C H A P T E R XI NUMERICAL ABILITY * A CONSTRUCT In the previous chapter the seed for the present test was described in the context of secondary schools in the State of Gujarat* The utility and limitations of the present work were also indicated* In this chapter the nature-nurture functions of mental abilities* models of'mental structure* and components of 'numerical ability complex* core described in turn. Nature-nurture interactions Mental measurements, at their best, are Indirect and largely hypothetical in nature. Whether the approach to test construction is rational or empirical, the nature of the 'entity* being measured is not discovered completely. The existence and extent of the measurement variable is just infered experimentally and expressed statistically* The ultimate reality of measurement variables, their origin, development, interactions and contribution of her&dity and environment are issues of great Interest but they are beyond the scope of the present work. Only a summary of generalizations from Gagne, (p.90) would suffice to provide a frame of reference* Human beings can display various functions such as discrimination and conceptualization. These functions develop
0 id within a framework of anatomical structures, which is unique for each individual* These functions are also shaped by learning that occures in varying social and cultural environ- ments. The inherent set of sensory effectors and neural structures set limit to the development of most of these functions for each human being* And therefore learning is also a matter of individual differences* To discover such inter-personal and intra-personal variations is a field of differential psychology* The mental functions are facilitated through the use of concrete* abstract and verbal imagery* Imagery depends on experiences available as well as receptivity and ajruity of the sensory systems* In this sente the verbal* numerical and mechanical abilities can be concieved to be highly cultural* They are largely developed by informal and formal experiences within the environment* Such abilities represent general potentialities, which can facilitate the learning of a variety of skills or job elements, in educational and vocational set-ups* Learning play a major role in influencing development of abilities, within the boundaries set b y heridity and environmental factors. According to Gagne,v the numerical ability continues to grow up to the age 22, remains almost stable up to 52 and then declines* (Gagne, p. 96)
0 i' 11 A c c o r d i n g t o G u i l f o r d t h e a v e r a g e c h i l d r e a c h e s e i g h t y p e r c e n t o f m a t u r i t y i n N u m e r i c a l f a c i l i t y a t t h e a g e o f s i x t e e n. ( G u i l f o r d. 3, p. 4 2 0 ) I n b o t h t h e r e f e r e n c e s r e f e r r e d t o a b o v e N u m e r i c a l F a c i l i t y i s i m p l i e d b y t h e t e r m n u m e r i c a l a b i l i t y. T h e c e n t a l i d e a i n h i s w o r k s ; o n P r i m a r y M e n t a l A b i l i t i e s h a s b e e n c o n c e p t u a l i z e d b y L. L. T h u r s t o n b y s t a t i n g t h a t. " t h e h u m a n i n t e l l e c t c o n s i s t s o f a f i n i t e n u n b e r o f d i s t i n g u i s h a b l e f u n c t i o n s w h i c h a r e c a l l e d p r i m a r y f a c t o r s o r a b i l i t i e s ". ( B a r n e t t e ( i d. ), p. 1 2 3 ) M o d e l s o f m e n t a l s t r u c t u r e E f f o r t s a r e b e i n g m a d e, c o n t i n u o u s l y, t o d i s c o v e r a n d d e s c r i b e t h e n a t u r e a n d o r g a n i z a t i o n o f m e n t a l f u n c t i o n s f r o m v a r i o u s a p p r o a c h e s. T w o o f t h e m o d e l s p r e s e n t e d b y p s y c h o - m e t r i c i a n s a r e b r i e f l y s u m m a r i s e d h e r e, t o t h e e x t e n t t h a t t h e y a r e r e l e v a n t t o t h e d e s i g n o f t h e p r e s e n t w o r k, a s w e l l a s t h e r a t i o n a l e b e h i n d t h e d e s i g n s o f s i m i l a r t e s t s o f r e p u t e. T h e d e s c r i p t i o n o f t h e t w o m o d e l s i n d i c a t e s h o w t h e h y p o t h e t i c a l c o n s t r u c t o f n u m e r i c a l a b i l i t y c a n b e v i e w e d a s a c o m p o n e n t o f ' g e n e r a l i n t e l l i g e n c e m o d e l s * A h i e r a r c h i c a l m o d e l. T h e o r g a n i z a t i o n a l h i e r a r c h i c a l m o d e l o f t h e B r i t i s h S c h o o l h a s b e e n w e l l p r e s e n t e d b y H u m p h r e y s.
At the top of the hierarchy is **gm the general factor depending on numerous overlap ing bonds rather than A unitary force or power within the person. There ore at^least two major group factors: the verbal, numerical educational factor and the mechanical spatial, practical factor, beneath "g", Below the broad group factors are the Thurstons*s primaries, Guilford's more numerous primaries are presumably still further down in the hierarchy. Specific factors are at the bottom of the hierarchy. It Is argued that the hierarchical model defines all factors in terms of original measurement variables. The model also indicates that the test con be constructed to measure a factor at any level of the hierarchy. One can move up or down the hierarchy to make the required kind of prediction. All ability tests*aptitude, ability and achievement-* are subsumed under the same model, (Gronlund (Ed,), p. 260) A morphological model. - The morphological or popularly called the ** Structure of Intellect N model, by Guilford, with its three parameters - operation, content and product - strives to develop unique definition of each cellular specific intellectuxal ability in terms of one of the five kinds of operations, confcined with one of the four kinds of content end with one of the six kinds of products. (Guilford, 3, p. 465) The three parameters or dimensions of Guilford's conception of the structure of Intellect are #
13 r e p r o d u c e d b y a s i m p l i f i e d f i g u r e b y H e l m e t a d t e r. ( p. 1 0 9 ) T h e s e m a n t i c c o n t e n t s h a v e t o d o w i t h l a n g u a g e. T h e s y n f c o l i c c o n t e n t s a r e d e l t w i t h a s n u m e r i c a l i d e a s a n d c o n c e p t s. T h e f i g u r a l c o n t e n t s a r e r e l a t e d t o v i s u a l, a u d i t o r y m a d o t h e r t y p e s o f c o n f i g u r a t i o n s, p a t t e r n s a n d s h a p e s, w h i l e t h e b e h a v i o u r a l c o n t e n t s a r e r e l a t e d t o t h e w a y i n w h i c h p e r s o n s b e h a v e i n s o c i a l c o n t e x t. G u i l f o r d * s p r o p o s a l, t h o u g h c o n s i d e r e d t o b e " r a n k s h e r e s a y * * b y G u i l f o r d h i m s e l f, s t r i v e s t o p r o v i d e a m e a n i n g * f u l a n d s y n t h e t i c c l a s s i f i c a t i o n o f f o r t y o d d f a c t o r s a l r e a d y r e c o g n i s e d. ( G r o n l u n d ( E d. ), p. 2 7 ^ ) I t a l s o s u g g e s t s a r e a s i n w h i c h n e w f a c t o r s m i g h t b e f o u n d. S u b s e q u e n t r e s e a r c h e s h a v e v e r i f i e d t h e e x i s t e n c e o f s o m e o f t h e s e a d d i t i o n a l f a c t o r s. H e l m s t a d t e r h a s r e c o g n i s e d t h e t r e m e n d o u s v a l u e o f G u i l f o r d ' s t a x o n o m i c w o r k I n t h e a r e a o f t e s t i n g, a s w e l l a s t h e u s e f u l n e s s o f t h e f a c t o r i a l a p p r o a c h t o c o n t e n t v a l i d i t y. ( H e l m s t a d t e r, p. I l l ) A t e x o n o m i c a l h i e r a r c h y c a n b e v i e w e d e v e n i n G u i l f o r d * # m o d e l. T h e d i m e n s i o n o f o p e r a t i o n c a n b e a s s u m e d t o b e r a n g i n g f r o m s i m p l e t o c o m p l e x a n d s p e c i f i c t o g e n e r a l a s o n e m o v e s f r o m c o g n i t i o n t o e v a l u a t i o n. T h e d i m e n s i o n o f p r o d u c t s c a n b e v i e w e d f r o m u n i t e t o i m p l i c a t i o n s a n d t h e d i m e n s i o n o f c o n t e n t f r o m f i g u r a l t o b e h a v i o u r ^. S u c h h i e r a r c h y c a n b e t e s t e d b y a d m i n i s t e r i n g a l a r g e n u m b e r o f i t e m s o n e a c h a s p e c t o f t h e s a m e d i m e n s i o n o n a s i n g l e s a m p l e o f e x a m i n e e s.
n Hie mental functions covered by some of the items of numerical ability tests can be properly accommodated in the lattice of the conversant production factors suggested in Guilford's Structure of Intellect model. The cells of the lattice are convergent symbolic units (NS0), - classes (NSC). relations (NSR), - systems (NSS), transformations (NST), and - implications (NSI). (Guilford, 3. p.i7l) There is scope for six factors for each of the other operations namely, cognition, memory, divergent thinking and evaluation. The hierarchical model accommodates numerical ability as a meidber of M the verbal, numerical educational group factor" below ng M and above the primary and specific factors in the organisation of mental abilities, A synthesis. * One has to strike a balance between the structural and functional approaches to achieve optimum validity and flexibility to serve the purpose of the test. The expectation that factoring will yield the capacities or unitary powers in absolute terms is farfetched due to lack of experimental controls on factor-analysis studies. The morphological model helps to decide the medium, level and form of the measurement tasks. The present test of Numerical Ability falls in the symbolic media in essence. The language Ability Test falls in semantic and the Pictorial Test of Hehanlcal comprehension in the figural
0 m e d i a. F r o m t h e h i e r a r c h i c a l m o d e l p o i n t o f v i e w t h e t h r e e t e s t s c a n b e a c c o m m o d a t e d a s m e m b e r s o f t h e m a j o r g r o u p f a c t o r s u n d e r g. T h e m o d e l i n d i c a t e s t h e h i e r a r c h i c a l r e l a t i o n s a m o n g s t c a t e g o r i e s o f f a c t o r s b u t i t i s s i l e n t o n t h e p r o d u c t s a n d o p e r a t i o n s o f t h e m e n t a l t a s k, I n d u c e d b y t h e t e s t i t e m s. T h e m u l t i p l e c h o i c e o b j e c t i v e t e s t i t e m s a r e c o n s i d e r e d s u i t a b l e f o r c o n v e r g e n t t h i n k i n g i n a n y m e d i u m. T h e n u m e r i c a l t e s t i t e m s g e n e r a l l y l e a d t o t h e t r a n s o r m a t l o n s a t t h e e n d o f t h e n u m e r i c a l o p e r a t i o n s. I n s h o r t, n u m e r i c a l a b i l i t y i s a n i m p o r t a n t c o m p o n e n t o f t h e s o c a l l e d g e n e r a l m e n t a l a b i l i t y. S t r u c t u r a l l y i t c a n b e f u r t h e r a n a l y s e d i n t o s p e c i f i c f a c t o r s. F u n c t i o n a l l y i t i s a p a r t o f s c h o l a s t i c a b i l i t y. B e f o r e t h e N u m e r i c a l A _ b i l i t y i s d e f i n e d t h e t a x o n o m i c a l r e l a t i o n s o f a b i l i t y w i t h a p t i t u d e a n d a c h i e v e m e n t a r e w o r t h c o n s i d e r i n g, A b i l i t y c o n t i n u u m I n t h e c o n t e x t o f p r o g r e s s a t e & i t h e l e v e l o f e d u c a t i o n, t r a i n i n g o r v o c a t i o n, a v a r i e t y o f c h a r a c t e r i s t i c s a r e a s s u m e d t o h e o p e r a t i n g. T h e s e c h a r a c t e r i s t i c s a r e i n c o n s i s t e n t l y r e f e r r e d t o a s a p t i t u d e s, a b i l i t i e s,f a c t o r s a n d t r a i t s. T h e d e f i n i t i o n o f t e r m s a p t i t u d e, a b i l i t y a n d a c h i e v e m e n t c a n b e b e t t e r e x p l a i n e d i f t h e y a r e t h o u g h t t o b e o n t h e s a m e v a r i a b l e c o n t i n u u m o f m a x i m u m p e r f o r m a n c e. I t i s c o n s i d e r e d i n a p p r o p r i a t e t o t h i n k o f t e s t s o f a c h i e v e m e n t,
' 16 ability and aptitude as discrete categories of tools, distinguishable by knowing or supposing whether they measure innate ability or influence of environment or a mixture of both. In general, aptitude tests can be thought of as those which measure mental functions that improve little with practice; ability tests as those which show some relationship to general enrichment of the environment; and achievement tests as those constructed specifically to measure the degree of accomplishment in specific training or subject area. (Helmstadter, p. 11) Numerical ability defined by others The test constructors are not unanimous on the defination of the term, Numerical Ability* They further vary on functional contents of tests perpe&tlng to measure the factor *N* under any name. Seven test batteries, all accommodating factor 'N», are presented by their authors and reviewed b y Super in a reprint series from the Personnel and Guidance Journal. ( Super, 2, pp. 9, 22, 32, 38, 58, 59, 69, 8l) Naming the tests. - Bennett, Crowder and Thorstone have prefered to label their subtests as tests of Numerical Ability. Dvorak has given a common label of Numerical Aptitude to his subfests II and VI. Guilford has called his sub test number III of Numerical operations measuring Number
# t 1? Factor. Segel has tagged his subtest number V arithmetic reasoning and sub test number VI arithmetic computation under Numerical Seasoning. Flanagan has referred to each of his IX measures arithmetic skill and subtest number XVI measures reasoning. Contents of the tests. So far as the functional contents of the tests are concerned, Crowder. Guilford and Thurstone have prefered to include the items of fundamental arithmetic operations or numerical facility alone. Dvorak. Segel and Flanagan have prefered to include items of numerical facility and numerical problemjsolving, in two subtests timed separately. The problems are cast in verbal sentences therein. Bennett has avoided the use of language in his test of Numerical Ability, which is a mixture of a few items on numerical facility and the rest on the handling of numerical concepts. The two test batteries, namely, Differential Aptitude Tests and The Academic Promise Tests, designed by Bennett and others, seem to be popular amongst school counsellors. D m analytical study of their subtests can lend some useful hints regarding the suitability of subtests and the type of items to be considered for inclusion in the tests for educational guidance.
The Differential Aptitude Tests Battery, which is adjudged by Super as 11 ready for use in educational counseling ** for grades 8 to 12, has adopted the definition of numerical ability as "understanding of numerical relation* ships and facility in handling numerical concepts". (Super, 2, pp. 9, 89) The subtest Numerical ability In the Academic Promise Tests by Bennette and others, is a mixture of items on number series (IS) and straight forward arithmetic computations and arithmetic reasoning items (45), on the lines of Otis and Thorndike, (Buros (Ed,), 2, p. 765) Factorial studies. - From a factor analytic study by Adkins, the Numerical Ability can be viewed as a group of related factors such as Arithmetic, Number series. Numerical operations I ( +, x), Numerical operations II (,*! and Numerical puzzles, which' are coded as 2, 32, 33, 34 and 35 in the report. The high inter-correlations among these factors Indicate a few things. It is difficult to construct factoriaily,pure tests. Secondly the numerical ability test could be constructed with closely related but identifiable subfactors. Thirdly Numerical operations II has larger overlapping than the Numerical operations 1. (Adkins, p. 108) -In Tharstone's first PMA analysis his four numerical operations tests and four others were found to be loaded by
19 N u m b e r f a c t o r a s f o l l o w s M u l t i p l i c a t i o n. 8 1 H u m b e r c o d e. 6 2 A d d i t i o n. 7 6 H i m. j u d g e m e n t. 4 3 S u b s t r a c t i o n. 6 7 T a b u l a r c o m p l e t i o n. 3 9 D i v i s i o n. 6 2 A r i t h. r e a s o n i n g. 3 8 ( G u i l f o r d, 3, p. 1 3 2 ) T h e t y p i c a l t r a d i t i o n a l n u m e r i c a l f a c i l i t y o r n u m b e r f a c t o r w a s s u p p o s e d t o o c c u p y t h e c e l l H S I c o n v e r g e n t p r o d u c t i o n o f s y m b o l i c i m p l i c a t i o n s, h u t r e c e n t s t u d i e d h a v e r a i s e d d o u b t s a b o u t t h e u n i v o c a l l t y o f t i n f a c t o r ' S ', I t h a s a l a r g e c o m p o n e n t o f s p e c i f i c p r a c t i c e. ( G u i l f o r d, 3, p. 1 3 4 ) T h u r s t o n e a l s o h a d d o u b t s r e g a r d i n g t h e u n d e r l y i n g n a t u r e o f t h e n u m b e r f a c t o r. H e r e p o r t s i A n o t h e r f a c t o r t h a t w a s I s o l a t e d e a r l y i n t h e f a c t o r i a l e x p e r i m e n t s w a s t h e n u m b e r f a c t o r ' V. T h i s f a c t o r r e p r e s e n t s f a c i l i t y i n d o i n g s i m p l e a r i t h m e t i c a l t a s k s b u t i t i s r a t h e r n a r r o w. T h e n u m b e r f a c t o r i s n o t h e a v i l y i n v o l v e d i n a r i t h m e t i c r e a s o n i n g o r i n m a t h e m a t i c s. I n f a c t i t w o u l d n o t b e s u r p r i s i n g t o f i n d g o o d m a t h e m a t i c i a n s w h o a r e r e l a t i v e l y p o o r i n t h i s f a c t o r. O n t h e o t h e r h a n d a c a s h i e r w o u l d s o o n q u i t t h i s j o b i f h e w e r e p o o r i n t h e n u m b e r f a c t o r. H e h a v e b e e n p u z z l e d a b o u t t h e u n d e r l y i n g n a t u r e o f t h e n u m b e r f a c t o r. I f w e a r e d e a l i n g w i t h d i f f e r e n c e s i n n a t i v e e n d f ^ b e n t s, t h e n i t s h o u l d b e p o s s i b l e t o d e s c r i b e t h e n a t u r e o f t h i s f a c t o r i n t e r m s o f f u n d a m e n t a l p r o c e s s e s r a t h e r t h a n i n s u c h t e r m s a s " n u m b e r ", w h i c h i s a c u l t u r a l c o n c e p t. E v e n t u a l l y, w h e n t h e n a t u r e o f t h i s f a c t o r i s m o r e c o m p l e t e l y u n d e r s t o o d, i t m a y t u r n o u t t o b e m e r e l y a h i s t o r i c a l a c c i d e n t t h a t t h e f a c t o r w a s i d e n t i f i e d i n t e s t s t h a t h a p p e n e d t o b e,
20 numerical in character. future to answer. This is a question for the ( Barnette (Ed.), p.116) The review of the standard tests and factor studies referred to above indicate that numerical ability, popularly known as numerical facility, continues to grow up t o.the age of 22, requiring 16 years to reach 80 per cent of maturity. The univocality of Number as a pure factor is seriously doubted b y some of the eminent psychometricians. Test items on numerical series, numerical concepts and numerical problems require mental operations related to a group factor of numerical ability but identifiable by experimental controls, as separate sub-factors. Numerical ability is considered to be the proper term to designate the measurement variable for the present work. The author* s construct of numerical ability and relevant terms are now defined at this stage, A construct of Numerical Ability The test author, on the basis of the survey of relevant literature referred to in this chapter could formulate a hypothetical construct of Numerical Ability for the design of the test. It was then decided to include the
0 i 21 f o l l o w i n g o p e r a t i o n a l l e v e l s, s a p a s p e c t s, o f n u m e r i c a l a b i l i t y I n h i s c o n s t r u c t J - 1, N u m e r i c a l f a c i l i t y m e a s u r e d t h r o u g h f o u r f u n d a m e n t a l o p e r a t i o n s i n v o l v i n g, p r i m a r i l y, n u m b e r f a c t s o f a d d i t i o n a n d m u l t i p l i c a t i o n * 2, N u m e r i c a l r e a s o n i n g ( e a s y, i n d u c t i v e, n o n - v e r b a l ) t h r o u g h n u m b e r s e r i e s, r e q u i r i n g t h e e x a m i n e e t o f i n d r e l a t i o n s, f o r m u l a t e t h e r u l e a n d s u p p l y t h e m i s s i n g n u m b e r i n t h e s e r i e s, 3, U s e o f b a s i c ( a r i t h m e t i c a l ) n u m e r i c a l c o n c e p t s ( r e l a t i v e l y e a s y ), t h r o u g h n o v e l i t e m s o f t r a n s f o r m i n g o n e n u m e r i c a l f o r m i n t o a n o t h e r, 4, U s e o f n u m e r i c a l c o n c e p t s ( a l g e b r a i c, r e l a t i v e l y d i f f i c u l t ), t h r o u g h l e a r n i n g - s e t i t e m s o f t r a n s f o r m i n g o n e n u m e r i c a l f o r m i n t o a n o t h e r. 5 * U s e o f n u m e r i c a l c o n c p p t s, ( d e d u c t i v e, r e l a t i v e l y d i f f i c u l t ) t h r o u g h s o l v i n g p r o b l e m s, c a s t i n v e r b a l f o r m, T h i s h y p o t h e t i c a l m o d e l w a s k e p t i n v i e w d u r i n g t h e p r o c e s s o f t h e t e s t d e v e l o p m e n t, X t w a s f o u n d t o b e w o r k i n g r e a s o n a b l y w e l l, a s r e v e a l e d b y d a t a o n i t e m a n a l y s i s a n d f a c t o r a n a l y s i s, a t a l a t e r s t a g e.
' 22 More terms defined The rest of the tends were adopted b y the author from Good who is quoted > Ability : the actual power present in an organism to carry to completion any given act* (Good, 1, p. i) Number concept : (l) the abstract ide of any order of multiplicity grasped in terms of meaningful symbols which may he manipulated to solve problems : the generalization which the learner gains involves a mental picture based upon his experience and understanding of the interrelationships of himself, objects, and symbols: the number concept of the modern Hindu-Arabic number system would encompass, among other specifics, the zero as a place holder, the base 10 idea, the unitary symbol representing a multiplicity of objects, the decimal point, signed numbers, and the paired opposite idea : (2) a general idea developed in regard to some procedure or basic process in arithmetic and identified by a particular word or phrase; for example, the general idea expressed by the word ''subtraction*1 is that or the division of a single group of objects into two smaller groups* (Good, 1, p* 118) Conceptual thinking : (l) the process of thinking in which abstract concepts are related without reference to concrete situations: (2) the manipulation of linguistic or mathematical symbols in terms of their inherent logical relations. (Good, 1, p. 570)
23 Problem solving ability s (math.) (l) the degree of capability necessary to find the correct solution to problem situations ; (2) an estimate of such ability derived by administering standardized tests* (Good* 1* p. 2) In this chapter* the contribution of nature and nurture in development of mental abilities was cursorily mentioned. It was followed by the presentation and synthesis of two models of mental organization* The concept of ability as a variable continuum and numerical ability complex in general were described* In the end the hypothetical construct of numerical ability adopted h for the present work was described* In the following chapter some test-literature is reviewed in brief. - - - o -