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Study of the relationship between perceptual training and arithmetic computation Gaskill, James Leslie 1971

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A STUDY OF THE RELATIONSHIP BETWEEN PERCEPTUAL TRAINING AND ARITHMETIC COMPUTATION by JAMES LESLIE GASKILL B.Sc, University of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics Education We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1971 In p r e s e n t i n g t h i s. thes.i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . James L e s l i e G a s k i l l Department o f Mathematics E d u c a t i o n The U n i v e r s i t y o f B r i t i s h C o l umbia Vancouver 8, Canada \ ABSTRACT An a n a l y s i s of the l i t e r a t u r e showed t h a t many c o n c l u s i o n s about the r e l a t i o n s h i p between p e r c e p t u a l s k i l l s and r e a d i n g were based on e i t h e r the measurement of c e r t a i n r e a d i n g s k i l l s which d i d not depend upon the p e r c e p t u a l s k i l l s t e s t e d , o r t r a i n i n g programs which were not matched wi t h the p e r c e p t u a l s k i l l s being s t u d i e d . T h i s , together with the f a c t t h a t r e s e a r c h i n the f i e l d o f mathematics has found t h a t many computational mistakes are made because of mistaken symbols, l e d the author to d e f i n e a p e r c e p t u a l s k i l l , the search mechanism, which was s p e c i f i c a l l y determined by the method of working a l g o r i t h m s . A p i l o t study was performed to e s t a b l i s h t e s t i n g procedures. The r e s u l t s o f t h i s study i n d i c a t e d t h a t there was a r e l a t i o n s h i p between the search mechanism and a r i t h m e t i c computation. The experiment c o n s i s t e d of a treatment group and a c o n t r o l . g r o u p . A l l s u b j e c t s were g i v e n p r e - and p o s t - t e s t s on each of four measures; the se a r c h mechanism, v e r t i c a l span, h o r i z o n t a l span and a r i t h m e t i c computation. The treatment group was g i v e n t r a i n i n g i n the search mechanism. The f o l l o w -i n g s t a t i s t i c a l r e s u l t s were e s t a b l i s h e d : there was a s i g n i f i c a n t difference between the control and experimental groups on a measure of change of search a b i l i t y ; there were no s i g n i f i c a n t differences between the control and experimental groups on measures of v e r t i c a l span, horizontal span and arithmetic computation. Using the post-test on the control group only, i t was established that: search a b i l i t y was correlated with v e r t i c a l span; search a b i l i t y was not cor-related with horizontal span; v e r t i c a l span was correlated with horizontal span; search mechanism, with the effects of v e r t i c a l span and horizontal span removed, was correlated with arithmetic computation. This l a t t e r finding means that the variance i n arithmetic scores accounted for by search mechan-ism, v e r t i c a l span, and horizontal span was s i g n i f i c a n t l y d i f f e r e n t from that accounted for by v e r t i c a l span and h o r i -zontal span alone. Two possible conclusions were suggested.. The f i r s t was that the t r a i n i n g period was too short for transfer from the search s k i l l to algorithmic performance to take place. The second was that the increase i n the search procedure test could be explained by v e r t i c a l span being used with increasing e f f i c i e n c y within the new context of search procedure t e s t i n g . Because a low co r r e l a t i o n was obtained between arithmetic and v e r t i c a l span the achieved s t a b i l i t y of the arithmetic scores was to be expected. I. DESCRIPTION OF THE STUDY 1 I. INTRODUCTION 1 I I . DEFINITIONS * . . . . 3 P e r c e p t i o n . . . . . . . 3 P e r c e p t u a l S k i l l . . . . . 4 A l g o r i t h m , 4 F i x a t i o n • 5 Span . 5 Search Procedure 5 A l g o r i t h m i c Performance 5 Causal R e l a t i o n 6 I I I . SIGNIFICANCE OF THE STUDY 6 A l g o r i t h m i c Performance 6 Reading and A r i t h m e t i c 9 IV. RELATED RESEARCH 11 Reading-Perception R e l a t i o n s h i p 12 P e r c e p t u a l T r a i n i n g f o r Reading S k i l l s . . . 13 V. DISCUSSION OF THE PROBLEM 17 VI. STATEMENT OF THE PROBLEM 19 V I I . QUESTIONS TO BE ANSWERED 20 I I . RESEARCH DESIGN 25 I. EXPERIMENTAL DESIGN . . . . . 25 Method of T r a i n i n g 26 Data A c q u i s i t i o n 26 I I . STATISTICAL DESIGN 27 A n a l y s i s of V a r i a n c e Hypotheses 28 C o r r e l a t i o n a l A n a l y s i s Hypotheses 30 I I I . ASSUMPTIONS AND LIMITATIONS 32 Study L i m i t a t i o n s . . . . . . 32 S t a t i s t i c a l Assumptions 33 I I I . RESULTS OF THE STUDY 36 I. AN ANALYSIS OF VARIANCE HYPOTHESES . . . . 36 Te s t s of S i g n i f i c a n c e Used 36-Hypothesis I 39 Hypothesis II . 40 Hypothesis I I I 40 Hypothesis IV 41 I I . CORRELATIONAL ANALYSIS . 43 Product Moment C o r r e l a t i o n s 43 M u l t i p l e C o r r e l a t i o n 44 IV. CONCLUSIONS AND HYPOTHESES FOR FURTHER RESEARCH . 49 I. CONCLUSIONS 49 I I . FURTHER RESEARCH 50 E d u c a t i o n a l Psychology . 51 BIBLIOGRAPHY 54 APPENDICES 56 I. TRAINING METHODS 57 I I . DESCRIPTION OF TESTS, PRESENTATION AND SCORING . 59 I I I . STATISTICAL ANALYSIS OF TEST RESULTS . . . 72 IV. PILOT STUDY 82 I. PLAN OF THE EXPERIMENTAL DESIGN AND ADMINISTRATION OF ARITHMETIC TESTS 27 I I . SOURCES OF VARIATION, DEGREES OF FREEDOM, AND EXPECTED MEAN SQUARES USED FOR THE ANALYSIS 29 I I I . MEANS OF ALL TEST ADMINISTRATIONS 37 IV. STANDARD DEVIATIONS OF ALL TEST ADMINI-STRATIONS . . . . . . . . . . 38 V. MEANS OF THE DIFFERENCE SCORES FOR THE FOUR GROUPS ON EACH OF THE FOUR MEASURES . . . . 38 VI. SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS I 39 V I I . SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS I I 40 V I I I . SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS I I I 41 IX. SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS IV 42 X. TABLE OF CORRELATIONS AND SIGNIFICANCE LEVELS. 43 XI. ITEMS COMPRISING THE VERTICAL AND HORIZONTAL SPAN TESTS 61 XII. ARRAY PRESENTED FOR THE SEARCH MECHANISM TEST WITH THE STARTING POINTS OF THE SEQUENCES USED . . . . . . . . . . 62 XI I I . SEQUENCES AND THE CORRECT RESPONSES USED FOR THE SEARCH MECHANISM TEST 63 XIV. SUMMARY OF THE ANALYSIS OF THE FOUR ARITHMETIC OPERATION SUBTESTS 6 6 XV. SUMMARY OF THE ANALYSIS OF THE VERTICAL AND HORIZONTAL SUBTESTS 6 6 XVI. SEARCH MECHANISM TEST RESULTS • 73 XVII. SUMMARY OF GROUP RESULTS 73 XVIII. VERTICAL SPAN TEST RESULTS . 76 XIX. SUMMARY OF GROUP RESULTS . . . . . . 76 XX. HORIZONTAL SPAN TEST RESULTS 78 XXI. SUMMARY OF GROUP RESULTS 78 XXII. ARITHMETIC COMPUTATION TEST RESULTS 80 XXIII. SUMMARY OF GROUP RESULTS 80 XXIV. SUMMARY OF PILOT STUDY RESULTS . . . . . . . . . . 83 DESCRIPTION OF THE STUDY I. INTRODUCTION There are many procedures which a f f e c t the ac q u i s i -t i o n and maintenance of computational or algorithmic s k i l l s . One of the purposes of research i s to f i n d those procedures or combination of procedures which most a f f e c t the ac q u i s i t i o n of s k i l l s . Various attempts have been made i n the realm of arithmetic research. One such procedure i s d r i l l . A student was believed to acquire the-solution procedure when required to make a given 1 2 3 response to a p a r t i c u l a r pattern or problem s i t u a t i o n . ' ' Once d r i l l was generally accepted i t then became important to increase the effectiveness of the drill:procedure. Motiva-t i o n a l techniques were employed with d r i l l procedures being accompanied by feedback materials i n the form of charts or graphs of programs. It was.Brownell and Chazal, however, who f i r s t emphasized the effects of d r i l l as a teaching procedure. They observed that the previous studies were related to improving, f i x i n g , maintaining, and r e h a b i l i t a t i n g computational s k i l l s ; and that the variables i n these studies were time, accuracy and both time and accuracy which measured the r e s u l t of the computation. That i s , these measures did not show the proced-ures used i n obtaining the answers. In t h e i r study performed on grade three students they found that even a f t e r one month of d r i l l , 5 minutes each day, on the items taught i n grades one and two that 24.4 percent of the responses were obtained by guessing, 22 percent by counting, 14.1 percent by i n d i r e c t solution and only 39.5 percent by memorization which was the 7 desired r e s u l t . They concluded, " . . . d r i l l i s exceedingly valuable for increasing, f i x i n g , maintaining, and r e h a b i l i -tating e f f i c i e n c y otherwise developed [emphasis not i n the g o r i g i n a l ] . " This implied that d r i l l was not as e f f e c t i v e as i t might have been considered because such a procedure neglects the i n i t i a l a c q u i s i t i o n of s k i l l s . It i s the purpose of t h i s study to i d e n t i f y a set of perceptual s k i l l s which either precede or are contempor-aneous with the i n i t i a l • a c q u i s i t i o n of algorithmic s k i l l s . I t w i l l be argued that there are c e r t a i n implications of studies done on algorithmic performance which indicate a dependence of the student upon s k i l l s analogous to some reading s k i l l s . Further, the studies c i t e d on the r e l a t i o n s h i p between reading and arithmetic do not account for the a b i l i t i e s or s k i l l s needed i n algorithmic performance. F i n a l l y , there i s a s p e c i f i c per-ceptual s k i l l , search procedure, which should be considered as a p o t e n t i a l l y s i g n i f i c a n t f a c t o r i n the a c q u i s i t i o n of a r i t h -metic s k i l l s . As the search procedure has not been i d e n t i f i e d i n the r e s e a r c h examined i t s development w i l l f o l l o w upon some i m p l i c a t i o n s of r e l a t e d r e s e a r c h done i n the f i e l d s of a r i t h -metic and r e a d i n g . P e r c e p t i o n must be i n t e r p r e t e d c a r e f u l l y f o r , as s t a t e d i n the E n c y c l o p e d i a of E d u c a t i o n a l Research, " P e r c e p t i o n " l i k e many o t h e r terms has a number of meanings and c o n n o t a t i o n s . . . . T h i s l a c k of r e s t r i e -tedness i s to be found even i n the usages g i v e n the term by those who study p e r c e p t i o n . 9 A l p o r t ' s book pr e s e n t s evidence of t h i s f a c t by t r a c i n g the h i s t o r i c a l development of t h e o r i e s of p e r c e p t i o n . " ^ However, he notes t h a t p e r c e p t i o n i s more than j u s t the awareness of a symbol but " . . . a l s o i n v o l v e s to some degree, an understanding awareness, a 'meaning' or a ' r e c o g n i t i o n ' of those objects.""'""'' Because of these v a r i o u s meanings and connotations i t i s neces-sary to i n t r o d u c e a s e t of d e f i n i t i o n s , not a l l of which p e r t a i n to p e r c e p t i o n , which w i l l be used throughout t h i s t h e s i s . I I . DEFINITIONS P e r c e p t i o n P e r c e p t i o n w i l l have been s a i d to take p l a c e i f the s u b j e c t can copy a v i s u a l symbol presented to him. For example, i f a student i s asked to write the tenth d i g i t appearing i n a sequence and he succeeds then perception i s said to have taken place. Perceptual S k i l l Perceptual s k i l l (syn. perceptual behaviour) i s a s k i l l which allows a more complex stimulus to be perceived i n a given period of time or allows a given stimulus to be per-ceived more rapidl y . Examples of these s k i l l s i n reading are scan and eye movement patterns. Algorithm An algorithm i s a procedure which, by correct, sequen-t i a l a pplication to a problem, i s believed to lead to a solution. Within the context of this study arithmetic algorithms w i l l r efer to the following methods: addition 13 27 63 + 42 145 subtraction 527 - 391 136 m u l t i p l i c a t i o n 256 x 42 512 1024 10752 d i v i s i o n 577 23)13271 115 177 161 161 161 F i x a t i o n Fixation i s that period of rest between successive eye movements when the eye focuses upon a set of s t i m u l i to be perceived. In reading there i s a successive series of eye movements each .followed by pauses during which the perception of symbols takes place. Span Span i s the width or length of symbolic material which can be perceived i n one f i x a t i o n of the eye. That i s the number of randomly ordered d i g i t s i n either a single h o r i -zontal number or i n a single v e r t i c a l column. Search Procedure Search procedure (syn. search mechanism, search a b i l i t y ) i s that procedure used to perceive a s p e c i f i c b i t of required material within a complex stimulus. In arithmetic terms i t i s the procedure used to select the d i g i t s required for the next step of an algorithm. Algorithmic Performance Algorithmic Performance (syn. arithmetic performance, arithmetic a b i l i t y , a b i l i t y to work an algorithm, computational a b i l i t y ) i s defined i n terms of the test given. In the case of this study the meaning of the above words w i l l be r e s t r i c t e d to algorithms of the whole numbers and the speed and accuracy of performance on the tests administered (see Appendix I I ) . Causal Relation A causal r e l a t i o n w i l l be said to ex i s t i f a change in one s k i l l i s accompanied by a change i n another. That i s , i f a change produced i n search procedure i s accompanied by a change i n arithmetic a b i l i t y then i t w i l l be said that there i s a causal r e l a t i o n between search procedure and arithmetic a b i l i t y . I I I . SIGNIFICANCE OF THE STUDY It w i l l be argued that some of the anomalies i n the results of several research studies on algorithmic performance may be accounted for by the consideration of the subjects per-ceptual s k i l l s - - t h a t these behaviours are si m i l a r to some of those considered i n reading research--that the l i t e r a t u r e c i t e d on the relat i o n s h i p of reading to arithmetic does not consider e x p l i c i t l y perceptual s k i l l s . I t i s upon these arguments that the s i g n i f i c a n c e of the study r e s t s . Algorithmic Performance Several studies which have t r i e d to i d e n t i f y s p e c i f i c d i f f i c u l t i e s i n algorithmic performance are related here. Withi some, perceptual d i f f i c u l t i e s are i m p l i c i t . Within others, p e r c e p t u a l d i f f i c u l t i e s are e x p l i c i t l y c o n s i d e r e d . None of the c i t e d s t u d i e s attempt to e x p e r i m e n t a l l y d e f i n e these per-c e p t u a l d i f f i c u l t i e s or t h e i r r e l a t i o n s h i p to a l g o r i t h m i c performance. Flournoy c o n s i d e r e d h i g h e r decade a d d i t i o n and 12 b r i d g i n g i n column a d d i t i o n . I t was found t h a t students were prone to count when numbers were arranged i n a column. T h i s study i n d i c a t e s t h a t the symbolic arrangement of the problem i n f l u e n c e d the c h i l d ' s method of performance. L e s s i n g e r i n v e s t i g a t e d d i s c r e p a n c i e s between a r i t h -metic achievement (low) and r e a d i n g (high) on mental age t e s t s He found t h a t because of f a u l t y r e a d i n g of symbols by students i n grades three through e i g h t , i n c o r r e c t o p e r a t i o n s were used and a l o s s of 6.1 months i n M.A. s c o r e r e s u l t e d . A f t e r the students were t r a i n e d i n r e a d i n g s k i l l s t h i s l o s s was reduced 13 to .7 months. The r e a d i n g program c o n s i s t e d of t r a i n i n g the students i n "...the a b i l i t y to focus a t t e n t i o n upon key words and on t e c h n i c a l s i g n s , the a b i l i t y to focus combinations i n 14 one eye pause and to read unimportant p a r t s h u r r i e d l y , e t c . " These s k i l l s are c l o s e to what are d e f i n e d above as search procedure and span together with a new s k i l l of i d e n t i f y i n g th important and unimportant s e c t i o n s of what was read. I t does not however i d e n t i f y any s k i l l s s p e c i f i c to a r i t h m e t i c p e r f o r -mance . The method of algorithmic performance, for example, adding upward or downward, has been examined i n order to f i n d the most e f f i c i e n t way. These studies are of i n t e r e s t primarily because some of the implications of these studies appear to have been overlooked. Brownell c i t e d a Scotish study which showed that a group taught downward addition and then upward addition were faster adding up and down but classes f i r s t taught upward addi-15 t i o n were superior on accuracy. Since the actual difference between the groups was small, he concluded that the s e l e c t i o n of methods used "ti.may be decided also on the basis of con-16 siderations other than speed and accuracy." A small difference could be expected because both groups had the same experiences with the exception of order. 17 Buckingham found that of 493 student teachers, 303 preferred to add up and 190 preferred the opposite. He suggested four p o s s i b i l i t i e s for preferring downward addition: f i r s t , i n downward addition the eye moves i n the d i r e c t i o n of ordinary reading; second, figures are commonly written downward; t h i r d , i n downward addition the eye successively fixes toward the point at which the answer i s written; fourth, i f the number of the columns are increased then more complex eye movements are needed i n upward addition."*"^ Buckingham, i n a l a t e r study which showed that grade two and three children who were taught down-ward addition achieved better results than those who were taught upward addition, made no attempt to experimentally re l a t e these findings to the reasons which he had proposed i n his previous a r t i c l e . A study by Cole on the same subject revealed more subtle differences i n behaviour. Subjects adding downward did so less rapidly but more accurately than those adding 20 upward. I t was also found that those counting objects to the l e f t were slower but more accurate than those counting to the r i g h t . These studies indicate that there i s some scanning process s i m i l a r to that taught i n reading which may make a difference i n the performance of algorithms. However, the following studies which re l a t e reading a b i l i t y to arithmetic a b i l i t y apparently neglected such perceptual s k i l l s . Reading and Arithmetic There are many studies dealing with the r e l a t i o n of reading s k i l l s and the a b i l i t y to solve word problems. Typical 22 studies have been done by Wilson, who suggested that acting out word problems gave greater meaning to the words read; 23 Stevens, who showed that tests of problem analysis seem to have higher correlations with problem solving than do tests of 24 25 general reading or of fundamental operations; Johnson, C a l l 2 6 and Pribnow, who demonstrated the p o s i t i v e r e l a t i o n between vocabulary and problem solving a b i l i t y . Those studies which attempted various methods of improving problem solving prescribed a c t i v i t i e s which would emphasize the r e l a t i o n between the words of the problem and the corresponding arithmetic operations. They do not however, attempt to t r a i n perceptual s k i l l s . Other studies which simply found a relat i o n s h i p between arithmetic and reading performance may have been accounting for that part of each common to general i n t e l l i g e n c e . I t i s possibly this which leads Ballow to conclude, That reading achievement would not be related to computational with the effects of i n t e l l i g e n c e controlled i s reasonable.... The subject need only read numbers and operation signs.27 This l a s t statement does not agree completely with Lessinger's findings c i t e d above, nor with a study done by Gilmary. She studied .the effects of remedial i n s t r u c t i o n i n both reading and arithmetic and found that when I.Q. was used 2 8 as a co-variate the former students made greater gains. Earp 1s examination of experimental studies dealing with teaching of reading s k i l l s i n mathematics did not f a c i l i -tate the development of teaching methods for teachers "...who continue to assert that most children have d i f f i c u l t y i n 29 reading content material." After c i t i n g Lessinger's study he concludes, "The teacher of mathematics at any l e v e l should also be a teacher of reading."^ 0 The above s t u d i e s have been c i t e d to show t h a t there i s a need f o r continued r e s e a r c h i n t o the r e l a t i o n s h i p between readi n g and a r i t h m e t i c - However, f u t u r e r e s e a r c h must account f o r p a r t i c u l a r p e r c e p t u a l d i f f i c u l t i e s which seem to have been i n d i c a t e d . I t i s c o n c e i v a b l e t h a t many of the t e s t s of r e a d i n g and a r i t h m e t i c a b i l i t i e s which have been used are so ge n e r a l as to mask the s p e c i f i c s k i l l s which need to be developed. I t i s f o r these reasons t h a t t h i s study i s s i g n i f i c a n t . I t has been e s t a b l i s h e d t h a t some of the p e r c e p t u a l s k i l l s needed may be s i m i l a r to those c o n s i d e r e d in.;the t e a c h i n g of r e a d i n g . I t w i l l now be necessary to examine the r e s e a r c h r e l a t e d t o those p e r c e p t u a l problems. IV. RELATED RESEARCH In the d i s c u s s i o n which f o l l o w s s t u d i e s t y p i c a l - o f r e s e a r c h i n t o the p e r c e p t u a l s k i l l s and t h e i r a s s o c i a t e d t r a i n i n g methods w i l l be examined. I t w i l l . b e shown t h a t although the e f f i c a c y of p e r c e p t u a l t r a i n i n g as a method of improving r e a d i n g performance has been questioned, many of the c o n f l i c t i n g r e s e a r c h r e s u l t s may be e x p l a i n e d i n terms of a l a c k of congruence between the s k i l l being taught and the t e s t used to e s t a b l i s h s k i l l performance. T h i s would imply t h a t percep-t u a l s k i l l s s p e c i f i c to the task d e s i r e d must be developed. Reading-Perception R e l a t i o n s h i p Glass c o r r e l a t e d r a t e of r e a d i n g with v a r i o u s academic, p e r s o n a l i t y , and p e r c e p t u a l measures. In h i s mul-t i p l e c o r r e l a t i o n a n a l y s i s he. found t h a t the three h i g h e s t beta weights were assigned to vocabulary (.431), r a t e of p e r c e p t i o n 31 (.221) and compulsiveness (.214). The v a l i d i t y of these 3 r e s u l t s c o u l d be que s t i o n e d i n t h a t a study done by M. Santoro showed t h a t v i s u a l p e r c e p t i o n t e s t s had a c o r r e l a t i o n of .438 with the O t i s Quick S c o r i n g Mental A b i l i t y t e s t s . G l a s s ' f i n d i n g s c o u l d then be i n t e r p r e t e d as p o s s i b l y j u s t measuring some p a r t of g e n e r a l i n t e l l i g e n c e which i s i n f l u e n c e d -by p e r c e p t i o n and r e a d i n g r a t e . A study done by Gates, however, c o n t r a d i c t s t h i s i n t e r p r e t a t i o n . Three p e r c e p t u a l t e s t s were g i v e n and compared to r e a d i n g s k i l l s . The t e s t s r e q u i r e d students to determine whether p a i r s of g e o m e t r i c a l f i g u r e s , d i g i t s , and words, were * d i f f e r e n t i n some s m a l l r e s p e c t . He found t h a t the t e s t with the g r e a t e s t r e l a t i o n to the r e a d i n g s k i l l s was the word d i f -f e r e n t i a t i o n t e s t . The c o r r e l a t i o n s between the word d i f f e r e n -t i a t i o n t e s t and s p e l l i n g , p r o n u n c i a t i o n , and s i l e n t r e a d i n g were .544, .555, and .69 r e s p e c t i v e l y with the i n f l u e n c e of 33 i n t e l l i g e n c e removed. He s t a t e d : These f i g u r e s imply t h a t the p e r c e p t i v e f a c t o r , i r r e s p e c t i v e of i n t e l l i g e n c e , i s more c l o s e l y a s s o c i a t e d with r e a d i n g and s p e l l i n g than a l l the other f u n c t i o n s embraced i n " i n t e l l i g e n c e " as measured.34 There i s evidence, then, that perceptual s k i l l s are related to reading s k i l l s . In addition Gates' study indicated that the perception of words rather than geometrical or numer-. i c a l differences i s linked with reading s k i l l s . Thus percep-tual s k i l l s may be quite s p e c i f i c to the behaviours tested. This i s borne out by analysis of attempts to improve reading through perceptual t r a i n i n g . Perceptual Training for Reading S k i l l s The present p o s i t i o n of reading teachers with respect to perceptual t r a i n i n g may be summed up by Witty. He conducted a remedial reading program for college students which consisted of reading with the assistance of a reading accelerator (a machine which forced the eye move more quickly over the printed page), materials of i n t e r e s t to the students, speed reading practice, tachistoscopic practice (ten minutes per session), vocabulary building, practice on reading s k i l l s , frequent testings, practice i n reading d i f f e r e n t kinds of materials, 35 frequent conferences, and help i n reading textbooks. Seven-teen years a f t e r this program he stated, ...at more advanced l e v e l s , pupils may derive some value from the enhanced i n t e r e s t r e s u l t i n g from the introduction of pacing devices. That such devices are necessary [emphasis not i n the o r i g i n a l ] for conducting a successful program has yet to be demonstrated.36 It should be noted that he did not state that pacing devices were useless but that they were just not "necessary." The implication i s that there may be alte r n a t i v e methods for developing the same s k i l l s . His own remedial program, for example, contained several methods of improving reading s k i l l s I t w i l l be argued that although there may be several ways of t r a i n i n g for perceptual s k i l l s , the a c q u i s i t i o n of these s k i l l s i s of importance. The following studies present c o n f l i c t i n g evidence for the importance of perceptual t r a i n i n g i n the a c q u i s i t i o n of reading s k i l l s . Grob used forced speeded o r a l reading rather than a pacing device because, by being pushed rapidly ahead the student's attention i s being focused, and this increases the accuracy of his perception.37 3 8 Freeburn examined the effects of perceptual span and perceptual speed upon reading a b i l i t y . The group was trained by systematically lengthening the phrases which the students were required to perceive and by decreasing the exposure times for the phrases during the reading period. He found that the span and speed had a c o r r e l a t i o n of .758 but that these perceptual s k i l l s and the results on reading were 39 not s i g n i f i c a n t l y r e l a t e d . Two studies done by Amble and Muehl on phrase t r a i n i by means of a tachistoscope had varying r e s u l t s . The f i r s t showed that the differences between the control and experimen-t a l groups on a measure of comprehension and on the Iowa Test of Basic S k i l l s were s i g n i f i c a n t at the .05 and .025 l e v e l s respectively. There was no difference between the groups i n 40 either rate of reading or vocabulary a c q u i s i t i o n . The results of the second study, however, indicated a two and one-half year gain on the Rate and Comprehension subtest of the Iowa Test 41 of Basic S k i l l s . This l a t t e r study then c o n f l i c t s with both the f i r s t study and that of Freeburn. 42 The F r o s t i g Visual Perceptual Training Program was 43 used by Jacobs i n his study on perception i n the primary grades The program consists of copying geometrical figures designed to give the student s k i l l s i n perceiving v e r t i c a l , horizontal and sloped l i n e s , and s p e c i f i c organized patterns. He found that although a l l grades improved, the higher grades improved the most. This was an unexpected r e s u l t i n that i t was suggested by Jacobs that the e a r l i e r grades would have shown the greatest increase because the t r a i n i n g was done at an e a r l i e r period of s k i l l a c q u i s i t i o n . A study, done by N. Santore, showed that i t was pos-s i b l e to d i r e c t the eyes from r i g h t to l e f t using the Controlled 44 Reader with Guided S l o t . The gains on reading achievement tests were markedly superior to those attained when the other methods of, Controlled Reader with no Guided Slot, Shadow scope, Rate-o-meter and timed reading were used. These results were maintained i n the retention test given several months l a t e r . I t i s i n t e r e s t i n g that one p a r t i c u l a r form of reading rate t r a i n i n g should be superior to a l l the others. Some of these apparent c b n f l i c t s may be resolved by a more ca r e f u l consideration of the above studies. Freeburn's study and the f i r s t study of Amble and Muehl agree that l i t t l e or no improvement i n reading resulted from perceptual t r a i n i n g . However, the second Amble and Muehl study contradicts that r e s u l t perhaps because of the more appropriate matching of material to the students a b i l i t i e s to read. That is , - t h e material matched the: performance behaviours of the students. Jacob's findings, that older children gained more than the younger, may be interpreted as supporting Bonsall and Dornbush who state, . . . i t appears that as the c h i l d matures, his reading a b i l i t y i s dominated by d i f f e r e n t functions; the exact course of these remains to be defined.4 5 In other words the F r o s t i g materials may better match the mental functions of the older c h i l d . Santore's findings that only one kind of device, namely the Controlled Reader with Guided Slot, produced s i g n i -f i c a n t improvement i n reading achievement further supports the present argument. There i s generaly agreement that i n reading a page of material the eye must go from the top l e f t hand corner to the bottom r i g h t hand corner of the page, and only the reader with guided s l o t trained this p a r t i c u l a r s k i l l . Santore conclu-des that the modification of performance s k i l l s were the most important e f f e c t s . ^ From these and previous considerations i t can be concluded that perceptual s k i l l s , analogous to those found i n reading, may be a s i g n i f i c a n t factor i n arithmetic computation, that there i s a r e l a t i o n between perceptual s k i l l s and reading achievement, and that these perceptual s k i l l s must be speci-f i c a l l y related to the behaviours required. As no analysis of the perceptual s k i l l s required i n arithmetic performance was found the following section w i l l present a discussion of the search procedure. V. DISCUSSION OF THE PROBLEM Arithmetic performance depends upon at least three a b i l i t i e s . . The student must know the steps of the algorithm and be able to perform them i n the proper sequence. He must either know or determine the basic number fa c t s . He must perceive the symbols involved i n the algorithm. I t i s this l a t t e r a b i l i t y which i s of i n t e r e s t i n this study. A discus-sion of the differences between the perceptual s k i l l s needed for word recognition and those needed for numeral recognition follows. An aid to the recognition of a word i s i t s shape. For example, the word "big" i s of d i f f e r e n t o v e r a l l shape than the word "bad." However, the numeral "279" i s of the same o v e r a l l shape as "384" and, i n fact, so are a l l three d i g i t numerals. Therefore, the student must be able to i d e n t i f y each s p e c i f i c d i g i t within the numeral i n order to be'able to read the numeral. In reading the word "idemnification" a l l the information i s present i n the order i n which i t i s used even i f the word i s broken down into s y l l a b l e s . On the other hand, in reading the numeral "378,521" i t i s f i r s t necessary to i d e n t i f y the number of d i g i t s and then, using this information, give,to each d i g i t the proper place value associated with i t . As well as lacking an i d e n t i f y i n g shape and requiring rescanning, the usual forms used for algorithmic working require even more complex s k i l l s . The horizontal writing of the i n d i -vidual numerals together with the fact that the performance of the algorithm proceeds from right, to. l e f t , might suggest that horizontal scanning and horizontal span should be considered i n the perceptual s k i l l s required. Likewise, the placement of the i n d i v i d u a l numerals i n the usual v e r t i c a l positions suitable for the algorithms may suggest that v e r t i c a l scanning and ver-t i c a l span should be considered. However, the required eye movements i n algorithmic per-formance, es p e c i a l l y for m u l t i p l i c a t i o n and d i v i s i o n obviate against any continuous scanning process, such as that required for reading a page of a novel. Further, the student must pick from the several numerals those d i g i t s with which he i s to operate. He must then r e t a i n the r e s u l t while searching for the next d i g i t or d i g i t s to combine with his r e s u l t , or trans-fer this r e s u l t to either i t s permanent or temporary p o s i t i o n required by the algorithmic procedure. This combination of scanning and s e l e c t i o n i s c a l l e d the search procedure. VI. STATEMENT OF THE PROBLEM The a b i l i t y to work algorithms may be increased by tr a i n i n g i n perceptual s k i l l s . Three have been i d e n t i f i e d : v e r t i c a l span, horizontal span, and search procedure. A preliminary study, as described i n Appendix IV, was performed for two purposes: to esta b l i s h standard t e s t procedures; to indicate the r e l a t i v e importance of the three perceptual s k i l l s i d e n t i f i e d . On the basis of a three week tr a i n i n g period i n which each of three groups was trained i n one of the perceptual s k i l l s , the search procedure t r a i n i n g seemed to produce the greatest improvements i n arithmetic achievements. Therefore, the primary problem i s , to what extent w i l l a period of t r a i n i n g i n the search procedure produce gains i n algorithmic perfor-mance? It i s quite conceivable that the changes so produced could be due to the increases of horizontal and v e r t i c a l span being developed during the search procedure t r a i n i n g session. Hence, two further questions are: Does v e r t i c a l span increase? and Does horizontal span increase? F i n a l l y , . t h i s study i s concerned with relationships between the various perceptual s k i l l s and arithmetic a b i l i t y . Therefore, a tes t for a li n e a r r elationship between arithmetic and each of the three perceptual s k i l l s as well as a test for a l i n e a r r elationship between algorithmic performance and search procedure with the effects of v e r t i c a l and horizontal span removed w i l l be made. VII. QUESTIONS TO BE ANSWERED The questions to be answered by thi s study are the following: Over a period of tra i n i n g does the search procedure improve? Over this same period do v e r t i c a l and horizontal spans improve? Over this period of t r a i n i n g does the speed and accuracy of aj-gorithmic performance improve? Is the search mechanism correlated with v e r t i c a l and horizontal span? Which of the perceptual s k i l l s i s more highly cor-related with algorithmic performance? Is the c o r r e l a t i o n between the search procedure and algorithmic performance s i g n i f i c a n t l y different;from the mul-t i p l e c o r r e l a t i o n -using algorithmic performance as the dependent variable and horizontal span, v e r t i c a l span and search procedure as the independent variables. Lowry W. Harding and Inez P. Bryant, "An E x p e r i -mental Comparison of D r i l l and D i r e c t Experience i n A r i t h m e t i c , L e a r n i n g i n Fourth Grade," J o u r n a l of Ed u c a t i o n Research, 37:321, January, 1944. 2 J.C. Brown, "An I n v e s t i g a t i o n of D r i l l Work i n the Fundamental Operations of A r i t h m e t i c , " J o u r n a l of E d u c a t i o n a l  Psychology, 2: 81-88, February, 1911. 3 J.C. Brown, "An I n v e s t i g a t i o n on the Value of D r i l l Work i n the Fundamental Operations of A r i t h m e t i c , " J o u r n a l of  E d u c a t i o n a l Psychology, 3: 485-492, November, 1912. 4 F . J . K e l l y , "The R e s u l t s of Three Types of D r i l l on the Fundamentals of A r i t h m e t i c , " J o u r n a l of E d u c a t i o n a l Research, 22: 381, November, 1920. 5 C.L. Kulp , "Study of the R e l a t i v e E f f e c t i v e n e s s of Two Types of Standard A r i t h m e t i c P r a c t i c e M a t e r i a l s , " J o u r n a l  of E d u c a t i o n a l Research, 22: 381-387, December, 1930. W i l l i a m A. Brownell and C h a r l o t t e B. Chazal, "The E f f e c t s of Pre-mature D r i l l . i n T h i r d Grade A r i t h m e t i c , " J o u r n a l of E d u c a t i o n a l Research, 29: 19, September, 1935. 7 I b i d . , p. 22. g I b i d . , p. 26. 9 S. Howard B a r t l e y , " P e r c e p t i o n , " E n c y c l o p e d i a of  E d u c a t i o n a l Research, Robert L. E b e l , ed., ( f o u r t h ed., London: C o l l i e r - M a c M i l l a n Co. 1969), p. 929. ~*"^ F. A l p o r t , T h e o r i e s of P e r c e p t i o n and the Concept  of S t r u c t u r e , (New York: John Wiley and Sons Inc., 1955). i : L I b i d . , p. 14 12 Frances Flournoy, "A C o n s i d e r a t i o n of the Ways C h i l d r e n Think when Performing Higher-Decade A d d i t i o n , " Elementary School J o u r n a l , 57: 204-208, January, 1957. W.E. L e s s i n g e r , "Reading D i f f i c u l t i e s i n A r i t h m e t i c Computation," J o u r n a l of E d u c a t i o n a l Research, 11: 288, A p r i l , 1925. 1 4 I b i d . , p. 289. 15 W.A. B r o w n e l l , R.A. Doty and W.C. Rien, " A r i t h m e t i c i n Grades I and I I , " Duke U n i v e r s i t y Research S t u d i e s i n E d u c a t i o n No. 6 (Dunbar, North C a r o l i n a : Duke U n i v e r s i t y P r e s s , 1941), p. 151, c i t i n g S t u d i e s i n A r i t h m e t i c , V o l . I, ( S c o t i s h C o u n c i l f o r Research i n E d u c a t i o n , X I I I ; London: U n i v e r s i t y of London Press L t d . , 1939) . ^ ^ I b i d . , p. 151. 17 B.R. Buckingham, "Adding Up or Down: A D i s c u s s i o n , " J o u r n a l of E d u c a t i o n a l Research, 12: 254, November, 1925. 1 8 I b i d . , p. 250. 19 B.R. Buckingham, "Upward versus Downward A d d i t i o n , " J o u r n a l of E d u c a t i o n a l Research, 16: 315-322, December, 1927. 20 Laurence E.W. Cole, "Adding Upward and Downward," J o u r n a l of E d u c a t i o n a l Psychology, 3: 90, February, 1912. 2 1 I b i d . , p. 93. 22 E s t a l i n e Wilson, "Improving the A b i l i t y to Read A r i t h m e t i c , " Elementary School J o u r n a l , 22: 380-386, January, 1922. 23 B.A. Stevens, "Problem S o l v i n g i n A r i t h m e t i c , " J o u r n a l of E d u c a t i o n a l Research, 25: 80, April-May, 193 2. 24 Harry C. Johnson, "The E f f e c t of I n s t r u c t i o n i n Mathematical Vocabulary upon Problem S o l v i n g i n A r i t h m e t i c , " J o u r n a l of E d u c a t i o n a l Research, 37: 97-110, October, 1944. 25 R.J. C a l l and M.A. Wiggm, . "Reading and Mathematics," The Mathematics Teacher, 59: 149-157, February, 1966. 2 6 J.R. Pribnow, "Why Johnny Can't Read Word Problems," School Science and Mathematics, 69: 591-598, October, 1969. Irving H. Balow, "Reading and Computation A b i l i t y as Determinants of Problem Solving," The Arithmetic Teacher, 11: 18, January, 1964. 2 8 S i s t e r Gilmary, "Transfer E f f e c t s of Reading Remed-i a t i o n to Arithmetic Computation when Intelligence i s Controlled and a l l Other School Factors are Eliminated," The Arithmetic  Teacher, 14: 17-20, January, 1967. 29 N. Wesley Earp, "Observations i n the Teaching.of Reading i n Mathematics," Journal of Reading, 13: 529, A p r i l , 1970. 30 Ibid., p. 531. 31 G.G. Glass, "Rate of Reading: A Correlation and Treatment Study," Journal of Reading, 11: 173, December, 1967. 32 Roseann Marie Santore, "The Relation of Reading Achievement to S p e c i f i c Measures of Visual Perception and Intelligence," (unpublished Doctoral Dissertation, Fordham University, New York, 1967). 33 A.I. Gates, "A Study of the Role of Vis u a l Perception Intelligence, and Certain Associative Processes i n Reading and Spel l i n g , " Journal of Educational Psychology, 17: 441, October, 1926. 3 4 I b i d . , p. 442. 35 Paul Witty, Theodore Stolarz and William Cooper, "Some Results of a Remedial Reading Program for College Students, School and Society, 76: 378-379, December, 1952. 3 6 Paul Witty, "Rate of Reading - A C r i t i c a l Issue,"-Journal of Reading, 13: 106, November, 1969. 37 J.A.. Grob, "Forcing Speed i n Oral Reading," Journal of Reading, 11: 624, May, 1968. C e c i l Max Freeburn, "The Influence of Perceptual Span and Perceptual Speed upon Reading A b i l i t y , " Journal of  Educational Psychology, 40: 334, October, 1949. Ibid., p. 340. B.R. Amble and S. Muehl, "Perceptual Span Training and Reading Achievement of School Children," Journal of  Educational Psychology, 157: 196-197, August, 1966. 41 B. R. Amble andS. Muehl, "Phrase Reading Training and Reading Achievement: A Replication Study," Journal of  Experimental Education, 35: 98, Winter, 1966. 42 Marianne F r o s t i g , D. Welty Lefever, and John R.B. Whittlesey, "A Developmental Test of Visual Perception for Evaluating Normal and Neurologically Handicapped Children," Perceptual and Motor S k i l l s , 12: 383-394, June, 1961. 43 J.N. Jacobs, "Evaluation of the F r o s t i g Visual Perceptual Training Program,"-Educational Leadership, 25: 332-340, January, 1968. 44 Nicholas Santore and Austin Gelzer, "A Comparison of Various Reading Improvement Approaches," Journal of Educational Research, 61: 269, February, 1968. 4 5 C. Bonsall and R.L. Dornbush, "Visual Perception and Reading A b i l i t y , " Journal of Educational Psychology, 60: 294, August, 1969. 46 Santore and Gelzer, op. c i t . , p. 271. CHAPTER II RESEARCH DESIGN I. EXPERIMENTAL DESIGN Selection of Classes and Assignment of Treatments Two schools were selected from the Vancouver, B r i t i s h Columbia School System. They were considered to be a random choice as far as treatment e f f e c t s were concerned. Each school had two grade seven classes, both of which were taught by the same teacher. In order to control the effects of a b i l i t y , the teachers were asked to select t h e i r class which performed the best i n arithmetic. The experimental treatment was randomly assigned to one of the two high a b i l i t y groups and to the low a b i l i t y group i n the other school. Thus i n one school the higher a b i l i t y group was the experimental and the lower a b i l i t y the control and i n the other the order was reversed. The Hawthorne effects were controlled i n the following ways: f i r s t , both experimental and control groups were told they were part of an experiment; second, each group i n the same school was taught by the same teacher; t h i r d , each group was taught t h e i r regular classwork with the overhead projector being used i n place of the chalk-board (this l a t t e r being a novel s i t u a t i o n for both groups). Method of T r a i n i n g The s u b j e c t m a t e r i a l d u r i n g t h i s p e r i o d c o n s i s t e d of geometry and a review of some of the g e n e r a l concepts covered d u r i n g the ye a r . The teachers were asked to s e l e c t t o p i c s which d i d not i n v o l v e much c a l c u l a t i o n . The c o n t r o l group was g i v e n no s p e c i a l t r a i n i n g . The experimental group r e c e i v e d a f i v e t o ten minute t r a i n i n g s e s s i o n each scheduled hour of a r i t h m e t i c over a p e r i o d of three weeks w h i l e c o n t i n u i n g t h e i r r e g u l a r work. T r a i n i n g f o r the search mechanism was accomplished by p r o j e c t i n g an 11 x 11 matrix of d i g i t s onto a scre e n w i t h an overhead p r o j e c t o r . The teacher read a sequence of d i g i t s from a g i v e n row, column, or d i a g o n a l and as the students i d e n t i f i e d the sequence they r e c i t e d along with the teacher. A complete d e s c r i p t i o n of the method of t r a i n i n g together with examples appears i n Appendix I. Data A c q u i s i t i o n There were p r e - and p o s t - t e s t s g i v e n on v e r t i c a l and h o r i z o n t a l span, search procedure, and a r i t h m e t i c speed and accuracy. The Form I to Form I I c o r r e l a t i o n s f o r the t e s t s were .76, .62, .81, and .65. A complete d e s c r i p t i o n of the t e s t c o n s t r u c t i o n and a d m i n i s t r a t i o n appears i n Appendix I I . To counter any syst e m a t i c e f f e c t s of the two p a r a l l e l forms of the a r i t h m e t i c t e s t the forms were c r o s s e d over the experimental-controls groups as can be seen from Table I. TABLE I PLAN OF THE EXPERIMENTAL DESIGN AND ADMINISTRATION OF ARITHMETIC TESTS School Group I II A b i l i t y l e v e l Higher Lower Experimental Pre-test Form I Form II Post-test Form II Form I Control A b i l i t y l e v e l Pre-test Post-test Lower Form II Form I Higher Form I Form II The perceptual tests were given as a group i n one period. In order to compensate for any s p e c i f i c learning associated with any of the tests the order of administration was changed from pre-test to post-test. I I . STATISTICAL DESIGN This study i s i n two parts. The f i r s t . i s designed to e s t a b l i s h whether or not there i s a causal r e l a t i o n s h i p between t r a i n i n g i n the search procedure on the one hand and v e r t i c a l span, horizontal span, and arithmetic ""computational a b i l i t y on the other.. I t w i l l be referred to as Analysis of Variance. The second part i s designed to t e s t for the existence of l i n e a r relationships between the various perceptual s k i l l s and computational a b i l i t y . I t w i l l be referred to as Correla-t i o n a l Analysis. Each of these headings refers to the s t a t i s -t i c a l tests made i n each. In both sections questions raised i n Chapter I w i l l be stated i n n u l l hypothesis form and a description of the appropriate tests w i l l be given. Analysis of Variance Hypotheses There i s no s i g n i f i c a n t difference between the control and the experimental:groups on the measure of the change i n search a b i l i t y . There i s no s i g n i f i c a n t difference between the control and experimental groups on the measure of change i n v e r t i c a l span. There i s no s i g n i f i c a n t difference between the control and the experimental groups on the measure of change i n h o r i -zontal span. There i s no s i g n i f i c a n t difference between the control and the experimental groups on the measure of change i n a r i t h -metic computational a b i l i t y . For convenience the tests of perceptual s k i l l s and arithmetic are numbered as follows: (1) search procedure, (2) v e r t i c a l span, (3) horizontal span, and (4) algorithmic perfor-mance. The groups are labeled (E) experimental and (C) c o n t r o l . The symbolic n u l l hypotheses are: H : E l - CI = 0 U l H : ,E3 - C3 = 0 U3 H : E2 - C2 = 0 U2 H : E4 - C4 = 0 U4 The experimental and c o n t r o l groups both c o n s i s t e d of two c l a s s e s of st u d e n t s . Since i t was not p o s s i b l e to ran-domly s e l e c t students to a s s i g n to each of the groups a nested d e s i g n w i t h c l a s s e s randomly chosen was employed."'" As the treatment e f f e c t s were random the term D i n the e x p r e s s i o n f o r the expected mean squares of the treatment e f f e c t (A i n Table II) approaches 1. T h e r e f o r e , the denominator f o r the F , . i s the pooled mean squares of the e f f e c t s of the c l a s s e s r a t i o c ^ (B i n Table I I ) . TABLE I I * SOURCES OF VARIATION, DEGREES OF FREEDOM, AND EXPECTED MEAN SQUARES USED FOR THE ANALYSIS Source of v a r i a t i o n df E (MS) A p-1 2 2 2 o„ + nD an + nqa E q 3 ^ a B (pooled) P(q-D 2 ^ 2 a E + n 3 Experimental e r r o r ( w i t h i n c e l l s ) pq(n-1) 2 a E *Reproduced from Winer, op. c i t . , p. 18 5. The tests of search mechanism, horizontal and v e r t i c a l span and arithmetic computation were independent measures and therefore the analysis outlined above was used to t e s t the difference between the experimental and control groups on each measure (hypotheses one, two, three, and four). The l e v e l of si g n i f i c a n c e was set at .05 for a l l tests. The F . • 3 p r o b a b i l i t i e s were.calculated by a standard program on the University of B r i t i s h Columbia computer. Cor r e l a t i o n a l Analysis Hypotheses There i s no s i g n i f i c a n t c o r r e l a t i o n between the tests of the search procedure and v e r t i c a l span. There i s no s i g n i f i c a n t c o r r e l a t i o n between the tests of the search procedure and horizontal span. There i s no s i g n i f i c a n t c o r r e l a t i o n between the tests of horizontal and v e r t i c a l span. There i s no s i g n i f i c a n t difference between the cor-r e l a t i o n of the search procedure with arithmetic a b i l i t y and the multiple c o r r e l a t i o n found by using algorithmic a b i l i t y . and the dependent variable and search procedure, v e r t i c a l span and horizontal span and the independent variables. Using the same notation as i n the previous section the hypotheses i n symbolic form are as follows: H : r = 0 H : r = 0 U5 C12 U7 C23 H : r =0 H : r =0 u6 ^13 u8 ^1(23)4 The correlations r e l a t i n g to hypotheses f i v e through eight were calculated only on the f i n a l r esults of the control group. The f i n a l tests were used because the students were capable of f a m i l i a r i t y with the testing procedure. The experi-mental group results were excluded i n case the treatment had any e f f e c t which would bias the c o r r e l a t i o n s . 2 The formula r N-2 t = — 1-r xy was used to transform the correlations into an approximate t-score with N-2 degrees of freedom. This was then used to test the n u l l c o r r e l a t i o n a l hypotheses. The l e v e l of s i g n i f i c a n c e was set at .05 for a l l tests. The F , , . -. . , . were computed p r o b a b i l i t i e s c by a standard program on the University of B r i t i s h Columbia computer. Hypothesis eight was tested i n the following way. A multiple c o r r e l a t i o n was calculated for v e r t i c a l and horizontal span and the search measures onto arithmetic scores. A second c o r r e l a t i o n was calculated ignoring the measure of search mech-anism. The s i g n i f i c a n c e of the difference i n the variances was-calculated by a computer program using the method of -Bottenberg and Ward. The technique i n v o l v e s the comparison of the v a r i a n c e accounted f o r by one m u l t i p l e c o r r e l a t i o n and the v a r i a n c e accounted f o r by another w i t h some independent v a r i a b l e s e x c e p t i n g the v a r i a b l e of i n t e r e s t . I I I . ASSUMPTIONS AND LIMITATIONS Study L i m i t a t i o n s The study has l i m i t a t i o n s r e s u l t i n g from the f o l l o w i n g c o n s i d e r a t i o n s . An optimum t r a i n i n g program may not have been developed because an e a s i l y f o l l o w e d program would be more l i k e l y . a c c e p t e d i n the classroom. The angle which the t r a i n i n g and t e s t i n g m a t e r i a l subtended was not a constant f o r a l l c h i l d r e n , but, by keeping the s e a t i n g arrangement the same from t e s t i n g to t e s t i n g the angle was kept constant f o r each student. The angle a t which the t r a i n i n g m a t e r i a l and percep-t u a l t e s t s were presented bore no r e l a t i o n to the angle a t which the students p e r c e i v e d t h e i r w r i t t e n a r i t h m e t i c t e s t s . The l i g h t i n g c o n d i t i o n s on a c h i l d ' s desk were q u i t e d i f f e r e n t from those of a p r o j e c t i o n screen.. Because the sequences were presented a u r a l l y t o , and then sought out v i s u a l l y by the s u b j e c t there may have been a confounding of the p e r c e p t u a l p r o c e s s e s . That i s , the c h i l d does not hear the number he i s to look f o r when he i s working an a r i t h m e t i c a l g o r i t h m . Thus the t r a i n i n g was not completely analogous to the search process used i n computation. The time p e r i o d may have been somewhat too s h o r t to d e t e c t even a r e l a t i v e l y low order change. A l s o the t r a i n i n g was l i m i t e d to one procedure. Perhaps some combination of t r a i n i n g procedures f o r the three p e r c e p t u a l s k i l l s c o u l d produce g r e a t e r change i n computation. The type of t e s t i n g m a t e r i a l was chosen to i n d i c a t e some change i n computational a b i l i t y . No attempt was made to t e s t other aspects such as the s o l u t i o n of problems r e q u i r i n g e x t e n s i v e r e a d i n g and t r a n s f e r r a l of i n f o r m a t i o n and numbers. For example, word problems o f t e n r e q u i r e r e w r i t i n g on a p i e c e of work paper e i t h e r the problem or an a s s o c i a t e d a l g o r i t h m o r both. S t a t i s t i c a l Assumptions Another s e t of assumptions made i n v o l v e the s t a t i s t i c s chosen f o r the study., The f i r s t i s t h a t the s t a t i s t i c a l e r r o r terms are independent. I t i s c l e a r t h a t as the students c o u l d not be chosen randomly there c o u l d be a s y s t e m a t i c v a r i a n c e due to being i n the same c l a s s f o r a year. I t was f o r t h i s reason t h a t the nested design was chosen. I t i s t h e r e f o r e assumed t h a t the e r r o r s are independent. The second i s the assumption of n o r m a l i t y . A, histogram was generated on a l l the scores and, as no gross departures from n o r m a l i t y o c c u r r e d , the assumption was accepted, The t h i r d assumption i s t h a t the v a r i a n c e s are homo-geneous. As the t e s t f o r s i g n i f i c a n c e i n v o l v e d an F ^. which r a t i o i s ,robust w i t h r e s p e c t to v i o l a t i o n of t h i s assumption when 4 sample s i z e s are equal, no t e s t was made. Footnotes - Chapter II B.J. Winer, S t a t i s t i c a l P r i n c i p l e s i n Experimental  Design, (New York: McGraw-Hill Book Company, Inc., 1962), p. 184. 2 William L. Hays, S t a t i s t i c s , (New York: Holt, Rinehart and Winston, 1963), p. 529. 3 R.A. Bottenberg and J.H. Ward, Applied Linear  Multiple .Regression, U.S. O f f i c e of Technical Services Technical Report PRL-TDR-63-6 (Washington: Department of Commerce, 1963). Rodger E. Kirk, Experimental Design: Procedures  for the Behavioral Sciences, ( C a l i f o r n i a : Brooks/Cole Publishing Company, 1968), p. 61. R E S U L T S O F T H E S T U D Y A s i n t h e p r e c e d i n g c h a p t e r i t w i l l b e c o n v e n i e n t t o p r e s e n t t h e a n a l y s i s o f d a t a i n t w o s e c t i o n s , o n e d e a l i n g w i t h q u e s t i o n s t o b e t e s t e d b y a n a l y s i s o f v a r i a n c e a n d t h e o t h e r w i t h c o r r e l a t i o n a l a n a l y s i s . T h e s e t w o s e c t i o n s r e s p e c -t i v e l y c o r r e s p o n d t o q u e s t i o n s w h i c h a t t e m p t t o e s t a b l i s h a c a u s a l r e l a t i o n s h i p a n d t h o s e w h i c h d e s c r i b e . I . A N A L Y S I S O F V A R I A N C E H Y P O T H E S E S T h e m e a n s a n d s t a n d a r d d e v i a t i o n s o n t h e p r e - a n d p o s t - t e s t o f e a c h m e a s u r e a r e p r e s e n t e d i n T a b l e s I I I a n d I V r e s p e c t i v e l y . G r o u p s I a n d I I I a r e f r o m o n e s c h o o l ; g r o u p s I I a n d I V a r e f r o m t h e o t h e r . T a b l e V c o n t a i n s t h e m e a n s o f t h e d i f f e r e n c e s c o r e s w h i c h w e r e u s e d t o c a l c u l a t e t h e c h a n g e o v e r t h e p e r i o d o f t h e e x p e r i m e n t i n e a c h g r o u p . I t i s t h e s e m e a n s w h i c h a r e u s e d i n t h e a n a l y s i s o f v a r i a n c e . T e s t s o f S i g n i f i c a n c e U s e d I n e a c h o f t h e f o u r h y p o t h e s e s t r e a t e d b y a n a l y s i s o f v a r i a n c e a n e s t e d d e s i g n w a s u s e d ( s e e C h a p t e r I I , s e c t i o n V , A n a l y s i s o f D a t a ) . T h e r e f o r e t h e . F , . w a s c a l c u l a t e d u s i n g J r a t i o ^ t h e m e a n s q u a r e s a s s o c i a t e d w i t h , t h e t r e a t m e n t i n t h e n u m e r a t o r and the mean squares a s s o c i a t e d with the c l a s s nested w i t h i n the treatment f o r the denominator. The degrees of freedom a s s o c i a t e d w i t h the numerator and denominator were 1 and 2 r e s p e c t i v e l y . As the alpha l e v e l chosen f o r each t e s t was .05 the F r a t i o r e c 3 u i r e d f o r s i g n i f i c a n c e was 18.5. For the d e t a i l e d c a l c u l a t i o n s see Appendix I I I . TABLE I I I MEANS OF ALL TEST ADMINISTRATIONS Group C o n t r o l Experimental T e s t I I I I I I i v Search pre 7. 04 9. 93 6. 67 11. 81 mechanism p o s t 7. 93 10. 44 11. 74 15. 85 V e r t i c a l pre 10. 78 11. 96 10. 07 11. 78 span p o s t 11. 07 11. 56 10. 33 11. 70 H o r i z o n t a l pre 12. 19 15. 33 10. 04 13 . 81 span p o s t 12. 62 11. 96 12. 25 11. 48 A r i t h m e t i c pre 15. 26 17. 96 12. 22 19. 19 computation p o s t 17. 26 17. 30 14. 25 18. 74 STANDARD DEVIATIONS OF ALL TEST ADMINISTRATIONS Group Control Experimental Test I • II III IV Search mechanism pre post 5.61 6. 18 5.13 5. 85 4.42 5.51 5.80 4.60 V e r t i c a l span pre post 4.63 3.90 3.77 3.92 3.40 3.25 3.38 4.00 Horizontal span pre post 4.62 5.12 5.02 5.09 3.60 3.64 5.64 4.82 Arithmetic computation pre post 5.22 5.73 5.75 4.88 5.77 5.69 4.52 4.52 TABLE V MEANS OF THE DIFFERENCE SCORES FOR THE FOUR GROUPS ON EACH OF THE FOUR MEASURES Group Test I II Com-bined III IV Com-bined Search mechanism -0.94 -0. 23 -0.77 -5. 26 -4.38 -4.8 V e r t i c a l span -0.31 0.42 0. 06 -0.27 0.15 -0.06 Horizontal span -0.46 3.2 1.35 -2.3 2.5 0.96 Arithmetic computation -2.1 0.69 -0. 69 -2.2 0.46 -0.85 Note: As the differences were calculated by taking the pos test scores from the pre-test scores a negative valu indicated an improvement. Hypothesis I Hypothesis I was that there would be no s i g n i f i c a n t difference between.the*experimental and control groups on a measure of change i n search a b i l i t y . The summary of the analy-sis of variance i s i n Table VI. On the basis of these results the n u l l hypothesis was rejected. Because the means of the experimental groups were s i g n i f i c a n t l y more negative than those of the control i t was concluded that the experimental groups improved more on the search mechanism than did the control . TABLE VI SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS I Source ss df MS Treatment 428.09 1 428.09 Classes within treatment 11.40 2 5.70 C e l l 1567.27 100 15.67 Total 2006.76 103 F , ,, '75.10 = .0225 prob.(1,2) r a t i o = 75.10 Hypothesis II Hypothesis II was that the experimental group would not b e , s i g n i f i c a n t l y d i f f e r e n t from the control group on a measure of change i n v e r t i c a l span. The summary of the analy-s i s of variance i s to be found i n Table VII. TABLE VII SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS II Source ss df MS Treatment 0.35 1 0.35 Classes within treatment 4.46 2 2. 23 C e l l 713.19 100 7.13 Total 718 .00 103 F . 0\ -1568 = .7229 prob(1,2) On the basis of these results the n u l l hypothesis was not rejected. There was no s i g n i f i c a n t difference between the control and the experimental groups on a measure of v e r t i c a l span. Hypothesis III Hypothesis III was that the experimental group would not be s i g n i f i c a n t l y d i f f e r e n t from the control group on a F , . = .1568 ratxo measure of change of horizontal span. Table VIII i s a summary of the analysis of variance for this hypothesis. On the basis of these results the n u l l hypothesis was not rejected. That i s , there was no s i g n i f i c a n t difference between the control groups and the experimental groups on a measure of change i n horizontal span. TABLE VIII SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS III Source ss df MS Treatment 42.34 1 42.34 Classes within Treatment 474.58 2 237.79 C e l l 1180.54 100 11.81 Total 1696.46 103 F , . = .178 F . O N.178 = .7079 r a t i o prob(l,2) Hypothesis IV Hypothesis IV stated that there would be no s i g n i f i c a n t difference between the control and experimental groups on a measure of change i n arithmetic computation. Table IX summaries the Analysis of Variance for this hypothesis. These results show that the n u l l hypothesis was not to be rejected. There was no s i g n i f i c a n t difference between the control and experimental groups on t h i s measure. TABLE IX SUMMARY OF ANALYSIS OF VARIANCE FOR HYPOTHESIS IV Source ss df MS Treatment .61 1 .61 Classes within treatment 188.62 2 24.31 C e l l 1129.23 100 11.29 Total 1318.46 103 F .. = .00636 F . ,, „, .00636 = .9013 r a t i o prob(l,2) I t should be noted that the F of hypotheses ra t i o s J t r I I , I I I , and IV are less than 1.0. F .. less than one may r a t i o s J indicate that the assumptions underlying the F d i s t r i b u t i o n are being v i o l a t e d . However, the p r o b a b i l i t y that a n F r a t ^ 0 would be below the ones calculated i s just 1-F , and-in this case J prob. i s above the l e v e l of s i g n i f i c a n c e established for the experi-ment. For example the lowest F. . . i s that associated with c r a t i o hypothesis IV (.00636). The calculated F p r o b i s .901. There-fore the p r o b a b i l i t y of getting an F r a t j _ 0 below this i s 1.0 -.901 = .099 which i s above the .05 l e v e l . The hypothesis, that the assumptions underlying the F d i s t r i b u t i o n are being v i o l a t e d , i s rejected. I I . CORRELATIONAL ANALYSIS The correlations for hypothesis V, VI and VII, together with the generated t-scores and p r o b a b i l i t i e s , are presented i n Table X. Hypothesis VIII w i l l be accorded a section i t s e l f because of the analysis used. For the equation used to transform the correlations to t-scores see Chapter I I , section V, Analysis of Data. TABLE X TABLE OF CORRELATIONS AND SIGNIFICANCE LEVELS Hypothesis Correlation t-score P r o b a b i l i t y : r 1 2 = 0 0.3697 7 .915 0.0069 : r 1 3 = 0 0 . 2460 3.219 0.075 : r 2 3 = 0 0.7604 68.553 0.000 Note: 1,2 and 3 denote search mechanism, v e r t i c a l span and horizontal span respectively. Only the post-test scores were used i n the calculations of the correlations be-cause i t was desired that the subjects become acquainted with the testing procedures. Product Moment Correlations The correlations reported were calculated by a stan-dard computer program. The p r o b a b i l i t i e s were calculated by the same program as was used f o r the F , . The 3 p r o b a b i l i t i e s l e v e l of 'significance reported i s a one-tailed p r o b a b i l i t y . Hypothesis V stated that the search mechanism would not be correlated with . v e r t i c a l span. A si g n i f i c a n c e l e v e l of .0069 was achieved and therefore the n u l l hypothesis was rejected. Hypothesis VI stated that the search mechanism would not be correlated with horizontal span. The l e v e l of s i g n i f i -cance was .075 and therefore the n u l l hypothesis was accepted. Hypothesis VII was that the v e r t i c a l span would not be correlated with the horizontal span. This n u l l hypothesis was rejected with a confidence l e v e l beyond .01. Multiple Correlation The f i n a l n u l l hypothesis to be tested was that search mechanism would not be related to arithmetic computation with the effects of horizontal and v e r t i c a l span removed. S p e c i f i c a l l y , i n t h i s case a multiple c o r r e l a t i o n was calculated with search procedure, horizontal span and v e r t i c a l span being the independent variables and arithmetic computation being the dependent vari a b l e . This c o r r e l a t i o n was .325. Then a second c o r r e l a t i o n with horizontal and v e r t i c a l span was c a l -culated. I t was .103. An F .. between the variances r a t i o accounted for the two equations was then calculated. The r e s u l t was 5.186 with a p r o b a b i l i t y of .027. The n u l l hypo-thesis was therefore rejected. That i s , the amount of addi-t i o n a l variance accounted for by the search mechanism score was s i g n i f i c a n t . I I I . INTERPRETATION OF RESULTS Introduction This study i s divided into two parts: causal ques-tions and r e l a t i o n a l questions; or, questions derived from past experimentation and questions to a s s i s t future experimentation. As these questions are answered by analysis of variance and c o r r e l a t i o n a l analysis, this section w i l l treat these separately. This section i s divided into two parts: Analysis of Variance and C o r r e l a t i o n a l Analysis. Analysis of Variance Hypothesis Null hypothesis I was rejected. There i s , then, a way i n which the teacher, using e a s i l y obtainable materials, can successfully t r a i n for search a b i l i t y as defined i n this study (see Chapter I, section I I , D e f i n i t i o n s ) . However, n u l l hypothesis IV, that such t r a i n i n g would also have no e f f e c t on arithmetic computation as defined, was not rejected. There-fore, i t i s to be concluded that t h i s perceptual t r a i n i n g prog-ram was unsuccessful i n increasing the a b i l i t y to work a l g o r i -thms. A causal l i n k was not established. As n u l l hypotheses I I and I I I , which s t a t e d t h a t v e r t i c a l and h o r i z o n t a l span would not be changed by the t r a i n i n g p e r i o d , were not r e j e c t e d i t c o u l d be concluded t h a t t r a i n i n g i n the search s k i l l s has no e f f e c t on v e r t i c a l and h o r i z o n t a l span. More w i l l be s a i d i n the s e c t i o n on the c o r r e l a t i o n a l a n a l y s i s . The c o n c l u s i o n ithat search t r a i n i n g has no e f f e c t upon a r i t h m e t i c computation must be approached w i t h c a u t i o n . The author c o n s t r u c t e d the t e s t s f o r a s p e c i f i c purpose and t h e i r c ontent d e f i n e s what was meant by a r i t h m e t i c computation. Examination of the t e s t s . w i l l r e v e a l t h a t the problems were of a most elementary type. A l s o , the t e s t s were b u i l t on the assumption t h a t the qu e s t i o n s were a measure of s k i l l and speed as evidenced by. the pre-experiment t e s t v a l i d a t i o n . However, there was an i n d i c a t i o n t h a t t h i s was not a t e s t of speed f o r the experimental.groups. One of the teachers mentioned t h a t about h a l f of h i s students were f i n i s h e d the t e s t s b e f o r e the end of the assigned time l i m i t . There i s c i r c u m s t a n t i a l evidence then, t h a t t h i s may not have been a t e s t of speed and t h a t the students were abl e to go back and made c o r r e c t i o n s of mistakes. Thus the t e s t was not as speeded as expected. Again, the a r i t h m e t i c t e s t s were of a s p e c i f i c type and the evidence of the p i l o t study i n d i c a t e s t h a t there may be more complex te s t situations i n which such a s k i l l as the search mechanism might be of importance. Co r r e l a t i o n a l Analysis Hypotheses The following discussion compares the correlations of v e r t i c a l span to search, and horizontal span to search, and then associates these with the r e s u l t s of the analysis of variance. I t was shown that the search mechanism was s i g n i f i -cantly correlated with arithmetic computation. I t was also shown that the search t r a i n i n g program did not produce any change i n arithmetic computation. One might conclude that these were two phenomena which did co-exist but had no causal r e l a t i o n s h i p . However, i t was also shown that v e r t i c a l span was correlated with,search mechanism and hence an a l t e r n a t i v e conclusion may be proposed: that the increase i n search a b i l i t y was caused by the use of v e r t i c a l span upon a new task, namely the performance on the search procedure t e s t . This conclusion i s further strengthened by,noting that the multiple c o r r e l a t i o n of v e r t i c a l and horizontal span to arithmetric computation was .1. Hence i t could be concluded that there was no new s k i l l taught. Since there i s l i t t l e c o r r e l a t i o n between ver-t i c a l span and the performance of algorithms an increase i n the use of span on the search procedure test should not neces-s a r i l y increase arithmetic a b i l i t y . This, i n f a c t , was the r e s u l t of the analysis of variance. Further, because the s i g n i f i c a n t c o r r e l a t i o n between search mechanism and a r i t h -metic computation was calculated on the control group only, i t i s possible to suggest that the search procedure i s an already established mechanism and that the t r a i n i n g period was too short to establish a detectable increment i n a l g o r i t h -mic performance. CONCLUSIONS AND HYPOTHESES FOR FURTHER RESEARCH I. CONCLUSIONS Part of this study involved an attempt to determine whether or not there exists a causal r e l a t i o n s h i p between search a b i l i t y and arithmetic computation. No such r e l a t i o n -ship was found. Nevertheless, the r e s u l t that search mechanism accounted for a s i g n i f i c a n t proportion of the variance of the arithmetic score must be explained. It has already been noted that the search mechanism was s i g n i f i c a n t l y , c o r r e l a t e d with v e r t i c a l span. The author suggests that those students who had an already well developed v e r t i c a l span were more l i k e l y to learn to perform well on the search procedure t e s t . Therefore, rather than developing a new s k i l l , the tra i n i n g period developed a new task perfor-mance (search procedure test) based upon an old or already developed s k i l l ( v e r t i c a l span) . That "is," i t may be concluded that the improvement i n the search mechanism was not the re s u l t of the development of the search a b i l i t y . Moreover, because the co r r e l a t i o n between v e r t i c a l span and arithmetic computation was very low then one should expect no increase i n arithmetic computation. The study findings bear this out. The results of the p i l o t study together with the results of t h i s study suggest two further conclusions. The f i r s t i s that the search mechanism may be a factor i n more complex arithmetic testing s i t u a t i o n s , and the second, that the t r a i n i n g for this mechanism may best be done at an e a r l i e r stage. II. FURTHER RESEARCH This study, indicates several p o s s i b i l i t i e s for future research. These questions may be divided into two areas: educational psychology and mathematics education. Educational Psychology Perception i s , i n the framework of e a r l i e r duscus-sions, a central issue of psychology. Therefore, there are several s p e c i f i c questions, related to the search mechanism, which are of concern to psychologists. Is search mechanism a compound perceptual s k i l l or i s i t a unique s k i l l independent of v e r t i c a l span, horizontal span, the a b i l i t y to distinguish items from a "noisy" background, and the a b i l i t y to perceive shapes? If the former i s the case can the search mechanism be best taught by t r a i n i n g designed for that s k i l l or can i t be best developed by t r a i n i n g those a b i l i t i e s of which i t i s composed? S p e c i f i c a l l y this study indicated that those students scored highest on a measure of v e r t i c a l span were also those who did well i n the search mechanism. Therefore, can the search mechanism be increased by t r a i n i n g i n v e r t i c a l span? A more general question would be are there components or other lower l e v e l perceptual s k i l l s which form the search procedure? Analysis of the t r a i n i n g method shows that other perceptual s k i l l s were involved i n the t r a i n i n g plan. They include auditory discrimination, aural to v i s u a l t r a n s l a t i o n , and the aural-visual alternation which provided i n t e r e s t and r e l i e f from the possible monotony of one kind of stimulus. Mathematics Education Although the results of this study indicate that arithmetic c a l c u l a t i o n was not increased by the search t r a i n i n g , i t should not be concluded that t r a i n i n g i n perceptual s k i l l s for an increase i n computational a b i l i t y i s of no value. The reasons for this are c l e a r . As this i s the f i r s t time that the search procedure has been defined i t cannot be assumed that the t r a i n i n g plan was the best that could be designed. During the administration of the tests the author became aware that i n order to have an e f f e c t i n an educational s i t u a t i o n one must t r a i n i n perceptual s k i l l s which exactly mirror the behaviour required of the c h i l d by that s i t u a t i o n . For example, i t i s useless to say that controlled readers are of no value i n helping the c h i l d a t t a i n speed and comprehension i n reading when only one or two actually t r a i n the c h i l d i n the required perceptual s k i l l , namely making the eye t r a v e l i n a general diagonal fashion,down the page. It i s here perhaps that the present t r a i n i n g procedure i s d e f i c i e n t : i t has not accurately p a r a l l e l e d what the c h i l d i s required to do when he works problems by himself. F i r s t the c h i l d hears few things which aid him to search for the required numbers to multiply together i n a m u l t i p l i c a t i o n problem. For example, the student while working the question 273 and having x96 8 placed the eight hears no voice saying, "Look for the seven." Future researchers must refi n e the t r a i n i n g procedures, and look for answers to the following questions. Is the search mechanism the only perceptual s k i l l r elated to mathematics? There i s a p o s s i b i l i t y that some per-ceptual s k i l l s are linked with certain geometric a b i l i t i e s . For example, the perception of s p a t i a l relationships may be a necessary s k i l l . Is the search mechanism only useful on s p e c i f i c kinds of mathematical problems? This study indicates that the search mechanism may be more useful i n complex situations or at least i n situations where the v i s u a l pattern of an algorithm i s presented for the f i r s t time. Is there a p a r t i c u l a r age at which the perceptual s k i l l s should be taught? Again, t h i s study indicates that either the t r a i n i n g period was too short to make any changes i n the t r a i n i n g mechanism or that i t would be best to t r a i n students at an e a r l i e r age. It i s to be hoped that future research w i l l not be hampered because of the f a i l u r e of this study to es t a b l i s h a s i g n i f i c a n t l i n k between a perceptual s k i l l and arithmetic computation. Caution must be taken though f o r even i f the search mechanism or any other perceptual s k i l l can be analyzed into i n d i v i d u a l components i t may s t i l l be most e f f i c a t i o u s both from educational and psychological viewpoints to consider these s k i l l s as a single u n i t . BIBLIOGRAPHY Alport, F. Theories of Perception and the Concept of Structure. New York: John Wiley and Sons Inc., 1955. Amble, B.R., and S. Muehl. "Perceptual Span Training and Reading Achievement of School Children," Journal of  Educational Psychology, 157: 196-197, August, 1966. . "Phrase Reading Training and Reading Achievement: A Replication Study," Journal of Experimental Education, 35: 98, Winter, 1966. Balow, Irving H. "Reading and Computational A b i l i t y as Determinants of Problem Solving," The Arithmetic Teacher, 11: 18, January, 1964. Bartley, S. Howard. "Perception," Encyclopedia,of Educational  Research, 4th ed. Robert L. Ebel. London: C o l l i e r -MacMillan Co., 1969. Bonsall, C. and R. L. Dornbush, "Visual Perception and Reading A b i l i t y , " Journal of Educational Psychology, 60: 294, August, 1969. Bottenberg, R.A. and J . H. Ward. Applied Linear Multiple Regression. Washington: Department of Commerce, U.S. O f f i c e of Technical Services Technical Report PRL-TDR-63-6, 1963 . Brown, J.C. "An Investigation of D r i l l Work i n the Fundamental Operations of Arithmetic," Journal of Educational Psychology, 2: 81-88, February, 1911. . "An Investigation of D r i l l Work i n the Fundamental Operations of Arithmetic," Journal of Educational Psychology, 3: 485-492, November, 1912. Brownell, William A. and Charlotte B. Chazal. "The Effects of Pre-mature D r i l l i n Third Grade Arithmetic," Journal of  Educational Research, 29: 19, September, 1935. , R. A. Doty and W. C. Rien. "Arithmetic i n Grades I and I t , " Duke University Research Studies i n Education, Dunbar, North Carolina: Duke University Press, 1941. Buckingham, B. R. "Adding Up or Down: A Discussion," Journal  of Educational Research, 12: 254, November, 1925. Buckingham, B. R. "Upward versus Downward Addition," Journal  of Educational Research, 16: 315-322, December, 1927. C a l l , R. J . and M. A. Wiggin. "Reading and Mathematics," The Mathematics Teacher, 59: 149-157, February, 1966. Cole, Lawrence E. W. "Adding Upward and Downward," Journal  of Educational Psychology, 3: 90, February, 1912. Earp, N. Wesley. "Observations i n the Teaching of Reading i n Mathematics," Journal of Reading, 13: 529, A p r i l , 1970. Freeburn, C e c i l Max. "The Influence of Perceptual Span and Perceptual Speed upon Reading A b i l i t y , " Journal of  Educational Psychology, 40: 334, October, 1949. Flournon, Frances. "A Consideration of the Ways Children Think when Performing Higher-Decade Addition," Elementary  School Journal, 57: 204-208, January, 1957. F r o s t i g , Marian, D. Welty Lefever, and John R. B. Wittlesey. "A Developmental Test of Visual Perception for Evaluating Normal and Neurologically Handicapped Children," Percep-tual and Motor S k i l l s , 12: 383-394, June, 1961. Gates, A. I. "A Study of the Role of Visual Perception, Intelligence, and Certain Associative Processes i n Reading and S p e l l i n g , " Journal of Educational Psychology, 17: 441, October, 1926. Gilmary, S i s t e r . "Transfer Effects of Reading Remediation to Arithmetic Computation when Intelligence i s Controlled and A l l other School Factors are Eliminated," The  Arithmetic Teacher, 14: 17-20, January, 1967. Glass, G. G. "Rate of Reading: A Correlation and Treatment Study," Journal of Reading, 11: 173, December, 1967. Grob, J . A. "Forcing Speed i n Oral Reading," Journal of  Reading, 11: 624, May, 1968. Harding, Lowry W. and Inez P. Bryant. "An Experimental Comparison of D r i l l and Direct Experience i n Arithmetic Learning i n Fourth Grade," Journal of Educational Research, 37: 321, January, 1944. Hays, William L. S t a t i s t i c s . New York: Holt, Rinehart and Winston, 1963. Jacobs, J . N. "Evaluation of the Fr o s t i g Visual Perceptual Training Program," Educational Leadership, 25: 332-340, January, 19 68. Johnson, Harry C. "The E f f e c t of Instruction i n Mathematical Vocabulary upon Problem Solving i n Arithmetic," Journal  of Educational Research, 37: 97-110, October, 1944. Kirk, Rodger E. Experimental Design: Procedures for the Behavioral Sciences"^ C a l i f o r n i a : Brooks/Cole Publishing Company, 1968. Kelly, F. J . "The Results of Three Types of D r i l l on the Fundamentals of Arithmetic, Journal of Educational  Research, 22: 381, November, 1920. Kulp, C. L. "Study of the.Relative Effectiveness of Two Types of Standard Arithmetic Practice Materials," Journal of  Educational Research, 22: 381-387, December, 1930. Lessinger, W. E. "Reading D i f f i c u l t i e s i n Arithmetic Compu-tat i o n , " Journal of Educational Research, 11: 288, A p r i l , 1925. Pribnow, J. R. "Why Johnny Can't Read Word Problems," School  Science and Mathematics, 69: 591-598, Oct o b e r 1 9 6 9 . Santore, Nicholas and Austin Gelzer. "A Comparison of Various Reading Improvement Approaches," Journal of Educational  Research, 61: 269, February, 1968. Santoro, Roseann Marie. "The Relation of Reading Achievement to S p e c i f i c Measures of Visual Perception and Inte l l i g e n c unpublished Doctoral d i s s e r t a t i o n , Fordham University, 1967 . Stevens, B.A. "Problem Solving, i n Arithmetic," Journal of  Educational Research, 25: 80, April-May, 1932. Wilson, E s t a l i n e . "Improving the A b i l i t y to Read Arithmetic," Elementary School Journal, 22: 380-386, January, 1922. Winer, B. J . S t a t i s t i c a l P r i n c i p l e s i n Experimental Design. New York: McGraw-Hill Book Company, Inc., 1962.. Witty, Paul, Theodore Stolarz and William Cooper. "Some Results of a Remedial Reading Program for College Student School and Society, 76: 378-379, December, 1952. . "Rate of Reading—A C r i t i c a l Issue," Journal of Reading, 13: 106, November, 1969. A P P E N D I X TRAINING METHODS In this study only t r a i n i n g i n the search mechanism was presented. However, the p i l o t study was made in order to suggest which of the three perceptual s k i l l s , search mechanism, v e r t i c a l span or horizontal span would produce the greatest change i n arithmetic s k i l l s . Therefore the procedures used for the three t r a i n i n g methods are presented below. V e r t i c a l Span The teacher was supplied with an overhead transparency of a 16 by 16 matrix of d i g i t s (black on white) and an adjustable paper mask. The large matrix was used to provide the teacher with a large number of d i g i t s which he could randomly select by moving the mask. During the t r a i n i n g period the following procedure was administered for f i v e to ten minutes d a i l y . The teacher covered the matrix with the mask and selected the width of numeral desired by opening the mask being careful to keep the numeral hidden with• his hand. The students were required to write the numeral they had seen and were then shown the correct answer. As the period of t r a i n i n g progressed the teacher would select larger-numerals and shorten exposure times. The s t u d e n t s were encouraged t o keep a r e c o r d o f t h e i r p r o g r e s s by c o l l e c t i n g each day's r e s u l t . H o r i z o n t a l Span The method o f t r a i n i n g f o r h o r i z o n t a l span was i d e n t i c a l t o t h a t f o r v e r t i c a l span e x c e p t f o r the p o s i t i o n i n g of the mask. Se a r c h Mechanism U s i n g t h e same type o f m a t r i x as mentioned above, the t e a c h e r was s u p p l i e d w i t h a mask which c o v e r e d a l l b u t a 12 by 12 d i g i t m a t r i x . T h i s a l l o w e d t h e s e l e c t i o n o f a d i f f e r e n t m a t r i x each day by s i m p l y moving t h e mask. The t e a c h e r would s e l e c t a row, column o r d i a g o n a l and b e g i n t o r e a d the d i g i t s b e g i n n i n g a t e i t h e r end of the row, column o r d i a g o n a l . The s t u d e n t s were t o f i n d the sequence and b e g i n t o r e a d o r a l l y w i t h the t e a c h e r . T h i s p r o v i d e d the s l o w e r s t u d e n t s time t o s e a r c h and f i n d t h e sequence. A t the b e g i n n i n g of the t r a i n i n g p e r i o d the t e a c h e r r e a d e i t h e r rows from l e f t t o r i g h t o r columns from t op t o bottom. As the p e r i o d p r o g r e s s e d he would r e v e r s e d i r e c t i o n more f r e q u e n t l y and r e a d more r a p i d l y . A t t h e end of each f i v e t o t e n minute s e s s i o n the t e a c h e r would g i v e a s h o r t t e s t (as d e s c r i b e d i n Appendix I I ) the s t u d e n t b e i n g encouraged to keep a r e c o r d o f the r e s u l t s . DESCRIPTION OF TESTS, PRESENTATION AND SCORING Horizontal and V e r t i c a l Span Only the horizontal span tests are described because the v e r t i c a l span tests are the same except for the d i g i t s being arranged i n columns. Material and content: Sl i d e s , of standard 35 mm color f i l m , were taken of f i v e 2, 3, 4, 5, 6 and 7 d i g i t rows. This gave a t o t a l of t h i r t y s l i d e s . Because of the experience of the p i l o t study the 7 d i g i t numbers were omitted when the tests were administered i n the f i n a l study. The numbers were photographed from the matrix supplied to the teachers i n order to keep the format of the numbers presented constant. The rows of d i g i t s were black on a white f i e l d and were masked with matt black construction paper. They were projected from a p o s i t i o n that would allow the 6 d i g i t row to almost f i l l a 50" by 50" screen at the front of the room. A standard s l i d e projector was used to project the s l i d e s . In order to control the duration of the v i s u a l pre-sentation a camera with a f o c a l plane shutter, back and lense removed, was placed i n front of the s l i d e projector lense. The shutter speeds of the camera were 1, 1/2, 1/4, 1/8, 1/15, 1/30, 1/60, 1/125, 1/500, and 1/1000 of a second. The l i s t of numbers and the speeds of presentation are given i n Table XI. Presentation: The sets of f i v e s l i d e s for 2, 3, 4, 5, 6 and 7 d i g i t numbers were arranged i n the order named i n Table XI. The numerals of length 2 and 3 were presented at 1/4, 1/8, 1/15, 1/30, and 1/60 of a second; of length 4 and 5 at 1/2, 1/4, 1/8, 1/15, and 1/3 0; of length 6 and 7 at 1, 1/2, 1/4, 1/8, 1/15. These speeds were selected because of the experience gained i n the preliminary study. Before each presentation the students were warned to watch the screen. Scoring: The number of correct answers for each of the tests was considered to.be the child's. score. The maximum score possible under this method would be 30 (25 i n the f i n a l administration). The order of the items was changed for the post-test. Test-retest correlations were calculated for v e r t i c a l span and horizontal span and were .76 and .62 res-pectively . Search Mechanism Test Materials and content: The materials consisted of an overhead projector transparency of an 11 x 11 matrix of d i g i t s written i n black. See Tables XII and XIII. ITEMS COMPRISING THE VERTICAL AND HORIZONTAL SPAN TESTS Number of Digits Duration V e r t i c a l Test Horizontal Test 25 125 066 033 017 25 74 59 41 67 63 87 36 17 94 25 125 066 033 017 942 281 572 418 374 713 138 326 879 956 .5 25 125 066 033 1253 2356 2547 1937 4136 2958 5817 7432 2637 3846 .5 .25 .125 .066 .033 57294 41896 87281 63654 32191 43138 13956 74129 38474 72581 1.0 .5 .25 .125 .066 937863 341815 896964 234395 374594 494558 562713 352158 326235 145367 TABLE XII ARRAY PRESENTED FOR THE SEARCH MECHANISM TEST WITH THE STARTING POINTS OF THE SEQUENCES USED 9 6/12 20 5 1 7 1 3 9. 5 6 9 3 8 1 5 4 1 8 6 9 8 7 9 4 2 4 9 7 3 4 5 3 2 6 2 3 5 1 9 11 3 6 1 4 9 4 5 5 8 9 7 13 5 1 9 1 4 5 3 6 7 3 6 2 5 8 6 2 1 4 9 2 7 5 4 3 7 3 5 2 1 5 8 8 4 2 3 8 3 6 4 5 8 5 1 6 7 3 6 4 5 5 7 3 1 1 9 3 1 7 8 4 4 5 9 5 6 7 6 3 17 16 4 3 8 7 2 6 4 9 3 9 8 18 8 7/15 19 14 Note: This table should be used as follows: To fi n d thirteenth sequence used locate the number 13 on the perimeter of the array and read the sequence beginning at that point; 51914. The correct res-ponse i s 53. Further, sequence 18 i s a diagonal sequence read from the lower l e f t hand corner; 48565. The correct response i s 13. SEQUENCES AND THE CORRECT RESPONSES USED FOR THE SEARCH MECHANISM TEST Number Sequence Response Number Sequence Response 1 86987 94 11 91532 62 2 38364 58 12 39519 67 3 76158 54 13 51914 53 4 51839 65 14 86951 14 5 84587 28 15 69352 15 6 16431 53 16 43872 64 7 25745 24 17 36765 95 8 38483 51 18 48565 13 9 76544 11 19 83174 56 10 59553 15 20 94253 41 The test consisted of c a l l i n g out a sequence of 5 d i g i t s , the f i r s t being located at one of the boundaries of the matrix. Thus, a sequence beginning at element 11,4 of the matrix (the eleventh row, the fourth column), would be read v e r t i c a l l y from bottom to top, whereas the sequence beginning at 5,11 would proceed ho r i z o n t a l l y from r i g h t to l e f t . The students responded to the sequence of f i v e d i g i t s by giving the sixth and seventh d i g i t s , i n order, of the sequence. Presentation: In order to administer the search mechanism test i d e n t i c a l l y to a l l groups and consistently from beginning to end, a tape recorder was used. Two sequences from l e f t to r i g h t , two from r i g h t to l e f t , two from top to bottom, two from bottom to top and two on the diagonals were presented at 7 second i n t e r v a l s . An additional ten sequencestwo from each of described directions were read i n random order. The sequences were changed for the post-test and the c o r r e l a t i o n between pre- and post-test was .81. Arithmetic Tests I n i t i a l s e l ection of items: ; In order to develop a suitable test for speed and accuracy i n arithmetic computation an i n i t i a l s e l e c t i o n of fo r t y addition, subtraction, m u l t i p l i -cation and d i v i s i o n questions were administered over an hour period to the grade Seven classes i n a Burnaby school. The tests were marked and items whose p value was less than .15 and greater than .85 were deleted. The i n i t i a l and f i n a l values of the mean, standard deviation and KR20 were 26.88, 7.48, and 0.8 8 and 23.34, 7.37, 0.88 respectively. Few of the students fi n i s h e d within an.hour and therefore the test was considered to be a good test of speed and accuracy. A c o r r e l a t i o n matrix was then calculated for the remaining items. Items were selected for the two p a r a l l e l forms of the f i n a l test on the basis of the operations involved, content and c o r r e l a t i o n . There were several questions which could not be matched on the basis of content. These were assigned to one or other of the tests using p values to bring the means close together. Items of p a r a l l e l content were then constructed and assigned to the alternate test . The f i n a l tests had s i x items for each of the four operations: three of each six were written h o r i z o n t a l l y and three v e r t i c a l l y . For example, compare questions 1 and 7 of test I at the end of this appendix. The tests were written i n the author's handwriting because the perceptual tests as well as the t r a i n i n g materials were so written. The tests had i d e n t i c a l format with the items from each of the eight categories randomly ordered. The f i n a l t e s t s : The f i n a l tests are included at the end of t h i s Appendix. Because they were used as pre- and post-tests for an experiment and the forms were alternated over the experimental group (see Table I) only the following administra-tions of the t e s t could be used for the c a l c u l a t i o n of the KR20: form I, pre-test group I I , pre-test group III and post-test group I; form I I , pre-test group I, pre-test group IV and post-test group I I . Note that only the pre-tests from the experi-mental groups were used i n case there was any.systematic e f f e c t of the treatment. The mean, standard deviation and KR20 for forms I and II were 16.58, 5.5, 0.88 and 17.91, 3.78, 0.73 respectively. The KR20 of form II was low when compared with that of form I and that of the i n i t i a l pool of items (.88). This may be attributed to two out of the four groups used for the analysis being from schools whose students were more homo-geneous and therefore achieving a lower test variance. The test was further broken into subtests i n order to see i f there.were any systematic differences between the four arithmetic operations, and between the v e r t i c a l l y and h o r i z o n t a l l y written questions. The summary of the analyses of the addition, subtraction, m u l t i p l i c a t i o n and d i v i s i o n sub-tests are i n Table XIV. In Table XV the v e r t i c a l and h o r i -zontal subtest analyses are summarized. Because the measures of consistency (KR20) f o r these subtests were so poor no attempt was made to apply any a p o s t e r i o r i tests of s i g n i -ficance to the differences between the experimental and con-t r o l groups on these measures. TABLE XIV SUMMARY OF THE ANALYSIS OF THE FOUR ARITHMETIC OPERATION SUBTESTS Subtest Addition Subtraction M u l t i p l i c a t i o n D i v i s i o n Form I II I II I II I II Mean SD. KR20 2.01 1.73 0.67 4.31 1.21 0.22 4.94 1.47 0.72 5.08 1.09 0.44 4.20 1.52 0.56 4.66 1.15 0. 29 3.50 1.95 0.76 3.86 1.82 0.71 TABLE XV SUMMARY OF THE ANALYSES OF THE VERTICAL AND HORIZONTAL SUBTESTS Subtest V e r t i c a l Horizontal Form I II I II Mean 8.59 8.89 7.89 9.03 SD 2.76 2.23 3.0 1.91 KR20 0.76 0.60 0.79 0.47 Another measure of r e l i a b i l i t y of concern i n the study was the inter-form r e l i a b i l i t y . Again, the post-tests of the experimental group could not be used because of any systematic variance induced by the treatment. A c o r r e l a t i o n of the control subjects performance of the two forms was found to be .65. This i s quite low fo r p a r a l l e l forms of a t e s t . I t can be attributed to the difference between the performance of the group used to establish p a r a l l e l items and that of the experi-mental group. As was noted above, the i n i t i a l form of the test was given to the two classes which formed the whole grade seven population of a school which came from the same kind of economic area as the schools used for the experiment. However comparison of the re s u l t s showed that the i n i t i a l group of students averaged about 65 percent whereas the experimental group averaged 70 percent. PAGE I 9 3 8 2 . 265 x 92 = 3 . 4138 - 509 765 476 832 516 318 42)7365 5 . 4001 6. 1376 * 71 = -292 56 + 42 + 81 + 39 + 62 + 17 + 83 = 8. 583 x29 37)7821 1 0 . 13726 - 4927 = 1 1 . 9 u 5 48 + 35 + 76 + 58 + 62 + 27 + 16 = PAGE II 1 3 . 9675 v 18 = 1 4 . 8221 1 5 . 512 16 . 583 , -597 890 x29 31 6 765 418 9 1 8 . 2005 - 947 = 1 9 . 527 x602 438 x 96 = 2 2 . 1865 v 32 = 1 7 . 63)2976 2 0 . 243 78 541 965 9 56 8 2 3 . 25965 -4103 24 . 413 + 876 + 531 + 29 + 871 + 13 + 9 = PAGE I 5 7 8 2 . 256 x 901 = 3 . 9763 - 407 362 985 413 794 185 431 87)5963 5. 7003 6. 2876 f 56 = -698 54 + 31 + 28 + 97 + 65 + 83 + 21 = 8. 385 x92 29)6821 1 0 . 13876 - 5487 = 45 + 63 + 67 + 82 + 75 + 26 + 61 = 1 1 . 806 x PAGE II 1 3 . 8374 - 95 = 14 . 17182 1 5 . 433 - 1 6 . 341 -9413 19 X 8 7 876 54 321 757 1 7 . 82)7136 1 8 . 5003 - 949 = 1 9 . 928 x95 20 . 324 2 1 . 403 x 96 = 2 2 . 2173 v 36 = 8 7 415 596 8 65 8 2 3 . 7863 -928 314 + 156 + 832 + 854 + 138 + 765 + 143 = STATISTICAL ANALYSIS OF TEST RESULTS As outlined i n Chapter I I , section V, the s t a t i s t i c a l design was h i e r a r c h i c a l with classes nested within treatment groups. A design of this kind i s to be considered with caution for although the nuisance variable of class e f f e c t i s parcelled out there i s a loss of degrees of freedom, which, unless the number of classes i s large, can be excessive. The calculations are summarized below and follow the outline of Kirk who uses the following notation: p l e v e l s of a^, where p = 2 q levels of b_. , where q = 4 q... l e v e l s of b. nested where q,.» =2 within A, n lev e l s of s , where n = 26 m "A" denotes the treatment groups, "B" the classes, "S" the subjects and [] a sum of squares. Hypothesis I The experimental group was s i g n i f i c a n t l y d i f f e r e n t from the control group on a measure of the search mechanism. The test r e s u l t s may be found i n Table XVI. The detailed c a l -culations follow. SEARCH MECHANISM TEST RESULTS Group I Group II Group III Group IV x l X2 d d2 X l X2 d d2 X l X2 d d2 X l X2 d d2 5 3 2 4 16 19 -3 9 2 11 • -9 81 11 16 -5 25 4 3 1 1 1 1 0 0 2 3 -1 1 18 20 -2 4 16 17 -1 1 *15 15 0 0 10 15 -5 25 17 20 -3 9 3 3 0 0 13 17 -4 16 5 17 -12 144 13 17 -4 16 3 1 2 4 4 4 0 0 7 16 -9 81 12 16 -4 16 11 16 -5 25 13 10 3 9 7 12 -5 25 4 18 -14 196 9 11 -2 4 8 16 -8 64 4 0 4 16 16 19 -2 4 2 2 0 0 13 15 -2 4 12 10 2 4 18 20 -2 4 6 11 -5 25 7 4 3 9 7 16 -9 81 10 16 -6 36 4 3 1 1 3 2 1 1 13 15 -2 4 11 ,16 -5 25 17 19 -2 4 4 3 1 1 17 18 -1 1 15 15 0 0 16 19 -3 9 9 7 2 4 3 9 -6 36 17 18 -1 1 3 10 -7 49 11 18 -7 49 6 16 -10 100 17 16 1 1 1 1 0 0 18 12 6 36 11 9 2 4 3 17 -14 196 1 3 -2 4 14 12 22 4 5 17 -12 144 14 17 -3 9 4 3 1 1 8 6 2 4 4 9 -5 25 12 19 -7 49 14 9 5 25 17 19 -2 4 5 15 -10 100 5 11 -6 36 5 10 -5 25 7 12 -5 25 1 12 -11 121 1 7 -6 36 9 12 -3 9 15 12 3 9 3 8 -5 25 16 15 1 1 14 6 8 64 5 7 -2 4 13 17 -4 16 *20 18 2 4 0 3 -3 9 13 11 2 4 9 9 0 0 7 18 -11 121 16 16 0 0 7 6 1 1 7 10 -3 9 16 20 -4 16 10 10 0 0 5 4 1 1 12 16 -4 16 16 19 3 9 6 6 0 0 18 13 5 25 0 12 -12 144 18 17 1 1 11 16 -5 25 11 15 -4 16 5 7 -2 4 3 7 -4 16 0 1 -1 1 8 14 -6 36 10 18 -8 64 12 20 -8 64 10 15 -5 25 14 17 -3 9 *These scores were randomly selected for deletion to equalize group s i z e . TABLE XVII SUMMARY OF GROUP RESULTS Group Treatment I II H I IV Totals Experimental Ed -137 -114 -251 £d2 1271 900 2171 Control Ed 24 -6 . z d 2 290 360 -40 650 Z ABS = 1291 1 n 2 £(ABS) = [ABS] = 2171 + 650 = 2821 n 2 (EABS) -_i = [X] = (-29?1j) = 14681 = 8 1 4 > 2 4 n P q ( i ) 26x2x2 104 P q o i _ J _ = [ A ] = (-251) + (-40) = 64601 = n q ^ j 2 6 x 2 52 P q ? £ £(AB) 1 1 _ r , R l _ (-24)^ + ( - 1 6 P + ( - 2 3 7 P + (-114)^ _ _ [AB] 2-g • = 3 2 6 9 7 = 1 2 5 3 - 7 3 -S S t o t a l = t A B S J " f x f = 2 8 2 1 - 814.24 = 2005.76 SS a = [A] - [X] = 1242.33 - 814.24 = 428.09 SS D _ = [AB] - [A] = 1253.73 - 1242.22 = 11.40 SS = [ABS] - [AB] = 2821 - 1253.73 = 1567 .27 w.A F r a t i o = ^rrnr= 7 5 - 1 0 F P r o b ( l , 2 ) ( 7 5 - 1 0 ) = - 9 6 5 2 The alpha l e v e l of the study was .05 and 1 - F , c J prob = .0347. Therefore the n u l l hypothesis was rejected. In other words the treatment groups was s i g n i f i c a n t l y d i f f e r e n t at the .0347 l e v e l of confidence. Hypothesis II The experimental group was s i g n i f i c a n t l y d i f f e r e n t from the control group on a measure of v e r t i c a l span. The test results may be found i n Table XVIII. The detailed calcula-tions follow, n E ABS = 0 1 n 2 E(ABS) = [ABS] = 718 n 2 (EABS)* = [ X ] = ^ 3 ± 3 = Q ^ P ^ i ) 26x2x2 p q ? E(EA) „ „ 1_1 = r A ] = (-3)^ + (3)^ = 18 = 3 5 n q ( i ) L A J 26 x 2 52 ' ° p q E E(AB) 9 9 9 9 i i _ r a R 1 _ <-7r + ( 4 r + ( - s r + g i r - [AB] ^ S S t o t a l = [ A B S ] " [ X ] = 7 1 8 - 0 = 718 SS A = [A] - [X] = .35- 0 = .35 SS_ . = [AB] - [A] = 4.81- .35 = 4.46 a . W . i\ SS, _ = [ABS] - [AB] = 718 - 4.81 = 713.19 W. A F r a t i o = 2TH = - 1 5 6 8 Vob(l,2) (' 1 5 6 8> = ' 2 1 The alpha l e v e l of the study was .05 and 1 - ( F p r o k ) = '79 Therefore the n u l l hypothesis was accepted. That i s , there was no s i g n i f i c a n t difference between the experimental and the control groups on a measure of v e r t i c a l span. TABLE XVIII VERTICAL SPAN TEST RESULTS Control Experimental Group I Group II Group III Group IV x l X2 d d2 X l X2 d d2 X l X2 d d2 X l X2 d d2 12 13 -1 1 11 11 0 0 7 9 -2 4 11 15 -4 16 15 14 1 1 7 8 -1 1 8 11 -3 9 11 16 -5 25 10 14 -4 10 *17 12 5 25 11 11 0 0 11 10 1 1 6 8 -2 4 16 15 1 1- 12 13 -1 1 14 14 0 0 8 9 -1 1 13 12 1 1 13 14 -1 1 6 5 1 1 13 15 -2 4 14 11 3 9 13 13 0 0 11 12 -1 1 7 9 -2 4 12 15 -3 9 8 10 -2 4 13 11 2 4 9 10 -1 1 15 10 5 25 14 11 3 9 11 9 2 4 8 12 -4 16 11 11 0 0 7 9 -2 4 16 19 -3 9 15 6 9 81 8 12 -4 16 10 6 4 16 12 12 0 0 18 17 1 1 9 10 -1 1 9 11 -2 4 11 12 -1 1 15 13 2 4 14 14 0 0 15 13 2 4 16 17 -1 1 8 12 -4 16 13 12 1 1 9 11 -2 4 13 5 8 64 2 2 0 0 15 18 -3 9 9 9 0 0 10 9 1 1 14 13 1 1 6 1 5 25 8 10 -2 4 12 15 -3 9 18 15 3 9 13 14 -1 1 12 10 2 4 11 13 -2 4 11 12 -1 1 15 16 -1 1 10 13 -3 9 11 9 2 4 16 13 3 9 14 12 2 4 12 13 -1 1 12 8 4 15 12 15 -3 9 10 10 0 0 13 9 4 16 15 14 1 1 12 14 -2 4 10 11 -1 1 10 14 -4 16 *10 11 -1 1 12 9 3 9 17 15 2 4 3 5 -2 4 12 11 1 1 7 11 -4 16 13 10 3 9 13 11 2 4 8 12 -4 16 10 9 1 1 12 12 -1 1 11 14 -3 9 15 14 1 1 16 14 2 4 17 15 2 4 9 5 4 16 13 14 -1 1 13 11 2 4 11 10 1 1 15 12 3 9 13 12 1 1 4 9 -5 25 12 12 0 0 11 12 -1 1 17 13 4 16 15 14 1 1 14 14 0 0 *These subjects were deleted at random to give equal group s i z e . TABLE XIX SUMMARY OF GROUP RESULTS Group Treatment I II III IV Total Experimental Ed -7 4 - 3 E d 2 153 198 351 Control Ed -8 11 E d 2 2 * 2 125 3 367 Although the F r a t i o i s below 1, which could indicate that the assumptions of the F,. . ., , . were v i o l a t e d , the d i s t r i b u t i o n ' p r o b a b i l i t y shows that the calculated r a t i o was not s i g n i f i c a n t l y d i f f e r e n t from 1. Hypothesis III The experimental group was s i g n i f i c a n t l y d i f f e r e n t from the control group on a measure of horizontal span. The tes t r e s u l t s may be found.in Table XX. The detailed c a l c u l a -tions follow. n EABS = 5 + 71 = 76 1 n 2 E(ABS) = [ABS] = 915 + 837 =1752 n 2 (EABS) 2 = [X] = o i Z l l o = 5776 = 55.54 n P q ( i ) 26x2x2 P q o E (EA) ±JL— , [A] - <7'" + ' 7> 2 = 5 0 9 0 . „ _ 8 8 nq 2.6 x 2. 52 p q E E(AB) 9 9 9 1 1 = [ A B ] _ (-60) Z + (65) Z + ( - 1 2 P + (83) n 1 J 26 14858 = 571.46 TABLE XX HORIZONTAL SPAN TEST RESULTS Group I Group II Group III Group IV x l X2 d d2 X l X2 d d2 X l X2 d d2 X l X2 d . d2 11 17 -6 36 14 11 3 9 7 14 7 49 15 12 3 9 16 12 4 16 6 7 -1 1 8 14 -6 36 16 13 3 9 13 15 -2 4 *18 8 10 100 11 12 -1 1 16 13 3 9 9 7 2 4 20 18 2 4 10 13 -3 9 17 15 2 4 14 11 3 9 15 14 1 1 14 16 -2 4 3 1 2 4 14 18 -4 16 13 10 3 9 16 14 2 4 10 7 3 9 10 6 4 16 15 14 1 1 9 12 -3 9 18 16 2 4 9 11 -2 4 15 14 1 1 12 13 -1 1 12 13 -1 1 12 12 0 0 13 14 -1 11 11 12 -1 1 20 17 3 9 15 8 7 49 17 9 8 64 10 13 -3 9 17 7 10 100 20 21 -1 1 13 15 -2 4 8 15 -7 49 15 13 2 4 15 18 -3 9 17 12 5 25 17 15 2 4 20 17 3 9 10 12 -2 4 14 6 8 64 9 12 -3 9 15 2 13 169 0 1 -1 1 20 11 9 81 9 13 -4 16 9 10 -1 1 16 15 1 1 5 1 4 16 7 10 -3 9 20 12 8 64 16 19 -3 9 18 18 0 0 12 17 -5 25 19 15 4 16 13 13 0 0 18 21 -3 9 12 14 -2 4 13 11 2 4 18 16 2 4 19 10 9 81 11 8 3 9 0 6 -6 36 13 18 -5 25 12 9 3 9 10 8 2 4 20 13 7 49 16 17 1 1 14 12 2 4 9 16 -7 49 *15 14 1 1 11 13 -2 4 21 16 5 25 5 8 -3 9 12 12 0 0 13 8 5 25 17 13 4 16 14 13 1 1 10 13 -3 9 11 12 -1 1 17 10 7 49 7 15 -8 64 15 14 1 1 13 14 -1 1 21 19 2 4 6 6 0 0 18 15 3 9 15 14 1 1 18 14 4 16 14 15 -1 1 12 12 0 0 6 13 -7 49 17 10 7 449 13 13 0 0 18 19 -1 1 21 19 2 4 14 11 3 9 *These subjects were deleted at random to give equal group s i z e . TABLE XXI SUMMARY OF GROUP RESULTS Group Treatment I II III IV Total Experimental Ed -69 65 5 Ed 2 376 539 915 Control Ed -12 83 71 Ed 2 290 547 837 1696.46 42.34 473.58 1180.54 .221 As the alpha l e v e l of the study was .05 and 1 -( F ^ ^ ) = .779, the n u l l hypothesis was accepted. There was not s i g n i f i c a n t difference between the experimental and the control groups on a measure of horizontal span. The F .. was not s i g n i f i c a n t l y below one and r a t i o 3 J therefore i t was concluded that the assumptions of the F c r a t i o were not v i o l a t e d . Hypothesis IV The experimental group was s i g n i f i c a n t l y d i f f e r e n t from the control group on a measure of arithmetic computation. The test results may be found i n Table XXII. The det a i l e d calculations follow. -80 604 + 776 = 1380 S S t o t a l = [ A B S ] " [ X ] = 1 7 5 2 " 5 5 - 5 4 SS A = [A] - [X] = 97.88-55.54 = SS,. „ , = [AB] - [A] = 571.46 - 97.88 = a. w. A . SS _ = [ABS] - [AB] = 1752 - 571.46 -w. A. 42.34 F r a t i o = 237.79 = • 1 7 8 F p r o b ( 1 , 2 ) * 1 7 8 = EABS = -44 + -36 = 1 n 2 £ (ABS) = [ABS] = TABLE XXII ARITHMETIC COMPUTATION TEST RESULTS x l x 2 d d2 X l S2 d d2 X l X2 d d2 X l X2 d d2 17 12 5 25 17 18 -1 1 14 18 -4 16 18 18 0 0 23 18 5 25 12 15 -3 9 6 10 -4 16 20 20 0 0 19 22 -3 9 21 18 3 9 3 6 -3 9 18 14 4 16 12 16 -4 16 23 17 6 36 12 21 -9 81 15 20 -5 25 9 16 -7 49 6 12 -6 36 12 11 1 1 22 20 2 4 12 18 -6 36 10 17 -7 49 15 6 9 81 18 16 2 4 18 17 1 1 21 20 1 1 14 17 -3 99 16 22 -6 36 18 23 -5 25 23 24 -1 1 5 10 -5 25 22 20 2 4 18 18 0 0 17 12 5 28 21 22 -1 1 22 19 3 9 13 19 -6 36 18 15 3 9 2 1 1 1 23 22 1 1 20 24 -4 16 22 21 1 1 18 18 0 0 17 21 -4 16 14 19 -5 25 15 15 0 0 10 17 -7 49 19 20 -1 1 18 19 -1 1 23 16 7 49 12 13 -1 1 22 24 -2 4 21 22 -1 1 23 22 1 1 11 17 -6 36 20 18 2 4 12 12 0 0 18 12 6 36 11 16 -5 25 23 19 4 16 13 10 3 9 23 19 4 16 17 14 3 9 24 23 1 1 18 20 -2 4 14 16 -2 4 13 14 -1 1 20 17 3 9 22 23 -1 1 22 23 -1 1 13 14 -1 1 16 15 1 1 16 23 -7 49 15 14 1 1 12 15 -3 9 20 21 -1 1 11 11 0 0 19 18 1 1 19 18 1 1 18 19 1 1 13 17 -4 16 24 21 3 9 14 8 6 36 22 22 0 0 12 22 -10" 100 18 16 2 4 11 15 -4 16 18 18 0 0 6 55 1 1 17 20 -3 9 9 14 -5 25 23 20 3 9 20 21 -1 1 23 22 1 1 24 22 2 4 21 19 2 4 16 21 -5 25 22 22 0 0 11 18 -7 49 19 15 4 16 21 20 1 1 19 20 -1 1 21 21 0 0 21 23 -2 4 21 20 1 1 19 20 -1 1 TABLE XXIII SUMMARY OF GROUP RESULTS Group Treatment I II III IV Totals Experimental d -56 -12 -44 d 2 418 186 604 Control d -54 18 -36 d 2 474 302 776 n ( Z A B S ) 1 npq (1) l X J 26x2x2 T04 ~ 6 1 , 5 4 p q 9 E ( Z A ) Z , 1 1 _ r , _ (-44) z + (-36) ^  _ 3232 nq (i) 26 x 3 52 = 62.15 P q ? £ £ ( A B ) 1 1 n = [AB] = (-56) 2 + (12) 2 + (-54) 2 + ( 1 8 ) 2 26 6520 26 = 250.77 S S t o t a l = [ A B S [ - [X] = 13.80 -• 61.54 1318.46 S S A : = [A] - [X] 62.15 --61.54 .61 S S B . w . A ss A w . A = [AB] -= [ABS] -[A] [AB] = 250.77 1380 -- 62.15 250.77 188.62 1129.23 F . . = r a t i o .61 94.31 .00646 F , ,, „, . 00646 prob (1,2) = .087 The alpha l e v e l for this test was set at .05 and as 1 - (Fpj-Qj-,) = .913, the n u l l hypothesis i s accepted. There was no significant-'di£flerence between the experimental and the control groups on a measure of computational a b i l i t y . As the F p r o k w a s n o t s i g n i f i c a n t l y d i f f e r e n t from one the assumptions underlying the F vi o l a t e d . r a t i o were taken to be not APPENDIX IV PILOT STUDY There were two major reasons for conducting a p i l o t study. The author wished to esta b l i s h which of the three methods of t r a i n i n g , v e r t i c a l span, horizontal span, or search t r a i n i n g would be most l i k e l y to be s i g n i f i c a n t i n demonstrating that perceptual t r a i n i n g i s a factor i n arithmetic computation. I t was also important to standardize the method of perceptual test presentation. An open area classroom was available i n which there were approximately s i x t y grade f i v e students enrolled. The class was randomly divided into three groups each of which were given pre- and post-tests on v e r t i c a l span, horizontal span, search a b i l i t y and the arithmetic sections of forms A and B of the.Canadian Test of Basic S k i l l s . Each of the three d i f f e r e n t t r a i n i n g methods was assigned to one of the groups. Because of the expected differences i n testing administration between pre-and post-test i t was decided to do a l l the testing i n one session. A l l three groups were scattered over the testing room and a note was made of where each student sat. The students were kept at the same seats f o r each testing session. Thus, although there were considerable differences between pre- and post-tests, these differences were kept constant for each of the t r a i n i n g groups. For the f i n a l system of testing and the tests see Appendix I I . After a t r a i n i n g period of three weeks i t was found that f i f t e e n students i n each group had scores for a l l the tes t administrations. The summary of the differences between pre- and post-tests w i l l be found i n Table XXIV. TABLE XXIV SUMMARY OF.PILOT STUDY RESULTS Test Groups A r i t h - A r i t h - V e r t i c a l Hori- Search metici metic2 zontal Horizontal X 18 12 -14 -49 40 tr a i n i n g X 2 1174 898 162 487 270 V e r t i c a l X 1 10 -6 40 26 tr a i n i n g X 2 791 1460 84 400 246 Search X 35 93 -24 -38 58 t r a i n i n g X 2 459 2654 238 400 372 F r a t i o .28 1.63 .54 .16 3 .23 A simple F , . was calculated for each of the te s t s . * r a t i o No levels of si g n i f i c a n c e are reported because of the d i f f e r -ences i n the tes t i n g procedures between pre- and post-tests. However, the trend i s c l e a r . The table shows that highest r a t i o s were for the search test and the arithmetic t e s t s . Moreover, the search trained group had the greatest difference scores on a l l but the horizontal t e s t . I t was on this basis that the search t r a i n i n g was chosen for the f i n a l study. 

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