Ability With The Mathematical Principles Governing The Operations Of Addition, Multiplication, Subtraction, And Division

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Content T his d isserta tio n has been 62-6061 m ic ro film ed ex a ctly as rec eiv e d HAMMOND, Robert L ee, 1928- ABILITY WITH THE MATHEMATICAL PRINCIPLES GOVERNING THE OPERATIONS OF ADDITION, MULTIPLICATION, SUBTRACTION, AND DIVISION. U n iv ersity of Southern C alifornia, E d .D ., 1962 Education, g en eral University Microfilms, Inc., Ann Arbor, Michigan C o p y r ig h t by ROBERT LEE H A M M O N D 1963 ABILITY WITH THE M ATHEM ATICAL PRINCIPLES GOVERNING THE OPERATIONS OF ADDITION, MULTIPLICATION, SUBTRACTION, A N D DIVISION by R o b ert Lee Hammond A D is s e r ta tio n P re se n te d to th e FACULTY OF THE SCHOO L OF EDUCATION UNIVERSITY OF SOU TH ERN CALIFORNIA In P a r t i a l F u lfillm e n t o f th e R equirem ents fo r th e Degree DO CTO R OF EDUCATION Ju n e 1962 This dissertation, written under the direction of the Chairman of the candidate’s Guidance Committee and approved by all members of the Committee, has been presented to and accepted by the Faculty of the School of Education in partial fulfillment of the requirements for the degree of D octor of Education. Date June,..I962.................................................. Dean Chairman TABLE OF CONTENTS Page LIST OF T A B L E S ....................................................... v C hapter I . PRESENTATION OF THE PROBLEM............................ 1 The Problem S tatem ent o f th e problem P urposes o f th e stu d y H ypotheses Q uestions to be answ ered Im portance o f th e Problem L im ita tio n s o f th e Study Scope o f th e in v e s tig a tio n In h e re n t lim ita tio n s o f th e stu d y D e fin itio n o f Terms M athem atical p r in c ip le s O p eratio n M athem atical o p e ra tio n s M athem atical s i t u a t i o n O rg a n iza tio n o f th e Rem aining C hapters I I . REVIEW OF THE LITERATURE........................... 17 The N atu re o f th e O b jects o f M athem atics M eaning and U nderstanding as They Apply to M athem atical O p eratio n s Concepts as A pplied to M athem atical O p eratio n s R esearch on th e E lem entary School C h ild 's Growth in U nd erstan d in g o f M athem atical O p eratio n s Summary ii iii C hapter I I I . IV . Page PROCEDURES A N D SOURCE OF DATA............................ 33 The Sample In stru m en ts and P ro ced u res Used in O b tain in g I n te llig e n c e and Achievement Data D e sc rip tio n o f th e Sample The T e st o f M athem atical O p eratio n s T reatm ent o f th e D ata Summary ABILITY WITH THE M ATHEM ATICAL PRINCIPLES GOVERNING THE OPERATIONS OF ADDITION, MULTIPLICATION, SUBTRACTION, AND DIVI SION, A N D ITS RELATION TO SELECTED FACTORS OF INTELLIGENCE, ACHIEVEMENT, A N D ALGEBRA APTITUDE ....................... 53 The R e la tio n s h ip Between Language IQ and A b ility in M athem atical O p eratio n s The R e la tio n s h ip Between Non-Language IQ and A b ility in M athem atical O p eratio n s The R e la tio n s h ip Between A rith m e tic R easoning and A b ility in M athem atical O p eratio n s The R e la tio n s h ip Between A rith m e tic Fun dam entals and A b ility in M athem atical O p eratio n s The R e la tio n s h ip Between A lg eb ra A p ti tu d e and A b ility in M athem atical O per a tio n s Summary AN ANALYSIS OF THE FACTORS RELATED TO ABILITY WITH M ATHEM ATICAL OPERATIONS . . 71 The T o ta l Sample F a c to r A F a c to r B F a c to r C iv Chapter Page A b ility L ev els as D escrib ed by a T est o f M athem atical O p eratio n s Sex D iffe re n c e s Summary V I. SU M M A R Y , CONCLUSIONS, A N D RECOM M ENDATIONS . . 103 Summary Summary o f F in d in g s C onclusions E d u ca tio n a l Im p lic a tio n s Recommendations f o r F u rth e r R esearch SELECTED BIBLIOGRAPHY .................................................................... 131 A PPEN D IX .....................................................................................................136 LIST OF TABLES T able Page 1. Means and S tan d ard D ev ia tio n s R e la te d to th e Language and Non-Language IQ o f th e Sample 37 2. Means, S tan d ard D e v ia tio n s , and Mean Grade Placem ent o f Raw Scores in A rith m e tic Achievem ent and A lgebra A p t i t u d e ..................... 39 3 . The C o e ffic ie n t o f M u ltip le C o rre la tio n and O ther S t a t i s t i c s R e la te d to th e C o rre la tio n o f th e M athem atical O p eratio n s and Language I Q ..................................................... ................................ 56 4 . The C o e ffic ie n t o f M u ltip le C o rre la tio n and O ther S t a t i s t i c s R e la te d to th e C o rre la tio n o f th e M athem atical O p eratio n s and Non- Language I Q ................................................................ 59 5 . The C o e ffic ie n t o f M u ltip le C o rre la tio n and O ther S t a t i s t i c s R e la te d to th e C o rre la tio n o f th e M athem atical O p eratio n s and A rith m e tic R easoning ......................................... 62 6 . The C o e ffic ie n t o f M u ltip le C o rre la tio n and O ther S t a t i s t i c s R e la te d to th e C o rre la tio n o f th e M athem atical O p eratio n s and A rith m e tic Fundam entals .................... 65 7. The C o e ffic ie n t o f M u ltip le C o rre la tio n and O ther S t a t i s t i c s R e la te d to th e C o rre la tio n o f th e M athem atical O p eratio n s and A lgebra A p titu d e . . . . . . . . . . . . . 68 8 . The C o rre la tio n M a trix , M eans, and S tandard D ev ia tio n s fo r th e T o ta l Group o f 300 . . 73 v vi Table Page 9 . R o tate d F a c to r M a trix f o r T o ta l Group o f 300 74 10. The C o rre la tio n M a trix , M eans, and S tan d ard D ev ia tio n s f o r th e Upper Group o f 100 . . 85 11. R o tate d F a c to r M a trix f o r th e Upper Group o f 1 0 0 86 12. The C o rre la tio n M a trix , M eans, and S tan d ard D ev ia tio n s fo r th e M iddle Group o f 100 . . 88 13. R o tate d F a c to r M a trix f o r th e M iddle Group o f 1 0 0 .............................................................................. 90 14. The C o rre la tio n M a trix , M eans, and S tan d ard D ev ia tio n s f o r th e Lower Group o f 100 . . 91 15. R o ta te d F a c to r M a trix f o r th e Lower Group o f 1 0 0 .................................................................... .... . 93 16. The C o rre la tio n M a trix , M eans, and S tan d ard D ev ia tio n s f o r 148 G ir ls ....................... 95 17. The C o rre la tio n M a trix , M eans, and S tan d ard D e v ia tio n s f o r 152 B o y s ...................... 96 18. R o ta te d F a c to r M a trix f o r 148 G ir ls . . . . 97 19. R o tate d F a c to r M a trix f o r 152 B o y s .. 98 * CHAPTER I PRESENTATION OF THE PROBLEM In re c e n t y e a rs r a d ic a l changes have been o c c u rrin g in th e te a c h in g o f a r ith m e tic . The new sc h o o ls o f th o u g h t w hich look upon a r ith m e tic as a r a t i o n a l p ro c e s s , a system f o r th in k in g , h o ld th a t th e tr u th s o f a r ith m e tic can be more e f f e c t i v e l y ta u g h t and b e t t e r r e ta in e d when th e mean in g o f th e p ro c e s se s i s s tr e s s e d . R esearch h as c o n s is te n tly su p p o rted th e b e l i e f th a t le a rn in g in a r ith m e tic i s m ost e f f e c t iv e when p re se n te d in a m ean in g fu l m anner. W ith t h i s "m eaningful" re v o lu tio n many o f th e problem s o f m ethodology have been overcom e. Y et, does a more e f f e c t iv e m ethod c o n ta in a l l th e answ ers n e c e s s a ry fo r e f f e c t iv e u n d e rsta n d in g and grow th in m athe m a tic s? What o f th e n a tu re o f m athem atics as a d is c ip li n e , and w hat e f f e c t does t h i s have on placem ent and c o n te n t a t th e v a rio u s g rad e le v e ls ? Does a lg e b ra need to be d e fe rre d to th e b eg in n in g y e a rs o f h ig h sc h o o l, o r can th e elem en ta r y c h ild d em o n strate a b i l i t y w ith m ath em atical p r in c ip le s th a t w a rra n t t h e i r in c lu s io n in th e elem en tary program? The answ ers to th e s e and many o th e r q u e s tio n s v i t a l to in s tr u c tio n in m athem atics have been l e f t unansw ered by re s e a rc h a t t h i s p o in t. Fundam ental to a l l a re a s o f m athem atics a re th e b a s ic o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , and d iv is io n . I t would be m ost b e n e f i c ia l to know w hat c h ild re n u n d ersta n d about th e s e o p e ra tio n s when re q u ire d to rea so n in term s more s u it a b le to th e advanced le v e ls o f a p p lie d m ath em atical re a s o n in g . The Problem T his in v e s tig a tio n was concerned w ith th e n a tu re o f c h i ld r e n 's a b i l i t y to d e s c rib e th e m ath em atic al p r in c ip le s governing th e o p e ra tio n s o f a d d itio n , m u ltip lic a tio n , su b t r a c t i o n , and d iv is io n th ro u g h th e u se o f symbols common to a lg e b ra . The in stru m e n t used to d e s c rib e t h i s a b i l i t y was based on t e s t item s r e q u ir in g a n a ly s is o f m ath em atical s itu a tio n s to d eterm in e th e o p e ra tio n s in v o lv e d . S tatem en t o f th e p roblem . —The problem o f t h i s stu d y was to a s c e r ta in th e u n d e rsta n d in g by 300 se v e n th - g rad e c h ild re n o f c e r ta in m ath em atical p r in c ip le s g overning th e o p e ra tio n s o f a d d itio n , m u ltip lic a ti o n , s u b tr a c tio n , and d iv is io n , and th e r e la tio n s h ip o f t h i s u n d e rsta n d in g to a r ith m e tic and m en tal a b i l i t y as m easured by sta n d a rd iz e d t e s t s . The m ath em atical p r in c ip le s s e le c te d fo r stu d y were b ased on th e ax io m atic method in m athem atics and in v o lv e d th e (1) c lo s u re law fo r a d d itio n , (2) c lo s u re law f o r m ul t i p l i c a t i o n , (3) com m utative law fo r a d d itio n , (4) commu t a t i v e law f o r m u ltip lic a tio n , (5) a s s o c ia tiv e law f o r m ul t i p l i c a t i o n , (6) a s s o c ia tiv e law f o r a d d itio n , (7) d e f i n i tio n o f d if f e r e n c e , (8) axiom o f s u b tr a c tio n , (9) d e f i n i tio n o f q u o tie n t, and (10) axiom o f d iv is io n . P urposes o f th e s tu d y .--T h e p u rp o ses o f th e stu d y w ere (1) to a s c e r ta in th e n a tu re o f u n d e rsta n d in g by 300 se v e n th -g ra d e c h ild re n o f th e o p e ra tio n s o f a d d itio n , m ul t i p l i c a t i o n , s u b tr a c tio n , and d iv is io n as d e s c rib e d by t e s t item s r e q u ir in g a n a ly s is o f m a th em atical s it u a ti o n s ; (2) to a s c e r ta in by th e c o r r e la tio n m ethod th e e x te n t o f r e la tio n s h i p th a t e x i s ts betw een th e t e s t o f m a th em atical o p e ra tio n s , and (a) a rith m e tic a b i l i t y as d e s c rib e d by th e a r ith m e tic s e c tio n o f th e C a lif o r n ia Achievement T e s t s . J u n io r High L ev el. G rades 7 - 8 -9 . Form X and th e Survey T e st o f A lg eb raic A p titu d e , (b) m en tal a b i l i t y as d e s c rib e d by th e C a lif o r n ia Short-Form T e s t o f M ental M a tu rity ; (3) to d isc o v e r by f a c to r a n a ly s is (a) th e r e la tio n s h ip o f th e m a th em atical p r in c ip le s to each o th e r and to th e f a c to r s o f a r ith m e tic achievem ent and m e n tal a b i l i t y , (b) th e r e l a tio n s h ip o f th e f a c to r s when th e sam ple was d iv id e d in to th r e e a b i l i t y le v e ls as d eterm in ed by sc o re s on th e t e s t o f m ath em atic al o p e ra tio n s , and (c) th e d if fe re n c e in f a c to r s f o r boys and g i r l s . H y p o th eses. --T he fo llo w in g h y p o th eses form ed th e b ases f o r th e fundam ental problem s to be checked by th e v a rio u s a n a ly se s o f th e d a ta c o lle c te d fo r u se in th e i n v e s tig a tio n : 1. A s ig n i f i c a n t r e la tio n s h i p e x i s ts between a b i l i t y d e s c rib e d by A T e s t o f M athem atical O p e ra tio n s . a t e s t developed e s p e c ia lly f o r t h i s stu d y , and m en tal a b i l i t y as in d ic a te d by language and n o n -lan g u ag e IQ determ ined from th e C a lif o r n ia Short-Form T e st o f Men t a l M a tu rity . 2= A s ig n i f i c a n t r e la tio n s h ip e x i s ts betw een a b i l i t y as d e s c rib e d by A T e s t o f M athem at i c a l O p e ra tio n s and a r ith m e tic a b i l i t y as in d ic a te d by sc o re s on th e a r ith m e tic re a so n in g and a r ith m e tic fu n d am en tals s e c tio n s o f th e C a lif o m ia A chievem ent T e s t, J u n io r High L e v e l. G rades 7 -8 -9 . Form X. 3 . A s i g n i f i c a n t r e la tio n s h i p e x i s ts betw een a b i l i t y as d e s c rib e d by A T e s t o f M athemat i c a l O p eratio n s and a lg e b ra a p titu d e as m easured by th e Survey T e s t o f A lg eb raic A p titu d e . 4 . A b ility as d e s c rib e d by A T e s t o f M athemat i c a l O p eratio n s a s s e s s e s c e r t a in f a c t o r s o f m ath em atical a b i l i t y d i f f e r e n t from th o se m easured by th e o th e r t e s t s se lec tied fo r u se in th e stu d y . 5 . The f a c to r s o f m ath em atical a b i l i t y d e sc rib e d by A T e st o f M athem atical O p e ra tio n s , when th e sam ple i s d iv id e d in to th r e e a b i l i t y le v e l s , w i l l v a ry from th e f a c t o r s d e s c rib e d f o r th e group as a w hole. 6 . The f a c to r s o f m ath em atical a b i l i t y d e s c rib e d by A T e st o f M athem atical O p e ra tio n s w i l l v ary f o r boys and g i r l s . Q u estio n s to be answ ered. - - I n t e s t i n g th e s e h y p o th e s e s , i t was n e c e ss a ry to seek answ ers to th e fo llo w in g q u e s tio n s : 1. To w hat e x te n t a r e n o n -lan g u ag e IQ and a b i l i t y d e s c rib e d by A T e st o f M athem at i c a l O p eratio n s r e la te d ? 2 . To w hat e x te n t a r e language IQ and a b i l i t y d e s c rib e d by A T e st o f M ath em atical O pera tio n s r e la te d ? 3 . To w hat e x te n t a re achievem ent in a r i t h m e tic re a so n in g and a b i l i t y d e s c rib e d by A T e st o f M athem atical O p e ra tio n s r e la te d ? 7 4 . To w hat e x te n t a re achievem ent in a rith m e tic fundam entals and a b i l i t y d e s c rib e d by A T e st o f M athem atical O p eratio n s r e la te d ? 5 . To what e x te n t a r e a lg e b ra a p titu d e and a b i l i t y d e s c rib e d by A T e st o f M athem atical O p eratio n s r e la te d ? 6 . Are c e r ta in f a c t o r s o f m ath em atic al a b i l i t y m easured by A T e s t o f M ath em atical O pera tio n s s i g n i f i c a n t l y d i f f e r e n t from th o se m easured by th e o th e r in s tru m e n ts used in t h i s study? 7. Do f a c to r s o f m ath em atical a b i l i t y m easured by A T e st o f M ath em atical O p eratio n s d i f f e r when th e sam ple i s d iv id e d in to th r e e a b i l i t y le v e ls b ased on sc o re s on th e t e s t ? 8 . Do f a c to r s o f m ath em atical a b i l i t y m easured by A T e st o f Mathematical O p eratio n s d i f f e r f o r boys and g i r l s ? 8 Im portance o f th e Problem In an a r t i c l e by G ibb, th e su g g e stio n was made th a t p re s e n t m ethods o f a r ith m e tic in s t r u c tio n need re-ex am in a- tio n to d is c o v e r new and b e t t e r means o f d ev elo p in g m athe m a tic a l c o n c e p ts . She w ro te: P a st e x p e rie n c e sh o u ld be a g u id in g s t a r , n o t a h itc h in g p o s t. In te a c h in g m ath em atics, p erh ap s more th a n in any o th e r s u b je c t, we need to keep t h i s epigram in m ind. In m ath em atics, we have developed and m a in ta in e d a t r a d i t i o n a l sequence o f s tu d y . Y et, we have v e ry l i t t l e e x p e rim en ta l ev id en ce to su p p o rt th e b e l i e f th a t t h i s sequence i s b e t t e r th a n o th e r p o s s ib le se q u e n c e s. The demands o f th e w o rld to d ay and th e ex p e c te d re q u ire m e n ts o f tomorrow have n e c e s s i ta te d a re -e x a m in a tio n o f th e t r a d i t i o n a l e le m e n ta ry -s c h o o l, se c o n d a ry -sc h o o l, and c o lle g e m athem atics c u rric u lu m s . As a r e s u l t , th e re i s now much a c t i v i t y and i n t e r e s t in th e d ev elo p m ent o f new program s in m ath em atics--p ro g ram s w hich re g a rd m athem atics as a stu d y and c l a s s i f i c a t i o n o f p o s s ib le s tr u c t u r e s and n o t a s a stu d y o f t r i c k s and r u l e s . (1 :6 5 ) The more advanced p h ases o f m ath em atic al re a so n in g r e q u ir e t h a t th e c h ild have a th o ro u g h u n d e rsta n d in g o f th e b a s ic o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , and d iv is io n . The p r in c ip le s th a t govern th e s e o p e ra tio n s h e lp to d e s c rib e th e elem en ts o f th e system o f m athem atics by p r e s c rib in g t h e i r b e h a v io r under c e r ta in c o n d itio n s . The p r in c ip le s g overning th e o p e ra tio n s t e l l a l l th a t th e elem ents can do; a l l t h e i r p r o p e r tie s a re consequent from th e s e fundam ental r u le s o f b e h a v io r. The in tr o d u c tio n to th e system o f m athem atics p ro v id e d by u n d e rsta n d in g o f th e o p e ra tio n s o f a d d itio n , sub tr a c t i o n , m u lt ip lic a ti o n , and d iv is io n p ro v id e s th e fo u n d a tio n n e c e ss a ry fo r developm ent o f th e more complex s t r u c tu r e s to be en co u n tered in l a t e r m ath em atic al stu d y . I t seems m ost im p o rta n t th a t we e v a lu a te th e p re s e n t c u r r i c ulum to d eterm in e w hether i t le a d s th e elem en tary c h ild to develop a knowledge o f th e p r in c .p ie s so b a s ic to th e s y s tem o f m a th e m a tic s. L im ita tio n s o f th e Study Scope o f th e in v e s t i g a t i o n . —T h is stu d y was lim ite d to an a n a ly s is o f u n d e rsta n d in g by 300 se v e n th -g ra d e s t u d e n ts o f th e m ath em atic al p r in c ip le s go v ern in g th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , and d i v i s io n , and th e r e la tio n s h i p o f t h i s u n d e rsta n d in g to sc o re s on t e s t s o f a r ith m e tic and m en tal a b i l i t y . The 300 s t u d e n ts w ere lo c a te d in a ju n io r h ig h sch o o l w ith in a p u b lic sch o o l d i s t r i c t in S outhern C a lif o r n ia . I n s tr u c tio n in 1 0 t h i s sch o o l was o rg a n iz e d on a s e lf - c o n ta in e d classro o m b a s is In h e re n t l i m i t a tio n s o f th e, s tu d y . —The in stru m e n t used to d e s c rib e th e se v e n th -g ra d e c h i l d 's u n d e rsta n d in g o f th e m ath em atical p r in c ip le s go v ern in g th e o p e ra tio n s o f a d d itio n , m u ltip lic a tio n , s u b tr a c tio n , and d iv is io n was developed s p e c i f i c a l l y fo r t h i s stu d y . The t e s t i s non s ta n d a rd iz e d and b ased on one p e r s o n 's co n cep t o f number. T h e re fo re , c a u tio n m ust be e x e rc is e d in m aking s p e c if ic sta te m e n ts about r e la tio n s h ip s and a b i l i t y . The c o r r e la tio n te c h n iq u e s used in t h i s stu d y to d e s c rib e r e la tio n s h i p s betw een th e t e s t o f m ath em atical o p e ra tio n s and th e o th e r v a r ia b le s do n o t n e c e s s a r ily im ply c a u s a tio n . C aution m ust be o b serv ed in m aking s p e c if ic s ta te m e n ts about cau se and e f f e c t r e la tio n s h i p s when two s e ts o f d a ta a re shown to be c o r r e la te d . B ecause o f th e c o n tro v e rsy o v er th e u se and a p p l i c a tio n o f f a c to r a n a ly s is , c a u tio n m ust be e x e rc is e d in m aking s p e c if ic s ta te m e n ts re g a rd in g th e n a tu re o f th e f a c to r s o f m ath em atical a b i l i t y d e s c rib e d by th e a n a l y s i s . The fin d in g s o f th e stu d y w ere f u r th e r lim ite d by th e v a l i d i t y o f th e in s tru m e n ts used to o b ta in th e d a ta 11 an aly zed in th e in v e s tig a tio n . The d a ta re p o r te d in th e stu d y m ust be in te r p r e te d in term s o f th e lim ita tio n s o f th e t e s t s employed to m easure th e v a rio u s t r a i t s because o f th e in h e re n t problem s o f m easurem ent. D e f in itio n o f Terms The s p e c if ic ways in w hich th e fo llo w in g term s w ere used in t h i s stu d y w a rra n t t h e i r d e f i n i t i o n . M athem atical p r i n c i p l e s . —T hroughout th e r e p o r t o f t h i s in v e s tig a tio n , th e term m a th em atical p r in c ip le s has been used to d en o te th e p r in c ip le s d e s c rib e d by th e a x io m a tic method in m a th em atic s. The p r in c ip le s a re : 1. The C lo su re Law f o r A d d itio n . An o p e ra tio n a d d itio n e x i s t s , d e s ig n a te d by +, w hich when a p p lie d to any two numbers a ,b o f th e system p roduces one and o n ly one number a + b , and t h i s number i s an elem ent o f th e sy stem . The number a + b i s c a lle d th e sum o f a and b; th e numbers a and b a r e c a lle d th e term s o f th e sum. We u se th e word "adding" to r e f e r to th e c a rry in g o u t o f th e a d d itio n o p e ra tio n , and we say t h a t a + b i s o b ta in e d by "adding a and b ." (21:18) 2. The C lo su re Law f o r M u lt ip lic a ti o n . An o p e r a tio n m u ltip lic a tio n e x i s t s , d e s ig n a te d by x, o r ', o r no symbol a t a l l , w hich when a p p lie d to any two numbers a ,b o f th e system produces one and o n ly one number a x b , o r a • b , o r sim ply ab, and t h i s number i s an elem ent o f th e system . 12 The number a x b , o r a • b , o r ab i s c a lle d th e p ro d u ct o f a and b; th e numbers a and b a re c a lle d th e f a c to r s o f th e p ro d u c t. We u se th e word " m u ltip ly in g ” to r e f e r to th e c a rry in g o u t o f th e o p e ra tio n m u ltip lic a tio n and say ab i s o b ta in e d by m u ltip ly in g a and b to g e th e r o r m u ltip ly in g a by b . (21:18) 3 . Commutative Law fo r A d d itio n . The o p e ra tio n a d d itio n has th e p ro p e rty t h a t th e sum o f two numbers i s in d ep en d en t o f th e s e q u e n tia l o rd e r o f i t s term s; in sym bols, a + b - b + a . (21:20) 4 . Commutative Law o f M u lt ip lic a ti o n . The o p e ra tio n m u ltip lic a tio n has th e p ro p e rty t h a t th e p ro d u ct o f two numbers i s in d e pendent o f th e s e q u e n tia l o rd e r o f i t s f a c to r s ; in sym bols, ab = b a . (21:20) 5 . A s s o c ia tiv e Law fo r A d d itio n . The o p e ra tio n a d d itio n h as th e p ro p e rty th a t th e sum o f th r e e numbers in a s ta te d s e q u e n tia l o rd e r i s in d ep en d en t o f th e g rouping o f two term s from th e th r e e ; in sym bols, (a + b) + c = (b + c ) . (21:22) 6 . A s s o c ia tiv e Law fo r M u lt ip lic a ti o n . The o p e ra tio n m u lt ip lic a ti o n h as th e p ro p e rty th a t th e p ro d u c t o f th r e e numbers in a s ta te d s e q u e n tia l o rd e r i s in d ep en d en t o f th e gro u p in g o f two f a c t o r s from th e th r e e ; in sym b o ls , (a b )c = a (b c) . (21:22) 7. Axiom o f S u b tr a c tio n . For any g iv en p a i r o f numbers a ,b o f th e system , th e re i s one and o n ly one number x o f th e system such th a t a + x = b; a ls o x + a = b . (21:26) 8 . D e fin itio n o f D if fe re n c e . The u n iq u e number x such th a t a + x = b i s c a lle d th e d i f f e r ence b - a (re a d "b m inus a " ) . The numbers 13 a and b a re c a lle d th e term s o f th e d i f f e r en ce. The d if f e r e n c e b - a i s a ls o d e s c rib e d a s th e number produced by s u b tr a c tin g a from b , and th e s ig n i s in te r p r e t e d as th e symbol o f an o p e ra tio n c a lle d s u b tr a c tio n . S in ce d eterm in in g x as b - a from th e r e l a tio n a + x = b i s th e problem o f fin d in g one term x o f a sum when th e sum b and th e o th e r term a a r e g iv e n , s u b tr a c tio n as an o p e ra tio n i s c a lle d th e in v e rs e o f a d d itio n . (21:26) 9 . Axiom o f D iv is io n . For any p a ir o f numbers a ,b o f th e system , o f w hich a 4 0 , th e r e i s one and o n ly one number x o f th e system such t h a t ax = b , a ls o xa = b . (21:38) 10. D e f in itio n o f Q u o tie n t. The u n iq u e number x such t h a t ax = b , w here a ^ 0 , i s c a lle d th e q u o tie n t ^ (re a d M b o v er a " ) . The numbers a ,b a r e c a lle d th e members o f th e q u o tie n t; b i s c a lle d th e n u m erato r, a th e denom inator o f th e q u o tie n t The denom inator o f a q u o tie n t m ust be a n o n -zero num ber. The q u o tie n t ^ i s a ls o d e s c rib e d as th e number o b ta in e d when b i s d iv id e d by a, o r by d iv id in g b by a , and a i s r e f e r r e d to as th e d iv is o r o f b . The b a r symbol w hich ap - . . * » -.* ! . J r p e a rs m th e q u o tie n t J 2 . i s in te r p r e t e d as d th e symbol o f an o p e ra tio n c a lle d d iv is io n . S in ce d eterm in in g x as § from ax * b , a 4 0 , i s fin d in g one f a c to r when th e p ro d u ct b and th e o th e r f a c t o r a a r e known, d iv is io n a s a p ro c e ss i s c a lle d th e in v e rs e o f m u ltip lic a t i o n . (21:39) O p e ra tio n . --T he term o p e ra tio n r e f e r s to th e p ro c e s s e s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , and 1 4 d iv is io n used to tra n sfo rm one form o f number in to a n o th e r form . M athem atical o p e r a tio n s . —The term m ath em atical o p e ra tio n s was used in th e in v e s tig a tio n to d en o te th e combined u se o f th e term s m ath em atical p r in c ip le s and o p e r a tio n s . A T e s t o f M ath em atical O p eratio n s was based on th e p r in c ip le s g overning th e o p e ra tio n s o f a d d itio n , m u l t i p l i c a tio n , s u b tr a c tio n , and d iv is io n . M athem atical s i t u a t i o n . — The term m a th e m atic al s i t - i u a tio n r e f e r s to th e v e r b a l s to r y c o n ta in in g th e m athem at i c a l o p e ra tio n s as used in A T e st o f M ath em atical O pera tio n s ■ S itu a tio n i s d i f f e r e n t from problem , as th e answer i s g iv en in th e s to r y . O rg a n iz a tio n o f th e Rem aining C h ap ters The rem ainder o f th e stu d y h as been o rg a n iz e d in to c h a p te rs ac co rd in g to th e fo llo w in g o u tlin e : C h ap ter I I c o n ta in s a review o f th e l i t e r a t u r e d e a lin g w ith (1) th e n a tu re o f th e o b je c ts o f m ath em atics, (2) m eaning and u n d e rsta n d in g as a p p lie d to m a th em atic al o p e r a tio n s , (3) co n cep ts as a p p lie d to m ath em atic al 15 o p e ra tio n s , and (4) re s e a rc h r e l a t e d to e v a lu a tio n o f th e elem en tary school c h i l d 's grow th in u n d ersta n d in g o f m ath e m a tic a l o p e r a tio n s . The d e t a i l s o f th e p ro ced u re and d esig n o f th e stu d y a re o u tlin e d in C hapter I I I . The problem has been a m p lifie d , th e sam ple more f u l l y d e s c rib e d , th e m easures o b ta in e d and th e in stru m e n ts used e la b o ra te d upon, and th e s t a t i s t i c a l te ch n iq u e s p re s e n te d in t h i s c h a p te r. C hapter IV p re s e n ts d a ta p e r ta in in g to a b i l i t y w ith th e m ath em atical p r in c ip le s g overning th e o p e ra tio n s o f a d d itio n , m u ltip lic a tio n , s u b tr a c tio n , and d iv is io n , arid th e r e l a t i o n o f th e s e d a ta to s e le c te d f a c to r s o f i n t e l l i gence, achievem ent, and a lg e b ra a p titu d e . C hapter V p re s e n ts th e d a ta r e s u lt in g from a n a ly s i s o f th e f a c to r s r e la te d to a b i l i t y w ith th e m ath em atical o p e ra tio n s fo r th e t o t a l sam ple o f 300 p u p ils , fo r th re e a b i l i t y groups determ ined from sc o re s on A T e st o f M athe m a tic a l O p eratio n s f o r each sex group. C hapter VI p re s e n ts th e summary, c o n c lu sio n s , and recom m endations o f th e stu d y . In t h i s c h a p te r, a summary o f th e in v e s tig a tio n and th e fin d in g s has been p re p a re d . F ollow ing th e summary i s a s e c tio n in w hich co n c lu sio n s 16 b ased upon th e fin d in g s o f th e stu d y have been fo rm u la te d . S e v e ra l e d u c a tio n a l im p lic a tio n s o f th e stu d y and recom m endations f o r f u r th e r re s e a r c h a re a ls o g iv en in th e c h a p te r . CHAPTER II REVIEW OF LITERATURE The hope f o r th e f u tu r e c o n s is ts in th e u n d er sta n d in g o f t h i s f a c t ; nam ely, t h a t we s h a ll alw ays be ru le d by th o se who r u le sym bols. . . . We would th e n demand t h a t our r u l e r s should be e n lig h te n e d and c a r e f u l ly s e le c te d . (2 :7 7 ) The l i t e r a t u r e review ed in t h i s c h a p te r d e a ls p r i m a rily w ith an a n a ly s is o f th e b a s ic s t r u c t u r e o f m athe m a tic a l o p e ra tio n s and th o se elem ents w hich govern t h e i r e x is te n c e and a p p lic a tio n . In clu d e d i s a review o f l i t e r a tu r e r e l a t e d to (1) th e n a tu re o f th e o b je c ts o f m athe m a tic s , (2) m eaning and u n d e rsta n d in g as a p p lie d to m athe m a tic a l o p e ra tio n s , (3) co n cep t as a p p lie d to m ath em atical o p e ra tio n s , and (4) re s e a r c h co v erin g e v a lu a tio n o f th e elem en tary sch o o l c h i l d 's grow th in u n d e rsta n d in g m athem at i c a l o p e ra tio n s . 17 18 The N atu re o f th e O b jects o f M athem atics A m ath em atic al o p e ra tio n i s governed by c e r t a in sig n s c a lle d o b je c ts o f m ath em atics. As K a tts o ff say s: In any d is c u s s io n o f th e o b je c ts o f m athe m a tic s , we m ust f i r s t c a r e f u lly s e p a r a te , in o u r m inds, m athem atics and th e m a th em atical t r e a t m ent o f c e r ta in e n t i t i e s . T h is means t h a t we m ust d is tin g u is h betw een two se m a n tic a l o b je c ts . In stu d y in g w hat i s commonly known a s p u re m athe m a tic s , th e se m a n tic a l o b je c t o f th e stu d y i s th e sig n i t s e l f . On th e o th e r hand, i f we c o n s id e r th e a p p lic a tio n o f m athem atics to a s p e c if ic f i e l d , th e n th e se m a n tic a l o b je c t o f m athem atics i s t h a t to w hich th e sig n r e f e r s , e . g . , to m o tio n , to tim e, e t c . (3 :1 6 ) Thus, i t ap p ears t h a t th e r e a re two d i s t i n c t a p pro ach es to th e stu d y o f o p e r a tio n s . For th e purpose o f t h i s stu d y th e approach to u n d e rsta n d in g o f th e o p e ra tio n s m ust be th o u g h t o f in term s o f m ath em atical a p p lic a tio n , n o t p ure m ath em atics. Among th e m ost im p o rta n t o b je c ts o f m athem atics a re th e numbers used to d e s c rib e a m ath em atical s i t u a t i o n . G o ttlo b F reg e, a p re d e c e ss o r o f R u s s e ll and W hitehead, d e s c rib e s number in th e fo llo w in g m anner. He say s: Number i s n o t a b s tra c te d from th in g s in th e way t h a t c o lo u r, w eig h t and h a rd n e ss a r e , n o r i s i t a p ro p e rty o f th in g s in th e se n se t h a t th ey are. . . . 19 Number i s n o t an y th in g p h y s ic a l, b u t n o r i s i t an y th in g s u b je c tiv e (an id e a ) . Number does n o t r e s u l t from th e annexing o f th in g to th in g . I t makes no d if fe re n c e even i f we a s s ig n a f re s h name a f t e r each a c t o f annexa tio n . . . . I t may throw some l i g h t on th e m a tte r to c o n sid e r number in th e c o n te x t o f a judgem ent w hich b rin g s o u t th e way in w hich i t i s in o r ig in a p p lie d . W hile lo o k in g a t one and th e same e x te rn a l phenomenon, I can say w ith eq u al t r u t h b o th " I t i s a copse" and " I t I s f iv e t r e e s , " o r b o th "Here a re fo u r com panies" and "Here a re 500 m en." Now what changes h e re from one judgem ent to th e o th e r i s n e ith e r any in d iv id u a l o b je c t, n o r th e w hole, th e agglom eration*'of them, b u t o n ly my te rm in o lo g y . But t h a t i s i t s e l f o n ly a sig n th a t one co n cep t has been s u b s titu te d fo r a n o th e r. T his su g g e sts . . ., th a t a sta te m e n t o f number c o n ta in s an a s s e r tio n about a c o n c ep t. T h is i s perhaps c l e a r e s t w ith th e number 0 . I f I say "Venus has 0 m oons," th e re sim ply does n o t e x i s t any moon o r ag g lo m eratio n o f moons f o r an y th in g to be a s s e r te d o f; b u t w hat happens i s th a t a p ro p e rty i s a ssig n e d to th e co n cep t "moon o f V enus," nam ely t h a t o f in c lu d in g n o th in g under i t . I f I say " th e K in g 's c a r r ia g e i s drawn by fo u r h o r s e s ," th en I a s s ig n th e number fo u r to th e concept "h o rse t h a t draws th e K in g 's c a r r ia g e ." (25:58-59) K a tts o ff summarized F re g e 's d e f in itio n o f number in th e fo llo w in g s ta te m e n t. He say s: Numbers, F rege p o in ts o u t, appear in la n guage fo r th e m ost p a r t as a d je c tiv e s and a t t r i b u te s in much th e same way as do h a rd , heavy, re d , e t c . , w hich d enote p r o p e r tie s o f e x te r n a l o b je c ts . But numbers d i f f e r from th e s e o th e r a t t r i b u t e s in th a t alth o u g h th e c o lo r o f an 20 o b je c t cannot be a l t e r e d no m a tte r w hat v iew p o in t i s ad o p ted , th e o b je c t may be co n sid e re d as b ein g e i t h e r (say ) one poem, o r tw e n ty -fo u r v e r s e s , o r a g r e a t number o f l i n e s . A gain, when we speak o f g reen le a v e s , we mean each le a f i s g reen , b u t i f we speak o f one hundred le a v e s , i t can n o t be s a id t h a t each le a f i s one hundred. The number o f a group o f o b je c ts depends upon th e p o in t o f view we adopt in lo o k in g a t th e o b je c ts . T h is i s o b v io u sly n o t th e c a se w ith c o lo r, w eig h t, e t c . , o f an o b je c t. T h e re fo re , a lth o u g h number ap p ears as an a t t r i b u t e , i t i s n o t an a t t r i b u t e w hich i s o b ta in e d by a b s tr a c tio n from e x te r n a l o b je c ts . Number i s a p ro p e rty b u t n o t a p ro p e rty o f s in g le o b je c ts . . . . F rege means t h a t number i s an o b je c tiv e a s p e c t o f a p l u r a l i t y . The number o f a p l u r a l i t y may be a l te r e d by changing o n e 's p o in t o f view ; b u t from a g iv en p o in t o f view i t i s p o s s ib le to p r e d ic a te o n ly one num ber. (3 :2 5 ) Number a s d e s c rib e d by F rege i l l u s t r a t e s th e f l e x i b i l i t y w ith w hich number may be a p p lie d , y e t p ro v id e s th e b u ild in g b lo c k s n e c e s s a ry f o r p u re m ath em atics. Number to F rege i s a c r e a tiv e to o l . He say s: I hope I may claim in th e p re s e n t work to have made i t p ro b ab le th a t th e laws o f a r i t h m e tic a re a n a ly tic judgem ents and co n seq u en tly a p r i o r i . A rith m e tic th u s becomes sim ply a developm ent o f lo g ic , and ev ery p ro p o s itio n o f a r ith m e tic a law o f lo g ic , a l b e i t a d e r iv a tiv e o n e. To ap p ly a r ith m e tic in th e p h y s ic a l s c ie n c e s i s to b rin g lo g ic to b e a r on observed f a c t s ; c a lc u la tio n becomes d e d u c tio n . The laws o f number w i l l n o t, . . ., need to sta n d up to p r a c t i c a l t e s t s i f th e y a re to be a p p lic a b le to th e e x te r n a l w orld; f o r in th e e x te r n a l w o rld , in th e w hole o f space and a l l th a t th e r e in i s , th e r e a re no c o n c e p ts, no p r o p e r tie s o f c o n c e p ts, no num bers. The laws o f number, th e r e f o r e , a re 21 n o t r e a l l y a p p lic a b le to e x te r n a l th in g s ; th e y a re n o t laws o f n a tu r e . What th e y do ap p ly to a re judgem ents ab o u t th in g s in th e e x te r n a l w orld: th e y a r e laws o f th e laws o f n a tu r e . They a s s e r t n o t connexions betw een phenomena* b u t connexions betw een judgem ents; and among judgem ents a r e i n clu d ed th e law s o f n a tu r e . (25:99) A lb e rt E in ste in * one o f th e m ost o u ts ta n d in g m athe m a tic ia n s o f t h i s o r any o th e r c e n tu ry , a s s e r ts t h a t " th e concept o f number i s a f r e e c r e a tio n o f thought* a s e l f - c re a te d to o l w hich s im p lif ie s th e o rg a n iz in g o f sen so ry e x p e rie n c e , b u t a to o l w hich cannot in d u c tiv e ly be g ain ed from sen se e x p e rie n c e " (2 6 :2 7 5 ). To a n a ly z e th e c h i l d 's u n d e rsta n d in g o f th e b a s ic o p e ra tio n s r e q u ir e s t h a t we have some system f o r o rg a n iz in g th e o p e ra tio n s in to laws o r r u le s th a t govern th e n a tu re o f t h e i r e x p e rie n c e . S in ce our u se o f a p p lie d m athem atics does n o t r e q u ir e a s p e c if ic s e t o f symbols we may u se w hat i s known as th e a x io m a tic method in m ath em atics. M aria says: A m ath em atical system d e s c rib e s i t s elem ents by s t a t i n g t h e i r p r o p e r tie s , t h a t i s , by p re s c rib in g t h e i r b e h a v io r under c e r t a in o p e r a tio n s , in a s e t o f a u t h o r it a t i v e s ta te m e n ts c a lle d axiom s. In d ev e lo p in g a m a th em atical system , we do n o t concern o u rs e lv e s w ith how th e axioms a re a r r iv e d a t —though c e r t a i n l y we u n d e rsta n d th ey a re g e n e r a liz a tio n s reac h e d by in d u c tiv e re a s o n in g from r a t i o n a l e x p e rie n c e --n o r why th e system 22 chooses th e s e p a r t i c u l a r p r o p e r tie s f o r c h a r a c te r iz in g i t s e lem e n ts. The axioms t e l l a l l t h a t th e elem en ts can do; a l l t h e i r p r o p e r tie s a re co n se quent from th e s e fundam ental r u le s o f b e h a v io r. The o n ly re q u ire m e n t on th e s e t o f axioms i s th a t th e y be f r e e o f c o n tr a d ic tio n . (2 1 :9 ) The ax io m atic method p e rm its us to d e s c rib e th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , and d iv is io n th ro u g h th e b eh a v io r o f t h e i r ele m e n ts. W ith t h i s lim ite d in tr o d u c tio n to th e o b je c ts o f m ath em atics, th e problem o f a n a ly z in g th e o p e ra tio n s was s e p a ra te d in to two d i s t i n c t a r e a s , one o f p u re m athem atics and one o f a p p lie d m ath em atics. For th e p urpose o f t h i s stu d y th e a p p lie d m ath em atical approach was u se d . T e st item s w ere d esig n ed by a p p ly in g F re g e 's d e f in i t i o n o f num b e r to th e m a th em atic al s i t u a t i o n , and M a ria 's d e f in it io n o f th e ax io m atic method to th e m u ltip le ch o ic e item s used to d e s c rib e th e o p e ra tio n . Number, as d e s c rib e d by F reg e, p o in ts o u t th e com p le x n a tu re o f i t s u se , and th e n e c e s s ity f o r some a g re e ment as to th e term s used in e v a lu a tin g i t s a p p lic a tio n . 23 Meaning and U nderstanding as T h e y A p p Iv to M athem atical O p eratio n s The c o n s ta n t u se o f th e te r n s "m eaning" and "u n d er s ta n d in g " in th e l i t e r a t u r e r e l a te d to elem en tary a r i t h m e tic has r e s u lt e d in much co n fu sio n ab o u t th e term s and some doubts as to w hether everyone i s ta lk in g ab o u t th e same th in g . Such term s as 'W a n in g ," " r e l a t i o n s h i p s ," and "u n d e rsta n d in g " a re fre q u e n tly used in te rc h a n g e a b ly as though th e y w ere synonym ous. H endrix (2 7 :3 3 4 -3 9 ) c a lle d a t te n t io n to th e problem by p o in tin g o u t th a t th e term s 'W a n in g " and "tin d e rstan d in g " a re n o t synonyms and, hence, to u se them in te rc h a n g e a b ly in th e d is c u s s io n o f m athem at i c a l o p e ra tio n s o n ly se rv e s to b lo c k th e com m unication p ro c e s s . Van Engen d e fin e s "m eaning" in th e fo llo w in g man n e r: A word o r a symbol i s n o t alw ays used in th e same way. A symbol may have a m eaning a c c o rd in g to th e way i t i s used in r e l a t i o n to o th e r w ords, o r a c co rd in g to how i t i s u sed in r e l a t i o n to o b je c ts , o r a c co rd in g to th e p u rp o se f o r w hich i t i s u se d . A cco rd in g ly , i t i s custom ary to - speak o f th e "dim ensions o f m ean in g ." They a re : The S y n ta c tic Dimension. Words and symbols have m eaning b ecau se o f th e way in w hich th ey 24 a re used in r e l a t i o n to th e o th e r w ords in a sen te n c e o r fo rm u la. The P ragm atic D im ension. Words and symbols w i l l v ary in m eaning acc o rd in g to t h e i r purpose and consequence so f a r as a p a r t i c u l a r in d iv id u a l o r o rg a n iz a tio n i s co n cern ed . . . . M athe m a tic s as m athem atics does n o t u se w ords o r symbols in th e prag m atic sen se b u t th e " f r in g e 1 1 m eanings o f th e symbol ’'m athem atics" f o r th e p u p il i s o f supreme i n t e r e s t to th e te a c h e r as an e d u c a to r. The Sem antic D im ension. T h is i s th e t h ir d dim ension o f m eaning o f prim ary im portance to th e te a c h e r o f m ath em atics, w h eth er an elem en ta r y te a c h e r o f m ath em atics, o r a secondary te a c h e r o f m a th em atic s. Of c o u rse , when th e c h ild le a rn s to combine symbols to ex p ress id e a s he i s em ploying words in th e s y n ta c ti c a l dimen s io n . However, any s e n s ib le th e o ry o f i n s t r u c tio n would i n s i s t t h a t th e c h ild f i r s t le a rn t h a t th e in d iv id u a l symbols r e p r e s e n t o b je c ts , o th e r sym bols, sim ple e v e n ts , o r m en tal con s t r u c t s . These o b je c ts , o r sym bols, a re c a lle d r e f e r e n ts o f th e g iv en word o r sym bol. (4 :7 0 -7 1 ) As s ta te d by Van Engen, th e te a c h e r i s e s p e c ia lly i n te r e s te d in th e s y n ta c tic and th e sem antic dim ensions o f m eaning. To apply th e s e d e f in itio n s to m ath em atical o p e r a tio n s would re q u ire b o th m eanings. The id e a s ex p ressed in a given problem would r e q u ir e (1) th e sem antic dim ension o f m eaning, " th e r e f e r e n ts fo r g iv en words and sym bols" used in th e problem , and (2) th e u se o f th e s y n ta c tic dim ension, " th e com bining o f words and symbols to ex p ress th e id e a s b ein g em ployed." 25 To a tte m p t e x p o s itio n o f a l l th e v a rio u s m eanings o f u n d e rsta n d in g would be o f l i t t l e v a lu e to any d is c u s sio n o f m ath em atical o p e r a tio n s . U n d erstan d in g o f a g iv en problem r e f e r s to som ething t h a t i s in th e p o s s e s s io n o f th e c h i ld . The c h ild who u n d e rsta n d s i s aw are o f a s a t i s fy in g f e e lin g , a p s y c h o lo g ic a l c lo s u re , w hich r e s u l t s from h av in g f i t t e d e v e ry th in g in to i t s p ro p e r p la c e . Of c o u rs e , t h i s p sy c h o lo g ic a l c lo s u re m ust be t e s te d b ecau se th e c h ild may th in k he u n d e rsta n d s when he r e a l l y does n o t ( 4 :7 5 ). The above d is c u s s io n may be summarized by sa y in g t h a t m eaning, a s a p p lie d to m ath em atical o p e r a tio n s , i s t h a t w hich i s re a d in to th e sym bols w ith in th e problem by th e c h i ld . He f i r s t r e a l i z e s t h a t th e symbols a r e a sub s t i t u t e f o r o b je c ts . F u rth e r u n d e rsta n d in g i s an o rg a n i z a tio n a l p ro c e s s . The c h ild who f u l l y u n d e rsta n d s a ls o i s in p o s s e s s io n o f th e cau se and e f f e c t r e la tio n s h i p o f an o p e ra tio n w ith in th e problem and th e lo g ic a l im p lic a tio n s and sequences o f th o u g h t t h a t u n ite two o r more sym bols o r s ta te m e n ts by means o f lo g ic . 26 C oncepts as A pplied to M athem atical O p eratio n s In s p i t e o f th e freq u en cy w ith w hich some edu c a to r s u se th e term concept* in elem en tary a r ith m e tic and o th erw ise* th e re i s s t i l l much to be le a rn e d ab o u t i t s n a tu re and a p p lic a tio n . A lthough th e re i s s t i l l much to be learned* th e re has been a g r e a t d e a l o f th o u g h t and re s e a rc h on th e s u b je c t. Venake say s: They (c o n c e p ts) m ust be re g a rd e d as s e le c t i v e mechanisms in th e m en tal o rg a n iz a tio n o f th e in d iv id u a l, ty in g to g e th e r sen so ry im p res sions* th u s a id in g in th e i d e n t i f i c a t i o n and c l a s s i f i c a t i o n o f o b je c ts . But co n cep ts in v o lv e more th an th e in te g r a tio n o f se n se im p ressio n s* a g a in s t th e background o f w hich re c o g n itio n o c curs* f o r th ey a r e lin k e d w ith sym bolic re sp o n se s w hich may be a c tiv a te d w ith o u t th e p h y s ic a l p re se n c e o f e x te r n a l o b je c ts . T hat is* co n c e p ts can be g iv en nam es—can be d etach ed from s p e c if ic in sta n c e s* by means o f a w o rd --an d used to m anip u l a t e ex p e rie n c e o v er and beyond th e more sim ple re c o g n itio n f u n c tio n . The sym bolic response* however* sta n d s f o r w hatever i t h as been lin k e d w ith in th e p re v io u s e x p e rie n c e o f th e organism and depends upon how th a t p a s t e x p e rie n c e i s o rg a n iz e d . (5 :5 ) As w ith any s u b je c t t h a t s t i l l r e q u ir e s a d d itio n a l p ro o f f o r th e n a tu re o f i t s e x is te n c e , ''co n ce p t" p ro v id e s a n o th e r a re n a f o r c o n tro v e rsy n o t o n ly f o r p sy ch o lo g y , b u t p h ilo so p h y as w e ll. Bruner* Goodnow* and A u stin sum up 27 th e e x is tin g c o n tro v e rs ie s by say in g : A c e r t a in co n fu sio n i s c o n trib u te d by th e p h ilo s o p h ic a l c o n tro v e rsy o ver th e n a tu re o f co n cep ts as u n iv e rs a ls : w hether a u n iv e r s a l i s som ething t h a t r e s id e s in o b je c ts and may be d i r e c t l y known o r w hether i t i s in a P la to n ic realm o f u n iv e r s a ls th a t can o n ly be "prehended" in c o rru p te d form o r w hether i t i s som ething th a t i s imposed on r e g u l a r i t i e s in n a tu re by a c o n c e p tu a liz in g m ind. And th e co n fu sio n comes p a r tly from c o n tro v e rs ie s w ith in p sy c h o lo g ic a l th e o ry . We f a i l to see th a t th e p h ilo s o p h ic a l c o n tro v e rsy has any b e a rin g on th e e m p iric a l stu d y o f c o n c e p tu a liz in g b e h a v io r. The p r i n c i p a l p s y c h o lo g ic a l c o n tro v e rsy has been between two v iew s. T here a re th o se who u rg e th a t a co n c ep t, p s y c h o lo g ic a lly , i s d e fin e d by th e com mon elem en ts sh a red by an a rra y o f o b je c ts and th a t a r r iv in g a t a co n cep t in d u c tiv e ly i s much l ik e " a r r iv in g a t " a com posite photograph by superim posing in s ta n c e s on a common p h o to g rap h ic p la te u n t i l a l l th a t i s id io s y n c r a tic i s washed o u t and a l l th a t i s common em erges. A second school o f th o u g h t h o ld s t h a t a concept i s n o t th e common elem en ts in an a r r a y , b u t r a th e r i s a r e l a t i o n a l th in g , a r e la tio n s h i p betw een c o n s t i t u en t p a r t p ro c e s s e s . (6 :243-244) C oncept as a p p lie d to m ath em atical o p e ra tio n s r e q u ire s a w orking d e f i n i t i o n . For th e p u rp o ses o f t h i s r e s e a rc h , a d e f in i t i o n as used by B ru n er, Goodnow, and A u stin has been h e lp fu l in d ev elo p in g a c o n s tru c t f o r d e s c rib in g knowledge o f m ath em atical o p e r a tio n s . They s t a t e : We have found i t more m ean in g fu l to re g a rd a co n cep t as a netw ork o f s ig n s ig n if ic a te [s ic ] 28 in fe re n c e s by w hich one goes beyond a s e t o f ob serv ed c r i t e r i a l p r o p e r tie s e x h ib ite d by an o b je c t o r ev en t to th e c la s s i d e n t i t y o f th e o b je c t o r ev en t in q u e s tio n , and th e n c e to a d d i tio n a l in fe re n c e s about o th e r unobserved p ro p e r t i e s o f th e o b je c t o r e v e n t. W e se e an o b je c t th a t i s re d , sh in y , and ro u n d ish and i n f e r th a t i t i s an a p p le , i t i s a ls o e d ib le , ju ic y , w il l r o t i f l e f t u n r e f r ig e r a te d , e t c . The w orking d e f in it io n o f a co n cep t i s th e netw ork o f i n f e r ences th a t a re o r may be s e t in to p la y by an a c t o f c a te g o r iz a tio n . (6:244) R esearch on th e E lem entary School C h ild ’s Growth in U n d erstan d in g o f M athem atical O p eratio n s R esearch in elem en tary a r ith m e tic h as v ery l i t t l e to o f f e r any d is c u s s io n d e a lin g w ith th e elem en tary c h i l d 's u n d e rsta n d in g o f th e m ath em atical o p e ra tio n s o f a d d itio n , m u ltip lic a tio n , s u b tr a c tio n , o r d iv is io n . P a st a tte m p ts have been based on s tu d ie s o f co m p u tatio n al s k i l l s and a n a ly s is o f v e rb a l problem s o lv in g , n e i th e r g iv in g much in s ig h t in to th e c h i l d 's a b i l i t y to re a so n th ro u g h th e u se o f m ath em atical o p e r a tio n s . A b u l l e t i n o f th e U. S. O ffic e o f E ducation summ arizes re c e n t re s e a r c h as i t a p p lie s to m athem atics a t th e elem en tary sch o o l l e v e l . I t s t a t e s : The in v e s tig a to r s o f problem s o lv in g in a r ith m e tic ag reed th a t many f a c t o r s w ere more 29 im p o rtan t th an co m p u tatio n al s k i l l . On th e o th e r hand, th e s e same in v e s tig a tio n s uncovered d i f f e r e n t s e ts o f f a c to r s w hich w ere im p o rtan t in p ro b le m -so lv in g . Would a d d itio n a l "n" in v e s tig a tio n s fin d "n" a d d itio n a l d is c r e te s e ts o f f a c to r s ? I s th e number o f v a r ia b le s in flu e n c in g su c cess in p ro b lem -so lv in g so la rg e th a t re c ip e s and in s t r u c tio n a l s t r a t e g i e s a r e bound to be f r u i t l e s s ? The re s e a rc h on m eanings and u n d ersta n d in g u s u a lly re v e a ls th a t p u p ils ta u g h t com putations o n ly d o n 't show much u n d e rsta n d in g . On th e o th e r hand, p u p ils o f av erag e IQ and above who re c e iv e in s tr u c tio n em phasizing m eaning and u n d e rsta n d in g n o t o n ly le a rn to compute j u s t as w e ll a s th o se ta u g h t com putations b u t a ls o come to u n d e rsta n d a r ith m e tic b e t t e r . The re s e a rc h o f 1957-58 does n o t d is p u te th e s e s ta te m e n ts . I t i s a p le a s a n t s u r p r is e when s tu d e n ts le a rn what th e te a c h e r has n o t ta u g h t them; i t i s h a rd ly sh o ck in g , when p u p ils do n o t le a rn what th e y have n o t been ta u g h t. (7 :1 8 ) The comments o f le a d e rs in th e f i e l d o f-e le m e n ta ry a r ith m e tic have le a d us to b e lie v e th a t th e elem en tary c h ild is n o t m ature enough to go beyond th e elem en tary p hases so c a r e f u lly l a i d o u t by th e e x p e rts o f th e a r i t h m e tic cu rric u lu m . T y p ic a l o f th e s e comments i s th e f o l low ing: The elem en tary sch o o l le a r n e r i s too imma tu r e to a p p re c ia te th e m a th e m a tic ia n 's s t a t e m ent, " s u b tra c tio n is d e fin e d as th e o p p o s ite o f a d d itio n ," o r "c - a = b i f th e sum o f a and b is c ," o r " th e d if fe re n c e betw een a minuend (c) and a su b trah e n d (a) i s th e number which added to a g iv es a sum o f c ." These 30 sta te m e n ts a re to o a b s t r a c t , to o rem ote from th e l e a r n e r 's e x p e rie n c e . Hence a slow , p lan n ed , s e q u e n tia l developm ent o f th e s e c o n cep ts and r e la tio n s h i p s i s e s s e n t i a l . (8 :7 2 ) A more r e c e n t approach seems to be t h a t th e le a r n e r sh o u ld be guided to d is c o v e r o r re d is c o v e r th e r e la tio n s h ip betw een th e v a rio u s o p e r a tio n s . A ty p ic a l example o f t h i s approach i s th e fo llo w in g : 5 . I f 17 + 28 * 45, th e n 45 - 28 - ? . and 45 - 17 - ? . Use e x e rc is e 5 to show th a t: The sum ( in a d d itio n ) co rre sp o n d s to th e minuend ( in s u b tr a c tio n ) One o f th e addends ( in a d d itio n ) becomes a su b trah e n d ( in s u b tr a c tio n ) The o th e r addend becomes a re m a in d e r. (8 :7 3 ) To d a te , l i t t l e o r no ev id en ce i s a v a ila b le w hich h as in any way in d ic a te d t h a t th e elem en tary sch o o l le a r n e r i s to o im m ature to u n d e rsta n d m ath em atics as d e s c rib e d by m a th e m a tic ia n s. The sta te m e n t t h a t " th e elem en tary sch o o l le a r n e r i s too im m ature to a p p r e c ia te m a th em atic al s t a t e m ents" seems based on an assu m p tio n n eed in g e x p e rim en ta l ev id en ce . Summary The review o f th e l i t e r a t u r e r e l a ti n g to m athem at i c a l o p e ra tio n s seems to p ro v id e th e b ases f o r th e fo llo w in g co n clu sio n s re g a rd in g th e o p e ra tio n s o f a d d itio n , m ul t i p l i c a t i o n , s u b tr a c tio n , and d iv is io n : 1. Any attem p t to m easure th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c tio n , o r d iv is io n m ust be approached th ro u g h a p p lie d m a th em atic s. Pure m athem atics re q u ir e s a fo u n d atio n beyond th a t p ro v id ed th e p re s e n t se v en th -g ra d e s tu d e n t. 2. The m eaning o f number as d e s c rib e d by F rege p o in ts o u t th e complex n a tu re o f i t s a p p li c a tio n to a p p lie d m a th em atic s. Concepts re g a rd in g th e a p p lic a tio n o f number need f u r th e r tre a tm e n t th a n th e scope o f t h i s stu d y p e r m its . 3 . "M eaning," as a term a p p lie d to m ath em atical o p e ra tio n s , m ust be approached w ith c a u tio n . R esearch in elem en tary a r ith m e tic has l i t t l e to o f f e r in any d is c u s s io n o f m a th em atical o p e r a tio n s . The approach used to d e s c rib e th e a b i l i t y a c h ild has w ith th e o p e ra tio n s i s b ased on knowledge a c h ild has o f th e number f a c t s and co m p u tatio n al s k i l l r a th e r th a n on u n d e rsta n d in g a s used in t h i s stu d y . CHAPTER III PROCEDURES A N D SOURCE OF D A TA In C hapter I I i t was p o in te d o u t t h a t p a s t re* se a rc h had l i t t l e b e a rin g on th e stu d y o f th e m a th em atic al p r in c ip l e s go v ern in g th e o p e ra tio n s o f a d d itio n , m u l t i p l i c a tio n , s u b tr a c tio n , and d iv is io n . In th e p a s t, r e s e a r c h r e la t e d to th e s e o p e ra tio n s h as d e a lt p r im a r ily w ith com p u ta tio n a l s k i l l s and a n a ly s is o f v e rb a l problem s o lv in g . The p re s e n t stu d y was d esig n ed to p ro v id e in fo rm a tio n r e l a t e d to f a c to r s w hich have been ig n o red o r in a d e q u a te ly tr e a te d in p re v io u s r e s e a r c h . The Sample The sam ple was drawn from an elem en tary sch o o l d i s t r i c t in S outhern C a lif o r n ia . A ll o f th e se v e n th -g ra d e p u p ils s tu d ie d came from one o f th e two ju n io r h ig h sc h o o ls in th e d i s t r i c t . A ll in s t r u c tio n in a r ith m e tic was 33 o rg an iz ed on a s e lf- c o n ta in e d classroom b a s is . The v a rio u s t e s t s used in t h i s stu d y were ad m in is te re d to a l l se v en th -g ra d e s tu d e n ts who w ere p re s e n t on th e days sch ed u led f o r t e s t i n g . T here w ere ap p ro x im ately 346 se v en th -g ra d e s tu d e n ts e n ro lle d in th e sc h o o l. S tu d en ts who w ere a b se n t fo r any reaso n when a t e s t o r t e s t s e c tio n was a d m in iste re d w ere p e rm itte d to make up th e t e s t w ith an o th er s e c tio n given on an o th e r day. One make-up t e s t was g iv en f o r th e a r ith m e tic achievem ent and m ath em atical o p e ra tio n s t e s t s . F if te e n s c o re s —a language and non language IQ, raw sc o re s f o r a rith m e tic re a so n in g and fu n d a m e n ta ls, a lg e b ra a p titu d e , and one raw sc o re f o r each o f th e te n m ath em atical p r i n c i p l e s —w ere re c o rd e d fo r each p u p il. Com plete d a ta w ere c o lle c te d on 148 g i r l s and 152 b o y s, t o ta lin g 300 p u p ils , o r ap p ro x im ately 86 p er c e n t o f th e t o t a l p o p u la tio n o f se v e n th -g ra d e s tu d e n ts in th e ju n io r h ig h sc h o o l. A ll d a ta w ere secu red d u rin g th e 1961-62 sch o o l y e a r. In stru m e n ts and P ro ced u res Used in O b tain in g I n te llig e n c e and Achievement D ata I n te llig e n c e , a r ith m e tic achievem ent, and a lg e b ra a p titu d e w ere determ ined through th e u se o f s ta n d a rd iz e d 35 group t e s t s . The p re s e n t s e c tio n o f t h i s c h a p te r d e s c rib e s th e in stru m e n ts and p ro ced u res used in o b ta in in g th e d a ta . The in stru m e n t used to m easure i n te lli g e n c e was th e C a lif o rn ia Short-Form T e st o f M ental M a tu rity . J u n io r High L e v e l. G rades 7 -8 -9 . 1957, S-Formu p u b lish e d by th e C a lif o rn ia T e st B ureau. Language and no n -lan g u ag e i n t e l l i gence q u o tie n ts w ere o b ta in e d from t h i s s ta n d a rd iz e d t e s t . The t e s t was a d m in iste re d d u rin g th e f i r s t two weeks o f O ctober by th e two co u n selo rs in th e ju n io r h ig h sc h o o l. The t e s t s w ere m achine sco red in th e d i s t r i c t o f f i c e . The in stru m e n t used to m easure a r ith m e tic a c h ie v e ment was th e a rith m e tic s e c tio n o f th e C a lif o r n ia A chieve m ent T e s t s . J u n io r High L e v e l. G rades 7 -8 -9 * Form X, pub lis h e d by th e C a lif o rn ia T e st B ureau. Raw sc o re s fo r a rith m e tic reaso n in g and fundam entals w ere o b ta in e d from th e t e s t . The t e s t was a d m in iste re d in th e f i r s t two weeks o f May by th e two c o u n se lo rs in th e ju n io r h ig h sc h o o l. The t e s t s w ere m achine sco red in th e d i s t r i c t o f f i c e . The in stru m e n t used to m easure a lg e b ra a p titu d e was th e Survey T e st o f A lg eb raic A p titu d e p u b lish e d by th e C a lif o rn ia T e st B ureau. Raw sc o re s f o r a lg e b ra a p titu d e 36 w ere o b ta in e d from th e t e s t . The t e s t was a d m in iste re d in th e l a s t two weeks o f A p ril by th e two c o u n se lo rs in th e ju n io r h ig h sc h o o l. The t e s t s w ere m achine sco red in th e d i s t r i c t o f f i c e . D e sc rip tio n o f th o Sample T able 1 p re s e n ts th e means and sta n d a rd d e v ia tio n s r e l a t e d to th e language and n on-language IQ o f th e sample by sex (152 boys and 148 g i r l s ) and th e t o t a l group o f 300 s tu d e n ts . E xam ination o f t h i s ta b le shows th a t th e mean language IQ o f th e boys was 104.29 w ith a sta n d a rd d e v ia tio n o f 1 5 .89. The mean n on-language IQ o f th e boys was 105.04 w ith a sta n d a rd d e v ia tio n o f 1 6 .6 9 . The ran g e in language IQ fo r th e boys was 59 to 141, and th e ran g e in non -lan g u ag e IQ was 62 to 143. The mean language IQ f o r th e g i r l s was 103.56 w ith a sta n d a rd d e v ia tio n o f 1 7 .7 5 . The mean non-language IQ fo r th e g i r l s was 104.58 w ith a s ta n d a rd d e v ia tio n o f 1 8.33. The ran g e in language IQ fo r th e g i r l s Was 71 to 145, and th e ran g e in non -lan g u ag e IQ was 63 to 154. The mean o f th e t o t a l group in language and non-language IQ was 103.56 and 104.42, r e s p e c tiv e ly . The sta n d a rd d e v ia tio n was 17.76 fo r language IQ and 18.47 37 TABLE 1 M EA N S A N D STA N D A RD DEVIATIONS RELATED TO THE LA N G U A G E A N D NON-LANGUAGE IQ OF THE SAM PLE Language 10 Non-Language 10 Group Range Mean S . D.* Range Mean S . D. M ales 59-141 104.29 15.89 62-143 105.04 16.69 Females 71-145 103.56 17.75 63-154 104.58 18.33 T o ta l Group 59-145 103.56 17.76 62-154 104.42 18.47 *S.D . * S tan d ard D e v ia tio n . 38 f o r non-language IQ. The ran g e in lan g u ag e IQ f o r th e t o t a l group was 39 to 145. In n o n -lan g u ag e IQ, th e ran g e o f th e t o t a l group was 62 to 154. T ab le 2 p re s e n ts th e m eans, s ta n d a rd d e v ia tio n s , and mean grade p lacem en ts, w here p o s s ib le , o f th e raw sc o re s in a rith m e tic re a s o n in g , a r ith m e tic fu n d am en tals, and a lg e b ra a p titu d e . A review o f th e s e d a ta in d ic a te s th a t th e mean raw sc o re s f o r a r ith m e tic re a s o n in g w ere 32.33 f o r th e boys, 30.51 f o r th e g i r l s , and 31.34 f o r th e t o t a l group. The sta n d a rd d e v ia tio n s f o r th e raw sc o re s o f a r ith m e tic re a so n in g w ere 1 0 .8 7 , 1 0 .5 5 , and 1 0 .8 4 , r e s p e c tiv e ly . The mean g rad e p lacem en ts in a r ith m e tic re a s o n in g w ere 8 .7 f o r th e b o y s, 8 .5 f o r th e g i r l s , and 8 .6 fo r th e t o t a l g ro u p . The mean raw sc o re s f o r a r i t h m e tic fundam entals w ere 52.24 f o r th e b o y s, 52.36 f o r th e g i r l s , and 52.13 fo r th e t o t a l g ro u p . The s ta n d a rd d e v ia tio n s f o r th e mean raw sc o re s o f a r ith m e tic fundam entals w ere 1 6 .7 2 , 1 6 .7 1 , and 16.94 r e s p e c tiv e ly . The mean g rad e placem ents in a r ith m e tic fu n d am en tals w ere 8 .1 f o r th e boys, 8 .1 f o r th e g i r l s , and 8 .1 f o r th e t o t a l g ro u p . The mean raw sc o re s f o r a lg e b ra a p titu d e w ere 70.07 f o r th e b o y s, 30.11 fo r th e g i r l s , and 3 0 .0 2 f o r th e t o t a l g ro u p . 39 TABLE 2 MEANS; STA ND A RD DEVIATIONS, A N D M E A N G RA D E PLACEM ENT OF R A W SCORES IN ARITHMETIC ACHIEVEMENT A N D ALGEBRA APTITUDE T e st M ales Fem ales T o ta l Group A rith m e tic R easoning Mean 32.33 30.51 3 1.34 S.D .* 10.87 10.55 10.84 M .G.P.* 8 .7 8 .5 8 .6 A rith m e tic Fundam entals Mean 52.24 52.36 52.13 S.D. 16.72 16.71 16.94 M.G.P. 8 .1 8 .1 8 .1 A lgebra A p titu d e Mean 30.07 30.11 30.02 S.D. 13.61 11.94 12.88 M.G.P. w *S.D . ■ S tan d ard D ev ia tio n ; M .G.P. » Mean Grade P lacem ent. 40 The sta n d a rd d e v ia tio n s f o r th e mean raw sc o re s o f a lg e b ra a p titu d e w ere 13.61, 11.94, and 12.88 r e s p e c tiv e ly . No g rad e placem ents w ere given, f o r a lg e b ra a p t it u d e . The n a tu re o f th e d is t r i b u t i o n o f sc o re s f o r boys and g i r l s ta k en in d iv id u a lly ten d s to produce means th a t appear s l i g h t l y abnorm al when compared w ith th e means o f th e t o t a l gro u p . These d iffe re n c e s may be accounted fo r by th e n a tu re o f th e sc o re s fo r g i r l s as compared w ith boys, th e s iz e o f th e sam ple, and th e s l i g h t d iffe re n c e s betw een m eans. The T est o f M athem atical O p eratio n s T h is s e c tio n o f C hapter I I I p re s e n ts a d e s c r ip tio n o f th e T e st o f M athem atical O p eratio n s d esig n ed f o r t h i s stu d y and th e p ro ced u res used in o b ta in in g th e d a ta from i t . The T e st o f M athem atical O p eratio n s was d esig n ed to t e s t th e se v en th -g ra d e c h i l d 's u n d ersta n d in g o f th e m athe m a tic a l p r in c ip le s governing th e o p e ra tio n s o f a d d itio n , m u ltip lic a tio n , s u b tr a c tio n , and d iv is io n th ro u g h th e u se o f t e s t item s re q u ir in g a n a ly s is o f word problem s to de term in e th e o p e ra tio n s in v o lv e d . The m ath em atical 41 p r in c ip le s in c lu d e d in th e t e s t w ere based on th e ax io m atic method as d e s c rib e d in C hapter I I . The m ath em atical p r in c i p le s , as d e fin e d in C hapter I , w ere: (1) th e c lo s u re law f o r a d d itio n , (2) th e c lo s u re law f o r m u lt ip lic a ti o n , (3) th e com m utative law fo r a d d itio n , (4) th e com m utative law fo r m u ltip lic a tio n , (3) th e a s s o c ia tiv e law fo r a d d itio n , (6) th e a s s o c ia tiv e law f o r m u lt ip lic a ti o n , (7) th e d e f i n i t i o n o f d if f e r e n c e , (8) th e axiom o f s u b tr a c tio n , (9) th e d e f in it io n o f q u o tie n t, and (10) th e axiom of. d iv is io n . Each p r in c ip le i s d e s c rib e d in d e t a i l in C hapter V. T e st ite m s .--A fu n c tio n a l d e f in it io n o f co n cep t as u sed by B runer, Goodnow, and A u stin was found h e lp f u l in th e s e le c tio n o f a p p ro p ria te s itu a ti o n s in w hich th e se v e n th -g ra d e c h ild m ight show h is u n d e rsta n d in g o f th e m athe m a tic a l p r in c ip le s in v o lv ed in th e o p e r a tio n s . They d e fin e d a concept as " th e netw ork o f s ig n i f ic a n t in fe re n c e s by w hich one goes beyond a s e t o f o b serv ed c r i t e r i a l p ro p e r t i e s e x h ib ite d by one o b je c t o r ev en t to th e c la s s id e n t i t y o f th e o b je c t o r ev en t in q u e s tio n and th en ce to a d d itio n a l in fe re n c e s about o th e r unobserved p r o p e r tie s o f th e o b je c t o r ev e n t" (6 :2 4 4 ). To ap p ly t h i s d e f i n i t i o n , e x e rc is e s based on e x tra p o la tio n w ere used to d eterm in e 42 w h eth er o r n o t th e s tu d e n t co u ld go beyond th e lim its o f th e d a ta o r in fo rm a tio n g iv en and make c o r r e c t a p p lic a tio n and e x te n s io n s o f th e d a ta in v o lv e d . A d d itio n a l re q u ire m e n ts f o r th e s e le c tio n o f ap p ro p r i a t e s i t u a t i o n s w ere (1) p ro b le m a tic s i t u a t i o n s d esig n ed to e lim in a te th e p o s s i b i l i t y o f r o t e knowledge o f number com binations b e in g used as th e key to a p p lic a tio n and use o f th e o p e r a tio n a l p r in c ip l e , and (2) p ro b le m a tic s i t u a t i o n s , to g e th e r w ith th e n e c e s s a ry elem en ts and an sw ers, d esig n ed to r e p r e s e n t problem s o r d i n a r i l y c o n sid e re d a t th e fo u rth -g ra d e le v e l o f d i f f i c u l t y . T h is l a t t e r p re c a u tio n was ta k en to p re v e n t problem d i f f i c u l t y from in flu e n c in g s c o re s . S in ce th e t e s t was d esig n ed as a power t e s t , no tim e lim its w ere im posed. E x tra p o la tio n a s d e s c rib e d in th e Taxonomy o f Edu c a tio n a l Obi e c tiv e s i s d e fin e d as " th e e x te n s io n o f tre n d s o r te n d e n c ie s beyond th e g iv en d a ta to d eterm in e im p lic a t i o n s , co nsequences, c o r o l l a r i e s , e f f e c t s , e t c . , w hich a re .in acco rd an ce w ith th e c o n d itio n s d e s c rib e d in th e o r ig in a l com m unication" (1 2 :2 0 5 ). T h is d e f i n i t i o n was a p p lie d to t e s t item s in th e fo llo w in g manner: 1. c + f ■ z Nan i s c y e a rs o ld . Her m other 2 . f - c = z i s f y e a rs o ld . Nan i s z y e a rs 3 . c - f * z younger th a n h e r m o th er. 4 . z - c * f 5 . none The c h ild was ask ed , "Which o f th e com b in atio n s on A% ! th e r i g h t g iv e s us th e same in fo rm a tio n found in th e s to r y I f none o f th e com binations g iv e s us th e same in fo rm a tio n th e n s e le c t number 5, la b e le d n o n e ." None was in c lu d e d in th e m u ltip le ch o ice ite m s to p ro v id e an a l t e r n a t i v e t h a t , i t was hoped, would e lim in a te as much g u e ssin g as p o s s ib le D esign o f th e problem s i t u a t i o n s a t th e f o u r th - g rad e le v e l re q u ire d an a n a ly s is o f th e c u r r e n t t e x t , The New L earn in g Numbers (1 3 ), u sed a t th e fo u rth -g ra d e le v e l in th e sch o o l d i s t r i c t from w hich th e sam ple was ta k e n . The v o c a b u la ry used in th e problem s i t u a ti o n s was lim ite d to th e v o c a b u lary used in t h i s t e x t . T e s t d i r e c t i o n s .--T h e d ir e c tio n s f o r A T e st o f M athem atical O p eratio n s w ere developed and based on con c lu s io n s by McGrath in a stu d y e n t i t l e d "Problem S o lv in g E ffic ie n c y as A ffe c te d by A ccessory R em arks." In th e con c lu s io n s o f th e stu d y , M cGrath sa y s: . A s in g le word o r p h ra s e —used e a r ly and on th e o u t s k i r t s o f o n e ' s a t t e n t i o n —m arkedly 44 r a i s e s and low ers o n e 's e f f ic ie n c y in a su b seq u en t p ro b lem -so lv in g ta s k . a . A g iv e n w o rd -la b e l may have a r e l a t i v e l y one-way e f f e c t on a s t u d e n t 's e f f ic ie n c y ; f o r exam ple, th e w o rd -la b e l "e asy " seemed to r a i s e e f f ic ie n c y f o r a l l e x p e rim e n ta l c o n d itio n s in w hich i t was u se d . b . A g iv e n w o rd -la b e l may have a v a r ia b le e f f e c t on o n e 's e f f ic ie n c y ; f o r exam ple, th e la b e l "v e ry d i f f i c u l t " r a is e d e f f i c ie n c y in th e e x p e rim e n ta l c o n d itio n c a lle d " p u z z le s" b u t low ered e f f ic ie n c y in th e e x p e rim en ta l c o n d itio n c a lle d "m a th e m a tic s," though n o t s i g n i f i c a n t l y . c . A g iv en w o rd -la b e l, when combined w ith a n o th e r w o rd -la b e l, can c o n tr ib u te an e f f e c t p re v io u s ly n o n - e x is te n t; f o r exam ple, w o rd -la b e ls "m athem atics" and " p u z z le s" in th em selv es caused no s i g n i f i c a n t a l t e r a t i o n in p ro b lem -so lv in g e f f ic ie n c y ; how ever, in co m bination w ith o th e r la b e l s , such as "v ery ea sy " and "v ery d i f f i c u l t , " t h e i r c o n trib u tio n suddenly n e a re d s ig n if ic a n c e . "M athe m a tic s " la b e ls th e n red u ced e f f ic ie n c y and " p u z z le s" r a is e d i t . (2 2 :7 8 -7 9 ) W ith th e above c o n c lu sio n s in m ind, th e t e s t admin i s t e r e d to th e se v e n th -g ra d e s tu d e n ts was t i t l e d "S o lv in g Easy A rith m e tic P u z z le s ." The word "problem " was n ev er u sed in th e d i r e c tio n s . R e l i a b i l i t y . —The r e l i a b i l i t y f o r th e T e s t o f Math e m a tic a l O p eratio n s was computed by th e K uder-R ichardson 45 Form ula 20 fo r th e upper and low er 27 p e r c e n t o f th e t o t a l number o f c a se s in th e se v e n th -g ra d e sam ple. The r e l i a b i l i t y c o e f f ic ie n t f o r th e t e s t was ,.94. I t should be n o te d t h a t a t r i a l ru n o f th e t e s t was made on a group o f 60 s ix th -g r a d e s tu d e n ts . An item -- a n a ly s is was made on th e t e s t by te c h n iq u e s su g g e sted by Ross and S ta n le y (2 4 :4 3 6 -4 5 3 ). The t e s t item s w ere th en ran k ed acco rd in g to d i f f i c u l t y in th e f i n a l form o f th e t e s t a d m in iste re d to th e se v e n th -g ra d e sam ple. R e l i a b i l i t y was com puted, u sin g th e f i r s t two s e c tio n s o f th e se v e n th - grade sam ple, t o t a l i n g 72 s tu d e n ts , th e same day th e t e s t was a d m in iste re d . The r e l i a b i l i t y c o e f f i c i e n t was found to be .9 3 . F u rth e r to s u b s t a n tia te r e l i a b i l i t y , th e K uder- R ich ard so n Form ula 21 was a p p lie d to th e same sam ple, g iv in g a r e l i a b i l i t y c o e f f i c i e n t o f .9 3 . The t e s t i n g was c o n tin u ed th e fo llo w in g day, w ith th e f i n a l r e s u l t s f o r r e l i a b i l i t y o f th e t e s t b ased on th e t o t a l sam ple as i n d i c a te d ab o v e. C o n stru c t v a l i d i t y . —A ccording to th e Cropbach and Meehl (23) stu d y , a c o n s tru c t i s some p o s tu la te d a t t r i b u t e o f p eo p le assumed to be r e f l e c t e d i n . t e s t p erfo rm an ce. In t e s t v a lid a tio n , th e a t t r i b u t e ab o u t w hich one makes 46 sta te m e n ts in in te r p r e tin g a t e s t i s a c o n s tru c t. Campbell (14:91) p o in te d o u t th a t: . . . A g iv en s c i e n t i f i c c o n s tru c t has m ul t i p l e p o te n tia l o p e r a tio n a l s p e c if ic a tio n s . I f , as sam pled, th e s e o p e ra tio n a l s p e c if ic a tio n s concur, th e c o n s tru c t and th e sam pled m easurem ent te ch n iq u e s have v a l i d i t y . The c o n s tru c t v a l i d i t y becomes th e c o r r e la tio n among two o r more in d e pendent m easures as c o n c e p tu a lly id e n tic a l in t h e i r r e f e r e n t as p o s s ib le . The c o r r e la tio n s between A T e st o f M athem atical O p eratio n s and th e a r ith m e tic s e c tio n o f th e C a lif o rn ia Achievement T e s ts . J u n io r High L e v e l. G rades 7 -8 -9 . Form X. and th e Survey T e st o f A lg eb ra ic A p titu d e w ere .77 and .7 1 , r e s p e c tiv e ly . These two in stru m e n ts a re as c o n c e p tu a lly id e n tic a l to th e o p e ra tio n s t e s t as was p o s s ib le to o b ta in . These c o r r e la tio n s w ould seem to d em o n strate some degree o f c o n s tru c t v a l i d i t y . The mean raw sc o re f o r A T e st o f M athem atical O per a tio n s was 26.74 fo r boys, 27.23 f o r g i r l s , and 26.90 fo r th e t o t a l gro u p . The sta n d a rd d e v ia tio n s w ere 1 2.11, 12.63, and 1 2.45, r e s p e c tiv e ly . The d if fe re n c e s between boys and g i r l s w ere n o t s t a t i s t i c a l l y s i g n i f i c a n t . O p e ra tio n s■ —A T e st o f M athem atical O p eratio n s was a d m in iste re d in th e f i r s t two weeks o f Hay. The t e s t was a d m in iste re d by th e p r in c ip a l o f th e ju n io r h ig h sc h o o l. S p e c ia l in s t r u c tio n s w ere g iv en as to th e m ethod o f adm in i s t e r i n g th e t e s t . S in ce no tim e elem ent was in v o lv e d , d ir e c tio n s f o r tim in g w ere n o t n e c e s s a ry . The t e s t s w ere sc o re d by m achine a t th e sch o o l d i s t r i c t o f f i c e . Eleven s e p a ra te raw sc o re s on t h i s t e s t w ere re c o rd e d f o r each s tu d e n t—one fo r each o f th e te n p r in c ip l e s , and one sc o re f o r th e t o t a l number c o r r e c t on th e t e s t . T reatm en t o f th e D ata T echniques o f m u ltip le c o r r e la tio n and f a c to r a n a l y s is w ere used as s t a t i s t i c a l to o ls in a n a ly z in g th e d a ta accum ulated in t h i s s tu d y . C o e ff ic ie n ts o f m u ltip le c o r r e l a t i o n w ere u t i l i z e d to d eterm in e th e r e la tio n s h ip b e tween th e p r in c ip le s in v o lv e d in A T e st o f M athem atical O p eratio n s and each o f th e v a r ia b le s o f IQ, a r ith m e tic achievem ent, and a lg e b ra a p titu d e . F a c to r a n a ly s is was u t i l i z e d to d e s c rib e th e dim ensions o f a b i l i t y d e s c rib e d by sc o re s on th e te n m a th em atical p r in c ip le s and each o f th e v a r ia b le s s e le c te d fo r stu d y . 48 M u ltip le c o r r e l a t i o n . —In o rd e r to d eterm in e th e r e la tio n s h ip th a t e x is te d betw een u n d e rsta n d in g o f th e m a th em atical p r i n c i p l e s , a s d e s c rib e d by A T e s t o f M athe m a tic a l O p e ra tio n s , and th e v a rio u s f a c t o r s o f IQ, a r i t h m e tic achievem ent, and a lg e b ra a p titu d e , th e raw sc o re s f o r each o f th e p r in c ip le s w ith in th e t e s t w ere tr e a te d as s e p a ra te s c o re s . M u ltip le c o r r e la tio n te c h n iq u e s su g g e s te d by G u ilfo rd (19:390-400) w ere u se d . The d a ta fo r each s u b je c t w ere tr a n s f e r r e d from th e o r i g i n a l d a ta c a rd to an IBM c a rd . By means o f e l e c t r o n ic equipm ent th e m eans, sta n d a rd d e v ia tio n s , c o r r e la tio n c o e f f i c i e n t s , m ul t i p l e c o r r e la tio n c o e f f i c i e n t s , and b c o e f f ic ie n ts w ere com puted. The B eta c o e f f ic ie n ts and p ro p o rtio n o f v a r ia n c e w ere computed by m achine c a l c u l a t o r . The m u ltip le c o r r e la tio n c o e f f ic ie n t and th e co e f f i c i e n t o f m u ltip le d e te rm in a tio n w ere used to i n t e r p r e t th e r e la tio n s h ip betw een s c o re s on th e t e s t o f th e p r i n c i p le s when tak en as a group and each o f th e o th e r f a c to r s in v o lv e d in th e s tu d y . The p e rc e n ta g e c o n trib u tio n o f each in d ep en d en t v a r ia b le and th e c o r r e la tio n c o e f f ic ie n ts u sed in th e m u ltip le r e g r e s s io n e q u a tio n w ere used to i n t e r p r e t th e r e la tio n s h i p o f th e in d ep en d en t v a r ia b le s to 49 th e dependent v a r ia b le . The i n te r p r e t a t i o n o f th e c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and c o r r e la tio n s was b ased upon th e fo llo w in g v e rb a l d e s c r ip tio n re p o rte d by G u ilfo rd (1 9 :1 4 5 ): L ess th an .20 .................. S lig h t; alm o st n e g lig ib le r e la tio n s h i p . 2 0 - . 4 0 .................. Low c o r r e la tio n ; d e f in i t e b u t sm all r e la tio n s h i p .4 0 -.7 0 .................. M oderate c o r r e la tio n ; s u b s t a n tia l r e la tio n s h i p .7 0 -.9 0 .................. High c o r r e la tio n ; marked r e la tio n s h i p .9 0 -1 .0 0 . . . . V ery h ig h c o r r e la tio n ; v e ry dependable r e l a t i o n sh ip F a c to r a n a l y s i s . —S e v e ra l w r ite r s have p re se n te d view s re g a rd in g th e aim s and o b je c tiv e s o f f a c to r a n a ly s is . For exam ple, C a rro l and Schw eiker w ro te: F a c to r a n a ly s is can be c ru d e ly d e s c rib e d as an e x te n sio n o f th e c o r r e la tio n m ethod. When s e v e ra l v a r ia b le s a re found to be r a t h e r h ig h ly c o r r e la te d , i t may be in f e r r e d t h a t th e y a re co n n ected in some way, p erh ap s by a common u n d e rly in g v a r ia b le w hich i s n o t iiim nediately p re s e n t in th e m easurem ents b u t w hich n e v e rth e le s s w ould acco u n t f o r them to a m ajor e x te n t. On th e o th e r hand, v a r ia b le s showing no r e l a tio n s h ip a re assumed to be m easuring d i f f e r e n t th in g s . When an in v e s tig a to r i s fac e d w ith a f a i r l y la rg e number o f v a r ia b le s i t becomes d i f f i c u l t to d is e n ta n g le th e r e la tio n s h i p i n h e re n t in them . I t i s th e p u rp o se o f f a c to r m ethods to en a b le th e in v e s tig a to r to f in d r e l a t i v e l y sim ple ways o f d e s c rib in g and acco u n tin g 50 fo r th e r e la tio n s h ip s betw een v a r ia b le s . (20:368) H o lzin g er and Herman d e fin e f a c to r a n a ly s is in th e fo llo w in g manner: F a c to r a n a ly s is i s a b ran ch o f s t a t i s t i c a l th e o ry concerned w ith th e r e s o lu tio n o f a s e t o f d e s c r ip tiv e v a r ia b le s in term s o f a sm all number o f c a te g o rie s o r f a c to r s . . . th e c h ie f aim i s to a t t a i n s c i e n t i f i c parsim ony o r economy o f d e s c r ip tio n . (1 5 :3 ) F a c to r a n a ly s is i s a condensed sta te m e n t o f r e l a tio n s h ip o b ta in e d betw een a s e t o f v a r ia b le s w hich can be used m a th em atic ally to sta n d f o r th e s e v a r ia b le s . F a c to r a n a ly s is , as any s t a t i s t i c a l to o l re q u ir in g in te r p r e t a ti o n , has en co u n tered much r e s is ta n c e among b o th s t a t i s t i c i a n s and p s y c h o lo g is ts . Hans E ysenick, in an a r t i c l e e n t i t l e d "The L o g ic a l B a sis o f F a c to r A n a ly s is ," sums up th e K elle y -T h u rsto n e d isag reem en t as fo llo w s: K elley and th o se who fo llo w --T h e f i r s t fu n c tio n o f s t a t i s t i c s i s to be p u re ly d e s c r ip tiv e , and i t s second fu n c tio n i s to en ab le a n a ly s is in harmony w ith h y p o th e ses, and i t s t h i r d fu n c tio n to su g g est by th e fo rc e o f i t s v ir g in d a ta a n a l y ses n o t e a r l i e r th o u g h t o f . . . . He who a s sumes to re a d more rem ote v a r i e t i e s in to th e f a c t o r i a l outcome i s c e r t a in ly doomed to d is a p pointm ent . T h u rsto n e, Spearman, and th o se who fo llo w — The c a u sa l im p lic a tio n c h a ra c te riz e s n o t o n ly th e in te r p r e t a ti o n o f f a c to r s as su g g e stiv e o f a h y p o th e s is , b u t a ls o th e n e x t le v e l o f f a c to r s 51 as p ro v in g a h y p o th e s is , and s in c e from th e psycho lo g ic a l p o in t o f view t h i s c a u sa l im p lic a tio n i s p r e c is e ly w hat le n d s i n t e r e s t and v a lu e to f a c to r a n a ly s is , i t may be op p o rtu n e h e re to g iv e a d e f i n i t i o n o f a f a c to r w hich b rin g s o u t t h i s elem en t. We may th e re fo r e o f f e r th e fo llo w in g d e f in itio n : A f a c to r i s a h y p o th e tic a l c a u sa l in flu e n c e u n d er ly in g and d eterm in in g th e o b serv ed r e la tio n s h ip betw een a s e t o f v a r ia b le s . (17:106-108) For th e pu rp o ses o f t h i s stu d y , f a c to r a n a ly s is was used to i n t e r p r e t th e r e la tio n s h ip betw een th e v a r ia b le s s e le c te d f o r stu d y as d e s c rib e d by sc o re s ach iev ed by a sam ple o f 300 se v en th -g ra d e p u p ils . The n a tu re o f t h i s stu d y does n o t w a rra n t sta te m e n t o f a cause and e f f e c t r e la tio n s h i p . In o rd e r to d eterm in e th e r e la tio n s h ip s among u n d e rsta n d in g s o f th e v a rio u s m ath em atical p r in c ip l e s , and to d e s c rib e f u r th e r th e r e la tio n s h ip o f u n d e rsta n d in g o f th e s e p r in c ip le s to language IQ, non-language IQ, a r i t h m e tic re a s o n in g , a rith m e tic fu n d am en tals, and a lg e b ra a p titu d e , m ethods o f f a c to r a n a ly s is w ere employed. The d a ta f o r each stu d e n t w ere tr a n s f e r r e d from th e o r ig in a l p u p il d a ta sh e e t to an IBM c a rd . By means o f e le c tr o n ic equipm ent, th e c o r r e la tio n m a tric e s and r o ta te d f a c to r m a tric e s w ere computed f o r th e t o t a l group, fo r th r e e a b i l i t y le v e ls determ ined by sc o re s on A T e st o f M athem at i c a l O p e ra tio n s, and f o r each se x . The r o ta ti o n o f 5 2 f a c to r s was b ased on a com puter program f o r th e varim ax method o f r o t a t i o n in f a c t o r a n a ly s is by K aiser (1 8 ). The f a c to r s fo r th e t o t a l group w ere an aly zed in d e t a i l , w h ile th e f a c t o r s f o r th e th r e e a b i l i t y groups o f 100 s tu d e n ts each w ere d e s c rib e d through t h e i r r e l a t i o n to th e f a c t o r s tr u c t u r e f o r th e t o t a l gro u p . The f a c to r s tr u c tu r e s f o r boys and g i r l s w ere compared w ith each o th e r and w ith th e s tr u c t u r e f o r th e t o t a l gro u p . Summary In t h i s c h a p te r, th e sam ple and th e so u rce s o f d a ta w ere d e s c rib e d , th e in stru m e n ts and th e m ethods o f o b ta in in g th e d a ta w ere p re s e n te d , and th e s t a t i s t i c a l te c h n iq u e s w ere b r i e f l y in d ic a te d . D ata on th e r e la tio n s h i p o f th e se v e n th -g ra d e c h i l d 's u n d e rsta n d in g o f th e m ath em atical p r in c ip le s i n v o lv ed in th e o p e ra tio n s o f a d d itio n , m u lt ip lic a tio n , sub t r a c t i o n , and d iv is io n to in t e l l i g e n c e , a r ith m e tic a c h ie v e m ent, and a lg e b ra a p titu d e a r e p re s e n te d in th e fo llo w in g two c h a p te r s . CHAPTER IV ABILITY WITH THE M ATHEM ATICAL PRINCIPLES GOVERNING THE OPERATIONS OF ADDITION, MULTIPLICATION, SUBTRACTION, A N D DIVISION, A N D ITS RELATION TO SELECTED FACTORS OF INTELLIGENCE, ACHIEVEMENT, A N D ALGEBRA APTITUDE T h is c h a p te r p r e s e n ts an a n a ly s is o f th e r e l a t i o n s h ip s found to e x i s t betw een th e a b i l i t y o f se v e n th -g ra d e s tu d e n ts w ith th e m a th em atic al p r in c ip l e s o f a d d itio n , m u lt ip lic a tio n , s u b tr a c tio n , and d iv is io n , and th e f a c to r s o f (1) language IQ, (2) n o n -lan g u ag e IQ, (3) a r ith m e tic re a s o n in g , (4) a r ith m e tic fu n d am en tals, and (5) a lg e b ra a p t i t u d e . The m ath em atical p r in c ip le s s e le c te d f o r stu d y w ere based on th e a x io m atic m ethod in m athem atics and i n v o lv e d th e (1) c lo s u re law f o r a d d itio n , (2) c lo s u re law f o r m u lt ip lic a ti o n , (3) com m utative law f o r a d d itio n , (4) 53 54 com m utative law f o r m u lt ip lic a ti o n , (5) a s s o c ia tiv e law f o r a d d itio n , (6) a s s o c ia tiv e law f o r m u lt ip lic a ti o n , (7) d e f i n i t i o n o f d if f e r e n c e , (8) axiom o f s u b tr a c tio n , (9) d e f i n i t i o n o f q u o tie n t, and (10) axiom o f d iv is io n . The s tr e n g th o f th e r e la tio n s h ip betw een a b i l i t y w ith th e m a th em atical p r in c ip le s and each o f th e f a c to r s was in v e s tig a te d by means o f th e m u ltip le re g r e s s io n equa t i o n . The c o e f f i c i e n t o f m u ltip le d e te rm in a tio n , th e p e r c e n ta g e c o n trib u tio n o f th e in d ep en d en t v a r ia b le s , and c o r r e la tio n c o e f f ic ie n ts w ere used to i n t e r p r e t th e m u lt i p le c o r r e la tio n c o e f f i c i e n ts . R e la tio n s h ip betw een th e in d ep en d en t and dependent v a r ia b le s was n o t presum ed to in d ic a te c a u s a l ity , b u t to show th e e x te n t to w hich c e r t a in a t t r i b u t e s o f th e compared v a r ia b le s w ere sh a re d . The R e la tio n s h ip Between Language IQ and A b ility in M athem atical O p eratio n s T h is s e c tio n o f C hapter IV p re s e n ts an a n a ly s is o f th e r e la tio n s h i p betw een language IQ and th e se v e n th -g ra d e s t u d e n t's a b i l i t y in m ath em atical o p e r a tio n s . The c o e f f ic ie n t o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een language IQ 55 m and th e b e s t w eig h ted co m b in atio n s o f th e m ath em atical p r in c ip l e s a re re c o rd e d in T ab le 3 . These d a ta r e v e a l a m u ltip le c o r r e la tio n o f .67 in d ic a tin g a s u b s t a n tia l r e l a tio n s h ip betw een know ledge o f th e m ath em atical' o p e ra tio n s and language IQ . The c o e f f i c i e n t o f m u ltip le d e te rm in a tio n o f .4524 in d ic a te s t h a t 4 5 .2 4 p e r c e n t o f th e v a r ia n c e in language IQ i s acco u n ted f o r by w h atev er i s m easured by th e te n p r in c ip l e s ta k e n to g e th e r , e lim in a tin g from d ouble c o n s id e ra tio n th in g s t h a t th e y have in common. The p e rc e n ta g e c o n trib u tio n o f th e in d ep en d en t v a r ia b le s in th e m u ltip le r e g r e s s io n e q u a tio n re v e a le d t h a t th e m a th em atical o p e ra tio n o f s u b tr a c tio n c o n trib u te d more th a n h a l f o f th e t o t a l p r e d ic te d v a r ia n c e . The c o r r e la tio n c o e f f ic ie n ts f o r th e p r in c ip l e s o f a d d itio n and m u lt ip lic a ti o n , shown in T ab le 8 , re v e a le d (1) an alm o st n e g l ig ib le r e la tio n s h i p to language IQ, (2) a m arked r e la tio n s h i p to each o th e r , and (3) a n e g lig ib le to sm all r e la tio n s h i p to th e p r in c ip le s o f s u b tr a c tio n and d iv is io n . I t would ap p ear t h a t th e f i r s t s ix p r in c ip l e s , as a group, c o n tr ib u te l i t t l e to th e s u b s t a n tia l r e l a t i o n sh ip found to e x i s t betw een th e m a th em atic al o p e ra tio n s and language IQ. 56 ' TABLE 3 THE COEFFICIENT OF MULTIPLE CORRELATION AND OTHER STATISTICS RELATED TO THE CORRELATION OF THE MATHEMATICAL OPERATIONS AND LANGUAGE IQ Variable Standard Deviation Ratio of Std. Deviation b Weight $ Weight Correlation Coefficient Proportion of Variance 1. Closure Lav Addition 7.998 .4503 ,0213 ,0096 -.220** -.0021 2. Closure Law Multiplication 8.547 .4811 .4610 .2218 -.207** -.0459 3. Commutative Law Addition 2.793 .1572 .9092 .1429 .050 .0071 4. Commutative Law Multiplication 3.932 .2214 .1867 .0413 -.070 -.0029 5. Associative Law Addition 6.112 .3441 *2.1040 *.7240 -.196** .1419 6. Associative Law Multiplication 2.411 .1357 .9689 .1315 .119* .0156 7. Definition of Difference 1.382 .0778 3.1643 .2462 .521** .1283 8. Axiom of Subtraction 1.580 .0890 2.7120 .2414 .499** .1205 9. Definition of Quotient 1.515 .0853 2,1644 .1846 .462** .0853 10. Axiom of Division 1.501 .0845 .1792 .0151 .307** .0046 (R2) Coefficient of Multiple Determination .4524 (100R2) Percentage of Variance 45.24 (R) Multiple Correlation Coefficient .67 (O' 11.12...10). Standard Error of the Estimate 13.37 *Significant at .05 Level **Significant at .01 Level 57 The c o r r e la tio n c o e f f ic ie n ts f o r th e p r in c ip le s o f s u b tr a c tio n and d iv is io n , shown in T ab le 8 , re v e a le d (1) a s u b s ta n tia l r e la tio n s h ip to each o th e r , (2) a n e g lig ib le to sm all r e la tio n s h ip to th e p r in c ip le s o f a d d itio n and m u lt ip lic a ti o n , (3) a d e f i n i t e b u t sm all r e la tio n s h ip b e tween th e axiom o f d iv is io n and language IQ, (4) a sub s t a n t i a l r e la tio n s h ip betw een th e p r in c ip le s o f s u b tr a c tio n and language IQ, and (5) a s u b s t a n tia l r e la tio n s h i p b e tween th e d e f in it io n o f q u o tie n t and language IQ. The p r in c ip le s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e la tio n s h i p found to e x i s t betw een th e te n p r in c ip le s and language IQ. An a n a ly s is s im ila r to th e p re c e d in g c o n s id e ra tio n o f language IQ was made w ith r e s p e c t to th e r e la tio n s h ip betw een a b i l i t y w ith th e m ath em atical o p e ra tio n s and non language IQ. The d a ta in v o lv e d in t h i s a n a ly s is a re r e p o rte d in th e fo llo w in g s e c tio n . The R e la tio n s h ip Between Non-Language 10 and A b ility in M athem atical O p eratio n s T h is s e c tio n o f C hapter IV i s concerned w ith an a n a ly s is o f th e r e la tio n s h ip betw een th e no n -lan g u ag e IQ 58 and th e se v e n th -g ra d e c h i l d 's a b i l i t y in m ath em atical op e r a tio n s . The c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een n o n -lan g u ag e IQ and th e b e s t w eig h ted com bination o f th e m ath em atical p r in c ip le s a re re c o rd e d in T ab le 4 . T hese d a ta r e v e a l a m u ltip le c o r r e la tio n o f .4 7 , in d ic a tin g a s u b s t a n tia l r e la tio n s h ip betw een knowledge o f th e m ath em atical o p e ra tio n s and n o n -lan g u ag e IQ. The c o e f f i c i e n t o f m u ltip le d e te rm in a tio n o f .2236 in d ic a te s t h a t 22.36 p e r c e n t o f th e v a ria n c e in n o n -lan g u ag e IQ i s acco u n ted f o r by th e te n p r in c ip le s ta k en to g e th e r , e lim in a tin g from double c o n s id e ra tio n th in g s t h a t th e y have in common. The p e r ce n tag e c o n trib u tio n o f th e in d ep en d en t v a r ia b le s in th e m u ltip le r e g re s s io n e q u a tio n re v e a le d th a t th e a s s o c ia tiv e law fo r th e o p e ra tio n o f a d d itio n c o n trib u te d more th an h a lf o f th e t o t a l p r e d ic te d v a r ia n c e . However, th e n a tu re o f th e n e g a tiv e b e ta w e ig h t does n o t p ro v id e a tr u e i n t e r p r e ta tio n o f th e r e la tio n s h i p o f th e v a r ia b le to non language IQ. The c o r r e la tio n c o e f f i c ie n ts f o r th e p r in c ip le s o f a d d itio n and m u lt ip lic a ti o n (a s shown in T ab le 4) TABLE 4 THE COEFFICIENT OF MULTIPLE CORRELATION AND OTHER STATIST CORRELATION OF THE MATHEMATICAL OPERATIONS AND NON- Variable Standard Deviation Ratio of Std. D eviation b Weight 1 . Closure Law Addition 7.998 .4327 - .1923 2. Closure Law M u ltip licatio n 8.547 .4624 .5363 3. Commutative Law Addition 2.793 .1511 .2131 4. Commutative Law M u ltip licatio n 3.932 .2127 .9197 5. A ssociative Law Addition 6.112 .3307 -2.5591 6 . A ssociative Law M u ltip licatio n 2.411 .1304 2.5976 7. D efin itio n of D ifference 1.382 .0748 .7794 8 . Axiom of Subtraction 1.580 .0855 1.0843 9. D efin itio n of Quotient 1.515 .0820 .4147 10. Axiom of D ivision 1.501 .0812 1.3220 (R2) C oefficient of M ultiple Determ ination (100R2) Percentage of Variance (R) M ultiple C orrelation C oefficient (O '1 1 .1 2 ...1 0 ) Standard Error of the Estimate * Significant at .05 Level ** Significant at .01 Level 59 TABLE 4 TIFLE CORRELATION AND OTHER STATISTICS RELATED TO THE HE MATHEMATICAL OPERATIONS AND NON-LANGUAGE IQ. R atio o f S td. D ev iatio n b Weight @ Weight C o rre la tio n C o e ffic ie n t P ro p o rtio n of V ariance .4327 - .1923 -.0832 -.222** .0185 .4624 .5363 .2480 -.204** -.0 5 0 6 .1511 .2131 .0322 -.037 -.0012 .2127 .9197 .1956 -.0 9 3 -.0182 .3307 -2.5591 -.8463 -.200** .1693 .1304 2.5976 .3387 .075 .0254 .0748 .7794 .0583 .272** .0159 .0855 1.0843 .0927 .299** .0277 .0820 .4147 .0340 .246** .0084 .0812 1.3220 .1073 .264** .0283 d e n t o f M u ltip le D eterm ination ventage o f V ariance : C o rre la tio n C o e ffic ie n t .0) S tandard E rro r o f th e E stim ate .2236 22.36 .47 16.55 60 in d ic a te an in v e rs e to n e g lig ib le r e la tio n s h i p to non language IQ. At th e same tim e , th e i n t e r - c o r r e l a t i o n s fo r th e s e p r in c ip le s (shown in T able 8) in d ic a te a marked r e la tio n s h ip to each o th e r . I t would ap p ear t h a t th e s e p r in c ip l e s , as a group, c o n trib u te l i t t l e to th e s u b s ta n t i a l r e la tio n s h ip found to e x i s t betw een th e m ath em atical o p e ra tio n s , tak en as a w hole, and n o n -lan g u ag e IQ. The c o r r e la tio n c o e f f ic ie n ts fo r th e s u b tr a c tio n and d iv is io n p r in c ip le s shown in T ab le 4 in d ic a te a d e f i n i t e b u t sm all r e la tio n s h ip to n o n -lan g u ag e IQ. The p r in c ip le s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e la tio n s h i p s found to e x i s t betw een non language IQ and th e b e s t w eig h ted com bination o f th e m ath e m a tic a l o p e r a tio n s . T h is r e la tio n s h i p i s f u r th e r sub s t a n t i a te d by th e n e g lig ib le to sm all r e la tio n s h i p found to e x i s t betw een th e m a jo rity o f th e a d d itio n and m u lti p lic a t io n p r in c ip le s and th o se o f d iv is io n and s u b tr a c tio n (a s re p o rte d in T able 8 ) . The n e x t phase o f th e in v e s tig a tio n to be re p o rte d in t h i s c h a p te r was concerned w ith an a n a ly s is o f th e s tr e n g th o f th e r e la tio n s h i p betw een a b i l i t y in m athem at i c a l o p e ra tio n s and a r ith m e tic re a s o n in g . 61 The R e la tio n s h ip Between A rith m e tic R easoning and A b ility in M athem atical O p eratio n s T h is s e c tio n o f C hapter IV p re s e n ts an a n a ly s is o f th e r e la tio n s h i p betw een a r ith m e tic re a so n in g and th e se v e n th -g ra d e s tu d e n t1s a b i l i t y in m ath em atical o p e r a tio n s . The c o e f f ic ie n t o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a te d to th e c o r r e la tio n betw een a r ith m e tic re a s o n in g and th e b e s t w eig h ted com bination o f th e m athe m a tic a l p r in c ip l e s a re re c o rd e d in T able 5 . T hese d a ta r e v e a l a m u ltip le c o r r e la tio n o f .7 7 , in d ic a tin g a m arked r e la tio n s h i p betw een know ledge o f th e m ath em atical o p e ra tio n s and a r ith m e tic re a s o n in g . The c o e f f ic ie n t o f m u lti p le d e te rm in a tio n o f .5998 in d ic a te d t h a t 59.88 p e r c e n t o f th e v a ria n c e in a r ith m e tic re a s o n in g i s acco u n ted f o r by w h atev er i s m easured by th e te n p r in c ip le s ta k en t o g e th e r, e lim in a tin g from double c o n s id e ra tio n th in g s t h a t th e y have in common. The p e rc e n ta g e c o n trib u tio n o f th e in d ep en d en t v a r ia b le s in th e m u ltip le r e g r e s s io n e q u a tio n re v e a le d t h a t th e c lo s u re laws f o r a d d itio n and m u l t i p l i c a tio n and th e a s s o c ia tiv e law f o r a d d itio n c o n trib u te d l i t t l e i f a n y th in g to th e r e la tio n s h ip betw een a r ith m e tic re a s o n in g and th e b e s t w eig h ted com bination o f th e TABLE 5 TH E COEFFICIENT O F M ULTIPLE C O R R ELA TIO N A N D O T H E R STATIST* C O R R ELA TIO N O F T H E M A T H E M A T IC A L O PERA TIO N S A N D ARITfiM E Standard R atio of Std. V ariable D eviation D eviation b Weight 1 . Closure Law Addition 7.998 .7373 - .2267 2. Closure Law M u ltip licatio n 8.547 .7880 .0686 3. Commutative Law Addition 2.793 .2575 .7814 4. Commutative Law M u ltip licatio n 3.932 .3625 .7764 5. A ssociative Law Addition 6.112 .5635 -1.0243 6 . A ssociative Law M u ltip licatio n 2.411 .2223 1.3064 7. D efin itio n of D ifference 1.382 .1274 1.4226 8. Axiom of Subtraction 1.580 .1457 1.4198 9. D efin itio n of Quotient 1.515 .1397 1.5361 10. Axiom of D ivision 1.501 .1384 .5844 (R2) C oefficient of M ultiple Determ ination (100R2) Percentage of Variance 5 (R) M ultiple C orrelation C oefficient (e f1 1 .1 2 ...1 0 ) Standard Error of the Estimate * Significant at .05 Level ** Significant at .01 Level 62 TABLE 5 FICIENT OF MULTIPLE CORRELATION AND OTHER STATISTICS RELATED TO THE ELATION OF THE MATHEMATICAL OPERATIONS AND ARITHMETIC REASONING Standard Deviation Ratio of Std. Deviation b Weight ? Weight Correlation Coefficient Proportion of Variance 7.998 .7373 - .2267 -.1671 -.058 .0097 8.547 .7880 .0686 .0541 -.029 -.0016 2.793 .2575 .7814 .2012 .269** .0541 3.932 .3625 .7764 .2814 .158** .0445 6.112 .5635 -1.0243 .5772 .004 -.0023 2.411 .2223 1.3064 .2904 .379** .1101 1.382 .1274 1.4226 .1812 .586** .1060 1.580 .1457 1.4198 .2069 .615** .1272 1.515 .1397 1.5361 .2146 .554** .1189 1.501 .1384 .5844 .0809 .399** .0323 (R2) Coefficient of Multiple Determination (100R2) Percentage of Variance (R) Multiple Correlation Coefficient ( e r 111.12... 10) Standard Error of the Estimate .5988 59.88 .77 6.99 m ath em atical o p e r a tio n s . The c o r r e la tio n c o e f f ic ie n ts f o r th e p r in c ip l e s o f a d d itio n and m u ltip lic a tio n and a r ith m e tic re a s o n in g i n d i c a te a n e g lig ib le to sm all r e la t i o n s h i p . The i n t e r c o r r e la t i o n s f o r th e s e f i r s t s i x p r i n c i p l e s , a s shown in T ab le 8 , in d ic a te a m arked r e la tio n s h ip to each o th e r . I t w ould appear t h a t th e f i r s t s ix p r in c ip l e s as a group c o n trib u te l i t t l e to th e s u b s t a n tia l r e la tio n s h i p found to e x i s t b e tween a r ith m e tic re a s o n in g and th e b e s t w eig h ted com bina tio n o f th e m ath em atical p r i n c i p l e s . The c o r r e la tio n c o e f f ic ie n ts f o r th e s u b tr a c tio n and d iv is io n p r in c ip le s in d ic a te a s u b s ta n tia l r e l a t i o n sh ip to a r ith m e tic re a s o n in g . The p r in c ip le s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e l a t i o n s h ip s found to e x i s t betw een a r ith m e tic re a s o n in g and th e m a th em atic al o p e r a tio n s . The n e x t phase o f th e in v e s tig a tio n to be re p o rte d in t h i s c h a p te r was concerned w ith an a n a ly s is o f th e s tr e n g th o f th e r e la tio n s h i p betw een a b i l i t y w ith th e m a th em atical o p e ra tio n s and a r ith m e tic fu n d am e n tals. 64 The R e la tio n s h ip Between A rith m e tic Fundamen t a l s and A b ility in M athem atical O p eratio n s T h is s e c tio n o f C hapter IV p re s e n ts an a n a ly s is o f th e r e la tio n s h i p betw een a r ith m e tic fundam entals and th e se v e n th -g ra d e s tu d e n t's a b i l i t y in m ath em atical o p e r a tio n s . The c o e f f ic ie n t o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een a r ith m e tic fundam entals and th e b e s t w eig h ted com bination o f th e m ath e m a tic a l p r in c ip le s a r e re c o rd e d in T ab le 6 . These d a ta r e v e a l a m u ltip le c o r r e la tio n o f .7 4 , in d ic a tin g a m arked r e la tio n s h ip betw een a b i l i t y in m ath em atic al o p e ra tio n s and a r ith m e tic fu n d am e n tals. The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .5521 in d ic a te s th a t 55.21 p e r c e n t o f th e v a ria n c e in a r ith m e tic fu n d am en tals i s acco u n ted f o r by w h atev er i s m easured by th e te n p r in c ip le s ta k en t o g e th e r, e lim in a tin g from double c o n s id e ra tio n th in g s t h a t th ey have in common. The p e rc e n ta g e c o n trib u tio n f o r th e p r in c ip le s o f s u b tr a c tio n and th e a s s o c ia tiv e law f o r m ul t i p l i c a t i o n ta k e n to g e th e r acco u n ted f o r more th an h a l f o f th e t o t a l p r e d ic te d v a r ia n c e . TABLE 6 THE COEFFICIENT OF MULTIPLE CORRELATION AND OTHER STATIST CORRELATION OF THE MATHEMATICAL OPERATIONS AND ARITHME Standard R atio of Std. V ariable D eviation D eviation b Weight 1 . Closure Law A ddition 7.998 .4719 - .5599 2. Closure Law M u ltip lic atio n 8.547 .5043 .2550 3. Commutative Law A ddition 2.793 .1642 1.0413 4. Commutative Law M u ltip lic a tio n 3.932 .2320 .6276 5. A ssociative Law A ddition 6.112 .3606 -1.3943 6. A ssociative Law M u ltip lic atio n 2.411 .1422 2.8708 7. D efin itio n of D ifference 1.382 .0815 2.7435 8. Axiom of S ubtraction 1.580 .0932 1.8784 9. D e fin itio n of Quotient 1.515 .0894 1.5216 10. Axiom of D ivision 1.501 .0886 .6245 (R2) C o efficien t of M ultiple D eterm ination (100R2) Percentage of Variance (R) M ultiple C o rrelatio n C o efficien t (ff 1 1 .1 2 .. .10) Standard E rror of the Estim ate * Significant at .05 Level ** Significant at .01 Level 65 TABLE 6 dPLE CORRELATION AND OTHER STATISTICS RELATED TO THE MATHEMATICAL OPERATIONS AND ARITHMETIC FUNDAMENTALS R atio of S td. D eviation b Weight £ Weight C o rrela tio n C o e ffic ie n t P ro p o rtio n of V ariance .4719 - .5599 -.2642 -.076 .0201 .5043 .2550 .1286 -.045 -.0058 .1642 1.0413 .1710 .241** .0412 .2320 .6276 .1456 .132* .0192 .3606 -1.3943 -.5028 -.004 .0020 .1422 2.8708 .4082 .385** .1576 .0815 2.7435 .2236 .582** .1301 .0932 1.8784 .1751 .573** .1003 .0894 1.5216 .1360 .496** .0675 .0886 .6245 .0553 .360** .0199 Lent of M u ltip le D eterm ination entage of V ariance C o rrela tio n C o e ffic ie n t 3) Standard E rror of the Estim ate .5521 55.21 .74 11.53 66 The c o r r e la tio n c o e f f ic ie n ts f o r th e c lo s u re laws o f a d d itio n and m u ltip lic a tio n and th e a s s o c ia tiv e law fo r a d d itio n in d ic a te an alm o st n e g l ig ib le r e la tio n s h i p to a r ith m e tic fu n d am en tals. At th e same time* th e i n t e r c o r r e l a t i o n s f o r th e s e p r i n c i p l e s , shown in T ab le 8 , in d ic a te a v e ry dependable r e la tio n s h ip among them . The c o r r e la tio n c o e f f ic ie n ts f o r th e d e f i n it i o n o f d if f e r e n c e , axiom o f s u b tr a c tio n , and d e f i n i t i o n o f quo t i e n t r e v e a l a s u b s ta n tia l r e la tio n s h i p to a r ith m e tic fu n d am en ta ls. The i n t e r c o r r e l a t i o n s f o r th e s e p r in c ip l e s , shown in T ab le 8 , in d ic a te a s u b s ta n tia l r e la tio n s h i p to each o th e r . The p r in c ip l e s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e la tio n s h i p s found to e x i s t betw een a r ith m e tic fundam entals and th e m a th em atical o p e r a tio n s , a lth o u g h sm all r e la tio n s h i p s a ls o w ere found betw een a r ith m e tic re a s o n in g and th e com m utative law o f a d d itio n , th e a s s o c ia tiv e law o f m u lt ip lic a ti o n , and th e axiom o f d iv is io n . The n e x t phase o f th e in v e s tig a tio n to be re p o r te d in t h i s c h a p te r was concerned w ith an a n a ly s is o f th e s tr e n g th o f th e r e la tio n s h i p betw een th e a b i l i t y w ith th e m ath em atical o p e ra tio n s and a lg e b ra a p titu d e . The R e la tio n s h ip Between A leeb ra A p titu d e and A b ility in M athem atical O p eratio n s T h is s e c tio n o f C hapter IV p re s e n ts an a n a ly s is o f th e r e la tio n s h ip betw een a lg e b ra a p titu d e and th e se v e n th - g rad e s tu d e n t 's a b i l i t y in m ath em atical o p e r a tio n s . The c o e f f ic ie n t o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een a lg e b ra a p titu d e and th e b e s t w eig h ted co m binations o f th e m athe m a tic a l p r in c ip le s a r e re c o rd e d in T ab le 7 . These d a ta r e v e a l a m u ltip le c o r r e la tio n o f .75, in d ic a tin g a m arked r e la tio n s h i p betw een a b i l i t y w ith th e m ath em atical o p e ra tio n s and a lg e b ra a p titu d e . The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .5683 in d ic a te d t h a t 56.83 p e r c e n t o f th e v a ria n c e in a lg e b ra a p titu d e i s acco u n ted f o r by w hat ev er i s m easured by th e te n p r in c ip le s ta k en to g e th e r , e lim in a tin g from double c o n s id e ra tio n th in g s t h a t th e y have in common. The p e rc e n ta g e c o n trib u tio n o f th e in d e pendent v a r ia b le s in th e m u ltip le r e g r e s s io n e q u a tio n r e v e a l t h a t th e c lo s u re law o f a d d itio n and th e c lo s u re and com m utative laws fo r m u ltip lic a tio n c o n tr ib u te l i t t l e i f a n y th in g to th e r e la tio n s h i p betw een a lg e b ra a p titu d e and th e b e s t w eig h ted com bination o f th e te n p r i n c i p l e s . TABLE 7 THE COEFFICIENT OF MULTIPLE CORRELATION AND OTHER STATIST CORRELATION OF THE MATHEMATICAL OPERATIONS AND ALGE Standard R atio of Std. V ariable D eviation D eviation b Weight 1 . Closure Law A ddition 7.998 .6206 .2409 2. Closure Law M u ltip lic atio n 8.547 .6632 .2893 3. Commutative Law A ddition 2.793 .2167 .3647 4. Commutative Law M u ltip lic atio n 3.932 .3051 .1446 5. A ssociative Law A ddition 6.112 .4743 -1.4983 6. A ssociative Law M u ltip lic a tio n 2.411 .1871 1.5442 7. D e fin itio n of D ifference 1.382 .1072 1.9614 8. Axiom of S ubtraction 1.580 .1226 2.3663 9. D e fin itio n of Q uotient 1.515 .1176 1.6446 10. Axiom of D ivision 1.501 .1165 .9311 (R2) C o e ffic ien t of M ultiple D eterm ination (100R2) Percentage of Variance (R) M ultiple C o rrela tio n C o effic ien t (d 1 1 .1 2 ...1 0 ) Standard E rro r of th e Estim ate * Significant at .05 Level ** Significant at .01 Level 68 TABLE 7 :iPLE CORRELATION AND OTHER STATISTICS RELATED TO THE IE MATHEMATICAL OPERATIONS AND ALGEBRA APTITUDE R atio o f S td. D ev iatio n b Weight $ Weight C o rre la tio n C o e ffic ie n t P ro p o rtio n of V ariance .6206 .2409 .1495 -.041 -.0061 .6632 .2893 .1919 -.023 -.0044 .2167 .3647 .0790 .229** .0181 .3051 .1446 .0441 .129* .0057 .4743 -1.4983 -.7106 -.003 .0021 .1871 1.5442 .2889 .337** .0974 .1072 1.9614 .2103 .582** .1224 .1226 2.3663 .2901 .636** .1845 .1176 1.6446 .1934 .524** .1013 .1165 .9311 .1085 .435** .0472 .ent o f M u ltip le D eterm ination n ta g e of V ariance C o rre la tio n C o e ffic ie n t 1 ) Standard E rro r o f th e E stim ate .5683 56.83 .75 .61 69 The c o r r e la tio n c o e f f i c i e n t s f o r th e p r in c ip l e s o f s u b tr a c tio n and d iv is io n in d ic a te a s u b s t a n tia l r e l a t i o n sh ip to a lg e b ra a p t itu d e . The c o e f f ic ie n ts o f c o r r e la tio n f o r th e com m utative law o f a d d itio n and th e a s s o c ia tiv e law o f m u ltip lic a tio n in d ic a te a low r e la tio n s h i p to a lg e b ra a p titu d e . The p r in c ip l e s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e la tio n s h ip s found to e x i s t betw een th e te n p r in c ip l e s and a lg e b ra a p titu d e . Summary T h is s e c tio n o f C hapter IV p r e s e n ts a summary o f th e f in d in g s in v o lv e d in th e a n a ly s is o f th e r e la tio n s h ip found to e x i s t betw een th e se v e n th -g ra d e s tu d e n t's a b i l i t y in m a th em atic al o p e ra tio n s and th e f a c t o r s o f language IQ, n o n -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r ith m e tic fu n d a m e n ta ls , and a lg e b ra a p titu d e . The c o e f f i c i e n t s o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e v e a l a m o d erate c o r r e la tio n betw een th e f a c t o r s o f language IQ and n o n -lan g u a g e IQ and th e b e s t w eig h ted co m b in atio n s o f th e m a th em atic al p r in c ip l e s , i n d i c a tin g a s u b s t a n tia l r e la tio n s h i p betw een dependent and in d ep en d en t v a r ia b le s . The p e rc e n ta g e c o n trib u tio n s and 70 c o r r e la tio n c o e f f ic ie n ts re v e a le d t h a t th e o p e ra tio n s o f s u b tr a c tio n and d iv is io n accounted f o r th e m a jo rity o f th e r e la tio n s h ip s betw een th e m ath em atical o p e ra tio n s and each o f th e dependent v a r ia b le s o f language and non-language IQ. C o rro b o ra tin g d a ta in C hapter V w i l l e s ta b lis h f u r th e r ev id en ce o f t h i s r e la tio n s h i p . The c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e v e a l h ig h m u ltip le c o r r e la tio n s betw een th e f a c to r s o f a r ith m e tic re a s o n in g , a r ith m e tic fundam entals, a lg e b ra a p titu d e , and th e b e s t w eig h ted com binations o f th e m ath em atical p r i n c i p l e s , in d ic a tin g a marked r e l a t i o n sh ip betw een th e dependent and in d ep en d en t v a r ia b le s . The p e rc e n ta g e c o n trib u tio n s and c o r r e la tio n c o e f f ic ie n ts r e v e a le d th a t th e o p e ra tio n s o f s u b tr a c tio n and d iv is io n accounted fo r th e m a jo rity o f th e r e la tio n s h ip s between th e m ath em atical o p e ra tio n s and th e th r e e f a c to r s . Cor ro b o ra tin g evidence in C hapter V w i l l e s ta b lis h f u r th e r ev id en ce o f t h i s r e la tio n s h i p . CHAPTER V A N ANALYSIS OF THE FACTORS RELATED TO ABILITY WITH MATHEMATICAL OPERATIONS T his c h a p te r p r e s e n ts an a n a ly s is o f th e f a c to r s c o n trib u tin g to a b i l i t y w ith th e m ath em atical p r in c ip le s go v ern in g th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , sub t r a c t i o n , and d iv is io n . To p ro v id e a more com plete p i c tu r e o f th e f a c t o r p a tte r n s f o r th e group o f 300 se v e n th - g rad e c h ild re n , a n a ly s is was made o f th r e e ty p e s o f group in g s: (1) th e t o t a l sam ple, (2) h ig h , medium, and low a b i l i t y le v e ls as d e s c rib e d by A T e s t o f M athem atical Op e r a tio n s . and (3) se x . The Total Sample T his s e c tio n o f C hapter V p re s e n ts an a n a ly s is o f th e f a c to r s c o n trib u tin g to a b i l i t y w ith th e m ath em atical p r in c ip le s g overning th e o p e ra tio n s o f a d d itio n , 71 72 m u ltip lic a tio n , s u b tr a c tio n , and d iv is io n f o r th e t o t a l group o f 300 sev en th g ra d e rs . The c o r r e la tio n m a trix , m eans, and sta n d a rd d e v ia tio n s a re p re se n te d in T ab le 8 . These d a ta re v e a l h ig h in te r c o r r e l a ti o n s f o r v a r ia b le s one th ro u g h s ix . The r o ta te d f a c to r m a trix f o r th e t o t a l group o f 300 i s p re se n te d in T ab le 9 . E ig h t f a c to r s w ere e x tra c te d 2 from th e f i f t e e n v a r i a b l e s . The column h r e p r e s e n ts th e com m unalities o f th e t e s t s o b ta in e d by summing th e sq u ares o f th e f a c to r lo a d in g s on each v a r ia b le . The communali t i e s a r e an in d ic a tio n o f th e c o n trib u tio n o f each f a c to r to common v a ria n c e on th e t e s t . A h ig h d eg ree o f communal- i t y i s in evidence on v a r ia b le s one th ro u g h s ix , e le v e n , and t h ir te e n th ro u g h f i f t e e n , in d ic a tin g a r e l a t i v e l y h ig h c o n trib u tio n o f each f a c to r to th e common v a ria n c e on each o f th e t e s t s . Of th e e ig h t f a c to r s a f t e r r o t a t i o n , th r e e have been given in te r p r e t a t i o n and f iv e a re reg ard ed a s r e s i d u a l s . A t e s t was l i s t e d under a f a c to r when i t was r e - \ garded a s " s i g n i f i c a n t ," th a t i s , when i t had a lo a d in g as la rg e o r la r g e r th an .3 0 . When a t e s t l i s t e d under a f a c to r had a lo a d in g o f .30 o r h ig h e r under an o th e r f a c to r , TABLE 8 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE TOTA Variable 1 2 3 4 5 6 7 8 9 10 11 1. Closure Law Add. .98 .81 .91 .96 .66 -.01 .09 -.07 -.03 -.22 2. Closure Law Multi. .98 .82 .93 .97 .70 -.00 .12 -.05 -.02 -.21 3. Commutative Law Add. .81 .82 .90 .84 .78 .27 .36 .15 .08 .05 4. Commutative Law Multi. .91 .93 .90 .93 .80 .14 .27 .07 .08 -.07 5. Associative Law Add. .96 .97 .84 .93 .77 .04 .15 -.02 -.01 -.20 6. Associative Law Multi. .66 .70 .74 .80 .77 .33 .42 .28 .17 .12 7. Definition of Difference -.01 -.00 .27 .14 .04 .33 .54 .46 .33 .52 8. Axiom of Sub. .09 .12 .36 .27 .15 .42 .54 .42 • 53 .50 9. Definition of Quotient -.07 -.05 .15 .07 -.02 .28 .46 .42 .25 .46 10. Axiom of Div. -.03 -.02 .08 .08 -.01 .17 .33 .53 .25 .31 11. Language IQ -.22 -.21 .05 -.07 -.20 .12 .52 .50 .46 .31 12. Non-Language IQ -.22 -.20 -.04 -.09 -.20 .08 .27 .30 .25 .26 .56 13. Arithmetic Reasoning -.06 -.03 .27 .16 .00 .38 .59 .61 .55 .40 .71 14. Arithmetic Funda mentals -.08 -.04 .24 .13 -.00 .38 .58 .57 .50 .36 .68 15. Algebra Apt. -.04 -.02 .23 .13 -.00 .34 .58 .64 .52 .43 .70 Mean 3.647 3.857 3.227 3.170 3.967 3.097 3.010 2.203 2.350 1.445 103.563 Standard Dev. 7.998 8.547 2.793 3.932 6.112 2.411 1.382 1.580 1.515 1.501 17.761 TABLE 8 IATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE TOTAL GROUP OF 300 73 Std. 4 5 6 7 8 9 10 11 12 13 14 15 Mean .91 .96 .66 -.01 .09 -.07 -.03 -.22 -.22 -.06 -.08 -.04 3.647 .93 .97 .70 -.00 .12 -.05 -.02 -.21 -.20 -.03 -.04 -.02 3.857 .90 .84 .78 .27 .36 .15 .08 .05 -.04 .27 .24 .23 3.227 .93 .80 .14 .27 .07 .08 -.07 -.09 .16 .13 .13 3.170 .93 .77 .04 .15 -.02 -.01 -.20 -.20 .00 -.00 -.00 3.967 .80 .77 .33 .42 .28 .17 .12 .08 .38 .38 .34 3.097 .14 .04 .33 .54 .46 .33 .52 .27 .59 .58 .58 3.010 .27 .15 .42 .54 .42 .53 .50 .30 .61 .57 .64 3.203 .07 -.02 .28 .46 .42 .25 .46 .25 .55 .50 .52 2.350 .08 -.01 .17 .33 .53 .25 .31 .26 .40 .36 .43 1.443 .07 -.20 .12 .52 .50 .46 .31 .56 .71 .68 .70 103.563 .09 -.20 .08 .27 .30 .25 .26 .56 .48 .49 .47 104.420 .16 .00 .38 .59 .61 .55 .40 .71 .48 .86 .85 31.337 .13 -.00 .38 .58 .57 .50 .36 .68 .49 .86 .81 52.133 .13 -.00 .34 .58 .64 .52 .43 .70 .47 .85 .81 30.017 .170 3.967 3.097 3.010 2.203 2.350 1.445 103.563 104.420 31.337 52.133 30.017 1.932 6.112 2.411 1.382 1.580 1.515 1.501 17.761 18.469 10.844 16.941 12.883 TABLE 9 ROTATED FACTOR MATRIX FOR TOTAL GROUP OF 300 A B C D E F G H h 2 1. C lo su re Law fo r a d d itio n .96 .13 .00 .08 .02 .16 .07 .00 .98 2. C lo su re Law M u ltip lic a tio n .98 .11 .01 .09 .01 .09 .08 .00 .99 3 . Commutative Law A d d itio n .88 .22 .03 .09 .08 .03 .19 .01 .87 4 . Commutative Law M u ltip lic a tio n .96 .08 .06 .1.1 .02 .04 .09 .00 .95 5 . A s so c ia tiv e Law A d d itio n .98 .07 .01 .08 .04 .05 .09 .01 .98 6 . A s s o c ia tiv e Law M u ltip lic a tio n .78 .35 .08 .06 .04 .30 .01 .01 .83 7. D e fin itio n o f D iffe re n c e .11 .63 .19 .19 .21 .06 .08 .04 .55 8 . Axiom o f S u b tra c tio n .20 .61 .48 .04 .07 .04 .07 .02 .65 9. D e fin itio n o f Q u o tien t .02 .60 .08 .02 .15 .09 .00 .11 .41 10. Axiom o f D iv isio n .01 .37 .50 .03 .03 .00 CM O • © o • .39 11. Language IQ (CTM M ) .12 .77 .08 .24 .13 .08 .04 .07 .70 12. Non-Language IQ (CTM M ) .13 .51 .09 .24 .31 o o • .00 .00 .44 13. A rith m e tic R easoning (CAT) .07 .93 .10 .07 .07 .01 .01 .05 .89 14. A rith m e tic Fundam entals (CAT) .06 .88 .06 .08 • o 0 0 .08 .01 .13 .82 15. A lgebra A p titu d e (AAS) .06 .88 .20 .03 .05 .06 .05 .14 .82;£ ■ p * 75 such lo a d in g s a r e shown in p a re n th e s e s . A d e s c r ip tio n o f each o p e ra tio n in term s o f th e ax io m atic m ethod, as d e s c rib e d by M aria, was in c lu d e d w ith th e o p e ra tio n as i t ap p eared in a f a c t o r . T h is , combined w ith a d e s c r ip tio n o f th e a p p lic a tio n o f th e p r in c ip l e to be a p p lie d to th e m a th em atic al s i t u a t i o n , was found n e c e s s a ry f o r a com plete i n t e r p r e t a t i o n o f th e f a c t o r . For co n v en ien ce, each f a c t o r , to g e th e r w ith th e t e s t s w ith s i g n i f i c a n t lo a d in g s , a r e shown in ta b u la r form . T his i s fo llo w ed by th e n e c e ss a ry d e s c r ip tio n s . t i o n . --T he c lo s u re p ro p e rty o f a d d itio n and m u ltip lic a tio n p ro v id e s th e laws go v ern in g th e com bination o f any two e le m e n ts . The C lo su re Law f o r A d d itio n . An o p e ra tio n a d d itio n e x i s t s , d e s ig n a te d by + , w hich when a p p lie d to any two numbers a ,b o f th e system g a c t p X . A 1 . C lo su re Law f o r A d d itio n 2. C lo su re Law f o r M u ltip lic a tio n 3 . Commutative Law f o r A d d itio n 4 . Commutative Law f o r M u ltip lic a tio n 5 . A s s o c ia tiv e Law f o r A d d itio n 6 . A s s o c ia tiv e Law f o r M u ltip lic a tio n .96 .98 .88 .96 .98 .78 ( .35B) The c lo s u re p ro p e rty o f a d d itio n and m u ltip lic a - 76 produces one and o n ly one number a + b , and t h i s number i s an elem ent o f th e sy stem . The number a + b i s c a lle d th e sum o f a and b; th e numbers a and b a r e c a lle d th e term s o f th e sum. W e u se th e word "adding" to r e f e r to th e c a rry in g o u t o f th e a d d itio n o p e r a tio n , and we say t h a t a + b i s o b ta in e d by "adding a and b ." (21:18) The C lo su re Law f o r M u lt ip lic a ti o n . An op e r a tio n m u ltip lic a tio n e x i s t s , d e s ig n a te d by x , o r •, o r no symbol a t a l l , w hich when a p p lie d to any two numbers a ,b o f th e system produces one and o n ly one number a x b , o r a • b , o r sim ply ab , and t h i s number i s an elem ent o f th e system . The number a x b , o r a • b , o r ab i s c a lle d th e p ro d u ct o f a and b; th e numbers a and b a re c a lle d th e f a c to r s o f th e p ro d u c t. W e u se th e word " m u ltip ly in g " to r e f e r to th e c a rry in g o u t o f th e o p e ra tio n m u lt ip lic a ti o n and say ab i s o b ta in e d by m u ltip ly in g a and b to g e th e r o r m ul tip ly in g a by b . (21:18) S e q u e n tia l o rd e r in a d d itio n and m u l t i p l i c a t i o n . - - The com m utative p ro p e rty f o r a d d itio n and m u ltip lic a tio n p ro v id e s th e laws g o v ern in g th e s e q u e n tia l o rd e r o f any two elem ents combined by th e law go v ern in g th e c lo s u re p ro p e rty . Commutative Law f o r A d d itio n . The o p e ra tio n a d d itio n has th e p ro p e rty t h a t th e sum o f two numbers i s in d ep en d en t o f th e s e q u e n tia l o rd e r o f i t s term s; in sym bols, a + b ® b + a . Commutative Law o f M u lt ip lic a ti o n . The op e r a tio n m u lt ip lic a ti o n has th e p ro p e rty t h a t 77 th e p ro d u ct o f two numbers i s independent o f th e s e q u e n tia l o rd e r o f i t s f a c to r s ; in sym bols, ab « b a . (21:20) Grouping in a d d itio n and m u l t i p l i c a t i o n . —The a s s o c i a t i v e p ro p e rty fo r a d d itio n and m u ltip lic a tio n p ro v id es th e laws f o r grouping any two o f th e th re e elem ents com b in ed by th e m ath em atical s i t u a t i o n . A s s o c ia tiv e Law fo r A d d itio n . The o p e ra tio n a d d itio n has th e p ro p e rty th a t th e sum o f th re e numbers in a s ta te d s e q u e n tia l o rd e r is independ e n t o f th e grouping o f two term s from th e th re e ; in sym bols, (a + b) + c * a + (b + c ) . A s s o c ia tiv e Law fo r M u lt ip lic a ti o n . The op e r a tio n m u ltip lic a tio n has th e p ro p e rty th a t th e p ro d u ct o f th re e numbers in a s ta te d s e q u e n tia l o rd e r i s independent o f th e grouping o f two f a c to r s from th e th re e ; in sym bols, (ab )c = a ( b c ) . ( 21: 22) The p r in c ip le s governing th e laws o f c lo s u re , s e q u e n tia l o rd e r, and grouping w ere a p p lie d to th e m athem at i c a l s it u a ti o n s in A T e st o f M athem atical O p eratio n s in th e fo llo w in g manner: C lo su re P ro p e rty 5 . Paul had e a i r p l a n e s . He made d more a i r p la n e s . He had e + d a irp la n e s in a l l . 1 . e + d S3 e %r * * d 2 . d m e + d 3 . e - d S3 e + d 4 . e + d S e + d 5. none 78 S e q u e n tia l O rder 1 1 . George had a w h ite nr a - b m ice. He s o ld a l l o f th e n a t b c e n ts U 2 . a X b * b - a e a c h . George r e 3 . a + b = a x b ce iv ed a x b c e n ts 4 . a X b is b x a f o r th e m ice. 5 . none G rouping 1. (b + e) + f 2. b x e x f 8 3 . (b + e) mm f 4 . b - e + f S 5 . none fo r e c e n ts , and eggs f o r f, c e n ts . 4. b-e + f = b + e + f M other sp e n t b + e f c e n ts in a l l . The b a s ic p r in c ip le s in v o lv ed w ere used in th e same manner f o r b o th a d d itio n and m u ltip lic a tio n . A com parison o f th e above laws g overning th e p r in c ip le s o f c lo s u re , s e q u e n tia l o rd e r, and grouping re v e a le d (1) ex cep t f o r symbolism and w ording, th e two o p e ra tio n s o f a d d itio n and m u ltip lic a tio n a re in d is tin g u is h a b le , fo r each has th e c lo s u re , com m utative, and a s s o c ia tiv e p ro p e r t i e s ; (2) th e w ording o f th e a p p lie d m ath em atical s it u a ti o n d eterm in es th e o p e ra tio n in v o lv e d . The answer g iv en w ith in th e a p p lie d s i t u a t i o n g iv e s a c lu e as to th e n a tu re o f th e o p e ra tio n w ith in th e s i t u a t i o n . In th e o p e ra tio n s o f sub t r a c t i o n and d iv is io n , t h i s i s n o t s o . T his could be one 79 o f th e reaso n s f o r s e p a ra te f a c to r lo a d in g s . The f a c to r lo a d in g s and o th e r s t a t i s t i c s r e la te d to th e d e s c rip tio n o f F a c to r A a r e d e s c rib e d in T ables 8 and 9. The f a c to r lo a d in g s o f .78 to .98, as compared to th e lo ad in g s o f .01 to .20 on th e o th e r v a r ia b le s , in d ic a te a h ig h degree o f r e la tio n s h ip between th e s ix p r in c ip le s lo a d in g on t h i s f a c t o r . A h ig h degree o f com munality is in evidence on v a r ia b le s one through s ix , in d ic a tin g a r e l a t i v e l y h ig h c o n trib u tio n o f w hatever i s m easured by F a c to r A to th e common v a ria n c e on each o f th e v a r i a b l e s . F a c to r B 6 . A s s o c ia tiv e Law f o r M u ltip lic a tio n .35 (.78A) 7. D e fin itio n o f D iffe re n c e .63 8 . Axiom o f S u b tra c tio n .61 ( .48C) 9. D e fin itio n o f Q u o tien t .60 10. Axiom o f D iv isio n .37 ( .50C) 11. Language IQ .77 12. Non-Language IQ .51 13. A rith m e tic R easoning .93 14. A rith m e tic Fundam entals .88 15. A lgebra A p titu d e .88 The In v e rse O p eratio n S u b tra c tio n . —The axiom o f s u b tra c tio n and th e d e f in it io n o f d if fe re n c e d e fin e th e o p e ra tio n s in v o lv ed in s u b tr a c tio n fo r th e t e s t o f m athe m a tic a l o p e r a tio n s . 80 Axiom o f S u b tra c tio n . For any g iv en p a ir o f numbers a ,b o f th e system th e re i s one and o n ly one number x o f th e system such th a t a + x » b; a ls o x + a > = b . D e f in itio n o f D if fe re n c e . The u n iq u e number x such th a t a + x ■ b i s c a lle d th e d if fe re n c e b - a (re a d "b m inus a " ) . The numbers a and b a re c a lle d th e term s o f th e d if f e r e n c e . The d if f e r e n c e b - a i s a ls o d e s c rib e d as th e number produced by s u b tr a c tin g a from b , and th e s ig n n“" i s in te r p r e t e d as th e symbol o f an o p e ra tio n c a lle d s u b tr a c tio n . S in ce d e term in in g x a s b * a from th e r e l a t i o n a + x = b i s th e problem o f fin d in g one term x o f a sum when th e sum b and th e o th e r term a a r e g iv e n , s u b tr a c tio n as an o p e ra tio n i s c a lle d th e in v e rs e o f a d d i t i o n . (21:26) The In v e rs e O p eratio n D iv is io n . —The axiom o f d i v i s io n and th e d e f in it io n o f q u o tie n t d e fin e th e o p e ra tio n s in v o lv ed in d iv is io n f o r th e t e s t o f m ath em atical o p e ra tio n s . Axiom o f D iv is io n . For any p a i r o f numbers a ,b o f th e system , o f w hich a ^ 0, th e r e i s one and o n ly one number x o f th e system such th a t ax = b , a ls o xa = b . D e f in itio n o f Q u o tie n t. The u n iq u e number x such th a t ax = b , w here a ^ 0 , i s c a lle d th e q u o tie n t (re a d "b o v er a ”) . The numbers a ,b a re c a lle d th e members o f th e q u o tie n t; b i s c a lle d th e n u m e rato r, a th e denom inator o f th e q u o tie n t The denom inator o f a q u o tie n t m ust be a n o n -zero num ber. 81 The q u o tie n t k I s a ls o d e s c rib e d as th e num b e r o b ta in e d when b i s d iv id e d by a , o r by d iv id in g b by a , and a i s r e f e r r e d to as th e d iv is o r o f b . The b a r symbol w hich ap p ears in th e quo t i e n t i s in te r p r e te d as th e symbol o f an op- c L e r a tio n c a lle d d iv is io n . S in ce determ ing x as ^ from ax - b , a 4 s 0 , i s fin d in g one f a c to r when th e p ro d u ct b and th e o th e r f a c to r a a r e known, d iv is io n as a p ro c e ss i s c a lle d th e in v e rs e o f m u lt ip lic a ti o n . (2 1 :3 8 -3 9 ) The p r in c ip le s go v ern in g th e o p e ra tio n s o f s u b tr a c tio n and d iv is io n w ere a p p lie d to th e m ath em atical s i t u a tio n s in th e fo llo w in g m anner: Axiom o f S u b tra c tio n 1 . z - f » e 29. M other baked f c h e rry t a r t s . 2. f x z « e The fam ily a t e e t a r t s f o r 3 . z + e * f lu n c h . T here w ere z t a r t s 4 . e x f » z l e f t . 5 . none D e f in itio n o f D iffe re n c e 1. b x z * * a 4 . D ic k 's allow ance i s a c e n ts 2. a + b - z a week. J im 's i s b c e n ts . 3 . a - b = z J im 's allow ance i s z, c e n ts 4 . b - z = a le s s th an D ic k 's . 5 . none Axiom o f D iv isio n 1. b + c * z 41. T here a re b desks in B i l l 's 2. z - c ■ b room. T here a r e c desks in 3 . c x z = b each row. T here a re z rows 4 . c x b * z in th e room . 5 . None D e f in itio n o f Q u o tien t 82 1. a + c » z 23. a boys sh a re d e q u a lly th e 2. a - c < ■ z c, d o lla r s th e y ea rn e d . Each 3 . £ = z boy re c e iv e d z d o l l a r s . , f " 4 . f ® c 5 . None A com parison o f th e p r in c ip le s g overning th e o p e r a tio n s o f s u b tr a c tio n and d iv is io n re v e a le d t h a t th e d e f i n i t i o n s o f d if f e r e n c e and q u o tie n t r e f e r to a d i r e c t op e r a tio n , w h ile th e axioms o f s u b tr a c tio n and d iv is io n r e p r e s e n t th e r e s u l t s when th e d ir e c tio n i s c a r r ie d o u t. A good example o f t h i s r e la tio n s h i p i s th e m ethod o f te a c h in g th e c h ild to check h is answ er by ad d in g th e minuend and d if fe re n c e in th e c a se o f s u b tr a c tio n and m u ltip ly in g th e d iv is o r tim es th e q u o tie n t in th e c a se o f d iv is io n . T his co u ld , in p a r t , acco u n t fo r th e s e p a ra te lo a d in g o f th e axioms o f s u b tr a c tio n and d iv is io n on F a c to r C. The complex n a tu re o f F a c to r B does n o t p ro v id e a sim ple e x p la n a tio n as to th e n a tu re o f i t s e x is te n c e . However, th e h ig h lo a d in g on language IQ, a r ith m e tic r e a so n in g , a r ith m e tic fu n d am en tals, and a lg e b ra a p titu d e in d ic a te a re a s o n in g f a c t o r . Of p rim ary i n t e r e s t i s th e r e la tio n s h i p o f th e m ath em atical o p e ra tio n s to th e f a c to r and th e d if f e r e n c e betw een t h i s s e t o f o p e ra tio n s and 83 th o se o f F a c to r A. F a c to r C 8 . Axiom o f S u b tra c tio n .48 (.6 1 ) 9. Axiom o f D iv isio n .50 (.3 7 ) As in d ic a te d under F a c to r B, th e axioms o f s u b tr a c tio n and d iv is io n r e p re s e n t th e r e s u l t s when th e d ir e c t o p e ra tio n o f s u b tr a c tio n and d iv is io n a re c a r r ie d o u t. When a p p lie d to th e m ath em atical s i t u a t i o n s , th e c h ild m ust be a b le to see th e r e la tio n s h ip o f th e elem ents w ith in th e s i t u a t i o n . A la c k o f ex p e rien ce w ith th e se concepts would cause th e c h ild to o v erlo o k th e r e la tio n s h ip i n v o lv e d . The n a tu re o f th e v a r ia b le s lo a d in g on t h i s f a c t o r m ight in d ic a te d ed u ctiv e re a s o n in g . However, more th a n th e above in fo rm a tio n i s needed to s u b s ta n tia te th is h y p o th e s is . A b ility L evels as D escrib ed bv a T est o f M athem atical O p eratio n s T his s e c tio n o f C hapter V p re s e n ts an a n a ly s is o f th e r e la tio n s h ip o f th e f a c to r lo a d in g s f o r th r e e s e p a ra te a b i l i t y le v e ls , c o n ta in e d w ith in th e t o t a l group, as com p ared to th e f a c to r lo a d in g s d e s c rib e d fo r th e t o t a l group. Group I . —This group i s composed o f th e to p 100 s tu d e n ts , w ith a raw sc o re ran g e o f 35 to 49, on A T est o f M athem atical O p e ra tio n s♦ The mean and sta n d a rd d e v ia tio n s on th e t e s t fo r t h i s group were 41.05 and 4 .3 3 , r e s p e c tiv e ly . The c o r r e la tio n m a trix , m eans, and sta n d a rd d e v ia tio n s fo r th e f i f t e e n v a r ia b le s a re p re se n te d in T able 10. These d a ta re v e a l a mean sc o re o f 4 .2 o r b e t t e r on th e f i r s t seven p r in c ip le s , in d ic a tin g a h ig h le v e l o f u n d er s ta n d in g . The c o r r e la tio n c o e f f ic ie n ts f o r th e f i r s t s ix p r in c ip le s have l o s t th e h ig h in te r - r e l a t i o n s h i p evidenced by th e a n a ly s is f o r th e t o t a l group. The r o ta te d f a c to r m a trix and o th e r s t a t i s t i c s r e la te d to th e f a c to r a n a ly s is a re p re se n te d in T able 11. The f i r s t s ix p r in c ip le s have s p l i t in to th r e e s e p a ra te f a c t o r s , in d ic a tin g th e complex n a tu re o f F a c to r A when d a ta f o r th e t o t a l group a re c o n sid e re d . P a rt o f th e reaso n f o r th e s p l i t in F a c to r A i s b e lie v e d to be caused by th e h ig h sc o re s on th e s e p r in c ip le s , th e re b y cau sin g a s h a tte r in g e f f e c t on t h i s f a c t o r . The low er c o e f f ic ie n ts 2 in th e column la b e le d h would tend to s u b s ta n tia te t h i s h y p o th e s is . TABLE 10 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE UPPER Variable 1 2 3 4 5 6 7 8 9 10 11 1. Closure Law Add. .04 .14 .12 -.22 -.04 .10 .12 .01 .17 .16 2. Closure Law Multi. .04 .16 .35 .06 .06 .04 .11 .04 .20 .16 3. Commutative Law Add. .14 .16 .26 .14 .30 .11 .21 .01 .09 .13 4. Commutative Law Multi. .12 .35 .26 .06 .20 .04 .06 .13 .12 .15 5. Associative Law Add. -.22 .06 .14 .06 .50 -.06 -.01 -.06 -.05 .02 6. Associative Law Multi. -.04 .06 .30 .20 .50 -.10 -.03 .11 -.13 .00 7. Definition of Difference .10 .04 .11 .04 -.06 -.10 .08 .18 .25 .25 8. Axiom of Sub. .12 .11 .21 .06 -.01 -.03 .08 .03 .47 .25 9. Definition of Quotient .01 .04 .01 .13 -.06 .11 .18 .03 .10 .25 10. Axiom of Div. .17 .20 .09 .12 -.05 -.13 .25 .47 .10 .28 11. Language IQ .16 .16 .13 .15 .02 .00 .25 .25 .25 .28 12. Non-Language IQ .03 .09 .11 .06 -.02 .11 .02 .21 .18 .24 .33 13. Arithmetic Reasoning .09 .12 .04 .17 -.07 -.02 .21 .10 .43 .26 .49 14. Arithmetic Funda mentals .16 .05 .11 .19 -.07 .02 .21 .16 .33 .34 .55 15. Algebra Apt. .21 .16 .16 .13 -.08 .02 .28 .29 .36 .36 .57 Mean 4.290 4.68 4.630 4.620 4.460 4.720 4.240 3.660 3.310 2.440 115.461 Standard Dev. 1.094 .548 .580 .678 .881 .683 .767 1.249 1.433 1.731 12.43' 85 TABLE 10 MEANS, AND STANDARD DEVIATIONS FOR THE UPPER GROUP OF 100 5 6 7 8 9 10 11 12 13 14 15 Mean Std. Dev. -.22 -.04 .10 .12 .01 .17 .16 .03 .09 .16 .21 4.290 1.094 .06 .06 .04 .11 .04 .20 .16 .09 .12 .05 .16 4.680 .548 .14 .30 .11 .21 .01 .09 .13 .11 .04 .11 .16 4.630 .580 .06 .20 .04 .06 .13 .12 .15 .06 .17 .19 .13 4.620 .678 .50 -.06 -.01 -.06 -.05 .02 -.02 -.07 -.07 -.08 4.460 .881 .50 -.10 -.03 .11 -.13 .00 .11 -.02 .02 .02 4.720 .683 -.06 -.10 .08 .18 .25 .25 .02 .21 .21 .28 4.240 .767 -.01 -.03 .08 .03 .47 .25 .21 .10 .16 .29 3.660 1.249 -.06 .11 .18 .03 .10 .25 .18 .43 .33 .36 3.310 1.433 -.05 -.13 .25 .47 .10 .28 .24 .26 .34 .36 2.440 1.731 .02 .00 .25 .25 .25 .28 .33 .49 .55 .57 : L15.460 12.439 -.02 .11 .02 .21 .18 .24 .33 .31 .33 .37 : 112.070 16.078 n* 0 • 1 -.02 .21 .10 .43 .26 .49 .31 .68 .71 41.020 6.884 -.07 .02 .21 .16 .33 .34 .55 .33 .68 .69 65.980 9.015 -.08 .02 .28 .29 .36 .36 .57 .37 .71 .69 41.230 10.260 4.460 4.720 4.240 3.660 3.310 2.440 115.460 112.070 41.020 65.980 41.230 .881 .683 .767 1.249 1.433 1.731 12.439 16.078 6.884 9.015 10.260 TABLE 11 ROTATED FACTOR MATRIX FOR THE UPPER GROUP OF 100 A B C D E F G H h2 1 . C lo su re Law f o r a d d itio n .11 .11 .10 .08 .05 .43 .01 .01 .23 2. C lo su re Law M u ltip lic a tio n .06 .05 .14 .50 .03 .01 .01 .02 .27 3 * Commutative Law A d d itio n .04 .34 .18 .23 .04 .29 .05 .02 .29 4 . Commutative Law M u ltip lic a tio n .12 .16 .00 .50 .11 .16 .05 .02 .33 5 . A s s o c ia tiv e Law A d d itio n .04 .60 .01 .06 .13 .23 .05 .04 .45 6 . A s s o c ia tiv e Law M u ltip lic a tio n .02 .69 .09 .11 .13 .01 .00 .04 .52 7. D e f in itio n o f D iffe re n c e .27 .08 .13 .06 .24 .11 .25 .03 .23 8 . Axiom o f S u b tra c tio n .10 .03 .60 .06 .01 .12 .01 .03 .39 9 . D e fin itio n o f Q u o tien t .46 .03 .05 .05 .12 .04 .23 .06 .29 10. Axiom o f D iv isio n .25 .12 .59 .17 .00 .06 .08 .06 .47 11- Language IQ (CTM M ) .61 .04 .25 .13 .06 .11 .06 .04 .47 12. Non-Language IQ (CTM M ) .36 .09 .27 .01 .22 .00 .10 .11 .28 13. A rith m e tic R easoning (CAT) .81 .07 .04 .12 .14 .01 .07 .03 .69 14. A rith m e tic Fundam entals (CAT) .77 .01 .15 .01 .12 .14 .01 .09 .66 15. A lgebra A p titu d e (AAS) .79 .00 .27 .07 .05 .18 .03 .02 .73 o o o» 87 The d e f in it io n o f q u o tie n t rem ained w ith th e v a r ia b le s o f language and non -lan g u ag e IQ, a r ith m e tic re a s o n in g , a rith m e tic fu n d am en tals, and a lg e b ra a p titu d e . The o th e r o p e ra tio n s o f s u b tr a c tio n and d iv is io n l o s t t h e i r s i g n i f i c a n t lo a d in g s w ith t h i s f a c t o r . Of th e th re e f a c to r s i d e n t i f i e d f o r th e t o t a l group, F a c to r C rem ained th e same f o r t h i s group, w ith la r g e r lo a d in g s th a n p re v io u s ly re c o g n iz e d f o r th e t o t a l group. T his would te n d to in c re a s e th e s t a b i l i t y o f t h i s f a c to r in th e perform ance o f t h i s subgroup. Group I I . - -T h is group i s composed o f th e m id d le 100 s tu d e n ts , w ith a raw sc o re ran g e o f 20 to 34 on A T est o f M athem atical O p e ra tio n s . The mean and s ta n d a rd d e v ia tio n s on th e t e s t f o r t h i s group w ere 27.04 and 4 .9 9 , r e s p e c tiv e ly . The c o r r e la tio n m a trix , m eans, and sta n d a rd d e v ia tio n s f o r th e f i f t e e n v a r ia b le s a re p re s e n te d in T ab le 12. These d a ta r e v e a l a mean s c o re o f 3 .3 6 o r h ig h e r on th e f i r s t s i x o p e ra tio n s , in d ic a tin g a r e l a t i v e l y h ig h le v e l o f tin d e rsta n d in g . The c o r r e la tio n c o e f f ic ie n ts f o r p r in c ip le s one th ro u g h s ix show a h ig h d eg ree o f i n t e r - r e l a tio n s h ip . The n e g a tiv e c o e f f ic ie n ts o f c o r r e la tio n s f o r TABLE 12 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR Variable 1 2 3 4 5 6 7 8 9 10 1. Closure Law Add. .99 .90 .94 .97 •77 -.15 -.03 -.19 -.10 2. Closure Law Multi. .99 .90 .96 .98 .81 -.18 -.02 -.19 -.09 3. Commutative Law Add. .90 .90 .91 .89 .73 -.19 -.04 -.15 -.20 4. Commutative Law Multi. .94 .96 .91 .94 .81 -.23 -.05 -.19 -.10 5. Associative Law Add. .97 .98 .89 .94 .84 -.19 -.06 -.19 -.12 6. Associative Law Multi. .77 .81 .73 .81 .84 -.22 -.07 -.09 -.13 7. Definition of Difference -.15 -.18 -.19 -.23 -.19 -.22 .10 •15 .05 8. Axiom of Sub. -.03 -.02 -.04 -.05 -.06 -.07 .10 .25 .36 9. Definition of Quotient -.19 -.19 -.15 -.19 -.19 -.09 .15 .25 .04 10. Axiom of Div. -.10 -.09 -.20 -.10 -•12 -.13 .05 .36 .04 11. Language IQ -.58 -.59 -.54 -.60 -.62 -.51 .20 .17 .31 .11 12. Non-Language IQ -.54 -.54 -.51 -.49 -.53 -.31 .04 .00 .17 .14 13. Arithmetic Reasoning -.37 -.35 -.28 -.31 -.38 -.23 .02 .25 .33 .20 14. Arithmetic Funda mentals -.40 -.38 -.31 -.33 -.38 -.16 T13 .17 .31 .14 15. Algebra Apt. -.27 -.26 -.24 -.26 -.29 -.15 .21 .33 .29 .19 Mean 4.01 4.59 3.66 3.55 4.02 3.36 2.95 2.03 2.30 .93 Standard Dev. 11.21 11.62 3.39 5.11 8.41 2.73 1.07 1.22 1.32 1.13 TABLE 12 MATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE MIDDLE GROUP OF 100 88 4 5 6 7 8 9 10 11 12 13 14 15 Mean Std. Dev. .94 .97 .77 -.15 -.03 -.19 -.10 -.58 -.54 -.37 -.40 -.27 4.01 11.21 .96 .98 .81 -.18 -.02 -.19 -.09 -.59 -.54 -.35 -.38 -.26 4.59 11.62 .91 .89 .73 -.19 -.04 -.15 -.20 -.54 -.51 -.28 -.31 -.24 3.66 3.39 .94 .81 -.23 -.05 -.19 -.10 -.60 -.49 -.31 -.33 -.26 3.55 5.11 .94 .84 -.19 -.06 -.19 -.12 -.62 -.53 -.38 -.38 -.29 4.02 8.41 .81 .84 -.22 -.07 -.09 -.13 -.51 -.31 -.23 -.16 -•15 3.36 2.73 .23 -.19 -.22 .10 •15 .05 .20 .04 .02 -.13 .21 2.95 1.07 .05 -.06 -.07 .10 .25 .36 .17 .00 .25 -.17 .33 2.03 1.22 .19 -.19 -.09 .15 .25 .04 .31 .17 .33 .31 .29 2.30 1.32 •.10 -•12 -.13 .05 .36 .04 .11 .19 .20 .14 .19 .93 1.13 -.60 -.62 -.51 .20 .17 .31 .11 .54 .55 .45 .58 103.11 16.84 -.49 -.53 -.31 .04 .00 .17 .14 .54 .39 .39 .37 103.81 19.11 -.31 -.38 -.23 .02 .25 .33 .20 .55 .39 .73 .75 31.25 7.46 -.33 -.38 -.16 :i3 .17 .31 .14 .45 .39 .73 .65 53.26 13.61 -.26 -.29 -.15 .21 .33 .29 .19 .58 .37 .75 .65 28.64 9.89 1.55 4.02 3.36 2.95 2.03 2.30 .93 103.11 103.81 31.25 53.26 28.64 ; .ii 8.41 2.73 1.07 1.22 1.32 1.13 16.84 19.11 7.46 13.61 9.89 89 th e o th e r v a r ia b le s (seven th ro u g h f i f t e e n ) in d ic a te an in v e rs e r e la tio n s h i p betw een th e f i r s t s ix p r in c ip le s and th e v a r i a b l e s , seven th ro u g h f i f t e e n . The r o ta te d f a c to r m a trix and o th e r s t a t i s t i c s r e l a t e d to th e f a c to r a n a ly s is a r e p re s e n te d in T ab le 13. These d a ta r e v e a l t h a t F a c to r A has r e ta in e d i t s i d e n t i t y w ith a d d itio n a l lo a d in g s on language and n o n -lan g u ag e IQ. F a c to r B has l o s t i t s o r i g i n a l i d e n t i t y w ith r e s p e c t to th e d e f in it io n o f d if f e r e n c e , axiom o f s u b tr a c tio n , and axiom o f d iv is io n . F a c to r C has s t i l l r e ta in e d i t s o r i g in a l i d e n t i t y . Group I I I .- -T h is group i s composed o f th e low er 100 s tu d e n ts , w ith a raw s c o re ran g e o f 2 to 18, on A T e st o f Mathematical O p e ra tio n s . The mean and s ta n d a rd d e v ia tio n s fo r t h i s group w ere 12.62 and 3 -9 8 , r e s p e c tiv e ly . The c o r r e la tio n m atrix ,, m eans, and s ta n d a rd d e v ia tio n s f o r th e f i f t e e n v a r ia b le s a re p re s e n te d in T ab le 14. These d a ta r e v e a l a mean s c o re o f 1.19 to 2.68 on th e f i r s t s i x p r in c ip l e s , in d ic a tin g a low er le v e l o f u n d er s ta n d in g th a n t h a t o f th e o th e r two g ro u p s . The c o e f f i c ie n ts o f c o r r e la tio n r e v e a l a h ig h i n t e r - c o r r e l a t i o n among th e f i r s t s i x p r in c ip l e s . TABLE 13 ROTATED FACTOR MATRIX FOR THE MIDDLE GROUP OF 100 A B C D E F G H h 2 1. C lo su re Law fo r a d d itio n .94 .23 .01 .20 .01 .03 .11 .02 .99 2. C lo su re Law M u ltip lic a tio n .95 .20 .01 .17 .05 .05 .09 .06 1.00 3 . Commutative Law A d d itio n .87 .12 .15 .27 .09 .09 .01 .12 .89 4 . Commutative Law M u ltip lic a tio n .93 .15 .02 .14 .18 .06 .08 .10 .96 5 . A s s o c ia tiv e Law A d d itio n .95 .23 .04 .07 .08 .08 .04 .07 .98 6 . A s s o c ia tiv e Law M u ltip lic a tio n .86 .04 .08 .16 .22 .02 .08 .04 .84 7. D e fin itio n o f D iffe re n c e .13 .06 .05 .09 .49 .00 .07 .00 .27 8 . Axiom o f S u b tra c tio n .02 .21 .52 .11 .14 .04 .19 .00 .38 9. D e fin itio n o f Q u o tien t .10 .30 .10 .02 .14 .03 .35 .00 .26 10. Axiom o f D iv isio n .08 .10 .57 .06 .02 .05 .04 .00 .35 11. Language IQ (CTM M ) .48 .51 .07 .03 .19 .35 .13 .02 .68 12. Non-Language IQ (CTM M ) .42 .36 .08 .28 .09 .37 .02 .01 .53 13. A rith m etic R easoning (CAT) .20 .85 .17 .07 .09 .03 .08 .00 .81 14. A rith m e tic Fundam entals (CAT) .18 .76 .07 .27 .05 .06 .07 .03 .70 15. A lgebra A p titu d e (AAS) .09 .78 .21 .01 .21 .14 .06 .03 « to 06 TABLE 14 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE LOWE Variable 1 2 3 4 5 6 7 8 9 10 11 1. Closure Law Add. .98 .88 .95 .97 .80 -.06 .24 -.15 -.13 oo 0 • 1 2. Closure Law Multi. .98 .87 .97 .98 .82 OO o • 1 .25 -.13 -.09 -.06 3. Commutative Law Add. .88 OO vl .87 .87 CM r^. -.04 .26 -.13 -.14 -.01 4. Commutative Law Multi. .95 .97 .87 .96 .80 i © oo .28 -.15 -.08 -.05 5. Associative Law Add. .97 .98 .87 .96 .86 -.06 .24 -.12 -.09 -.06 6. Associative Law Multi. .80 .82 .72 .80 .86 -.09 .20 .01 .01 -.01 7. Definition of Difference -.06 -.08 -.04 -.08 -.06 -.09 -.00 .15 -.14 .15 8. Axiom of Sub. .24 .25 .26 .28 .24 o CM • 0 o 1 -.00 .09 .27 9. Definition of Quotient -.15 -.13 -.13 -.15 -.12 .01 .15 -.00 -.03 .23 10. Axiom of Div. -.13 -.09 -.14 -.08 -.09 .01 -.14 .09 -.03 .04 11. Language IQ -.08 -.06 -.01 -.05 -.06 -.01 .15 .27 .23 .04 12. Non-Language IQ .01 .02 .05 .04 .01 .03 .00 .18 -.08 .09 .54 13. Arithmetic Reasoning -.03 -.02 .00 CM 0 • 1 -.01 .07 .16 .33 .18 .10 .60 14. Arithmetic Funda mentals .02 .03 .02 .00 .03 .10 .16 .23 .10 .01 .56 15. Algebra Apt. -.05 i • © ■p* OO © • 1 -.07 -.07 -.05 .09 .31 © CM • .06 .50 Mean 2.68 2.31 1.39 1.34 1.68 1.19 1.86 .91 1.44 .94 92.08 Standard Dev. 8.04 9.04 2.49 3.80 6.07 1.82 1.06. .82 1.17 1.01 15.69 TABLE 14 MATRIX, MEANS, AND STANDARD DEVIATIONS FOR THE LOWER GROUP OF 100 4 5 6 7 8 9 10 11 12 13 14 15 Mean Std. Dev. .95 .97 .80 -.06 .24 -.15 -.13 -.08 .01 -.03 .02 -.05 2.68 8.04 .97 .98 .82 -.08 .25 -.13 -.09 -.06 .02 -.02 .03 -.04 2.31 9.04 .87 .87 .72 -.04 .26 -.13 -.14 -.01 .05 .00 .02 -.08 1.39 2.49 .96 .80 -.08 .28 -.15 -.08 -.05 .04 -.02 .00 -.07 1.34 3.80 .96 .86 -.06 .24 -.12 -.09 -.06 .01 -.01 .03 -.07 1.68 6.07 .80 .86 -.09 .20 .01 .01 -.01 .03 .07 .10 -.05 1.19 1.82 .08 -.06 -.09 -.00 .15 -.14 .15 .00 .16 .16 .09 1.86 1.06 .28 .24 .20 -.00 -.00 .09 .27 .18 .33 .23 .3i .91 .82 .15 -.12 .01 .15 -.00 -.03 .23 -.08 .18 .10 .20 1.44 1.17 .08 -.09 .01 -.14 .09 -.03 .04 .09 . 1 0 .01 .06 .94 1.01 .05 -.06 -.01 .15 .27 .23 .04 .54 .60 .56 .50 92.08 15.69 .04 .01 .03 .00 .18 -.08 .09 .54 .43 .44 .34 97.37 17.29 CM o • -.01 .07 .16 .33 .18 .10 .60 .43 .76 .70 21.74 8.03 .00 .03 .10 .16 .23 .10 .01 .56 .44 .76 .70 37.16 13.38 -.07 -.07 -.05 .09 .31 .20 .06 .50 .34 .70 .70 20.20 8.41 .34 1.68 1.19 1.86 .91 1.44 .94 92.08 97.37 21.74 37.16 20.20 1.80 6.07 1.82 1.06 .82 1.17 1.01 15.69 17.29 8.03 13.38 8.41 92 The r o ta te d f a c to r m a trix and o th e r s t a t i s t i c s r e la t e d to th e f a c to r a n a ly s is a r e p re s e n te d in T ab le 15. The f i r s t s i x o p e ra tio n s have r e ta in e d t h e i r i d e n t i t y w ith F a c to r A f o r th e t o t a l g ro u p . F a c to r B i s now composed o f th e axiom o f s u b tr a c tio n , language IQ, n o n -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r ith m e tic fu n d am en tals, and a lg e b ra a p titu d e , p ro v id in g s t i l l f u r th e r ev id en ce o f th e complex n a tu r e o f th e f a c to r s f o r th e t o t a l g ro u p . F a c to r C has l o s t i t s i d e n t i ty f o r th e f i r s t tim e , in d ic a tin g th e p o s s i b i l i t y o f a complex s t r u c t u r e w ith in t h i s f a c t o r . Sex D iffe re n c e s T his s e c tio n o f C hapter V p re s e n ts an a n a ly s is o f th e r e la tio n s h ip o f th e f a c to r lo a d in g s f o r sex d if fe re n c e s d e s c rib e d by a t e s t o f m ath em atical o p e r a tio n s , language and n o n -lan g u ag e IQ, a r ith m e tic re a s o n in g and fu n d am en tals, and a lg e b ra a p titu d e . R esearch has c o n s is te n tly in d ic a te d t h a t boys and g i r l s d i f f e r in a b i l i t y w ith r e s p e c t to m ath em atic s. W atters (28) found d i f f e r e n t f a c t o r p a tte r n s f o r h ig h school, boys and g i r l s in m athem atics achievem ent, and th e s tu d ie s by Doyle (2 9 ), D r is c o ll (3 0 ), R o e s s le in (3 1 ), TABLE 15 ROTATED FACTOR MATRIX FOR THE LOWER GROUP OF 100 A B C D E F G H h2 1 . C lo su re Law f o r A d d itio n .97 .01 .12 .11 .08 .08 .02 .05 .98 2. C lo su re Law M u ltip lic a tio n .98 .00 .08 .13 .00 .06 .05 .00 .99 . 3 . Commutative Law A d d itio n .87 .00 .15 .08 .11 .08 .14 .04 .82 4 . Commutative Law M u ltip lic a tio n .96 .01 .11 .09 .02 .01 .10 .01 .96 5 . A s s o c ia tiv e Law A d d itio n .99 .01 .03 .01 .02 .02 .00 .05 .99 6 . A s s o c ia tiv e Law M u ltip lic a tio n .87 .04 .13 .16 .11 .07 00 o • .02 .82 7 . D e fin itio n o f D iffe re n c e .07 .13 .17 .08 .34 .01 .03 .04 .17 8 . Axiom o f S u b tra c tio n .25 .35 .03 .19 .16 .04 .22 .00 .29 9 . D e fin itio n o f Q u o tien t .01 .13 .50 o o • .10 .01 .00 .00 .28 10. Axiom o f D iv isio n .08 .08 .00 .07 .40 .02 .03 .03 .18 11. Language IQ (CTM M ) .05 .67 .17 .16 .02 .30 .03 .00 .60 12. Non-Language IQ (CTM M ) .01 .55 .16 .12 .09 .33 .04 .00 .46 13. A rith m e tic Reasoning (CAT) .01 .87 .15 .13 .01 .01 .10 .01 .80 14. A rith m e tic Fundam entals (CAT) .04 .84 .03 .11 .11 C M O • .08 .02 .74 15. A lgebra A p titu d e (AAS) .06 .78 .14 .10 .02 .20 .01 .02 .69*, u > 94 R uszel (3 2 ), and Donohue (33) in d ic a te th a t th e two sexes may u se t h e i r m en tal a b i l i t i e s in d i f f e r e n t com binations on c e r ta in ty p es o f p ro b lem s. The c o r r e la tio n m a trix , m eans, and sta n d a rd d e v ia tio n s a re p re s e n te d in T able 16 f o r th e g i r l s and T able 17 fo r th e b o y s. These d a ta re v e a l r e l a t i v e l y h ig h i n t e r c o r r e la tio n s f o r th e g i r l s on th e f i r s t s i x v a r ia b le s , w h ile th e boys do n o t show th e same c o n s is te n c y . The r o ta te d f a c to r m a trix and o th e r s t a t i s t i c s r e l a t e d to f a c to r a n a ly s is a re s e t f o r th in T able 18 fo r g i r l s and T able 19 fo r b o y s. These d a ta re v e a l fo u r f a c to r s lo a d in g s i g n i f i c a n t ly f o r g i r l s and s ix fo r boys, in d ic a tin g a d if fe re n c e in f a c to r s tr u c t u r e f o r boys and g i r l s . Summary T his s e c tio n o f C hapter V p re s e n ts a summary o f th e fin d in g s in v o lv ed in th e a n a ly s is o f th e f a c to r s con t r i b u t i n g to th e se v e n th -g ra d e c h i l d 's a b i l i t y w ith th e m ath em atical p r in c ip le s governing th e o p e ra tio n s o f a d d i tio n , m u ltip lic a tio n , s u b tr a c tio n , and d iv is io n . TABLE 16 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FO Variable 1 2 3 4 5 6 7 8 9 10 1 1. Closure Law Add. .97 .74 .88 .94 .55 .01 .14 -.07 -.02 .0 2. Closure Law Multi. .97 .75 .91 .96 .60 .00 .16 -.04 -.02 .0 3. Commutative Law Add. .74 .75 .85 .80 .77 .32 .44 .20 .11 .2 4. Commutative Law Multi. .88 .91 .85 .92 .78 .18 .34 .10 .12 .1 5. Associative Law Add. .94 .96 .80 .92 .71 .09 .22 .01 .00 .0 6. Associative Law Multi. .55 .60 .77 .78 .71 .38 .49 .36 .18 .3 7. Definition of Difference .01 .00 .32 .18 .09 .38 .61 .44 .35 .5 8. Axiom of Sub. .14 .16 .44 .34 .22 .49 .61 .40 .54 .4 9. Definition of Quotient -.07 -.04 .20 .10 .01 .36 .44 .40 .22 .4 10. Axiom of Div. -.02 -.02 .11 .12 .00 .18 .35 .54 .22 .2 11. Language IQ .01 .00 .27 .15 .03 .33 .51 .48 .41 .25 12. Non-Language IQ .03 .05 .21 .17 .06 .29 .24 .32 .20 .24 .5 13. Arithmetic Reasoning .03 .06 .41 .29 .13 .52 .63 .62 .57 .41 .6 14. Arithmetic Funda mentals .04 .08 .40 .29 .16 .55 .62 .57 .50 .33 .6 15. Algebra Apt. .01 .03 .33 .22 .08 .44 .63 .66 .54 .44 .7 Mean 3.64 3.78 3.13 3.07 3.44 3.11 3.17 2.26 2.32 1.41 103.5 Standard Dev. 6.62 7.40 2.46 3.43 5.11 2.21 1.35 1.58 1.53 1.52 17.7 TABLE 16 CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR 148 GIRLS 95 3 4 5 6 7 8 9 10 11 12 13 14 15 Mean Std. Dev. .74 .88 .94 .55 .01 .14 -.07 -.02 .01 .03 .03 .04 .01 3.64 6.62 .75 .91 .96 .60 .00 .16 -.04 -.02 .00 .05 .06 .08 .03 3.78 7.40 .85 .80 .77 .32 .44 .20 .11 .27 .21 .41 .40 .33 3.13 2.46 .85 .92 .78 .18 .34 .10 .12 .15 .17 .29 .29 .22 3.07 3.43 .80 .92 .71 .09 .22 .01 .00 .03 .06 .13 .16 .08 3.44 5.11 .77 .78 .71 .38 .49 .36 .18 .33 .29 .52 .55 .44 3.11 2.21 .32 .18 .09 .38 .61 .44 .35 .51 .24 .63 .62 .63 3.17 1.35 .44 .34 .22 .49 .61 .40 .54 .48 .32 .62 .57 .66 2.26 1.58 O CM • .10 .01 .36 .44 .40 .22 .41 . 2 0 .57 .50 .54 2.32 1.53 .11 .12 .00 .18 .35 .54 .22 .25 .24 .41 .33 .44 1.41 1.52 .27 .15 .03 .33 .51 .48 .41 .25 .55 .68 .67 .72 103.56 17.75 .21 .17 .06 .29 .24 .32 .20 .24 .55 .45 .46 .45 104.58 18.33 .41 .29 .13 .52 .63 .62 .57 .41 .68 .45 .84 .84 30.51 10.55 .40 .29 .16 .55 .62 .57 .50 .33 .67 .46 .84 .83 52.36 16.72 .33 .22 .08 .44 .63 .66 .54 .44 .72 .45 .84 .83 30.11 11.94 J .13 3.07 3.44 3.11 3.17 2.26 2.32 1.41 103.56 104.58 30.51 52.36 30.11 1.46 3.43 5.11 2.21 1.35 1.58 1.53 1.52 17.75 18.33 10.55 16.72 11.94 TABLE 17 THE CORRELATION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR Variable 1 2 V- 3 4 5 6 7 8 9 10 11 1. Closure Law Add. .59 .54 .40 .37 .40 .48 .35 .27 .25 .38 2. Closure Law Multi. .59 .64 .69 .62 .68 .57 .51 .38 .34 .49 3. Commutative Law Add. .54 .64 .76 .53 .56 .57 .54 .38 .21 .48 4. Commutative Law Multi. .40 .69 .76 .57 .64 .55 .54 .40 .29 .48 5. Associative Law Add. .37 .62 .53 .57 .79 .48 .42 .38 .26 .36 6. Associative Law Multi. .40 .68 .56 .64 .79 .52 .51 .42 .31 .41 7. Definition of Difference .48 .57 .57 .55 .48 .52 .47 .47 .32 .55 8. Axiom of Sub. .35 .51 .54 .54 .42 .51 .47 .45 .52 .57 9. Definition of Quotient .27 .38 .38 .40 .38 .42 .47 .45 .27 .51 10. Axiom of Div. .25 .34 .21 .29 .26 .31 .32 .52 .27 .37 11. Language IQ .38 .49 .48 .48 .36 .41 .55 .57 .51 .37 12. Non-Language IQ .29 .32 .27 .32 .22 .32 .28 .30 .26 .28 .45 13. Arithmetic Reasoning .45 .58 .57 .60 .47 .58 .56 .63 .53 .38 ■74 14. Arithmetic Funda mentals .41 .54 .55 .57 .47 .59 .54 .59 .48 .38 .68 15. Algebra Apt. .46 .56 .50 .52 .42 .52 .54 .63 .50 .42 .70 Mean 2.93 3.14 3.13 2.94 2.80 2.93 2.87 2.14 2.39 1.47 104.29 Standard Dev. 1.66 1.72 1.82 1.83 1.73 1.89 1.39 1.59 1.50 1.48 15.89 96 TABLE 17 ,TION MATRIX, MEANS, AND STANDARD DEVIATIONS FOR 152 BOYS 4 5 6 7 8 9 10 11 12 13 14 15 Mean Std. Dev. .40 .37 .40 .48 .35 .27 .25 .38 .29 .45 .41 .46 2.93 1.66 .69 .62 .68 .57 .51 .38 .34 .49 .32 .58 .54 .56 3.14 1.72 .76 .53 .56 .57 .54 .38 .21 .48 .27 .57 .55 .50 3.13 1.82 .57 .64 .55 .54 .40 .29 .48 .32 .60 .57 .52 2.94 1.83 .57 .79 .48 .42 .38 .26 .36 .22 .47 .47 .42 2.80 1.73 .64 .79 .52 .51 .42 .31 .41 .32 .58 .59 .52 2.93 1.89 .55 .48 .52 .47 .47 .32 .55 .28 .56 .54 .54 2.87 1.39 .54 .42 .51 .47 .45 .52 .57 .30 .63 .59 .63 2.14 1.59 .40 .38 .42 .47 .45 .27 .51 .26 .53 .48 .50 2.39 1.50 .29 .26 .31 .32 .52 .27 .37 .28 .38 .38 .42 1.47 1.48 .48 .36 .41 .55 .57 .51 .37 .45 .74 .68 .70 104.29 15.89 .32 .22 .32 .28 .30 .26 .28 .45 .49 .47 .47 105.04 16.69 .60 .47 .58 .56 .63 .53 .38 •74 .49 .88 .86 32.33 10.87 .57 .47. .59 .54 .59 .48 .38 .68 .47 .88 .79 52.26 16.72 .52 .42 .52 .54 .63 .50 .42 .70 .47 .86 .79 30.07 13.61 .94 2.80 2.93 2.87 2.14 2.39 1.47 104.29 105.04 32.33 52.26 30.07 .83 1.73 1.89 1.39 1.59 1.50 1.48 15.89 16.69 10.87 16.72 13.61 TABLE 18 ROTATED FACTOR MATRIX FOR 148 GIRLS A B C D E F G H h2 1. C lo su re Law f o r A d d itio n .08 .96 .01 .01 .18 .00 .05 .01 .97 2. C lo su re Law M u ltip lic a tio n .06 .98 .01 .02 .09 .08 .02 .03 .98 3 . Commutative Law A d d itio n .31 .81 .07 .10 .12 .16 .01 .07 .82 4 . Commutative Law M u ltip lic a tio n .13 .94 .12 .09 .13 .02 .03 .07 .95 5 . A s s o c ia tiv e Law A d d itio n .03 .98 .00 .02 .05 .00 .04 .07 .97 6 . A s s o c ia tiv e Law M u ltip lic a tio n .46 .68 .08 .12 .39 .02 .00 .00 .85 7. D e fin itio n o f D iffe re n c e .68 .06 .29 .05 .00 .20 .00 .02 .59 8 . Axiom o f S u b tra c tio n .56 .21 .55 .12 .05 .13 .03 .01 .69 9 . D e fin itio n o f Q u o tien t .63 .00 .07 .02 .11 .03 .11 .00 .43 10. Axiom o f D iv isio n .28 .00 .58 .10 .00 .03 .01 .00 .43 11. Language IQ (CTM M ) .62 .03 .09 .53 .09 .07 .04 .01 .70 12. Non-Language IQ (CTM M ) .30 .07 .12 .55 .04 .01 .01 .01 .41 13. A rith m e tic R easoning (CAT) .84 .11 .21 .28 .04 .05 .06 .05 .84 14. A rith m e tic Fundam entals (CAT) .81 .13 •13 .32 .06 .03 .16 .01 .82 15. A lgebra A p titu d e (AAS) .81 .06 .29 .32 .07 .05 .03 .01 .86 v o TABLE 19 ROTATED FACTOR MATRIX FOR 152 BOYS A B C D E F G H h 2 1 . C lo su re Law f o r A d d itio n .24 .22 .59 .13 .18 .07 .00 .51 2. C lo su re Law M u ltip lic a tio n .27 .50 .49 .23 .31 .09 .04 .72 3 . Commutative Law A d d itio n .26 .35 .39 .09 .60 .15 .04 .74 4 . Commutative Law M u ltip lic a tio n .29 .45 .23 .17 .60 .11 .05 .75 5 . A s s o c ia tiv e Law A d d itio n .17 .75 .18 .14 .18 .14 .01 .70 6 . A s s o c ia tiv e Law M u ltip lic a tio n .29 .77 .18 .18 .20 .07 .01 .79 7. D e fin itio n o f D iffe re n c e .33 .32 .36 .18 .25 .35 .01 .55 8 . Axiom o f S u b tra c tio n .39 .25 .14 .52 .30 .18 .07 .64 9. D e fin itio n o f Q u o tien t .37 .25 .09 .19 .14 .37 .02 .41 10. Axiom o f D iv isio n .26 .14 .12 .52 .04 .06 .03 .39 11. Language IQ (CTM M ) .63 .12 .21 .27 .19 .33 .06 .68 12. Non-Language IQ (CTM M ) .48 .11 .17 .17 .06 .01 .16 .33 13. A rith m e tic R easoning (CAT) .82 .27 .18 .20 .25 .15 .09 .91 14. A rith m e tic Fundam entals (CAT) .77 .32 .15 .18 .23 .10 .07 .82 15. A lgebra A p titu d e (AAS) .74 .21 .25 .31 .15 .16 .08 98 o 0 0 • 99 The t o t a l sam p le.--T h re e f a c to r s f o r th e t o t a l group o f 300 s tu d e n ts w ere g iv en i n t e r p r e t a t i o n . F a c to r A was composed o f th e p r in c ip le s fo r a d d itio n and m u lt ip lic a t i o n . F a c to r B was composed o f th e s u b tr a c tio n and d i v i sio n p r in c ip l e s , language IQ, non -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r ith m e tic fu n d am en tals, and a lg e b ra a p titu d e . F a c to r C was composed o f th e axioms o f d iv is io n and su b t r a c t i o n . The f a c to r lo a d in g s and o th e r s t a t i s t i c s r e la te d to F a c to r A re v e a le d lo a d in g s o f .78 to .9 8 . These lo a d in g s , as compared to th e lo a d in g s o f .01 to .02 on th e o th e r v a r ia b le s , in d ic a te a h ig h d eg ree o f r e la tio n s h i p betw een th e s ix p r in c ip le s lo a d in g on t h i s f a c t o r . A h ig h d eg ree o f com m unality i s in ev id en ce on v a r ia b le s one th ro u g h s ix , in d ic a tin g a r e l a t i v e l y h ig h c o n trib u tio n o f w hatever i s m easured by F a c to r A to th e common v a ria n c e on each o f th e v a r i a b l e s . A com parison o f th e v a r ia b le s lo a d in g on t h i s f a c t o r re v e a le d t h a t , ex c ep t f o r sym bolism and w ording, th e two o p e ra tio n s o f a d d itio n and m u ltip lic a tio n a r e in d is tin g u is h a b le , f o r each has th e c lo s u re , commuta t i v e , and a s s o c ia tiv e p r o p e r tie s . 100 The f a c t o r lo a d in g s and o th e r s t a t i s t i c s r e la te d to F a c to r B re v e a le d lo a d in g s o f .63 to .93 f o r n in e o f th e v a r ia b le s . These lo a d in g s , as compared to th e lo a d in g s o f .07 to .35 on th e f i r s t s ix v a r ia b le s , in d ic a te a h ig h d eg ree o f r e la tio n s h ip betw een th e n in e p r in c ip le s lo a d in g on t h i s f a c t o r . The complex n a tu re o f F a c to r B does n o t p ro v id e a sim ple e x p la n a tio n as to th e n a tu re o f i t s e x is te n c e . How e v e r, th e h ig h lo a d in g s on language IQ, a r ith m e tic re a s o n in g , a r ith m e tic fu n d am en tals, and a lg e b ra a p titu d e in d ic a te a re a s o n in g f a c t o r . A com parison o f th e m ath em atical p r in c ip le s lo a d in g on t h i s f a c t o r re v e a le d t h a t th e d e f in itio n s o f d if fe re n c e and q u o tie n t r e f e r to a d ir e c t o p e ra tio n , w h ile th e axioms o f s u b tr a c tio n and d iv is io n r e p r e s e n t th e r e s u l t s when th e o p e ra tio n i s c a r r ie d o u t. The f a c to r lo a d in g s and o th e r s t a t i s t i c s r e la te d to F a c to r C re v e a l lo a d in g s o f .48 and .5 0 . These lo a d in g s , as compared to th e lo a d in g s o f .01 to .20 on th e o th e r v a r ia b le s , in d ic a te a s u b s t a n t ia l d eg ree o f r e la tio n s h ip betw een th e two p r in c ip le s lo a d in g on t h i s f a c t o r . A com p a ris o n o f th e v a r ia b le s lo a d in g on t h i s f a c to r re v e a le d t h a t the. axioms o f s u b tr a c tio n and d iv is io n r e p r e s e n t th e 101 r e s u l t when th e o p e ra tio n s o f d iv is io n and s u b tr a c tio n a re c a r r ie d o u t. T his could account f o r th e s e p a ra te lo a d in g s o f th e s e p r in c ip le s from th o se d e fin in g q u o tie n t and d i f fe re n c e . A b ility le v e ls as d e s c rib e d by A T e st o f M athemat- ic a ^ O g e r a tio n g .—The f a c to r lo a d in g s f o r th r e e s e p a ra te a b i l i t y le v e ls d e s c rib e d by A T e st o f M athem atical O pera tio n s compared to th e f a c t o r lo a d in g s f o r th e t o t a l group re v e a le d : 1. F a c to r A f o r th e t o t a l group s p l i t in to th r e e s e p a ra te f a c to r s f o r group I and r e ta in e d i t s i d e n t i t y f o r groups I I and I I I ; 2. F a c to r B f o r th e t o t a l group s p l i t in to s e p a ra te f a c to r s fo r groups I , I I , and I I I ; 3. F a c to r C f o r th e t o t a l group r e ta in e d i t s i d e n t i ty f o r groups I and I I , and s p l i t in to s e p a ra te f a c to r s f o r group I I I . A n aly sis o f f a c to r s f o r th re e s e p a r a te a b i l i t y groups re v e a le d th e complex n a tu re o f th e f a c to r s d e s c rib e d f o r th e t o t a l group. Of th e th r e e f a c to r s d e s c rib e d fo r th e t o t a l group, A and C ap p ear to be more s ta b le th a n B. 102 Sex d if f e r e n c e s ♦—The f a c t o r m a tric e s r e v e a l fo u r f a c to r s lo a d in g s i g n i f i c a n t l y f o r g i r l s and s ix fo r boys, in d ic a tin g a d if fe re n c e in f a c to r s tr u c t u r e f o r boys and g i r l s . These d a ta f u r th e r e s ta b li s h th e complex n a tu re o f th e f a c to r s d e s c rib e d f o r th e t o t a l group. CHAPTER VI SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS In t h i s f i n a l c h a p te r, a summary o f th e in v e s tig a tio n and th e fin d in g s has been p re p a re d in th e f i r s t two s e c t io n s . F ollow ing th e summary is a s e c tio n in w hich con c lu s io n s based upon th e fin d in g s o f th e stu d y have been fo rm u la te d . In th e fo u rth s e c tio n o f th e c h a p te r, s e v e ra l e d u c a tio n a l im p lic a tio n s o f th e stu d y a r e s t a t e d . Recom m endations f o r f u r th e r re s e a rc h a re p re s e n te d in th e l a s t s e c tio n . Summary S tatem en t o f th e p ro b lem . —The problem o f t h i s stu d y was to a s c e r ta in th e u n d e rsta n d in g by 300 se v e n th - g rad e c h ild re n o f c e r t a in m a th em atical p r in c ip le s g overning th e o p e ra tio n s o f a d d itio n , m u ltip lic a ti o n , s u b tr a c tio n , and d iv is io n , and th e r e la tio n s h i p o f t h i s u n d e rsta n d in g to 103 104 a r ith m e tic and m en tal a b i l i t y as m easured by sta n d a rd iz e d t e s t s . The m ath em atical p r in c ip le s s e le c te d f o r stu d y w ere based on th e ax io m atic m ethod in m athem atics and in v o lv ed th e (1) c lo s u re law f o r a d d itio n , (2) c lo s u re law f o r m ul t i p l i c a t i o n , (3) com m utative law f o r a d d itio n , (4) commuta t i v e law f o r m u ltip lic a tio n , (5) a s s o c ia tiv e law fo r m u lti p l i c a t i o n , (6) a s s o c ia tiv e law f o r a d d itio n , (7) d e f in it io n o f d if f e r e n c e , (8) axiom o f s u b tr a c tio n , (9) d e f in it io n o f q u o tie n t, and (10) axiom o f d iv is io n . S u b je c ts . —The sam ple was drawn from th e se v en th g rad es o f a ju n io r h ig h sch o o l in an elem en tary sch o o l d i s t r i c t in S outhern C a lif o r n ia . In clu d e d in th e stu d y w ere 300 p u p ils o r ap p ro x im ately 86 p e r c e n t o f th e t o t a l popu la t i o n o f th e se v e n th -g ra d e s tu d e n ts in th e ju n io r h ig h sc h o o l. Methods o f g a th e rin g d a t a . —The in stru m e n t u sed to m easure m en tal a b i l i t y was th e C a lif o r n ia Short-Form T e st o f M ental M a tu rity . J u n io r H ieh L e v e l. G rades 7 -8 -9 . 1957. S-Form . The a r ith m e tic s e c tio n o f th e C a lif o r n ia A chieve ment T e s ts . J u n io r High L e v e l. G rades 7 - 8 -9 . Form X was 105 u t i l i z e d to m easure th e e d u c a tio n a l developm ent o f th e sam ple in th e a re a o f a r ith m e tic . The in stru m e n t u sed to m easure a lg e b ra a p titu d e was th e Survey T e s t o f A lg e b ra ic A p titu d e . A T e st o f M athem atical O p eratio n s was developed s p e c i f i c a l l y f o r t h i s stu d y by th e in v e s tig a to r . A ll t e s t s w ere a d m in iste re d by th e p r in c ip a l and g u id an ce c o u n s e lo rs . C lassroom te a c h e rs d id n o t p a r t i c i p a t e in th e t e s t ad m in is t r a t i o n . S coring was done by e le c tr o n ic equipm ent. M ethods o f s t a t i s t i c a l tre a tm e n t o f th e d a t a . - -The r e la tio n s h i p between a b i l i t y w ith th e m ath em atic al p r i n c i p le s governing th e o p e ra tio n s o f a d d itio n , m u lt ip lic a tio n , s u b tr a c tio n , and d iv is io n and th e f a c to r s o f language IQ, n o n -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r ith m e tic fundamen t a l s , and a lg e b ra a p titu d e was an aly zed by m u ltip le c o r r e l a t i o n te c h n iq u e s . The r e la tio n s h ip o f th e f a c to r s o f m ath em atical a b i l i t y a s s e s s e d by th e t e s t o f m a th em atic al o p e ra tio n s to language IQ, no n -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r i t h m e tic fu n d am en tals, and a lg e b ra a p titu d e was e x p lo re d by f a c t o r a n a ly s is . 106 Summar-v of F in d in g s The fin d in g s p re s e n te d in t h i s s e c tio n o f C hapter VI r e l a t e to th e h y p o th eses s ta te d in C hapter I . Each o f th e h y p o th eses i s s ta te d and fo llo w ed by a d is c u s s io n o f th e fin d in g s r e la te d to i t . H ypothesis number o n e . — "A s i g n i f i c a n t r e la tio n s h ip e x i s ts betw een a b i l i t y d e s c rib e d by A T e st o f M athem atical O p e ra tio n s . a t e s t developed e s p e c ia lly f o r t h i s stu d y , and m en tal a b i l i t y as in d ic a te d by language and n o n -lan g u ag e IQ d eterm in ed from th e C a lif o r n ia T e st o f M ental M a tu rity ." The r e la tio n s h ip betw een language IQ and th e b e s t w eighted com bination o f th e m ath em atical p r in c ip le s govern in g th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , s u b tr a c ti o n , and d iv is io n was d eterm in ed by m u ltip le c o r r e la tio n te c h n iq u e s . The fin d in g s in t h i s r e s p e c t a r e summarized below . T hese d a ta r e v e a l: 1. The m u ltip le c o r r e la tio n d a ta ten d to su p p o r t th e f i r s t h y p o th e s is . 2. A m u ltip le c o r r e la tio n o f .67 in d ic a te d a s u b s t a n tia l r e la tio n s h i p betw een a b i l i t y w ith th e m ath em atical p r in c ip le s and la n guage IQ. The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .4524 in d ic a te s t h a t 4 5 .2 4 p e r c e n t o f th e v a ria n c e in language IQ i s acco u n ted f o r by w hatever i s m easured by th e te n m ath em atical p r in c ip le s tak en to g e th e r , e lim in a tin g from double c o n s id e ra tio n th in g s t h a t th e y have in common. An ex am in atio n o f th e c o n trib u tio n o f th e in d ep en d en t v a r ia b le s in d ic a te s t h a t th e p r in c ip le s r e l a t e d to s u b tr a c tio n accounted f o r more th a n h a l f o f th e t o t a l p re d ic te d v a ria n c e . The c o r r e la tio n c o e f f ic ie n ts f o r th e p r i n c ip le s o f a d d itio n and m u ltip lic a tio n i n d i c a te an alm o st n e g lig ib le r e la tio n s h i p to language IQ. The in t e r - c o r r e l a ti o n s f o r th e p r in c ip le s o f a d d itio n and m u ltip lic a tio n in d ic a te a m arked r e la tio n s h i p to each o th e r . The c o r r e la tio n c o e f f ic ie n ts in d ic a te a d e f i n i t e , b u t sm a ll, r e la tio n s h i p o f th e axiom o f d iv is io n to language IQ, w h ile 108 th e rem ain in g p r in c ip le s f o r d iv is io n and s u b tr a c tio n in d ic a te a s u b s ta n tia l r e l a t i o n s h ip to language IQ. 8 . The i n t e r - c o r r e l a t i o n s f o r th e p r in c ip le s o f s u b tr a c tio n and d iv is io n in d ic a te a d e f i n i t e to s u b s ta n tia l r e la tio n s h ip to each o th e r . 9. The c o r r e la tio n s f o r th e r e la tio n s h ip o f th e p r in c ip le s o f a d d itio n and m u ltip lic a tio n to th o se o f s u b tr a c tio n and d iv is io n re v e a l a n e g a tiv e to sm all r e la tio n s h ip f o r th e m a jo rity o f th e p r in c ip l e s . 10. The p r in c ip le s o f s u b tr a c tio n and d iv is io n account f o r th e m a jo rity o f th e r e l a t i o n sh ip found to e x i s t betw een th e te n p r i n c i p le s ta k en as a group and language IQ. The m u ltip le c o r r e la tio n c o e f f ic ie n ts and o th e r s t a t i s t i c s r e la t e d to th e c o r r e la tio n betw een n o n -lan g u ag e IQ and th e b e s t w eighted com bination o f th e m ath em atical p r in c ip le s a re summarized below . These d a ta re v e a l: 1. The m u ltip le c o r r e la tio n d a ta ten d to su p p o r t th e f i r s t h y p o th e s is . 2. A m u ltip le c o r r e la tio n o f .47 in d ic a te d a s u b s t a n tia l r e la tio n s h ip betw een a b i l i t y w ith th e m ath em atical p r in c ip le s and non- language IQ . 3 . The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .2236 in d ic a te d th a t 22.36 p er c e n t o f th e v a ria n c e in n o n -lan g u ag e IQ i s a c counted fo r by w hatever is m easured by th e te n m ath em atical p r in c ip le s ta k en t o g e th e r, e lim in a tin g from double c o n s id e r a tio n th in g s t h a t th e y have in common. 4 . The c o r r e la tio n c o e f f ic ie n ts f o r th e p r i n c ip le s o f a d d itio n and m u ltip lic a tio n in d ic a te an in v e rs e to n e g lig ib le r e l a tio n s h ip to n o n -lan g u ag e IQ. 5 . The c o r r e la tio n c o e f f ic ie n ts f o r th e su b t r a c t i o n and d iv is io n p r in c ip le s in d ic a te a d e f i n i t e , b u t sm a ll, r e la tio n s h i p to n o n -lan g u ag e IQ. 6 . The p r in c ip le s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e l a t i o n - sh ip found to e x i s t betw een th e te n ♦ 110 p r in c ip l e s , tak en as a group, and language IQ. H ypothesis number tw o. —"A s ig n i f i c a n t r e la tio n s h i p e x i s ts betw een a b i l i t y as d e s c rib e d by A T e st o f M athemat i c a l O p eratio n s and a r ith m e tic a b i l i t y as in d ic a te d by sc o re s on th e a rith m e tic re a so n in g and a r ith m e tic fundamen t a l s s e c tio n s o f th e C a lif o rn ia Achievem ent T e s t. J u n io r High L ev el. G rades 7 -8 -9 . Form X. M The c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e la te d to th e c o r r e la tio n betw een a r ith m e tic re a so n in g and th e b e s t w eighted com bination o f th e m athe m a tic a l p r in c ip le s a re summarized below . These d a ta r e v e a l: 1. The m u ltip le c o r r e la tio n d a ta te n d to su p p o r t th e second h y p o th e s is . 2. A m u ltip le c o r r e la tio n o f .77 in d ic a te d a m arked r e la tio n s h ip betw een knowledge o f th e m ath em atical o p e ra tio n s and a r ith m e tic re a s o n in g . 3 . The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .5988 in d ic a te d t h a t 59.88 p er c e n t o f th e v a ria n c e in a r ith m e tic re a so n in g is Ill accounted fo r by w hatever i s m easured by th e te n p r in c ip le s tak en to g e th e r , e lim in a tin g from double c o n s id e ra tio n th in g s th ey have in common. 4. The p e rc e n ta g e c o n trib u tio n o f th e in d e pendent v a r ia b le s in th e m u ltip le re g re s s io n eq u a tio n in d ic a te d th a t th e c lo s u re laws f o r a d d itio n and m u ltip lic a tio n , and th e a s s o c ia tiv e law fo r a d d itio n c o n trib u te d l i t t l e i f a n y th in g to th e r e la tio n s h i p b e tween a r ith m e tic re a so n in g and th e b e s t w eighted com bination o f th e m ath em atical o p e r a tio n s • 5- The c o r r e la tio n c o e f f ic ie n ts f o r th e p r in c ip le s o f a d d itio n and m u ltip lic a tio n in d ic a te a n e g lig ib le to sm a ll r e l a t i o n sh ip to a r ith m e tic re a s o n in g . 6 . The c o r r e la tio n c o e f f ic ie n ts f o r th e su b t r a c t i o n and d iv is io n p r in c ip le s in d ic a te a s u b s t a n tia l r e la tio n s h i p to a rith m e tic re a s o n in g . 112 7. The p r in c ip le s o f s u b tr a c tio n and d iv is io n acco u n t f o r th e m a jo rity o f th e r e l a t i o n sh ip found to e x i s t betw een a r ith m e tic re a s o n in g and th e jn a th e m a tic a l o p e r a tio n s . The c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een a r ith m e tic fundam entals and th e b e s t w eighted com bination o f th e m ath em atical p r in c ip le s a r e summ arized below . These d a ta r e v e a l: 1. The m u ltip le c o r r e la tio n d a ta te n d to su p p o r t th e second h y p o th e s is . 2. A m u ltip le c o r r e la tio n o f .74 in d ic a te d a marked r e la tio n s h ip betw een a b i l i t y w ith th e m ath em atical o p e ra tio n s and a r ith m e tic fu n d a m e n ta ls. 3 . The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f .5521 in d ic a te d t h a t 55.21 p e r c e n t o f th e v a ria n c e in a r ith m e tic fundam entals i s acco u n ted f o r by w hatever i s m easured by th e te n p r in c ip le s ta k en to g e th e r , e lim in a tin g from dou b le c o n s id e ra tio n th in g s th e y have in common. 113 4 . The p e rc e n ta g e c o n trib u tio n f o r th e p r i n c i p le s o f s u b tr a c tio n and th e a s s o c ia tiv e law f o r m u ltip lic a tio n accounted f o r more th an h a l f o f th e t o t a l p re d ic te d v a r ia n c e . 5 . The c o r r e la tio n c o e f f ic ie n ts f o r th e c lo s u re laws o f a d d itio n and m u lt ip lic a ti o n , and th e a s s o c ia tiv e law f o r a d d itio n in d ic a te an alm o st n e g lig ib le r e la tio n s h i p to a r ith m e tic fu n d a m e n ta ls. 6 . The c o r r e la tio n c o e f f ic ie n ts f o r th e d e f i n i tio n o f d if f e r e n c e , axiom o f s u b tr a c tio n , and d e f in it io n q u o tie n t r e v e a l a s u b s t a n tia l r e la tio n s h i p to a r ith m e tic fu n d a m e n ta ls. H ypothesis number t h r e e *— "A s i g n i f ic a n t r e l a t i o n s h ip e x i s ts betw een a b i l i t y , as d e s c rib e d by A T e st o f M athem atical O p e ra tio n s . and a lg e b ra a p titu d e , "as m easured by th e Survey T e s t o f A lg eb raic A p titu d e . 1 1 The c o e f f ic ie n ts o f m u ltip le c o r r e la tio n and o th e r s t a t i s t i c s r e l a t e d to th e c o r r e la tio n betw een a lg e b ra a p t i tu d e and th e b e s t w eighted com bination o f th e m ath em atical p r in c ip le s a re summarized below . These d a ta r e v e a l: 114 1. The m u ltip le c o r r e la tio n d a ta te n d to su p p o rt th e t h i r d h y p o th e s is . 2. A m u ltip le c o r r e la tio n o f .75 in d ic a te d a marked r e la tio n s h ip betw een a b i l i t y w ith th e m ath em atical o p e ra tio n s and a lg e b ra a p titu d e . 3 . The c o e f f ic ie n t o f m u ltip le d e te rm in a tio n o f •5683 in d ic a te d th a t 56.83 p e r c e n t o f th e v a ria n c e in a lg e b ra a p titu d e i s accounted fo r by w hatever i s m easured by th e te n p r i n c ip le s ta k en to g e th e r , e lim in a tin g from double c o n s id e ra tio n th in g s th a t th e y have in common. 4 . The p e rc e n ta g e c o n trib u tio n o f th e independ e n t v a r ia b le s in th e m u ltip le re g re s s io n eq u a tio n r e v e a l th a t th e p r in c ip le s o f a d d i tio n , and th e c lo s u re and com m utative laws fo r m u ltip lic a tio n c o n trib u te l i t t l e i f any th in g to th e r e la tio n s h i p betw een a lg e b ra a p titu d e and th e b e s t w eig h ted com bination o f th e te n p r i n c i p l e s . 5. The c o r r e la tio n c o e f f ic ie n ts f o r th e p r i n c ip le s o f s u b tr a c tio n and d iv is io n in d ic a te 1 1 5 a s u b s ta n tia l r e la tio n s h ip to a lg e b ra a p t i tu d e . 6 . The c o e f f ic ie n ts o f c o r r e la tio n f o r a d d itio n and m u ltip lic a tio n in d ic a te a low to s l i g h t r e la tio n s h i p to a lg e b ra a p titu d e . H ypothesis number f o u r . - - " A b il ity as d e s c rib e d by A T e st o f M athem atical O p eratio n s a s s e s s e s c e r t a in f a c to r s o f m ath em atical a b i l i t y d i f f e r e n t from th o s e m easured by th e o th e r t e s t s s e le c te d fo r u se in th e s tu d y ." The r o ta te d f a c to r m a trix and o th e r s t a t i s t i c s r e l a t e d to th e a n a ly s is o f f a c to r s c o n trib u tin g to a b i l i t y w ith th e m ath em atical p r in c ip le s and t h e i r r e la tio n s h i p to o th e r v a r ia b le s s e le c te d f o r stu d y a re summ arized below . These d a ta r e v e a l: 1. The f a c t o r a n a ly s is d a ta ten d to su p p o rt th e fo u rth h y p o th e s is . 2. Of th e e ig h t f a c to r s a f t e r r o t a t i o n , th r e e w ere g iv en i n t e r p r e t a t i o n and f iv e w ere re g a rd e d as r e s i d u a l s . 3 . F a c to r A re v e a le d lo a d in g s o f .78 to .9 8 . T hese lo a d in g s , as compared to th e lo a d in g s o f .01 to .02 on th e o th e r v a r ia b le s , i n d i c a te a h ig h d eg ree o f r e la tio n s h i p between th e s ix p r in c ip le s lo a d in g on t h i s f a c t o r . F a c to r A re v e a le d s i g n i f i c a n t lo a d in g s fo r th e c lo s u re laws o f a d d itio n and m u l t i p l i c a tio n , com m utative laws f o r a d d itio n and m u lt ip lic a ti o n , and a s s o c ia tiv e laws fo r a d d itio n and m u lt ip lic a tio n . A h ig h d eg ree o f com m unality i s in ev id en ce on v a r ia b le s one th ro u g h s ix , in d ic a tin g a r e l a t i v e l y h ig h c o n trib u tio n o f w hatever i s m easured by F a c to r A to th e common v a r ia n c e on each o f th e v a r i a b l e s . A com parison o f th e v a r ia b le s lo a d in g on F a c to r A re v e a le d t h a t , ex cep t f o r symbo lism and w ording, th e two o p e ra tio n s o f a d d itio n and m u ltip lic a tio n a re in d is tin g u is h a b le , fo r each has th e c lo s u re , com m utative, and a s s o c ia tiv e p r o p e r tie s . F a c to r B re v e a le d s ig n i f i c a n t lo a d in g s fo r th e a s s o c ia tiv e law o f m u ltip lic a tio n , d e f in it io n o f d if f e r e n c e , axiom o f s u b tr a c tio n , d e f in it io n o f q u o tie n t, axiom o f d iv is io n , language IQ, n o n -lan g u ag e IQ, a r ith m e tic re a s o n in g , a r ith m e tic fundamen t a l s , and a lg e b ra a p titu d e . 8 . F a c to r B re v e a le d lo a d in g s o f .63 to .93 f o r n in e o f th e v a r ia b le s . T hese lo a d in g s , as compared to th e lo a d in g s o f .07 to .35 on th e f i r s t s ix v a r ia b le s , in d ic a te a h ig h d eg ree o f r e la tio n s h i p betw een th e n in e p r in c ip le s lo a d in g on t h i s f a c t o r . 9. A com parison o f th e m ath em atical p r i n c i p le s lo a d in g on F a c to r B re v e a le d t h a t th e d e f in itio n s o f d if f e r e n c e and q u o tie n t r e f e r to a d ir e c t o p e ra tio n , w h ile th e axioms o f s u b tr a c tio n and d iv is io n r e p r e s e n t th e r e s u l t s when th e o p e ra tio n is c a r r ie d o u t. 10. F a c to r C re v e a le d s i g n i f i c a n t lo a d in g s o f .48 f o r th e axiom o f s u b tr a c tio n and .50 fo r th e axiom o f d iv is io n . These lo a d in g s , as compared to th e lo a d in g s o f .01 to .20 on th e o th e r v a r ia b le s , in d ic a te a 118 s u b s t a n tia l d eg ree o f r e la tio n s h ip betw een th e two p r in c ip le s lo a d in g on t h i s f a c t o r . 11. A com parison o f th e v a r ia b le s lo a d in g on F a c to r C re v e a le d th a t th e axioms o f sub t r a c t i o n and d iv is io n r e p r e s e n t th e r e s u l t when th e d i r e c t o p e ra tio n o f s u b tr a c tio n and d iv is io n i s c a r r ie d . Hvpothefli-q number f i v e . —"The f a c to r s o f m athem at i c a l a b i l i t y d e s c rib e d by A T e st o f Mathematical O p eratio n s when th e sam ple i s d iv id e d in to th r e e a b i l i t y le v e ls w il l v ary from th e f a c to r s d e s c rib e d f o r th e group as a w h o le." The r o ta te d f a c t o r m a tric e s and o th e r s t a t i s t i c s r e l a t e d to th e a n a ly s is o f th e m ath em atical a b i l i t y d e s c rib e d by th r e e a b i l i t y le v e ls on A T est o f M athem atical O p eratio n s a r e summarized below . These d a ta re v e a l: 1. The f a c to r a n a ly s is d a ta ten d to su p p o rt th e f i f t h h y p o th e s is . 2. Group 1 i s composed o f th e to p 100 s tu d e n ts , w ith a raw sc o re ran g e o f 35 to 49 o u t o f 50 ite m s, a mean o f 4 1 .0 5 , and a s ta n d a rd d e v ia tio n o f 4.33 on A T e st o f M athem atical 119 O p e ra tio n s . 3 . Group 11 i s composed o f th e m id d le 100 s t u d e n ts , w ith a raw sc o re ran g e o f 20 to 34 o u t o f 50 item s* a mean o f 27.04* and a sta n d a rd d e v ia tio n o f 4 .9 9 on A T e st o f Math e m a tic a l O p e ra tio n s. 4 . Group 111 i s composed o f th e low er 100 s t u dents* w ith a raw s c o re ran g e o f 2 to 18 o u t o f 50 item s* a mean o f 12.62* and a sta n d a rd d e v ia tio n o f 3 .9 8 on A T e st o f M athem atical O p e ra tio n s . 5 . F a c to r A, fo r th e t o t a l group o f 300* s p l i t in to th r e e s e p a ra te f a c to r s fo r group 1* and r e ta in e d i t s i d e n t i t y f o r group I I and I I I . 6 . F a c to r B, f o r th e t o t a l group o f 300* s p l i t in to s e p a r a te f a c to r s f o r groups I* II* and I I I . 7 . F a c to r C, f o r th e t o t a l group, r e ta in e d i t s i d e n t i t y f o r groups I and II* and s p l i t in to s e p a ra te f a c to r s f o r group I I I . 8 . A n a ly sis o f f a c to r s fo r th r e e s e p a ra te 120 a b i l i t y groups re v e a le d th e complex n a tu re o f th e f a c to r s d e s c rib e d f o r th e t o t a l g ro u p . 9. Of th e th r e e f a c to r s d e s c rib e d f o r th e t o t a l group, A and C appear to be more s ta b le th a n B. H ypothesis number s i x .--"T h e f a c to r s o f m athem at i c a l a b i l i t y d e s c rib e d by A T e s t o f M athem atical O p eratio n s w il l v a ry f o r boys and g i r l s . " The r o ta te d f a c t o r m a tric e s and o th e r s t a t i s t i c s r e l a t e d to an a n a ly s is o f th e r e la tio n s h i p o f th e f a c to r lo a d in g s f o r sex d if fe re n c e s d e s c rib e d by A T e st o f M athe m a tic a l O p e ra tio n s , language and n on-language IQ, a r i t h m e tic re a so n in g and fu n d am en tals, and a lg e b ra a p titu d e a re summarized below . These d a ta re v e a l: 1. F a c to r a n a ly s is d a ta te n d to su p p o rt th e s ix t h h y p o th e s is . 2. Four f a c to r s loaded s i g n i f i c a n t l y fo r g i r l s and s ix f o r boys, in d ic a tin g a d i f fe re n c e in f a c to r s tr u c t u r e f o r boys and g i r l s . C onclusions The fin d in g s o f th e p re s e n t stu d y in d ic a te t h a t c e r t a in f a c to r s a re s i g n i f i c a n t ly a s s o c ia te d w ith th e se v e n th -g ra d e c h i l d 's a b i l i t y w ith th e m ath em atical o p e ra t i o n s . In view o f th e s e fin d in g s i t was p o s s ib le to form u l a t e s e v e ra l c o n c lu s io n s . However, u n t i l more ev id en ce i s o b ta in e d th e c o n c lu sio n s in t h i s stu d y m ust be tr e a te d w ith c a u tio n . T his i s a p i l o t stu d y in v o lv in g few c a s e s , u sin g a n o n -sta n d a rd iz e d t e s t , in a s p e c if ic sc h o o l, based on one p e rs o n 's concept o f number, and u sin g o n ly one c r i te r io n t e s t o f a r ith m e tic . The te n m ath em atic al p r in c ip le s d iv id e d them selves in to th r e e f a c to r s c o n trib u tin g to a b i l i t y w ith th e m athe m a tic a l o p e ra tio n s . The c o n c lu sio n s t h a t fo llo w a re based on f a c to r lo a d in g s f o r th e t o t a l sam ple o f 300 se v e n th - g rad e c h ild re n . The r e la tio n s h i p betw een th e te n p r i n c i p le s and each o f th e v a r ia b le s o f IQ, a r ith m e tic a c h ie v e m ent, and a lg e b ra a p titu d e d e s c rib e d by m u ltip le c o r r e la tio n te c h n iq u e s and th e f a c to r a n a ly se s f o r th r e e a b i l i t y le v e ls w ere used to d e s c rib e th e f a c to r s lo a d in g s i g n i f i c a n tly f o r th e t o t a l group. 122 F a c to r A. —This f a c to r i s composed o f th e s ix p r in c ip le s governing th e o p e ra tio n s o f a d d itio n and m u ltip lic a tio n . 1. Except fo r symbolism and w ording, th e two o p e ra tio n s o f a d d itio n and m u ltip lic a tio n a re in d is tin g u is h a b le , fo r each has th e c lo s u re , com m utative, and a s s o c ia tiv e p ro p e r t i e s . 2. The n a tu re o f th e lo a d in g s on t h i s f a c to r , th e r e la tio n s h ip o f th e p r in c ip le s to each o th e r and to o th e r v a r ia b le s s e le c te d fo r stu d y in d ic a te a re a so n in g f a c to r re q u ir in g th e a b i l i t y to re c o g n iz e th e r e la tio n s h ip o f two o r more elem ents combined by a m athe m a tic a l s i t u a t i o n . 3. The a b i l i t y to re c o g n iz e th e r e la tio n s h ip o f two o r more elem ents combined in a m athe m a tic a l s it u a tio n ap p ears to be w e ll d e v e l oped by th e m a jo rity o f th e sam ple and independent o f IQ, a rith m e tic achievem ent, and a lg e b ra a p titu d e as m easured by s ta n d a rd iz e d t e s t s . 123 4 . The u se o f symbols o th e r th an num erals does n o t a f f e c t th e c h i ld r e n 's a b i l i t y to re c o g n iz e th e r e la tio n s h i p o f two o r more e l e m ents combined in a m ath em atical s i t u a t i o n . F a c to r B. —T his f a c to r i s composed o f th e fo u r p r in c ip le s g overning th e o p e ra tio n s o f s u b tr a c tio n and d iv is io n , language and n o n -lan g u ag e IQ, and a lg e b ra a p t i tu d e . 1. The n a tu re o f th e lo a d in g s on t h i s f a c to r by th e m ath em atical p r in c ip l e s , th e r e l a tio n s h ip o f th e p r in c ip le s to each o th e r and to o th e r v a r ia b le s s e le c te d f o r stu d y in d ic a te a re a s o n in g f a c t o r r e q u ir in g th e a b i l i t y to re c o g n iz e th e r e la tio n s h i p o f two elem ents to a t h i r d elem ent when th e t h i r d elem ent i s d e s c rib e d as a com bina tio n o f th e o th e r two, e . g . , a + x = b, b - x = a , b - a = x. 2. The r e la tio n s h i p o f th e p r in c ip le s lo a d in g on t h i s f a c to r to IQ, a r ith m e tic achievem ent, and a lg e b ra a p titu d e in d ic a te 124 a complex re a s o n in g f a c t o r . 3 . The r e la tio n s h ip o f th e m ath em atical p r in c ip le s lo a d in g on t h i s f a c to r to th o s e o f a d d itio n and m u ltip lic a tio n in d ic a te t h a t th e o p e ra tio n s o f s u b tr a c tio n and d iv is io n a re more d i f f i c u l t and complex th an th o se o f a d d itio n and m u ltip lic a tio n when a p p lie d to m ath em atical s it u a ti o n s in v o lv in g th e u se o f sym bols o th e r th an n u m erals. F a c to r C. —T his f a c t o r is composed o f th e axioms o f s u b tr a c tio n and d iv is io n . 1. The n a tu re o f th e lo a d in g s on t h i s f a c to r by th e m ath em atical p r in c ip l e s , th e r e l a tio n s h ip o f th e p r in c ip le s to each o th e r and o th e r v a r ia b le s in d ic a te a re a so n in g f a c to r r e q u ir in g th e a b i l i t y to re c o g n iz e th e com bination o f two elem ents as d e s c rib e d by a t h i r d elem ent in th e m ath em atical s i t u a t i o n , e . g . , a + x = b . 2. The r e la tio n s h i p o f th e p r in c ip le s lo a d in g on t h i s f a c to r to th e o th e r p r in c ip le s 125 I n d ic a te s t h a t th e axioms f o r s u b tr a c tio n and d iv is io n r e q u ir e u n d e rsta n d in g s d i f f e r e n t from th o se o f th e d e f in it io n o f d i f f e r ence and q u o tie n t when a p p lie d to th e same m ath em atical s i t u a t i o n s . Sex d if f e r e n c e s . —F a c to r lo a d in g s fo r boys and g i r l s show s im ila r p a tte r n s o f a b i l i t y in th e o p e r a tio n s . However, th e lo a d in g s a re more complex f o r boys th an g i r l s . 1. The extrem e d if fe re n c e s betw een th e s ta n d a rd d e v ia tio n s fo r boys and g i r l s on th e f i r s t s ix v a r ia b le s co u ld acco u n t fo r th e d if f e r e n c e in f a c t o r lo a d in g s . 2. As a group, th e g i r l s ap p ear to show g r e a te r extrem es in a b i l i t y th a n th e b o y s. E d u ca tio n a l Im n lic a tio n s The fin d in g s o f th e p re s e n t stu d y in d ic a te th a t c e r t a in f a c to r s a r e s ig n i f i c a n t l y a s s o c ia te d w ith th e se v e n th -g ra d e c h i l d 's a b i l i t y w ith th e m a th em atic al o p e ra tio n s d e s c rib e d by t e s t item s in v o lv in g th e .a n a ly s is o f m athe m a tic a l s i t u a t i o n s . In view o f th e s e fin d in g s and o th e r c o n s id e ra tio n s , c e r ta in e d u c a tio n a l im p lic a tio n s seem to The se v e n th -g ra d e c h ild i s ca p ab le o f m athe- m a tic a l re a s o n in g on th e more advanced le v e l d e s c rib e d by th e ax io m atic m ethod o f m athe m a tic s . On t h i s b a s i s , th e i n s t r u c tio n a l program sh o u ld in c lu d e m a te r ia l d esig n ed to develop t h i s a b i l i t y . I n s tr u c tio n program s f o r m ath em atic al re a s o n ing sh o u ld be p ro v id ed fo r a l l le v e ls o f a b i l i t y , w ith ample o p p o r tu n itie s f o r th e more cap ab le s tu d e n ts to advance a t a r a t e determ ined by t h e i r in d iv id u a l a b i l i t i e s . The r e s u l t s o f t h i s stu d y in d ic a te th a t th e a b i l i t i e s in v o lv ed in th e u se o f th e p r in c ip le s o f a d d itio n and m u ltip lic a tio n a re such th a t th e p re s e n t program o f in s t r u c tio n p ro v id e s o p p o r tu n itie s fo r d ev elo p in g th e s k i l l s n e c e ss a ry to ap p ly th e s e p r in c ip le s . The a b i l i t y to ap p ly th e p r in c ip le s govern ing th e o p e ra tio n s o f s u b tr a c tio n and d i v i sio n appear to show a c lo s e r e la tio n s h ip betw een f a c to r s o f IQ, a r ith m e tic achievem ent, and a lg e b ra a p titu d e . To c a re f o r th e w ide ran g es o f d if fe re n c e s in a b i l i t y , th e r e sh o u ld be a d i f f e r e n t i a t e d program o f i n s tr u c tio n in th e p r in c ip le s governing th e s e o p e r a tio n s . R easoning in term s o f a p p lie d m athem atics i s one method t h a t ap p ears to be s u c c e s s f u l f o r p ro v id in g th e te c h n iq u e s n e c e ss a ry fo r d e s c rib in g th e m ath em atical p r in c ip le s g overning th e o p e r a tio n s . The a p p lie d m ath em atical s i t u a t i o n does n o t need to be h ig h ly co m p licated to d e t e r m ine m ath em atical re a s o n in g a b i l i t y . M athem atical p r in c ip le s g o v ern in g th e o p e r a tio n s can d e riv e t h e i r m eaning from con c r e te e x p e rie n c e s . T h e re fo re , i t would seem t h a t any i n s t r u c t i o n a l program sh o u ld c a p i t a l i z e on c o n c re te e x p e rie n c e s to develop th e p r in c ip le s in v o lv ed in th e o p e r a tio n s . 128 Recommendations f o r F u rth e r R esearch The need f o r f u r th e r re s e a rc h p e r ta in in g to th e elem en tary c h i l d 's a b i l i t y to re a so n in term s more s u it a b le to advanced le v e ls o f m athem atics seems to be in d ic a te d by th e p re s e n t stu d y . P re se n te d below a re s e v e ra l a re a s r e c ommended fo r in v e s tig a tio n : 1. The p re s e n t stu d y sh o u ld be expanded to in c lu d e (1) sam ples ta k en from a much b ro ad er a r e a , (2) a l l g rad e le v e ls , and (3) a l l p r in c ip le s b a s ic to th e o p e ra tio n s o f a d d itio n , m u lt ip lic a ti o n , su b t r a c t i o n , and d iv is io n . 2 . R esearch i s needed to d e s c rib e th e r e l a tio n s h ip o f s p e c if ic f a c to r s o f i n t e l l i gence to th e f a c to r s d e s c rib e d by A T e st o f M athem atical O p e ra tio n s . 3 . R esearch i s needed~to d e s c rib e th e r e l a tio n s h ip o f s p e c if ic f a c to r s o f a r ith m e tic to th e f a c to r s d e s c rib e d by A T e st o f Mathemat i c a l O p e ra tio n s . 4 . The te c h n iq u e s o f e v a lu a tio n used in th e p re s e n t stu d y need to be im proved upon and expanded to in c lu d e a l l m a th em atical p r i n c ip le s a p p lic a b le to a p p lie d m ath em atical s i t u a t i o n s . SELECTED BIBLIOGRAPHY SELECTED BIBLIOGRAPHY G ibb, E. G. "Some A pproaches to M athem atics Con c e p ts ," J o u rn a l o f The N a tio n a l E d u catio n A sso c i a t i o n . XLVIII (November, 1959), 65-66. K orzybski, A lfre d . S cien ce and S a n ity . 3 rd ed. The I n te r n a tio n a l N o n -A ris to ta lia n L ib ra ry P u b lis h in g Company. D is tr ib u te d by th e I n s t i t u t e o f G en eral S em an tics, L a k e v ille , C o n n e c tic u tt, 1948. K a tts o ff , L ouis 0 . A P h ilo so p h y o f M a th em atics. Ames, Iowa: The Iowa S ta te C o lleg e P re s s , 1948. Van Engen, H enry. "The Form ation o f C o n c e p ts," The L earn in g o f M ath em atics. T w e n ty -F irst Yearbook The N a tio n a l C ouncil o f T eachers o f M athem atics W ashington, D .C ., 1953. V inacke, W. E dgar. "The I n v e s tig a tio n o f Concept F o rm a tio n ," P sy c h o lo g ic a l B u lle tin , XLVIII (Jan u a ry , 1951), 1 -3 1 . B ru n er, Jerom e S ., J a c q u e lin e J . Goodnow, G eorge A. A u s tin . A Study o f T h in k in g . New York: John W iley and Sons, I n c ., 1956. Brown, K enneth E. and John J . K in s e lla . "A n aly sis o f R esearch in th e T eaching o f M athem atics, 1957 and 1 9 5 8 ," U n ited S ta te s D epartm ent o f H e a lth , E ducation and W e lfa re , O ffic e o f E d u catio n , B u lle tin 1960. No. 8 . W ashington, D .C .: U. S. Government P rin tin g O ffic e , 1960. C la rk , John R. "G uiding th e L ea rn er to D isco v er and G e n e ra liz e ." I n s tr u c tio n in A rith m e tic . The T w en ty -F ifth Y earbook, The N a tio n a l C ouncil o f T eachers o f M athem atics, W ashington, D. C ., 1960. 132 9. S u lliv a n , E liz a b e th T ., W illis W. C lark , and E rn e st W. T ie g s. M anual. C a lif o rn ia Short-Form T e st o f M ental M a tu rity . Los A ngeles: C a lif o rn ia T e st B ureau, 1957. 10. T ie g s, E rn e st W . and W illis W. C la rk . M anual. C a li f o r n ia Achievement T e s t. Complete B a tte r y . Los A ngeles: C a lif o rn ia T e st B ureau, 1957. 11. D inkel, R obert E. M anual. Survey T est o f A lg eb raic A p titu d e . Los A ngeles: C a lif o r n ia T est B ureau, 1959. 12. Bloom, Benjam in S. Taxonomy o f E d u ca tio n al O bjec t i v e s . The C la s s if ic a tio n o f E d u ca tio n al G o a ls. Handbook I : C o g n itiv e Domain. New York: Long mans, Green and C o., I n c ., 1956. 13. B rueckner, Leo J . , Edna L. M erton, and F o s te r E. G ro s sn ic k le . The New L earning o f Numbers. Sacram ento: C a lif o rn ia S ta te D epartm ent o f Edu c a tio n , 1957. 14. Cam pbell, D. T. and B. B. T y le r. "The C o n stru c t V a lid ity o f Work Group M orale M easu res," J o u rn a l o f A pplied Psychology, XLI (A p ril, 1957), 91-92. 15. H o lz in g er, K. J . , and H. H. Harman. F a c to r A n a ly s is : A S y n th esis o f F a c to r ia l M ethods♦ Chicago: U n iv e rs ity o f Chicago P re s s , 1941. 16. F ru c h te r, Benjam in. In tro d u c tio n to F a c to r A n a ly s is . New York: D. Van N ostrand C o., I n c ., 1954. 17. E ysenick, Hans J . "The L o g ical B asis o f F a c to r A nal y s i s , " American P s y c h o lo g is t. V III (March, 1953), 105-114. 18. K a ise r, H. F. "Computer Program fo r Varimax R o ta tio n in F a c to r A n a ly s is ," E ducation and Psycholog i c a l Measuranpnt. V ol. 19, No. 3, 1959. 19. 2 0. 2 1. 22 . 23. 24. 25. 26. 27. 28. 133 G u ilfo rd , J . P. Fundam ental S t a t i s t i c s in Psychology and E d u c a tio n . New York: M cGraw-Hill Book Com pany, I n c ., 1956. C a rro l, Hohn B ., and R obert F. Schwecker. "F a c to r A n aly sis in E d u ca tio n al R e se a rc h ," Review o f E d u ca tio n al R e se a rc h . XXI (December, 1951), 368-388. M aria, May H ickey. The S tru c tu re o f A rith m e tic and A le e b ra . New York: John W iley and Sons, I n c ., 1958. McGrath, W illiam H aro ld . "Problem S olving E ffic ie n c y as A ffec ted by A ccessory R em arks." U npublished D o c to ral d i s s e r t a t i o n , U n iv e rs ity o f S outhern C a lif o r n ia , Los A ngeles, 1956. Cronbach, L ., and R. E. M eehl. "C o n stru ct V a lid ity in P sy ch o lo g ical T e s ts ," P sy ch o lo g ic al B u l le t i n . 1955, 52:281-302. R oss, C. C ., and J u lia n C. S ta n le y . M easurement in Today’s S ch o o l. New York: P re n tic e -H a ll, I n c ., 1954. F reg e, G. F oundations o f A rith m e tic . O xford: B a s il B lack w ell, 1953. E in s te in , A lb e rt. "Remarks on B e rtra n d R u s s e ll's Theory o f Knowledge," The P hilosophy o f B e rtra n d R u s s e ll. V ol. V o f The L ib ra ry o f L iv in g P h ilo s ophers . E d ite d by P aul A rth u r S c h ilp p . Evan s to n , I l l i n o i s : N o rth w estern U n iv e rs ity P re s s , 1944. H endrix, G e rtru d e . " P re re q u is ite to M eaning," The M athem atics T ea c h er. X LIII (November, 1950), 334-39. W a tte rs, Loras J . F a c to rs in Achievem ent in M athemat ic s . W ashington, D. C .: The C a th o lic U n iv e rs ity o f America P re s s , 1954. 134 29. 30. 31. 32. 33. D oyle, Andrew M. Some A spects o f A b ility and A chieve ment in High School G i r l s . W ashington, D.C.: The C a th o lic U n iv e rs ity o f America P re s s , 1952. D r is c o ll, J u s t in A. F a c to rs in I n te llig e n c e and A chievem ent. W ashington, D .C .: The C a th o lic U n iv e rs ity o f Am erica P re s s , 1952. R o e s s le in , C h arles G. D if f e r e n tia l P a tte rn s o f I n t e l - lic e n c e T r a its Between High A chieving and Low A chieving High School B ovs. W ashington, D .C.: The C a th o lic U n iv e rs ity o f Am erica P re s s , 1953. R u szel, Humphrey. T e st P a tte rn s in I n t e l l i g e n c e . W ashington, D .C.: The C a th o lic U n iv e rs ity o f America P re s s , 1952. Donohue, James C. F a c to r ia l Comparison o f A rith m e tic P roblem -Solving A b ility o f Bovs and G ir ls in Seventh G rad e. W ashington, D .C .: The C a th o lic U n iv e rs ity o f Am erica P re s s , 1957. 135 A P P E N D I X SOLVING EASY ARITHMETIC PUZZLES C A N Y O U SOLVE THE PUZZLE? READ THE FOLLOW ING DIRECTIONS: W e a r e going to work a new and i n t e r e s ti n g k in d o f a r ith m e tic p u z z l e . The p u z z le s a r e easy b ecau se th e answ ers a r e g iv e n . One o f th e answ ers ( 1 ,2 ,3 ,4 , o r 5) on th e r i g h t , i s to be m atched w ith th e s to r y on th e l e f t . B lacken th e sp ace o f th e c o r r e c t answ er co rresp o n d in g to th e number on your answ er s h e e t. HERE IS A SAM PLE PUZZLE USING N U M BERS 5 1 . Ja n e earn ed 15 c e n ts . She p u t 10 2. 15 + 10 = 5 c e n ts o f t h i s money in th e bank. 3 . 10 - 5 - 15 She had 5 c e n ts l e f t . 4 . 15 - 10 « 5 5. none L et us look a t each o f th e p o s s ib le answ ers to th e p u z z le . Number 1 t e l l s us t h a t i f we d iv id e 15 (th e money Ja n e earn ed ) by 10 ( th e money she p u t in th e bank) we w i l l g e t 5 ( th e money sh e had l e f t ) . Remember t h a t 15 o v er 10 t e l l s us to d iv id e 15 by 10. Number 2 t e l l s us th a t i f we add 15 (th e money Ja n e earn ed ) to 10 ( th e money she p u t in th e bank) we w i l l g e t 5 ( th e money sh e had l e f t ) . Number 3 t e l l s us th a t i f we s u b tr a c t 5 (th e money Ja n e had l e f t ) from 10 ( th e money sh e p u t in th e bank) we w i l l g e t 15 ( th e money she e a rn e d ). 137 138 Number 4 t e l l s us th a t I f we s u b tr a c t 10 (th e money Jan e p u t in th e bank) from 15 (th e money she earned) we g e t 5 (th e money she had l e f t ) . Number 5 t e l l s us t h a t none o f th e above answ ers a re c o r r e c t. Which o f th e com binations on th e r ig h t g iv e s us th e same in fo rm atio n found in th e s to ry ? I f none o f th e com binations g iv e us th e same in fo rm a tio n th en s e le c t number 5 la b e le d "n o n e." Blacken th e space o f th e c o r r e c t answer to sam ple number 1 on your answer s h e e t. HERE IS THE SA M E PUZZLE USING LETTERS OF THE ALPHABET In a l l th e p u zz les you a re about to do, l e t t e r s o f th e a lp h a b e t have been s u b s titu te d f o r num bers. a 1 . b - z 2. Ja n e earn ed a c e n ts . She p u t b c e n ts 2. a + b “ z o f t h i s money in th e b an k . She had 3 . b - z ■ a z c e n ts l e f t . 4 . 5. a - b = none z L et us look a t each o f th e p o s s ib le answ ers to th e p u z z le . W e m ust look a t each answer v ery c a r e f u lly to avoid c a re le s s m is ta k e s . Number 1 t e l l s us th a t i f we d iv id e a (th e money Ja n e earned) by b ( th e money she p u t in th e bank) we w i ll g e t z (th e money she had l e f t ) . Remember th a t S L o v er b t e l l s us to d iv id e a by b . Number 2 t e l l s us th a t i f we add & (th e money Ja n e earned) to b (th e money she p u t in th e bank) we w ill g e t jj, (th e money Jan e had l e f t ) . Number 3 t e l l s us th a t i f we s u b tr a c t & (th e money Jan e had l e f t ) from b (th e money she p u t in th e 139 bank) we w i l l g e t a ( th e money she e a rn e d ). Number 4 t e l l s us t h a t I f we s u b tr a c t b ( th e money Ja n e p u t in th e bank) from & ( th e money she earn ed ) we g e t £ ( th e money she had l e f t ) . Number 5 t e l l s us th a t none o f th e above answ ers a re c o r r e c t . Which o f th e com binations on th e r i g h t g iv e us th e same in fo rm a tio n found in th e s to ry ? I f none o f th e com binations g iv e us th e same in fo rm a tio n th en s e l e c t number 5 la b e le d "n o n e." B lacken th e sp ace o f th e c o r r e c t answer to sam ple num b e r 2 on your answer s h e e t. HERE IS A SAMPLE PUZZLE , d 1. e » w 3 . D ic k 's m other gave him c e n ts . He sp e n t £ c e n ts f o r gum. D ick had w 2. e - d = w c e n ts l e f t . 3 . d + e 4 . w + e 5 . none * w “ d Which o f th e com binations on th e r i g h t g iv e us th e same in fo rm a tio n found in th e s to ry ? I f none o f th e com binations g iv e us th e same in fo rm a tio n th en s e l e c t number 5 la b e le d "n o n e." B lacken th e sp ace o f th e c o r r e c t answer to sam ple num b e r 3 on your answer s h e e t. THINGS TO REM EM BER You may have a l l th e tim e you n eed , so do n o t h u rry . Read each s to r y and s e t o f answ ers v e ry c a r e f u l ly . The p u z z le s a re ea sy , b u t i t is s t i l l e a s ie r to make c a r e le s s m is ta k e s . Do n o t mark on th e book o f p u zz les p ap er i f you need i t . Answer a l l th e p u z z le s . D O Y O U R BEST W O R K 140 You have s c ra tc h 1. Tom had a bag o f d. m a rb le s . He bought £ more m a rb le s . Tom had d + e m arb les in a l l . 1 . d - ♦ 3 . 4 . 5 . e d + e d + e — = d e none d d d e + + 2. L arry saved d d o lla r s to spend f o r v a c a tio n fu n . F i r s t he bought a b a s k e t b a l l fo r £ d o l l a r s . L a rry had w d o lla r s l e f t . 1. e - w 2. w x e 3 . e x d 4 . d - e 5 . none d d w w 1. a - b = a x b 3 . T here a re a boys and b g i r l s in J a c k 's room. 2. a x b = a + b 3 . a + b = * c T here a r e a + b c h ild re n 4 . a + b « a - b in J a c k 's room. 5 . none D ic k 's allow ance i s a c e n ts a week. J im 's i s b c e n ts . J im 's allo w an ce i s z c e n ts le s s th an D ic k 's . 1. b x z = a 2. a + b = z 3 . a - b = z 4 . b - z * = a 5. none 5 . P aul had £ a i r p la n e s . He made d more a i r p l a n e s . He ha<i e + d a ir p la n e s in a l l . 1. 2 . 3. 4 . 5. e + d = * d = e + e - d = e + d « none e x d d e + d e + d 141 1. a + z « b 6 . At th e Lake S chool, th e re 2. b x a = z a re b c h ild re n . T here a re 3 . z x b « a a boys and & g i r l s • 4 . z + b “ a 5 . none 7 . In a d a r t game, N ick made 1. f + g + h ■ f X g X h & p o in ts , g, p o in ts and h 2. f + g - h p o i n t s . N ick made f + e + h 3 . f + g + h - f + g + h p o in ts in a l l . 4 . f - h 5 . none - f + g + h 8 . Jim had £ m arb le s h o o te rs 1 . a » a + b and b o th e r m a rb le s. Jim 2. a - b B a + b had a + b m arb les o f b o th 3 . a + b m a + b k in d s . 4 . a + b m a X b 5 . none 9 . At summer camp, £ boys can l . f - C X f s le e p in each t e n t . T here 2. c - f B c X f a r e f t e n t s . c x f bovs 3 . c x f SB C c X f can s le e p in te n ts a t th e 4 . c + f m c X f camp. 5 . none 10. At th e sch o o l c a f e t e r i a , 1 . c x d X e ss c + d + e B illy bought a sandw ich fo r 2. d + e + e s e + c + d c. c e n ts , he a ls o bought a 3 . c - d - e s c + d + e bowl o f soup f o r £ c e n ts 4 . c + d + e 8 d and m ilk f o r £ c e n ts . B illy 5. none SDent c + d + e c e n ts a t th e c a f e t e r i a . 11. George had a w h ite m ice. 1. a x b b b - a He s o ld a l l o f them a t b 2. S “ a - b c e n ts each. George r e 3. a + b = a x b c e iv ed a x b c e n ts fo r 4. a x b ’ b x a th e m ice. 5. none 142 12. Bob ch arg es < £ c e n ts f o r ra k in g a law n. He rak ed & law n s. He earn ed d x e c e n ts f o r ra k in g th e law n s. 13. Ann re a d one s to r y £ pages lo n g . She re a d a n o th e r s to r y £ pages long and a t h i r d s to r y j; pages lo n g . Ann re a d d + e + f pages in a l l . 14. M other bought a melon f o r b c e n ts , co o k ies fo r e c e n ts , and eggs f o r f. c e n ts . M other sp e n t b + e + f c e n ts in a l l . 15. In one s e c tio n o f th e rodeo s ta n d Jack counted d, rows w ith e s e a ts in each row. T here w ere J E s e c tio n s in th e rodeo s ta n d . A ll s e c tio n s w ere th e same s i z e . T here w ere d x e x f s e a ts in th e rodeo s ta n d . 1. d - e ® d x e 2. d x e d x e 3 . d x e * d + e 4 . — = > d x e e 5 . none 1. d x e x f * d + e + f 2. d » d + e + f 3. f + e + d = e + f + d 4. f - d « * d + e + f 5 . none 1. (b + e) + f “ b + (e + f ) 2. b x e x f « b + e + f 3. (b + e) - f = b + e + f 4- b - e + f “ b +e +f 5 . none 1. d + e + f = d x e x f 2. d x e x f - e x d x f 3. d x e - f 4. f + d- e = d x e x f 5 . none 16. Howard bought a t a b l e t fo r 1. a + b + c ^ b + c + a S i c e n ts , a r u l e r f o r b 2 . a + b + c « a x b x c c e n ts , and a draw ing pen- 3 . a - c *=a + b + c c i l f o r £ c e n ts . Howard 4 . a*=a + b + c sp e n t a 4- b + c c e n ts in 5 . none a l l . 17. A DC-6 flew a t an av erag e speed o f £ m ile s an h o u r. I t tr a v e le d f o r & hours a t t h i s r a t e . The DC-6 flew a x b m il e s . 1. b x a ” a x b 2. ll « a x b 3 . £ + a = * a x b 4 . a - b * * a x b 5 . none 143 18. On a t e s t , S a lly had £ examples r i g h t in a d d i t i o n . She had & exam p le s r i g h t in s u b tr a c ti o n , h r i g h t in m u lti p l i c a t i o n , and i, r i g h t in d iv is io n . S a lly had f+g+h+i r i g h t in a l l . 1. f+ g+ i * f+g+h+i 2. fx g x h x i = f+g+h+i 3 . f+g+h+i * h + i 4 . f+g+i+h = > h+i+f+g 5 . none 19. Nan i s £ y e a rs o ld . Her 1. c + f = z m other i s £ y ea rs o ld . 2. f - c * z Nan i s £ y e a rs younger 3 . c - f = z th a n h e r m o th er. 4 . 5. 3 - C none - f 20. Sam made £ p o in ts in h is 1. e x g « e + f + g f i r s t game o f b a s k e tb a ll. 2. f + e + g « g + f + e He made £ p o in ts in th e 3 . f - e - g ® e + f + g second game, and jg p o in ts 4 . e x f x g « e + f + g in th e t h i r d game. Sam 5 . none made e + f + g p o in ts in a l l th r e e gam es. 21. C a rl c u t d badges from a p ie c e o f rib b o n . Each badge was £ in ch es lo n g . C a rl used d x e in ch es o f rib b o n f o r th e badges d x d - d SB, 1 . 2 . 3 . 4 . 3 . none % + “ e x d = » d x e x e = e x d 22. Mac p ick ed b m elons from 1. b + c = b x c h is g ard en . He s o ld th e 2. ^ = b x c m elons f o r £ c e n ts 3 . b x c = b - c a p ie c e . He re c e iv e d 4 .bx c= bx c b x c c e n ts f o r th e 5 . none m elo n s. 144 23. £ boys sh a re d e q u a lly th e £ d o lla r s th e y ea rn e d . Each boy re c e iv e d d o l l a r s . 1. 2 . 3 . 4 . a + c - z a - c * * z £ b z a I - c 5 . none 24. The f ly in g d is ta n c e from 1. Chicago to Denver i s £ 2. m il e s . The Jo n es fa m ily 3 . made th e t r i p in f h o u rs . 4 . The p la n e averaged z. 5. m ile s p e r h o u r. f = z f x e f = 2 e + f none z z 25. Roger s o ld h is p e t w h ite 1. a x b = a x b m ice fo r £ c e n ts a p ie c e . 2. a + b = a x b He s o ld b w h ite m ice. 3 . b - a = a He re c e iv e d a x b c e n ts 4 . a x b « * b fo r th e m ic e . 5. none 26. J a n e t worked £ a rith m e 1. a - b - z t i c exam ples. b o f them 2. a + b * = z w ere a d d itio n . The r e s t 3 • a x b = z w ere s u b tr a c tio n . J a n e t L a B S tf. _ = z worked z. s u b tr a c tio n 5 • none exam ples. 1. w x b ■ c 27. F red saved b d o lla r s in 2. b + c = w ^ w eeks. He saved w 3 . — * = w d o lla r s each week. 4 . c £ B w none b . 28. M rs. N elson bought d ch ick en s a t £ c e n ts a pound. Each ch ick en weighed a po u n d s. M rs. N elson p a id d x e x a c e n ts f o r th e c h ic k e n s . 1. d x e = a 2. (d x e) x a « d x (e x a) 3 . d + e + a « - d x e x a 4- d - e = a 5 • none 145 29. M other baked £ c h e rry 1 . z - f * e t a r t s . The fam ily a te £ 2 . f x z ■ e t a r t s f o r lu n c h . There 3 . z + e » f w ere z t a r t s l e f t . 4 . e x f = z 5 . none 30. Edward bought a box o f 1. j * b b c o lo re d p e n c ils f o r £ 2. b x a = z c e n ts . Each p e n c il c o s t 3 . ^ = z z c e n t s . 4 . a - z = b 5. none 31. The Browns a re ta k in g a 1. a + b ■ > w t r i p to th e la k e . The 2. a x b = w la k e is a m ile s from 3 . w x b « a home. They have tr a v e le d 4 . b - w « a £ m ile s . They have w 5 . none m ile s l e f t to g o . 32. L essons a t th e swimming 1. (a x b) x c = a x (b x c) p o o l c o s t £ d o lla r s a 2. a .+ b + c * = a x b x c w eek. ]£. c h ild re n took 3 . a - c - b « = a + b + c le sso n s f o r £ w eeks. The 4 . a x b = c le sso n s fo r th e c h ild re n 5. none c o s t a x b x c d o l l a r s . 33. The c h ild re n to o k th r e e 1. a + (b + c) * (a + b) + c s p e llin g t e s t s . Ja ck had 2. a + b + c * a x b x c a r i g h t on th e f i r s t 3 . a - b - c = a + b + c t e s t . He had b r i g h t on 4 . a + b + c = a + w + c th e second t e s t and c. 5. none r i g h t on th e t h i r d t e s t . Ja ck had a + b + c r i g h t on a l l th r e e t e s t s . 34. J a n e t worked £ a d d itio n 1. c - d + e = c + .d + e pro b lem s, cl s u b tr a c tio n 2. c + (d + e) » (c + d) + p ro b lem s, and e d iv is io n 3 . c x d x e = c + d + e problem s. J a n e t worked c + d + e oro^lem s in a l l . 4 . d + e “ c + 5 . none d + e 146 35. A lic e w eighs £ pounds. A lic e m ust g a in £ pounds b e fo re she w i l l w eigh d p ounds. 1 . c + d = z 2 . d - c = z 3 . c - z » d 4 . c x z * d 5 . none 36. J im 's c la s s has r a is e d £ 1 . a + b + c + d * b + c + a d o lla r s from cookie 2 . a + b + c = b + c + d s a l e s . For b d o lla r s , 3 . a « b + c + d th e y bought a p a i r o f 4 . (b + c) + d S 3 b + (c + d) h a m ste rs, a p e t pen f o r 5 . none c. d o l l a r s , and a tr e a d - w heel f o r d d o l l a r s . They sD ent b + c + d d o lla r s in a l l . 37. Mike has d exam ples to do 1 . c + d = z f o r h is a r ith m e tic l e s 2. c + z = d so n . He has worked c. 3 . z - c = d ex am p les. Mike has z 4 . z x d = c exam ples l e f t to w ork. 5. none 38. James had a b o x e s. In 1 . a x b * c each box he p u t b e g g s . 2. a + b + c = a x b x c He s o ld th e eggs f o r c 3. a x (b x c) as (a x b) x c c e n ts a p ie c e . He re c e iv e d 4 . a - b - c = a x b a x b x c c e n ts f o r th e 5. none eg g s. 39. C arl c u t § . badges from a 1 . f = z rib b o n b in ch es lo n g . 2. a x b < = z The badges w ere a l l th e 3 . ! -■ * same le n g th . Each badge 4 . C L b - z = a was £ in ch es lo n g . 5. none 40. Farm er N e ls o n 's hens l a id 1 . ( f + a) + b a f + (a + b) J, eggs one week. They 2. f - a - b = f + a + b l a id £ eggs th e second 3. f + b * = f x b week and b eggs th e th i r d 4 . f + a + b » a x b week. The hens la id 5. none 147 f 4- a 4 b eggs in th e th re e w eeks. 41. T here a r e b desks in B i l l 's room. T here a r e £ desks in each row. T here a r e £ rows in th e room. 42 . T ic k e ts f o r th e puppet show w ere d c e n ts a p ie c e . £ c h ild re n s o ld t i c k e t s each . They re c e iv e d d x e x f c e n ts f o r th e t i c k e t s s o ld . 43 . J o e 's f a th e r p a id £ c e n ts f o r £ t i c k e t s to th e b a s k e tb a ll game. He sp e n t e x f c e n ts f o r th e t i c k e t s . 44. M iss Sm ith gave each o f £ c h ild re n a s h e e t o f p a p e r. On each s h e e t w ere b rows o f c sq u ares each . T here w ere e x b x c sq u a re s in th e s h e e ts o f p a p e r. 45. A lad y buys d pep p ers fo r £ c e n ts . A ll th e p ep p ers a re th e same p r ic e . Each pepper c o s t £ c e n t s . 1. b 4- c 8 z 2. z - c = b 3 . c x z « b 4 . c x b = z 5. none 1. d + e + f = d x e x f 2. d - e 8 f 3 • d x (e x f ) ■ (d x e) x f 4 . d x f x e 8 f 5 * none , e _ 1. f = e x f 2. e + f = e x f 3 . e x f 8 f x e 4 . e x f « f 5 . none 1. (e x b) x c = e x (b x c) 2. e x c = e x b x c 3. e + b + c = » e x b x c 4 . e- "..ft 8 e x b x c c 5 . none 1. z x e 8 d 2. e x d 8 z 3 . d x z 8 e 4 . d + e 8 z 5. none 148 46. B i l l bought a to p s a t b 1. a + b - a x b c e n ts a p ie c e . 411 th e 2. a x b a x b to p s c o s t th e same. 3 . b - a x s a x b B i l l n a id a x b c e n ts 4 . | - a X b f o r th e to p s . 5. none 47. K enneth s e l l s tom atoes 1. d x w m f f o r d c e n ts a box. He 2. f - w a d packs £ tom atoes in each 3 . d x f as w box. Each tom ato c o s ts 4 . w x f cs d w c e n ts . 3. none 48. N ick has £ p ic tu r e s o f 1. z + b m a b a s e b a ll p la y e r s . He 2. z + a SB b tra d e d b o f them w ith 3 . z - a a b Andy fo r a k n if e . N ick 4 . a x b m z had £ b a s e b a ll p ic tu r e s 5. none l e f t . 49. £ c h ild re n w ished to 1. b - a = w s h a re e q u a lly £ candy 2. w ® b x a s t i c k s . Each c h ild r e 3 . w + b = a ce iv ed w candy s t i c k s . 4 . a x w « b 50. The c h ild re n bought £ 1. b - a - z p la n ts f o r window boxes 2. z x b ■ a f o r th e s c h o o l. I t took 3 . a x z = b b p la n ts to f i l l each 4 . a + b = z box. The c h ild re n f i l l e d 5. none jg . window b o x e s.

Asset Metadata

Creator Hammond, Robert Lee (author)

Core Title Ability With The Mathematical Principles Governing The Operations Of Addition, Multiplication, Subtraction, And Division

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Education

Degree Program Education

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Education, general,OAI-PMH Harvest

Language English

Advisor Naslund, Robert A. (committee chair), Metfessel, Newton S. (committee member), Perry, Raymond C. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-265867

Unique identifier UC11359050

Identifier 6206061.pdf (filename),usctheses-c18-265867 (legacy record id)

Legacy Identifier 6206061.pdf

Dmrecord 265867

Document Type Dissertation

Rights Hammond, Robert Lee

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA

Linked assets

University of Southern California Dissertations and Theses