classrooms. The A r i t h m e t i c  Teacher, 1977, 25, 40-42. Tomlinson-Keasey, C , and K e l l y , Ronald R. The deaf c h i l d ' s symbolic world. American Annals of the Deaf, 1978, 123, 452-459. Treacy, John P. The r e l a t i o n s h i p of reading s k i l l s to the a b i l i t y to solve a r i t h m e t i c problems. J o u r n a l of Educational Research, 1944, 38, 86-96. Trybus, Raymond J . , and Karchmer, Michael A. School achievement scores of hearing impaired c h i l d r e n : n a t i o n a l data on achievement s t a t u s and growth p a t t e r n s . American Annals of the Deaf, 1977, 122, 62-69. U n d e r h i l l , Robert G. Teaching word problems to f i r s t graders. The  A r i t h m e t i c Teacher, 1977, 25, 54-56. VanderLinde, Louis F. Does the study of q u a n t i t a t i v e vocabulary improve problem s o l v i n g ? Elementary School J o u r n a l , 1964, 65^, 143-152. Vernon, McCay, and Koh, Soon D. E a r l y manual communication and deaf c h i l d r e n ' s achievement. American Annals of the Deaf, 1970, 115, 527-536. Vernon, McCay, and Koh, Soon D. E f f e c t s of o r a l preschool compared to e a r l y manual communication on education and communication i n deaf c h i l d r e n . American Annals of the Deaf, 1971, 116, 569-574. Washburne, C a r l e t o n , and Osborne, Raymond. Sol v i n g a r i t h m e t i c problems. Elementary School J o u r n a l , 1926, 27, 219-226. Webb, Leland F., and Ost, David H. Real comprehensive problem s o l v i n g as i t r e l a t e s to mathematics teaching i n the secondary schools. School Science and Mathematics, 1978, 78, 197-206. 63 Webb, Norman L. Processes, conceptual knowledge and mathematical problem s o l v i n g a b i l i t y . J o u r n a l f o r Research i n Mathematics Education, 1979, 10, 83-93. Wilson, John W. The r o l e of s t r u c t u r e i n v e r b a l problem s o l v i n g . The  A r i t h m e t i c Teacher, 1967, 14, 486-497. Wrigley, Jack. The f a c t o r i a l nature of a b i l i t y i n elementary mathematics. B r i t i s h J o u r n a l of E d u c a t i o n a l Psychology, 1958, 28[, 61-78. APPENDIX A COMPUTATION TASK Name: B i r t h d a t e : School:_ Teacher: Examples: C i r c l e the r i g h t answer. A) 6 + 4 = B) 3 - 1 = 2 6 8 10 1 2 3 4 C) 4 X 3 = D) 10 * 5 = 1 7 12 15 1 2 5 15 64 C i r c l e the r i g h t answer. A) 6 + 2 = 3 4 8 12 B) 6 x 3 = 2 3 9 18 C) 3 x. 3 = 0 1 6 9 D) 6 * 3 = 2 3 9 18 E) 6 + 3 = 2 3 9 18 F) 9 - 3 = 3 6 12 27 G) 10 * 2 = 5 8 12 20 H) 9 x 3 = 3 6 12 27 I) 10 x 2 = 5 8 12 20 J) 8 x 2 = 4 6 10 16 K) 9 * 3 = 3 6 12 27 L) 6 - 2 = 3 4 8 12 C i r c l e the r i g h t answer. M) 9 + 3 = 3 6 12 27 N) 8 * 2 = 4 6 10 16 0) 3 + 3 = 0 1 6 9 P) 10 - 2 = 5 8 12 20 Q) 8 + 2 = 4 6 10 16 R) 6 - 3 = 2 3 9 18 S) 8 - 2 = 4 6 10 16 T) 10 + 2 = 5 8 12 20 U) 6 x 2 = 3 4 8 12 V) 3 * 3 = 0 1 6 9 W) 6 - 2 = 3 4 8 12 X) 3 - 3 = 0 1 6 9 67 WORD PROBLEM TASK C i r c l e the r i g h t answer. Name: A) John had 6 toys. B i l l had 2 toys. How many toys i n a l l ? 3 4 8 12 B) John had 6 boxes. John had 3 toys i n each box. How many toys i n a l l ? 2 3 9 18 C) John had 3 boxes. John had 3 toys i n each box. How many toys i n a l l ? 0 1 6 9 D) John had 6 toys. B i l l put 3 toys i n each box. How many boxes? 2 3 9 18 E) Susan had 6 boxes. Susan found 3 boxes. How many boxes were there? 2 3 9 18 F) John had 9 toys. John l o s t 3 toys. How many toys l e f t ? 3 6 12 27 G) B i l l had 10 toys. B i l l had 2 boxes. How many toys i n each box? 5 8 12 20 H) B i l l had 9 boxes. Susan saw 3 times as many boxes. How many boxes d i d Susan see? 3 6 12 27 68 C i r c l e the r i g h t answer. I) John had 10 boxes. John had 2 toys i n each box. How many toys i n a l l ? 5 8 12 20 J) B i l l had 8 boxes. Susan saw 2 times as many boxes. How many boxes d i d Susan see? 4 6 10 16 K) John had 9 toys. B i l l put 3 toys i n each box. How many boxes? 3 6 12 27 L) John had 6 toys. B i l l put 2 toys i n each box. How many boxes? 3 4 8 12 M) Susan had 9 boxes. Susan found 3 boxes. How many boxes were there? 3 6 12 27 N) B i l l had 8 toys. B i l l had 2 boxes. How many toys i n each box? 4 6 10 16 0) Susan had 3 boxes. Susan found 3 boxes. How many boxes were there? 0 1 6 9 P) Susan had 10 toys. John had 2 boxes. How many more toys than boxes? 5 8 12 20 69 C i r c l e the r i g h t answer. Q) John had 8 toys. B i l l had 2 toys. How many toys i n a l l ? 4 6 10 16 R) John had 6 toys. John l o s t 3 toys. How many toys l e f t ? 2 3 9 18 S) Susan had 8 toys. John had 2 boxes. How many more toys than boxes? 4 6 10 16 T) John had 10 toys. B i l l had 2 toys. How many toys i n a l l ? 5 8 12 20 U) B i l l had 6 boxes. Susan saw 2 times as many boxes. How many boxes d i d Susan see? 3 4 8 12 V) B i l l had 3 toys. B i l l had 3 boxes. How many toys i n each box? 0 1 6 9 W) John had 6 toys. John l o s t 2 toys. How many toys l e f t ? 3 4 8 12 X) Susan had 3 toys. John had 3 boxes. How many more toys than boxes? 0 1 6 9 

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