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Computer-paced versus self-paced arithmetic drill-and-practice Dyck, Anthony Carey 1971

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COMPUTER-P&CED VERSUS S E L F - P A C E D A R I T H M E T I C DRILL—AND—PRACTICE by ANTHONY CAREY DYCK B . S c . , 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 , 1969 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L L M E N T OF MASTER OF ARTS i n t h e D e p a r t m e n t o f E d u c a t i o n We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE U N I V E R S I T Y OF B R I T I S H COLUMBIA J U N E , 1971 In p r e s e n t i n g t h i s t h e s 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 a t 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 t h e 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 t h e 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 . Department The U n i v e r s i t y o f B r i t i s h C olumbia V a n c o u v e r 8, Canada Date Q.^^ AH/-? / An a n a l y s i s of the l i t e r a t u r e showed that there i s very l i t t l e agreement on when and how a computer program should branch a student through a CAI program. T h i s , together with the f a c t t h at r e s e a r c h i n the f i e l d of a r i t h m e t i c has shown t h a t d r i l l should f o l l o w e f f e c t i v e t e a c h i n g of concepts, l e d the author to i n v e s t i g a t e whether students working on a r i t h m e t i c d r i l l - a n d - p r a c t i c e would do b e t t e r on a COMPUTER—PACED program or a SELF-PACED program. COMPUTER-PACED was de f i n e d to be where the computer program determined when the students should be branched to more or l a s s d i f f i c u l t q u e s t i o n s . SELF-PACED was de f i n e d t o be where the s t u d e n t s determined when they were presented more or l e s s d i f f i c u l t questions by pushing one of the two marked keys on the computer t e r m i n a l . The e v a l u a t i o n was done by comparing the achievement of the COMPUTER-PACED and the SELF-PACED groups. For the length of the study the two groups of grade s i x s t u d e n t s had a d a i l y a r i t h m e t i c l e s s o n followed by a s e s s i o n at a computer t e r m i n a l t o work on a r i t h m e t i c d r i l l - a n d - p r a c t i c e programs. The r e s u l t s of the p o s t - t e s t (adjusted by using a p r e - t e s t as a c o v a r i a t e ) showed that t h e r e was no s i g n i f i c a n t d i f f e r e n c e between the two s e l e c t i o n mechanisms. Fur t h e r a n a l y s i s showed that t h e r e was no s i g n i f i c a n t d i f f e r e n c e between the males and females performance and that there was no s i g n i f i c a n t i n t e r a c t i o n (sex X groups) e f f e c t . The r e s u l t s of the study i n d i c a t e that when working with a r i t h m e t i c d r i l l - a n d — p r a c t i c e , students w i l l do as we l l i f the computer program c o n t r o l s when to branch as they would i f the students c o n t r o l when to branch to a d i f f e r e n t l e v e l of d i f f i c u l t y . Chapter Page I. THE PROBLEM , 1 INTRODUCTION 1 STATEMENT OF THE PROBLEM .... 4 I I . REVIEW OF THE PERTINENT LITERATURE AND THE DEFINITIONS OF TERMS 5 INTRODUCTION 5 DRILL IN ARITHMETIC 5 DECISION STRUCTURES 7 DEFINITIONS -.11 HYPOTHESES 13 Hypothesis 1 13 Hypothesis 2. 13 Hypothesis 3. 13 I I I . EXPERIMENTAL DESIGN 15 INTRODUCTION 15 PILOT STUDY 15 FORMATION OF GROUPS ....... 17 MATERIAL 18 Levels 18 Computer Terminals 19 Test 19 Chapter Page PROCEDURE 19 STATISTICAL ANALYSIS 23 Data 23 Design 24 IV. ANALYSIS OF RESULTS 25 TESTING OF HYPOTHESES 25 Hypothesis 1. 26 Hypothesis 2. 27 Hypothesis 3. 27 INTERPRETAION OF RESULTS ............ .... 27 ANALYSIS OF ADDITIONAL DATA 28 V. CONCLUSION AND SUGGESTIONS FOR FURTHER RESEARCH 3 1 SUMMARY .... 3 1 DISCUSSION 32 LIMITATIONS OF THE STUDY 33 SUGGESTIONS FOR FURTHER RESEARCH 34 BIBLIOGRAPHY 36 APPENDIXES ....... . . 4 1 A. LEVELS 41 B. EASIER LEVELS 46 C. TEST.. 48 D. SAMPLE PRINTOUTS 52 E. ANALYSIS OF A STUDENT'S WORK 55 F. EXPERIMENTAL DATA 57 LIST OF TABLES Table Page 1 . THE NUMBER OF SUBJECTS IN THE TWO FACTOR DESIGN . 24 2. ANALYSIS OF VARIANCE TABLE , . 25 3. ADJUSTED EXPECTED MEANS 26 THE PROBLEM INTRODUCTION A modern approach to teaching arithmetic i s characterized by meaningful d r i l l - a n d - p r a c t i c e along with the development of arithmetic concepts. after the teacher presents the student with a c t i v i t i e s and i l l u s t r a t i o n s on the concepts, d r i l l - a n d - p r a c t i c e i s given to reinforce the facts and processes. The task of adequately developing concepts, furnishing meaningful d r i l l suited to the in d i v i d u a l needs, and the checking of results to diagnose weaknesses in understanding i s often beyond time l i m i t a t i o n s of the teacher. The computer lends i t s e l f well to the task of dr i l l - a n d - p r a c t i c e . It can present the exercises suited to the a b i l i t y of the student, check responses, and i d e n t i f y weaknesses very quickly. D r i l l - a n d - p r a c t i c e , the simplest form of computer assisted i n s t r u c t i o n (CAI), i s the type of computer int e r a c t i o n that i s of interest i n t h i s study. The role of the computer i s to provide regular review and practice to supplement the e s t a b l i s h e d c u r r i c u l u m . There are two other types of CAI. The t u t o r i a l system i s the second and more complex l e v e l of i n t e r a c t i o n between the s t u d e n t and computer program. Here the computer program a c t s as t u t o r . The t h i r d type of CAI i s the dialogue system where the computer c a r r i e s on a c o n v e r s a t i o n with the student. The t h i r d l e v e l i s s t i l l i n the p l a n n i n g stages. The S t a n f o r d group headed by Suppes has pioneered the work of d r i l l - a n d - p r a c t i c e programs i n a r i t h m e t i c fundamentals. By the 1969-70 s c h o o l year over 8,000 st u d e n t s were t a k i n g a r i t h m e t i c l e s s o n s i n the S t a n f o r d d r i l l - a n d - p r a c t i c e programs. More d e t a i l s about the S t a n f o r d programs may be found i n the book by Suppes, Jerman, and B r i a n (42) . In S t a n f o r d ' s program, l i k e most other d r i l l - a n d - p r a c t i c e programs, the teacher or the program determines when the student moves from one l e v e l of d i f f i c u l t y to the next. At the present time we are moving the s t u d e n t s up and down the l e v e l s of d i f f i c u l t y on the b a s i s of the p r e v i o u s days's performance. I f more than 80 per cent of the e x e r c i s e s are c o r r e c t , the student moves up one l e v e l , unless he i s already at the top l e v e l . I f l e s s than 60 per cent of the e x e r c i s e s are c o r r e c t , the student moves down a l e v e l , u n l e s s he i s a l r e a d y a t the bottom. I f h i s percentage of c o r r e c t answers f a l l s between 60 per cent and 80 per cent he s t a y s a t the same l e v e l . I t should be emphasized t h a t the s e l e c t i o n of e x a c t l y f i v e l e v e l s and of the percentages 60 and 80 has no f i r m t h e o r e t i c a l b a s i s but i s based on p r a c t i c a l - p e d a g o g i c a l judgments. As s y s t e m a t i c data are accumulated, we expect to modify our c h o i c e s i n the l i g h t of experience ( Suppes. 41:15). G e n t i l e s a i d t h a t there i s a g r e a t need f o r more r e s e a r c h i n the area of CAI (15:24). Most of the people that are using CAI to-day decide on a c r i t e r i o n f o r branching from one l e v e l to the next, with l i t t l e or no r e s e a r c h i n the area of how the student l e a r n s . As i s pointed out i n Chapter II there i s very l i t t l e agreement i n the r e s e a r c h s t u d i e s on what d e c i s i o n s t r u c t u r e s t o use. G e n t i l e (15:23-24) s t a t e d that p r a c t i c a l l y a l l CAI support funds go i n t o the development of systems, equipment, and courses and not i n t o r e s e a r c h on l e a r n i n g v i a CAI. The computer i s capable of f o l l o w i n g the most complicated d e c i s i o n making s t r u c t u r e i f the d e c i s i o n making c r i t e r i a can be s t a t e d i n a simple o b j e c t i v e manner. As poi n t e d out by Suppes, he has no fi r m t h e o r e t i c a l b a s i s f o r branching the st u d e n t s to d i f f e r e n t l e v e l s of d i f f i c u l t y i f he answers l e s s than 60 per cent or more than 80 per cent of the q u e s t i o n s c o r r e c t . H o p e f u l l y , with the use of computers i n r e s e a r c h on l e a r n i n g , a t h e o r e t i c a l b a s i s can be made f o r branching the students through a CAI program. The major j u s t i f i c a t i o n f o r CAI has been the i n d i v i d u a l i z a t i o n made p o s s i b l e by the computer. The big qu e s t i o n now i s who d e c i d e s what i n s t r u c t i o n or m a t e r i a l i s a p p r o p r i a t e f o r the student's e d u c a t i o n a l needs. Does the student d e c i d e or does some educator decide what i s b e s t f o r the student? Because of the many factors that influence a person such as age l e v e l , a b i l i t y l e v e l , attention span, attitude, sex, anxiety, etc. u n t i l more research i s done on i n d i v i d u a l differences (ID), possibly the student should have control of the path taken through a CAI course. Because there are no t h e o r e t i c a l grounds on when the program should branch a student, research on decision making structures has only started, and there are so many factors that influence a student, i t i s the concern of t h i s study to test to see i f the student should control the branching from one l e v e l of d i f f i c u l t y to another when working on a CAI program. STATEMENT OF THE PROBLEM The present investigation attempts to answer the following question: Do students who have control over the l e v e l of d i f f i c u l t y i n a learning sequence achieve higher scores than students who do not have control over the lev e l of d i f f i c u l t y when working on an arithmetic d r i l l - a n d - p r a c t i c e program? A formal statement of the hypotheses i s stated at the end of Chapter I I . REVIEW OF THE PERTINENT LITERATURE AND THE DEFINITION OF TERMS INTRODUCTION The following review of the l i t e r a t u r e summarizes the research that i s applicable to the problem being investigated. It i s important to note that the term PACED as used in COMPUTER—PACED and SELF-PACED does not refer to the speed at which the frames or material are presented as i n Programmed Instruction. PACED here refers to the choice of the LEVELS of d i f f i c u l t y or the d i f f i c u l t y of the questions presented not on how much time a student has to answer a question or frame. DRILL IN ARITHMETIC Many people are s t i l l confused with respect to the use of d r i l l i n the classroom to-day. When commenting on readiness for d i v i s i o n Brownell (5) stated that i f childr e n f i n d the topics d i f f i c u l t , many times i t i s due to inadequate mastery of the s k i l l s and basic f a c t s needed. Jerman (23) cited a study by Anaspaugh in which 93 percent of the e r r o r s made i n long d i v i s i o n of decimals and i n common f r a c t i o n s i n grades 4,5 and 6 were due t o the l a c k of mastery of number f a c t s r a t h e r than the number processes. Brownell and Chazel (6) pointed out the dangers of t e a c h i n g by the d r i l l method alone . They found t h a t a f t e r grade three students were gi v e n f i v e minutes of d r i l l each day f o r a month on items taught i n grade one and two that 15.4 percent of the responses were ob t a i n e d by guessing, 19.3 percent by counting, 18.7 percent by i n d i r e c t s o l u t i o n and 52.5 percent by immediate r e c a l l . They concluded that e f f e c t i v e t e a c h i n g must precede d r i l l , as d r i l l o n l y r e i n f o r c e s the procedure the student has learned to o b t a i n an answer. T h e i r study p o i n t e d out t h a t d r i l l can be most e f f e c t i v e l y used to overcome a l a r g e percentage of t y p i c a l e r r o r s i n a r i t h m e t i c a f t e r the concepts are i n t r o d u c e d and discussed by the classroom teacher. The S t a n f o r d a r i t h m e t i c d r i l l - a n d - p r a c t i c e programs are the most widely known and used of any of the d r i l l - a n d — p r a c t i c e programs. As pointed out i n Chapter I, i n the S t a n f o r d d r i l l — a n d — p r a c t i c e programs the student i s moved up and down the LEVELS of d i f f i c u l t y on the b a s i s of the previous day's performance. In the S t a n f o r d case the s t u d e n t s had no c o n t r o l over whether they moved up or down the LEVELS of d i f f i c u l t y . I t i s assumed, i n programs l i k e the S t a n f o r d a r i t h m e t i c d r i l l - a n d - p r a c t i c e programs, that the i n s t r u c t o r knows what i s best f o r the s t u d e n t . The s t u d i e s c i t e d above suggest that e f f e c t i v e t e a c h i n g should be f o l l o w e d by d r i l l . Suppes and o t h e r s have d e f i n i t e l y shown that the computer i s capable of p r e s e n t i n g d r i l l t o the s t u d e n t s , as well as checking responses and summarizing the students' work f o r the t e a c h e r . How the q u e s t i o n i s ; "what i s the d e c i s i o n s t r u c t u r e t h at i s t o be used that determines how the student w i l l be branched through a CAI course?" DECISION STRUCTURES Small wood (35) i n A D e c i s i o n S t r u c t u r e For Teaching. Machines developed a model f o r a d e c i s i o n system t h a t can use past i n p u t s t o the system i n d e c i d i n g among v a r i o u s a l t e r n a t e p r e s e n t a t i o n s of the m a t e r i a l . He attempted to o r g a n i z e h i s d e c i s i o n process so as to be s i m i l a r to that of a p r i v a t e t u t o r . I f t h i s d e c i s i o n process i s t o be u s e f u l i t must have the a b i l i t y to adapt to students and to improve i t s e f f e c t i v e n e s s with experience. I f a student i s very slow and needs many v i s u a l a i d s or i f he learns more q u i c k l y than o t h e r s , the teaching machine should detect these c h a r a c t e r i s t i c s i n the student and take advantage of them by branching the student through more a p p r o p r i a t e b l o c k s of m a t e r i a l . I t i s p o s s i b l e f o r a t e a c h i n g machine to give the more i n t e l l i g e n t students a deeper and f u l l e r p r e s e n t a t i o n of the s u b j e c t matter while p r e s e n t i n g a slower student with a l e s s r i g o r o u s treatment of t h e same m a t e r i a l . Smallwood pointed out that "a good t e a c h i n g machine should be capable of improving i t s d e c i s i o n processes as i t ' l e a r n s ' more about the e f f e c t s t h a t are caused by the d e c i s i o n s " (36:2). Smallwood had the computer c o l l e c t and use i n f o r m a t i o n t o r e - e s t i m a t e the parameters used i n making the branching d e c i s i o n s as the computer taught m i n i a t u r e geometry to twenty Massachusetts I n s t i t u t e of Technology s t u d e n t s . He succeeded i n demonstrating that h i s model d i d adapt the d e c i s i o n r u l e as more data was used to estimate the parameters of the model. He pointed out at the end of the study that even though h i s model would adapt the d e c i s i o n r u l e he d i d not know i f the students l e a r n e d any b e t t e r with h i s model or one t h a t d i d not adapt to past i n f o r m a t i o n . Stolurow has been c l o s e l y a s s o c i a t e d with another i n s t r u c t i o n a l system, SOCRATES, t h a t was designed at the U n i v e r s i t y Of I l l i n o i s . Stolurow and h i s a s s o c i a t e s where attempting to c o n s t r u c t a d e c i s i o n making system that, g i v e n a l l the previous i n f o r m a t i o n p o s s i b l e on the student, c o u l d p r e d i c t where the s t u d e n t should s t a r t a CAI course. Once the student s t a r t e d the course the program was to adapt to responses of the student and a p p r o p r i a t e l y branch the student through the CAI course. The problem of attempting to s o l v e the best way of using a l l the i n f o r m a t i o n a v a i l a b l e about a student i n order to optimize the t e a c h i n g s t r a t e g y used with him i s very s i m i l a r t o the problem t h a t Smallwood was working with, Stolurow (37) pointed out that much more r e s e a r c h must be done i n the area of d e c i s i o n s t r u c t u r e s before we w i l l have a s a t i s f a c t o r y model. Other s h o r t term o p t i m i z a t i o n s t r a t e g i e s were discussed by Atkinson (2:143-165). Ha has worked on some d e c i s i o n s t r a t e g i e s in r e a d i n g programs f o r elementary s c h o o l c h i l d r e n . Atkinson p o i n t s out that "even i f short—term o p t i m i z a t i o n s t r a t e g i e s can be d e v i s e d which are e f f e c t i v e , a t o t a l r eading c u r r i c u l u m that i s optimal s t i l l has not been achieved (2:163). Stolurow and Davis (38) reviewed s t u d i e s of i n t e r a c t i o n of i n d i v i d u a l d i f f e r e n c e s (ID) v a r i a b l e s with methods of i n s t r u c t i o n and concluded that such i n t e r a c t i o n s occur i n a v a r i e t y of i n s t r u c t i o n a l s e t t i n g s and methods. They f i n i s h e d t h e i r paper by suggesting that CAI w i l l be a tremendous a i d i n conducting research i n ID-method i n t e r a c t i o n s and i n implementing i n d i v i d u a l i z e d i n s t r u c t i o n . Two years l a t e r D avis, Denny and Harzocco (9) reviewed theory and e m p i r i c a l research on i n d i v i d u a l d i f f e r e n c e s i n l e a r n i n g and r e p o r t e d r e s e a r c h on the i n t e r a c t i o n of ID and method v a r i a b l e s i n CAI and programmed i n s t r u c t i o n (PI) i n a c o l l e g e - l e v e l remedial mathematics course. The ID i n c l u d e d numerousness a b i l i t y , a t t i t u d e , and i n t e r e s t t e s t s . They concluded t h a t the ID v a r i a b l e s had no r e l a t i o n s h i p with the treatments and were of no value i n p r e s c r i b i n g i n s t r u c t i o n a l treatments. I t should be c l e a r t h a t there i s l i t t l e agreement on what v a r i a b l e s , i f any, should be i n c l u d e d i n a d e c i s i o n making model f o r CAI. U n t i l there i s some agreement on what v a r i a b l e s are important i t i s i m p o s s i b l e t o decide on a decison making model to c o n t r o l the i n s t r u c t i o n a l s t r a t e g y . Because of the lack of agreement as to the make up of a d e c i s i o n making model the author suggests that a studen t , assuming h i s b e t t e r self-awareness of a l l h i s i n t e r n a l mental processes and immediate s t a t e s of awareness, can best s e l e c t h i s own s t r a t e g y f o r a c q u i r i n g a s e t of concepts. Gay (14) has done some r e s e a r c h t h a t suggests that males w i l l do b e t t e r i f they have c o n t r o l over the l e v e l of d i f f i c u l t y while females w i l l do b e t t e r i f the computer c o n t r o l s the l e v e l of d i f f i c u l t y . Gay found t h a t i n a CAI course on polynomial equations, boys achieved b e t t e r r e s u l t s when they c o n t r o l l e d t h e i r own l e v e l of d i f f i c u l t y while g i r l s a chieved b e t t e r r e s u l t s when the number of questions that they were given at any one LEVEL was based on t h e i r memory r e t e n t i o n . LEVELS: A series of problems or types of questions arranged sequentially according to the order of d i f f i c u l t y as determined by the author, other teachers, and the professors consulted. See Appendix A for a l i s t i n g of the 60 LEVELS used. PATH: a record of the branches to EASIER LEVELS. The f i r s t time the student signs onto the computer terminal his PATH i s n u l l , and i t w i l l stay n u l l u n t i l the program branches to an EASIER LEVEL. For example i f the program branched to an EASIER LEVEL, e.g. 34, from LEVEL 38 PATH would be the vector of one element,38. Now i f the program branches again to an EASIER LEVEL, say 32, from LEVEL 34 PATH would now be the vector PATH=34,38. Now when the program branches to a HARDER LEVEL from LEVEL 32 the program w i l l branch to LEVEL 34 not the next LEVEL,33. HARDER: a higher LEVEL. In most cases the program w i l l branch to the next higher LEVEL whan a HARDER LEVEL i s requested. There are two exceptions. The f i r s t one i s obvious in that i f the program i s at LEVEL 60 and a HARDER LEVEL i s requested the program cannot branch to LEVEL 61 since LEVEL 61 does not e x i s t . In this case the program stays at LEVEL 60. The other exception i s when PATH i s not the n u l l vector, the program has reached the current LEVEL by branching to an EASIER LEVEL. If PATH i s not n u l l then the program w i l l branch to the f i r s t element of the PATH vector (see d e f i n i t i o n of PATH). EASIER: a lower LEVEL of that operation wherever possible. If a student i s at an addition question then an EASIER LEVEL would ba an addition question that i s at a lower LEVEL. There i s a l i s t of the EASIER LEVELS used for a l l sixt y LEVELS in Appendix B. Note that i n some cases the EASIER LEVEL i s of a d i f f e r e n t operation: the EASIER LEVEL f o r the lowest LEVEL of m u l t i p l i c a t i o n i s an addition LEVEL. COMPUTER-PACED: the program w i l l branch to a HARDER or an EASIER LEVEL depending on the number of questions the student has answered co r r e c t l y at any given LEVEL. The frequency or the number of questions given at any one LEVEL was i n i t i a l i z e d to 2. The freguency would remain at two u n t i l the program branched to an EASIER LEVEL i n which case the frequency would be increased by two to a maximum of ten. If the student answered more than one-half the questions i n c o r r e c t l y at any given LEVEL then the program w i l l assume that the student does not understand the concept and branch to an EASIER LEVEL. The program w i l l branch to a HARDER LEVEL i f the number correct i s greater than one-half the freguency at any given LEVEL. SELF-PACED: the student determines when he w i l l branch to a HABDEB or an EASIEB LEVEL. When the student i s presented a que s t i o n he may push the key marked 'H» f o r HABDER or a key marked 'E» f o r EASIEB i n s t e a d of answering the question. When the student pushed the key marked 'H* the student was give n the message " IF YOU ANSWER THIS QUESTION CORRECTLY YOU MAY GO ON (to the next LEVEL)." then i f the student answered the q u e s t i o n c o r r e c t l y the computer branched to a HARDER LEVEL. When the student pushed the key marked *E 1 i n s t e a d of answering the g u e s t i c n the program immediately branched to an EASIER LEVEL and presented the student with a guest i o n from the EASIER LEVEL. HYPOTHESES On the b a s i s of the reviewed l i t e r a t u r e the author expects the f o l l o w i n g hypotheses t o be t r u e : H1. Students who have c o n t r o l over the l e v e l of d i f f i c u l t y (group S) w i l l achieve higher p o s t - t e s t s cores than students who do not have c o n t r o l over the l e v e l of d i f f i c u l t y (group C) . H2. Males w i l l achieve higher p o s t - t e s t s c o r e s than females when working on a r i t h m e t i c d r i l l - a n d - p r a c t i c e . H3. There w i l l be an i n t e r a c t i o n e f f e c t between the groups and sex. The author i s assuming t h a t the i n t e r a c t i o n t h a t w i l l occur i s as f o l l o w s : (1) the males i n the SELF-PACED group w i l l achieve higher p o s t - t e s t s c o r e s than the females i n the SELF-PACED group and (2) the females i n the COMPUTER—PACED group w i l l achieve higher p o s t - t e s t s c o r e s than the males i n the COMPUTER-PACED group. In more o p e r a t i o n a l terms, the students t h a t stay at the same l e v e l of d i f f i c u l t y (LEVEL) u n t i l they push a key marked HARDER or EASIER w i l l achieve higher s c o r e s on an a r i t h m e t i c t e s t than students t h a t have no c o n t r o l over t h e i r l e v e l of d i f f i c u l t y . EXPERIMENTAL DESIGN INTRODUCTION The r a t i o n a l e f o r having a SELF-PACED program was that i t would allow the students freedom i n s e l e c t i n g the d i f f i c u l t y of q u e s t i o n s presented to them and as a r e s u l t of t h e freedom these s t u d e n t s would master the m a t e r i a l b e t t e r than students who were COMPUTER-PACED, The h y p o t h e s i s was t e s t e d by comparing the performance of two groups of s t u d e n t s answering questions concerning the m a t e r i a l presented. One group of students had c o n t r o l over the d e c i s i o n of when t o t r y a HARDER or an EASIER LEVEL, and f o r the other group the computer program determined when the LEVEL should be changed. PILOT STUDY A p i l o t study was conducted with two above average grade s i x s t u d e n t s . The main o b j e c t i v e s of the p i l o t study were to determine whether the program was working c o r r e c t l y , the i n s t r u c t i o n s were c l e a r enough f o r the s t u d e n t s to f o l l o w without any d i f f i c u l t y , and f i v e twenty minute s e s s i o n s on the d r i l l - a n d - p r a c t i c e program were reasonable. The two s t u d e n t s were brought out to the U n i v e r s i t y Of B r i t i s h Columbia to work on the d r i l l - a n d - p r a c t i c e programs f o r t h r e e h a l f - d a y s . The computer t e r m i n a l used was the same type as used i n the main study, a t e l e t y p e w r i t e r connected to the u n i v e r s i t y ' s IBM 360/67 computer by telephone l i n e s . The two students a l t e r n a t e d working a t the computer t e r m i n a l . The g i r l working on the COMPUTER—PACED program had her twenty minute s e s s i o n f i r s t , then the boy on the SELF-PACED program took h i s twenty minute s e s s i o n . Both students were encouraged to ask qu e s t i o n s while they were working a t the computer t e r m i n a l and a f t e r they had f i n i s h e d t h e i r t u r n . Both s t u d e n t s asked some gu e s t i o n s while working at the t e r m i n a l but saved most of t h e i r g uestions u n t i l t h e i r s e s s i o n was f i n i s h e d - While the one student was on the computer t e r m i n a l the other student was ab l e t o ask the author guestions or t o engage i n other a c t i v i t i e s except watching the other student working at the computer t e r m i n a l . There were very few t e c h n i c a l problems during the p i l o t study. The computer shut down onca d u r i n g one of the s e s s i o n s but the author was ab l e to r e s t a r t the student at the same p o i n t i n the program. There was a l s o some i n t e r f e r e n c e on the telephone l i n e s but t h i s caused very few problems even though i n some cases the student would have to ret y p e h i s answer. The students had two twenty minute s e s s i o n s each day f o r three days f o r a t o t a l of s i x s e s s i o n s each. One student reached the LEVEL 53 while the other student reached LEVEL 51. An a n a l y s i s of the s t u d e n t s progress i n d i c a t e d t h a t too much c a l c u l a t i o n was i n v o l v e d i n some of the higher LEVELS, and as a r e s u l t the s t u d e n t s were making mistakes even when they understood the concept i n v o l v e d . The LEVELS i n v o l v e d were changed so as to n e c e s s i t a t e l e s s computation. As a r e s u l t of the p i l o t study the author concluded that the i n s t r u c t i o n s were c l e a r enough f o r the s t u d e n t s to f o l l o w and to understand. The author a l s o f e l t t h a t s i n c e some of the LEVELS were changed t o i n v o l v e l e s s computation that f i v e twenty minute s e s s i o n s would be an a p p r o p r i a t e amount of time f o r the m a t e r i a l presented. The students both i n d i c a t e d t h a t they f e l t t h a t the twenty minute s e s s i o n s were not too long and one student s t a t e d t h a t he f e l t the length of time per s e s s i o n should be i n c r e a s e d . FORMATION OF GROUPS A grade s i x c l a s s was s e l e c t e d from a p a r o c h i a l s c h o o l i n Vancouver, B r i t i s h Columbia. The s e l e c t e d s c h o o l i s s i t u a t e d i n a lower—middle c l a s s d i s t r i c t where most of the old homes are being r e p l a c e d by h i g h - r i s e apartment b u i l d i n g s . The c l a s s c o u l d be considered r e p r e s e n t a t i v e f o r the type of d i s t r i c t the s c h o o l i s i n . The number of c h i l d r e n i n the s c h o o l i s d e c l i n i n g every year because many of the new apartment b u i l d i n g s w i l l not take c h i l d r e n . The Canadian B a s i c S k i l l s Test In Mathematics was the most r e c e n t s t a n d a r d i z e d t e s t t h at these students had taken. T h i s t e s t was wr i t t e n i n the f o u r t h month of the s i x t h year and the students averaged a grade e q u i v a l e n t of s i x years e i g h t month with a range from f i v e years zero months to ei g h t years f o u r months. The c l a s s was d i v i d e d i n t o f o u r groups f o r t h i s s tudy. The f o u r t e e n boys were separated from the ten g i r l s , then the boys were randomly assigned to the SELF-PACED and the COMPOTER-PACED groups. The g i r l s were s i m i l a r l y assigned to the COMPOTER-PACED and SELF-PACED groups. The random assignment t o groups aided i n making the groups f a i r l y equal but a p r e - t e s t was used as a c o v a r i a t e t o a d j u s t f o r any remaining d i f f e r e n c e s . MATERIAL L e v e l s The two groups, the SELF-PACED and the COMPUTER-PACED, both worked on the same d r i l l - a n d - p r a c t i c e m a t e r i a l c o n s i s t i n g of questions i n v o l v i n g the f o u r b a s i c o p e r a t i o n s i n whole numbers and i n decimal f r a c t i o n s . A complete l i s t of the s i x t y d i f f e r e n t LEVELS or types of problems used can be found i n Appendix A. Computer TerminaIs Two t e l e t y p e w r i t e r s were i n s t a l l e d i n the s c h o o l where the twenty f o u r grade s i x students worked on the d r i l l - a n d - p r a c t i c e q u e s t i o n s . The t e l e t y p e w r i t e r s were connected by telephone l i n e s to the U n i v e r s i t y of B r i t i s h Columbia IBM 360/67 computer. Both computer te r m i n a l s were the same but one t e r m i n a l was always used by the COMPUTER-PACED group and the other computer t e r m i n a l was always used by the SELF—PACED group. The ENTER and DECIMAL keys were c l e a r l y marked with p l a s t i c tape so that the students would be a b l e to f i n d these keys e a s i l y . The SELF-PACED computer t e r m i n a l had two a d d i t i o n a l keys marked with p l a s t i c tape, one marked H f o r HARDER and the other marked E f o r EASIER . Test A l l the s t u d e n t s were given a p r e - t e s t (see Appendix C) and a t the end of the study they were g i v e n the same t e s t as a p o s t - t e s t . The t e s t was c o n t r u c t e d by having the computer program generate one question from each LEVEL. PROCEDURE A l l the s t u d e n t s i n the study were taken out to the U n i v e r s i t y of B r i t i s h Columbia f o r a tour of the u n i v e r s i t y ' s Computing Centre and to see the IBM 360/67 computer so that they would have some idea of what a computer i s , The s t u d e n t s were given a chance t o play games such as TICTACTOE and COINFLIP with the IBM 360/67 computer so they a l l had some f a m i l i a r i t y with pushing the keys on computer t e r m i n a l s before they s t a r t e d using the t e r m i n a l s at t h e i r s c h o o l . The study s t a r t e d on a F r i d a y whan the s t u d e n t s were t o l d that they would be s t a r t i n g to do t h e i r a r i t h m e t i c e x e r c i s e s on computer t e r m i n a l s the f o l l o w i n g week. Af t e r the students had been given o p p o r t u n i t y to ask questions they were given one hour t o complete the p r e — t e s t . The stu d e n t s were gi v e n e x t r a paper where they were asked to do a l l c a l c u l a t i o n s . At the end of the hour the t e s t s and a l l the papers were c o l l e c t e d . The students were not given the r e s u l t s on the t e s t ; nor were they given t h e i r t e s t s back u n t i l a f t e r the end of the study. The f o l l o w i n g Monday was a s c h o o l h o l i d a y so Tuesday was the f i r s t day t h a t the students worked at the computer t e r m i n a l s . The o n l y i n i t i a t i o n that the students had other than the p l a y i n g of games on the computer t e r m i n a l s a t the u n i v e r s i t y was t h e i r f i r s t twenty minute s e s s i o n when the author e x p l a i n e d how to enter t h e i r answers on the computer t e r m i n a l s . None of the students had any t r o u b l e a f t e r they were helped e n t e r i n g the f i r s t two or three answers. The SELF-PACED students were shown how to request a HARDEE LEVEL and an EASIER LEVEL only a f t e r they demonstrated that they were not having any d i f f i c u l t y entering their answers. This usually took about two or three minutes. The students might have needed a longer introduction period i f the beginning questions had been more d i f f i c u l t but since the f i r s t LEVEL contained questions l i k e 3 + U = ? , the only d i f f i c u l t y with the f i r s t LEVEL was getting used to the computer terminal. For the length of t h i s study the author taught a t h i r t y minute arithmetic lesson to the students at 9 A.M. every morning. The lessons consisted of a review of the four basic operations in whole numbers and in decimal fr a c t i o n s which included a l l the sixty LEVELS in the d r i l l - a n d - p r a c t i c e program l i s t e d in Appendix A. After t h e i r arithmetic lesson the students continued with their normal classes. The students names were l i s t e d on the blackboard i n the order that they were to have their d r i l l - a n d — p r a c t i c e session at the computer terminal. There was one l i s t of names for each terminal. The two terminals were marked 'COMPUTES-PACED• and 'SELF-PACED' as were the two l i s t s on the blackboard in the classroom. The students on the SELF-PACED l i s t always worked on the computer terminal marked 'SELF-PACED* and the students on the COMPUTER—PACED l i s t always worked on the t e r m i n a l marked 'COMPUTER-PACED', T h i s proved to be very h e l p f u l i n that the students soon knew e x a c t l y which t e r m i n a l to go to and no student was ever g i v e n the wrong program. The very f a c t t h a t the students knew t h a t they were working on a d i f f e r e n t program from the s t u d e n t s i n the other group may have had some e f f e c t on the outcome but they were both experimental groups so the e f f e c t should have been the same f o r both groups. When the f i r s t student on a l i s t f i n i s h e d h i s l e s s o n on the computer t e r m i n a l he would n o t i f y the next student who would then q u i e t l y leave the classroom f o r h i s s e s s i o n on the t e r m i n a l . T h i s process continued u n t i l a l l the st u d e n t s had t h e i r t u r n . Each l i s t was r o t a t e d by two st u d e n t s each day so t h a t the s t u d e n t s would not be working at the t e r m i n a l the same time every day and thus miss time i n the same s u b j e c t each day. Because of the number of sudants i n v o l v e d per computer t e r m i n a l i t was necessary to have the students continue through t h e i r r e c e s s and t h e i r lunch breaks. The st u d e n t s even agreed to stay a f t e r s c h o o l i f a l l the students on a l i s t d i d not f i n i s h . One student c o u l d not take h i s turn i f i t happened to f a l l during the lunch hour so the p o s i t i o n of the names on one l i s t had t o be a l t e r e d at times. The l a s t s t udent f i n i s h e d h i s turn at about 3:10 p.m. on Monday, the f i f t h s c h o o l day that the s t u d e n t s had been working at the computer t e r m i n a l s . Tuesday morning the s t u d e n t s were gi v e n one hour to complete the t e s t again. Eleven calendar days had passed s i n c e the students f i r s t wrote the t e s t . The same t e s t was used as the p r e — t e s t and the p o s t — t e s t only because i t was extremely u n l i k e l y t h a t any student would remember any of the q u e s t i o n s . The s t u d e n t s had no idea t h a t the same t e s t would be used. A l l the papers that the students had used f o r c a l c u l a t i o n s while w r i t i n g the t e s t were c o l l e c t e d . None of the q u e s t i o n s on the t e s t were ever d i s c u s s e d with the c h i l d r e n , and the s t u d e n t s had c a l c u l a t e d a great number of problems between the two a d m i n i s t r a t i o n s of the t e s t s i m i l a r t o those on the t e s t . A f t e r the s t u d e n t s had w r i t t e n the p o s t - t e s t they were asked i f they r e c o g n i z e d any of the q u e s t i o n s . About one-fourth of the s t u d e n t s s a i d t h a t they thought they had seen some of the g u e s t i o n s before and only one student s a i d t h a t he was sure t h a t i t was the same t e s t that they had w r i t t e n b e f o r e . STATISTICAL ANALYSIS Data For each of the students two scores were obtained. The f i r s t was h i s s c o r e on the p r e — t e s t and the second was h i s score on the same t e s t used as a p o s t - t e s t . The LEVEL that each student achieved d a i l y was recorded. T h i s data i s i n Appendix E. THE NUMBER OF SUBJECTS IN THE TWO FACTOR DESIGN USED N = 7 N=7 N=5 N = 5 i In order to make i t easier for labeling the diagrams the groups were labeled as M (male), F (female), S (SELF-PACED), and C (COMPUTER—PACED) , A standard analysis of covariance program at the University of B r i t i s h Columbia (BMDX64) was used to analyze the data for this two factor fixed design. An alpha l e v e l of 0.05 was selected. The c r i t i c a l value f o r F with one and nineteen degrees of freedom f o r th i s alpha l e v e l i s 4.38. ANALYSIS OF RESULTS TESTING OF HYPOTHESES A summary of the analysis of the post-test scores using the pre—test scores as a covariate may be found in the following table. TABLE 2 ANALYSIS OF VARIANCE TABLE SOURCE SUM OF SQUARES D.F. MEAN SQUARE F MEAN GROUPS SEX GROUP X SEX COVS COV. 1 ERROR 158.38185 35.86396 67.08819 1.29 310 1252.86695 1252.86694 634.27590 19 158. 381 85 3 5.863 95 67.08818 1.293 10 1252. 86694 1252. 86694 33. 3 82 93 4.74440 1.07432 2.00965 0.03874 37.53015 37.53015 Table 3 contains the expected scores for each of the four c e l l s of two by two f a c t o r i a l design when the pre—test scores were used as a covariate. See Appendix F for the observed means for both the pre-test and the post-test. T A B L E 3 ADJUSTED EXPECTED MEANS M F i 1 S | 46.83 | 49.74 | 48.29* C | 43.84 | 47.72 | 45.78* 45.34* 48. 73* 47.04** * THE AVERAGE OF THE TWO MEANS ** THE EXPECTED GRAND MEAN S.229.%.h§sis I (H 1) Since the F value of 1.07 was less than the c r i t i c a l value of 4.38 H1 was rejected. This indicates that there was no s i g n i f i c a n t difference i n achievement of post-test scores between thoses students who had control over the le v e l of d i f f i c u l t y (group.S) and those that did not have control over the l e v e l of d i f f i c u l t y (group C). Since the F value of 2.01 was less than the c r i t i c a l value of 4.3 8 H2 was rejected. This means that there was no s i g n i f i c a n t difference i n achievement of post—test scores between males and females when working on arithmetic d r i l l - a n d - p r a c t i c e . I l f o t h e s i s 3 (H3) The hypothesis that there would be a s i g n i f i c a n t interaction e f f e c t between groups and sex, H3, was rejected because the F value of 0.04 i s less than the c r i t i c a l value of 4,38. INTERPRETATION OF RESULTS The expected values i n TABLE 3 indicate that the students on the COMPUTER—PACED program scored higher than those on the SELF-PACED program though not s i g n i f i c a n t l y higher. Even though the females did not achieve s i g n i f i c a n t l y higher scores than the males i t i s intere s t i n g to note that the g i r l s in both the COMPUTER—PACED and the SELF-PACED groups did better on the post—test. It was expected that the females would do r e l a t i v e l y better on the COMPUTER—PACED than the SELF-PACED program but i t was not expected that the females would do better on the COMPUTER—PACED than the SELF-PACED program. ANALYSIS OF ADDITIONAL DATA The students in the SELF-PACED group achieved higher LEVELS on the average than did the students in the COMPUTER-PACED group every day except on day one (see Appendix F). It appeared to take some time for the SELF-PACED students to become fam i l i a r with how to ask for questions from HARDER or EASIER LEVELS. The LEVELS achieved on the f i f t h day correlate f a i r l y well with the c r i t e r i o n scores on the post—test. This was expected since the test was constructed by taking one question from each LEVEL. The SELF-PACED students answered approximately the same number of questions as did the COMPUTER—PACED group but the number of questions that they answered at each LEVEL varied a great amount. Some of the students from the SELF-PACED would answer more questions on the d i f f i c u l t LEVELS and only one from LEVELS that they considered t r i v i a l . This was the behavior the author hoped for but there were about four students i n the SELF-PACED group that did just the opposite. When these students came to a LEVEL that was easy for them they would stay on that LEVEL for about ten guestions before moving on to a HARDER LEVEL. When they were presented a question from a LEVEL that they considered d i f f i c u l t they would request an EASIER LEVEL or else request a HARDER LEVEL and guess at the answer just so that they could get to another LEVEL that they considered easy. Student #15 would not have ventured much past LEVEL one or two had i t not been f o r the pressure exerted by the other students. Each student was given h i s p r i n t o u t from the computer t e r m i n a l when he f i n i s h e d h i s t u r n . . The f i r s t time student #15 brought h i s p r i n t o u t back t o t h e classroom he bragged about how many qu e s t i o n s he had done. The other s t u d e n t s q u i c k l y looked a t h i s p r i n t o u t to see the guestions t h a t he d i d and then teased him about doing q u e s t i o n s l i k e 7 - 3 = ?, which i s LEVEL two. Student #15 was r e a l l y t h r i l l e d with doing the d r i l l - a n d - p r a c t i c e e x e r c i s e s at the computer t e r m i n a l f o r the f i r s t two days but a f t e r the second day the pressure from the other s t u d e n t s f o r c e d him ahead to g u e s t i o n s where he had to c a l c u l a t e the answers on the scrap paper provided and t h i s became too much work f o r him. He i s a very slow student, day dreams a great deal and i s the only one in the c l a s s t h a t w i l l repeat grade s i x next year. Other than student #15 there was nothing but excitement and enthusiasm shown toward the d r i l l - a n d - p r a c t i c e e x e r c i s e s . As shown i n Appendix F the s t u d e n t s averaged a gain of e i g h t marks on the p o s t - t e s t over the p r e - t e s t . T h i s i s a gain of 20 per c e n t . Some of the gain i s because of the novelty e f f e c t of having a d i f f e r e n t teacher f o r a r i t h m e t i c , some because they were a b l e to use computer t e r m i n a l s and some because they were re-taught the material ana were given questions to do s i m i l a r to those on the test. CONCLUSION AND SUGGESTIONS FOB FURTHER RESEARCH SUMMARY This study was designed to determine whether or not i t makes any difference i f the student controlled when the computer program branched to a different LEVEL or i f the computer program controlled when i t branched to a d i f f e r e n t LEVEL. There was no s i g n i f i c a n t difference between the two methods of the selection of diff e r e n t LEVELS. The results of t h i s study indicate that when working with arithmetic d r i l l - a n d - p r a c t i c e , students w i l l do as well i f the computer program controls when to branch as they would i f the students control when to branch to a d i f f e r e n t l e v e l of d i f f i c u l t y . Further analysis showed that there was no s i g n i f i c a n t difference between the males and females performance and that there was no s i g n i f i c a n t interaction (group X sex) ef f e c t . The author of t h i s study i s optimistic about the future of the computer i n the classroom especially for arithmetic d r i l l - a n d - p r a c t i c e . The students seemed to enjoy working at the computer terminals and they had very l i t t l e trouble getting used to the computer terminals. The f a c t that one can summarize a student's work for the day, as shown in appendix F, or for the week or month and see exactly where the student i s having d i f f i c u l t y i s probably the most important aspect of computerized d r i l l - a n d - p r a c t i c e . The result that the females scored higher, though not s i g n i f i c a n t l y higher, on both the COMPUTER—PACED and the SELF-PACED programs i s contrary to the results that Gay (14) found when he had students working at a CAI t u t o r i a l program written to teach f i r s t year college students polynomial equations. The reason f o r the d i f f e r e n t r e s u l t s could have been because the students in t h i s study were much younger than those i n Gay's study. Silberman pointed out that "undoubtedly there w i l l be an age gradient i n determining the extent to which the student should control his own i n s t r u c t i o n , younger children w i l l reguire more structure" (33:51). another reason for the d i f f e r e n t r e s u l t s could have been that the material was different. LIMITATIONS OF THE STUDY The students were a l l t o l d that there were 60 LEVELS. They were a l s o t o l d t h a t everyday a f t e r they f i n i s h e d t h e i r s e s s i o n at the computer t e r m i n a l they would be t o l d the LEVEL achieved only i f they asked. T h i s was a perso n a l t h i n g between the author and the stude n t , the student would only be t o l d h i s own LEVEL. For many of the stu d e n t s i t was a c o m p e t i t i o n to see i f they could reach a hi g h e r LEVEL than t h e i r f r i e n d . Some of the st u d e n t s set t h e i r goal at LEVEL 60 before the f i v e s e s s i o n s were f i n i s h e d . I f the stu d e n t s had not been t o l d that there were 60 LEVELS and i f they had not been t o l d t h e i r own LEVEL at the end of each s e s s i o n the r e s u l t s might have been d i f f e r e n t . A grade s i x c l a s s of twenty f o u r s t u d e n t s was s e l e c t e d from a p a r o c h i a l s c h o o l . The r e s u l t s of the study may have been d i f f e r e n t i f a l a r g e c l a s s i n a p u b l i c s c h o o l had been s e l e c t e d . The students that attend p a r o c h i a l s c h o o l s may not be r e p r e s e n t a t i v e of a l l s t u d e n t s . The s u b j e c t s chosen were from a s m a l l c l a s s of twenty f o u r students. The c l a s s was very c l o s e i n t h a t they always pl a y e d together at r e c e s s , noons, and a f t e r s c h o o l as a group with vey few o u t s i d e r s . T h i s c l o s e n e s s would r e s u l t i n more i n t e r a c t i o n between the students about the experimental program than i f the c l a s s were not so c l o s e . The d e c i s i o n model used i s onl y one of an i n f i n i t e number of p o s s i b l e d e c i s i o n models. I f a d i f f e r e n t d e c i s i o n model was used f o r the students i n the COMPUTES—PAC ED group the r e s u l t s may have been very d i f f e r e n t . The s i t u a t i o n of the teacher t e a c h i n g concepts f o l l o w e d by d r i l l - a n d - p a r c t i c e was not r e a l l y achieved i n that the concepts presented were not new concepts to the students. The st u d e n t s had p r e v i o u s l y been taught how to do a l l the m a t e r i a l covered by the d r i l l - a n d — p r a c t i c e programs. The students were re-taught, or given a review o f , the concepts and the review was f o l l o w e d by d r i l l - a n d - p r a c t i c e . SUGGESTIONS FOR FURTHER RESEARCH A study should be conducted using the same m a t e r i a l with a d i f f e r e n t d e c i s i o n model f o r the COMPUTER-PACED group. The study c o u l d have many d e c i s i o n making models i f the study i n v o l v e d enough students to make more groups. The study may show that one d e c i s i o n making model that was used was s u p e r i o r t o the o t h e r s or i t may show t h a t i t make very l i t t l e d i f f e r e n c e which d e c i s i o n making model i s used. The SELF-PACED group may achieve higher p o s t - t e s t score than some of the COMPUTER-PACED groups. Another study s i m i l a r t o t h i s study should be conducted with s t u d e n t s over many grades. P o s s i b l y the r e s u l t s may be very d i f f e r e n t f o r students i n grade three, s i x , and nine. There i s a need f o r more re s e a r c h i n t o the ways a person l e a r n s . As more r e s e a r c h i s done with d e c i s i o n making models p o s s i b l y man w i l l l e a r n much more of how he l e a r n s . Once more knowledge about l e a r n i n g i s known then the d e c i s i o n can be made of whether the student o r the computer can best guide the student through the course m a t e r i a l . BIBLIOGRAPHY 1. Atkinson, R. C. "Computer Instruction and the learning Process," American Psychology^ 23, (1968), 225. 2. Atkinson, R. C. and H. A. Wilson (eds.). Comguter Assisted Instruction: A Book of Readings. New York: Academic Press, 1969. 3. Baker, Frank B. "Computer-Based I n s t r u c t i o n a l Management Systems: A F i r s t Look," Review of Educational Research, Feb. 19 71. Pp. 51-70. 4. Belcastro, F. P. "Programmed Learning and In t e l l i g e n c e , " School Science and Mathematics, 1966. Pp. 29-36. 5. Brownell, W. A. "Arithmetic Readiness as a P r a c t i c a l Classroom Concept," Elementary School Journal^ 52, (1952), 15-22. 6. 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APPENDIX A LEVELS A,B AND 7 ABE WHOLE NUMBERS FOR LEVELS #1 TO #24 1 A + B = 7 4 + 2 = ? 0<A, B<10 2 A — B = 7 8 - 3 = ? 0<A, B<10; B<A 3 A + B = 12 + 46 = ? 10<A,B<100; NO CARRYING 4 A — B = 7 347 - 221 = ? 10<A,B<1000; NO BORROWING 5 A + B = 7 709 + 231 = ? 10<A,B<1000; CARRYING ON DIGIT 1 6 A — B = 7 870 - 454 = ? 100<A<1000 B<=A; 8 A + B = ? 61 + 146 = ? A — B = ? 745 - 684 = ? 9 A + B = ? 98+ 665 = ? 10 A — B = ? 910 + 641 = ? BORROWING ON DIGIT 1 10<A,B<1000; CARRYING ON DIGIT 2 100<A<1000; B<=A; BORROWING ON DIGIT 2 10<A,B<1000; CARRYING ON DIGITS 1 AND 2 100<A<1000; B<=A; BORROWING ON DIGITS 1&2 11 A X B = 7 3 X 4 = 7 0<A, B<10 12 A X B 7 54 X 10 - 7 0<A<100; B=10 OR 100 13 A X B = 7 12 X 27 = 7 0<A, B< 100 14 A / B = ? 72 / 8 = 7 0<B# ? < 1 0 15 A / B = 7 994 / 7 = 7 10<A<1000 16 A / B 7 100 / 5 — 7 A = 1 0 , 100,OR 1000 17 A + 7 B 751 + ? = 780 ONE OF #4,#6,#8,OR #10 18 7 + A = B ? + 38 = 379 ONE OF #4,#6,#8,OR #10 19 A — 7 •= B 902 - ? = 23 ONE OF #4,#6,#8,OR 20 7 — A = B 7 _ 72 = 21 ONE OF #3,#5,#7,OR #9 21 A X 7 - B 14 X ? = 84 10<A<99; 2<?<19 22 •> X A = B ? X 13 = 65 10<A<99; 2<?<19 23 A / 7 = B 156 / ? = 13 10<B<99; 2<?<19 24 7 / A = B ? / 1 2 = 6 REFER TO #13 A,B AND 7 ABE DECIMAL FRACTIONS UNLESS OTHEBWIS E SPECIFIED 25 A + B - 7 0.3 + 0.4 = ? 0 . 0 < A , B < 1 . 0 , 1 DEC PL; NO CARRYING 26 A — B 7 0.7 - 0.2 = ? 0.0<A,B<1.0,1 DEC PL; NO BORROWING 27 A + B = 7 2. 1 + 7. 8 = ? 1. 0<A, B<10.0, 1DEC PL; NO CARRYING 28 A - B 7 2.5 - 1.4 = ? 1.0<Ar B<10.0, 1DEC PL; NO BORROWING 29 A + B - 7 8.81 + 1.16 = 7 1.0<A,B<10.0,2 DEC PL; NO CARRYING 30 A — B 7 8.88 - 3.62 = 7 1.0<ArB< 10.0,2 DEC PL; NO BORROWING 31 A + B 7 0.7 + 0.02 = 7 0.0<A,B<1.0; A 1 DEC PL; B 2 DEC PL; NO CARRYING 32 A — B — 7 0.5 - 0.34 = 7 0.0<A,B<1.0; A 1 DEC PL; B 2 DEC PL; NO BORROWING 33 A + B = 7 151. 48 + 833. 34 - 7 100<A,B<1000,2 DEC PL; CARRYING ON D1 34 A - B •= 7 772. 81 - 562. 77 = 7 100<A,B<1000,2 DEC PL; BORROWING ON D1 35 A + B = 7 4 + 8.2 = ? A IS A WHOLE NUMBER; 1.0<B< 10.0,1 DEC PL 36 A - B = 7 8 - 1.6 = ? A IS A WHOLE NUMBER; 1.0<B<10.0,1 DEC PL 37 A + B 7 23.57 + 104.2 = 7 A HAS 2 OR 3 DEC PL; B HAS 1 DEC PL 3 8 A - B = ? 267.8 - 63.37 = ? A HAS 1 DEC PL; B HAS 2 OR 3 DEC PL 3 9 A X B = ? 2 X 0.3 = ? A IS A WHOLE # <10; 0. 0<E<0.7,1 DEC PL 40 A X B = ? 4 X 1.2 = ? A IS A WHOLE # <10; 1.0<B<10.0,1 DEC PL 41 A X B = ? 46 X 4. 8 = ? 10<A IS A WHOLE #<100; 1.0<B<10.0,1 DEC PL 42 A / B = ? 62.4 / 2.6 = ? 10<? IS A WHOLE #<100; 1.0<B<10.0; 43 A / B = ? 34.8 / 0.29 = ? 100<? IS A WHOLE #<1000; A HAS 1 DEC PL; 0. 1<B<1.0,2 DEC PL 44 ? X A = B ? X 0.4 = 2.4 ? IS A WHOLE #<10; 1.0<=A<=9.0,1 DEC PL 45 ? X A = B ? X 1.6 = 36.8 10<? IS A WHOLE #<100; 1.0<A<2.0,1 DEC PL 46 A X ? = B 1.8 X ? = 111.6 10<? IS A WHOLE #<100; 1.0<A<2.0,1 DEC PL 47 A X B = ? 86.6 X 0.344 = ? A=0. 1**N X T WHERE 10<T<1000,N=1,2, OR 3 B=0.1**N X T WHERE 10<T<1000,N=1,2, OR 3 48 A X B = ? 10 X 3.4 = ? A=10r100, OR 1000; 1. 0<B< 100.0, 1 DEC PL 49 A X B = ? 0.01 X 3.41 = ? A=0. 1,0.01, OR 0.001 ; B=0. 1**N X T WHERE 10<T<1000,N=1 OR 2 50 A X B = ? 7.62 X 0.01 = ? B=0. 1, 0. 0 1,0.00 1 ; A=0, 1**S X T WHERE 10<T<1000,N=1 OR 2 51 A + ? = B 38.37 +? = 892.7 REFER TO #38 52 ? + A = B ? + 6.653 = 974.7 REFER TO #38 53 A - ? = B 896.3 - ? = 7.873 REFER TO #38 54 ? — A = B ? - 4.2 = 1.6 ONE OF f27,#29,#31,OR #33 55 ? / A = B 56 A / B = ? 57 A / B = ? 58 ? X A = B 59 A X ? = B 60 A / ? = B ? / 2. 88 = 0.352 16.984 / 4.4 = ? 1.8122 / 8. 2 = ? ? X 4.6 = 2. 1068 6.2 X ? = 3.9928 6.0918 / ? = 7. 1 REFER TO #47 1.0<B<10.0, 1 DEC PL; ACB ?=0.1**N X T WHERE 10<T<1000, N=0 OR 1 1.0<B<10.0 f1 DEC PL; A<B; ?=0.1**N X T WHERE 10<T<1000,N=3 OR 4 ONE OF #56 OR #57 ONE OF #56 OR #57 ONE OF #56 OR #57 APPENDIX B EASIER LEVELS A LISTING OF THE EASIEB LEVELS FOB EACH OF THE 60 LEVELS LEVEL EASIER LEVEL EASIEB 1 1 31 29 2 2 32 30 3 1 33 31 4 2 34 32 5 3 35 27 6 4 36 34 7 5 3 7 33 8 6 38 34 9 7 39 25 10 8 40 39 11 9 41 40 12 11 42 16 13 11 43 42 14 13 44 43 15 14 45 44 16 15 46 45 17 10 47 41 1 8 10 48 12 19 10 49 48 20 9 50 49 21 15 51 17 22 21 52 51 23 15 53 19 24 13 54 53 25 1 55 47 26 2 56 43 27 25 57 56 28 26 58 57 29 27 59 58 30 28 60 2 3 APPENDIX TEST 1. 8 + 9 = ? 2. 8 - 2 = ? 3. 41 + 57 = ? 4. 459 - 12 = ? 5. 706 + 246 = ? 6. 291 - 285 = ? 7. 371 + 458 = ? 8. 709 - 518 = ? 9. 293 + 628 = ? 10. 815 - 68 = ? 11. 6 x 5 = ? 12. 59 x 100 = ? 13. 82 x 36 = ? 14. 24 / 3 = ? 15. 558 / 93 = ? 16. 100 / 4 = ? 17. 334 + ? =411 18. ? + 63 = 886 19. 512 - ? = 233 20. ? - 129 = 243 21. 25 x ? = 75 22. ? x 66 = 462 23. 468 / ? = 26 24. ? / 79 = 97 25. 0.1 + 0.7 = ? 26. 0.9 - 0.1 = ? 27. 8.1 + 1.8 = ? 28. 6.5 - 5.4 = ? 29. 1.11 + 8.83 = ? 30. 4. 76 - 4.55 = ? 31. 0.4 + 0.48 = ? 32. 0. 8 - 0.17 = ? 33. 1 11. 49 + 786. 31 = ? 34. 778.56 - 115.39 = ? 35. 9 + 4.1 = ? 36. 9 - 4.6 = ? 37. 52. 92 + 411.2 = ? 38. 391. 9 - 17.38 = ? 39. 1 x 0. 1 = ? 40. 5 x 7.8 = ? 41. 83 x 5.9 = ? 42. 275.4 / 3.4 = ? 43. 210.8 / 0.62 = ? 44. ? x 0.8 = 2.4 45. ? x 1.8 = 97.2 46. 1.3 x ? =33.8 47. 0.541 x 7.28 = ? 48. 1000 x 57.7 = ? 49. 0. 01 x 8.88 = ? 50. 7.52 x 0.001 = ? 51. 67. 94 + ? = 698.6 52. ? + 7.587 = 795.7 53. 892. 9 - ? = 7.839 54. ? - 0.1 = 0.61 55. ? / 0.72 = 0.373 225. 05 / 3.5 •= ? 0.22661 / 4.3 = ? ? x 2.1 = 0.8715 3.1 x ? = 2.4397 129.6 / ? =1.5 APPENDIX D SAMPLE PRINTOUTS SAMPLE OF A SELF-PACED PRINTOUT NOTE THE USE OF THE »H» ANS 'E« RESPONSE HI GOOD LOCK IN YOOR WORK TO-DAY PLEASE TYPE IN YOUR I.D. NOMBER #: 14 TYPE 1 IF YOUR NAME IS WENDY OTHERWISE TYPE 0 #: 1 GOOD NOW ON WITH TO-DAYS QUESTIONS 658.71 - 618.32 = ? ?= (34)* #: 40.39 JOLLY GOOD SHOW 879.95 - 317. 18 = ? ?= (34) #: H ANSWER THIS QUESTION CORRECTLY THEN YOU WILL GO ON #: 562.77 O.K. 3 + 3.2 = ? ?= (35) #: H ANSWER THIS QUESTION CORRECTLY THEN YOU WILL GO ON #: 6.2 O.K. 9 - 3.3 = ? ?= (36) #: 12.3 TRY AGAIN 9 - 3.3 = ? ?= #: 5.7 THAT IS CORRECT WENDY 6 - 5.9 = ? ?= (36) #: H ANSWER THIS QUESTION CORRECTLY THEN YOU WILL GO ON #: 0. 1 O.K. •22.76 + 727. 1 = ? ?= (37) #: E 152. 56 + 41 1.37 = ? ?= (33) #: 563.93 GOODBYE FOR NOW, SEE YOU AGAIN WENDY * THE NUMBER WITHIN THE PARENTHESES WERE ADDED LATER ONLY TO INDICATE THE LEVEL SAMPLE OF A COMPUTER-PACED PRINTOUT NOTE THE 'TRY AGAIN' AND 'THE ANSWER IS 101.75' RESPONSES HI GOOD LUCK IN YOUR WORK TO-DAY PLEASE TYPE IN YOUR I.D. NUMBER #: 7 TYPE 1 IF YOUR NAME IS JOHN OTHERWISE TYPE 0 #: 1 GOOD NOW ON WITH TO-DAYS QUESTIONS 0.6 - 0. 15 = ? ?= (32) #: 0.55 TRY AGAIN 0.6 - 0. 15 = ? ?= #: 0.45 THAT IS RIGHT JOHN 356.09 + 423.81 = ? ?= (33) #: 779. 90 GOOD WORK 232. 38 + 24 1.47 = ? ?= (33) #: 473. 85 GOOD NOW TRY THE NEXT ONE 879.94 - 778. 19 = ? ?= (34) #: 101.85 TRY AGAIN 879.94 - 778.19 = ? ?= #: 100.85 THE ANSWER IS 101.75 0.9 - 0.85 = ? ?= (32) #: 0.05 CORRECT ANSWER 0.6 - 0.29 = ? ?= (32) #: 0.31 GREAT 996.91 - 231.32 = ? ?= (34) #: 765.59 FANTASTIC JOHN GOODBYE FOR NOW, SEE YOU AGAIN JOHN * THE NUMBERS WITHIN THE PARENTHESES WERE ADDED LATER ONLY TO INDICATE THE LEVEL APPENDIX E ANALYSIS OF A STUDENT'S WORK SAMPLE ANALYSIS OF A STUDENTS WORK FOR THE DAY ANALYSIS OF CORRECTLY SOLVED PROBLEMS FOR DARRELL I. D. NO,2 DATE JUNE 2, 1971 PROBLEM RESPONSE NO. OF LEVEL TIME TRIALS 483.07 + 212.77 — 7 48. 3 33 514.09 + 2 84.71 = 7 48.9 1 33 121.25 + 553.28 - 7 35.0 1 33 161.56 + 737. 17 •= ? 31.8 1 33 214.08 + 552.52 = 7 28.7 1 33 1.092 + 815.7 = 7 42.9 1 37 20.51 + 355.1 = 7 30.7 1 37 72.74 + 213.2 = 7 33. 2 1 37 3. 384 + 256.4 = 7 38.2 1 37 772.3 — 51.41 = 7 54.0 1 38 693.7 — 5. 167 = 7 43.9 1 38 572.8 — 15.66 = 7 4 1.4 1 38 988.9 — 1.799 = 7 34.4 1 38 4 X 0.2 - 7 7.9 1 39 3 X 0. 6 = 7 8.2 39 1 X 0.5 - 7 7.6 1 39 4 X 0.2 - 7 6.7 1 39 8 X 8.7 = 7 52.9 1 40 7 X 4. 8 - 7 23.4 1 40 5 X 1.2 — 7 15.8 40 9 X 7.2 - 7 13.6 1 40 28 X 8.6 = 7 53.9 1 41 85 X 2. 7 = ? 29.6 1 41 22 X 8,4 = 7 3 1.7 1 41 87 X 7.6 - 7 64. 1 2 41 1000 / 5 = ? 17.5 2 16 MEDIAN RESPONSE TIME: 31.8 SECONDS ANALYSIS OF UNSOLVED PROBLEMS: PROBLEM RESPONSE ANSWER LEVEL TIME GIVEN 426.23 + 233.38 = ? 66.1 77.4 / 1.8 = ? 11.3 MEDIAN RESPONSE TIME: 11.3 SECONDS 66000 37. 3 33 42 APPENDIX F EXPERIMENTAL DATA SUMMARY OF THE LEVELS ACHIEVED AND THE TEST SCORES FOR THE COMPUTER-PACED GROUP COMPUTER—PACED MALES STUDENT LEVELS ACHIEVED ON DAY PRE—TEST POST-TEST B-A NO. 1 2 3 4 5 SCORE (A) SCORE (B) 1 21 31 37 42 42 38 55 17 2 19 25 3 3 37 42 30 45 15 3 21 37 45 49 54 47 50 3 4 23 34 41 45 47 36 46 10 5 8 17 21 29 36 31 34 3 6 22 38 42 43 45 38 52 15 7 23 4 1 45 51 55 52 5 1 -1 AVERAGES 19. 6 3 1.9 37.8 42. 3 45.9 38.9 47.6 8. ' COMPUTER—PACED FEMALES STUDENT LEVELS ACHIEVED ON DAY PRE—TEST POST-TEST B-A NO. 1 2 3 4 5 SCORE (A) SCORE (B) 8 24 38 38 42 43 34 47 13 9 27 37 42 47 52 45 55 10 10 10 10 13 14 19 23 34 1 1 1 1 20 30 35 41 43 36 50 14 12 17 14 15 21 28 36 48 12 AVERAGES 19. 6 25.8 28.6 33.2 37.0 34. 8 46.8 12. ' SUMMARY OF THE LEVELS ACHIEVED AND THE TEST SCORES FOR THE SELF-PACED GROUP SELF-PACED MALES STUDENT LEVELS ACHIEVED ON DAY PRE—TEST POST-TEST B-A NO. 1 2 3 4 5 SCORE (A) SCORE (B) 13 6 15 23 39 46 37 39 2 14 13 29 37 44 53 40 53 13 15 2 12 1 4 20 30 24 16 -6 16 8 13 23 31 37 37 47 1 0 17 11 23 35 42 46 36 44 8 18 14 25 33 40 48 41 47 6 19 15 34 44 57 60 + 47 55 8 AVERAGES 9. 9 21. 6 29.9 39.0 45.8 37.5 43.3 5. 9 SELF-PACED FEMALES STUDENT LEVELS ACHIEVED ON DAY PRE—TEST POST-TEST B-A NO. 1 2 3 4 5 SCORE (A) SCORE (B) 20 15 29 37 43 53 37 49 12 21 11 40 57 60 + 6 0+ 56 60 4 22 15 33 45 59 60+ 50 56 6 23 7 16 24 29 33 25 33 8 24 22 37 45 57 60 + 37 54 17 AVERAGES 14. 0 31.0 41.6 59. 6 53.2 41.0 50.4 9.4 NOTE: 60+ INDICATES THAT THE STUDENT HAS REACHED LEVEL 60 AND HAS STARTED OVER AT LEVEL 25. 

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