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Evaluation of a computer-administered challenging teaching strategy. Floyd, Ann Rosalind 1970

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AN EVALUATION OF A COMPUTER-ADMINISTERED CHALLENGING TEACHING STRATEGY by . ANN ROSALIND FLOYD M.A., U n i v e r s i t y of Cambridge, 1963 A THESIS SUBMITTED IN PARTIAL FUIFILMENT OF THE, REQUIREMENTS FOR THE, DEGREE OF MASTER OF ARTS i n the Department of Education We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f 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 Co lumb i a , I a g ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d 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 o f The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT This study was motivated by the b e l i e f that teaching a student i n a c h a l l e n g i n g way would increase h i s a b i l i t y to apply what he had learned to new, though r e l a t e d , problems. A s p e c i f i c c h a l l e n g i n g teaching strategy was chosen, which attempted to challenge a l l students a p p r o p r i a t e l y , and to give the minimum amount of help. I t was administered by the computer, which considerably f a c i l i t a t e d the use of such an i n d i v i d u a l i s e d s t r a t e g y . The e v a l u a t i o n was done "by comparing the e f f e c t s of the c h a l l e n g i n g teaching strategy w i t h those of a l i n e a r program, a l s o computer-administered. A l i n e a r program was considered to exemplify an unchallenging approach. Both programs taught elementary "base f i v e a r i t h m e t i c to Grade S i x students, the students being, assigned to the programs at random. The e f f e c t s of the two s t r a t e g i e s were then measured by means of a post-t e s t . This aimed at ev a l u a t i n g both the grasp of the basic m a t e r i a l and the a b i l i t y to extrapolate from i t to solve mew problems i n the same general subject area. The r e s u l t s of the p o s t - t e s t showed that both s t r a t e g i e s succeeded i n teaching the basic m a t e r i a l equally w e l l , so that n e i t h e r strategy gave the student an advantage i n t h i s respect. However, the challenged group of students showed f a r greater a b i l i t y to extrapolate from the m a t e r i a l than d i d the l i n e a r program group, w i t h an average, score over i i i 45% better. This was s i g n i f i c a n t at the .007 l e v e l . These results suggest that further investigation of the merits and application of a challenging teaching strategy-should be eminently worthwhile. TABLE OE CONTENTS CHAPTER PAGE I. THE PROBLEM 1 Background 1 Statement of the problem 3 Review of the Literature 4 I I . DESIGN OE THE STUDY ' 12 Introduction 12 D e f i n i t i o n of Terms 13 Formation of the Groups . 14 Development of Materials . . . . . 15 Content , 15 The challenging teaching strategy 16 The l i n e a r program 19 The post-test 20 S t a t i s t i c a l Analysis 21 Data 21 Statement of hypotheses 21 S t a t i s t i c a l treatment of the data 22 I I I . ANALYSIS OF THE RESULTS 24 Testing of Hypotheses 24 The straightforward scores 24 The extrapolation scores 25 Conclusions 25 CHAPTER PAGE Analysis of Additional Data 26 The time taken to complete the program . . . . 26 The paths taken through the challenging program 26 IV. IMPLICATIONS OE THE STUDY 28 Introduction . 28 The Need f o r P a r a l l e l Studies 29 A wider sample 29 Removal of Hawthorne effects 29 Teaching of other material . . . . . 30 Further Development of the Strategy 30 Summary 34 BIBLIOGRAPHY 35 APPENDIX A. The Linear Program 38 APPENDIX B. The Challenging Program 61 APPENDIX C. The Post-test 69 APPENDIX D. The Experimental Data 79 LIST OF TABLES TABLE PAGE I. Straightforward Scores for the Linear Program Group 80 I I . Straightforward Scores f o r the Challenged Group 81 I I I . Extrapolation Scores for the Linear Program Group 82 IV. Extrapolation Scores f o r the Challenged Group 83 V. Time Taken to Complete Program 84 VI. Paths Through Challenging Program 85 LIST OF FIGURES FIGURE PAGE 1. Flow-chart of the Challenging Strategy 18 CHAPTER I THE PROBLEM I. BACKGROUND Today i t i s more important than ever before that education should not only provide the student with a s o l i d base of information and s k i l l s , but should also make him capable of adapting what he has learned to new situations and new problems, as they arise i n a rapidly changing world. Much of the information and many of the s k i l l s he learns during his schooldays w i l l be obsolete long before the end of his working l i f e , and his education should make i t possible for him to cope with t h i s . Hence any teaching strategy that fosters t h i s a b i l i t y has evident educational value. Piaget's model of i n t e l l e c t u a l development provides considerable guidance i n the design of such a strategy.'' In the Piagetian model, the a b i l i t y to adapt to a new situ a t i o n depends on the complexity of the i n t e l l e c t u a l structure that has already been developed. The development of th i s i n t e l l e c t u a l structure i s stimulated by demands being made upon i t ; such development i s consolidated by practice i n ^John H. F l a v e l l , The Developmental Psychology of  Jean Piaget (New York: Van Nostrand Reinhold, 1963) 2 t h e use o f t h e s k i l l s i t makes p o s s i b l e . S i m o n ' s i n f o r m a t i o n -p r o c e s s i n g model o f p r o b l e m - s o l v i n g l e a d s t o s i m i l a r c o n c l u s i o n s . ^ I t f o l l o w s , t h e n , t h a t a t e a c h i n g s t r a t e g y w h i c h f a c i l i t a t e s a s t u d e n t ' s a d a p t a t i o n o f what he has l e a r n e d t o new s i t u a t i o n s needs t o have two p r i n c i p a l c h a r a c t e r i s t i c s . F i r s t o f a l l , i t must make demands on a s t u d e n t ' s a b i l i t i e s and must make h i m t h i n k f o r h i m s e l f ; i t s h o u l d n o t g i v e t h e s o l u t i o n t o any d i f f i c u l t y t h a t may a r i s e b e f o r e t h e d i f f i c u l t y has a r i s e n . The demands must be s u f f i c i e n t l y t a x i n g as t o r e q u i r e some t h o u g h t on h i s p a r t , bu t must n o t be so extreme as t o be i m p o s s i b l e f o r h i m t o meet . The a p p r o p r i a t e n e s s o f t h e demands made on each i n d i v i d u a l s t u d e n t i s c r u c i a l . S e c o n d l y , s u c h a t e a c h i n g s t r a t e g y must c o n s o l i d a t e t h e p r o g r e s s t h a t a s t u d e n t has made, by g i v i n g h i m p r a c t i c e i n t h e s k i l l s he has j u s t l e a r n e d . W i t h o u t s u c h r e i n f o r c e m e n t , t h e development t h a t has o c c u r r e d may be o n l y t e m p o r a r y . An a p p r o a c h w i t h t h e s e two c h a r a c t e r i s t i c s w i l l be c a l l e d a c h a l l e n g i n g t e a c h i n g s t r a t e g y . The e s s e n t i a l l y t u t o r i a l n a t u r e o f t h i s a p p r o a c h makes i t s i m p l e m e n t a t i o n i n the c l a s s r o o m e x t r e m e l y d i f f i c u l t . However , c o m p u t e r - a s s i s t e d i n s t r u c t i o n does make s u c h an i n d i v i d u a l i s e d a p p r o a c h f e a s i b l e , and as i t i s a method whose ^ H e r b e r t A . S imon, The S c i e n c e s o f t h e A r t i f i c i a l ( C a m b r i d g e , M a s s : M . I . T . P r e s s , 1969) 3 costs should eventually come within the range of the educational budget, its use has practical significance. It is also a much easier way of investigating the merits of teaching strategies than is trying them out in the classroom, as, unlike even the best of teachers, i t is consistent in its treatment of different students. Furthermore, the evaluation of a teaching strategy is not bedevilled by imponderables such as teacher-student interactions. In fact, Stolurow argues that the principal use of computer-assisted instruction at the present time should be in educational research.5 Hence the aim of this study is to use a challenging teaching strategy to teach a small amount of mathematics to a group of students, by means of computer-assisted instruction, and to see what effect the strategy has on their ability to adapt what they have learned to new problems in the same subject area. Statement of the problem Does teaching a student in a challenging way increase his ability to adapt what he has learned to new, though related problems ? ^Lawrence M. Stolurow, "Some Factors in the Design of Systems for Computer-Assisted Instruction," Computer-Assisted  Instruction: A Book of Readings, ed. Richard C. Atkinson and H. A. Wilson. (New York: Academic Press, 1 9 6 9 ) , p. 9 1 . I I . REVIEW OF THE LITERATURE 4 P i a g e t d e s c r i b e s i n t e l l e c t u a l development i n terms of two i n t e r w o v e n p r o c e s s e s , which he c a l l s a s s i m i l a t i o n and accommodation.4 J u s t as the a b i l i t y t h a t an organism has t o a s s i m i l a t e f o o d i s governed by t h e d i g e s t i v e powers i t p o s s e s s e s , so the a s s i m i l a t i o n o f i n t e l l e c t u a l s t i m u l i by a p e r s o n i s l i m i t e d by t h e i n t e l l e c t u a l s t r u c t u r e t h a t has been de v e l o p e d . The a s s i m i l a t i o n o f an i n t e l l e c t u a l s t i m u l u s , i n o t h e r words i t s r e c o g n i t i o n and u n d e r s t a n d i n g , does not depend on whether the p e r s o n has a l r e a d y a s s i m i l a t e d one e x a c t l y l i k e i t on a p r e v i o u s o c c a s i o n , but on whether the s t i m u l u s i s s u f f i c i e n t l y c l o s e t o e a r l i e r ones s u c c e s s f u l l y a s s i m i l a t e d by the p e r s o n . The i n t e l l e c t u a l s t r u c t u r e i s c a pable of a d a p t i n g , i n a s m a l l way, t o the s p e c i a l c h a r a c t e r i s t i c s o f a n o v e l s t i m u l u s , and t h i s a d a p t a t i o n P i a g e t c a l l s accommodation. T h i s accommodation cor r e s p o n d s t o g r a d u a l m o d i f i c a t i o n s i n the i n t e l l e c t u a l s t r u c t u r e , and t h i s i s how i n t e l l e c t u a l development o c c u r s . A l l such m o d i f i c a t i o n s need t o be r e i n f o r c e d by f r e q u e n t use, i f the development i s t o be m a i n t a i n e d . 5 Thus i n t e l l e c t u a l growth r e s u l t s from demands b e i n g made on the p r e s e n t c a p a b i l i t i e s o f a s t u d e n t , p r o v i d e d t h a t the c h a l l e n g e t h e y r e p r e s e n t i s not beyond h i s powers; the growth i s c o n s o l i d a t e d by p r a c t i c e 4 F l a v e l l , op_. c i t . , pp. 46 - 5 0 , 237-249. 5 I b i d . , p. 5 7 . 5 i n i t s u s e . Simon d e s c r i b e s the c o g n i t i v e powers a pers o n p o s s e s s e s i n a d i f f e r e n t way, drawing an a n a l o g y w i t h the computer.^ He d i s c u s s e s t h e problem i n terms of s h o r t - t e r m and l o n g - t e r m memory, t h e s h o r t - t e r m memory b e i n g t h a t p a r t of t h e mind t h a t i s d e a l i n g w i t h the immediate s i t u a t i o n , w h i l s t the l o n g - t e r m memory c o n t a i n s the accumulated s t o r e s of e x p e r i e n c e . The r e l a t i o n s h i p between the two can be compared w i t h t h a t between the core of a computer and i t s f i l e s t o r a g e . The memory s t o r e s i t s knowledge i n u n i t s which Simon c a l l s 'chunks', and t h e c u r r e n t e n v i r o n m e n t a l s i t u a t i o n causes the r e l e v a n t chunks t o be t r a n s f e r r e d from s t o r a g e i n l o n g - t e r m memory i n t o t h e s h o r t - t e r m memory.'7 Here t h e y can be put t o work; the p r o c e s s i s r a t h e r l i k e d rawing on a l i b r a r y of computer programs. The l i m i t i n g f a c t o r s i n t h i s p r o c e s s a re f i r s t l y , t he s m a l l number of chunks, or programs, t h a t the s h o r t - t e r m memory can handle s i m u l t a n e o u s l y , and s e c o n d l y , the s o p h i s t i c a t i o n of the programs th e m s e l v e s . Program development o n l y o c c u r s when the need i s f e l t f o r i t , and so a person's a b i l i t y t o t a c k l e a new problem s u c c e s s f u l l y depends on the e x p e r i e n c e he has p r e v i o u s l y had. New programs w i l l o n l y be deve l o p e d , or e x i s t i n g programs combined i n t o more v e r s a t i l e ones, when the person f i n d s i t ^Simon, o_p_. e x t . , pp. 33-34. ^ I b i d . , p. 34. 6 necessary. Hence the educator must provide a challenging environment i n which the appropriate needs are f e l t , i f any development i s to occur. Pask adopts a challenging approach i n the development of his teaching-machine programs.8 His i s a cybernetician's point of view, and to him the relationship between teacher and taught i s one i n which a teaching-machine and student are coupled to form a single system. At the beginning of the learning process, the material to be taught i s contained e n t i r e l y within the machine; the i n s t r u c t i o n a l aim i s to transfer t h i s from the machine to the student i n as e f f i c i e n t a way as possible. The machine continually attempts to challenge the student, i n what Pask describes as an ' i n t e l l e c t u a l donkey and carrot race'.9 By giving him hints and help where necessary, i t functions as an extension of the student's brain, enabling him to perform tasks that he would not otherwise be capable of performing. The emphasis i s on challenging the student and keeping him working at his optimum l e v e l . The instruction i s prepared by analysing the material taught into components; lack of mastery of any one of these w i l l lead to errors i n certain problems related to ^Gordon Pask, "Theory and Practice of Adaptive Teaching Systems," Teaching Machines and Programmed  Learning II, ed. Robert Glaser. (National Education Association of the United States, 1965) pp. 213-266. 9 l b i d . t pp. 231-2. 7 the topic. These components Pask refers to as error factors.10 Once the error factors i n the material have been i d e n t i f i e d , i t i s possible to design questions for the machine to ask the student, the answers to which w i l l demonstrate whether or not the student i s i n need of help with s p e c i f i c error factors. Thus, as teaching proceeds, the machine can keep track of the student's mastery of the material, and can correct such error factors as need correction. The system has the additional capability of having more sophisticated information at i t s disposal, i n the form of interrelationships that may weel exist between the error factors. These can be investigated i n an i n i t i a l stage of program development, and such relevant facts as 'removing error factor 2 helps to remove error factor 5, but not vice versa' can be taken into account i n the system's f i n a l decision s t r u c t u r e . ^ It would also be possible to refine and update the decision structure as necessary, using information gained from students learning the material i n t h i s way.''^  Smallwood's work on appropriate decision structures for teaching machines suggests how a challenging teaching strategy might be refined i n the l i g h t of experience with 1 0Pask, op_. c i t . , p. 226. 1 1 Ibid., p. 228. 1 2 I b i d . , p. 229. 8 s t u d e n t s . ^ His study was e s s e n t i a l l y an e x p l o r a t o r y one, w i t h the p r i n c i p a l aim of i n v e s t i g a t i n g the f e a s i b i l i t y of a p a r t i c u l a r approach to the problem. He drew an analogy w i t h the way i n which a human t u t o r might behave. I n i t i a l l y , when the t u t o r i s confronted w i t h a new student, he w i l l adopt a d e f i n i t e approach, which may or may not s u i t the student i n v o l v e d . The t u t o r w i l l search f o r ways of e x p l a i n i n g t h i n g s , u n t i l he meets wi t h success. As he gains experience with t e a c h i n g t h i s s ubject matter to s u c c e s s i v e students, he becomes more and more e f f i c i e n t at f i n d i n g the most ap p r o p r i a t e approach as q u i c k l y as p o s s i b l e . Smallwood's system i s designed to l e a r n i n the same kind of way, by experience w i t h students. He summarises the b a s i c c o n f i g u r a t i o n of h i s system i n the f o l l o w i n g way: 1. The decomposition of the subject matter i n t o a set of concepts that the educator would l i k e to teach to the student. 2. A set of t e s t questions f o r each concept, that adequately t e s t s the student's understanding of the concept. 3. An a r r a y of i n f o r m a t i o n blocks f o r each concept that can be presented to the student i n some order - to be. determined by the t e a c h i n g machine - and thus provide a course of i n s t r u c t i o n to the student on the concept. 4. A model that can be used to estimate the p r o b a b i l i t y t h a t a g i v e n student w i t h a p a r t i c u l a r past h i s t o r y w i l l respond to a g i v e n block or t e s t q u e s t i o n w i t h a p a r t i c u l a r answer. 5. A d e c i s i o n c r i t e r i o n upon which to base the d e c i s i o n s mentioned i n 3 . ^ ^5Richard D. Smallwood, A D e c i s i o n S t r u c t u r e f o r  Teaching Machines, (Cambridge, Mass: M.I.T. Press, 1962). 1 4 l b i d . . p. 27 9 He took a short topic, a miniature geometry, and used his program to teach i t to twenty M.I.T. students. He discusses several possible models for estimating the probability of success at any point i n the program, and explains the rationale for the one he f i n a l l y chose to use. The effect of the procedure was to teach each student as fast as possible, subject to the r e s t r i c t i o n that his expected number of errors be below an arbit r a r y maximum. With a sample of only twenty students, he was limited i n the conclusions he was able to draw. For the f i r s t f i v e students the program used a set of a p r i o r i p r o b a b i l i t i e s for i t s decisions, corresponding to the i n i t i a l approach decided upon by the tutor. For the six t h and subsequent students the experience with e a r l i e r students was taken into account. The machinB was using i t s experience just as a tutor would do. Smallwood was able to show that, even with such a small number of students, the a p r i o r i p r o b a b i l i t i e s were modified. Thus the structure he had devised was capable of adaptation i n the l i g h t of experience. His p r i n c i p a l aim was to demonstrate t h i s , and, as he points out, more questions were raised than were answered.15 The m 0 s t important question i s whether or not an approach such as thi s i s worthwhile; w i l l a structure of thi s kind r e a l l y tend to an optimum teaching strategy or n o t . ^ The changes that were made i n the decision 1 ^ Smallwood, op_. c i t . , p. 2. 1 ^ Ibid., pp. 106-7 10 c r i t e r i a were not necessarily changes for the better, l e t alone a foolproof procedure for a r r i v i n g at the best of a l l possible systems.'''7 Stolurow lays particular emphasis on the use of the computer for educational research, stressing the usefulness of the r e p l i c a b i l i t y i t p e r m i t s . ^ Using a computer, i t i s possible to evaluate alternative i n s t r u c t i o n a l strategies, with the ultimate aim of developing a meaningful and useful theory of teaching. A teaching system with which he has been . closely associated i s the SOCRATES system, at the University of I l l i n o i s . This system has three l e v e l s , only two of which are relevant here.^ 9 They are: 1. P r e t u t o r i a l : at th i s l e v e l the system has to decide how to i n i t i a t e the teaching process, given certain information about the student. The problem i s to decide just what information might be relevant; aptitude scores, personality test scores, reading rate, and knowledge of prerequisite material are a l l p o s s i b i l i t i e s . Neither Pask nor Smallwood attempts to tackle t h i s problem. Both of them start a l l students i n the same way, and then adapt to t h e i r subsequent needs. ^Smallwood, op_. c i t . , p. 1 0 3 . 1 8Stolurow, 0 £ . c i t . , pp. 6 5 - 9 3 . 1 9 I b i d . , pp. 7 2 - 7 3 . 11 2. T u t o r i a l : at th i s l e v e l the problems are the same as those of Pask and Smallwood, to both of whom Stolurow re f e r s . The system reacts to the responses the student makes, and attempts to provide the student with the most appropriate i n s t r u c t i o n . The t h i r d l e v e l i s the administrative one. The problem that Stolurow i s attempting to solve i s that of finding the best way of using a l l the information available about a student i n order to optimise on the teaching strategy used with him. He concludes t h i s a r t i c l e by suggesting that the main contribution of computer-assisted ins t r u c t i o n i s to enable us to investigate these very problems, and to make our understanding of the conditions for learning more precise. CHAPTER II DESIGN OF THE STUDY I. INTRODUCTION The rationale for using a challenging teaching strategy was that i t would generate greater a b i l i t y to solve new problems i n the same subject area than an unchallenging strategy would. Such problems w i l l be called extrapolation questions. Therefore the v a l i d i t y of the argument can be tested by comparing the a b i l i t y to solve extrapolation questions shown by two groups of students, one taught the material by means of a challenging teaching strategy, and the other by means of an unchallenging one. The unchallenging strategy chosen was a li n e a r program. Of necessity, such programs lead students step by step through the material, at the d i f f i c u l t y l e v e l of the least able among them. They must cater for a l l possible errors and misconceptions, though hardly any student w i l l need a l l the help provided. The subject matter chosen was half an hour to an hour's instruction i n elementary base f i v e arithmetic, and two computer programs were written to teach the material. One used a challenging strategy and the other taught by means of a li n e a r program. 29 Grade Six students were brought to the university, to work through one or other of these programs, 13 at a teletypewriter connected to the U.B.C. 360/67. The students were assigned to the programs at random, using a table of random numbers. Immediately after completing his program each student wrote a post-test. Some of the post-test questions covered the s p e c i f i c material taught, and these w i l l be called straightforward questions. The other questions were extrapolation questions. The entire process took from one hour to one and a half hours for each student, so that i t was possible for three students to complete both program and test i n the morning, and for two students to complete them i n the afternoon. For each student there were two post-test scores, one obtained on the straightforward items, and the other obtained on the extrapolation items. The mean scores of the challenged group and the li n e a r program group were compared, using two-sample t - t e s t s . I I . DEFINITION OF TERMS (a) challenging teaching .strategy: a teaching strategy that attempts to make demands on a student, and to make him think for himself, giving hints and help only when necessary. It seeks to consolidate the student's progress by giving him practice i n the s k i l l s he has just acquired. 14 (b) r e i n f o r c i n g questions: those questions that the student i s asked during the teaching of the material, when he has responded successfully to a new challenge. They give him practice i n the new s k i l l s he has just worked out for himself. (c) straightforward questions; those post-test questions which measure grasp of the s p e c i f i c s k i l l s taught - a l l these questions are similar to ones that students have been taught how to solve, and have had practice i n solving. (d) extrapolation questions: those post-test questions which involve the s k i l l s the student has been taught, but which are di f f e r e n t from any he has solved hitherto. They require him to use what he has learned i n a new way. II I . FORMATION OF THE GROUPS The population chosen was Grade Six students, from Vancouver schools. Grade Six students could be expected to have s u f f i c i e n t background for elementary base f i v e arithmetic. However, they would not normally have encountered i t , as number bases are usually taught i n Grade Seven. This was certai n l y true of a l l the schools involved, both i n the main study and i n the p i l o t stages. Due to transportation and administrative d i f f i c u l t i e s , the sample ultimately had to be confined to a l l the Grade Six students from a single school. 15 The students were assigned to the two programs at random, using a table of random numbers, so that the groups could be assumed to be of equivalent a b i l i t y . The only r e s t r i c t i o n was that the proportion of boys and g i r l s i n each group should be approximately equal. There were nine g i r l s and six boys i n the challenged group, and eight g i r l s and six boys i n the li n e a r program group. IV. DEVELOPMENT OF MATERIALS Content The s p e c i f i c topics covered by both the challenging and the l i n e a r program were: 1. Quick review of place value i n base ten. 2. Given a set of objects thus * * * .... * * * , how to express the number of objects i n dif f e r e n t bases. A l l bases are less than ten, and no numeral has more than two d i g i t s . 3. Discussion of the symbols required i n base f i v e , and base f i v e counting. 4 . Development of base f i v e addition f a c t s . 5. The use of a base f i v e addition table. 6. Addition of two two-digit base f i v e numerals, with no carrying required. 7. Addition of two two-digit base f i v e numerals, with carrying required. 8 . M u l t i p l i c a t i o n by two of two-digit base f i v e numerals. 1 6 The challenging teaching strategy The content taught consists of eight sections, and the challenging teaching strategy was used on a l l but the f i r s t and f i f t h sections. Both these sections were taught to the students i n the challenged group i n the same way as they were taught to the students i n the lin e a r program group. The f i r s t section was taught i n thi s way because i t was review, and contained no new material. Teaching i t i n t h i s way had the added advantage of getting the student used to the teletypewriter without his having to solve challenging questions at the same time. The f i f t h section was taught i n th i s way as i t involved a purely technical s k i l l . For a l l the sections taught using the challenging teaching strategy, the following material was prepared: An i n i t i a l challenge: This was for a l l students. It presented a minimal amount of information and a question on the content of the section, so that i f a student was capable of working i t out for himself he had the chance to do so. A hint: This was only for the students who f a i l e d to respond correctly to the i n i t i a l challenge. It consisted of some additional information and a question intended to guide the student along the right l i n e s . A second challenge: This was for those students who had f a i l e d with the i n i t i a l challenge, but had successfully answered the 'hint* question. It was similar to the i n i t i a l challenge, but with di f f e r e n t numbers. 1 7 Reinforcing questions: These were questions for those students who had responded correctly to either the i n i t i a l or the second challenge. They were similar to the challenging questions, but with different numbers. A l i n e a r program version of the material: This was for those students who f a i l e d to give the correct answer to both the i n i t i a l challenge and the 'hint' question, or who answered the 'hint' question correctly but could not so answer the second challenge. It was the same version of the material as was taught to a l l students i n the li n e a r program group. The strategy for each section can be summarised thus. Present a l l students with an i n i t i a l challenge, and give the successful students some reinforc i n g questions before going on to the next section.Give those students that are not successful with the i n i t i a l challenge some help, and a question to set them thinking along the correct l i n e s . If a student cannot answer th i s question correctly, refer him to the l i n e a r program version of the material for the section. Give those students that do respond correctly to the hint question a second challenge. Present the successful students with some reinforc i n g questions before going on to the next section, and the unsuccessful ones with the lin e a r program version of the material. The student was given a second chance with a l l questions he was asked, except when he was referred to the FIGURE 1. 18 FLOW-CHART OF THE CHALLENGING STRATEGY Start of Section \ / I n i t i a l Challenge Yes No V Second Chance Hint and Hint Question Second Chance V Second Challenge Yes V Yes V Reinforcing Questions Second Chance No End of Section Linear Program Version 19 l i n e a r program; t h i s has been shown to lead to success f o r a l a r g e number of students, without any a d d i t i o n a l help.'' Each student's path through the m a t e r i a l was recorded, as was the time taken f o r him to complete the program. The c h a l l e n g i n g strategy i s i l l u s t r a t e d i n flow-chart form i n Eigure 1. D e t a i l s of the i n i t i a l challenge, h i n t , second challenge, r e i n f o r c i n g questions, and the l i n e a r program v e r s i o n of the m a t e r i a l , f o r each s e c t i o n , can be found i n Appendix B. Tables V and VI i n Appendix D contain the time each student took t o complete the program and the paths of the students through the m a t e r i a l , r e s p e c t i v e l y . The l i n e a r program This program was t e s t e d , i n book form, i n a p i l o t study. F i r s t of a l l , four Grade S i x students, whose mathematical a b i l i t y was below average, worked through the program, one a f t e r the other. Each student was t o l d to c a l l a t t e n t i o n to anything he d i d not understand. In t h i s way ambiguities were c l a r i f i e d , and o v e r - d i f f i c u l t frames s i m p l i f i e d . A f t e r each student had f i n i s h e d , the m o d i f i c a t i o n s suggested by h i s experience w i t h the program were made before the next student began. By the time the t h i r d and f o u r t h students worked through the program, no ^John J . Schurdak, "An Approach to the Use of Computers i n the I n s t r u c t i o n a l Process, and an Eval u a t i o n , " American Educational Research J o u r n a l , 4: 72, 1967. 20 further alterations were necessary. The program was then tested on 137 Grade Six students, 80 from Coquitlam schools and 57 from a Vancouver school. These students also wrote the post-test, immediately on completing the program. The results of t h i s test demonstrated that the program was teaching the material s a t i s f a c t o r i l y . The entire program can be found i n Appendix A. The post-test This was validated and item-analysed i n conjunction with the p i l o t test of the l i n e a r program. A r e l i a b i l i t y c o e f f i c i e n t (Kuder-Richardson Formula 20) of 0.92 was obtained. The test contained 12 straightforward questions and 30 extrapolation ones. The straightforward questions tested the s p e c i f i c content taught. The extrapolation questions f a l l into the following categories: 1. Addition i n base f i v e of (a) three two-digit numerals (b) two three-digit numerals 2. M u l t i p l i c a t i o n i n base f i v e of (a) three-digit numerals by two (b) two-digit numerals by numbers greater than two 3. Counting (a) i n base f i v e beyond 30 (b) i n base four 21 4. Using a base eight addition table to (a) add two two-digit numerals (b) multiply two-digit numerals by two 5. The symbols that are used i n base s i x . 6. Deduction of the base being used. 7 . Conversion from one base to another. 8 . Development of a base four addition table. 9. Subtraction i n base f i v e . 1 0 . Development of a base f i v e m u l t i p l i c a t i o n table. The complete post-test can be found i n Appendix C. V. STATISTICAL ANALYSIS Data For each student two sub-scores on the post-test were obtained. The f i r s t was his score on the 12 straightforward questions and the second was his score on the J>0 extrapolation questions. The time that each student took to complete his program was recorded, as was the path taken through the material by each student i n the challenged group. A l l t h i s data can be found i n Appendix D. Statement of hypotheses Two things were expected to happen, namely: 1. Both teaching strategies would result i n the same mastery of the s p e c i f i c s k i l l s taught, so that both groups would solve the straightforward questions equally well. 22 2. The challenged group would be much better at solving extrapolation questions than the li n e a r program group. Stated as n u l l hypotheses, these were: 1. There i s no s i g n i f i c a n t difference i n the a b i l i t y to solve straightforward questions between the challenged group and the l i n e a r program group. 2. There i s no s i g n i f i c a n t difference i n the a b i l i t y to solve extrapolation questions between the two groups. It was expected that the f i r s t hypothesis would be accepted and the second one rejected. S t a t i s t i c a l treatment of the data Both the straightforward sub-scores and the extrapolation sub-scores were analysed i n the same way. The following s t a t i s t i c s were calculated i n both cases: Linear Program Group Challenged Group Variance Mean Score X Number i n Group 2 The two-sample t-value was then computed thus: t X 1 " X2 and compared with the tabulated value for N^  + N degrees of freedom. - 2 23 The means of the straightforward scores were compared using a two-tailed test, as what was being tested was whether there was any difference between the groups. The means of the extrapolation scores were compared using a one-tailed test, as the issue here was whether the challenged group was better than the l i n e a r program group. CHAPTER III ANALYSIS OF THE RESULTS I. TESTING OF HYPOTHESES The straightforward scores The hypothesis concerning the straightforward scores was that there would "be no s t a t i s t i c a l l y s i g n i f i c a n t difference between the li n e a r program group and the challenged group. The table below summarises the results obtained on the twelve straightforward questions. Each question was worth one mark, so the maximum score possible was twelve. Linear Program Group Challenged Group  Mean Score 9.93 10.67 Number i n Group 14 15 The t-value obtained was 1.35, which, with 27 degrees of freedom and a two-tailed test, i s s i g n i f i c a n t only at the .186 l e v e l . It i s reasonable to conclude that there was no s t a t i s t i c a l l y s i g n i f i c a n t difference between the groups as regards performance on the straightforward questions. This confirms the hypothesis. The extrapolation scores In the case of the extrapolation scores, i t was expected that the n u l l hypothesis of no s i g n i f i c a n t difference between the groups would be rejected, and that the challenged group would obtain appreciably higher scores than the l i n e a r program group. The table below summarises the results obtained on the t h i r t y extrapolation questions. Each question was worth one mark, except for two questions which were worth two marks each, so the maximum score possible was thirty-two. Linear Program Group Challenged Group  Mean Score 16.07 23.33 Number i n Group . 14 15 The t-value obtained was 2.62, which, with 27 degrees of freedom and a one-tailed test, i s s i g n i f i c a n t at the .007 l e v e l . Thus the challenged group performed s i g n i f i c a n t l y better on the extrapolation questions, as was expected. I I . CONCLUSIONS There was no s t a t i s t i c a l l y s i g n i f i c a n t difference between the l i n e a r program group and the challenged group as regards th e i r performance on the straightforward questions. That both groups averaged scores of over 80% on these questions showed that the basic subject matter had been 26 s u c c e s s f u l l y taught; that the groups were so close together demonstrated that both the l i n e a r program and the c h a l l e n g i n g program had done t h e i r teaching equally w e l l . E v i d e n t l y , teaching these students t h i s m a t e r i a l i n a c h a l l e n g i n g way was n e i t h e r a handicap nor a help to them v i s - a - v i s mastery of the s p e c i f i c subject matter taught. While the l i n e a r program group averaged a score of 16.07 on the e x t r a p o l a t i o n questions, the challenged group averaged 23.33, more than 45% b e t t e r . E v i d e n t l y , teaching these students t h i s m a t e r i a l i n a c h a l l e n g i n g way was a considerable advantage to them as regards s o l v i n g the e x t r a p o l a t i o n questions. I I I . ANALYSIS OP ADDITIONAL DATA The time taken to complete the programs The students i n the challenged group completed the program more q u i c k l y than d i d the students i n the l i n e a r program group, as can be seen from Table V i n Appendix D. The average time f o r the challenged group was 36.9 minutes, while the l i n e a r program group averaged 60.7 minutes. The paths taken through the c h a l l e n g i n g program Table VI i n Appendix D summarises the routes taken by the challenged group. The f o l l o w i n g p o i n t s are of i n t e r e s t : 1. In each s e c t i o n some students were able to respond c o r r e c t l y to the i n i t i a l c h a l l e n g i n g question, i n d i c a t i n g that the challenges were not too d i f f i c u l t . 2. In a l l hut the fourth section some students were able to respond correctly to the second challenging question, after having been given a hint, so that the hints appear to have been relevant and h e l p f u l . 3. In each section at least one student needed the l i n e a r program version of the material, showing that the challenging questions r e a l l y were challenges, and were not so easy that every student could do them immediately. 4. A l l students responded successfully to at least one i n i t i a l challenging question, demonstrating that each student found something he could work out for himself, even though he might have needed considerable help with other parts of the material. 5. The students needed help at different points, so that no single presentation of the material would have been appropriate for a l l students. Each individual obtained his own presentation of the material, and t h i s varied considerably from student to student. CHAPTER IV IMPLICATIONS OF THE STUDY I. INTRODUCTION The extrapolation questions were intended to test a student's a b i l i t y to extend the s p e c i f i c material he had learned to solving new problems i n the same general subject area. If i t i s accepted that the questions did indeed perform t h i s function, then the considerable difference between the challenged group and the l i n e a r program group provides strong support for the general hypothesis discussed i n Chapter I. Teaching a student i n a challenging way makes him better at adapting his knowledge to new situations and new problems than does an unchallenging teaching strategy. In addition, i t i s reasonable to claim that t h i s study has demonstrated the f e a s i b i l i t y of using a computer to administer a challenging teaching strategy of the kind chosen. Nevertheless, one swallow does not make a summer, and one study alone does not confirm a theory. Clearly, therefore, there i s a need for further studies confirming the results obtained, and extending them beyond the particular circumstances of t h i s study. 29 I I . THE WEED FOR PARALLEL STUDIES This study had some unavoidable lim i t a t i o n s , and a l l of these suggest interesting p a r a l l e l studies. A wider sample As was mentioned i n Chapter II, the students involved a l l came from a single school. This school generally has students of above average a b i l i t y , so that the participants were probably not a representative sample of the Grade Six population. It would be very interesting to see i f similar results were obtained with less able students. It i s quite possible that a challenging teaching strategy i s less suitable for such students, especially i n the form used i n the study. Removal of Hawthorne effects There was only one teletypewriter available f o r t h i s study, which meant that the students had to work through t h e i r programs i n succession, instead of simultaneously. Consequently there i s a r e a l p o s s i b i l i t y that some of the l a t e r students heard about what they were going to do from t h e i r predecessors, since a l l of them were from the same school. This could have affected t h e i r performances on the questions, and hence the r e s u l t s obtained. Obviously, then, there i s a need for further studies that circumvent t h i s problem, either by using a large number of teletypewriters, or by using students from di f f e r e n t schools, some distance apart. 30 Teaching of other m a t e r i a l The t o p i c chosen f o r t h i s study was base f i v e a r i t h m e t i c . The choice of t h i s m a t e r i a l made i t p o s s i b l e to formulate a l l the questions asked, i n the course of e i t h e r program, i n such a way that the answers were always numeric, never v e r b a l . In the absence of a language s p e c i f i c a l l y designed f o r computer-assisted i n s t r u c t i o n , numeric answers s i m p l i f i e d the programming considerably. There i s no reason to suppose that there was any i n t e r a c t i o n between the choice of t o p i c and the cha l l e n g i n g teaching s t r a t e g y . There would appear to be nothing about base f i v e a r i t h m e t i c that would make teaching i t i n t h i s way s i n g u l a r l y appropriate. However, ob t a i n i n g s i m i l a r r e s u l t s from p a r a l l e l s t u d i e s , using d i f f e r e n t m a t e r i a l , would enhance the s i g n i f i c a n c e of the r e s u l t s . I I I . FURTHER DEVELOPMENT OE THE STRATEGY Although t h i s study demonstrated some of the f l e x i b i l i t y of the computer, i t d i d not take f u l l advantage of i t s p o t e n t i a l . The cha l l e n g i n g strategy adapted to the responses of the i n d i v i d u a l student w i t h i n the confines of each s e c t i o n of the m a t e r i a l , but i t d i d not take h i s performance i n e a r l i e r s e ctions i n t o account. Nor were the d e c i s i o n r u l e s of the strategy s u s c e p t i b l e to change; f o r example, the i n i t i a l challenge was always presented at the 31 beginning of a section, regardless of whether a student had f a i l e d with a l l previous i n i t i a l challenges, and a second chance was always given, though never a t h i r d . The strategy did not modify i t s e l f i n the l i g h t of i t s accumulated experience with successive students. Such a teaching strategy can be considered to be i n the f i r s t stage of development, characterised by two p r i n c i p a l properties: 1. A student's performance i n e a r l i e r sections does not affect the presentation of the material he receives i n l a t e r sections. 2. The decision structure i s immutable; the teaching strategy does not learn from i t s experience what approach i s l i k e l y to succeed, and what i s not, and so never changes. These properties suggest two further stages of development which could be undertaken. The second stage of development would make the strategy more f l e x i b l e by removing the r e s t r i c t i o n imposed by the f i r s t property. The information on the basis of which decisions are made about the most appropriate next step f o r a given student, would be extended to include h i s responses to the questions of e a r l i e r sections, as well as his performance i n the current section. The decision structure would s t i l l be fixed, as, for a given set of responses, the most appropriate next step decided upon would be the same for the hundredth student as 32 for the f i r s t . However, i t would he making use of considerably-more information. It would be adapting to the individual student to a f a r greater extent than before. By keeping track of his d i f f i c u l t i e s , the teaching strategy would be able to offer the most appropriate challenges i n l a t e r sections, and could also cause a student to repeat a section, i f his inadequate grasp of i t was re s u l t i n g i n an i n a b i l i t y to perform successfully i n other sections. The main d i f f i c u l t y i n implementing the second stage of development would be determining just what use should be made of the additional information available. Pask's concept of error factors, and his ideas about finding relationships between them, as discussed i n Chapter I, could be very useful i n t h i s context. By recording the paths of students through the material, when taught by the strategy i n i t s f i r s t stage, relationships between performance at diff e r e n t points i n the material.could be investigated. Por instance, i t might be found that f a i l u r e to respond correctly to the i n i t i a l challenge i n Section 2 meant that there was a 95% chance of f a i l i n g with the i n i t i a l challenge i n Section 4 . The teacher-programmer would then have to make a qualit a t i v e judgement as to how to incorporate t h i s relationship i n the teaching strategy;'whether the decision structure should never offer the i n i t i a l challenge i n Section 4 to a student who f a i l e d with the i n i t i a l challenge of Section 2, whether i t should 33 o f f e r i t to such students as meet c e r t a i n other s i m i l a r requirements, or whether i t should always he o f f e r e d . Whatever the teacher-programmer decided to do i n such circumstances, each student w i t h the same response p a t t e r n would get i d e n t i c a l treatment, even though, i n p r a c t i c e , the d e c i s i o n turned out to be unsuccessful. The t h i r d stage of development of the teaching st r a t e g y would be one i n which the d e c i s i o n s t r u c t u r e was no longer f i x e d , so removing the r e s t r i c t i o n imposed by the second property of the f i r s t stage. No longer would unsuccessful d e c i s i o n r u l e s remain i n v i o l a t e ; once shown to be mistaken, they would be changed. In other words, the strategy would l e a r n from i t s experience. Unsuccessful r u l e s could be eliminated manually from time to time, on the basi s of student records, but the ul t i m a t e aim would be to design a strategy that l e a r n t continuously from i t s experience. Such a strategy would t e s t the e f f e c t i v e n e s s of a l t e r n a t i v e d e c i s i o n r u l e s by t r y i n g them out, i n much the same way that a good teacher would. Smallwood's system, as discussed i n Chapter I , was a very simple v e r s i o n of such an approach. The p r i n c i p a l problem w i t h t h i s would be t h a t , j u s t as i n the case of the good human teacher the strategy i s seeking to emulate, a great d e a l of experience i s required before a r e a l l y s a t i s f a c t o r y s trategy can be evolved. As a r e s u l t , t h i s approach w i l l only be r e a l l y feasible when computer-assisted i nstruction i s used extensively, and on a regular basis. IV. SUMMARY This study has shown that a challenging teaching strategy merits further investigation. Suggestions have been made for p a r a l l e l studies, whose success would reinforce the conclusions drawn here, and for development of the teaching strategy u n t i l i t was capable of systematically r e p l i c a t i n g many of the important characteristics of a good teacher. The writer considers that there i s tremendous scope for useful research i n t h i s area, and that the educational benefits accruing from i t would be well worth the time and e f f o r t expended. BIBLIOGRAPHY MAIN REFERENCE 36 Entelek Incorporated, Computer-Assisted Instruction: A Survey  of the Literature, Fourth E d i t i o n . Newburyport, Mass: Entelek Incorporated, 1969. Flave11, John H., The Developmental Psychology of Jean Piaget. New York: Van Nostrand Reinhold, 1963. Markle, Susan M., Good Frames and Bad. New York: Wiley, 1964. Nelson, Charles W., Programmed Supplement: Exploring Modern  Mathematics, Book 1. New York: Holt, Rinehart & Winston, 1 963. Pask, Gordon, "Theory and Practice of Adaptive Teaching Systems," Teaching Machines and Programmed Learning I I , Robert Glaser, editor. National Education Association of the United States, 1965. Pp. 213-266. Schurdak, John J., "An Approach to the Use of Computers i n the Instructional Process, and an Evaluation," American  Educational Research Journal 4: 59-73> 1967. Seltzer, Morton, Bases and Numerals: An Introduction to Numeration. New York: Macmillan, 1963. Simon, Herbert A., The Sciences of the A r t i f i c i a l . Cambridge, Mass: M.I.T. Press, 1969. Smallwood, Richard D., A Decision Structure for Teaching  Machines. Cambridge, Mass: M.I.T. Press, 1962. Stolurow, Lawrence M., "Some Factors i n the Design of Systems for Computer-Assisted Instruction," Computer-Assisted  Instruction: A Book of Readings, Richard C. Atkinson and H.A. Wilson, editors. New York: Academic Press, 1969. Pp. 65-93. ADDITIONAL REFERENCE Atkinson, Richard C. and Wilson H. A.,(eds.), Computer- Assisted Instruction: A Book of Readings. New York: Academic Press, 1969. Bacon, Glenn C , "Roles and Directions i n CAI," Computer- Assisted Instruction and the Teaching of Mathematics. U.S.A.: National Council of Teachers of Mathematics, Inc., 1969. Pp. 67-80. 37 Bruner, Jerome S., The Process of Education. Cambridge, Mass: Harvard University Press, 1962. DeCecco, J . P. (ed.), Educational Technology. New York: Holt, Rinehart & Winston, 1964o Hansen, Duncan N., "Development of CAI Curriculum," Computer- Assisted Instruction and the Teaching of Mathematics. U.S.A.: National Council of Teachers of Mathematics, Inc., 1969. Pp. 127-133. Komoski, Kenneth, "Teaching Machines and Programmed Instruction Today," Automated Education Handbook, E. H. Wilson,editor. Detroit: Automated Education Center, 1965. II A, pp. 1-8. Michael, Donald N., "Cybernation and Changing Goals i n Education," The Computer i n American Education, Don D. Bushnell and Dwight W. Allen, editors. New York: Wiley, 1967. Pp. 3-10. Pask, Gordon, An Approach to Cybernetics. London: Hutchinson, 1961 . Suppes, Patrick, and Momingstar, Mona, "Computer-Assisted Instruction," Science 166: 343-350, 1969. Suppes, Patrick, Jerman, Max, and Brian, Dow, Computer- Assisted Instruction: Stanford's 1965-66 Arithmetic  Program. New York: Academic Press, 1968. Suppes, Patrick, "On Using Computers to Individualize Instruction," The Computer i n American Education, Don D e Bushnell and Dwight W. Allen, editors. New York: Wiley, 1967. Pp. 11-24. Sutter, Emily G., and Reid, Jackson B., "Learner Variables and Interpersonal Conditions i n Computer-Assisted Instruction," Journal of Educational Psychology 60: 153-157, June 1969. Taber, Ju l i a n I., Glaser, Robert, and Schaefer, Halmuth H., Learning and Programmed Instruction. Reading, Mass: Addison-Wesley, 1965. Wertheimer, Max, Productive Thinking. New York: Harper, 1945. Zinn, Karl L., "Functional Specifications f o r Computer^ Assisted Instruction Systems," Automated Education  Handbook, E. H. Wilson, editor. Detroit: Automated Education Center, 1965. IV A, pp. 21-33. A P P E N D I X A T H E L I N E A R P R O G R A M FRAME NUMBER FRAME 39 CORRECT ANSWER 1. In the number 26, which of these does the 2 mean ? 2 20 200 2000 20 2. In 26 the 2 means 20. What does the 6 mean ? 6 3 . So 26 = 20 + 6 which we can write as 26 = 2 x 1 0 + 6 So the 2 i n 26 t e l l s you there are 2 ....'s. 10 4. Look at the number 84. 84 = 8 x .... + 4 10 5. So 84 = 8 x 1 0 + 4 You can see how important 10 i s i n our counting system. Our counting system i s based on i t , and we say that we count i n base 10. How many fingers have you got ( including thumbs ) ? 10 6. A l o t of people count on th e i r fingers. That probably explains why we count i n base 10 FRAME NUMBER FRAME 40 CORRECT ANSWER 7. Before someone thought of number bases they had to write numbers l i k e t h i s . ////////////////////////// Would t h i s take longer than our usual way of writing numbers ? 1 . yes 2. no Type 1 or 2, whichever answer you think i s correct. 1 8. Then a clever person invented a short code to save a l o t of time. He decided to count i n tens and see how many groups of ten he could make. Here i s that number again. ////////////////////////// How many groups of ten could he make ? 2 9. How many would be l e f t over ? 6 10. Because he could make 2 groups of ten and then had 6 l e f t over his code for the number was 26. In his code the 2 i n 26 means 2 x 10 FRAME NUMBER FRAME 4 1 CORRECT ANSWER 11. Everybody who knew his code knew that when he wrote 57 the 5 meant 5 x 10 12. Of course, i f you didn't know his code you wouldn't know what he was tal k i n g about. You might be used to a different code. Suppose, instead of counting i n tens, you decided to use a code based on eight. Then you would make as many groups of as you could. 8 13. Here i s that number again. ////////////////////////// How many groups of eight can you make ? 3 14. How many are l e f t over ? 2 15. Because you could make 3 groups of 8 and there were 2 l e f t over, you would write 32 i n th i s code. Say th i s to yourself as three-two. Don't say thirty-two because i t does not mean that. In thi s code the 3 i n 32 means 3 • • • • 1 s. 8 16. When you use base 8 code you make as many groups of .... as you can. 8 42 FRAME FRAME CORRECT NUMBER ANSWER 17. Here i s another number. ////////////////////// What i s i t i n base 8 code ? 26 18. The base 8 code number 26 t e l l s you that there were 2 groups of 8 19. I f we were using base f i v e code we would make as many groups of .... as we could. 5 20. Write t h i s number i n base 5 code. ///////////////////// 41 21. Write t h i s number i n base 5 code. //////////////////////// 44 22. Instead of tal k i n g about base 5 code we w i l l just say base 5 from now on. If 42 means 4 x 5 + 2 we are using base ..... 5 23. I f 42 means 4 x 1 0 + 2 we are using base 10 24. I f 32 means 3 x 8 + 2 we are counting i n base 8 FRAME NUMBER FRAME 43 CORRECT ANSWER 25. If we are counting i n base 5, 43 means 4 x .... + 3 5 26. If we are counting i n base 8, 43 means 4 x .... + 3 8 27. Do 43 in. base 5 and 43 i n base 8 mean the same ? 1 . yes 2. no Type 1 or 2, whichever answer you think i s correct. 2 28. So i n order to t e l l the difference between the two 43's we need to know what bases are being used. Suppose you were watching someone counting some things, and to help himself he was arranging them l i k e t h i s . • * * * * * * * * * - - X -What number base would you guess he was using ? 10 44 FRAME FRAME CORRECT NUMBER ANSWER 29. Base 10 would be the obvious one to assume because he has arranged as many of them as possible i n groups of 10 3 0 . How many complete groups of ten are there i n question 28 ? 2 31. How many are l e f t over ? 3 3 2 . So the number of things i s 2 x 1 0 + 3 which i s written i n base 10 as 23 33. Here are the same things, arranged d i f f e r e n t l y . * * * * * * * * * * * * * * * * * * * * * * * What base do you think t h i s person i s using ? 6 34. It looks as though he i s using base 6 because he has arranged as many of them as possible i n groups of 1 6 35. How many are l e f t over ? 5 FRAME NUMBER 36. 37. 38. 39. 40. FRAME 45 CORRECT ANSWER 41 How many complete groups of 6 are there ? The number of things i s 3 x 6 + 5 » so that a base six person would write: there are things. In 35 i n base 6 the 3 means 3 ....'s. Here are the same stars, not arranged. Suppose you were used to counting i n base 5. The f i r s t thing you would do would be to make as many complete groups of .... as you could. To help you count, mark off the stars i n fiv e s , l i k e t h i s . and so on. With the stars i n question 39, bow many complete groups of f i v e can you make ? How many are l e f t over ? 35 6 4 3 FRAME NUMBER FRAME 46. CORRECT ANSWER 42. So a base f i v e person might arrange the stars l i k e t h i s : * * * * * * * # # * * * * * * * * * # * * * * i n 4 groups of 5 with 3 l e f t over. He would write: there are .... stars. 43 43. In base 5 the 4 i n 43 means 4 ....'s. 5 44. Here i s another c o l l e c t i o n of stars. * * * * * * * * * * * * * * * A base 7 person would write: there are .... stars. 21 45. The 2 i n 21 i n base 7 t e l l s you that you were able to make 2 groups of 7 46. Here are the same stars. * * * * * * * * * * * * * * * Write the number of stars i n base 8. 17 47. It i s 17 because when you have made 1 group of 8 there are .... l e f t over. 7 FRAME NUMBER FRAME 47 CORRECT ANSWER 48. So 17 i n base 8 and 21 i n base 7 both mean the same thing, namely the number of stars. Do 17 and 21 normally mean the same thing ? 1 • yes 2. no 2 49. So i t i s important to know what base i s being used. Now we w i l l work i n base 5 for a while and see how t h i s changes our arithmetic. When we count i n base 5 we make as many groups of .... as we can. 5 50. Here are some more stars. * * * * * * * * * * * * * * What i s the number of stars i n base 5 ? 24 51. Counting can be i l l u s t r a t e d t h i s way. * * * * * * * * * * * * * * * You can write each of these as base 5 numbers. What i s * * * i n base 5 ? 3 52. What i s * * * * * i n base 5 ? 10 53. What i s * i n base 5 ? 1 FRAME NUMBER FRAME 4 8 CORRECT ANSWER 5 4 . So our counting so f a r i s * * * -x-** * * - * • * * * * * * 1 3 10 There are two spaces here. What goes i n the f i r s t one ? 2 5 5 . What goes i n the second space ? 4 5 6 . So we have * 1 * * 2 * * * 5 * * * * 4 * # * * -x- ] o * * * * * # •x- -x- * -x- * * * ##•*•****# • X - - X - - X - - X - * * * * * * Let's f i l l i n some of these spaces. The f i r s t space i s opposite * * * * * * What i s the base 5 number for t h i s ? 11 5 7 . The t h i r d space i s opposite * * * * * * * * * What i s the base 5 number for thi s ? 13 FRAME NUMBER FRAME 49 CORRECT ANSWER 58. The f i f t h space i s opposite * * * * * * * * * * What i s the base 5 number for t h i s ? 20 59. So our counting i s 1 2 3 4 10 11 .. 13 .. 20 What comes after the 11 ? 60. What comes after the 13 ? 61. So base 5 counting looks l i k e t h i s . 1 2 3 4 1 0 1 1 1 2 1 3 H 20... When we count i n base 5 do we use the symbol 6 ? 1 . yes 2. no 62. When we count i n base 5 do we use the symbol 5 ? 1 . yes 2. no 12 14 63. So when we count i n base 5 we only use the symbols 0,1,2,3, and 4. In base 5 we only use those symbols that are less than 5 FRAME NUMBER 64. 65. 66. FRAME 67. Here i s a base 5 question: 3 + 4 We can write t h i s as * * * + * * * * _ * * * * * * * 3 + 4 What i s the answer ? Remember, t h i s i s base 5. In base 5, * * * * * * * = * * * * */* * which i s 12 . So, i n base 5, 3 + 4 = 12 Here i s another question i n base 5. * * + * * * * = * * * * * * 2 + 4 What i s the answer ? So, i n base 5, 2 + 4 = 11 Try these base 5 questions. Draw stars to help you i f you l i k e . 1 + 1 = ... 1 + 2 = ... 50 CORRECT ANSWER 12 11 2 3 68. 1 + 3 = ... 4 FRAME NUMBER 69. 70. 71 . 72. 73. FRAME Remember, you are using base 5. 1 + 4 = ... The answer to question 69 cannot be 5, since we don't use the symbol 5 i n base 5 code. Our counting went 1 2 3 4 10 11 ... In stars the question 1 + 4 can be written # + -X- tt * * _ # * * * # 1 + 4 What i s * * * * * i n base 5 ? So we have 1 + 1 = 2 1 + 2 = 3 1 + 3 = 4 1 + 4 = 10 Now t r y some more. 2 + 1 = ... 2 + 2 = ... 2 + 3 = ... 51 CORRECT ANSWER 10 10 3 4 10 52 FRAME FRAME CORRECT NUMBER ANSWER 74. 2 + 4 = ... 11 75. So we have 2 + 1 = 3 2 + 2 = 4 2 + 3 = 10 2 + 4 = 11 Now t r y these. Remember, a l l t h i s i s i n base 5. 3 + 1 = ... 4 76. 3 + 2 = ... 10 77. 3 + 3 = ... 11 78. 4 + 1 = ... 10 79. 4 + 2 = ... 11 8 0 . 4 + 4 = . . . 13 81. Now we can summarise these base 5 addition facts i n a table. In front of you i s a blue folder, and inside i t i s a base 5 addition table. Take i t out, so you can use i t . FRAME NUMBER FRAME 53 CORRECT ANSWER To show how i t works, you w i l l f i n d the answer to 2 + 3 Look down the left-hand column to 2, and put your finger there. Keep that finger where i t i s and look across the top row to 3 and put another finger there. Move the 2 finger across, and the 3 finger down, u n t i l they meet, which should he at 10 This t e l l s you that 2 + 3 = 1 0 Now use the table to answer 2 + 4 = ... 11 82. Use the table to answer 4 + 1 10 83. Use the table to answer 3 + 3 11 84. Use the table to answer 4 + 3 12 85. Use the table to answer 1 + 3 4 FRAME NUMBER FRAME 54 CORRECT ANSWER 8 6 . This table can help you to do harder addition problems i n base 5 . Use the table whenever you l i k e . Look at th i s question. It i s a base 5 question. 41 ±22. The f i r s t thing to do i s add 1 and 2 . What i s 1 + 2 i n base 5 ? 3 8 7 . So the f i r s t step i s 41 +32 3 Next we.add 4 and 3 • What i s that i n base 5 ? 12 8 8 . So the answer to the question i s 41 +32 1 23 Is t h i s the same answer as you would get i n base 10 ? 1 . yes 2 . no 2 FRAME NUMBER FRAME 55 CORRECT ANSWER 8 9 . The answer i s d i f f e r e n t "because numbers l i k e 41 mean d i f f e r e n t things i n base 5 and base 10. In base 5 the 4 i n 41 means 4 . . . . ' s . 5 9 0 . In base 10 the 4 i n 41 means 4 ....'s. 10 91 . In base 8 the 4 i n 41 would mean 4 . . . . ' s . 8 92. Here i s another base 5 question. 32 +22 Which i s the f i r s t t h i n g to do ? 1 . 3 + 2 2. 2 + 2 2 93. You always add the right-hand column f i r s t . What i s 2 + 2 i n base 5 ? Don't f o r g e t , you can use the t a b l e whenever you l i k e . 4 94. So the question begins l i k e t h i s . 32 +22 4 FRAME NUMBER FRAME 56 CORRECT ANSWER The next step i s to add 3 and 2. What i s t h i s i n base 5 ? 10 95. So the question i s 32 +22 1 0 4 Now t r y these base 5 questions. Use the table whenever you l i k e . 21 +42 1 1 3 96. 40 +34 124 97. 43 +31 124 98. 3 2 +32 1 1 4 99. 3 0 +20 1 0 0 FRAME NUMBER FRAME 57 CORRECT ANSWER 100. This one needs some care. 23 +14 The f i r s t thing to do i s to add 3 and 4. What i s that i n "base 5 ? 12 101. So you have to write down 2 and carry 1 . You have not had to do any carrying before i n base 5, but i t works just the same way as usual. So what i s the answer to the question ? 23 +14 42 102. Here are some more questions i n which you w i l l have to do some carrying. They are a l l base 5 questions. 13 +24 42 103. 14 +24 43 104. 14 +2 21 FRAME NUMBER FRAME 58 CORRECT ANSWER 105. 23 +12 40 106. 13 +23 41 107. Some of these questions involve carrying, and some do not. They are a l l base 5 questions. 20 +34 104 108. 1;2 +24 41 109. 23 +34 112 110. 43 +44 142 111. Most of these answers are diff e r e n t from the ones you would usually get because t h i s i s base .... arithmetic. 5 112. When we count i n base 5 we arrange as many things as we can i n groups of ..... 5 59 FRAME FRAME CORRECT NUMBER ANSWER 113. For instance, 34 i n "base 5 means that there were .... groups of 5 and 4 l e f t over. 3 114. Which of these i s the same as 2 x 41 ? 1 . 2 + 41 2. 41 x 41 3. 41 + 41 3 115. So the mul t i p l i c a t i o n question 41 x2 and the addition question 41 ±ii mean the same and so w i l l have the same answer. So you can fi n d the answer to 41 x2 "by working out 41 +41 What i s the answer to thi s question ? Remember, base 5. 132 60 FRAME FRAME CORRECT NUMBER ANSWER 116. Here i s another base 5 m u l t i p l i c a t i o n question. 32 x2 Write i t as an addition question on a piece of paper i f you l i k e . You only have an addition table, not a mu l t i p l i c a t i o n table, so addition i s probably easier for you. What i s the answer to the question ? 114 117 . Try t h i s one. 23 x2 101 118. Try t h i s one. 33 x2 121 119. Try t h i s one. 24 _x2 103 120 . So now you know how to count i n base 5 and how to do some addition and mu l t i p l i c a t i o n . This i s the end of the lesson. Go and t e l l the teacher you have finished. Goodbye. APPENDIX B THE CHALLENGING PROGRAM The material covered by the programs has 8 sections. The students i n the challenged group were taught i n the following way: Section 1 : Linear program, frames 1-11. Section 2: Challenging program. Section 3: Challenging program. Section 4: Challenging program. Section 5: Linear program, frames 81-85. Section 6: Challenging program. Section 7: Challenging program. Section 8: Challenging program. Section 1 i s a review of place-value i n base ten. Section 5 i s the section which teaches the use of a base 5 addition table. The challenging program functions on sections 2,3,4,6,7, and 8 of the content taught. Summarised below, for each of these sections, are: 1. The i n i t i a l challenging question. 2. The hint that w i l l be given, i f necessary. 3. The second challenging question (similar to the f i r s t , but with di f f e r e n t numbers). 4. Details of the re i n f o r c i n g questions. 5. The l i n e a r program to which the student w i l l be referred, i f necessary. 6 3 Section 2: Expressing a number of objects i n diff e r e n t  bases, a l l numerals being less than 3 d i g i t s long, and  a l l bases being less than ten. 1. I n i t i a l challenge: There i s nothing special about ten. We are just used to using i t as a base. Eight would have done just as well. Try writing the number of stars as a base eight number. * * * * * * * * * * * * * * * * * * * 2. Hint: When you count i n base ten you make as many groups of ten as you can. So, when you count i n base eight, you make as many groups of .... as you can. 3. Second challenge: Now have a t r y at t h i s question. Write t h i s number as a base eight number. * * * * * * * * * * * * * 4. Reinforcing questions: Writing numbers * * * * * * i n d i f f e r e n t bases, u n t i l 5 are correct, or 8 have been t r i e d . 5. Linear program: Let's have a think about t h i s . Here i s a number. ////////////////////////// How many groups of eight can you make ? Continue with l i n e a r program, frames 14-50. 69 S e c t i o n 3: The symbols used i n base 5, and base 5 c o u n t i n g . 1i. I n i t i a l c h a l l e n g e : C o u n t i n g can be i l l u s t r a t e d t h i s way. * * * * * * * * * * * * * * * * * and i n base t e n we would w r i t e 1 2 3 4 5 6 t o d e s c r i b e t h i s . I n base 5 c o u n t i n g we b e g i n 1 2 3 4 .... What comes next ? 2. H i n t : When you use base f i v e numbers you arrange as many t h i n g s as p o s s i b l e i n groups o f f i v e . What i s * * * * *' as a base f i v e number ? 3. Second c h a l l e n g e : Now we have 1 2 3 4 10 .... so f a r f o r our base f i v e c o u n t i n g . What comes next ? 4. R e i n f o r c i n g q u e s t i o n s : (a) Here i s some base f i v e c o u n t i n g , w i t h some gaps. 1 2 3 4 1 0 1 1 12.. 14.. What goes i n the f i r s t gap ? (b) What goes i n the second gap ? (c) I n base t e n we use the symbols 0,1,2,3,4,5,6,7,8,9. What symbols do we use i n base f i v e ? 1. 0,1,2,3,4,5,6,7,8,9 2. 0,1,2,3,4,5 3. 0,1,2,3,4 5. L i n e a r program: Now you saw t h a t c o u n t i n g c o u l d be i l l u s t r a t e d w i t h s t a r s . Continue w i t h l i n e a r program, frames 56-63. 65 Section 4 : Base 5 addition of single d i g i t numbers, 1 . I n i t i a l challenge: Here i s a base 5 sum. 3 + 4 In base ten the answer would be 7. What i s the answer i n base 5 ? 2 . Hint: You can write t h i s sum as + * * * * 3 + 4 and the answer i n stars i s * * * * * * * What i s * * * * * * * as a base f i v e number ? 3 . Second challenge: So the answer to the base f i v e sum 3 + 4 i s 1 2 . What i s the answer to t h i s base f i v e sum ? 2 + 3 4 . Reinforcing questions: Similar questions u n t i l 5 are correct or 8 have been t r i e d . 5. Linear program: Look at t h i s sum 3 + 4 again. * * * 4. * * * * — * * * * * * * 3 + 4 What i s the answer ? Remember, thi s i s base f i v e . Continue with l i n e a r program, frames 65-80. 66 Section 6: Addition of two two-digit base f i v e numerals,  with no 'carrying' required. 1. I n i t i a l challenge: Here i s a base f i v e addition sum. What i s the answer ? You can use the table to help you. 41 ±22 2» Hint: Perhaps you forgot that t h i s was base f i v e . In base f i v e 3 + 4 i s not 7, but 3. Second challenge: This makes the answer to the question 123, as you were t o l d . Now try t h i s one. Remember, base 5. 21 +33 4. Reinforcing questions: Similar questions u n t i l 3 are correct or 5 have been t r i e d . 5. Linear program: Let's have a closer look at that e a r l i e r question. 41 +52 The f i r s t thing to do i s add 1 and 2. What i s 1 + 2 i n base f i v e ? Continue with l i n e a r program, frames 87-99. 67 Section 7: Addition of tsro two-digit numerals i n base f i v e , with 'carrying' required. 1. I n i t i a l challenge: Be careful with t h i s one. 23 +14 2. Hint: This was the f i r s t question i n which you had some carrying to do. This one requires carrying too. See i f you can do i t . 12 +14 3. Second challenge: Now t r y t h i s one. 24 ±11 4. Reinforcing questions: Similar questions, some with 'carrying' and some without, u n t i l 5 are correct or 8 have been t r i e d . 5. Linear program: Let's have another look at that e a r l i e r question. 23 ±11 The f i r s t thing to do i s to add 3 and 4. What i s that i n base f i v e ? Continue with l i n e a r program, frames 1 0 1 - 1 1 3 . 68 Section 8: M u l t i p l i c a t i o n by two of two-digit base f i v e numerals. 1. I n i t i a l challenge: You have just done quite a l o t of base f i v e addition. Now here i s a mu l t i p l i c a t i o n question, s t i l l i n base f i v e . 41 x2 2. Hint: There i s an easy way for you to do these questions, by changing them to addition questions. Which of these means the same as 41 x 2 ? 1 . 41 x 41 2. 41 + 41 3. 41 + 2 3. Second challenge: So 41 x 2 and 41 + 41 mean the same and so have the same answer. Now t r y t h i s one. Write i t as an addition question on a piece of paper, i f you l i k e , and then do i t . 31 x2 4. Reinforcing questions: Similar questions u n t i l 3 are correct or 5 have been t r i e d . 5. Linear program: Let's have another think about 41 x 2 Continue with l i n e a r program, frames 115-120. APPENDIX C THE POST-TEST 70 A l l questions were worth 1 mark, except for Nos. 38 and 41, which were worth 2 marks each. Base f i v e and base eight addition tables were provided. Included i n t h i s appendix i s a complete copy of the test, and copies of the two addition tables provided. The questions cover the following content: The 12 straightforward questions 1. Recognition of the base being used. (No. 1) 2. Expression of a number of objects i n dif f e r e n t bases. (Nos. 2-6) 3. Base f i v e counting. (No. 7) 4. Addition of two two-digit base f i v e numerals. (Nos. 8-10) 5. M u l t i p l i c a t i o n by two of two-digit base f i v e numerals. (Nos. 11,12) The 50 extrapolation questions 1 . Addition i n base f i v e of three two-digit numerals, and of two three-digit numerals. (Nos. 13-17) 2. M u l t i p l i c a t i o n i n base f i v e of three-digit numerals by two, and of two-digit numerals by numbers greater than two. (Nos. 18-21) 3. Counting i n base f i v e beyond 30, and counting i n base four. (Nos. 22,23) 4. Using a base eight addition table to add two two-digit base eight numerals, and to multiply two-digit base eight numerals by two. (Nos. 24-29) 5. The symbols that are used i n base s i x . (No. 30) 6. Deduction of the base being used. (Nos. 31-34,37) 7. Conversion from one base to another. (Nos. 35,36,42) 8. Development of a base four addition table. (No. 38) 9. Subtraction i n base f i v e . (Nos. 39,40) 10. Development of a base f i v e m u l t i p l i c a t i o n table. (No. 72 NAME: 1 . If 27 means 2 x 9 + 7 , what number base i s being used ? 2. Write the number of stars i n base 7. * * * * * * * * * * * * * * * * * * 3. Write the number of stars i n base 6. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4. Write the number of squares i n base 8. • O D P D D D D D d D D D D D D D 5. Write t h i s sum as a base 5 sum. * * * * + * * * * * * * = * * * * * * * * * * * + = 6. Write t h i s sum as a base 4 sum. * * * * * + * * * * * * = * * * * * * * * * * * + = 7. Count i n base 5 from 1 to 20. (both 1 and 20 are base 5 numbers) 73 A l l the questions on t h i s page are base 5 questions. You may use your base 5 addition table whenever you l i k e . 8. Add i n base 5 . 10 1 5 . Add i n base 5 . 321 240 9. Add i n base 5 . 13 16. Add i n base 5 . 24 33 42 10. Add i n base 5 . 32 17. Add i n base 5 . 432 524 11 . Multiply i n base 5 . 32 x2 18. Multiply i n base 5 . 321 x 2 1 2 . Multiply i n base 5 . 24 x2 19. Multiply i n base 5 . 234 x 2 1 3 . Add i n base 5 . 21 31 AO 20. Multiply i n base 5 . 21 M 1 4 . Add i n base 5 . 412 22i 21 . Multiply i n base 5 . 34 22 74 2 2 . Count i n base 4 from 1 to 12. (both. 1 and 12 are base 4 numbers) 23. Count i n base 5 from 32 to 44 . (both 32 and 44 are base 5 numbers) The next six questions (numbers 24 to 29) are base 8 questions. Use the base 8 table provided whenever you l i k e . 24. Add i n base 8. 24 H 27. Multiply i n base 8. 63 x2 25. Add i n base 8. 36 24 28. Multiply i n base 8. 35 x2 26. Add i n base 8. 47 29. Multiply i n base 8. 57 x2 30. I f you were counting i n base 6, what symbols would you use ? 75 31. What hase would a person he using i f he wrote: I have 14 toes ? (in fact he has the same number of toes as everyone else) 32. What hase i s t h i s person counting i n ? 33. What hase i s being used here ? 4 + 3 = 10 34. Here i s an addition problem: 34 + 62 Could t h i s be a base f i v e sum ? Why ? 35. If Ann writes: "I have 18 d o l l a r s " , when she i s counting i n base ten, then i f she were using base f i v e she would write: "I have d o l l a r s " . 36.. Pete and B i l l have the same number of books. Pete counts his i n base f i v e and writes that he has 43 books. B i l l counts his i n base ten and writes that he has books. 76 3 7 . What i s the smallest possible base a person could be using i f he wrote down the sum 35 + 2 3 ? 3 8 . Here i s part of a + base four addition ^ table. F i l l i n the ^ spaces. ^ 1 2 3 3 9 . Subtract i n base 5 . 3 3 z i 4 0 . Subtract i n base 5 . 23 =11 41. Here i s part of a x base f i v e m u l t i p l i c a t i o n ^ table. F i l l i n the ^ spaces. ^ 4 1 2 3 4 4 2 . If I have 2 3 d o l l a r s i n base 6 , how many do I have i n base 5 ? 77 BASE E I V E ADDITION TABLE + 1 2 3 4 1 2 3 4 1 0 2 3 4 1 0 11 3 4 1 0 11 12 4 1 0 11 12 13 78 BASE EIGHT ADDITION TABLE + 1 2 3 4 5 6 7 1 2 3 4 5 6 7 10 2 3 4 5 6 7 10 11 3 4 5 6 7 10 11 12 4 5 6 7 10 11 12 13 5 6 7 10 11 12 13 14 6 7 10 11 12 13 14 15 7 10 11 12 13 14 15 16 APPENDIX D THE EXPERIMENTAL DATA 80 TABLE I STRAIGHTFORWARD SCORES FOR THE LINEAR PROGRAM GROUP SCORES FOR STUDENT NUMBER EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 . 1 1 1 1 1 0 1 0 0 1 0 1 1 1 2. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4. 1 1 1 1 1 1 1 1 1 0 1 1 1 1 . 5. 1 1 1 0 0 0 1 1 0 0 1 0 1 1 6. 1 1 1 1 0 1 0 0 0 0 0 1 1 1 7. 1 .1 1 1 1 1 1 0 1 1 1 1 0 1 8. 1 1 1 1 1 1 1 1 0 1 1 1 1 1 9. 1 1 1 1 1 0 1 1 1 1 1 1 1 1 10. 1 1 1 1 1 0 1 1 1 1 1 1 1 1 11 . 1 1 0 1 1 1 1 1 0 1 1 1 1 1 12. 1 1 0 1 1 1 1 1 1 0 0 0 1 0 TOTAL SCORE 12 12 10 11 10 8 11 9 7 8 9 10 11 11 81 TABLE II STRAIGHTFORWARD SCORES FOR THE CHALLENGED GROUP SCORES FOR STUDENT NUMBER EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 . 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 2. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5. 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 6. 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 7. 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 8. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 10. 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 11 . 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 12. 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 TOTAL SCORE 12 12 11 12 12 9 8 11 11 12 8 11 10 11 10 82 TABLE III EXTRAPOLATION SCORES POR THE LINEAR PROGRAM GROUP SCORES FOR STUDENT NUMBER EACH ITEM 1 2 3 4 5 6 7 8 9 10 11 12 13 14 13. 0 0 1 1 1 0 1 1 1 1 1 14. 1 1 1 .1 1 1 1 0 0 1 1 1 1 1 15. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16. 1 1 0 0 0 0 0 0 0 1 1 0 0 0 17. 1 1 1 1 0 0 0 1 1 1 1 1 1 1 18. 1 1 0 1 1 1 1 1 0 1 0 1 1 1 19. 1 1 0 0 1 1 1 0 0 1 1 0 1 0 20. 1 1 0 1 1 1 1 1 0 1 1 0 1 0 21 . 1 1 0 0 0 0 1 0 0 1 0 0 0 1 22. 1 1 0 0 0 0 0 0 0 0 1 1 0 0 23. 1 1 0 0 1 1 1 0 0 1 1 1 0 0 24. 1 1 1 1 1 1 1 0 1 1 1 0 1 1 25. 1 1 1 1 1 1 0 0 1 1 1 0 1 0 26. 1 1 1 1 0 1 1 0 1 0 1 0 1 1 27. 1 1 0 0 1 1 1 0 0 1 1 0 0 1 28. 1 0 0 0 0 1 0 0 0 1 0 0 0 1 29. 1 0 0 0 1 1 0 0 0 1 0 0 1 1 30. 1 1 0 0 1 1 1 0 0 1 0 1 0 0 31 . 1 1 0 0 1 0 1 0 0 0 0 1 0 1 32. 1 1 0 0 1 1 1 0 0 0 1 1 1 0 33. 1 1 0 0 0 1 1 0 0 0 0 0 1 0 34. 1 1 0 0 1 0 1 0 0 0 0 0 0 0 35. 1 1 0 0 1 1 1 0 0 0 0 1 1 0 36. 1 1 0 0 1 0 1 0 0 0 0 0 1 0 37. 1 1 0 1 0 0 1 0 0 0 0 0 0 0 38. 2 2 0 0 0 0 0 0 0 0 0 0 0 0 39. 1 1 0 0 0 1 0 0 0 0 1 0 1 0 40. 1 0 0 0 0 1 0 0 0 0 1 0 1 0 41 . 2 2 0 0 2 2 2 0 0 2 2 1 0 0 4 2 . 1 1 0 0 0 0 1 0 0 0 0 0 0 0 TOTAL SCORE 32 29 6 9 19 21 22 5 5 18 18 12 17 12 83 TABLE IV EXTRAPOLATION SCORES POR THE CHALLENGED GROUP SCORES FOR EACH ITEM 1 2 3 4 5 STUDENT NUMBER 6 7 8 9 10 11 12 13 14 15 13. 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 14. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 15. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 16. 1 0 1 1 1 1 1 1 1 1 0 0 1 1 0 17. 1 1 1 1 1 1 1- 1 1 1 0 1 1 0 1 18. 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 19. 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 20. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 21 . 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 22. 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 23. 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 24. 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 25. 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 26. 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 27. 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 28. 1 1 1 0 1 0 1 1 1 1 0 0 1 1 1 29. 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 30. 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 31 . 1 1 0 1 0 0 0 1 1 1 0 1 0 0 1 32. 1 1 1 1 1 0 0 1 1 1 0 1 1 1 1 33. 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 34. 1 0 1 0 0 0 0 1 0 0 0 0 0 1 1 35. 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 36. 0 1 1 1 1 0 0 1 0 0 0 1 1 0 0 37. 0 1 1 1 1 0 0 . 1 1 0 0 0 1 1 1 38. 1 2 2 0 2 2 2 2 2 2 0 2 2 2 1 39. 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 40. 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 41. 2 2 0 2 0 2 2 2 0 2 0 0 2 2 2 42. 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 TOTAL SCORE 25 27 29 27 25 15 21 31 24 24 6 20 27 21 28 84 TABLE V TIME TAKEN TO COMPLETE PROGRAM LINEAR PROGRAM GROUP CHALLENGED GROUP STUDENT TIME TAKEN STUDENT TIME TAKEN NUMBER (MINUTES) NUMBER (MINUTES) 1 . 45 - 1 . 50 2. 50 2. 35 3. 65 3. 40 4. 77 4. 21 5. 49 5. 36 6. 58 6. 35 7. 59 7. 33 8. 75 8. 22 9. 59 9. 63 10. 62 10. 31 11 . 64 11 . 54 12. 60 12. 49 13. 70 13. 25 14. 57 14. 31 15. 28 85 TABLE VI PATHS THROUGH CHALLENGING PROGRAM STUDENT LEVEL AT WHICH SECTION WAS COMPLETED NUMBER SECTION SECTION SECTION SECTION SECTION SECTION 2 3 4 6 7 8 1. 1 1 3 3 1 2 2. 1 1 1 1 1 2 3. 2 1 1 1 • 1 1 4. 1 1 1 1 1 1 5. 2 3 1 1 1 1 6. 1 2 1 1 2 1 7. 1 2 1 1 2 1 8. 1 1 1 1 1 1 9. 3 3 3 1 1 2 10. 2 1 1 1 1 2 11 . 2 1 1 2 3 1 12. 3 1 1 1 1 2 13. 1 1 1 1 1 1 14. 1 1 1 1 1 3 15. 2 3 1 1 1 1 Level 1: I n i t i a l challenging question correct. Level 2: Second challenging question correct (after hint was given). Level 3: Linear program required. 

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