A STUDY OF COMPUTER-ASSISTED INSTRUCTIONAL STRATEGIES AND LEARNER CHARACTERISTICS by DAVID M. KAUFMAN M.Eng., McGill University, 1970 A DISSERTATION SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION in the Faculty o t Graduate Studies We accept t h i s d i s s e r t a t i o n as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of €-t) U C^ATt The University of British Columbia Vancouver 8, Canada Date Oct. / f / Z J ? i i A STUDY OF COMPUTER-ASSISTED INSTRUCTIONAL STRATEGIES AND LEARNER CHARACTERISTICS Supervisor: Dr. David R o b i t a i l l e ABSTRACT This study was undertaken to investigate the use of computer-assisted i n s t r u c t i o n as an i n s t r u c t i o n a l laboratory. The concept of an i n s t r u c t i o n a l l o g i c was defined as an algorithm followed by the computer program for each i n s t r u c t i o n a l unit. This step-by-step logic was repeated for each i n s t r u c t i o n a l unit but with d i f f e r e n t content. This procedure permitted the c o n t r o l l e d manipulation » of the variable of correctional feedback. Thiee forms of co r r e c t i o n a l feedback were defined by varying the information content of the feedback. These were response-sensi t i v e c o r r e c t i o n a l feedback, response-insensitive c o r r e c t i o n a l .feedback and no correctional feedback (only that the answer was i n c o r r e c t ) . The i n t e r a c t i o n of c o r r e c t i o n a l feedback with selected learner c h a r a c t e r i s t i c s was examined as ' v ^ l l , These learner t r a i t s were mathematical a b i l i t y , p rerequisite knowledge i i i and state anxiety. The e f f e c t of c o r r e c t i o n a l feedback and i t s i n t e r a c t i o n with these variables was examined. Subjects of the study were a representative sample of sixty-three preservice elementary school teachers from f i v e sections of a mathematics course given i n a large education f a c u l t y . These subjects were randomly assigned to the three treatment conditions, although they selected the CAI experimental periods i n which they would p a r t i c i p a t e . The test of mathematical a b i l i t y used was the Cooperative Sequential Test of Educational Progress, Mathematics Form 2A (STEP). The state anxiety instrument used was the State-Trait Anxiety Inventory (STAI) and the f i v e iteir short form used by O'Neil (1972) was administered twice. The eighteen item posttest was constructed by the experimenter and the measure of pr e r e q u i s i t e knowledge used was a nine item prelesson with a possible mark of 0, I or 2 on each item. The mathematics lesson was a topic i n introductory calculus dealing with the concept of d e r i v a t i v e . The topic was treated from a physical point of view, using concepts of distance, speed and time to i l l u s t r a t e the mathematical concepts. The main objectives were to show that the derivative i s a l i m i t and to show how to use t h i s l i m i t d e f i n i t i o n to c a l c u l a t e the derivative of a function at a point. i v The CAI lesson was programmed using an author language developed by the experimenter as a vehicle for implementing the i n s t r u c t i o n a l logic and varying the c o r r e c t i o n a l feedback. The language i s l i m i t e d in use but has the advantage of requiring e s s e n t i a l l y no computer experience of an i n s t r u c t i o n a l designer. The main l i m i t a t i o n of the language as implemented at the University of B r i t i s h Columbia i s the cost, which li m i t e d the sample size i n this experiment. The r e s u l t s of the study were generally i n the expected d i r e c t i o n but the e f f e c t s were not as pronounced as had been hypothesized. The most important f i n d i n g was the s i g n i f i c a n t difference i n proportion of errors on the main lesson between the response-insensitive {T^) and no co r r e c t i o n a l feedback ( T 3 ) groups. The importance of t h i s f i n d i n g was then increased by the s i g n i f i c a n t r e l a t i o n s h i p found between immediate learning and proportion of errors with the e f f e c t of learner t r a i t s and treatment e f f e c t s s t a t i s t i c a l l y removed. The most e f f e c t i v e variable i n p r e d i c t i n g performance i n the experiment was mathematical a b i l i t y and i t s e f f e c t was s t a t i s t i c a l l y c o n t r o l l e d when te^tina the e f f e c t of the other variables. p r e r e q u i s i t e knowledge was alsc important i n p redicting performance and was s t a t i s t i c a l l y c o n t r o l l e d as well. State anxiety was s i g n i f i c a n t i n pr e d i c t i n g V proportion of errors but not response latency. However, s i g n i f i c a n t treatment-by-A-State interactions were observed for posttest and for response latency. The three treatment groups d i f f e r e d i n the expected d i r e c t i o n on most of the important variables but the differences were not s t a t i s t i c a l l y s i g n i f i c a n t . In p a r t i c u l a r , the A-State l e v e l s for the three groups were ordered as expected, but the differences were not large enough to cause the hypothesized i n t e r a c t i o n s . The r e s u l t s of the study p a r t i a l l y supported the hypothesis of the important role of c o r r e c t i o n a l feedback in i n s t r u c t i o n and i t s i n t e r a c t i o n with in d i v i d u a l t r a i t s of the learner. F i n a l l y , the variable of state anxiety was examined and i t was found that higher l e v e l s of state anxiety led to longer response latencies. Also, state anxiety increased when no cor r e c t i o n a l feedback was provided to the students as well as when the content became more d i f f i c u l t . This f i n d i n g confirmed the expected r e l a t i o n s h i p between state anxiety and task d i f f i c u l t y . ACKNOWLEDGEMENT S I wish to express my sincerest thanks to the following people: Dr. David R o b i t a i l l e , for his personal and professional guidance and without whom this d i s s e r t a t i o n would not have been produced. Dr. Robert Conry, for his personal and professional guidance throughout my doctoral programme. Dr. J . S h e r r i l l , Dr. J. Kennedy and Dr. S.S. Lee, members of the d i s s e r t a t i o n committee, for their extra e f f o r t s i n making i t possible for me to complete this d i s s e r t a t i o n on time. Mrs. M e l l e t t , from the Faculty of Graduate Studies, f o r her e f f o r t s above and beyond the c a l l of duty. Katherine Li-dderdale, f o r her e f f o r t s i n typing, proofreading, e d i t i n g and preparing copies of t h i s d i s s e r t a t i o n under great pressure of time. Brina Aronovitch, for helping to prepare the f i r s t draft of this d i s s e r t a t i o n under great pressure of time. Others, too numerous to mention i n the Faculty Education, who provided advice, assistance and comput time whenever i t was needed. P a t r i c i a , f o r understanding during the more d i f f i c u l t periods. v i i i TABLE OF CONTENTS Page ACKNOWLEDGEMENTS :vi LIST OF TABLES xi-LIST OF FIGURES *ii.„ Chapter I. THE PROBLEM Overview of the Problem 1 Introduction to CAI 3 Organizational Scheme for CAI 7 Discussion of the Problem 11 Statement of the Problem 18 Research Questions . . . . 19 Importance of the Study 21 II. REVIEW OF RELATED LITERATURE AND RESEARCH HYPOTHESES Introduction 23 Lit e r a t u r e on Knowledge of Results . . . 23 Summary 36 Li t e r a t u r e on State-Trait Anxiety . . . 40 Summary 44 Lite r a t u r e on Aptitude-Treatment Interactions . . . . . 46 Summary ' 52 Liter a t u r e on T u t o r i a l Computer -As s i s t e d Instruction . 52 Summary 67 Summary of Research Hypotheses . . . . 67 ix Chapter Page III. METHOD Subjects • • 70 Experimental Procedure • • 71 Design • • 72 Instruct i o n a l Logic for Prelesson 74 Inst r u c t i o n a l Logic f o r Main Lesson . . . • • 77 Operational D e f i n i t i o n of Treatments . . . • • 79 CAI Author Language • • 80 Inst r u c t i o n a l Materials • • 80 Measurement Instruments . . 88 Posttest . 88 Mathematical A b i l i t y Test . 90 State-Trait Anxiety Inventory 91 Prelesson . 92 Main Lesson . 93 IV. ANALYSIS AND RESULTS Method of Analysis . • • 95 Results of Analysis-Means 97 Results of Analysis - Hypothesis Testing . . . 100 Posttest (Y±) 101 Proportion of Errors ) 105 Average Latency (Y3) 107 Relationship Between Process and Product . . 110 Results of Post hoc Analysis 115 Summary of S t a t i s t i c a l Results 126 V. DISCUSSION AND CONCLUSIONS Summary of Study -*-3-1-Discussion of Findings 1 3 5 Correctional Feedback 135 A-State 1 3 6 Mathematical A b i l i t y 1 3 9 Prerequisite Knowledge . . . . 139 Relationship Between Process and Product • • 140 Post hoc Analysis Results 14 L X Page APPENDICES Appendix A - Users Guide for CAI Author Language 156 Appendix B - Users Guide f o r Prelesson Language 170 Appendix C - CAI Prelesson L i s t i n g . . . . 178 Appendix D - CAI Main Lesson L i s t i n g (versions T]_ > T 2 » T 3 ) • • • 1 8 8 Appendix E - L i s t i n g for One CAI Student Session . . 218 Appendix F - Posttest 227 Appendix G - State-Trait Anxiety Inventory 237 Appendix H - Tables of Co r r e l a t i o n C o e f f i c i e n t s . . . . . . . . .240 xi LIST OF TABLES Table Page 1. Experimental Procedures 71 2. Prerequisites for Main Lesson 84 3. I n s t r u c t i o n a l Objectives f o r Main Lesson . . . 86 4. Posttest Data 88 5. Item Analysis Information f o r Posttest . . . . 89 6. Anxiety Test Data 92 7. Prelesson Data 92 8. Item Analysis Information for Prelesson . . . . 93 9. Main Lesson Data 94 10. Means of Variables f o r Combined Groups and f o r T±, T2, T3 99 11. Symbols Used i n S t a t i s t i c a l Analysis 100 12. Results of Regression Analysis for Posttest . . 102 13. Results of Regression Analysis for Proportion of Errors 105 14. Results of Regression Analysis f o r Average Latency . . . , 108 15. Results of Regression Analysis for Hypothesis Five I l l 16. Average Number of Responses i n Each Response Class for Main Lesson 125 17. I n t e r c o r r e l a t i o n Matrix for Combined Groups . . 241 18. Comparison of Selected C o r r e l a t i o n C o e f f i c i e n t s for T 1, T 2, T 3 . 242 z i i LIST OF FIGURES Figure Page 1. Organizational Scheme for CAI 8 2. Prelesson I n s t r u c t i o n a l Logic 75 3. Main Lesson I n s t r u c t i o n a l Logic 76 4. Detailed View of Prelesson 83 5. Detailed View of Main Lesson 87 6. Regression Lines of Posttest and Main Lesson A-State for T1, T 2, T 3 113 7. Regression Lines of Main Lesson Latency and A-State for T 1 ? T 2, T 3 . . . 114 8. Graph of Errors on Ins t r u c t i o n a l Units f o r Combined Groups 119 9. Graph of Errors on Ins t r u c t i o n a l Units for T l f T 2, T 3 120 10. Graph of Latencies on Ins t r u c t i o n a l Units fo r T 1, T 2, T 3 123 11. Graph of A-State Levels During Experiment . . . . 124 CHAPTER I THE PROBLEM Overview of the Problem One goal of t h i s study was to develop a methodology which would demonstrate the power of u t i l i z i n g a computer to a s s i s t i n the development of models for i n s t r u c t i o n . This methodology was developed i n the context of the computer acting as personal tutor for in d i v i d u a l students. By using the computer i n t h i s manner, a laboratory to study i n s t r u c t i o n was created. This laboratory permitted the gathering of r e l i a b l e and v a l i d data under c a r e f u l l y c o n t r o l l e d and r e p l i c a b l e conditions, using materials having some educational s i g n i f i c a n c e to the students. The major goal of t h i s study was to apply the methodology to a research study which would make a contribution to the area of i n s t r u c t i o n a l theory. A' p a r t i c u l a r l y controversial problem was examined: the role of c o r r e c t i o n a l feedback i n the i n s t r u c t i o n a l process. Several steps were involved i n the attainment of these two goals. F i r s t , the methodology was developed. A computer language was written by the experimenter to 2 implement a p a r t i c u l a r i n s t r u c t i o n a l logic which permitted lessons to be programmed for the computer by novice computer users. The language implemented a p a r t i c u l a r l y f l e x i b l e teaching logic which was followed by the computer many times during an i n s t r u c t i o n a l session. The second step involved the development of a teaching unit (lesson) dealing with a topic i n elementary calculus. This unit i s presently being used i n an alte r e d form as a t u t o r i a l a i d by college students i n an introductory physics course (Kalman et. a l . , 1972). The t h i r d step consisted of developing an organizational scheme for studying the i n s t r u c t i o n a l process using the above methodology. Several variables i n the organizational scheme were examined. These variables r e f l e c t e d some unresolved nroblems i n the l i t e r a t u r e on i n s t r u c t i o n . The e f f e c t of c o r r e c t i o n a l feedback on learning was assessed by manipulating this v a r i a b l e i n a controlled experiment. The i n t e r a c t i o n of c o r r e c t i o n a l feedback and selected learner c h a r a c t e r i s t i c s v/as also examined. The use of the computer permitted the variable of c o r r e c t i o n a l feedback to be examined under c o n t r o l l e d conditions and provided data to c l a r i f y the e f f e c t s of t h i s variable on learning. 3 Although the experiment focused on a p a r t i c u l a r aspect of the i n s t r u c t i o n a l process, the methodology that was developed could be used by others to examine other variables of i n t e r e s t i n the organizational scheme. Introduction to CAI The term computer-assisted i n s t r u c t i o n (CAI) describes an i n s t r u c t i o n a l s i t u a t i o n where a student works at a terminal connected e l e c t r o n i c a l l y to a computer. An i n s t r u c t i o n a l program i s stored i n the computer. The program comprises the complete package of information, instructions and l o g i c with which the student w i l l i n t e r a c t during his learning session. The terminal, which serves as the interface between the computer and the student, usually consists of a typewriter keyboard and either a paper r o l l of TV-like screen, c a l l e d a cathode-ray tube, upon which the communications to and from the computer are displayed. In order to have CAI, the computer must act u a l l y i n s t r u c t the student through the program and net just be used as a tool to a s s i s t i n the solut i o n of problems or the r e t r i e v a l of information. By d e f i n i t i o n , there must be two-way human-to-computer communication i n which there occurs a stimuxus-response-feedback r e l a t i o n s h i p producing learning (Silvern, 1967). 4 Atkinson (1968a) commented that i n recent years a great number of a r t i c l e s and news releases dealing with CAI has been published. He observed that few of such reports are based on substantial experience and research, but that the majority consist of vague speculations and conjectures with l i t t l e , i f any, data or r e a l experience to substantiate the claims for CAI. Bundy (1967) noted that, with few exceptions, a v a i l a b l e information about CAI consists of descriptive accounts of what a p a r t i c u l a r i n s t i t u t i o n i s doing, or short statements of research f i n d i n g s . He also noted that CAI has not reached the kind of maturity that programmed learning has attained as an area of endeavor. Some recent CAI work in the l a s t two years indicates that t h i s s i t u a t i o n i s changing (e.g. Judd et a l . , 1973; Keats and Hansen, 1972; O'Neil, 1972b; Tobias, 1973). Suppes (1966) distinguished three lev e l s of i n t e r a c t i o n between the student and the computer program. At the most s u p e r f i c i a l , and also the most economical l e v e l , are d r i l l - a n d - p r a c t i c e systems. At. t h i s l e v e l , the student i s presented with examples on which he needs prac t i c e , and d r i l l e d on those to which he f a i l s to respond c o r r e c t l y (Suydam, 1969). Instructional programs that f a l l under t h i s heading are merely supplements to a regular curriculum taught by a teacher. The next l e v e l of i n t e r a c t i o n includes t u t o r i a l systems which are more complex than dri11-and-practice systems. The computer-teacher i n i t i a t e s the question, f o r which answers are stored i n advance and a r e s t r i c t e d kind of dialogue between the student and the computer i s achieved. Suppes (1966) claims that the aim here i s to i n d i v i d u a l i z e i n s t r u c t i o n and to free the teacher from many classroom r e s p o n s i b i l i t i e s so that he w i l l have more time to i n d i v i d u a l i z e his own i n s t r u c t i o n a l e f f o r t s . At the t h i r d and deepest l e v e l of student-computer i n t e r a c t i o n are systems that allow a complex dialogue between the student and the computer (Suppes, 1966). Dialogue systems exist only as rudimentary prototypes. Before t h i s l e v e l of i n t e r a c t i o n can be achieved, the computer w i l l require the c a p a b i l i t y of in t e r p r e t i n g any response and question given by the student, either i n writing or o r a l l y . A search of the l i t e r a t u r e indicated that there are three major arguments used in support of CAI. These are not independent cf one another, but they do serve to indicate the major views of CAI which are currently prevalent.' The f i r s t argument i n support of CAI i s that 6 CAI i s a medium used to i n d i v i d u a l i z e i n s t r u c t i o n . Suppes (1966) claimed that this i s the single most important a p p l i c a t i o n of CAI. He wrote that the p r i n c i p a l obstacles to widespread implementation of CAI are not technological but pedagogical: how to devise means of i n d i v i d u a l i z i n g i n s t r u c t i o n and how to design a curriculum that i s suited to i n d i v i d u a l s instead of groups. Many researchers (Atkinson, 1968b; Bundy, 1967; Seidel, 1969; Stolurow, 1962) have claimed that a second ap p l i c a t i o n of CAI provides the most valuable contribution to education. These writers refer to the use of CAI as a laboratory for research i n learning and i n s t r u c t i o n . With the computer, i t i s possible to be quite e x p l i c i t about a teaching method and to reproduce the conditions as often as desired. Added to t h i s i s the c a p a b i l i t y of the computer f o r stori n g and manipulating data. The data can be used by the researcher to modify the presentation i n progress and the data can be manipulated l a t e r i n many d i f f e r e n t ways to provide information about the i n s t r u c t i o n a l variables under study. Variables such as errors i n learning, response latency and effectiveness of diagnostic materials can be examined i n d e t a i l . Stolurow (1962) and Atkinson (1968c) predicted that the computer w i l l contribute to the emergence of one or more theories of i n s t r u c t i o n supported by r e l i a b l e and v a l i d data. A t h i r d argument used to support CAI i s that i t i s a medium which can change the r o l e of the teacher and the school environment. Computers have been used to a l i m i t e d extent for d i r e c t i n s t r u c t i o n i n a few selected schools i n the United States of America (Atkinson, 1968d; Pressman, 1970; Suppes, 1966). S t a n s f i e l d (1968) cautioned that CAI i s not ready for the schools and that the schools are not ready for CAI. Hicks and Hunka (1972) made the following assumptions about CAI: Assumption I. Computer-assisted i n s t r u c t i o n w i l l surely come into general use in the schools probably within the next decade, and possibly before either the schools or manufacturers of CAI systems can ensure i t s wise use (1972, p. 69). Assumption III. Computer-assisted i n s t r u c t i o n i s capable of becoming a widely used, v e r s a t i l e and e f f e c t i v e educational t o o l , but i t must overcome many handicaps impeding i t s development (1972, p. 23). 3efore this a p p l i c a t i o n of CAI becomes widespread, f a c t o r s such as cost-effectiveness and acceptance of CAT by school personnel remain to be solved. Organizational Scheme for CAI i n This author has proposed an organizational scheme Figure 1 for research into a p a r t i c u l a r area, of CA T. The scheme assumes that the important r o l e of CAI at the present time i s that of an i n s t r u c t i o n a l laboratory, although questions r e l a t i n g to i n d i v i d u a l i z i n g i n s t r u c t i o n may also a r i s e from the model. LEARNER -cognitive variables -personality variables -psychomotor variables CAI LESSON •instructional variables -machine variables -nature of d i s c i p l i n e OUTPUT -process variables -product variables Figure 1 Organizational Scheme for Research Into CAI as an Instructional Laboratory The organizational scheme shown in Figure 1 suggests that a learner i n t e r a c t s with a CAI lesson and ex h i b i t s behaviour which i s observable and from which one can make inferences about his learning. The basic assumption underlying the l u o d e l i s that c e r t a i n learner variables and c e r t a i n CAI lesson variable? w i l l i n t e r a c t to produce d i f f e r e n t i a l e f f e c t s on the output variables. Learner variables have also been c a l l e d personological variables (Bracht, 1970) and these may be considered as measures of i n d i v i d u a l student c h a r a c t e r i s t i c s . Cognitive variables can be considered as general or s p e c i f i c i n t e l l e c t u a l a b i l i t i e s of the learner. Examples of cognitive variables are IQ, mathematical a b i l i t y and p r e r e q u i s i t e knowledge. Personality variables r e f e r to variables such as attitudes, anxiety or motivation. Psychomotor variables refer to the manipulative or m o t o r - s k i l l area, e.g., typing speed and accuracy. A CAI Lesson comprises three d i s c e r n i b l e variables. The i n s t r u c t i o n a l variables refer to those variables which characterize the i n s t r u c t i o n a l strategy, not the content, and these can be manipulated by the i n s t r u c t i o n a l programmer. Examples of i n s t r u c t i o n a l variables are step s i z e , type or frequency of feedback and type of branching. The machine variables r e f e r to both hardware and software variables which a f f e c t the CAI lesson. Hardware variables involve c h a r a c t e r i s t i c s of the computer terminal used, typing rate for example. Software variables are c h a r a c t e r i s t i c s of the CAI author (programming) language used to write the lesson, branching c a p a b i l i t i e s for example. Rogers (1966) cautioned that these computer c h a r a c t e r i s t i c s can impose severe l i m i t a t i o n s upon both the materials which can be presented to the learner and the responses which the learner can be required and allowed to make. The f i n a l dimension of the CAI lesson refers to the nature of the subject being.taught. Hicks and Hunka (1972) suggested some subject matter d i s t i n c t i o n s : loosely structured versus highly structured, e x p e r i e n t i a l versus r a t i o n a l , value-laden versus neutral. Suppes (1966) remarked that well-structured subjects such as reading and mathematics can e a s i l y be handled by t u t o r i a l as well as by d r i l l - a n d - p r a c t i c e systems. The output variables represent observable behaviors exhibited by the learner and from which learning may be i n f e r r e d . These can be considered as either process or product variables. Process variables, such as the number or type of errors made during learning and response latency, give an i n d i c a t i o n of the student's performance during the lesson. These may be considered as representing either accuracy (errors) cr e f f i c i e n c y (latency) cf the student's learning process. Product variables, such as immediate learning, retention, transfer and attitudes, are the r e s u l t of what the student has gained from the completed CAI lesson. 11 Discussion of the Problem The present study made use of the organizational scheme described e a r l i e r to examine the e f f e c t on the output variables of learner variables, CAI lesson variables and their i n t e r a c t i o n . The r e l a t i o n s h i p between the process and product variables at the output of the model was also examined. In the CAI lesson component of the model, i n s t r u c t i o n a l variables were under consideration and three d i f f e r e n t i n s t r u c t i o n a l strategies were examined. Stolurow (1969) explained that strategy can be thought of as a set of rules and the f i r s t task of the teacher or educational theorist i s to reduce the strategy to an e x p l i c i t algorithm so that i t can be programmed for implementation by a CAI system. Sherman (1971) pointed out that most of the time required to construct conversational exchanges between student and the computer i s taken up by two tasks: designing the i n s t r u c t i o n a l logic and programming the conversational network. He suggested that by u t i l i z i n g a s p e c i f i c predetermined i n s t r u c t i o n a l strategy, which he c a l l e d a template, the time spent on the second task could be reduced s u b s t a n t i a l l y . The reason for t h i s i s that the template, which may be compared to a standard subroutine or 12 procedure i n computational programs, allows the lesson designer to merely supply the content and not the l o g i c elements to the CAI lesson. A lesson consists of a p a r t i c u l a r inter-connection of one or more of these templates. Two templates were designed and implemented for t h i s study: one f o r the main lesson and one for the prelesson. These templates are discussed i n Chapter I I I . The three i n s t r u c t i o n a l strategies considered i n this study were three variations of the template designed for the main lesson. The variable which distinguished these strategies from one another was the extent of knowledge of r e s u l t s or, more s p e c i f i c a l l y , the c o r r e c t i o n a l feedback given to the student after each incorrect response. The c o r r e c t i o n a l feedback given i n the f i r s t i n s t r u c t i o n a l strategy was characterized as response-sensitive, i . e . , c o r r e c t i o n a l feedback given to the student was appropriate to the type of error that he made i n his response. The second strategy was response-insensitive, i . e . , the corr e c t i o n a l feedback given consisted of a hint which was constant regardless of the nature of the error made on that response. The f i n a l strategy involved informing the student whether his response was correct or not, but no c o r r e c t i o n a l feedback cr hint was given. The t h i r d strategy served as a control condition since no 13 assistance was provided to the student when he responded i n c o r r e c t l y . Suppes (1966, 1967) concluded that there i s c o n f l i c t i n g evidence regarding the e f f e c t s of immediately informing the student each time he makes a mistake. He stated that a ce n t r a l weakness of t r a d i t i o n a l psychological theories of reinforcement i s that too much of.the theory has been tested by experiments i n which the content of the information transmitted i n the feedback procedure i s e s s e n t i a l l y very simple. As a r e s u l t , the information content of feedback has not been s u f f i c i e n t l y emphasized in t h e o r e t i c a l discussions. Many i n s t r u c t i o n a l systems or theories contain knowledge of r e s u l t s as a major component although the precise roles of thi s variable are unclear. Stolurow (1969) and M o r r i l l (1961) noted that although knowledge of r e s u l t s appears to be e f f e c t i v e i n the learning process, t h i s problem contains many facets which need empirical study. Along these l i n e s , Gilman (1967) wrote that his r e s u l t s suggest that less elaborate c o r r e c t i o n a l feedback procedures are as e f f e c t i v e as the more elaborate prompting, response-contingent feedback, and overt-correct i o n procedures. He suggested that his re s u l t s should be checked with other subject matters and other students to e s t a b l i s h their degree of generality. The present study examined the knowledge-of-results variable under con t r o l l e d conditions made possible by the use of CAI and provided empirical data to c l a r i f y the e f f e c t of this variable. Suppes (1967) pointed out that a troublesome issue has arisen i n recent research. Should d i f f e r e n t kinds of reinforcement, of which knowledge-of-results may be one kind, and d i f f e r e n t sorts of reinforcement schedules be given to children with d i f f e r e n t p e r s o n a l i t i e s ? This issue r e l a t e s to the problem of i n d i v i d u a l i z i n g i n s t r u c t i o n . Bracht (1970) suggested that i t i s possible that no sing l e i n s t r u c t i o n a l process provides optimal learning p o s s i b i l i t i e s f o r a l l students. However, Bundy (1967) pointed out that to date we s t i l l do not understand the learning process s u f f i c i e n t l y to make t r u l y self-adaptive learning programs. Bundy (1967) stated that we might be able to make self-adaptive learning programs i n the future and the vehicle for accomplishing this may wall be CAI, viewed not as an i n s t r u c t i o n a l t o o l , but as an i n s t r u c t i o n a l laboratory. The present study contributed to an understanding of the i n s t r u c t i o n a l process by examining aptitude - treatment interactions (ATI). The goal of research on ATI i s to f i n d s i g n i f i c a n t i n t e r a c t i o n s between a l t e r n a t i v e treatments and i n d i v i d u a l c h a r a c t e r i s t i c s of the learner (Bracht, 1970). Subsequently, a l t e r n a t i v e i n s t r u c t i o n a l programs may be developed so that optimal educational benefits are obtained when students are assigned to the a l t e r n a t i v e programs. Gentile (1967) emphasized that i f i t i s claimed that adapting to i n d i v i d u a l differences through CAI would improve some aspect of learning, then parametric studies of variables considered to be important should be undertaken. He found that these parametric studies are scarce because almost a l l of the funds a l l o t t e d to CAI projects are being spent on the development of courses or equipment to the exclusion of research on teaching-learning variables, where research i s needed most. Dick (1965) noted that the matter of personality-computer i n t e r a c t i o n remains to be studied. This d i s s e r t a t i o n study focused on the i n s t r u c t i o n a l variables i n the CAI lesson and their i n t e r a c t i o n with selected c h a r a c t e r i s t i c s of i n d i v i d u a l learners. The learner c h a r a c t e r i s t i c s examined i n t h i s study f e l l into two of the three categories i n the model, cognitive and personality. Psychomotor variables were not considered. The cognitive variables were mathematical a b i l i t y and prerequisites required f o r the main lesson. These cognitive variables ranged from general to s p e c i f i c in terms of their r e l a t i o n s h i p to the content and s k i l l s required for the CAI lesson. The personality variable under consideration was anxiety. O'Neil e_t al_. (1969a,b; 1972a,b) and Spielberger (1972) examined the e f f e c t of anxiety i n a CAI context and found s i g n i f i c a n t e f f e c t s on student performance. They found a d i s o r d i n a l i n t e r a c t i o n between state anxiety and d i f f i c u l t y of the material presented i n the CAI lesson. O'Neil (1972a) pointed out that some studies i n the l i t e r a t u r e have suggested that the r e l a t i o n s h i p between anxiety and learning i s d i f f e r e n t f o r men and women. This issue was not considered i n thi s study but a representative sample of students was employed to increase the g e n e r a l i z a b i l i t y of the r e s u l t s . The r e l a t i o n s h i p between the process and product var i a b l e s i s important. If a well-defined r e l a t i o n s h i p e x i s t s , then process variables such as errors or latencies may be u t i l i z e d to make decisions during learning i n order to maximize the product variables of learning. This l a t t e r question was. also examined i n the present study. The questions under consideration i n this study appear to be important t h e o r e t i c a l issues. An important p r a c t i c a l reason e x i s t s as well for examining the i n s t r u c t i o n a l variable of c o r r e c t i o n a l feedback. The issue i s a cost-effectiveness one. Rogers (1968) estimated that approximately 100 hours of analysis, programming and e d i t i n g e f f o r t s are required to produce programmed i n s t r u c t i o n a l (PI) material which occupies the student for one hour. He estimated that one or two orders of magnitude separate the CAI lesson from the PI lesson i n terms of the time required f o r production of a lesson. A major portion of the time required i n developing a CAI t u t o r i a l lesson i s spent i n a n t i c i p a t i n g a l l possible student responses and providing response-sensi t i v e c o r r e c t i o n a l feedback. The computer software required becomes more complex and computer execution time, and cost, increase i f t h i s response-sensitive c o r r e c t i o n a l feedback i s implemented. If t h i s i n s t r u c t i o n a l variable or i t s i n t e r a c t i o n with learner c h a r a c t e r i s t i c s has no e f f e c t on the output variables i n the model, then the time, e f f o r t and extra cost required to include t h i s feature i n the CAI lesson are being poorly spent. Bracht (1970) advised that experimenters should begin to formulate hypotheses about ATI with administrative f a c t o r s , such as cost, i n mind. The r e s u l t s of t h i s study have cost-effectivenes.s implications because the costs f o r providing the three types of feedback range from expensive to inexpensive. Statement of the Problem The s p e c i f i c tasks proposed for t h i s d i s s e r t a t i o n were: 1. Development of an organizational scheme fo r research into CAI as an i n s t r u c t i o n a l laboratory. 2. Design and implementation of an empirical study to examine several variables i n the organizational scheme. The i n s t r u c t i o n a l variable under consideration was c o r r e c t i o n a l feedback. Its i n t e r a c t i o n with the learner variables of anxiety, p r e r e q u i s i t e knowledge and mathematical a b i l i t y was examined. 3 . Development of a CAI author language permitting a high degree of response-sensitive feedback to be supplied to the student. This language served as a vehicle for the study and implemented a p a r t i c u l a r i n s t r u c t i o n a l l o g i c , permitting data regarding the student's performance tc be recorded for subsequent analysis. 4. Development of a CAI module dealing with some 19 elementary calculus concepts and programmed using the above-mentioned author language. The module included a CAI prelesson dealing with the prerequisites i d e n t i f i e d f o r the main lesson. Research Questions The several terms which were used in the following research questions are explained below. • The learning process variables were proportion of errors, i . e . t o t a l errors divided by t o t a l responses, and average response latency. The learning product variable was immediate learning, as measured by a posttest. Because of the nature of the i n s t r u c t i o n a l l o g i c used i n t h i s study, i t was f e l t that proportion of errors would r e f l e c t performance better than t o t a l errors. The reason f o r not using t o t a l errors was that two errors of the same type would cause the computer to produce the correct answer and would end that i n s t r u c t i o n a l unit. Therefore, a student who immediately made two similar errors i n a row would terminate that i n s t r u c t i o n a l unit without being seriously penalized, even though he had not produced the correct response. The use of proportion of errors corrected this s i t u a t i o n . The average latency variable was obtained by c a l c u l a t i n g the average of the t o t a l response latencies for a l l i n s t r u c t i o n a l u n i t s . It was f e l t that t o t a l latency on an i n s t r u c t i o n a l unit would better r e f l e c t performance than the latency f o r the f i r s t response on that unit. The reason for t h i s i s that t o t a l latency r e f l e c t e d the eff e c t of the c o r r e c t i o n a l feedback given during that unit. The research questions were: 1. What i s the e f f e c t of c o r r e c t i o n a l feedback on the learning process and product? 2. What i s the e f f e c t of cor r e c t i o n a l feedback on the learning process for students with d i f f e r e n t l e v e l s of anxiety? 3. What i s the e f f e c t of c o r r e c t i o n a l feedback on the learning process and product f o r students with d i f f e r e n t levels of prerequisite s k i l l s ? 4 . What i s the e f f e c t of c o r r e c t i o n a l feedback on the learning process and product for students with d i f f e r e n t l e v e l s of mathematical a b i l i t y ? 5. What i s the r e l a t i o n s h i p between the learning process variables and the product variables, independent of the e f f e c t of che other variables, i . e . anxiety, prere q u i s i t e knowledge, mathematical a b i l i t y and treatment? 21 Importance of the Study This study was important because of i t s contribution to several d i f f e r e n t areas of knowledge. The use of CAI as a laboratory to study the i n s t r u c t i o n a l process has been discussed. An organizational scheme f o r CAI as an i n s t r u c t i o n a l laboratory has been developed and several aspects of the model are under consideration. Some empirical evidence was provided regarding the e f f e c t s of c o r r e c t i o n a l feedback on learning under c a r e f u l l y c o n t r o l l e d conditions made possible by the use of CAI. Higgins (1973) suggested that one of the variables with greatest potential for contributing to the design of e f f e c t i v e i n s t r u c t i o n i s the amount of information contained i n the feedback stimulus. The r e s o l u t i o n of t h i s controversial issue has t h e o r e t i c a l s i g n i f i c a n c e f o r i n s t r u c t i o n a l theory, p r a c t i c a l s i g n i f i c a n c e f o r classroom i n s t r u c t i o n and implications f o r subsequent development of CAI software and lessons. The area of 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 i s currently receiving much attention from educators. A pressing need e x i s t s to devote more research a c t i v i t y to the study of i n t e r a c t i o n between the conditions of i n s t r u c t i o n and the nature of the learner (Sutter and Reid. 1969). This study examined several learner variables and their i n t e r a c t i o n with the d i f f e r e n t i n s t r u c t i o n a l strategies. The r e s u l t s would have implications for 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 2 2 since the presence of s i g n i f i c a n t d i s o r d i n a l i n t e r a c t i o n s makes the assignment of i n d i v i d u a l s to d i f f e r e n t i n s t r u c t i o n a l treatments desirable i n order to produce optimal learning for i n d i v i d u a l learners. The variable of anxiety i s currently receiving much attention i n the l i t e r a t u r e and empirical evidence i s required to further develop the State-Trait Anxiety Theory (Spielberger, 1971). This study also contributed to t h i s area by c l a r i f y i n g some of the e f f e c t s of anxiety on learning. The r e l a t i o n s h i p between the process variables, such as errors during learning and response latency, and the product variables, such as immediate learning, i s important because of the p o s s i b i l i t y of using the learning process variables to optimize the learning product for i n d i v i d u a l s . This optimization could be accomplished by basing i n s t r u c t i o n a l decisions on these process variables during learning. An important contribution was the development of a methodology which implemented the concept of CAI as a laboratory to 9 t u d y learning and i n s t r u c t i o n , The methodology used i n t h i s study may be u t i l i z e d by others or may serve <? s a model f o r other researchers to follow i n the development of new methods of applying the computer to the s o l u t i o n of educational problems. CHAPTER II REVIEW OF RELATED LITERATURE AND RESEARCH HYPOTHESES Introduc tion This chapter contains a survey of the l i t e r a t u r e r e l a t e d to the areas under consideration i n t h i s study. The topics included are: Knowledge-of-Results, St a t e - T r a i t Anxiety, Aptitude-Treatment Interactions and T u t o r i a l CAI. L i t e r a t u r e on Knowledge of Results Annett (1964, p.280) defined knowledge-of-results as " . . . knowledge which an i n d i v i d u a l or group receives r e l a t i n g to the outcome of a response or group of responses.' Higgins (1972) categorized feedback into three common forms that provide d i f f e r e n t amounts of information. The three forms were knowledge-of-results (KR), knowledge-of-correct-response (KCR) and i n s t r u c t i o n a l feedback. Knowledge-of-results, the form containing the. least information, indicates only whether a response i s correct or i n c o r r e c t . If the response i s incorrect, knowledge-of-r e s u l t s does not indicate what the correct response i s . Knowledge-of-correct-response d i f f e r s from knowledge-of-r e s u l t s i n that knowledge-of- correct-response always 24 indicates the correct response. I n s t r u c t i o n a l feedback, the form containing the most information, indicates the correct response and provides an explanation of why that response i s correct. Higgins (1972) wrote that i t appears from the research l i t e r a t u r e that p o t e n t i a l contributions to the design of e f f e c t i v e i n s t r u c t i o n can be generated by research e f f o r t s contrasting the e f f e c t s of various combinations of the above forms of feedback. Higgins (1972) pointed out that one factor influencing the effectiveness of feedback i s the nature of the learner. S p e c i f i c a l l y , i t appears that feedback i s more e f f e c t i v e when the learner possesses a low l e v e l of competence with regard to the i n s t r u c t i o n a l task. Annett (1964) observed that probably the biggest issue i n the area of feedback i s that of motivation versus information. He concluded from his study of the area that the information content rather than the motivation content of feedback i s important to learning while motivation content of feedback i s an important variable i n performance. However, i n his recent book (1969, p.169), Annett wrote: . . . to say that knowledge-of-results provides motivation i s misleading. The so-called incentive function of knowledge-of-results seems to involve both providing the subject with a performance standard to aim f o r and information necessary for corrective action. Annett (1964) concluded that none of the generalizations which have been made i n the past about knowledge-of-r e s u l t s can be accepted at face value. Annett (1969) reviewed the psychological l i t e r a t u r e i n the area of feedback. He concluded that the role of r e s u l t s or consequences i n behaviour seems to have been underrated. He thought of knowledge-of-results as the manipulation of an external feedback loop r e l a t i n g to c e r t a i n aspects of a subject's performance. M o r r i l l (1961) reached e s s e n t i a l l y the same conclusion as Annett when he stated: Because of t h e i r controvertible r e s u l t s , the above studies demonstrate that, although immediate feedback appears to be'effective in the learning process, t h i s problem contains many facets which need more empirical, data. Higgins (1972) pointed out that the research most frequently c i t e d i n describing the r o l e of feedback i n i n s t r u c t i o n has come from three sources of experimentation: laboratory studies of human learning, studies employing e x i s t i n g conventional i n s t r u c t i o n a l materials which i n their o r i g i n a l form do not require overt learner response, and investigations with programmed i n s t r u c t i o n a l materials. Most research from the f i r s t two sources has very l i m i t e d a p p l i c a b i l i t y tc the design of i n s t r u c t i o n a l materials because of the difference between the materials and procedures employed i n these studies and those used i n systematically designed i n s t r u c t i o n , such as i n the present study. Higgins (1972) pointed out that i n most laboratory studies of human learning, the learner receives no i n s t r u c t i o n before being asked to respond. He stated that t h i s procedure i s i n e f f e c t i v e i n s t r u c t i o n a l l y when contrasted with procedures i n which the learner receives i n i t i a l i n s t r u c t i o n designed to enable him to respond c o r r e c t l y . Higgins (1972) f e e l s that the l i m i t e d a p p l i c a b i l i t y to i n s t r u c t i o n a l design of laboratory studies and of inv e s t i g a t i o n s involving conventional materials leaves the studies using programmed i n s t r u c t i o n a l materials as the basis f o r useful research up to this point i n time. Obviously, studies u t i l i z i n g CAI f a l l into t h i s category and may a c t u a l l y provide stronger evidence due to increased control of extraneous varia b l e s . The variable of i n s t r u c t i o n a l feedback i s an important component of most i n s t r u c t i o n a l models (Stolurow, 1971; M e r r i l l , 1971). Stolurow (1971) defined three c r i t i c a l system functions i n his i n s t r u c t i o n a l approach: a. The cue function provided by the program of in s t r u c t i o n , i . e . the stimulus to which each c r i t e r i o n response i s attached; b. the motivation function, i . e . e l i c i t i n g the desired performance; and c. the feedback function, i . e . providing immediate knowledge-of-results. Knowledge-of-results or feedback i s defined as one of Stolurow's ten c r i t i c a l requirements of a teaching machine. . Stolurow described his conception of knowledge-of-results i n a teaching program. I f the learner selects the correct a l t e r n a t i v e among a preprogrammed set of choices, he i s tol d that he was correct and given a d d i t i o n a l information. If he sel e c t s an incorrect a l t e r n a t i v e , he i s t o l d that he i s i n c o r r e c t , the computer provides c o r r e c t i v e and supplementary information, and he then makes another choice. This process continues u n t i l the correct choice has been made or u n t i l the computer provides the correct answer. Feedback implemented by the program in this way i s thus s e n s i t i v e to i n d i v i d u a l differences i n lea.rning. Stolurow claimed that t h i s type of feedback i s often preferred ever the simple right/wrong feedback when the implementation of the respective functions takes the same amount of time. Otherwise, an immediate right/wrong feedback i s preferred. This study examined t h i s claim from a cost-benefit viewpoint, i . e . the corrective feedback function which provides information may be superior to simple right/wrong feedback, 28 even when the former i s more expensive and d i f f i c u l t to implement i f t h i s feedback s i g n i f i c a n t l y improves student learning. Geis and Chapman (1971) noted that knowledge-of-re s u l t s i s the most frequently c i t e d r e i n f o r c e r i n the l i t e r a t u r e on s e l f - i n s t r u c t i o n a l systems, e s p e c i a l l y programmed i n s t r u c t i o n . They found that most studies are not d i r e c t l y aimed at investigating whether or not answers are r e i n f o r c e r s . The question usually being attached i s a broader one. Does feedback i n some way a f f e c t performance during and a f t e r programmed s e l f - i n s t r u c t i o n ? Geis and Chapman (1971) reviewed the l i t e r a t u r e i n t h i s area and r e s t r i c t e d themselves to s e l f - i n s t r u c t i o n a l s i t u a t i o n s . The authors cautioned that research using other than s e l f - i n s t r u c t i o n a l materials d i f f e r so much from s e l f - i n s c r u c t i o n that extrapolation i s u n j u s t i f i e d . In an i n v e s t i g a t i o n of the r e l a t i o n s h i p between test anxiety and feedback i n programmed i n s t r u c t i o n , Campeau (1968) found that feedback was a s i g n i f i c a n t variable i n the performance of. grrde-school g i r l s . P o s t - i n s t r u c t i o n a l test scores were higher for those high-anxiety g i r l s who had .feedback during learning. Low-anxiety female students who had no feedback had higher posttest scores than high-anxiety females who had no feedback. No s i g n i f i c a n t differences were found between low and high anxiety students under feedback conditions. Male students showed no s i m i l a r r e g u l a r i t y . Cronbach and Snow (1969) reported that t h i s f i f t h grade study by Campeau used programmed i n s t r u c t i o n , giving one group feedback to a s s i s t i n the c o r r e c t i o n of responses, and the other group no feedback. The number of subjects was small (36 boys, 44 g i r l s ) e s p e c i a l l y since the analysis was performed within sexes. Only persons at the extreme of the anxiety d i s t r i b u t i o n were used, the dependent variable was a posttest score with i n i t i a l IQ p a r t i a l l e d out. For g i r l s there was a s i g n i f i c a n t i n t e r a c t i o n , with those high on test anxiety doing d i s t i n c t l y better when given feedback and d i s t i n c t l y worse than low anxious when given no feedback. This was also found on a retention test. For boys, the r e l a t i o n s h i p s were not s i g n i f i c a n t , and on the immediate posttest there was e s s e n t i a l l y no e f f e c t . Reporting i s inadequate. It i s uncertain that the low anxious toys are s i m i l a r to the low anxious g i r l s . It i s often found that g i r l s are considerably higher i n test anxiety and i t may be that a g i r l ' s low score matches a boy's high score. Campeau's (1968) i n t e r p r e t a t i o n was that witholding feedback i n t e n s i f i e s motivation by maintaining a c e r t a i n incompleteness. That i s to say, the no feedback s i t u a t i o n 30 i s more challenging and more s t r e s s f u l . A l t e r n a t i v e l y , one could perhaps say that the provision of feedback provides greater structure, leaving the person much l e s s on his own resources. The e s s e n t i a l l y negative re s u l t f o r boys was not explained. Wittrock and Twelker (1964) found an i n t e r e s t i n g r e l a t i o n s h i p between knowledge-of-results and rules. While rules alone proved most e f f e c t i v e i n teaching subjects to decode ciphered sentences, knowledge-of-results was e s p e c i a l l y useful when rules were not supplied. It did not add to teaching effectiveness when supplied i n conjunction with rules, supporting the authors' contention that knowledge-of-results enhances learning, retention and transfer when the information i t contains i s not greatly redundant. Geis and Chapman (1971) found c o n f l i c t i n g r e s u l t s among studies that examined variable schedules of reinforcement and variable delays i n confirmation of correct or incorrect responses. No conclusions were reached about these variables. Anderson et_ al_. (1972) used several feedback arrangements involving a computer and a program on diagnosing myocardial i n f r a c t i o n . In on? experiment using several groups, they presented the correct response (1) only aft e r a correct response had been emitted, or (2) only aft e r a wrong response had been emitted, or (3) always (100 percent), or (4) never (0 percent), or (5) a f t e r a correct response, but the subject had to "loop" back to the same frame after the wrong response. C r i t e r i o n test scores were higher for the 100 percent feedback and the "looped" groups. The only group with s i g n i f i c a n t l y lower test scores was the no feedback group (0 percent). Of int e r e s t here i s the f a c t that no s i g n i f i c a n t difference was found between the 100 percent and the "looped" group, although the l a t t e r students underwent much more elaborate feedback procedures. There was no evidence i n t h i s study that knowledge-of-results functioned as co r r e c t i v e feedback. The group receiving knowledge-of-results only when errors were made did not perform s i g n i f i c a n t l y better on the c r i t e r i o n test than the other knowledge-of-results groups. The knowledge-of-results-only-when-correct group performed at the seme l e v e l as the other groups. The function of knowledge-of-results was not c l a r i f i e d by t h i s experiment. In a recent CAI study, Keats and Hansen (1972) examined the e f f e c t s of c o r r e c t i o n a l feedback on learning. They noted that the precise form and content for co r r e c t i o n a l messages that w i l i maximize learning remains a clouded issue, although the requirement for 32 c o r r e c t i o n a l feedback to wrong answers i n order to f a c i l i t a t e a c q u i s i t i o n has been well established. This study compared the e f f e c t s of using verbal d e f i n i t i o n s and numerical examples as CAI c o r r e c t i o n a l feedback i n a program involving mathematical proofs. Ss were f o r t y - f i v e ninth grade . students. Students were only required to state the rule which was applied to each step Of the proof. The feedback chosen f o r a p a r t i c u l a r step varied according to rules derived by l o g i c a l analysis c a r r i e d out by an experienced mathematics i n s t r u c t o r . The very low r e l i a b i l i t y of the instruments prevented conclusions from being drawn about the s i m i l a r i t y i n posttest and retention scores. However, i n terms of errors during the program, providing c o r r e c t i o n a l feedback i n the form of a verbal d e f i n i t i o n was more b e n e f i c i a l to the learner. Gilman (1967) investigated the e f f e c t of various kinds of feedback i n a computer a s s i s t e d i n s t r u c t i o n (CAI) system. He wrote that ". . . i f there were no purpose to feedback other than to provide the student with reinforcement, statements such as 'you are correct' should prove equally-e f f e c t i v e as confirmation of a correct answer." U n i v e r s i t y upperclassmen were taught t h i r t y general science concepts by means of a CAI s e l f - i n s t r u c t i o n a l system, using a multiple choice format. Various modes of feedback were used: no 33 feedback, "correct" or "wrong", feedback of correct response, feedback appropriate to the student's response, and a combination of the three l a t t e r modes. Students repeated items which were missed u n t i l a perfect run through was obtained. The no feedback group and the "correct" or ''wrong" group performed l e s s well on the program, making a s i g n i f i c a n t l y greater number of responses and requiring a greater number of i t e r a t i o n s of the program i n order to reach c r i t e r i o n . On the posttest, the combination feedback group scored s i g n i f i c a n t l y higher than did the others. This study suggests that more elaborate feedback may be more e f f e c t i v e i n changing student behaviour. Hernandez and Gilman (1969) compared the effectiveness of several feedback modes for correcting errors i n CAI. Seventy-five u n i v e r s i t y upperclassmen were taught t h i r t y general science concepts. The frames were multiple choice items and f i v e d i f f e r e n t conditions were administered, from no feedback up to feedback appropriate to the student's response. The r e s u l t s indicated that the most s i g n i f i c a n t f a c t o r i n rate of error correction i s guiding the subject to the correct response. The most s i g n i f i c a n t factor i n immediate learning i s amount of feedback information the subject receives. The author made an important point by noting that p r i o r studies i n programmed learning have not 3 4 been able to compare the effectiveness -of the several modes of feedback i n correcting the student errors because these studies u t i l i z e d low error rate, l i n e a r type programs. Since few incorrect responses are made by a student i n t h i s type of learning s i t u a t i o n , l i t t l e i s presently known concerning how feedback can be used to correct student errors. Van Dyke and Newton (1972) examined the e f f e c t of immediate and delayed knowledge-of-response i n a CAI task. Response differences between the sexes were investigated as well as attitudes toward CAI. Ss were college students taking an Introductory Psychology course. They concluded that short i n t e r v a l s of delay with CAI had no s i g n i f i c a n t e f f e c t on the learning or test performance of the in d i v i d u a l s used i n t h i s study. Delay of knowledge-of-r e s u l t s did r e s u l t i n a t t i t u d e differences, and t h i s e f f e c t varied with the sex of the learner. However, i t should be noted that the eleven item achievement test used i n the study had a K-R 20 r e l i a b i l i t y c o e f f i c i e n t of .44. Narva (1970) reported a study with c h a r a c t e r i s t i c s similar to the present study. He compared two types of CAI programs, one employing branching end the othar one linear i n format. He also varied the c o r r e c t i o n a l feedback given the student i n both types of programs. He used resporse-sens.it ive and response-insensitive c o r r e c t i o n a l 35 feedback i n a way similar to the present study except that the response-insensitive feedback provided only the correct answer. He used thirty-two subjects i n a 2 x 2 f a c t o r i a l design. The question was raised about whether t a i l o r i n g the program to i n d i v i d u a l students i s worth the cost. He found that p r e - i n s t r u c t i o n scores were not re l a t e d to l a t e r performance but that aptitude scores were s i g n i f i c a n t l y c o r r e l a t e d to posttest scores. He also found no s i g n i f i c a n t differences between either the two sequencing programs or between the two types of feedback. However, when an analysis unadjusted for the influence of the aptitude score was performed, a s i g n i f i c a n t difference i n favor of response-sensitive feedback was found. Finer grained analysis of student behaviour and knowledge-of-results begin to reveal s p e c i f i c conditions under which knowledge-of-results seems to be acting as re i n f o r c e r . A few studies scattered throughout the l i t e r a t u r e report on manipulation of subject and test variables and of kinds of feedback. The r e s u l t s of these studies suggest that knowledge-of-results may well be a re i n f o r c e r when uncertainty or p r o b a b i l i t y of emitting an incor r e c t response i s high, or where confidence i s low. Geis ana Chapman (1971) concluded that the weight of evidc-nce from global studies comparing programs with and 36 without feedback i s that feedback did not enhance learning, as measured.by immediate posttest scores or by retention scores. These authors conclude that: . . . one might jump into broader questions such as how, when and why information on one's own performance i n a learning s i t u a t i o n becomes r e i n f o r c i n g and contributes to more e f f e c t i v e learning. Along these l i n e s , several studies have recently concerned themselves with the e f f e c t of d i f f e r e n t types of c o r r e c t i o n a l feedback. These studies were reported above. The present study examines t h i s question as well. Summary The exact r o l e of c o r r e c t i o n a l feedback on the learning process and product i s unclear. Stolurow (1971), M e r r i l l (1971) and Annett (1969) have made a strong case f o r the importance of knowledge-o f - r e s u l t s i n the i n s t r u c t i o n a l process. Some experimenters have shown that c o r r e c t i o n a l feedback has a p o s i t i v e e f f e c t on learning and that t h i s e f f e c t depends on the nature of the learner. Campeau (1967) provided some evidence for a feedback-by-anxiety i n t e r a c t i o n on posttest scores, although t h i s study was c r i t i c i s e d by Crcnbach and Snow (1969). Wittrock and Twelker (1964) showed that knowledge-of-results enhanced learning, 3 7 retention and transfer when the information i t contained was not redundant. Keats and Hansen,(1972) performed a CAI study which provided some evidence that fewer errors were made during learning when co r r e c t i o n a l feedback i n the form of a verbal d e f i n i t i o n was supplied. Gilman (1967) performed a CAI experiment which suggested that more elaborate feedback might be more e f f e c t i v e i n changing student behaviour. Hernandez and Gilman (196 9) found that the most s i g n i f i c a n t f a c t o r i n immediate learning was the amount of feedback information the subject received. Narva (1970) c a r r i e d out a CAI experiment which showed that aptitude scores were re l a t e d to posttest scores and that response-sensitive feedback was superior to response-i n s e n s i t i v e feedback i n terms of posttest scores (unadjusted for aptitude). Despite the conclusion of Geis and Chapman (1971) that feedback d i d not enhance learning i n most studies reviewed, there i s enough evidence i n favour of c o r r e c t i o n a l feedback to question t h i s conclusion. If the information content of c o r r e c t i o n a l feedback is important to learning, then differences i n performance between the three treatment groups would be expected i n t h i s study. It was expected that providing response-s e n s i t i v e c o r r e c t i o n a l feedback (T^) would provide more 38 information to the student than providing response-i n s e n s i t i v e c o r r e c t i o n a l feedback (T 2) and that the T-^ treatment condition would then be less d i f f i c u l t than the T 2 treatment condition. Therefore, i t was hypothesized that i n terms of proportion of errors, average latency and immediate learning, T-[ students would perform s i g n i f i c a n t l y better than To students. Similar reasoning l e d to the conclusion that providing information i n the form of response-insensitive c o r r e c t i o n a l feedback (T 2) w i l l make the lesson less d i f f i c u l t than providing no c o r r e c t i o n a l feedback other than information that the student's response was correct ( T Q ) . Therefore, i t was hypothesized that i n terms of proportion of errors, average latency and immediate learning, T2 students would perform s i g n i f i c a n t l y better than T^ students. Higgins (1972) pointed out that feedback appears to be more e f f e c t i v e when the learner possesses a low l e v e l of competence with regard to the i n s t r u c t i o n a l task. If the i n s t r u c t i o n a l program i s e f f e c t i v e , then providing the student with corr e c t i o n a l feedback (T^ or T 2 ) would allow a l l students to a t t a i n the objectives of the unit. However, i f no c o r r e c t i o n a l feedback i s provided (T3), the lack of information would render the material too d i f f i c u l t for students who were lacking the p r e r e q u i s i t e s k i l l s or who had l i t t l e mathematical a b i l i t y . Students who had attained the prerequisites or who were high on mathematical a b i l i t y would be able to compensate fo r the lack of information provided and would often be able to provide their own explanation for their incorrect responses. Therefore, i t was hypothesized that i n terms of proportion of errors, average latency and immediate learning, students who are high on prerequisite s k i l l s or mathematical a b i l i t y would perform better than students low i n prerequisite s k i l l s or mathematical a b i l i t y when no c o r r e c t i o n a l feedback i s provided (Tg). The c r i t i c a l c r i t e r i o n of success i s the product of an i n s t r u c t i o n a l session, which i n this study was immediate learning. The process variables such as proportion of errors and response latency may be useful for basing decisions during i n s t r u c t i o n i n order to maximize immediate learning. However, a r e l a t i o n s h i p would have to be established between the process and product varia b l e s . Therefore, i t othesized that there would be a s i g n i f i c a n t l i n e a r r e l a t i o n s h i p between the process variables, i . e . proportion of errors and average latency and the product variable, i . e . immediate learning, with the e f f e c t s of treatment and learner c h a r a c t e r i s t i c s removed. X 40 Lit e r a t u r e on State-Tr a i t Anxiety Spielberger (1966) pointed out that since 1950 over 1,500 studies indexed under "anxiety" have been reported i n Psychological Abstracts. Despite so much e f f o r t i n the area, attempts to define anxiety have met with d i f f i c u l t i e s owing to widely d i f f e r e n t conceptions of anxiety. Szetela (1970) described some of the problems with the anxiety construct. He mentioned differences i n d e f i n i t i o n , l i m i t a t i o n s of paper and p e n c i l t e s t s , the unidimensional versus multidimensional question, and the state versus t r a i t conceptions of anxiety as unresolved issues. O'Neil e_t al_. (1969) noted that most studies concerning the e f f e c t s of anxiety on learning have originated either in a r t i f i c i a l laboratory settings or in r e a l i s t i c but poorly c o n t r o l l e d natural settings. CAI systems provide a convenient netting in which i t i s possible to evaluate the learning process under c a r e f u l l y c o n t r o l l e d conditions with materials that are relevant to the learner. O'Neil e_t a l . (1969) observed that research on anxiety and learning has suffered from ambiguity with regard to .the status of anxiety as a th e o r e t i c a l concept. Spielberger (1971, 1972) recently emphasized the need to d i s t i n g u i s h between anxiety conceptualized a.s a tra n s i t o r y state or condition of the organism and as a r e l a t i v e l y stable personality t r a i t . State anxiety (A-State) consists of fe e l i n g s of apprehension and heightened autonomic nervous system a c t i v i t y that vary i n in t e n s i t y and flu c t u a t e over time. T r a i t anxiety (A-Trait) r e f e r s to i n d i v i d u a l differences i n anxiety proneness, that i s , to d i f f e r e n t i a l tendencies among individuals' to respond with d i f f e r e n t levels of A-State i n si t u a t i o n s that are perceived as threatening. Persons high i n A-Trait are also more disposed to see certa i n types of situations as more dangerous, p a r t i c u l a r l y s i t u a t i o n s that involve f a i l u r e or some threat to that i n d i v i d u a l ' s self-esteem. F a i l u r e to make this d i s t i n c t i o n has led to the inappropriate use of operational measures of A-State and A-Trait, and this has contributed to the inconsistent and contradictory findings i n investigations of anxiety. The most serious methodological flaws are the use of A-Trait scales to measure tr a n s i t o r y anxiety and the common practice of f a i l i n g to obtain measures of A-State to corroborate the e f f e c t s of experimental manipulations designed to be s t r e s s f u l (Spielberger, 1972; O'Neil, 1972a). O'Neil e_t al_. (1969) pointed out that i n order to study the e f f e c t s of anxiety on learning, a theory of learning i s needed that s p e c i f i e s the complex r e l a t i o n s h i p between anxiety and behaviour. According to the Drive 42 Theory proposed by Spence and Taylor (Spielberger, 1972) the performance of high anxious students would be i n f e r i o r to that of low anxious students on complex or d i f f i c u l t tasks and superior on easy tasks. Much empirical support has.been provided f o r t h i s theory (O'Neil, 1969a, 1972a). O'Neil e_t a_l. (1969) investigated the r e l a t i o n s h i p between A-State and performance on a CAI task for college males with extreme scores on the A-Trait scale of the State-Trait Anxiety Inventory. D i f f i c u l t and easy CAI learning materials were presented by an IBM 1500 system which also presented the State-Trait Anxiety Inventory (STAI ) A-State scale, before, during and a f t e r the learning task. The findings of an e a r l i e r study (O'Neil e_t al_. , 1969) were confirmed i n that (a) A-State scores increased while subjects worked on d i f f i c u l t materials and decreased when they responded to easy materials; and (b) high A-State subjects made s i g n i f i c a n t l y more errors on the d i f f i c u l t materials than low A-State subjects. While there was no r e l a t i o n between A-T r a i t and performance, high A - T r a i t subjects responded throughout the learning task with higher l e v e l s of A-State than low A - T r a i t subjects. In a later study, O'Neil (1972aj again used a CAI task with college students i n an introductory psychology course. Subjects i n the stress condition were given negative feedback regarding their performance on a CAI 43 learning task, whereas subjects i n the non-stress condition received neutral feedback. Negative feedback about performance i n the stress condition led to greater i n i t i a l increments i n A-State for high A - T r a i t subjects than for low A-Trait subjects. This did not occur i n the non-stress group. These r e s u l t s are consistent with State-Trait Anxiety theory. This study produced d i f f e r e n t r e s u l t s with regard to the r e l a t i o n s h i p between A-State and errors because the high A-State subjects made s i g n i f i c a n t l y more errors than low A-State subjects on the easy materials. This r e s u l t was not consistent with the pre d i c t i o n from Drive Theory. One possible explanation of the inconsistent r e l a t i o n between A-State and errors i n O'Neil's two studies i s that the l a t t e r study used only female subjects and the former used males. O'Neil pointed out that the l i t e r a t u r e suggests that r e l a t i o n s h i p between anxiety and learning i s d i f f e r e n t f o r men and women. He noted that sex and anxiety interactions probably r e f l e c t s p e c i f i c s i t u a t i o n a l variables which influence learning and which may have d i f f e r e n t i a l s i g n i f i c a n c e for men and women. He suggests that caution must be exercised i n making generalizations concerning the rel a t i o n s h i p between anxiety, sex and learning. Wine's (1971) leview of the test anxiety l i t e r a t u r e has indicated that on d i f f i c u l t tasks i n which evaluative stress i s present, low anxiety (LA) students tend to achieve more than high anxiety (HA) i n d i v i d u a l s . Wine suggested that HA students, compared to LA, focus a greater proportion of their attention on personal preoccupations and less to task relevant problems. Tobias (1973) wrote that previous CAI research i n which state anxiety was assessed while students were working on i n s t r u c t i o n a l programs have indicated higher l e v e l s of anxiety for constructing responses compared to reading the program. He examined test anxiety i n a CAI study and found no e f f e c t on learning and concluded that i f anxiety i s to exercise d e b i l i t a t i n g e f f e c t s , more d i f f i c u l t content than presently used i s required. Hensen (1972) found that r e a l reductions i n A-State were obtained through increased use of information feedback, although th i s reduction d i d not necessarily r e s u l t i n higher levels of performance. Summary The findings from anxiety studies have been contradictory for reasons described by Spielberger (1966) and Szetela (1970). Recent work done with the State-Trait Anxiety Inventory (Spielberger e_t al_. , 1970) has shown promise i n leading to better understanding of the e f f e c t s 45 of anxiety on learning. O'Neil (1969, 1972a) has done some CAI research which has produced evidence to support Drive Theory, i . e . that the performance of high anxious students would be i n f e r i o r to that of low anxious students on complex or d i f f i c u l t tasks and superior on easy tasks. They also found that higher lev e l s of A-State were induced i n s i t u a t i o n s where stress was induced i n the students, which meant that the s i t u a t i o n was perceived as threatening. Wine (1971) also found evidence to suggest an anxiety by d i f f i c u l t y i n t e r a c t i o n for s t r e s s f u l tasks. As discussed i n the l a s t section, the three treatment groups i n this study should be ranked i n terms of d i f f i c u l t y from easy to d i f f i c u l t (T^ easier than T 2, T2 easier than T 3 ) . However, i t would appear that the large d i f f i c u l t y gap would occur between the T 2 and T3 groups. The T-^ or To treatments would probably be perceived as r e l a t i v e l y easy i n comparison to the more d i f f i c u l t T3 task, where no corr e c t i o n a l feedback i s provided. The students have had no experience i n i n t e r a c t i n q with a computer and t h i s factor combined with the strange content and surroundings would seem to provide enough stress to induce high levels of anxiety. Therefore, i n l i n e with recent evidence i t was hypothesized that students who are high in A-State w i l l have a s i g n i f i c a n t l y higher proportion of errors and average latency than low A-State students 46 when no co r r e c t i o n a l feedback i s provided (Tg) and this r e l a t i o n s h i p would be reversed when co r r e c t i o n a l feedback i s provided (To). Liter a t u r e on Aptitude-Treatment Interactions Bracht (1970) described the goal of research on aptitude-treatment interactions (ATI) as finding s i g n i f i c a n t d i s o r d i n a l i n t e r a c t i o n s between a l t e r n a t i v e treatments and personological variables, i . e . development of a l t e r n a t i v e i n s t r u c t i o n a l programs so that optimal educational benefits are obtained when students are assigned 'differently to the al t e r n a t i v e programs. The personological variable i n ATI research was defined as any measure of i n d i v i d u a l c h a r a c t e r i s t i c s , e.g. IQ, s c i e n t i f i c i n t e r e s t or anxiety. Cronbach and Snow (1969) have written a major report in the area of ATI. They defined "aptitude" as any c h a r a c t e r i s t i c of the i n d i v i d u a l that increases (or impairs) his p r o b a b i l i t y of success i n a given treatment, and includes personality c h a r a c t e r i s t i c s . They agreed +hat the immediate objective of current ATI work i s to match s p e c i f i c i n s t r u c t i o n a l methods or materials to selected learner c h a r a c t e r i s t i c s . But more broadly, they argued that ATI research i s concerned with theory to overarch diverse ideas such as the branching rules and strategies required i n CA.I. 47 As Bracht (1970) pointed out, there i s an increasing i n t e r e s t i n the topic of ATI among educational researchers, but very l i t t l e empirical evidence has been provided to support the concept. He conducted a systematic analysis of research studies to investigate the r e l a t i o n s h i p of treatment tasks, personological variables and dependent variables to the occurrence of ATI. He gave a flowchart of a procedure f o r testing an ATI i n a treatment-by-levels f a c t o r i a l design, since most studies used th i s design with analysis of variance. He suggested that t h i s method i s less powerful than regression analysis since creating a r t i f i c i a l l evels of a continuous personological variable tends to increase the error component i n the analysis. Regression analysis, however, was used i n r e l a t i v e l y few ATI studies. From his analysis, Bracht (1970) found d i s o r d i n a l i n t e r a c t i o n s i n only f i v e of the 103 studies. The r e s u l t s tend to indicate that there i s a r e l a t i o n s h i p between d i s o r d i n a l interactions and the degree of control over the treatment tasks of the experiment. Controlled treatment tasks may be a necessary but c e r t a i n l y not a s u f f i c i e n t requirement for d i s o r d i n a l i n t e r a c t i o n s . Measures of s p e c i f i c a b i l i t i e s , i n t e r e s t s , personality t r a i t s were c l a s s i f i e d as f a c t o r i a l l y simple. A personological variable was c l a s s i f i e d as f a c t o r i a l l y complex i f i t was judged to have a substantial loading on many facto r s i n the imaginary 48 \ factor matrix, e.g. general a b i l i t y . Bracht's (1970) r e s u l t s lend some support to the r e l a t i o n s h i p between ATI and the degree of f a c t o r i a l s i m p l i c i t y of the personological variable, although f a c t o r i a l s i m p l i c i t y c e r t a i n l y i s not a s u f f i c i e n t requirement f o r ATI. Many experiments using IQ as a variable found no evidence to suggest that the IQ score and similar measures of general a b i l i t y . a r e useful variables for d i f f e r e n t i a t i n g a l t e r n a t i v e treatments for subjects i n a homogeneous age group (Bracht, 1970). However, Cronbach and Snow (1969) recommended the development of a l t e r n a t i v e treatments on the basis of genera! a b i l i t y . They f e e l that general a b i l i t y seems to be nearly synonomous with ' a b i l i t y to learn', when that term i s given i t s usual common sense i n t e r p r e t a t i o n . One treatment should be designed to r e l y heavily on general a b i l i t y , and the other treatment should be designed to achieve the same objectives without r e l y i n g on general a b i l i t y . This a p r i o r i s p e c i f i c a t i o n of treatments had not been done i n past research. The reason for a p p l i c a t i o n of a broad, loose construct i s that at present there i s no evidence to support a more r e f i n e d one. Bracht (1970) learned very l i t t l e from his analysis about the r e l a t i o n s h i p between the dependent variable and the occurrence of ATI because the dependent variable was 49 most often errors i n learning or posttest score. Webb (1972) suggested that ATI studies include as many d i f f e r e n t c r i t e r i o n measures as possible, e.g. learning tests, retention tests, transfer tests, time to completion, etc. He f e e l s that the current state of the theory of ATI i s not well enough developed to determine the s p e c i f i c dependent variables that should or should not be included. Bracht found that ATI i s more l i k e l y to occur when two personological variables have been included i n the experimental design. One variable i s judged to correlate s u b s t a n t i a l l y with, success i n one treatment and the other judged to co r r e l a t e s u b s t a n t i a l l y with success i n the second treatment. The c o r r e l a t i o n between the two personological variables must be moderately low or nonsignificant for the d i s o r d i n a l i n t e r a c t i o n to occur. F i n a l l y , Bracht suggested that experimenters should begin to formulate hypotheses about ATI with administrative f a c t o r s , such as cost, i n mind. Hence, even ordinal i n t e r a c t i o n s may lead to decisions about d i f f e r e n t i a l assignment to treatments when administrative f a c t o r s are taken into account. Campeau (1963) found a s i g n i f i c a n t test anxiety by feedback i n t e r a c t i o n for g i r l s i n a study described e a r l i e r i n the section dealing with l i t e r a t u r e on knowledge-of-r e s u l t s . O'Neil (1972) found A s i g n i f i c a n t State Anxiety 50 by task d i f f i c u l t y i n t e r a c t i o n i n another study described e a r l i e r i n the section dealing with l i t e r a t u r e on anxiety. Dick and Latta (1970) compared a programmed i n s t r u c t i o n version with a comparable CAI version of the same material using grade eight mathematics students. The si x t y - f o u r students were randomly assigned to programmed i n s t r u c t i o n and CAI programs on the topic of s i g n i f i c a n t f i g u r e s . The r e s u l t s indicated that the students using programmed i n s t r u c t i o n performed s i g n i f i c a n t l y better than those using CAI. This difference was a t t r i b u t e d primarily to the very poor performance by the low a b i l i t y students in the CAI group. There was also a s i g n i f i c a n t a b i l i t y e f f e c t . The posttest and retention test r e s u l t s , as well as number of errors i n the actual learning sequence, indicated that there was a trait-by-treatment i n t e r a c t i o n which was interpreted p r i m a r i l y as a very poor performance by low a b i l i t y students on CAI, with almost equal performance by high and low a b i l i t y students u t i l i z i n g programmed i n s t r u c t i o n a l materials, The conjecture may be made that the low a b i l i t y students are unable to cope with the continuous flow of information as presented by the cathode-ray tube (CRT) without the • a b i l i t y to return to the information previously provided them Becker (1970) wrote that there e x i s t s a large number of studies that are i n d i r e c t l y relevant to aptitude-treatment i n t e r a c t i o n research but that few studies have been designed to investigate i n t e r a c t i o n between aptitude and i n s t r u c t i o n . Becker (1970) discussed the idea that various a b i l i t i e s might play an important r o l e i n determining what method of i n s t r u c t i o n i s prescribed for a p a r t i c u l a r learner or group of learners. For example, ind i v i d u a l s who are high on mathematical a b i l i t y should p r o f i t more from i n s t r u c t i o n that provides mathematical mediators than verbal ones. Becker (1970) pointed out some po t e n t i a l d i f f i c u l t i e s i n aptitutde-treatment i n t e r a c t i o n research. He concluded that the s e l e c t i o n of aptitude measures that w i l l i n t e r a c t with methods of i n s t r u c t i o n may be d i f f i c u l t . In general, aptitude measures may be needed that get at s p e c i f i c aspects of mental a b i l i t i e s and with acceptable l e v e l s of r e l i a b i l i t y . Determining the length of the treatment period i s another problem. The problem e x i s t s of assessing the impact of past i n s t r u c t i o n on subjects involved i n current research. F i n a l l y , i t i s d i f f i c u l t to determine the type of achievement ( c r i t e r i o n ) measures from which we are l i k e l y to derive i n t e r a c t i o n s . 52 Summary This b r i e f review of the ATI l i t e r a t u r e indicates that interactions between aptitude and treatment have been rare and that more work i s needed. The evidence from most ATI research i s not d e f i n i t i v e enough to provide guidance f o r i n d i v i d u a l i z i n g i n s t r u c t i o n . This study has attempted to provide evidence fo r the existence of interactions between c o r r e c t i o n a l feedback and anxiety, c o r r e c t i o n a l feedback and prerequisite s k i l l s and c o r r e c t i o n a l feedback and mathematical a b i l i t y . L i t e r a t u r e on T u t o r i a l Computer-Assisted Instruction (CAI) The most relevant CAI studies have already been reviewed i n previous sections of t h i s chapter. Some of the other l i t e r a t u r e i n this area w i l l now be examined. Stolurow (1968) i d e n t i f i e d several modes of CAI: problem solving, d r i l l and p r a c t i c e , inquiry, simulation and gaming, t u t o r i a l i n s t r u c t i o n and author mode. This study i s concerned with CAI t u t o r i a l mode. In t h i s racde, the i n s t r u c t i o n a l programmer takes r e s p o n s i b i l i t y for the student's i n s t r u c t i o n on the system. The l o g i c of i n s t r u c t i o n must be formalized i n d e c a i l and entered into the system. This mbc'e is i n a p r i m i t i v e state of development, It i s primarily conciotually rather than emp i r i c a l l y based, and i t s ultimate form w i l l i n e f f e c t be a theory of teaching. 53 The theory w i l l depend for development upon s p e c i f i c studies designed to generate data on how to use d i f f e r e n t rules for i n s t r u c t i n g students with d i f f e r e n t response h i s t o r i e s and unique c h a r a c t e r i s t i c s . Unfortunately, the above theory development has not progressed as well as Stolurow had envisioned. Most discussions of CAI are concerned with computer hardware and software problems. A few are concerned with decision l o g i c s and i n s t r u c t i o n a l strategies that primarily involve explanation of the functions involved i n t r a n s f e r r i n g from i n s t r u c t i o n a l block to i n s t r u c t i o n a l block. Fewer discussions involve d e t a i l about the c h a r a c t e r i s t i c s of the teaching sequence within these blocks i n terms of the stimulus display, the student response or the feedback information required (Glaser, 1971). The i n t e r a c t i o n of these variables with c h a r a c t e r i s t i c s of the learner s t i l l remains to be studied. Gentile (1967) and Dick (1965) reached the same conclusion. The matter of personality-computer i n t e r a c t i o n remains to be studied. One program structure w i l l not be appropriate foa. a i l l e v e l s of a b i l i t y and various subject matter areas (Dick, 1965). Gentile (1967) wrote that almost a l l funis a l l o t t e d to CAI projects are being spent cn the development of courses or equipment to the exclusion of research on teaching and learning , variables, where 54 research i s needed most. Sherman (1971) distinguished two tasks i n developing a CAI lesson. The f i r s t task i s one of designing the pedagogical exchanges, i . e . the sequence of exchanges that conversations may follow. The second task i n developing CAI conversations i s that of programming the conversational network so that a given computer can play i t s r o l e i n the dialogue. He proposed reducing the time spent on the second task by the use of programmed templates. Sherman (1971) defined the work "template" as: . . . a sequence of instructions to the computer for processing a p a r t i c u l a r network of conversational exchanges and connotes a p a r t i c u l a r pattern f o r the presentation of questions, answers and other basic elements of t u t o r i a l conversation. The pattern determines only the l o g i c of the con-versational exchanges, not the content of the conversations themselves. Sherman (1971) developed a template and used i t to program a dialogue to teach some college physics concepts. Investigators i n several major CAI centres are curren t l y conducting research and development a c t i v i t i e s i n the area of t u t o r i a l mode CAI. Several CAI projects were reported by Bande.rsor. et _al, (1968) at the University of Texas at Austin o.nd Hansen (1971) at F l o r i d a State Un i v e r s i t y . In a project at the University of Texas at Austin (Judd e_t al_. , 1970) a CAI course was developed to provide diagnosis of d e f i c i e n c i e s i n s k i l l s p r e r e q u i s i t e to freshman science courses. The CAI students achieved well and demonstrated superior performance to the non-CAI students on the posttest. They also found that i n some cases program control was superior to learner control and i n other cases some form of learner control was superior. Bunderson (1971) f e l t that the r e s u l t s of t h i s study are so complicated by i n t e r a c t i o n s with pretest score, terminal type, amount of p r a c t i c e and topic as to make generalizations r i s k y . Other CAI programs at Texas are used to teach English, Computer Programming, Business Management, Astronomy Mathematics and Science, Arabic Writing System and Chemistry. The r e s u l t s from these projects are not conclusive. The projects are regarded as exploratory i n nature a.nd provide information and methods to ease the problem of designing high q u a l i t y i n s t r u c t i o n a l programs; the costs of preparation maintenance, and r e v i s i o n are the primary variables. Minimum l e v e l s of student performance, time to complete and a t t i t u d e ratings are used as design c r i t e r i a . Several recent studies have u t i l i z e d t u t o r i a l mode CAI to study the i n s t r u c t i o n a l process. Some of these studies (Gilman, 1967; Keats and Hansen, 1972; V^n Dyke, 1972) have been described i n the e a r l i e r sections of this chapter. Other studies i n the areti arc described below. Sutter and Reid (1969) showed that the effectiveness of CAI i n teaching a course i n problem solving i s the same for the student working alone with the machine as for the student working with a partner at the machine, except when conditional upon c e r t a i n personality t r a i t s . College students were used and high test anxiety was associated with negative attitudes toward CAI i n both the paired and alone groups. A s i g n i f i c a n t i n t e r a c t i o n was obtained between test anxiety and achievement for the two groups. Students high i n test anxiety achieved better working alone, while those low i n test anxiety achieved better working with a partner. An i n t e r a c t i o n was also found between s o c i a b i l i t y and achievement for the two groups. Bunderson and his colleagues (Bunderson, 1971) developed an imaginary science task c a l l e d a Xenograde system. The Xenograde system has the h i e r a r c h i c a l structure of concepts and quantitative rules c h a r a c t e r i s t i c of many topics i n science education. The greatest advantage of i t s imaginary character has been to enable researchers to concentrate on design variables - structure, display, etc.. rather than subject matter variables. Several CAI studies of learning have been conducted with t h i s topic. O l i v i e r (1971) compared the performance of students i n the learner control mode with students for whom the sequence was con t r o l l e d by the program. He matched students in both 57 groups and so the data were not treated s t a t i s t i c a l l y but are suggestive only. The conclusion i s that except for a small number of exceptional students, learner control of the sequence of lessons i n a hierarchy may be less e f f e c t i v e for learning new material than a r a t i o n a l l y planned, c a r e f u l l y designed program-controlled sequence. Bunderson (1971) wrote that i n subsequent studies using p a r t i a l l y f a m i l i a r material the r e s u l t s were less clear cut. M e r r i l l (1971) investigated the e f f e c t s that a v a i l a b i l i t y of behaviourally stated objectives would have on the learning process. The Xenograde science was used with 130 college students. The r e s u l t s showed that objectives s i g n i f i c a n t l y reduced the number of examples required to learn the task. Objectives did not reduce t o t a l latency but did reduce test-item response latency. No s i g n i f i c a n t differences were found between treatments on the posttests or retention tests, but a s i g n i f i c a n t r u l e e f f e c t was found i n favour of the groups which received the rule as opposed to examples. S i g n i f i c a n t a b i l i t y - b y -treatment i n t e r a c t i o n s were obtained using test-item-response latency as c r i t e r i o n and reasoning f a c t o r scores, plus i n d i v i d u a l reasoning tests as covariables. On the basis of the r e s u l t s of t h i s study i t was concluded that objectives have or i e n t i n g and organizing e f f e c t s which dispose students 58 to attend to, process and structure relevant information in accordance with the given objectives. Lorber (1970) demonstrated that CAI i s an e f f e c t i v e means by which to teach the basic elements of tests and measurements and pup i l evaluation of prospective secondary school teachers. His r e s u l t s showed that the CAI group scored higher on the posttest than the conventional classroom group and needed less i n s t r u c t i o n a l time. Attitudes towards CAI were favourable. T i r a (1970) produced a CAI program dealing with the Product-Moment f a c i l y of c o r r e l a t i o n s . This non-experimental study had four phases: (1) d e f i n i t i o n of requirements, (2) design of lesson content, (3) production of CAI dialogue from content, (4 ) evaluation of the program. The program was judged to be successful and the author recommended that the entering c h a r a c t e r i s t i c s of the learner be i d e n t i f i e d and paired with a program sequence which would best f a c i l i t a t e the meaningful learning of the concepts on the CAI course. Ibrahim (1970) compared CAI with other i n s t r u c t i o n a l methods i n the teacning of the concepts of l i m i t s i n freshman calculus. The other i n s t r u c t i o n a l methods were (1) the.instructor centred or t r a d i t i o n a l approach and (2) a combination of t r a d i t i o n a l and CAI. The findings were that the CAI students did s i g n i f i c a n t l y better than t r a d i t i o n a l l y taught students on immediate achievement but no s i g n i f i c a n t differences were found i n retention, a t t i t u d e s toward CAI and toward mathematics. CAI was as e f f e c t i v e as other methods i n teaching the concepts of l i m i t s . Castleberry (1970) developed and evaluated CAI programs on selected topics i n introductory college chemistry. These topics were a combination of t u t o r i a l d r i l l and simulation modules designed solely as supplementary study aids. The conclusions were that the students acquired the behavioural objectives of the modules to varying degrees. The modules allowed for a range of i n d i v i d u a l differences based on a b i l i t y , p rior learnings and s e l f - e v a l u a t i o n . The CAI programs had a s i g n i f i c a n t e f f e c t on achievement as measured by the f i n a l examination. F u l l scale implementation of a course with these modules would be f e a s i b l e . Cropley and Gross (1970) compared a group of u n i v e r s i t y students who received i n s t r u c t i o n i n FORTRAN . programming via CAI through a remote terminal connected to a computer more than 400 miles away with a programmed i n s t r u c t i o n group and a conventional lecture group. A l l three methods proved to be equally e f f e c t i v e and no serious 60 negative f e e l i n g s about being taught by machines were reported by the students. CAI compared favourably with t r a d i t i o n a l i n s t r u c t i o n as f a r as time was concerned. Their f i n d i n g s suggest that CAI i s a r e a l i s t i c and acceptable a l t e r n a t i v e to t r a d i t i o n a l classroom procedures i n appropriate s i t u a t i o n s . Oldehoeft and Conte (1971) developed a computer system at Purdue University to teach portions of an undergraduate course i n numerical methods. Each i n s t r u c t i o n a l unit or lesson was divided into three modes of i n s t r u c t i o n which allowed the student to progress from a computer-controlled presentation to a student-controlled i n v e s t i g a t i o n . These were t u t o r i a l mode, problem mode and in v e s t i g a t i o n mode. The system was designed as a classroom independent course of study, and had been used for two semesters by students i n place of conventional classroom i n s t r u c t i o n . The program consisted of twenty-five lessons. The t u t o r i a l mode was si m i l a r to l i n e a r programmed i n s t r u c t i o n where the student responded to a mixture of multiple choice and constructed response type items. The i n s t r u c t i o n a l strategy here was sim i l a r to that developed by Bork (1971). A major factor which l i m i t s the o v e r a l l effectiveness i s a general lack of anticipated incorrect responses b u i l t into the program. The o v e r a l l i n d i c a t i o n was that the average CAI student performed 61 as well as he would have under the conventional system. The o v e r a l l attitude toward CAI was favourable. Fenichel e_t a l . (1970) developed the TEACH system at MIT to ease the cost and improve the r e s u l t s of elementary i n s t r u c t i o n in programming. To the student, TEACH o f f e r s loosely guided experience with a conversational language which was designed with teaching i n mind. Faculty involvement i s minimal. The evaluation of the project indicated that the technical aim of completing the system and bringing i t into production and smooth operation has been met. No tests have been made to determine the e f f i c a c y of the teaching method, although interviews with the students determined that the students had i n f a c t learned the material. The authors distinguished between semantic and syntactic errors. The technical issue of error detection was r a i s e d and the authors described the problem of semantic error detection as that of informing a student of what error he has made and o f f e r i n g him suitable help at the appropriate time. The system mu?t be able to go beyond s u p e r f i c i a l l y erroneous input to determine whether i t was a simple s l i p or whether i t was the r e s u l t of some deep conceptual misunderstanding. This task i s presently being tackled by the experimenters. Bork and Sherman (1971) and l a t e i Kalman, Kaufman and 62 Ladd (1971) used a t u t o r i a l program to teach the proof of the conservation of energy to college students. The CAI lesson proved successful, a c t i v e l y involving the students in a t u t o r i a l lesson and teaching the concepts involved. Several problems were noted. The d i f f i c u l t y of using standard Calculus notation on a computer terminal and the i n a b i l i t y of the computer to recognize a l l correct and incorrect responses were notable l i m i t a t i o n s . Hansen (1966) reviewed CAI t u t o r i a l applications and found that the most consistent f i n d i n g i s the marked saving i n i n s t r u c t i o n a l time with no loss i n p o s t - i n s t r u c t i o n achievement test performance. Kromhout, Edwards and Schwartz (1969) described the use of computers i n physics i n s t r u c t i o n . They presented a d e s c r i p t i o n of representative CAI projects i n physics and concluded that most projects were s t i l l i n the early testing stage. They also concluded that the varied and imaginative uses of the computer were encouraging and that introduction of the computer into physics i n s t r u c t i o n represented a s i g n i f i c a n t development. Atkinson (1968) described a CAI t u t o r i a l project to teach reading to f i r s t grade c u l t u r a l l y disadvantaged children. An experiment was conducted which compared a CAI group to a t r a d i t i o n a l group taught by a teacher i n the 63 classroom. The two groups were not s i g n i f i c a n t l y d i f f e r e n t at the s t a r t of the year, but at the end of the year the group that received computer-assisted reading i n s t r u c t i o n performed s i g n i f i c a n t l y better on almost a l l of the reading achievement tests. He concluded that from the standpoint of both the rate of progress and the t o t a l number of problems completed during the year, the computer curriculum appeared to be quite responsive to i n d i v i d u a l differences. Few sex differences were found and the r e s u l t s suggest that with CAI, sex difference i s minimized as the emphasis moves toward analysis and away from rote memorization. The one kind of problem on which the g i r l s achieved s i g n i f i c a n t l y higher scores than the boys, word-test learning, i s e s s e n t i a l l y a memorization task. Students, teachers and parents reacted favourably to the introduction of CAI to the classroom. Various optimization routines were evaluated and Atkinson believes that these evaluations have suggested other experiments and analysis which could lay the grcundword for a theory of i n s t r u c t i o n t r u l y useful to the educator. Cartwright and M i t z e l (1971) developed a mobile CAI laboratory at Pennsylvania State University u t i l i z i n g an IBM 1500 i n s t r u c t i o n a l system. A CAI course was developed which would provide intensive training in s p e c i a l education 64 concepts to sparsely populated counties. The course was c a l l e d Computer-Assisted Remedial Education (CARE). P i l o t groups and other formative evaluation procedures were used to produce a CAI course which was i n t e r n a l l y v a l i d and error f r e e . Summative evaluations have shown that students who took the CAI course scored s i g n i f i c a n t l y higher i n achievement and used about one-third less time to cover the r same objectives than students instructed i n the conventional lecture-discussion method. Lower (1971) recently reported on CAI a c t i v i t i e s at Simon Fraser University (SFU) i n B r i t i s h Columbia. He reported that more than eighty CAI courses have been authored at SFU but that most of these have been experimental or exploratory i n nature. Only about ten have ac t u a l l y received substantial use i n connection with regular univer s i t y cr high school courses. Most of the courses are i n chemistry, but there has been some authoring a c t i v i t y within the areas of physics, mathematics, biology and economics, while the high school courses cover a wider sphere. He notes that students have played an important r o l e i n programming of the courses. An important conclusion i s the idea that CAI i s only one component of an i n s t r u c t i o n a l system and that computer managed i n s t r u c t i o n may be more e f f e c t i v e i n prac t i c e . 65 The most sophisticated and advanced CAI system i s currently the PLATO system at the University of I l l i n o i s (Bitzer, 1967; B r a u n f i e l d et a l . , 1962; Alpert et a l . , 1970). Hammond (1972) wrote that the PLATO system i s one of the most ambitious time sharing systems ever attempted. Much of the hardware, a new programming language adapted for teaching, and economical new techniques f o r l i n k i n g remote terminals to a cen t r a l computer were designed s p e c i f i c a l l y for educational use. The student terminals are perhaps the most sophisticated and expensive devices ever developed for communicating with a computer. For authoring new courses, a programming language (Tutor) based on English grammar and syntax i s designed f o r use by teachers with no knowledge of computers. Some 200 such teachers of varying backgrounds have created courses with Tutor and e a r l i e r versions of PLATO. Bi t z e r e_t a l . (1967) described the ap p l i c a t i o n of the PLATO system to science education. Lessons on genetics, chemistry, physics, engineering and elementary school topics are being used. Braunfield and Fosdick (1962) pointed out that the power of the computer based teaching system stems from i t s a b i l i t y to pose complex questions, judge the students' answers to these questions and take an appropriate course of action on the basis of student responses. The computer also 66 keeps d e t a i l e d and accurate records of student performance, which are useful guides to improving course content. The authors reported a study using PLATO to teach nine under-graduate students a portion of a course i n computer programming. The teaching l o g i c was defined and the authors presented tables to indicate the most useful types of data which can be gathered about the i n s t r u c t i o n a l process. They concluded that the students found the PLATO system very easy to use and that the system did not d i s t r a c t the students' attention from the lessons themselves, Alpert and B i t z e r (1970) reported on the PLATO IV system, which i s the l a t e s t version of t h i s system. They reported a study i n which a c l a s s of twenty students i n a medical science course was taught for a semester e n t i r e l y with the PLATO IV system. When compared with a control group i n a n a t i o n a l l y administered test, the i.tudents taught with the PLATO IV system were found to have scored as well in grade performance even though they had required only one-third to one-half as many student contact hours of i n s t r u c t i o n as those taught in the conventional classroom. Subsequent measurements extending over a twenty-six week period indicated that the PLATO group showed H greater retention over that i n t e r v a l . 67 B i t z e r et. al. (1967) claimed that CAI was e f f e c t i v e i n teaching e l e c t r i c a l engineering. Students taught by the inquiry method showed greater problem solving a b i l i t y than those taught by the t u t o r i a l method. He further claimed to have gained some insight into the learning process, thereby improving the material presented i n both inquiry and t u t o r i a l modes. He concluded that the PLATO-CAI system can both teach and explore physical and behavioural experiments, thus he described his system as v e r s a t i l e and f l e x i b l e . Summary The purpose of t h i s review of t u t o r i a l CAI was to provide a variety of examples which would demonstrate the po t e n t i a l contribution of CAI to education. The studies relevant to the present study were discussed i n e a r l i e r sections of the chapter. A wide range of a c t i v i t y i s ongoing i n the area of t u t o r i a l CAI with regard to hardware (Hammond, 1972), software (Fenichel et a l . , 1970), i n s t r u c t i o n a l courses (Bitzer e_t aJ. , 1967; Cartwright and Metzel, 1971; Lower, 1971), i n s t r u c t i o n a l strategies (Bork, 1971; Oldehoeft and Conte, 1971). Summary of Rcser.rch Hypotheses The research questions of int e r e s t were raised i n Chapter 1. The questions can be translated into s p e c i f i c hypotheses as follows: l a . In terms of proportion of err o r s , average latency and immediate learning, students provided with response-sensitive c o r r e c t i o n a l feedback (T^) w i l l perform s i g n i f i c a n t l y better than students provided with response-insensitive c o r r e c t i o n a l feedback ( T 2 ) . lb. In terms of proportion of errors, average latency and immediate learning, students provided with response-insensitive c o r r e c t i o n a l feedback (T 2) w i l l perform s i g n i f i c a n t l y better than students provided with no c o r r e c t i o n a l feedback ( T 3 ) . 2. Students who are high i n A-State w i l l have a s i g n i f i c a n t l y higher proportion of errors and average latency than low A-State students when no co r r e c t i o n a l feedback i s provided (T3) and this r e l a t i o n s h i p w i l l be reversed when co r r e c t i o n a l feedback i s provided (T 2) 3. In terms of proportion of errors, average latency and immediate learning, students who are high i n prer e q u i s i t e s k i l l s w i l l perform better than students low i n prereq u i s i t e s k i l l s when no c o r r e c t i o n a l feedback i s provided ( T 3 ) 6 9 4. In terms of proportion of errors, average latency and immediate learning, students who are high i n mathematical a b i l i t y w i l l perform better than students low i n mathematical a b i l i t y when no co r r e c t i o n a l feedback i s provided ( T 3 ) . 5. There w i l l be a s i g n i f i c a n t l i n e a r r e l a t i o n s h i p between the process variables (proportion of errors and average latency) and the product variable (immediate learning) with the e f f e c t s of treatment and learner c h a r a c t e r i s t i c s (A-State, prerequisite s k i l l s , mathematical a b i l i t y ) removed. CHAPTER III METHOD Subjects The experimental subjects consisted of sixty-three preservice elementary school teachers i n the Faculty of Education at the University of B r i t i s h Columbia. These students were following a compulsory course i n methods of teaching mathematics and they were awarded cr e d i t towards their f i n a l course grade f o r p a r t i c i p a t i n g i n the experiment. A p i l o t study c a r r i e d out i n the developmental stage of the materials suggested that the content i s most suitable for undergraduate students having some mathematical or s c i e n t i f i c background (Kalman, Kaufman and Smith, 1972). Students from the Faculty of Education were u t i l i z e d i n t h i s study since these students had studied the necessary secondary-school mathematics, but would probably f i n d the material f a i r l y d i f f i c u l t . It was assumed that these students would make more errors during learning. These incorrect responses would provide data for evaluation of the e f f e c t s of c o r r e c t i o n a l feedback on student performance and would provide a v a l i d test of the program's teaching effect i v e n e s s . 70 71 Experimental Procedures The experimental procedure was divided into one day periods: (1) i n i t i a l lecture and t e s t i n g session;- (2) task period on CAI computer terminal followed by posttest and debriefing. The procedure i s l i s t e d below i n Table 1. TABLE 1 EXPERIMENTAL PROCEDURE 1 A c t i v i t y ! Variables Measured Approximate Time Required (min.) DAY 1 1. Short lecture 15 given to subjects. 2. Tests State and t r a i t 80 adminis tered anxiety, to subjects. mathematical a b i l i t y . DAY 2 3. CAI prelesson Errors, l a t e n c i e s , 30 taken by o v e r a l l score, subject s. state anxiety. 4. CAI main . Errors, latencies, 90 lessen taken state anxiety. by subjects. 5. Posctest Immediate 35 administered learning. to subjects. 6. Short - 5 debriefing given to subjects. 72 The experimental subjects were tested i n three separate groups during DAY 1 except for six students who arranged to write the tests at separate times. The DAY 1 testing sessions were also u t i l i z e d to schedule students fo r the DAY 2 computer sessions. The DAY 2 sessions were held Monday to Friday at 9:00 a.m. and 5:00 p.m. for most students, although sixteen students completed these sessions on either Saturday or Sunday morning or afternoon. The experimental subjects defined e a r l i e r were run i n groups of three to f i v e persons on IBM 2741 computer terminals i n an i s o l a t e d room above the Computing Centre at the University of B r i t i s h Columbia. Subjects from each of the three treatment groups worked simultaneously i n order to control for extraneous environmental and time variables. The CAI sessions were interrupted only once when the computer became inoperational fo r f i f t e e n minutes near the start of the prelesson. Four students were required to r e s t a r t t h e i r CAI lessons and to re-enter t h e i r responses up to the point of the i n t e r r u p t i o n . A few minor d i f f i c u l t i e s were encountered with the terminals but these were e a s i l y remedied by replacing the type-ball or by t r a n s f e r r i n g the student to another terminal. Design The subjects were randomly assigned to three treatment 73 groups and a balanced design was obtained with twenty-one students i n each group. Each group received the same content and i n s t r u c t i o n a l l o g i c f or the CAI lesson. The three treatment groups d i f f e r e d i n terms of the i n s t r u c t i o n a l strategy employed to achieve the main lesson i n s t r u c t i o n a l objectives. The difference was s p e c i f i c a l l y i n terms of the d i f f e r e n t type of c o r r e c t i o n a l feedback provided to the learner. A l l three treatment groups received i d e n t i c a l information regarding the correct response together with the explanation at some point during the implementation of the p a r t i c u l a r i n s t r u c t i o n a l strategy. The key independent variable i n the experiment was co r r e c t i o n a l feedback and i t was the informational content of the feedback that was being varied. The other independent variables represented both cognitive and personality c h a r a c t e r i s t i c s of the students. The cognitive variables d i f f e r e d i n terms of the i r r e l a t i o n s h i p of the s p e c i f i c CAI task. These were mathematical a b i l i t y and a b i l i t y to perform on the prelesson. Cronbach and Snow (1969) have suggested that the pretest score i s an aptitude and should be treated along with other aptitudes. The personality variables were both t r a i t - and state-anxiety. State-anxiety was measured at three points throughout the experiment and Trait-anxiety was measured at the beginning. The dependent or c r i t e r i o n variables i n t h i s study were posttest score, proportion of errors during learning and average response latency. The l a t t e r two were the process variables and the former was the product variable. Suppes (1967) claimed that response latencies are more se n s i t i v e measures of s k i l l mastery and depth of learning than the response errors themselves and response latencies were examined as well. Unanticipated student responses and the number of times the student asked for help were recorded for post hoc analysis, as well as means and correlations of selected experimental variables. I n s t r u c t i o n a l Logic f o r Prelesson The template used i n the prelesson i s shown i n Figure 2. The flowchart reveals how t h i s CAI t u t o r i a l lesson served as a pretest of p r e r e q u i s i t e s . A student was f i r s t tested on the p r e r e q u i s i t e concept and then given i n s t r u c t i o n i n that concept only i f he f a i l e d i n i t i a l l y to respond"correctly. After the concept was taught, the student was again tested, but on a d i f f e r e n t example. As 75 Question i s asked Can ftudent answer on f i r s t s t r y ? no Assistance i s given to student Student i s asked to answer once again Qi Question i s asked s i m i l a r to above Q±+i Next question yes *Mark = 2 7 i s asked * Mark re f e r s to the grade assigned to the student for a p a r t i c u l a r i n s t r u c t i o n a l unit, or item, Q^ . Figure 2 Prelesson I n s t r u c t i o n a l l o g i c Student i s •)) asked a question HE1V is given 37 W Write correct answer ' and Comments ' Student's answer not recognized NOMATCH - What is val ue of GX? T 7 V i • Write "Write Write Write "ok" |"gocd" "Excel- other lent" comment Comment Comment no no . \ Comment Comment made made w * Qj/ C^, Qm, Qj +^f a l l represent different instructional units (IU) in the lesson. . [i>means the n t h time through the block. Figure 3 Main Lesson Instructional Logic 77 shown i n the flowchart, a student received a mark of 0, 1, or 2 on each i n s t r u c t i o n a l unit, based on his previous knowledge and his a b i l i t y to learn the concepts new to him. The prelesson had three functions: (a) to provide a measure of the student's knowledge or a b i l i t y to learn this p a r t i c u l a r domain of content i n t h i s p a r t i c u l a r medium; (b) to ensure that the student had attained the necessary prerequisites before proceeding to the main lesson; and (c) to provide p r a c t i c e i n working on the CAI terminal. I n s t r u c t i o n a l Logic for Main Lesson A flowchart of the i n s t r u c t i o n a l l o g i c used for the three treatment groups on each i n s t r u c t i o n a l unit i s shown i n Figure 3. An explanation of the flowchart l o g i c w i l l now be given. The student i s f i r s t asked a question by the computer and he responds either by asking f o r help or by attempting to answer the question. If he types HELP, a hint i s given and he must respond again. I f he types HELP a second time, the student i s given the correct answer with an explanation before proceeding to the next i n s t r u c t i o n a l unit. The student who responds to the o r i g i n a l question may respond c o r r e c t l y . •In other words, h i s answer matches a predetermined correct answer keyword. When thi s match occurs, the computer types: an appropriate encouraging comment 78 such as "good". The wording of the comment depends on the value of a counter, GX, which keeps track of the number of consecutive correct r e p l i e s up to four i n a row. After the encouraging comment, the computer provides the correct answer with an explanation before proceeding to the following i n s t r u c t i o n a l unit. The student may respond i n c o r r e c t l y to the o r i g i n a l question. This means that h i s answer matches a predetermined incorrect answer keyword. The computer may branch to another question for remedial assistance or may provide c o r r e c t i o n a l feedback, depending upon the pr i o r decision of the i n s t r u c t i o n a l programmer. Making two incorrect responses of the same type, that i s , f a l l i n g i nto the same wrong answer c l a s s twice, causes the correct answer and an explanation to be provided before proceeding to the following i n s t r u c t i o n a l unit. An incorrect response r e - i n i t i a l i z e s the counter, GX, to zero. The student's response may not match any of the correct or incorrect keywords and his response w i l l not be recognized by the computer. One or two NOMATCH responses cause a comment to be provided and the student i s asked to try again. A t h i r d NOMATCH response causes the computer to provide the student with the correct answer and an explanation before proceeding with the subsequent i n s t r u c t i o n a l u n i t . The option also e x i s t s f o r branching to other i n s t r u c t i o n a l units i f a correct or a p a r t i c u l a r incorrect response i s recognized. Operational D e f i n i t i o n s of Treatments The f i r s t treatment group (T^) was instructed p r e c i s e l y according to the i n s t r u c t i o n a l l o g i c shown i n Figure 3. The deepest l e v e l of i n t e r a c t i o n was attained between the learner and the computer through the i n s t r u c t i o n a l program because the nature of the student's incorrect response was used as the c r i t e r i o n for providing response-sensitive c o r r e c t i o n a l feedback. This feedback was determined by pr i o r analysis of each question by the i n s t r u c t i o n a l designer. The second treatment group (T 2) received response-i n s e n s i t i v e c o r r e c t i o n a l feedback. Each T0 subject was provided with a hint when he responded i n c o r r e c t l y . This hint was predetermined and was provided to each T2 subject regardless of the nature of his i n c c r r e c t response. An important c r i t e r i o n for choosing the hint i n T^ was that no more information would be provided than could be obtained from a l l the hints f o r the corresponding i n s t r u c t i o n a l unit i n T^. The comment blocks which followed the wrong answers 8 0 a l l contained i d e n t i c a l comments f o r a p a r t i c u l a r i n s t r u c t i o n a l u n i t (see Figure 3 ) . The t h i r d treatment group (T3) received no co r r e c t i o n a l feedback information. Each T3 subject was merely informed about the incorrectness of his response and no remedial information was provided. The comment blocks which followed the wrong answers contained no hints, but contained only information t e l l i n g the T^ subject that his answer was incor r e c t . The reader can refer to Appendix D i n order to compare the three versions of the main lesson, T-^ , T 2 and T 3 . CAI Author Language The CAI author, or programming, language was written by the experimenter i n FORTRAN IV and requires a minimum amount of computer knowledge and experience on the part of a user. A User's Guide for the language along with a source l i s t i n g of the program i s provided i n Appendix A. This language was modified i n order to implement the i n s t r u c t i o n a l l o g i c f or the prelesson. These modifications are described and a l i s t i n g of the modified source program i s provided i n Appendix B. Inst r u c t i o n a l Materials The procedure ui-ed i n designing the i n s t r u c t i o n a l 81 materials was described i n a review of modular i n s t r u c t i o n by Goldschmid and Goldschmid (1971). The procedure consisted of the following steps: 1. I d e n t i f i c a t i o n of the subject matter to be taught 2. D e f i n i t i o n of a set of objectives 3. Deciding upon the hierarchy of objectives which i n • turn describes the sequence of i n s t r u c t i o n 4. I d e n t i f i c a t i o n of prerequisites 5. Development of a pretest 6 . P r o v i s i o n of i n s t r u c t i o n a l options 7. Design of a posttest The subject matter consisted of the concept of the de r i v a t i v e i n elementary calculus and the r e l a t i o n s h i p s between the mathematical concepts and the ph y s i c a l concepts of distance, speed and time. The lesson could be characterized as a mathematical derivation supplemented by numerical problems which are solved during the lesson. This topic was chosen to s a t i s f y the requirement for a module to teach this material to a group of undergraduate science students at Loyola College i n Montreal, Quebec (Kalman, Kaufman, Smith, 1972). The CAI module i s currently f u l f i l l i n g t h i s role but i n a d i f f e r e n t form than the one used i n t h i s experiment. Cronbach and Snow (1969) have suggested that the ideal treatment-set for ATI research i s l i k e l y to consist i n applications of some regular i n s t r u c t i o n a l material. This suggestion was followed i n t h i s study. The prerequisites f o r the main lesson are given i n Table 2. These were i d e n t i f i e d by a l o g i c a l a n a lysis of the content area performed by the investigator with the assistance of two college i n s t r u c t o r s who had been teaching mathematics and science courses f o r at least three years (Kalman, Kaufman, Smith, 1972). The prerequisites were subsequently modified af t e r a p i l o t test of the program. The prelesson dealt with these prer e q u i s i t e s . An overview of the ccntent of the prelesson i s given i n Table 2. and a d e t a i l e d view of the prelesson i s shown i n Figure 4. If a subject had acquired the p r e r e q u i s i t e objective, he proceeded immediately to the following item. If not, he was instructed and l a t e r retested on this p r e r e q u i s i t e , as indicated e a r l i e r i n Figure 2. A d e t a i l e d l i s t i n g of the CAI prelesson i s given i n Appendix C with the accompanying graphs. Reading a graph I Evaluating the value of a function at a point Using distance=speed x time to f i n d distance given speed and time »k Using DEL(S) notation to f i n d change IX Calculating average speed given distance, and time sk Knowledge of the meaning of the term "instantaneous speed" Five item A-State instrument i s given sJe , Calculating the slope of a st r a i g h t l i n e graph Reducing an algebraic expression such as [(2+x) 2-2 2]/C(2+x)-2] I Reducing same algebraic expression as above but using DEL(S) i n place of x Signoff Figure 4 Detailed View of Preleirson 84 TABLE 2 PREREQUISITES FOR MAIN LESSON 1. A b i l i t y to read a graph, i . e . t o . f i n d a value at a point. 2 . A b i l i t y to calculate the value of a function at a point, given the equation. 3. Knowledge of the terms "average speed" and "instantaneous speed". 4. A b i l i t y to calculate distance when given speed and time, using the formula, distance = (speed) (time). 5. Knowledge of the concept of change i n distance or time. 6. A b i l i t y to apply the notation, DEL(S) and DEL(T) to ca l c u l a t e change. 7. Knowledge of the d e f i n i t i o n of "slope". 8. A b i l i t y to calculate the slope of a straight l i n e . 9 . A b i l i t y to expand a binomial which i s squared and to f a c t o r , i . e . a b i l i t y to reduce algebraic expressions such as |72 + x ) 2 - 2^] / | 2 + x) - 2J to simplest form. 1 0 . A b i l i t y to perform the above algebraic manipulations using the cumbersome notation used on a computer terminal, e.g. ( 2 + DEL(S)) * 2 - 2 * 2 / ( 2 + DEL(S ) ) - 2. The objectives for the main lesson were determined and these provided the rationale for the l o g i c of the i u a i n ienson content. These i n s t r u c t i o n a l objectives are given i n Table 3 and a d e t a i l e d view of the content o.~ the mnin lesson i s given i n Figure ~. Detailed l i s t i n g s of the main 85 lesson, versions T l 3 T 2 and are provided i n Appendix E. The f i f t e e n i n s t r u c t i o n a l units analyzed i n the study are the i n s t r u c t i o n a l units with the following number i n Appendix E: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 23, 24. These units were the ones given to a l l students i n the experiment. Units 15, 16 and 17 comprised the optional section on l i m i t s . Units 18 to 22 comprised the A-State scale and Unit 10 was an exposition of some content followed by the question, "Do you understand?" A posttest was developed and i s given i n Appendix F. The items of the test r e f l e c t e d the s p e c i f i c behaviours implied by tt-.e i n s t r u c t i o n a l objectives. 8 6 TABLE 3 INSTRUCTIONAL OBJECTIVES FOR MAIN LESSON 1. Rec a l l the r e l a t i o n s h i p between the following: (a) slope of secant and average speed (b) slope of tangent and instantaneous speed (at a point) (c) average speed and instantaneous speed (d) slope of tangent and d e r i v a t i v e (at a point) 2. Calculate average speed from a graph of distance vs. time for both linear and non-linear graph. 3. Calculate instantaneous speed from a graph of distance vs. time with tangent to the curve drawn at a point on the graph. 4. Calculate simple l i m i t s , e.g. l i m i t (6 + 3/\t) At-* 0 5. Define instantaneous speed at any time t as a function of s and t, i . e . v = l i m i t As At-* 0 A t 6 . Calculate average speed and instantaneous speed at a point, given the equation of s as a function of t. The student muct use basic p r i n c i p l e s , i . e . l i m i t d e f i n i t i o n . 7. Calculate the de r i v a t i v e (dy./dx) at a point given the equation of y = f ( x ) , from basic p r i n c i p l e s , i . e . l i m i t def i n i t i o n . no ino ' \ 'Has he\ krevipwect -nee? [yes Relationship between slope and average speed of straight, line graph of. S vs.. T... Shown, to. te. constant X ^ Relationship between slope of secant and average speed for non-linear graph of S vs. T. Not constant Relationship between elope of tangent and instantaneous speed at a point Relationship between average speed and instantaneous speed, i.e., instantaneous speed is limit of average speed no Examples of finding instantaneous speed given non-linear graph of S vs. T with tangent drawn on it.. Can student do. i t ? . . ; X yes—" -• Notation and definition of derivative are given Example of computing average speed from equation of S vs. T at a point Finding instantaneous speed by taking limit of average speed. Can he do i t ? •no A-State instrument .-. five items I Another example of computing instantaneous speed from equation of S vs. T. Can he do i t ? 1 ^,£3 Signoff Evaluating limits Figure 5 Detailed View of Main Lesson 88 Measurement Instruments The analysis of a l l measurement instruments was performed using the PIA and TIA test analysis programs ava i l a b l e in the Faculty of Education at the University of B r i t i s h Columbia. Posttest. The eighteen-item multiple choice posttest was designed to measure student performance on the objectives given i n Table 2. This instrument was intended to serve as a cri t e r i o n - r e f e r e n c e d test and a copy i s given i n Appendix F. Content v a l i d i t y was assured f o r thi s test by generating s p e c i f i c items from the l i s t of i n s t r u c t i o n a l objectives with due regard given to the r e l a t i v e emphasis given to these objectives i n the main lesson. Test analysis data f o r thi s instrument i s given i n Tables 4 and 5, TABLE 4 POSTTEST DATA Sample size Mean Standard Deviation 63 8. 95 2.89 89 TABLE 5 ITEM ANALYSIS INFORMATION FOR POSTTEST Item Proportion Answering Correctly ( D i f f i c u l t y ) Pooled T l T2 T3 1 .54 .62 .38 . .62 2 .44 .57 .24 .52 3 .78 .67 .86 .81 4* .71 .67 .76 .71 5 . 94 . 95 .90 . . 95 6 .40 .38 .38 .43 7 .49 .48 ,5.7 .43 8 .40 .33 .52 .33 9 .84 .81 .86 .86 10 . 78 .76 . 62 . 95 11 .51 .52 .57 .43 12 .68 .81 .67 .57 13 . 19 .24 .24 . 10 14 .33 .48 ( .24 .28 15 .16 .24 . 14 .10 16 .43 .48 .48 .33 17 . 08 .00 . 19 . 05 18 .25 .33 .24 . 19 *The seventeen subjects who answered t h i s item before a typographical error was corrected on che posttest were assigned a value of "1" for the purpose of using the item analysis program. The r e s u l t s shown i n Table 5 indicate tnat there were no meaningful differences between the three groups on the in d i v i d u a l items, but that cer t a i n items t?ere too d i f f i c u l t and should be revised. The d i f f i c u l t items were 13, 15, 17 and 18. 90 Mathematical a b i l i t y test. The Cooperative Sequential Test of Educational Progress, Mathematics Form 2A (1957), was administered to a l l subjects at the beginning of the experiment. The norm sample for t h i s form of the test consisted of students i n grades ten, eleven and twelve and the test was found to be of appropriate d i f f i c u l t y l e v e l for students p a r t i c i p a t i n g i n t h i s experiment. The i n t e r n a l consistency c o e f f i c i e n t , K-R 20, for t h i s form of the test normed on grade eleven students was reported to be .84. The concurrent v a l i d i t y c o e f f i c i e n t was defined as the c o r r e l a t i o n with the SCAT-Quantitative test and was reported as .70 for a grade eleven sample and .76 f c r a grade twelve sample. The test consisted of f i f t y items and was administered i n seventy minutes. For the t o t a l sample of sixty-three students, the mean was 31.1, the standard deviation was 7.6 and the c o e f f i c i e n t of i n t e r n a l consistency, K-R 20, was .85. A l o g i c a l analysis of the test items seems to suggest that higher-order mental processes, such as problem solving, were being measured and not only r e c a l l of information or a p p l i c a t i o n of algorithms. The l a t t e r two processes were emphasized i n the prelesson. 91 State-Trait anxiety inventory. The State-Trait Anxiety Inventory (STAI) was u t i l i z e d i n order to measure both A-State and A-Trait (Spielberger e_t al_. , 1970). The twenty-item A-State and A - T r a i t four point L i k e r t scales were administered at the beginning of the experiment. In addition, a short form of the A-State scale (O'Neil, 1972), consisting of the f i v e items with the highest item-remainder correlations i n the STAI normative sample were given during the prelesson and during the main lesson. These f i v e items were administered by the computer during the CAI lesson. The twenty-item A-State and A - T r a i t scales have been shown to have high value of r e l i a b i l i t y , i . e . Cronbach's alpha, and evidence of construct v a l i d i t y has been provided (Spielberger et a l . , 1970). O'Neil (1972) reported r e l i a b i l i t y (alpha) c o e f f i c i e n t s for the five-item scale ranging from .83 to .93 i n seventeen administrations. The test s t a t i s t i c s for t h i s study are reported i n Table 6. 92 TABLE 6 ANXIETY TEST DATA Test Mean S.D. Cronbach Alpha A-T r a i t (20 items) 41. 9 8.5 .88 A-State (20 items) 39.1 9.4 .89 A-State pretest (5 items) 9.6 3.8 . 92 A-State main (5 items) 12.0 4.5 . 92 Prelesson. The grading scheme used for the prelesson was discussed e a r l i e r . A student received a grade of 0, 1, or 2 on a single i n s t r u c t i o n a l unit and there were nine i n s t r u c t i o n a l u n i t s i n the lesson. The prelesson was considered as a test of a student's knowledge and a b i l i t y to learn the prerequisites required for the main lesson as taught by the computer. The prelesson data are given i n Tables 7 and 8. TABLE 7 PRELESSON DATA Sample Size Mean Standard Deviation 63 14, 4 1.84 TABLE 8 93 ITEM ANALYSIS INFORMATION FOR PRELESSON Item Mean Standard Deviation * r t o t a l 1 1.73 .48 .35 2 1. 78 .42 .32 3 1. 92 .27 .52 4 1.90 .29 .17 5 1.87 .38 .22 6 1.92 .27 .29 7 . 1.73 .44 .34 8 0.68 .64 .73 9 0. 90 .81 .71 t o t a l represents the c o r r e l a t i o n c o e f f i c i e n t of that item with the t o t a l test score. Table 7 indicates that the prelesson was not d i f f i c u l t for t h i s group since the mean score for the t o t a l group was 14.4 on a possible score of 18. The l a s t two items were the only ones which the students found d i f f i c u l t . Main lesson. The main lesson was considered as a measure of a student's a b i l i t y to perform on a CAI terminal. The student's errors and latencies were recorded for subsequent ana l y s i s . Each i n s t r u c t i o n a l unit was also considered as a test item with possible scores of 0 or 1. The student was 94 assigned a value cf 0 for an item i f the computer provided the correct answer before he answered c o r r e c t l y . If the student answered c o r r e c t l y before being given the answer, he was assigned a grade of 1 f o r that item. Analysis information for the main lesson regarded as a test i s given below i n Table 9. Table 9 indicates that the mean score f o r the group was 10.9 out of a possible 15. This finding shows that most students were able to produce the correct answer before the computer d i d so, and the lesson was a r e l a t i v e l y e f f e c t i v e i n s t r u c t o r . TABLE 9 MAIN LESSON DATA Sample Size Number of Items Mean Standard Deviation 63 15 10. 9 2.0 CHAPTER IV ANALYSIS AND RESULTS Method of Analysis In t h i s study two phases were involved i n the analysis of data. The f i r s t phase consisted of a regression analysis procedure designed to test the research hypotheses stated i n Chapter II. The second phase involved some post hoc analysis of the data i n order to gain a d d i t i o n a l insight into the r e s u l t s of the f i r s t phase as well as to examine several a l t e r n a t i v e questions. This second phase involved the examination of i n t e r c o r r e l a t i o n s f o r a variety of measures, which were examined for the t o t a l group as well as separately for the three experimental groups. Graphs of latencies and errors f o r the main lesson were also examined for meaningful information. The regression analysis approach employed fo r t h i s analysis has been described by many writers (Bottenberg and Ward, 1963; Cohen, 1968; Overall and Spiegal, 1970; Walberg, 1971; Kaufman and Sweet, 1973). The advantages of this'method of data analysis over the conventional ANOVA approach have been discussed i n d e t a i l by these writers. 95 9 6 Cronbach and Snow (1969) recommended the use of the regression analysis method for testing i n t e r a c t i o n terms i n ATI studies. Separate stepwise univariate regression analyses were performed for each of the three c r i t e r i o n variables i n order to test the f i r s t four hypotheses in t h i s study. Hypothesis 5 was tested using a regression analysis approach with posttest score as the dependent variable. The proportion of errors, average latency and learner c h a r a c t e r i s t i c s defined e a r l i e r served as independent variables. This analysis technique permitted the learner variables and treatment e f f e c t s to be s t a t i s t i c a l l y removed ( p a r t i a l l e d ) i n order to test the r e l a t i o n s h i p s between posttest score and the other two variables. A l l analyses were performed at the University of B r i t i s h Columbia Computing Centre. The regression analyses were performed using the MULTIVAR program (Finn, 1968) and the BMD02R program (Dixon, 1968). The means and i n t e r -c o r r e l a t i o n s were calculated with the STRIP program av a i l a b l e at the Computing Centre (Seagraves, 1971). The p r o b a b i l i t y levels (p) for s i g n i f i c a n c e of the F - r a t i o s were calculated using the l o c a l FPROB program (Dempster, 1969). A l l hypotheses having a s i g n i f i c a n c e l e v e l of less than .07 were used for substantive discussion and i n t e r p r e t a t i o n of the 97 r e s u l t s . The hypotheses given at the end of Chapter III were translated into s t a t i s t i c a l terms. The symbols used to represent the variables i n t h i s study are given i n Table 11. Results of Analysis - Means Table 10 shows the means of the variables observed in the study for the t o t a l group and for the treatment groups taken separately. Although the difference between the means on most of the variables was not s t a t i s t i c a l l y s i g n i f i c a n t , a consistent pattern was evident. The treatment groups were ranked i n the hypothesized order (T-^>^2i> T3) o n t l i e posttest scores, t o t a l errors, t o t a l responses, proportion of e r r o r s , t o t a l correct i n main lesson, number taking optional l i m i t section, enjoyment and A-State main lesson. The main lesson latency v/as the exception, with the T^ group having a higher average response latency than the other two groups. The r e s u l t s suggest that with the exception of latency, the c o r r e c t i o n a l feedback variable had an e f f e c t on performance and learning i n the expected d i r e c t i o n , but • that a more d e f i n i t i v e r e s u l t may have been obtained over a longer period of time or with a more d i f f i c u l t lesson. The time of the CAI experimental session was coded '1' for the mornihg (9:00 a.m.), '2' for evening (5:30 p.m.) and '3' for the weekend session. The means indicate that the three treatment groups were well balanced i n each session except that the students i n the t h i r d treatment group ( T 3 ) attended s l i g h t l y more experimental sessions ( 2) i n the morning and the T 2 group attended the least sessions i n the morning. TABLE 10 MEANS OF VARIABLES FOR COMBINED GROUPS AND FOR T, T T„ Variable Pooled 1. Posttest 2. Time of Experimental Session* 3. Total Errors - main lesson 4. Total Responses - main lesson 5. Proportion of Errors - main lesson 6. Total Correct - main lesson 7. Do you understand? - main lesson 8. Time to answer (7) above - main lesson 9. Had l i m i t section - main lesson 10. Average F i r s t Latency - main lesson 11. Average Total Latency - main lesson 12. Enjoyment - main lesson 13. Prelesson Score 14. Prelesson Correct on f i r s t try 15. Average F i r s t Latency - prelesson 16. Average Total Latency - prelesson 17. Math A b i l i t y - f i r s t testing 18. F u l l A-State - f i r s t testing 19. A-Trait - f i r s t testing 20. Short A-State - f i r s t testing 21. A-State - prelesson 22. A-State - main lesson 8.84 1. 76 16.63 27.60 .58 10. 90 .76 83.7 .79 116.4 189. 0 .63. 14.46 6.25 90.3 119.4 31.08 46. 9 41. 8 9.6 4.9 11.9 9. 19 1.86 14. 71 26.48 .53 11. 62 .71 84.0 . 62 118. 0 177. 5 .67 14. 71 6.24 81. 9 108.2 30.86 47.3 41.3 8.8 4. 6 11.2 8. 74 1.81 17. 14 28. 10 .57 10. 71 .76 45.9 .81 123. 7 202.7 . 62 . 14.28 6.10 99. 9 134.0 30.24 46.5 43. 1 10.2 4.7 12.0 8.59 1.62 18.05 28.24 .63 10.38 .81 121. 1 .95 107.4 186. 9 .62 14. 38 6.43 89.2 115.8 32. 14 46.8 41.2 9.8 5.3 12.5 *This i s a nominal variable. It i s technically incorrect to calcu l a t e the mean and th i s value i s merely intended here to serve as a crude shorthand comparison.. 100 Results of Analysis - Hypothesis Testing TABLE 11 SYMBOLS USED IN STATISTICAL ANALYSIS Symbol Variable Represented Y l Posttest Score Proportion of errors i n main lesson Y 3 Average latency i n main lesson F i r s t contrast - mean of T^ vs. mean of T 2 x 2 Second contrast - mean of T 2 vs. mean of T^ X 3 Mathematical a b i l i t y test score X 4 Prelesson A-State score X 5 i Main lesson A-State score X 6 Prelesson score x 7 Prelesson average latency Four separate regression equations were defined for the analysis, one for each of the three dependent measures and a l a s t equation to test Hypothesis 5. The independent variables i n these equations r e f l e c t the factors which 101 this investigator considered as important i n this study. An ordering l o g i c was defined i n testing the terms i n the equations i n a stepwise manner (Overall and Spiegal, 1970). The l o g i c used involved entering learner c h a r a c t e r i s t i c terms f i r s t into the regression equation followed by treatment terms and then i n t e r a c t i o n terms. This means that treatment and i n t e r a c t i o n e f f e c t s were tested with the e f f e c t of learner c h a r a c t e r i s t i c s being c o n t r o l l e d s t a t i s t i c a l l y . The four equations and the corresponding tables are given i n the next section. Posttest (Yi ) T-L = - 2.80 X X + 2.32 X 2 Treatment Math A b i l i t y + . 1 4 X 3 + Prelesson .26 X, 6 A-State Treatment x Math A b i l i t y • 12 X ^ X ^ — .05 X 2 X ^ *5 + Treatment x Prelesson . 13 X-^X^ - .22 X ^ X ^ + Treatment x A-State .11 X 1 X 5 + .16 X 2 X 5 error + e TABLE 12 1 0 2 RESULTS OF REGRESSION ANALYSIS FOR POSTTEST Source of XAR2 Degrees of 2Fobs 3 P < V a r i a t i o n Freedom * 3 . 1 6 6 1 1 3 . 92 . 0 0 1 X 6 . 0 3 2 1 2 . 6 8 . 1 0 X 5 . 0 4 4 1 3 . 6 9 . 0 6 X l . 0 0 0 1 . 0 0 • -A X 2 . 0 0 5 1 . 4 2 . 5 3 - ( x 1 } x 2 ) . 0 0 5 2 . 2 1 . 8 1 X l x 3 . 0 2 0 1 1 . 6 8 . 2 0 X 2 X 3 . 0 1 9 1 1 . 5 9 . 2 1 ( X 1 X 3 , X 2 X 3 ) . 0 3 9 2 1 . 6 4 . 2 0 X X X 6 . 0 1 0 1 . 8 4 . 3 7 x 2 x 6 . . 0 1 6 1 1 . 3 4 . 2 5 <x x x 6 x x 6 ) . 0 2 6 2 1 . 0 9 . 3 4 x l X s . 0 5 3 1 4 . 4 4 . 0 4 X 2 X 5 . 0 2 7 1 2 . 2 6 . 1 4 ( X 1 X 5 , X 2 X 5 ) . 0 8 0 2 3 . 3 6 . 0 4 t o t a l . 3 9 2 11 error . 6 0 8 1 5 1 . . . i £\R represents the increment i n R 2 given previous terms already entered i n the equation. Fobs - R /df-, j where d f = number of degrees of freedom / i D 2 / J ^ T for corresponding term(s) (!-R f u l l / t e r r o r ) being tested d f f u i i = number of degrees of freedom for f u l l model above 1 - R 2 f u l l = A R 2 e r r o r a s defined i n table above 3 P r o b a b i l i t y of making a Type I error by r e j e c t i n g n u l l hypothesis, i . e . , claiming s t a t i s t i c a l s i g n i f i c a n c e . 4 Indicates'that the i n d i v i d u a l terms i n parentheses are tested as a set. 103 The r e s u l t s i n Table 12 indicate that several variables were s t a t i s t i c a l l y s i g n i f i c a n t i n accounting for v a r i a t i o n i n the subjects' posttest scores (Y-^). The students' mathematical a b i l i t y (X3) was a highly s i g n i f i c a n t variable (p<.001) in explaining posttest r e s u l t s . The main lesson A-State l e v e l ( X 5 ) was s t a t i s t i c a l l y s i g n i f i c a n t , (p ^*. 06 ) c o n t r o l l i n g for the previous two v a r i a b l e s . This finding would seem to indicate that a high l e v e l of anxiety may have existed even a f t e r the students completed the main lesson on the terminal. This was not surprising since the posttest was written immediately at the completion of the main lesson. A main lesson A-State-by-treatment i n t e r a c t i o n (p<^.04) was obtained for posttest scores. In t h i s case, the e f f e c t of anxiety (A-State) was d i f f e r e n t f o r subjects i n T-L as compared with T 2 ( X -^X^ ) . A more d e t a i l e d examination of t h i s f i n d i n g i s given i n figure 6. The r e s u l t s f o r the hypotheses stated i n Chapter III are given below f o r the posttest: Hypothesis l a : No s i g n i f i c a n t difference was found i n immediate learning between students provided with response-sensitive c o r r e c t i o n a l feedback ( T ^ ) and students provided with response-i n s e n s i t i v e c o r r e c t i o n a l feedback ( T 2 ) . 1 0 4 Hypothesis l b : No s i g n i f i c a n t difference was found i n immediate learning between students provided with response-insensitive c o r r e c t i o n a l feedback (T 2) and students provided with no co r r e c t i o n a l feedback ( T 3 ) Hypothesis 3: No s i g n i f i c a n t i n t e r a c t i o n was found i n terms of immediate learning between the T 2 versus T3 groups and prereq u i s i t e s k i l l s . Hypothesis 4 : No s i g n i f i c a n t i n t e r a c t i o n was found i n terms of immediate learning between the versus T*3 groups and mathematical a b i l i t y . 105 Proportion of Errors (Y 2) A-State Treatment Math A b i l i t y Prelesson Y 2 = .22 X : - .06 X 2 - .004 X 3 - .005 X 4 .Treatment A-State Main Prelesson x Math A b i l i t y + .01 X 5 - .03 X 6 + .000 X±X3 + .002 X0X3 Treatment Treatment \. x Prelesson x A-State Main error + .01 X 1X 6 + .01 X 2X 6 + .002 X-,X5 - .004 X 2X 5 + e TABLE 13 RESULTS OF REGRESSION ANALYSIS FOR PROPORTION OF ERRORS Source of A * 2 Degrees of Fobs P < V a r i a t i o n Freedom X3 .094 1 7.53 .01 x 6 .096 1 7.69 .01 x 4 . 008 1 .64 .43 X 5 . 068 1 5.23 .02 X l .004 1 .32 .58 x 2 .059 1 4.72 .03 ( x x x 2 ) . 063 2 2.02 . 14 X-, x „ . 006 1 .48 .50 X 2 X 3 . 007 1 .56 .46 ( X! X3,X2X 3) .013 2 .52 .60 X 1 X 6 .019 1 1.52 .22 x 2 x 6 . 007 1 .56 .46 ( X1 X6 X 2 X 6 ) . 026 2 1 . 04 .36 X 1 X 5 .000 1 - -X 2 X 5 . 008 1 .64 .43 ( X 1 X 5 , X 2 X 5 ) . 008 2 .32 .73 t o t a l .376 12 error .624 50 1 106 It should be noted that HELP and NOMATCH responses were counted as errors and responses. Also, responses that were made when a subject repeated a section of the main lesson were ignored except for the l a s t i n s t r u c t i o n a l unit i n the sequence. The r e s u l t s of the students' performance on t h i s l a s t i n s t r u c t i o n a l unit were combined for the two times that the student attempted the unit. The r e s u l t s given i n Table 13 for proportion of errors (Y 2) iridicate that mathematical a b i l i t y and prelesson score were both s t a t i s t i c a l l y s i g n i f i c a n t at the .01 l e v e l . The main lesson A-State l e v e l was s t a t i s t i c a l l y s i g n i f i c a n t (p<^.02) i n pr e d i c t i n g proportion of errors on the main lesson. A s i g n i f i c a n t treatment e f f e c t was found i n comparing the T 2 and T^ group means (p<.03) with regard to proportion of e r r o r s . There were no s i g n i f i c a n t trait-treatment i n t e r a c t i o n s . The r e s u l t s of the hypotheses stated i n Chapter III are given below for proportion of errors: Hypothesis l a : No s i g n i f i c a n t difference was found in proportion of errors between students proviaed with response-sensi t i v e c o r r e c t i o n a l feedback ( T i ) and students provided with response-i n s e n s i t i v e c o r r e c t i o n a l feedback (Tp). 107 Hypothesis lb: A s i g n i f i c a n t difference was found i n proportion of errors between students provided with response-insensitive co r r e c t i o n a l feedback (T 2) and students provided with no cor r e c t i o n a l feedback (T 3) Hypothesis 3: A s i g n i f i c a n t i n t e r a c t i o n was found i n terms of proportion of errors between the T 2 versus groups and prerequisite s k i l l s . -Hypothesis 4: No s i g n i f i c a n t i n t e r a c t i o n was found in terms of proportion of errors between the T 2 versus groups and mathematical a b i l i t y . Average Latency (Y 3) Treatment Y 3 = 14.92 X± - 122.54 X 2 Math A b i l i ty 1.47 X„ Prelesson A-State 6.66 X 4 A-State Main + .50 X 5 Prelesson Prelesson Score Latency + + 9.22 X, Treatment x Math A b i l i t v Treatment x Prelesson + 1.63 X XX3 10 X 2X 3 Treatment x A-State Main 6.32 X X . +4.96 X 0 X A 1 o do Treatment x Prelesson Latency error 78 XnX 3 + 4.72 X 2 X 5 + .24 X±X7 - .0.1 X 2 X 7 + e 108 TABLE 14 RESULTS OF REGRESSION ANALYSIS FOR AVERAGE LATENCY Source of R 2 Degrees of Fobs P< V a r i a t i o n Freedom X 3 . 156 1 13.57 . 001 X 6 . 028 1 2.30 . 13 x 4 .061 1 5. 02 . 03 X7 . 128 1 11.14 .002 X5 . 001 1 .09 .76 X l .006 1 .52 .48 x2 . 002 1 .18 . 68 (x x 2 > . 008 2 .35 . 71 X 1 X 3 . 003 1 .26 .02 x 2x 3 . 001 1 .09 .76 ( X1 X3» X2 X3) . 004 2 . 17 .84 X 1 X 6 . 009 1 .78 .39 X 2 X 6 . 005 1 .43 .52 ( X1 X6' X2 X6 ) . 014 2 .61 .55 x xx 5 . 006 1 .52 .48 X 2 X 5 . 048 1 4. 18 .04 (Xix5,x2x5) . 054 2 2.35 . 10 X 1 X 7 . 007 1 .61 .44 X 2 X 7 . 000 1 . 00 -( X1 X7' X 2 X 7 ) . 0C7 2 .30 .67 t o t a l .461 15 error . 539 : i 47 109 The r e s u l t s shown i n Table 14 indicate the mathematical a b i l i t y (p<^.001), prelesson A-State (p<-.03) and prelesson average latency (p<^".002) are a l l s t a t i s t i c a l l y s i g n i f i c a n t i n terms of predicting average main lesson latency (Y ). 3 A s i g n i f i c a n t main A-State-level-by-treatment i n t e r a c t i o n was obtained (p<-^ ".04) fo r the T 2 and T^ treatment groups. This f i n d i n g i s examined i n more d e t a i l i n f i g u r e 8. The r e s u l t s of the hypotheses stated i n Chapter III are given below for average response latency: Hypothesis l a : No s i g n i f i c a n t difference was found i n average response latency between students provided with response-s e n s i t i v e c o r r e c t i o n a l feedback (T^) and students provided with response-i n s e n s i t i v e c o r r e c t i o n a l feedback (T^) Hypothesis l b : No s i g n i f i c a n t difference was found i n average response latency between students provided with response-i n s e n s i t i v e c o r r e c t i o n a l feedback (T^) and students provided with no co r r e c t i o n a l feedback (T^). 110 Hypothesis 2: A s i g n i f i c a n t i n t e r a c t i o n (p .04) was found i n terms of average response latency between the T 2 versus T 3 groups and A-State. See f i g u r e 7. Hypothesis 3: No s i g n i f i c a n t i n t e r a c t i o n was found i n terms of average response latency between the versus groups and prerequisite s k i l l s . Hypothesis 4: No s i g n i f i c a n t i n t e r a c t i o n was found in terms of average response latency between the T versus T groups and mathematical a b i l i t y . Relationship 3etween Process and Product Treatment Math A b i l i t y Prelesson A-State Y1 = -.05 Xi - .03 X 2 + .09 X 3 - .01 X 4 Main A-State Prelesson Score .10 X 5 + .09 X 6 Prelesson Latency .01 X 7 Proportion of Errors Main Latency. error 5.56 + .005 Y 3 e I l l TABLE 15 RESULTS OF REGRESSION ANALYSIS FOR HYPOTHESIS 5 Source of V a r i a t i o n AR 2 Degrees of Freedom Fobs P < ( X l , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 ) .261 7 2.62 Y2 .042 1 3.22 . 07 Y 3 .006 1 0.46 .46 ( Y2, Y 3) .048 2 1.82 . 17 t o t a l .309 9 error .691 53 The r e s u l t s i n Table 15 indicate that proportion of errors (Y 2) was l i n e a r l y r e l a t e d (p«^.07) to posttest score (Y-^ ) with the e f f e c t of learner c h a r a c t e r i s t i c s and treatment difference s t a t i s t i c a l l y c o n t r o l l e d . No r e l a t i o n s h i p was found between average latency (Y 2) and posttest (Y-^ ) Hypothesis 5: A s i g n i f i c a n t 1inear r e l a t i o n s h i p (p<^.07) was found between immediate learning and proportion of errors with the e f f e c t of learner c h a r a c t e r i s t i c s and treatment s t a t i s t i c a l l y removed. 112 The graph shown i n figur e 6 i l l u s t r a t e s the d i f f e r e n t i a l e f f e c t of A-State on posttest (raw) scores for the three treatment groups. An increase i n A-State caused a decrement i n posttest performance. However, the li n e a r r e l a t i o n s h i p between posttest score and A-State was only s t a t i s t i c a l l y s i g n i f i c a n t for the T^ treatment group. The i n t e r a c t i o n between the A-State and c o r r e c t i o n a l feedback was merely ordinal, i . e . the regression l i n e s did not cross. The graph shown i n figure 7 i l l u s t r a t e s the r e l a t i o n s h i p between average latency i n the main lesson and main A-State. For the T-^ and T^ treatment groups, an increase i n A-State was not s t a t i s t i c a l l y s i g n i f i c a n t . for the T group (no co r r e c t i o n a l feedback) an increase 3 i n A-State l e v e l l e d to a s i g n i f i c a n t increase i n response latency. The i n t e r a c t i o n between A-State and c o r r e c t i o n a l feedback was merely o r d i n a l , i . e . the regression l i n e s d i d not cross. 113 1 0 T -t-8 10 12 14 A-State (X 5) * T n : Y- = 9 . 2 - . 0 9 Xc T 2: YT = 8 . 7 - . 44 X, T 3 : Y ^ = 8 . 6 - . 0 5 X^ Figure 6 Regression Lines of Posttest (Y-j_) and Main Lesson A-State (X 5) for T l f T 2, T 3 114 150 -+-6 10 12 14 *TX:. T 2 : T 3 : Y3 = 177.5 2.02.7 186.9 .68 X 5 1.25 X 5 5.55 X^ A-State ( X J Figure 7 Regression l i n e s of Average Main Lesson Latency (Y 3) an Main A-State (X 3) for the Three Treatment Groups 115 Results of Post hoc Analysis Table.17 i n Appendix H l i s t s the c o r r e l a t i o n c o e f f i c i e n t estimates between a l l p a i r s of variables observed i n the study f o r the t o t a l sample of sixty-three subjects. The reader should note that a c o r r e l a t i o n c o e f f i c i e n t which i s s t a t i s t i c a l l y s i g n i f i c a n t provides' a d d i t i o n a l insight into the data but does not allow causal inferences to be made. The extremely high c o r r e l a t i o n s (> .95) of proportion of errors (4) with t o t a l errors (2) and with t o t a l responses (3) suggests that e i t h e r of the l a t t e r two variables could have been used as c r i t e r i o n measures i n place of proportion of errors (2 divided by 3) with no important differences i n r e s u l t s . The extremely high c o r r e l a t i o n (.94) between latency for the f i r s t response of an i n s t r u c t i o n a l unit i n the prelesson (14) and t o t a l latency for thct i n s t r u c t i o n a l unit (15) indicates that either of these two measures could have been used as independent variables i n the experiment. The t o t a l latency (15) was actually used. S i m i l a r l y , the high c o r r e l a t i o n (.81) between f i r s t time latency (9) i n the main lesson and t o t a l latency (10) indicates that either variable could have been used as a c r i t e r i o n v a r i a b l e . The t o t a l latency was a c t u a l l y used. 116 The r e s u l t s i n Table 17 show that mathematical a b i l i t y (16) i s the most important variable. This variable (16) i s correlated s i g n i f i c a n t l y with posttest (1), errors (2,4,5), latency i n the prelesson (14, 15) and latency i n the main lesson (9, 10). At one point i n the main lesson the student was asked the question, "Do you understand?" The c o r r e l a t i o n between the response to t h i s question (6) and time to answer the question (7) was s i g n i f i c a n t (-.58) which indicated that people who answered "yes" took less time to respond. A s i g n i f i c a n t c o r r e l a t i o n (-.35) was found between the main lesson A-State (21) and the students' enjoyment of the lesson (11) as measured by a question at the end, 'Did you enjoy t h i s method of learning?" Lower A-State l e v e l was rela t e d to more enjoyment of the CAI experience. A p o s i t i v e c o r r e l a t i o n (.33) was found between proportion of errors (4) and whether the students went through the optional section on l i m i t s (8). Students with a higher proportion of errors tended to opt for taking the l i m i t section during the main lesson. Surprisingly, there was no r e l a t i o n s h i p found between t r a i t anxiety, A-Trait (18) as measured i n the f i r s t t e s t i n g session, and the state anxiety, A-State during the prelesson (20) or the main lesson (21). However, a s i g n i f i c a n t l i n e a r r e l a t i o n s h i p (.38) was found between A- T r a i t and the five- i t e m A-State score (19) measured on the DAY 1 paper and pen c i l test. Table 18 i n Appendix H l i s t s a comparison of co r r e l a t i o n c o e f f i c i e n t s between selected variables f o r the three treatment groups considered separately. The Chi-square value indicates whether the three c o r r e l a t i o n c o e f f i c i e n t s are s i g n i f i c a n t l y d i f f e r e n t at the"five percent l e v e l . The only s t a t i s t i c a l l y s i g n i f i c a n t f i n d i n g was the c o r r e l a t i o n between posttest score and the t o t a l number of main lesson correct responses given by the student before being given the answer by the computer. A p o s i t i v e c o r r e l a t i o n (> .55) was found for the two groups receiving feedback information (T-^ and T 2 ) but no r e l a t i o n s h i p (r = .07) f o r the group receiving no feedback information ( T 3 ) . The graph i n figure 8 i l l u s t r a t e s the average number of errors made by the t o t a l group on each i n s t r u c t i o n a l unit i n the main lesson. The re s u l t s indicate the students made r e l a t i v e l y few errors during the main lesson. The subjects averaged more than two errors only on one i n s t r u c t i o n a l unit (9) and on units 12, 13 and 14, the students averaged almost two errors. This f i n d i n g indicate 118 that the c o r r e c t i o n a l feedback variable may not have been potent enough to s i g n i f i c a n t l y d i f f e r e n t i a t e between the T and T_ treatment groups. The reason for t h i s i s that 1 ^ feedback was r a r e l y received from more than one or two wrong answer categories before the computer provided the answer to the student. Also, i n some cases many students provided the correct response without making an error, as i n units 2, 3, 4, 5, 6, 10 and 11. However, a higher error rate program would have been se l f - d e f e a t i n g i n terms of the goal of e f f i c i e n c y i n learning. Another trend was noted i n that the d i f f i c u l t y l e v e l s of the i n s t r u c t i o n a l units were not evenly d i s t r i b u t e d since the lesson f l u c t u a t e d several times from easy to d i f f i c u l t . The graph i n Figure 9 shows the average number of errors made on each i n s t r u c t i o n a l unit f o r the three treatment groups taken separately. On the more d i f f i c u l t units such as 7, 9 and 12, the s u p e r i o r i t y of the T^ group was evident. On i n s t r u c t i o n a l unit 9, which was the only unit where a l l groups averaged more than two errors, the ranking of average number of errors was as hypothesized-( T ^ T 2 Tu ). This f i n d i n g suggests that the r e s u l t s expected i n the study may have been more pronounced i f the error rate of the lesson had been higher. 119 Graph of Errors on In s t r u c t i o n a l Units for Combined Groups 120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Instr u c t i o n a l Unit Figure 9 Graph of Errors on Instructional Units for Ti T 2, T3 121 The graph i n figure 10 shows the average t o t a l response latency on each i n s t r u c t i o n a l unit f o r the three treatment groups. Although no obvious trend seemed to have occurred, one clear f i n d i n g emerged. The subjects in the second treatment group (T 2) took more time to respond on nearly a l l d i f f i c u l t i n s t r u c t i o n a l units from the seventh unit onward. For example, examining i n s t r u c t i o n a l units 7, 9, 12, 13 and 14, the T 2 subjects had larger average latency values. As noted e a r l i e r , the T2 group attended the fewest morning sessions and so i t i s possible that a f t e r a day of classes, a fatigue e f f e c t became predominant i n the second half of the main lesson. The graph i n f i g u r e 11 shows the A-State lev e l s for the three treatment groups at d i f f e r e n t points i n time during the study. The graph i l l u s t r a t e s that on DAY 1 and during the prelesson, the A-State l e v e l of T 2 subjects was higher than that of T^ subjects who, i n turn, had higher A-State l e v e l s than the T subjects. This seems to indicate that the T 2 subjects were more anxious than the others. However, during the main lesson, the r e l a t i v e A-State l e v e l s were a l l s i g n i f i c a n t l y higher and the ranking i n this case was as expected ( T - ^ 1^ .^T^ ). A group of f i f t e e n subjects reviewed a portion of the main lesson and was presented with the A-State instrument 122 twice. Six subjects were i n the group, f i v e were i n the T 2 group and four were i n the T3 group. The mean A-State score of these students the f i r s t time through the . main lesson was 13.1. Afte r reviewing a section of the lesson and repeating the A-State questionnaire, t h e i r A-State l e v e l was 14.6. This indicates that A-State l e v e l was increasing with time at this stage of the experiment and was probably quite high when the students wrote the posttest. Unfortunately, the A-State l e v e l was not obtained during the posttest but the fatigue e f f e c t evident f o r the T 2 group i n terms of latencies may have caused a reversal of the r e l a t i v e order of A-State for the three groups. It i s evident that more d i f f i c u l t items would have increased the A-State l e v e l even more and probably would have better separated the three groups i n terms of anxiety. 123 \ 1 1 1 1 1 1 1 1 ) 1 1 1 ' I 1' L 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I n s t r u c t i o n a l Unit Figure 10 Graph of Latencies on I n s t r u c t i o n a l Units 124 Paper and - Prelesson Main Pe n c i l Day 2 Lesson Day 1 Day 2 Time of Testing Figure 11 Graph of A-State Levels During Experiment ,125 TABLE 16 AVERAGE NUMBER OF RESPONSES IN EACH RESPONSE CLASS FOR MAIN LESSON Response Average Number of Responses Class T l T2 T3 • HELP 0.8 °2.9 2.6 Correct Answer 11.6 10.6 10. 4 Wrong Answer 1 3.5 3. 9 3.6 Wrong Answer 2 3.2 3.1 3. 9 Wrong Answer 3 1.3 1.5 1. 2 NOMATCH 5.4 5.8 6.3 The r e s u l t s shown i n Table 16 indicate that the T^ group asked for help on the average less than once i n the main lesson. This was s i g n i f i c a n t l y fewer times than the other two groups (T 2, T^). The average number of correct responses made were ranked i n the expected order, although the differences between the three groups were not s i g n i x i c a n t . It i s inter e s t i n g to note that as less c o r r e c t i o n a l feedback was provided from T-^ to T^, the number of unanticipated (NOMATCH) responses increased s l i g h t l y . The average number of unanticipated responses for the f i f t e e n unit main lesson was approximately six. This i s a very respectable f i g u r e i n view of the f a c t that many of the errors were typing or notation errors. Therefore, l i m i t a t i o n s of the CAI author language did not seriously hamper the recognition of student responses i n th i s experiment. A f a i r balance was achieved for the wrong answer classes except that few responses f e l l into the t h i r d c l a s s . Some improvement could be made here.in better a n t i c i p a t i n g student incorrect responses. Summary of S t a t i s t i c a l Results The regression analysis produced several findings. The r e s u l t s of the posttest analysis indicate that mathematical a b i l i t y (p<^.001) was s i g n i f i c a n t i n pre d i c t i n g immediate learning. However, when the e f f e c t of t h i s variable and the other variables stated i n Hypothesis 5 was removed, the proportion of errors f o r the main lesson was also s i g n i f i c a n t fp^.07) i n predicting posttest r e s u l t s . The expected difference between the three treatment groups receiving d i f f e r e n t types of c o r r e c t i o n a l feedback was not observed on the posttest. An unexpected f i n d i n g occurred i n that the main lesson A-State Jevel was 127 s i g n i f i c a n t i n p r e d i c t i n g posttest performance (p<^.06) and an A-State-by-treatment i n t e r a c t i o n was found (p<".04) in p r e d i c t i n g posttest scores. The e f f e c t of state anxiety on posttest performance was d i f f e r e n t for students i n the T^ group compared to students i n the T 2 group. Graphical a n a l y s i s showed that a s i g n i f i c a n t decrement i n posttest performance occurred when A-State increased, but only f o r the T 2 group. The r e s u l t s of the regression analysis f o r proportion of e r r o r s indicated that mathematical a b i l i t y ( p ^ . O l ) and prelesson score (p <~\ 01} were both s t a t i s t i c a l l y s i g n i f i c a n t . Also, the A-State l e v e l was s i g n i f i c a n t (p .02) i n p r e d i c t i n g the main lesson proportion of e r r o r s . No s i g n i f i c a n t i n t e r a c t i o n s were observed. The expected treatment e f f e c t was observed. A s i g n i f i c a n t d i f f e r e n c e i n proportion of errors ( p 0 3 ) was found between the T 2 and T^ treatment groups. This d i f f e r e n c e was i n the expected d i r e c t i o n , i . e . the T^ group had a higher proportion of e r r o r s than the T 2 group. The r e s u l t s o:' the regression analysis for average response latency i n d i c a t e d that, once aqain, mathematical a b i l i t y was highly s i g n i f i c a n t (p <-^. 001) i n p r e d i c t i n g performance. The A-State l e v e l that was reached during the prelesson was s i g n i f i c a n t in p r e d i c t i n g response latency 128 during the main lesson (p<.03). As would be expected, the prelesson average latency was s t a t i s t i c a l l y s i g n i f i c a n t (p<^.002) i n pre d i c t i n g the main lesson average latency. The expected treatment e f f e c t was not observed since no s i g n i f i c a n t d i f ferences were found between the three treatment groups on response latency. However, the expected A-State-by-treatment i n t e r a c t i o n for latency was observed (P<-04). The e f f e c t of state anxiety on response latency was d i f f e r e n t for students i n the T2 group compared to students in the T3 group. Graphical analysis showed that a s i g n i f i c a n t increase i n average response latency occurred with an increase i n state anxiety f o r the T3 treatment group only. The post hoc analysis provided some useful information for i n t e r p r e t a t i o n of the r e s u l t s and suggested some other questions. The table of means further indicated that the T-, group attended the fewest morning sessions and the T,, group attended the most. This f i n d i n g suggests that the longer latencies may have been due to a fatigue e f f e c t . This would have been caused by the s l i g h t predominance of evening sessions f o r the T2 group. An i n t e r e s t i n g f i n d i n g was the s i g n i f i c a n t c o r r e l a t i o n (r = - .35) between enjoyment of the main lesson and le v e l 129 of A-State. As might be expected, the low A-State students enjoyed this method of learning more than the high A-State students. The s i g n i f i c a n t c o r r e l a t i o n between proportion of errors and whether or not students had the optional l i m i t section (r = .33) suggests that students r e a l i z e d when they required extra assistance and that, therefore, more learner control could be b u i l t into the CAI lessons. An examination of the HELP and NOMATCH options showed that the T^ group asked f o r he.lp s i g n i f i c a n t l y fewer times than the other two groups (T 2, T^). This f i n d i n g again provides support f o r more learner control of i n s t r u c t i o n . The number of unrecognized responses (NOMATCH) were ranked as expected (T^ T 2 T 3 ) . Therefore, increased information i n the c o r r e c t i o n a l feedback seems to provide a p a r t i a l s olution to the problem of a n t i c i p a t i n g a l l possible incorrect responses The ta.ble of means showed that the three treatment groups d i f f e r e d on nearly a l l of the important variables i n the d i r e c t i o n expected, but that the e f f e c t s were not s t a t i s t i c a l l y , s i g n i f i c a n t . The graphs of errors showed that the program had a low error rate despite e f f o r t s by the experimenter to u t i l i z e a sample that would f i n d the material d i f f i c u l t . 130 Only one i n s t r u c t i o n a l unit had an average of more than two errors for the t o t a l group. It i s i n t e r e s t i n g to note that a large difference was found i n the expected d i r e c t i o n i n terms of errors on this i n s t r u c t i o n a l u n i t among the three groups. The graph of latencies indicated that the units were uneven i n d i f f i c u l t y and that the T 2 group took consistently longer on the i n s t r u c t i o n a l units i n the second half of the main lesson. The graph of A-State over time indicated that the expected pattern i n anxiety d i d occur but that the difference between the three groups was not as pronounced as expected. The t o t a l group, however, did increase i n A-State during the main lesson, as expected. CHAPTER V SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary of Study This study was undertaken to investigate the use of computer-assisted i n s t r u c t i o n as an i n s t r u c t i o n a l laboratory. The concept of an i n s t r u c t i o n a l l o g i c was defined as an algorithm followed by the computer program for each i n s t r u c t i o n a l unit. This step-by-step l o g i c was repeated for each i n s t r u c t i o n a l unit but with d i f f e r e n t content., This procedure permitted the c o n t r o l l e d manipulation of the variable of c o r r e c t i o n a l feedback. Three forms of corr e c t i o n a l feedback were defined by varying the information content of the feedback. These were response-sensitive c o r r e c t i o n a l feedback, response-insensitive c o r r e c t i o n a l feedback and no co r r e c t i o n a l feedback (only that the answer was i n c o r r e c t ) . The i n t e r a c t i o n of c o r r e c t i o n a l feedback with selected learner c h a r a c t e r i s t i c s was examined as y/ell. These learner t r a i t s were mathematical a b i l i t y , p r e r e q u i s i t e knowledge and state anxiety. The e f f e c t of co r r e c t i o n a l feedback and i t s .131 132 i n t e r a c t i o n with these variables was examined. Subjects of the study were a representative sample of sixty-three preservice elementary school teachers from f i v e sections of a mathematics course given i n a large education f a c u l t y . These subjects were randomly assigned to the three treatment conditions, although they selected the CAI experimental periods i n which they would p a r t i c i p a t e , The test of mathematical a b i l i t y used was the Cooperative Sequential Test of Educational Progress, Mathematics Form 2A (STEP). The state anxiety instrument used was the State-Trait Anxiety Inventory (STAI) and the f i v e item short form used by O'Neil (1972) was administered twice. The eighteen item posttest was constructed by the experimenter and the measure of prerequisite knowledge used was a nine item prelesson with a possible mark of 0, 1 or•2 on each item. The mathematics lesson was a top i c i n introductory calculus dealing with the concept of d e r i v a t i v e . The topic was treated from a physical point of view, using concepts of distance, speed and time to i l l u s t r a t e the mathematical concepts. The main objectives were to show that the de r i v a t i v e i s a l i m i t and to show how to use t h i s l i m i t d e f i n i t i o n to ca l c u l a t e the derivative of a function at a point. 133 The CAI lesson was programmed using an author language developed by the experimenter as a vehicle for implementing the i n s t r u c t i o n a l l o g i c and varying the c o r r e c t i o n a l feedback. The language i s l i m i t e d i n use but has the advantage of requiring, e s s e n t i a l l y no computer experience of an i n s t r u c t i o n a l designer. The main l i m i t a t i o n of the language as implemented at the University of B r i t i s h Columbia i s the cost, which severely l i m i t e d the sample size i n t h i s experiment. The r e s u l t s of the study were generally i n the expected d i r e c t i o n but the e f f e c t s were not as pronounced as had been hypothesized. The most important f i n d i n g was the s i g n i f i c a n t difference i n proportion of errors on the main lesson between the T 2 and T 3 groups. The importance of t h i s f i n d i n g was then increased by the s i g n i f i c a n t r e l a t i o n s h i p found between immediate learning and proportion of errors with the e f f e c t of learner t r a i t s and treatment e f f e c t s s t a t i s t i c a l l y removed. The most e f f e c t i v e variable i n predicting performance in the experiment was mathematical a b i l i t y and i t s e f f e c t was s t a t i s t i c a l l y c o n t r o l l e d when test i n g the e f f e c t of the other variables. Prerequisite knowledge was also important i n p r e d i c t i n g performance and was s t a t i s t i c a l l y c o n t r o l l e d as well. State anxiety was s i g n i f i c a n t i n pr e d i c t i n g 134 response latency but not i n pr e d i c t i n g errors. S i g n i f i c a n t treatment by A-State interactions were observed for posttest and for response latency. The three treatment groups d i f f e r e d i n the expected d i r e c t i o n on most of the important variables but the differences were not s t a t i s t i c a l l y s i g n i f i c a n t . In p a r t i c u l a r , the A-State lev e l s for the three groups were ordered as expected, but the differences were not large enough to cause the hypothesized in t e r a c t i o n s . The r e s u l t s of the study p a r t i a l l y supported the hypothesis of the important r o l e of c o r r e c t i o n a l feedback i n i n s t r u c t i o n and i t s i n t e r a c t i o n with i n d i v i d u a l t r a i t s of the learner. F i n a l l y , the p o s s i b i l i t y of a confounding variable became evident. A fatigue e f f e c t for the T 2 group seemed possible i n l i g h t of the s l i g h t predominance of evening sessions coupled with the larger response latencies for this group. In general, the two goals of t h i s project were attained. A methodology was developed for CAI as an i n s t r u c t i o n a l laboratory and t h i s methodology was used succ e s s f u l l y to perform a c o n t r o l l e d experiment. This experiment provided evidence that c o r r e c t i o n a l feedback leads to improved performance during learning compared with 135 feedback t e l l i n g a student merely that his response i s incorrect. A s i g n i f i c a n t r e l a t i o n s h i p was found between proportion of errors and posttest score which suggests that a number of errors made during a lesson i s a measure that can be used to maximize immediate learning. F i n a l l y , the variable of state anxiety was examined and was found to negatively a f f e c t response latency. Also, state anxiety increased when no c o r r e c t i o n a l feedback was provided to the students as well as when the content became more d i f f i c u l t . This f i n d i n g confirmed the expected r e l a t i o n s h i p between state anxiety and task d i f f i c u l t y . Discussion of Findings Correctional feedback. The c o r r e c t i o n a l feedback variable had some e f f e c t i n t h i s experiment. A s i g n i f i c a n t difference (p^.03) i n proportion of errors was observed between the group receiving response-insensitive c o r r e c t i o n a l feedback (T 2 ) and the group receiving no c o r r e c t i o n a l feedback (T-j). The raw means of the three groups on most of the variables examined i n the study were ranked i n the expected order, i . e . T -^T^^T^. However, the d i f f e r e n c e s were not s t a t i s t i c a l l y s i g n i f i c a n t . The r e s u l t s suggest that the response-insensitive c o r r e c t i o n a l feedback (T?) may be optimal since some hel p f u l information i s provided to the student af t e r an 136 incorrect response but p r i o r analysis of a l l possible responses (and keywords) i s not required. This form of c o r r e c t i o n a l feedback requires f a r less time to prepare by the i n s t r u c t i o n a l programmer and much less computer time and storage i s used. More d e f i n i t i v e r e s u l t s may have been obtained with a lesson having a higher error r a t e , but this would have been self-defeating since the aim of the program was to help the student succeed. These r e s u l t s suggest that i n terms of using CAI i n the schools, programmed i n s t r u c t i o n a l materials can emulate the strategy quite e a s i l y and cheaply. U n t i l e f f e c t i v e response-sensitive feedback can be developed which can be demonstrated to cause improved performance and learning, the use of CAI in the classroom i s not j u s t i f i e d . However, using CAI as an i n s t r u c t i o n a l laboratory permits research to be c a r r i e d out that could not be done using programmed-instructional materials. A-State. As expected, the r e l a t i v e d i f f i c u l t i e s of the three treatment conditions caused corresponding differences i n A-State l e v e l during the lesson. However', these A-State l e v e l s were not s i g n i f i c a n t l y d i f f e r e n t for the three treatment groups. For a l l groups, the A-State l e v e l did increase i n the main lesson compared to the written test and prelesson. Also, a few students i n each group 137 repeated a section and were given the A-State questions again. The mean A-State l e v e l for these twelve students again increased s u b s t a n t i a l l y suggesting that A-State l e v e l s were increasing towards the end of the CAI session. The hypothesized A-State-by-correctional-feedback i n t e r a c t i o n did not occur f o r proportion of errors, but an i n t e r a c t i o n did occur for the other two c r i t e r i o n variables. An increase i n A-State f o r the T3 group (no c o r r e c t i o n a l feedback) was coupled with a s i g n i f i c a n t increase i n response latency. There was no signifi c a n t - r e l a t i o n s h i p between A-State and response latency f o r the T^ and T^ groups. This f i n d i n g again provided evidence i n favor of response-insensitive c o r r e c t i o n a l feedback (T 2) as the optimal condition. An unexpected f i n d i n g occurred for the T 2 group. An increase i n main lesson A-State f o r t h i s giovp was r e l a t e d to a decrease i n posttest performance. Unfortunately, t h i s f i n d i n g i s d i f f i c u l t to in t e r p r e t because the ac tue.l A-State l e v e l during the written posttest was not obtained. It i s possible that these A-State l e v e l s may have be^n s u b s t a n t i a l l y d i f f e r e n t from the main lesson since A-State anxiety l e v e l s were increasing towards the end of the main lesson, as pointed out above. The explanation offered at t h i s point f o r this f i n d i n g i s that a fatigue e f f e c t may have 138 been present and may have affected the T 2 group. The reason f o r th i s statement i s that students i n the T 2 group took consistently longer to respond during the second half of the main lesson and were probably most fatigued. A possible explanation for th i s i s the fa c t that s l i g h t l y more T 2 students took the CAI lesson and posttest after classes i n the evening, despite e f f o r t s by the experimenter to control for this f a c t o r . The main lesson A-State was s i g n i f i c a n t i n pr e d i c t i n g proportion of errors for the t o t a l group. Also, the prelesson A-State was s i g n i f i c a n t (p<.03) i n pre d i c t i n g the main lesson response latency. Therefore, the r e l a t i o n s h i p between anxiety and performance for the whole group was well established i n this study. Surprisingly, no r e l a t i o n s h i p was found between A-Trait and main lesson A-State l e v e l s . Tobias (1973) suggested that i t may be that the variable of anxiety, while useful in other areas, has limi t e d u t i l i t y i n the area of 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 . The reason f o r th i s statement i s that i n 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 a l contexts an attempt i s made to minimize d i f f i c u l t y i n order to have a high r a t i o of success. Even when i n s t r u c t i o n a l materials are experimentally a l t e r e d to increase their d i f f i c u l t y , these a l t e r a t i o n s are often 139 i n s u f f i c i e n t to both evoke and maintain l e v e l s of anxiety s u f f i c i e n t to exert s i g n i f i c a n t d e b i l i t a t i n g e f f e c t s on achievement. This may explain the lack of stronger findings i n t h i s p a r t i c u l a r study. Mathematical a b i l i t y . The variable of mathematical a b i l i t y had the most evident e f f e c t i n the study. This variable was highly s i g n i f i c a n t i n predicting immediate learning, proportion of errors and response latency. However, the hypothesized c o r r e c t i o n a l feedback-by-mathematical -abi l i t y i n t e r a c t i o n was not observed for any of the c r i t e r i o n variables. This f i n d i n g suggests that mathematical a b i l i t y may not be a useful variable f o r i n d i v i d u a l i z i n g i n s t r u c t i o n and that more s p e c i f i c a b i l i t i e s known to be important to task performance are required. Prerequisite knowledge. The variable of p r e r e q u i s i t e knowledge, as measured on the prelesson, was found to be s i g n i f i c a n t i n p r e d i c t i n g proportion of errors (p<".01) only. This f i n d i n g i s reasonable since i t would seem that a good predictor of main lesson performance on a CAI lesson during the main lesson would be the performance on CAI prelesson. In l i n e with t h i s reasoning, the prelesson average response latency was s i g n i f i c a n t (p <-•- . 002) i n p r e d i c t i n g the main lesson average response latency. 140 The hypothesized prerequisite knowledge-by-cor r e c t i o n a l feedback i n t e r a c t i o n was not observed for any of the c r i t e r i o n variables. This f i n d i n g seems to have been caused by the lack of variance i n the prelesson scores f o r subjects i n the experiment. Most students already had attained the prereq u i s i t e objectives and the prelesson served merely as a warm-up f o r them, Relationship between process and product. A s i g n i f i c a n t l i n e a r r e l a t i o n s h i p (p^.07) was observed between,the process variable (proportion of errors) and the product variable (immediate learning). No such r e l a t i o n s h i p was observed between response latency and immediate learning. However, there i s some suggestion i n the l i t e r a t u r e (Judd e_t al_. , 1973) that response latency may have an e f f e c t on retention. The above s i g n i f i c a n t r e l a t i o n s h i p was observed a f t e r the e f f e c t of a l l learner variables i n the study and treatment had been s t a t i s t i c a l l y removed ( p a r t i a l l e d out). The f i n d i n g that providing response-insensitive c o r r e c t i o n a l feedback (T 2) i s better than providing no cor r e c t i o n a l feedback (T 3 ) f or reducing proportion of errors and that a s i g n i f i c a n t r e l a t i o n s h i p (negative c o r r e l a t i o n ) e x i s t s between proportion of errors and immediate learning has important implications. The f i n a l goal of the 141 i n s t r u c t i o n i s to maximize the learning product which i s what the student takes with him when he leaves the CAI terminal. The evidence from this study suggests that response-insensitive c o r r e c t i o n a l feedback i s the optimal condition to achieve this goal at the least cost. Post hoc analysis r e s u l t s . The post hoc analysis provided some useful information f o r i n t e r p r e t a t i o n of the r e s u l t s and suggested some further questions. The c o r r e l a t i o n a l analysis also indicated that mathematical a b i l i t y had the strongest e f f e c t i n this experiment. It was noted that a s i g n i f i c a n t c o r r e l a t i o n (r = - . 35) occurred between main lesson A-State and enjoyment of t h i s method of learning. Students with lower A-State lev e l s tended to enjoy the lesson more. Two findings that have implications for future CAI work were: (1) there was a s i g n i f i c a n t c o r r e l a t i o n (r = .33 between proportion of errors and whether or not students chose the optional section on l i m i t s , and (2) the number of students i n each group that had the l i m i t section increased from T^ to T 2 to T^ and the 1^ group asked f o r help s i g n i f i c a n t l y fewer times than the- other two groups. These findings suggest that students r e a l i z e when they require extra assistance and that more learner control of i n s t r u c t i o n could be b u i l t into CAI lessons. 142 The only c o r r e l a t i o n a l c o e f f i c i e n t s i g n i f i c a n t l y d i f f e r e n t for the three groups taken separately was the c o r r e l a t i o n between posttest score and number of main lesson correct responses given by the student before the computer provided the answer. The c o r r e l a t i o n (r> .55) for the T 1 and T^ groups suggests that students who f i n d the correct response themselves during the lesson w i l l perform better on the posttest. This r e s u l t suggests that c o r r e c t i o n a l feedback information i s an important part of this learning process. This f i n d i n g also r a i s e s the question about the motivational properties of co r r e c t i o n a l feedback since no s i g n i f i c a n t c o r r e l a t i o n was found when correc t i o n a l feedback was not provided ( T 3 ) . The table of means indicates that the three treatment groups d i f f e r e d on nearly a l l of the important variables i n the expected d i r e c t i o n , but that e f f e c t s were not s t a t i s t i c a l l y s i g n i f i c a n t . The variables r e f e r r e d to are posttest score, t o t a l e r r o r s , t o t a l responses, proportion of e r r o r s , t o t a l correct, number of students taking optional l i m i t section, enjoyment and main lesson A-Sta^e l e v e l . The graphs of errors show that the program had a low error rate despite e f f o r t s by the experimenter to u t i l i z e a sample of students who would make many errors. Only one i n s t r u c t i o n a l unit had an average of more tha.n 143 two errors for the t o t a l group. This i n s t r u c t i o n a l unit produced large differences i n the expected d i r e c t i o n between the three groups, suggesting that a more d i f f i c u l t lesson may have produced more d e f i n i t i v e r e s u l t s . A more d i f f i c u l t lesson could be designed by dealing with more content i n each i n s t r u c t i o n a l unit than was dealt with i n this study. The graph of latencies shows that the i n s t r u c t i o n a l units were uneven i n d i f f i c u l t y and that the T 2 group took consistently longer on the units i n the second half of the main lesson, thereby producing a fatigue e f f e c t that may have affected the r e s u l t s . Limitations of the Study A most apparent l i m i t a t i o n of t h i s study, and other studies i n the area of CAI at the present time, i s the cost f a c t o r . This factor was the major constraint in l i m i t i n g the sample size i n th i s study to sixty-three students. Unless an i n s t i t u t i o n i s w i l l i n g to invest funds i n t h i s type of research, i t i s probably wiser to l i m i t CAI research to i n s t i t u t i o n s having the s p e c i a l i z e d hardware and software f a c i l i t i e s required for e f f i c i e n t CAI. The biggest d i f f i c u l t y f o r the students seemed to be the notation used by the computer. S i n c i a standard typewriter b a l l was used, symbols f o r change ( A ) and a 144 normal d i v i s i o n sign could not be used. Also, variables could not be r a i s e d to a power using the usual notation. An examination of the students work sheets indicated that they often did their c a l c u l a t i o n s using their own notation and then entered the answer using the computer's cumbersome notation. Therefore, the hardware l i m i t a t i o n s probably had an e f f e c t on the experimental r e s u l t s . Bork and Sherman (1971) also noted t h i s l i m i t a t i o n . The students i n the experiment were preservice elementary school teachers and so the g e n e r a l i z a b i l i t y of the r e s u l t s i s l i m i t e d to t h i s or s i m i l a r populations of students. The content of the CAI lesson dealt with a well structured mathematical algorithm which was derived i n the lesson and applied to concrete examples. Since content may act to moderate the e f f e c t s of other variables, the g e n e r a l i z a b i l i t y of the r e s u l t s i s l i m i t e d to material having similar structure. An obvious l i m i t a t i o n was the short term nature of the experiment. The differences that d i d occur may not have been observed over a longer period of time. Also, a novelty e f f e c t was c e r t a i n l y present and may have affected the r e s u l t s . Once again, t h i s i s a cost l i m i t a t i o n of much of the present CAI research because the time required to develop a f u l l term CAI course i s enormous fo r a single person. 1 4 5 An unexpected fatigue e f f e c t seemed to have affected the r e s u l t s . This e f f e c t was not observed i n the p i l o t group of college students i n Montreal but seemed to be a f a c t o r i n t h i s study, p a r t i c u l a r l y during the posttest. Recommendations for Further Research Although some treatment e f f e c t s were observed i n t h i s study, these were not d e f i n i t i v e due to the lack of large enough differences i n treatment between the T^ and T 2 groups. A study similar to the present one should be conducted with a more d i f f i c u l t lesson and t h i s could be done by increasing the material treated i n each i n s t r u c t i o n a l u n i t . An attempt should be made to design a longer term study where a student would come more often to the CAI terminal, but f o r less time i n order to minimize fatigue e f f e c t s . Possibly a team e f f o r t would best f a c i l i t a t e t h i s type of study. The organizational scheme given i n Chapter I generates many questions. The independent variables may have an e f f e c t on retention and transfer, and these should also serve as c r i t e r i o n variables for further studies. Other learner variables should be examined, such as motivational variables and verba.l aptitude. These should come from an adequate p r i o r conceptual analysis of the 146 treatment. Other CAI lesson variables should be studied, such as types of branching and degree of learner control. Similar studies u t i l i z i n g d i f f e r e n t populations, p a r t i c u l a r l y elementary and secondary school students, would c e r t a i n l y be desirable as would be studies i n a variety of subject areas. This would increase the g e n e r a l i z a b i l i t y of the r e s u l t s . A CRT (graphics) terminal would possibly reduce the fatigue e f f e c t i n further studies. An attempt should be made to assess the e f f e c t s of a vari e t y of CAI terminal types. It has been noted that sex may be an important variable i n anxiety studies. Further studies should attempt to look at the e f f e c t of t h i s variable. F i n a l Comment As with many other studies, the present one had weaknesses and produced some unanticipated r e s u l t s . However, a major goal of the study was achieved: the demonstration that CAT i s a powerful and valuable means to examine i n s t r u c t i o n f o r the purpose of producing a s c i e n t i f i c a l l y - b a s e d theory of i n s t r u c t i o n . It i s hoped that t h i s study w i l l serve as an example of the power of the computer and that others w i l l accept the challenge of overcoming i t s l i m i t a t i o n s . 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Paper presented at the annual convention of the American Educational Research Association, New Orleans, La., February, 1973. Kalman, C., D. Kaufman, and H. Ladd. "Evaluation of a Computer-Based Dialogue." Unpublished manuscript, Montreal, 1971. Kalman, C., D. Kaufman, and R. Smith. "CAI Calculus Dialogue for Freshman Physics." Unpublished manuscript, Montreal, 1972. Kaufman, D., and R. Sweet. "Contrast Coding i n Least-Squares Regression An a l y s i s . " Unpublished manuscri University of B r i t i s h Columbia, Vancouver, 1973. O l i v e r , W.P.. "Learner and Program-Controlled Sequences of Computer-Assisted In s t r u c t i o n . " Paper presented at the annual meeting of the American Educational Research Association, New York, February, 1971. O'Neil, H.F., J r . "Anxiety Reduction and Computer-As s i s t e d Learning." Paper presented at the American Educational Research Association, Honolulu, 1972b. Szetela, W. "The E f f e c t s of Test Anxiety and Success -F a i l u r e on Mathematics Performance i n Grade Eight," Doctoral D i s s e r t a t i o n , University of Georgia, August, 1970. 155 Tests Educational Testing Service. Cooperative Sequential Tests of Educational Progress, Form 2A. Princeton, N.J., 1957. Spielberger, CD., R.L. Gorsuch, and R.W. Lushene. Manual for the State-Trait Anxiety Inventory. Palo A l t o , C a l i f . , Consulting Psychologist Press, 1970. Computer Programs Dempster, J.R.H. P r o b a b i l i t y i n an F or t D i s t r i b u t i o n (UBC FPROB ). The University of B r i t i s h Columbia, Vancouver, November, 1969. Dixon, W.J., ed. BMP: Biomedical Programs. Berkeley and Los Angeles: University of C a l i f o r n i a Press, 1968. Finn, J.D. Mu1tivariance - Univariate and M u l t i v a r i a t e Analysis of Variance, Covariance and Regression, (version 4). State University of New York, Buffalo, June, 1968. Seagraves, Paul. Small Triangular Regression Package (UBC STRIP). The University of B r i t i s h Columbia, Vancouver, August, 1971. APPENDIX A User's Guide for CAI Author Language Source L i s t i n g of Program This program i s written i n FORTRAN IV and requires a minimum of computer knowledge and experience on the part of the user. Introduction A lesson consists of a series of i n s t r u c t i o n a l units with each unit having the form shown on the next page i n f i gure 1. These i n s t r u c t i o n a l units may be presented to the student i n a sequential manner or i n an order determined i n advance by the lesson designer. A t y p i c a l lesson could take the form shown i n fi g u r e 2. IU 7 IU 1 IU 2 \ IU 3 A IU 4 —IK IU. 5 7 7 1 ) Typical Lesson Figure 2 User's Guide for CAI Author Language I GX=0 | Student i s asked a ques'.-.ion ^ S ' t ' u d e n t r e s p o n d s , Student's answer not recognized NOMATCH Comment Comncnt X to a»o no no Comment Comment made made w w i Write correct answer | and comments ' ,' a l l represent different instructional son. ^>means the n t n time through the block. Q i ' Q j ' Qk' <V Qj+i' units (10) in the lesson. in Co Figure 1 Main Lesson Instructional logic Using the Program Each i n s t r u c t i o n a l unit (Question) i s coded i n e s s e n t i a l l y the same manner as shown below. CARD 1 Col. 1-2: Question number, right j u s t i f i e d . Must be integer number between 1 and 30. CARD 2 Col. 2: No. of response classes. Maximum i s 4 Col. 4: No. of keywords i n response class 1. Maximum i s 8 Col. 6: No. of keywords i n response c l a s s 2. Maximum i s 8 Col. 8: No. of keywords i n response class 3. Maximum i s 8 Col.10: No. of keywords i n response class 4. Maximum i s 8 CARD 3 Col. 1-80: Any comment that the lesson designer wishes to have the computer make to the student. This comment always begins the unit and w i l l contain a question to the student. SIX cards maximum. Last card must contain a $END i n c o l . 77-80. CARD 4 Col. 1-80: Any comment that the lesson designer wishes to have the computer make to che student i f the student asks f o r help. THREE cards maximum. Last card mu.^ t contain a $END i n c o l . 77-80. 160 CARD 5 Col. 1-10: The f i r s t keyword accepted as correct answer. Should begin in c o l . 1 and end with a $ sign. Col.11-20: The second keyword accepted as correct answer. Should begin in c o l . 11 and end with a $ sign. Col.21-30, 31-40, 41-50, 51-60, 61-70, 71-80: Other keywords accepted as correct answer. Should begin i n appropriate column and end with a $ sign. The number of keywords on this card should be the same as the number indicated in c o l . 4 of card 2. CARD 6 Cel. 1-80: Any comment that the lesson designer wishes to have the computer make to the student i f ; (1) the student responds c o r r e c t l y , i . e . response wishes a keyword on l a s t card, (2) the student asks for help more than once, (3) the student responds more than once to the same wrong answer class, (4) the student's response i s not recognized more than twice. FIVE cards maximum. Last card must contain a $END i n c o l . 77-80. This card may also be a GOTO command. See l a s t section. CARD 7 Col. 1-10: F i r s t keyword accepted as wrong answer.' Should begin in c o l 1 and end with a $ sign. Col.11-80: Same as on card 5. Other keywords accepted as wrong answer. Number of keywords on th i s card should be the same as the number indicated in c o l . 6 of card 2. 161 CARD 8 Col. 1-80: Any comment that the lesson designer wishes to have the computer make to the student i f his response matches one of the keywords given on card 7. FIVE cards maximum. Last card must contain a $END i n c o l . 77-80. This card may also be a GOTO command. See l a s t section. CARD 9 Same form as card 7 CARD 10 Same form as card 8 CARD 11 Same form as card 7 CARD 12 Same form as card 8 The t o t a l number of keyword cards ( i . e . cards 5, 7, 9, 11) must be the same as the numbers i n c o l . 2 of card 2. This number i s limi t e d to a minimum of one and a maximum of four. Note: In order to construct a lesson having more than t h i r t y i n s t r u c t i o n a l units, s p e c i a l techniques must be used. See the author for d e t a i l s . GOTC Command Branching to any point i n the i n s t r u c t i o n a l lesson i s possible by using a GOTO command. This command may replace any comment card ( 5 ) such as cards 6, 8, 10 or 12. If an i n s t r u c t i o n a l unit has been presented twice to the student, a GOTO command to that i n s t r u c t i o n a l unit w i l l be ignored and the computer w i l l proceed to the next question. 162 A GOTO command must begin i n c o l . 1 and has the form GOTOXX where XX i s the i n s t r u c t i o n a l unit number, r i g h t -j u s t i f i e d , where the program w i l l branch i f that response class i s chosen by the student. System Commands The Michigan Terminal System (MTS) i s currently in use at the University of B r i t i s h Columbia. The user should create a desk f i l e to store the i n s t r u c t i o n a l program, e.g. CAI. The CAI program may then be run by entering the following commands, s t a r t i n g i n column 1: $RUN CAI 5=LESSON 9=*MSOURCE oi;fSTiON no. '*• • SAMPLE OuTSTION THTJJ. i ' i •: 1 i h 2 2 1 ?; : L E T ' S PFVLFV PI'R RAS'lC FACTS. TODAY,WE V W A T IS ' |/ x R ? J'CV? MUCH 15' SEVSfc MULYIPLIFn 8Y EIGHT? 'T»;I 'S- CAM S E V : R I T T : N AS 7 ! : • per, . • i F I F T Y .MX i I , i?xi--=r 6 AN P. Px7"f.B. P.:MFMPFi; THAT ML.LT I PI- 1/VkTIOtl IS; n ; . ; ! I i I'LL1 OO MULTIPLICATION i , . 1J : JUST nrPEATEO ; PPITIOM. !|F VP TAKr' 7 jPfOt'PS. OF' 8' THIJ|PS_OP 8 rPQUPS Or 7 THI TPS, . ; ' ; • ' ; jFIFTFFf'$ pO TO •? ' | : , . . . : |78*.' : ; i i 1 i I i ! ;i:o; 7? M.FA'.'S e w*$ /»wv 7 TF.MS •AoniTioii. :?5.i i i i i ! IF V.'E TAKE 7 fPOliPS' 5 7.$;' \ i • i : 58< • :, : . I !' . THI.VK OFi N U L T I PL I CAT I OM AS P.f P C A T F P OF ? t p n r s . i ' o w MAKY! T H I N G S ' on wi HAVE.?' 111 ;'T I i r i : I I ' 1 ! I I 11 II ji-ji 1 ii i i i i Hi 1 ; i •jT'"' 5.i.$..: ^ I i 1 ' I j j . i . L l . ; ! ii!:! |.i I j i I 11 I I ! 1 ! IT i':i i i i i ! •Ii Hi! P 11 111 55x.: i ' i i: : i I i ..! 5.9$. i- . I • I : I ;; :•;;;< 6it$ 1—>-i)8« I! Mi I t • r i 164 r3 copt 9 x x x x x x x x x x x x x x x x x i x x x x x x x x x x i x x x i x x x x i . x x x x x x x x x i i x x x x x x x x x x x x x x x x x / . x x x x i S BO. 04991S OBIVERSITY OF B C COMPUTING CENTRE HTS(OG073) IG L404 PRINT=TH COPIES=10 PAGES=100 t i S T SIGNON WAS: 13:37:52 unit oooooooo _. i n 1111 0000000000 1444 n 44 00 00 41 44 i i 44 00 00 44 44 i i 44 00 00 44 44 14141414!) tit 00 00 44444444441 4Q4441444U44 00 00 444444444444 44 00 00 11 44 00 00 41 44 00 00 11 LLLLLLLLLL 44 oooooooooo 44 LLLLLLLLLL 41 oooooooo 11 ZCCCCCCCC NS BH TTTTTTTTTTTT RRRRRRRRRRB CCCCCCCCCC NNH HH TTTTTTTTTTTV RRRRRRRBR8RB CC Nt'HB BB TT RR BR KK HK HN T T BB BE HH SH MB T T RR BB KB BH MH T T BRRRRRRRRR3R HH HH HN T T RRRRRRRBRRR HH BB HH TT BR BR HH BHNB T T GR BB CC HH HKB T T BR BB CCCCCCCCCC NN HN T T " BR BB c c c c c c c c c KB H T T BB BB SER_"L404"_SIGHED J)H » T _ 15:15:24 ON HOB AOG 27/73 BE -CAI" " " IL15 "-CAI" HiS BEEN CREATED. 08 IB 1ST -CAI 1 C DAVID KAUFMAN PROGRAM FOR MAIN CAI LESSOB 2 C THE KAIiJ PRCGRAh READS THE QUESTION NUMBES [IQ) , HOMBF.R OF RESPONSE 3 C CLASSES(KC) , NUMBER OF KEY WORDS IN EACH CLASS (KTOT),ALL KEYWORDS(KEY) , H C ALL QUESTIONS (Q"I;S) . HELT COMMENTS (QMOD) , COMr, ENTSFOR E..CH i. JESTIOH (TEXT) . 5 C THE MAIN PROGRAM ">!LS6" CO ilTROLS THE" FLU W" O F TK E LESSON" IN A 6 C SEQUENTI'.L Oi> NON-SEQUENTIAL MANNER AS SPZCIFrEO IN ADVANCE BY THE LESSOR 7 C DESIGNER.THE STATISTICS FOR EACH QUESTION IN THE LESSON,I.E.,NUKE^R 8 (&\ AT THE END C.F T/fE LESSON.?;IIS R ECOR D-K E E P I S F C T I O K PECl'ISES DIS? SPACE 9 *t> OF RESPONSES MADE BY THE STUDENT IN EACH RESPONSE CLASS (I* RES) ARE WRITTEN 10 C_ WHERE THE R ESU LTS_S AY_BE _W R I TT E N. R E S P C 5 E LATENCIES ARE ALSO WRITTEN 11 C FOR KAX::" RESPONSE OH EACH INSTRUCTION AL UNIT. 12 C A LESSON IS LIKITED TO 30 INSTRUCTIONAL UNITS 13 C 14 C" 15 IH?EGER*2 ANS(£0) .KEY (30,M,3,10) 16 _ IKTEGER QUSS(30,l20),gHOD(30,60),TEXT(je,4,100) ,STHO(30,6), 165 17 1 COD NT (30) .CLASS (30) 18 DIMENSION NUH (30) , KTOT (30,4),NRES (3, 30, 6) ,BAME (30) , LQ (30) ,LH (30) , 19 IB (30,1) .SEC (30, 10) ' 20 COMMON TEXT,QUES,QMOD.STNO.IP,IGX.NRES,COUNT,LQ,LH,B 21 1,ANS,CLASS,KTOT,KEY,SEC,IRES 22. DATA I EN D, IB, IGO/J SEND' , • S i G O T O V 23 DATA NUH(1),NUM(2),NUM(3),NUN(<4),NUH(5),NUM(6),NUM(7),NUH(8), 2<t 1NUH(9),NUM(1.9),NUH(11),NUH(12).NUM(13),NUH(14),NUM(1S),NUH(16), _ 2 5 2K0M (17) ,NUM (18) . NUM (19) , NUH (20)/' 1 «,« 2 «,« 3 1 «, . 26 3" 5 ".« 6 ',« 7 ',» 8 ',' 9 «.«10 ',«11 «,«12 ', • 13 27 I ' l l •, • 15 ','16 '.'17 <,*18 ','19 '.^O •/ _ 2 j l DATA _HOfl_[2J ) , NUM (22) , NU_M_(23J , MUM t2U_L,_NU_M_(251, NOM (26) . 29 1 NUM (27) , NUM (2B) ,NUM (29) ,NUM (30)/'21 *,*22 •, ' 23 •, 30 2«24 ','25 '.'26 •,'27 ','26 ','29 ••,•30 •/ 31 C INITIALIZE COUNTEF.S TO ZERO 32 IGI=0 33 DO 91 1=1,30 30 CPU NT (I)=0 ] 35 DO 41 J=1.6 36 DO 4 2 L=1,3 _ 3 7 02 NRES(L,I,J)=0 . _ . _ 38 41 STKO (I,J) =0 39 VRITE(6,13) HQ 13 FORMAT (' PLEASE ENTER TOUH FIRST AND LAST NAME') ; 11 C 2EAD-1N STUDENT NAME 42 BEAD (9, 11) (NAME (I) , 1=1, 10) 03 11 FORMAT (10A4) _ 44 DO'33" IH-1,31 45 L1=1 16 L2 = 20 ; 17 L3=1 18 L4=20 49 C BEAD NUMBER OF THE INSTRUCTIONAL UNIT BEING PREPARED 50 READ (5, i;EKD=20) IQ 51 1 FORMAT (12) 52 C READ TH E NO. OF RF.SPONSE CLASHES AND THE SO. OF KEYWORDS IB 53 C EACH" RESPONSE CLASS 54 READ(5,2)KC, (KTOT (IQ , J) , J = 1 , KC) 55 CLASS (IQ) = KC . 56 "2 FORMAT (912) 57 C BEAD THE QUESTION TO BE PRESENTED TO THE STUDENT 56 3_READ(5j«) (QUES (IQ. J) ,J = L1,L?) 59 4 FORMAT (20AU ) 60 C CHECK FOR THE END OF A COMMENT 61 IF QUES (TO, L2) . EQ. IENDJGO TO 5 _ 62 C "CHECK THAT THE INPUT IS CORRECT.IF NOT.GIVE ERROR MESSAGE TO AUTHOR 63 L1=L1+20 64 L2=L 2*20_ 65 IF(L2.LE.120)GO TO 3 O 66 BRITE (6,90) IQ 67 90 FORMAT (' TOO MAN T C A B DS__ 0_R_ b* O J E N_D IB FIRST PART 'JF Q'JES. BO.'. 68 113) 69 STOP 70 5 QUES (IQ.L21 -IB 71 LQ(IQTSL2 72 C BEAD-IN HELP CORKEHT 73 11 READ (5,4) (QHOD (IQ, J) , J = L3. L4) 74 C CHECK TH.'.T INPUT IS CORRECT.IF NOT.GIVE EBROi. RESSAiE 75 ' IP (QMOD ;IQ,L4) . EQ. IEND) GO TO 12 76 L3=L3*20 166 77 L4=L4*20 78 IF(L4.LE.60)GO TO 11 79 VRITE (6,91) IQ 80 91 FORMAT (' TOO BANT CARDS OR NO J END IN A COBHENT OF QOES. HO.', 81 113) 82. STOP 83 12 QBOD (IQ,L4)=IB 81 Ltl(IQ)=L1 85 DO 66_II=1.KC __ . 86 H1=1 87 M2 = 20 86 C READ ALL KEYWO R_D S_ IN E A CH RESPONSE C L A.SS ' 89 EEAD(5,6) ( (KEY (IQ, I I , J , K) , K= 1, 10) , J= 1, 8) 90 6 FOR MAT(80A1) 91 C READ COMMENTS TO BE GI'.EN TO STUDENTS FOR EACH RESPONSE CLASS 92 8 READ(S,4) (TEXT (IQ, I I , J) ,J = M1,M2) .93 C CHECK FOR A BRANCH TO ANOTHER INSTRUCTIONAL UNIT 91 IFJTEXT (IQ,I_I,_1) .NS.IGOIGO TO 10 95 C CHECK i n AT THE INPUT IS CORRECT.IF NOT,GIVE ERROR MESSAGE TO AUTHOR 96 DO 60 IK=1,30 97 IF (TEXT (IQ, 11,2) . EQ. NUM (IK) ) GO TO 30 98 60 CONTINUE 99 WRITE (6.93) II,IQ 100 93 FORMKT (INCORRECT GOTO STATEMENT IN CLASS'. 13. 2X , 101 1'QUESTION NO. *,I3) 102 STOP 103 30_ STflO CIQ.II) =IK _ V)1 TEXT (IQ, I I . i) =IE • 05 TEXT (IQ,II,2)=IB _106__ G0_T0_6^ 107 C CHECK FOR THE END OF A COMMENT 108 10 IF (TEXT (IQ, I I , M2) • EQ. IEND) GO TO 9 __109 «1=*1»20 110 H2=M2*20 111 C CHECK THAT THE INPUT IS CORRECT.IF NOT.GIVE ERROR MESSAGE TO A0T30R 112 IF [US.. LE. 100) GS TO 8 113 WRITE (6,91) IQ 111 STOP / 115 9 TEXT (IQ,II,B2) = ID f 116' ""H(IQ. II) =H2 117 66 CONTINUE 11B L?_CONTINU_E ; 119 20 M?OT=IH-1 120 m=^ _121 C KEEP COUNT ON THE !.'0. OF THE INSTRUCTIONAL UNIT BEI3G EXECUTED _ 122 >OOK-" (III) =C<Vv.'NT (IH) *1 123 CALL Q(IN.XOUT) _120 C WRITE RFSPONSE 1ATENCIES ON A FILE 125 WRIT -J'(i, 105) IN, {SBC (IN.K) , K= 1, IRES) 126 105 FOP.rATilH ,I5.10F8.1> 127 _GO TO (50,70),II- ____ '__ "128 70 IF (COUNT (IOUT) .GE.2) 10UT=IH*1 129 50 IF (ICOT.GT. MTOT) CO TO 55 130 IN=ICUT 131 GO TO 13 132 C WRITE STUDENT NAME ON A FILE _133 55 WHITE (1, 102) (NAME (I) , 1-1, 10) _ 131 102 FOKMA1 (IH . 1C..4J 135 C WRITE TAnLE 0? JI1MEER OF RESIOHSSS IH EACH CLASS WITH HEADINGS OH FILE 136 WHITE . 100) 137 100 FORMAT (' QUES',5X. 'CLASSES',IX. "TIMES',51,•HELP•,2X,'CL 1 ', 138 121,'CL2*,2X,*CL3',2X,'CL4',2X,•KOHATCH'//) 139 DO 103 NQ =1,MTOT 100 KCO=COUNT (NQ) 101 KCL=CLASS (NQ) _142 103 BRITE (4,101) NQ.KCL.KCO, ( (S3ES (L.NQ.J) ,J=1.6) ,L=1.KC0) 103 101 FORM AT (4 X, 12.3(91,11), 4 (ix". I1),6X. I 1/(3 IX, 5 (4X, 11) » 61, 11) ) 100 STOP JOS... END 106 C 107 C _10 8 SOBR OJDTIN E Q (NQ, IOUT) 109 C THIS SUBROUTINE CONTROLS THE PROCESSING FOR EACH INSTRUCTIONAL UNIT 150 C UNTIL THAT UNIT IS COMPLETED.CONTROL IS THEN RETURNED TO THE MAIN PROGRAM. _151 C__THIS SUBROUTINE GIVES THE APPROPRIATE COMMENT TO THE STUDENT BASKD ON THE 152 C RESPONSE CLASS INTO WHICH FELL HIS RESPONSE. COUNTERS FOR THE NUMBER OF. '.53 C ERRORS ON EACH INSTRUCTIONAL UNIT ARE KEPT TRACK OF IN THIS SUBROUTINE. ".50 c : 155 INTEGER*2 A NS (80) , KEY (30 , 0 , 8, 10) 156 INTEGER QUES (30, 120) ,QMOD (30,6C) .TEXT (30,1,100) ,STHO (30,6) , _J57 1COUNT (30) ,CLASS (30) 158 DIMENSION K (6) .KTOT (30,4) ,NRES (3,30,6) ,LQ (30) , LM (30) , M (30,1) 159 1,SEC(30,10) _160 COMMON TEXT , QU ES, QM0D ,_STN0, IP , IGX, NRES, COUNT , LO, LH,B 161 1, AN S.C LASS, KTOT, KEY",'~SZC.~, I RES 162 C INITIALIZE COUNTERS TO ZERO 163 " IKES = 0 : " 160 ~K1 = 0 165 K2=0 166 K6=0 167 DO 77 L=3,5 ?68 77 K<L)=0 169 L2=LQ(NQ) 170 C PRESENT THE QUESTION TO THE STUDENT 171 VRITE (6,1) (QUES (HQ, I) ,7=1,L2) 172 C KEEP A TIBER FOR RESPONSE LATENCY \ 173 3 CALL TIME (0) 170 C READ-IN STUDENT KESPONSE.FIRST 10 CHARACTERS ARE READ 175 READ (9,2) (ANS (I) ,1=1,10) 176 CALL TISE(2,C.NLAT) 177 IRES=IBES*1 178 SEC (NQ, IRES) =NLAT/1000. • 179 CALL MATCH(KQ,NEXT) 180 C KEEP A COCH1ZP, FOR THE NUMBER OF ERRORS IN EACH FES ?ONL" E CLASS 181 NRES (COUNT (NQ) , NQ, NEXT) = NRES (COUNT (NQ) , NQ,NEXT)_+1 182 G0~T0 (10,20,30,30.30,98),NEXT " 183 C HKJ.r SECTION 180 10 K1-K 1 + 1 185 IF (K1. EQ. 2) GO TO 99 186 L1 = LH (NQ) _10V VrtlTE ( 6 , J J J Q a 0 D .^fi.if?.) : 188 GO TO 3 1B9 C CORRECT RESPONSE SECTION 190 20 K2=K2 + 1 191 12 IOUT=KQ*1 192 IF(IGX.EQ.O) VRITE(6,21) _193 IF (IGX.EQ. 1) VRITE (6,22) ' 190 If(IG".EQ.J)WRITE (5 ,23) 195 1V (IGX .GE. 3) WRITE (6, 24) 196 IGX=1GX*1 . 168 197 IF (STNO (HQ, NEXT-1) . EQ. 0) GO TO 99 198 IOUT=STNO(NQ,NEXT-1) t99 IP = 2 200 RETURN 201 C WRONG ANSWER SECTIOH _20_2 30 K (NEXT) =K (NEXT) »1 203 204 IF (STNO (NO, NEXT-1) . EQ. 0)GO TO 13 IO0T=STNO(NQ,NEXT-1) 205 IP = 2 206 RETURN 207 13 IO0T=NQ*1 208 IP f K (NEXT) . EO. 2) GO TO 99 209 NN=KEXT-1 210 211 HH=M (NQ.NN) WHITE(6.1) (TEXT (NO, NNf J) .J=1.BH) 212 GO TO 97 213 c BOBATCH SECTIOH 214 98 K6=K6+1 215 IF (K6. EQ. 1) WRITE (6,92) 216 217 IP(K6.EQ.2) WRITE(6 r93) IF (K6. EQ. 3) GO TO 99 218 97 IGX=0 219 GO TO 3 220 99 H1 = B(NQ,1) 221 IOUT=KQ*1 222 223 iRITE(6,1) (TEXT (HQ. I.J) ,J=1.B1) IP=t 221 RETURN 225 22C 1 FORK AT (/, (IH ,20A4)) 2 FORK AT (80 M) 227 21 FORMAT (' OK') 228 22 FORMAT(• GOOD.') 229 23 FORR AT (1 EXCELLENT!•) 230" 24" FORMAT ( • EXCELLENT! KEEP P? THE GOOD WORK. ') " " " " ~ ~ 231 92 FORMAT(• I DON T RECOGNIZE YOUR RESPONSE. TRY AGAIN') 232 93 FORMAT (• BE CAREFUL.I STILL DON«'T READ YOU.ANSWER AGAIH") 233 £110 / 231 c • 235 c 236 SOBROUTIKE MATCH(NQ.NEXT) 237 c SUBROUTINE TO MATCH STUDENT'S RESPONSE TO KEYWORD 238 c THIS SUBROUTINE IS A CHARACTER MATCHING ROUTINE THAT LOOKS FOR A HATCH OF 239 c THE STUUENT'T RESPONSE WITH ANY OF THE KEYVORDS,WITH A BLANK RESPONSE OR 240 c WITH THE HOED HELP. THIS ROUTINE RETURNS T.iT. VALOB OF THE gESPSSSE CLASS 241 c TI'AT THE STUDtiNT HAS HIT WITH HIS RESPONSE TO THE SUBROUTINE Q. 202 c 243 INTEGER *k »KS(30),KEr(30,4,8,10),IB,IFIS,I1,I2,I3,I4 20* INTT^ER QUESOO,120),QSOD(30,60),TEXT'30,4,100),ST.NO(30.6», 245 1 COUNT (30) .CLASS (30) 246 DIMENSION KTOT (30,H) , NRES (3.30,6) ,LQ (?0) ,LH (30) ,H (3C, 4) 2«7 . 1,SEC(30, 10) 218 COBHOH TEXT.QUES.QKOD.STNO, IP,IGX,NRES, COUNT. LQ, LH, H 249 1,ANS,CLASS.KTOT,KEY,SEC,TRES 5>50 DATA IB,IFI.",I1 ,12. 13,I«." •, • S • , • H « , ' E« , ' L • , ' P • / 251 c INITIALIZE COUNTERS 252 KLASS=1 253 K=1 254 1=1 255 LL=1 256 1=1 169 257 C CHECK FOR HELP RESPONSE 258 1 I F (HNS ( I ) . NE. 11) GO TO 3 259 I F (ANS (1*1) . EQ. 12. AND. ANS (1*2) .EQ.I3. AHD. AHS (1*3). EQ. II) GO TO 10 260 3 1=1*1 261 I F J I . L E . 4 0 ) GO TO 1 _26.2 do.. 3 0 _ l = 1., u 0_ 263 C CHECK FOR BLANK RESPONSE 2 6 0 I F (ANS (I ) . HE. IB) GO TO 22 ..265 J O . C O N T I N U E 266 GO TO 10 2 6 7 22 HCHA=0 2 6 8 D 0_2 0 _ I L=J_._10 2 6 9 h'CHA=NCHA»1 2 7 0 C CHECK FOR END OF KEYWORD($) _271 I F ( K E Y (NQ,KLASS,K,IL) . E Q . I F I N J G O TO 21 272 20 CONTINOE 2 7 3 21 HAX=NCHA-1 2 7 1 C CHECK FOR KEYWORD AND f- NSWER M ATCH 2 7 5 2 I F (KEY (NQ.KLASS.K.LL) . EQ. ANS (L) ) GO TO 4 276 21 L=L*1 _277 C _ 0 N L I F I R S T 40 CHARACTERS OF RESPONSE ABE CHECKED 2 7 8 LL=1 2 7 9 I F ( L . GT. 4 0) GO TO 5 _ 2 8 0 I F (ANS ( L - 1 ) . EQ. IB) GQ TO 2 281 GO TO 24 282 1 L=L*1 _283 LL=LJ.*1 _ 284 ' C CHECK I F A L L L E T T E R S IN RES1OHSE HAVE BEEN HATCHED ~ 285 I F ( L L . I . E . H A X ) G O TO 2 _286_ CO TO 12 287 5 L=1 238 L L = 1 2 8 9 K=K»1 ' 2 9 0 C CHECK THAT A L L KEYWORDS I N THAT C L A S S HAVE BEEN" L O O K E D V T 291 I F (K.LE.KTOT (NQ.KLASS) ) GO TO 22 2 9 2 L=1 293 L L = 1 2 9 4 K=1 _ 2 9 5 K L A S S = K L A S S * 1 _ ' 2 9 6 C HAVE A L L RESPONSE C L A S S E S BEEN LOOKED AT 2 9 7 I F ( K L A S S . L E . C L A S S ( N Q ) ) G O TO 22 298 GO_TO 11 2 9 9 C 'NEXT* I S " T H E VALUE OF T H E RESPONSE CLASS WHERE ST^'DEHT'S 3 0 0 C RESPONSE F E L L . RETURN T H I S VALUE TO SUBROUTINE Q •. _ 3 0 1 1_0_HEXT=1 ; 3 0 2 RETURN"" 3 0 3 11 HEXT=6 304 RETURN ; 3 0 5 12 NF.XT = Ki ASS* 1 3 0 6 RETURN 3 0 7 . END ' IG 170 APPENDIX B User's Guide f o r Prelesson Author Program Source L i s t i n g of Program 171 . .Q±. Question i s asked Assistance i s given to student Student i s asked to answer once again — — — 1 I i Question i s j asked s i m i l a r to above * Mark r e f e r s to tne grade assigned to the student f o r a p a r t i c u l a r i n s t r u c t i o n a l u n i t , or item, Q^ . Figure / Prelesson I n s t r u c t i o n a l Logic 172 Description of CAIPRB, Pretest Program This program i s e s s e n t i a l l y the same as the program for the main lesson (CAIPRE). The s t a t i s t i c s section has been removed. Modifications have been made so that the program follows the lo g i c of the pretest - template. Precautions have been taken to insure that the student cannot "sneak" through a question, i n f a c t , he may be caught i n a loop i f he refuses to follow the in s t r u c t i o n s given to him. The l o g i c i s changed so that a l l NOMATCH answere are channelled to the f i r s t wrong answer cl a s s . This means that comments which the student receives i f his answer matches a key word i n the f i r s t wrong answer class w i l l also be given i f h i s answer i s not recognized. The l o g i c used here allows for easier coding of lessons. It i s usue*,l to use only two answer classes, a correct class with a l l acceptable keywords and a wrorig c l a s s with one keyword i s only required since NOMATCH answers go to th i s class anyway. See fi g u r e 1. 173 (LIST CAIPRE 1 C_ DAVID KAOFHAR PHOGRAB TO RON PRETEST LESSOH 2 - c - - - - - " -3 C 4 C BAIN PROGRAB DETERMINES BRANCHING LOGIC 5 INTEGER+2 ANS (80) , KEY (30,it ,8, 10) 6 INTEGER QOES (30, 120) ,QMOD (30,60) .TEXT (30, fl, 100) ,STNO (30,6) , 7 1 COUNT (30) .CLASS (30) 8 DIMENSION NUK (30) , KTOT (30,4) , NRES (3, 30, 6)., NAME (30) ,LQ (30) ,LB (30), 9 1B(30,<!) ,SEC (30, 10) 10 COB HON T E XT_ , Q U E S , Q B 0 D , S T K O , IP, TG X, HR ES, CPU NT. LQ, LB, H 11 1.ANS, CLASS, KTOT'VKE?; r.TOTTSEC, IRES 12 DATA IEND,IB,IGO/'SEND1,• '.•GOTO'/ 13 DATA NUK(1),NUf;(2),NUK(3),NUM(il),NUM(5),NUr(6),NUM(7),NUB(8), 14 1HUn(9),MUH(19),HUB(U),NUH(12),MUB(13),NUB(1l»),SUB(15),HDN(16), i 15 2KUK (17) ,NUM (18) .HUM (19) ,NUB (20)/' 1 ',* 2 •,.' 3 ',• 4 •, : 16 3« 5 ' t«_6 L«J _I_ '. ' 8 ', ' 9 ','10 ' ,J 11 '.'12 '.'13 ', 17 4'1« *15 ','16 ' , ' 1 7 «,«16 '.'19 ','20 '/ 18 DATA HUB (21) , NUH (22) , HUM (23) ,N0fl (2U) ,NU« (25) , HUM (26) , NUK (27) , -' 19 1HUB (28) , NUB (29) ,NUfl (30)/' 21 ','22 ','23 '.'2<4 ','25 ', 20" "2'26 ','27 ','28 ','29 ','30 •/ 21 IGX>=0 ; 2 i DO 41 i^iL,_30 . 23 COUNT (I) =0 ? 24 DO 11 J= 1,6 2_5 DO 12 L=1,3 _ 26 12 HRES (L, I, J) =0 " " ' " " " ~ 27 11 STHO(I,J)=0 28 WRITE (_6._1 3)_ 29 13 F0RH"AT("' PLEASE ENTER YOUR FIRST ANu LAST NAME') 30 READ (9, IH) (NAME (I) ,1=1 , 10) / 31 . 14 FORMAT (10A4) _ •' 32 DO 33 IH=1,31 """ " " " 33 L1=1 34 1.2=20 174 35 13=1 36 t4=20 37 READ(5.1,END=20)IQ 38 1 FORMAT (12) 39 READ(S,2)KC, (KTOT (IQ, J) , J= 1, KC) HO CLASS (IQ) = KC 41 2 FORMAT (912) 42 3 READ(5,4) (QOES(IQ,J) ,_ = L1,L2) 8.3 4_FORMAT (20AU) _ ___ ._ J. _. 44 IF (QUES (IQ.L2) . EQ. IEND) GO TO 5 45 L1=L1»20 46 L2=L2+20 47 IF (L2.LE. 120) GO TO 3 48 WHITE (6,90) IQ 49 90 FORMAT (• TOO MAHI CARDS OR NO_$END IH FIRST PART OF QOES. HO. ', 50 113) 51 STOP 52 5 QUES(IQ.L2)= IB 53 LQ (IQ) =L2 54 11 READ(S,4) (QMOD (IQ, J) , J = L3 ,14) 55 IF (QKOD (IQ.L4) . EQ. IEND) GO TO 12 _ _ _ _ _ _ 56 L3=L3+20 " " " 57 L4=L4*20 58 IFJL4. LE. 60) GO TO 11 59 WRITE (6,9'. j IQ 60 91 FORMAT (' TOO -ANY CARDS OR HO SEND IH A COMMENT OF QUES. HO.', 61 113) 62 STOP ~ 63 12 QKOD(IQ,L4)=IB 64 __tiQJ_ L_ : 65 DO'66 II=1,KC 66 111 = 1 67 112 = 20 68 BEAD(5,6) ( (K EY (IQ^II." 3 . K) . K= 1 , 110) , J= 1 , 8) 69 6 FORMAT(BOA 1) 70 8 READ (_ , 4) (T E X TJ1Q, 11,_J_ , J =H1.H2) 71 IF (TEXT (IQ7i'l, 1) . H2. IGOi GO TC 10 72 DO 60 IK=1,30 / 73 _IF (TEXT (IQ, I I , 2) . EQ. HUM (IK) ) GO TO 30 ' 74 60 CONTINUE 75 WRITE (6.93) I I , IQ 76 93 FORMAT (• IN CORRECT GOTO STATEMENT IH CL ASS ' , 13 . 21. 77 I'QUESTIOH NO. ',13) 78 STOP 79 __STNu (lQ.T.I) =IK • 80 TEXT (IQ, IT, 1) =IB 81 TEXT(IQ,II,2)=IB 82 _?_TO 66 83 10 IF (TEXT (IQ, I I , H2) . EQ. I END) GO TO 9 84 H1 = ri1+20 85 H2=K2«20 _ 86 IF(K2.LE. IOC) GO TO U 87 WRITE (6,9 1) IQ 88 S TOP 89 9 TE.'T(IQ,1-,»2;^IB 90 n(IQ,II)=K2 9 J _ _6_COHTINUE 92 33 CONTINUE "~ _ t .3 20 ETOT=IU-1 94 IH=1 175 95 43 COUHT (IN) =COUNT (IN) »1 96 CALL Q(IN.IOUT) 97 c 98 c 99 c 100 c 101 WHITE (4, 18) IH. (SEC (IN, J) ,J= 1,1 RES) 102 18 FORMAT (1H' ,I5,10F8.1) 103 IF (IOUT.GT. HTOT) GO TO 55 101 IN=IOUT 105 GO TO 4 3 106 55 CONTINUE 107 STOP 108 END 109 SUBHOUTIKE Q(NQ.IOUT) 110 INTEGER*2 A NS (80) .KEY (30,4,8,10) 111 c 112 c 113 INTEGER QUES(30,120),QUOD(30,60).TEXT(30,4,100),STHO(30,6), 114 1COUHT (30) .CLASS (30) 115 c 116 DISSENSION K (6) .KTOT (30, 4) , NRES (3 , 30 , 6) » LQ (30) ,LB (30 ) , H (30,4) 117 1..SEC (30,10) 118 c 119 COHHOH TEXT,Q3ES,QHOD,STNO,IP,IGX,HRES,COUNT,LQ,LH,H 120 1,ANS,CLASS,KTOT,KEY,HTOT,SEC,IRES 121 IRES=0 122 K1 = 0 123 K2 = 0 124 c 125 K6 = 0 126 KAHS=0 127 DO 77 L=3,5 128 c / 129 77 K(L)=0 / 130 L2=LQ (NQ) 131 WRITE(6, 1) (OUES (NQ.I) ,I=1,L2) 132 3 CALL TIHE(O) I " 3 BEAD(9,2) (Atl-> !I) ,1=1,40) 134 CALL TIME (?,0.N T >.T) 135 IRES=IRES*1 136 SEC (NQ, IRES) = NUT/1000. 137 <Zl>l,L CATCH (NQ, NEXT) 138 HBEJ (COUNT (NQ) , HQ, NEXT) =NRES (C'JUNT (NO) , NQ, NEIT) «1 139 GO TC (10,20,30,30,30,98),NEXT liiO ~ i d K1=K1*1 141 I f (CI. EQ. 2) GO TO 99 142 L4 = LH (NQ) 1X3 WBITE(6,1) (QKGD (NQ,I) ,1= 1,L4) GO TO 3 105 20 K2=K2*1 146 12 IOUT=NQ* 1 " " - • - • 117 IF (IGX. iQ.O) WRVTE (6,21) 148 IF (IGX. EQ. 1) KhITE(6,22) 149 IF(IGX. EQ.2)WSITE(6,23) 150 ir <iiX.GE.3) WRITE(6,24) 151 IGX=IGX*'< 15? IF (KANS. GE. 1) GO TO 99 i s : IF (STNO (KQ, NEXT-1) . EQ.O) GO TO 99 154 IOUT=STNO (NQ.NEXT-1) 176 155 RETURN 156 30 K (NEXT) =K (H EXT) *1 157 KAHS=KANS* 1 158 IF(STNO(NQ,NEXT-1).EQ.0)GO TO 13 159 IOUT=STN0 (NQ,MEXT-1) 160 IP = 2 161 RETURN 162 13 I0UT-NQ+1 163 IF(K(NEXT) . EQ. 2)GO TO 999 164 BN=N EXT-1 165 HH = B (NQ, NH) 166 WRITE [6. 1) [TEXT (HQ.HN.J1 .J=1.HB1 167 GO TO 97 168 98 K6=K6*1 169 IF (K6. EQ. 1) WRITE (6.92) 170 IF (K6. EQ. 2)WRITE(6,93) 171 IF (K6. EQ. 3) GO TO 99 172 97 IGX=0 173 GO TO 3 174 99 H1 = B(NQ,1) 175 I0UT=NQ*1 176 BRITE(6,1) (TEXT(NQ, I.J) ,J=1,H1) 177 IP=1 178 RETURN 179 999 IO0T=NQ+1 180 RETURN 181 1 FORHAT (/. (1H .20A4) ) 182 2 FOR BAT(80A 1) 183 21 FORHAT (' OK') Ifo 22 FCPflATC GOOD.') 185 23 FCKMATC EXCELLENT! •) 186 24 FORBAT(• EXCELLENT! KEEP UP THE GOOD WORK.') 187 92 FORMAT (' I DON T RECOGNIZE YOUR RESPONSE. TRY AGAIN *) 188 C 189 93 FORBAT (' BE CAREFUL.I STILL DOR'*T READ YOU.ANSWER AGAIH•) 190 END 191 SUBROUTINE BATCH ( NQ,NEXT) 192 C SUBROUTINE TO MATCH STUDENT'S RESFONSE TO KEYWORD 193 INTEGER»2 AHS(80),K£Y(30,4,8,10),IB,IFIH,I1,I_,I3,I<i 194 INTEGER QUES(30,120) ,QBOD(3C,60) , T- XT (30,4 , 100) , STHO (_0, 6) , 195 1CCCNT (30) .CLASS(30) 196 DIMENSION KTOT (30,H),NRES(3,30,6),LQ(3C),LH(3?),fl(30,4j 197 1 , SEC (30. 10) 198 COMMON TEXT,QUES,CH0D.STN0,IP,IGX,Nfl_2,CC-;.T,LC,Lfl,B 199 1,A»S,CLASS, KTOT,KEY, MTOT , S EC , I R SS 200 DATA IB,IFIN,I1,_2,I3,I«/' ','$'. 'H ",' E ', 'L',':-" / 201 KLJSS=1 202 K='. 203 I.= 1 204 _L=1 205 1=1 206 1 IF (AHS (I) . HE. 11) Co TO 3 207 IF(INS(1*1) .EQ.I7.-ND.&NS (1*2) .EQ.I3. AND. AHS (I*3).EQ. 14) CO TO 10 ?on 3 I = I * i 209 IF(I.LE.40) r,o TO 1 210 DO 30 1=1,40 211 IF (ANf (I) . NE. IE) GO TO 22 212 30 CONTINUE 213 GO TO :0 214 HCHA=0 215 DO 20 1L=1, 10 216 RCHA=NCHA*1 217 IF(KEY(NQ,KLASS,K#IL).EQ.IFIN)GO TO 21 218 20 CONTINUE 219 21 BAX=NCHA-1 220 2_IF_(KEY (NQ.KLA_SSjJK._LIO . EQ. ANS (L) ) GO TO 0 221 20 L=L*1 222 LL=1 223 IF {I. GT.10) GO TO 5 _ _ 221 IF (ANS (L- 1) . EQ. IB) GO TO 2 225 CO TO 20 226 0 L=L*1 227 LL=LL*1 228 IF (LL. LE. HAX) GO TO 2 22? GO_TO__12 . 230 5 L=1 231 LL=1 232 K=K»1 233 IF(K.LE.KTOT(NQ,KLASS))GO TO 22 231 L=1 235 LL=1 236 K=1 237 KLASS=KLASS«1 238 IF (KLASS. LE.CLASS (NQ) ) GO TO 22 239 GO TO 1 i 200 10 NEXT=1 201 RETURN . 202 11 EEXT=3 . 203 RETURN 20 0 12 NEXT =KLASS* 1 205 RETURN 206 END EBP OF FILE $DES BIBLIO , DOBR. tSIG 178 APPENDIX C CAI Prelesson L i s t i n g ^ M tr g -r A 179 £ H £ F T ft 180 °i'H F F T C. 181 182 FBETES • . „ _ ._ 1 1 2 3 3 8 3 3 BI. I'B TOUR PBRSOBAL TDTOR FOR TODAT.LET'S START Bit DOIHO A LESSOH OS. SOBg 4 OF THE THINGSTHAT 'YOU SHOULD"KNOW BEFORE DOING THE MAIN LESSON..... 5 LOOK AT THE GRAPH SHOWN IN SHEET A. 6 WHAT IS THE VALUE OF Y AT THE POINT X=1? ._SE«D 7 BEAD THE VALUE OFF THE GRAPH. SEND 8 10$ 10$ TENS 9 GOTO 3 0 5$ i~5l 20$ T I 2$ 3$ 5$ 81 1 HO. FIHD THE POINT ON THE CURVE WHICH CORRESPONDS TO A VALUE OF X= 1 2 BI DRAWING A VERTICAL LINE UP FROM X=1 UNTIL IT MEETS THE CURVE.TBEH 3 DRAW A HORIZONTAL LINE TO THE LEFT FROM THAT POINT UNTIL IT HEETS THE 4 T-AXIS AT ONE POINT. WHAT IS THE VALUE OF I AT THAT POINT? $EHD 5 10$ LOS 10$ 6 TOU SHOULD USE NUMERALS INSTEAD OP LETTERS TO REPRESENT NUMBERS. 7 TOU ARE CORRECT IF YOU MEANT TO TYPE 10 8 PLEASE ENTER THIS NUMBER AGAIH CORRECTLY. SEHD 9 2 0 2 3 7 ' 1 LOOK AT THE GRAPH IH SHEET A AGAIN. 2 WHAT IS THE VALUE OF i AT THE POIST 1=37 SEND 3 BEAD THE VALUE OFF THE GRAPH. SEHE 4 20$ 20* TWEH$ 5 GOTO 3 6 5$ 15$ 1$ 2$ 3$ «$ OS 7 GOTO 3 8 3 9 2 2 8 0 SUPPOSE THAT TOO ARE TOLD THAT S IS A FUHCTION OF T. _ _ _ T" ' HHAT IS THE VALUE OF S WHEN T=3,IF S AND TARE RELATED BY THE EQOATIOS 2 S=2T*2 ,tRTAD THIS AS TWO TIMES (T SQUARED). SEHD 3 SDBSTITUTE THE VALUE CI- T=3 INTO THE EQUATION. SEHC 183 4 18$ EIGHTEEN $ 5 GOTO 5 6 2$ MS 1 $ 12$ 8$ 9$ 16$ 29$ 7 HO. TOO SHOULD SUBSTITUTE THE VALUE OF T,WHICH IS 3.IHTO THE EQOATIOH 8 AHD OBTAIH S= 2T*2=2 (3) *2 = 2 (9) = ? 3 WMAT IS TBE_¥ALUE OF S HHEH T=3? S-Ht 0 4 1 2 2 7 2 HOW_SUPPOSE_THAT_WE ABE GIVES THE EQUATIOH S.= 3T*2 .HEAD AS 3 TIRES 3 I SQUARED. 4 WHAT IS THE VALUE OF S WHEN T=2? $EHD 5 SUBSTITUTE THE VALUE OF T=2 INTO THE EQUATIOH. SEND 6 12$ T- ELS 7 GOTO 5 8 3$ 6$ 9$ 15$ 2S 27$ 18$ 9 GOTO 5 0 5 J 2 3 4 2 SUPPOSE THAT A CAB IS TRAVELLING AL08G A HIGHWAY AT A SPEED OF 6G BILES 3 PEB HOUB. * HOW FAR WILL THE CAR TRAVEL IH 2 HOURS? $EHD 5 WHAT WILL T H E DISTAHCE BE IF THB SPEED IS 60 BILES/BOOB AHD THE~ TIBE 6 IS 2 HOURS? $EH0 7 _H>$ ED AHD TWSDRED T-ENS 8 GOTO 7 9 60S 2S 30S 12$ 0 BO. BEHEHBER THAT DISTAHCE .SPEED AHD TIHF ARE BELATED BI THE EQUATIOH _ _ T D I S T A 3 C E = (SPEED) ( T I B E ) 2 BE WILL WRITE THIS AS S=VT. THEREFORE, S= (601 (2)=? 3 WHAT IS THE DISTANCE IS TWO HOURS? $EHD 4 6 5 2 2 4 6 SUPPOSE ANOTHER CAR IS TRAVELLING ALOHG A HIGHWAY AT A SPEED OF 7 50~BILES/HOUR. HOW'FAR WILL THE CAR TRAVEL IH 3 HOURS? " ' " "' " ~*?HD 8 WHAT WILL T H E DISTAHCE BE IF THE SPEED IS 50 BILES/HOUB AHD THE TIB. 9 IS 3 HOURS? SEHD 0 150$ BED AHD FS 1 GOTO 7 2 50 S 3$ 100$ 200$ .* 3 GOTO 7 4 7 5 2 3 6 ; 6 A CAR P>._3ES T H E V>0 B I L E POST OH A HIGHWAY AT 12 NOOS.'THREE HOUBS LATER, 7 THE CAR PASSES THE 250 K I L E POST.WE USE T H L SYMBOL D_L (SI TO RECORD 8 THE CHANGE III DIS_.\NCE,THAT IS, DEL (S) = (FINiL DISTAHCE) (INITIAL DISTAHCE) 9 W H A T ' I S D E L (S) IN T H I S C A S E ? SEND 0 SUBTRACT THE DISTANCE AT THE 100 BILE POST FROB THE DISTAHCE 3 HOURS LATER SEND 1 15_$ ON- FIFTY$RS0 AHD FS 2 GOTO 9 3 ~$ 100$ 2505 50$ 350$ 3$ HO. DEL (S) = (FlHAL DISTAHCE) - jIHITIAt DISTAHCE) . S " " =250-100=? _ " ' " ~ 6 WHAT IS DEL(S) IH THIS CASE? $EHD 7 e ; •• 8 2 3 4 9 SUPPOSE THAT THE _ABE C»R THAT PASSED THE 1?0 BILE POST AT HOOH PASSES 0 THE 400 BLIE : OST S I I HOURS LATER. 1 " ""WHAT IS !)EL(S? _» THIS C A S E ? ' " • — - ;-.J EKJ) 2 DEL (S) = (FIHAL DISTANCE) - (INITIAL DISTAHCE). 3 WHAT IS PEL(S) IH THIS CASE? SEND 184 « 300$ T U B E S HUHSTHREEHUHS 5 GOTO 9 6 100$ 400$ 500$ 4S • 7 8 GOTO 9 9 9 2 2 2 0 SOPPOSE THAT A CAB TRAVELS FBOH MONTREAL TO TORONTO,A DISTANCE OF 1 3 5 0 BILES,AND THE DRIVER STOPS SEVERAL TIMES FOR FOOD AND GAS. 2 WHAT SPEED MUST THE CAR AVERAGE IN ORDER TO MAKE THE TRIP I N 7 HO0BS7 SEND 3 T H E AVERAGE SPEED IS THE SPEED WHICH THE CAR WOULD HAVE TO TRA V E L IN « OBDER TO TRAVEL 350 MILES IN SEVEN HOURS WITHOUT STOPPING. 5 S I N C E DISTANCE=.SPEED) (TIME) .WHAT IS THIS AVERAGE SPEED? SEND 6 50$ F I F T Y S 7 GOTO11 e 3 5 0 $ 7 $ 9 BO. T H E CAB STOPPED SEVERAL TIMES AND DID SOT KEEP A CONSTANT S P E E D . 0 T H E AVERAGE SPEED IS THE SPEED AT WHICH T H E CAB WOULD HAVE TO THAVEL 1 IH ORDER TO TRAVEL 350 MILES IN 7 HOURS. 2 AVERAGE SPEED=DISTANCE/TIHE=350/7=? 3 •BAT IS T H E AVERAGE SPEED? SEND 4 10 5 2 2 2 6 DURING ITS' JOURNEY.THE SAME CAR PASSED KINGSTON,A D I S T A N C E OF 7 160 BILES FROM MONTREAL,AFTER 4 HOURS. 8 WHAT SPEED DID THE CAR AVERAGE? SEND 9 D I S T A N C E = (SPEED) (TIME) .WUAT IS THE AVE MAGE SP E E D ? SEHD 0 40$ F O B T Y $ 1 GOTO11 2 160 $ 4S 3 COT011 4 11 5 2 2 4 6 SOPPOSE THAT ANOTHER CAR TRAVELLING FROM MONTREAL TO TOROHTO(350 MILES) 7 ROVES AT 70 MILES/HOUR FOR THE FIRST 3 HOURS AND AT 60 H/HR FOR THE 8 BEXT 4 HOURS. WHAT IS THE SPEED OF THIS CAR EXACTLY 50 MINUTES 9 AFTER LEAVING MONTREAL? THE SPEED AT A PARTICULAR TIME IS CALLED 0 T H E INSTANTANEOUS SPEED. SEHD / t T H E CAR TRAVcLS AT 70 MIL2S/H0UR FOR THE F I 2 S T 3 HOURS. HOW FAST l l I S I T MOVING EXACTLY 50 MINUTES AFTER LEAVING MONTREAL? SEHD 3 7 0 $ SEVENTY* 4 GOT013 .5 6 0 $ 501 3$ 4$ 6 BO. THE SPEED CF THE CAR EXACTLY 50 MINUTE-* AFTER LEAVING MONTREAL 7 I S CALLED THE INSTANTANEOUS SPEED AT THAT POINT. THE CAR IS MOVING 8 AT 7 0 :V:!B AT THIS PARTICULAR TIME. WHAT IS THE INSTANTANEOUS SPEED? SEHD 9 12 0 2 2 C 1 WHAT IS T H E INSTANTANEOUS SPEED OF T H E CAR EXACTLY 4 HOURS AFTER L E A V I N G 2 SOSTREAL? SEND .3 T H E CAR IS TRAVELLING AT 61? R/KR DURING T H E LAST FOUR HOURS CF T H E 4 T R I P . HOW FAST IS IT GOING 4 HOJRS AFTER LEAVING MONTREAL? SEHD & 60$ SIXTYS 6 G0TO13 7 4$ 7 0 S 5 0 $ 3$ 8 GOTO13 9 13 0 2 2 8 "•1 LET'S TAKE A SHORT bhEAK FROM THE LESSON.I'D LIKE i6 KNOW HOW TOO FEEL... •2 . . B I G H T NOW. WHICH OF THE CATEGORIES BELOW DESCRIBE BEST IODB REACTION..NOW. •3 TO T H E STATEMENT I AH TENSE. 185 .» A MOT IT ILL C HODEHATELI SO •5 B SOMEWHAT D VERI BOCH SO •6 PLEASE AHSWER A,B,C,OR D. SEBD .7 ARSWER A,B.C.OR D TO DESCRIBE TOOB BEACTIOH BIGHT BOB TO THE STATEHEHT... •8 I AH TENSE. SEBD •9 BOS IESS •0 1 GOTO13 AS BS CS DS SOTS SOBES BODERS VERTS 2 GOTO10 3 in « 2 2 8 5 BHICH CATEGORY BELOW..A.B.C.OB D..BEST DESCRIBES.TOOB BEACTIOH TO THE STATEBENT. 6 I FEEL AT EASE. 7 A HOT AT ALL C BODERATELT SO 8 B SOMEWHAT D VERT KOCH SO AHSWER 1.B.C.OB 0. SEHD 9 PLEASE A,B.C.08 D. SEHD 0 BOS TESS 1 GOT01U 2 AS BS CS DS IOTS SOBES EODEBS VESTS 3 GOTO15 « 15 S 2 2 8 6 I AH RELAXED. 7 _ BOT AT ALL C BODERATELT SO 8 B SOBEWHAT D VERT BOCH SO 9 PLEASE AHSWER A,B.C.OB D. SEBD 0 AHF-EB A,B.C.OR D...TO THE STATEMENT.... I AH BELAXED. SEHD •1 BOS IESS 2 3 G0TO15 AS BS CS DS ROTS SOBES rlODEBS VESTS « GOT016 5 16 6 2 2 8 7 I FEEL CALB. 8 A BOT AT ALL C BODERATELT SO 9 B SOMEWHAT D VERY BOCH SO 0 ANSWER A,B.C. OR D. SEHD 1 AHSWER A,B.C.OR D TO THE STATEBENT... I FEEL CALB. SEHD 2 BOS TESS 3 « GOT016 AS BS CS DS IOTS SO-ES BODEBS TESTS 5 GOT017 6 17 7 2 2 _ 8 I AM JITTERY. 9 A ROT AT A"LL C BODERATELT SO 0 B SOBEWHAT D VERY BOCH SO 1 ANSWER A.B.C.OR D. SEBD 2 AHSWEB A.B.C.OR D TO THE STATEMENT...I AH JITTERY. SEHD 3 TESS HOS 0 GOT017 5 AS BS CS DS HOTS SOBES HODS TESTS 6 G0TO18 7 18 8 2 6 8 9 -OW,LET'S GET BACK TO THE LESSON... 0 LOOK AT THE GRAlMI OF S VS. T SHOWN IN SHEET B. 1 WHAT IS THE SLOPE OF THE LINE INDICATED O N T I E GfAPH? SEBD 2 SLOPE IS JUST THfc CHANGE IH S DIVIDED BY THE CHANGE IH T. 3 OSIHG THE NOTATION THAT WE BIFINED EASi LIER , SLOP E = DEL (S)/DEL (T) . 186 4 - PI C K TWO POINTS ON T H E L I N E AND C A L C U L A T E THE SLOPE. $END 5 4 $ FOUR* 2 0 / 5 $ 4 0 / 1 0 $ 60/15$ 8 0 / 2 0 $ 6 . GOTO20 7 20$ 4 0 $ 6 0 $ 80$ 5$ 10$ 15$ 20$ 8 VO. THE SLOPE IS DEFINED AS SLOPE=(CHANGE IN D I S T A N C E ) / ( C H A R G E I I T I H E ) 9 = DEL (S) /DEL (T) . 0 P I C K TWO POINTS ON T H E GRAPH.SAY T=5 AND T= 10. THEN , 1 SLOP E= ( 4 0 - 2 0 ) / ( 1 0 - 5 ) =20/5 = ? 2 WHAT IS THE S L O P E ? $ESD 3 19 « 2 6 8 5 LOOK AT THE GRAPH OF S VS.T SHOWN IR SH E E T C. 6 WHAT I S THE S L O P E OF T H E L I N E I N D I C A T E D ON THE GRAPH? SEHD 7 SLOPE I S J U S T T H E CHANGE IN S DIVIDED BY T H E CHANGE IN T. 8 OSINGOUR NOTATION,SLOPE = DEL ( S ) / D E L (T) BETWEEN TWO POINTS CN THE L I K E . 9 C A L C U L A T E T H I S S L O P E . SEBD 0 5$ F I V E S 10/2$ 20/4 $ 30/6$ 40/8$ 1 GOTO20 2 10$ 20$ 30$ 40$ 2$ 4$ 6$ 8 $ 3 GOTO20 4 20 S 2 6 2 6 LET'S REVIEW SOME B A S I C ALGEBRA. SAI THAT YOU ABE GIV2N T H E E X P R E S S I O N 7 3 (1 * 1 ) *2-3 (1) *2 / (1 »X) -1 8 WHAT IS T H E S I M P L I F I E D FORM OF T H I S EXHKESSION? SESD 9 EXPAND 3 ( 1 * X ) * 2 I H THE NUMERATOR AND THEN M P L I F Y THE HOMERATOB. 0 THEH S I M P L I F Y T H E DENOMINATOR. F I N A L L Y , C A H C E L X FROM BOTH KOMERATOS 1 ABD DENOMINATOR. SEND 2 6*3X$ 6 • 3X$ 3X*6S 3X • 6$ 3 ( 1 * 2 ) $ 3(2*1)$ 3 GOT022 4 2$ /$ 5 BO. THE NUMERATOR BECOMES 3 (1 *X) *2-3 (1) *2= 3 (1 *2X*T»2>-3 6 = 6X*3X»2 = X (6*3X) 7 T H E DENOMINATOR BECOMES ( 1 * X ) - 1 = X 8 THEN, T H E EXPR E S S I O N EQUALS X ( 6 * 3 X ) / X = ? 9 WHAT IS THE F I N A L FORM OF T H E E X P R E S S I O N ? SEHD 0 21 1 2 5 2 •2 HOW IOU ABE GI V E N AH EX P R E S S I O N ( 2 * X ) » 2 - 2 » 2 / ( 2 * X ) - 2 •3 WHAT I S THE S I M P L I F I E D FORM OF T H I S EXPRESSION.' SEBD •4 PROCEED EXACTLY AS ABOVE. SEBD -5 4*1$ X*4$ 4 * XS X • 4$ 4 *X$ •6 GOT022 ' .7 2$ / * •8 GOT022 •9 22 0 2 5 2 .1 LET'S DC TflE SAME EXAMPLE AS BEFORE.BUT HOW KE' L L USE THE S I B 3 0 L S THAT •2 H I L L BE USED IN THE MAIN LESSON. .3 G I V E N T H E E X P R E S S I O N 3 (1 • OEL (T) ) • '2-3 (1) »2 / ( 1 * D E L ( T ) ) - 1 .4 WHAT I S T H E S I M P L I F I E D FORM OF T H I S E X P R E S S I O N ? SEND >i LOOK AT THE PREVIOUS PROBLEM. THE PROCEDURE I S THE SAME AS iEFOHE .6 E X C E P T THAT VE NOW HAVE "SED DEL (T) IN PL A C E OF X. SEHD ;7 6*3DEL* 3DEL(T)*6*6 • D E L I 3 ( 2 * D E L $ 3 ( D E L ( T ) * $ .8 GOT024 .9 1 $ /$ •0 HO. T H E SOLUTION I S T H E SAME AS BEFORE . M THE NUMERATOR BECOMES 3 (1 • DEL (T) ) "2-3 ( 1) * 2= J ("ODE! (*') •D'vL (T) • 2 ) - 3 '2 •=3*6DEL (T) • 3 DEL (T) *2-3 = bDEL (T) ODEL (T) »2=DEL T) (6*3DEL (Ti ) '3 THE DENOMINATOR BECOMES (1 • DEL (T) ) - 1 = DZL (T J 187 '« THEN T H E E X P R E S S I O N BSCORES D E I (T) (6 + 3DEL ( T ) ) / D E L (T) = ??? SEHD '5 23 ' 6 2 3 2 '7 ROW,TOO ABE G I V E N AS EXP R E S S I O N '6 (2*DEL (T) ) »2-2*2 / (2*DEL (T) )-2 '_9_ BHAT. iS_T..HE—SI?PLIFIED__F0RFL OF THIS__E_.P.BESSIOB7 _S__SD 10 PBOCEED EXACTLY AS BEFORE. °° SEBD 11 « « D E L S II • DELS D E L ( T ) » V S _ GOTO2U _ ; 13 2$ / % 1% G0T021 15 2B 16 2 2 2 ;7 TOO S E E N TO UNDERSTAND T H E CONCEPTS HEEDED TO TAKE T H E HAIB L E S S O B . ;8 TOO MAT TAKE_A SHORT BREAK OR YOU CAH START T H E HAIH LESSOR •9 B I G H T AH AY. DO YOU WANT TO TA K E A BREAK? . SEHD 0 AHSWER HO OB T E S . SEHD '1 BOS NO.S ; • 2 TBEH T T P E . . . S S O U R C E LESSOR SEHD 3 T E S * OKS » T H E - TAKE t SHORT BEST AHD WHEH TOU'BE B E A D Y _ T T P E . . . S S O U R C E L E S S O B SEHD r ntE 188 APPENDIX D CAI Main Lesson L i s t i n g (versions T , T 2, T ) 189 s t 190 4 5 T H _ £ T 2. £»n« Us) io- ! .— • ' -w _ • * ? . / / / / / * / i r t 191 SHEET 3 C Aon. J) S H E E T 4-6fMPH O F D I . - M A J - . _ - t/g. 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SEHD 6 HEHEBBER THAT DISTANCE 3 (S PEED) (TIME).WRITE AN EQUATION FOR S IN TERMS OF T. SEBD 7 10TS 10*T$ 10 TIMES $10XT$ S=10TS S=T10J T10S T»10S 8 THE EQOATION WHICH DESCRIBES TEE MOTION 0P_THE„'OA'T_JS_ S=10T. 9 THIS IS SIB PL I THE FAMILIAR DfSTANCE= "(SPEED)'(TIME") EQUATION, WITH 0 SPEED V BEING CONSTANT \7 10 MILES/HOU".. fEBD . 1 10-3/T S T=10/S $ 2 ' THIS IS THE EQUATION RELATING DISTANCE, SPEED, AND TIME. NOW RF» '.RANGE 3 IT TO GET DISTANCE S ALONE OH THE LEFT HAND SIDE. SEBD 4 10 S T S 5 THIS IS FART OF THE ANSWER. REMEMBER THAT DISTAPC3= (StTED) (TIME) AND 6 TOO HAVE ONLY LOO.-.ED AT ONE OF THESE QUA1IITIES. TRY AGAIN. SBSD 7 S S DISS 10SS 8 THE DISTAfCE ^ GIVEN BY S.NOW,WHAT IS TUZ EQUATION IC? S 9 IN TERMS OF T7 SEBD 0 2 195 1 M e u 2 LOOK AT SHEET 1 THAT WAS GIVEN TO TOU.THE GRAPH INDICATES 3 HOB FAR THIS BOAT IS FROM THE COCK AT ANY TIHE. 4 WHAT IS THE SLOPE OF THIS STRAIGHT LINE REPRESENTING THE BOAT'S 5 BOTIOH IH A TIME INTERVAL T=2 TO T=6 HOURS? SEHD 6. TOU _NEED__Tp__OBTAIN THE CHANGE__IN DISTANCE AND THE CHANGE IH TIHE BETWEEN 7 THESE TWO POINTS IN ORDER TO C A L C U L A T E THE S L O P E . GO AHEAD... SEHD 8 10S 1 0 / 4 $ 0 - 2 0 ) / ( 6 - $ 6 0 - 2 0 / 6 - 2 $ SI T H E SLOPE IS THE CHANGE IN DISTANCE DIVIDED BY THE CHANGE I H T I H E . 0 WE WRITE THIS AS S LOPE= DEL ( S ) / D E L (T) WHICH IS EQUAL TO 1 < 6 0 - 2 0 ) / ( 6 - 2 ) = 1 0 HILES/HOUR. SEBD 2 1_1_Q$ .__1_$ 4/40$ 2/20$ 6/60J$_ e/_B0S 6 - 2 / 6 0 - 2 0 S 6 - 2 ) / (60 - S 3 ALMOST RIGHT.YOU'VE CALCULATED THE CHANGE IN DISTANCE AND THE CHARGE * IB T I H E CORRECTLY BUT YOU'VE DIVIDED THEM INCORRECTLY. S_ T H E SLOPE IS THE CHANGE IN DIS T A N C E DIVIDED BY CHANGE IH T I B E . T B T AGAIH. SEBD 6 60S 6 0 / 6 $ 2 0$ 20 / 2 $ 7 TOO'BE ON T H E RIGHT TRACK SINCE YOU'RE DIVIDING DISTANCE BY TIHE, 8 BUT YOU'RE ONLY LOOKING AT ONE END OF THE INTERVAL FROM T=2 TO T = 6. 9 LOOK AT BOTH ENDS OF THE" INTERVAL AND D 1 V I D E T H E CHANGE IH DISTANCE 0 8 1 THE CHANGE IH TIHE. SZIE 1 8 0 / 8 $ S/T$ 2 TOO'RE PARTLT RIGHT SINCE YOU'RE DIVIDING DISTANCE BY TIHE,BUT """ 3 TOO'BE ONLY LOOKING AT ONE POINT OH THE LIRE.LOOK AT THE CHiSGE IB 4 DISTANCE AND DIVIDE BT THE CHANGE IN TIHE FROH T = 2 TO T=6. SEHC S 3 6 4 5 8 4 2 7 WHAT IS THE SLOPE OF THE LINE IN THE TIHE INTERVAL T=1 TO T « 8 HOURS? SEBD 8 TOU HEED TO OBTAIN THE CHANGE It." DISTANCE AND T H E CHANGE I H TIHE BETWEEN" 9 THESE TWO POINTS IN ORDER TO CA L C U L A T E THE 3L O P E . GO AHEAD... SEHD 0 10$ 1 0 / " $ 0-0 0) / ( 8 - $ 8 0 - 4 0 / 8 - 4 SCONSTA NT$ 1 T H E SLOPE IS CONSTANT AT 10 HILES/l-ODR. THIS IS" ALWAYS THE CASE WHEN 2 T H E GRAPH OP THE MOTION IS A STRAIGHT LINE.THE GRAPH IS JUST A . 3 PICTURE OF THE EQUATION OF MOTION S=10T,AND_SO THE SLOPE OFTHE LIBE_ 4 IS" THE SPEED OF THE BOAT. ' " " S E H D $ 1/10$ . 1 $ 4 / 4 0 J 2/20$ 6/60$ 8/80$ 8 - 4 / 8 0 - 1 0 $ 8 - 4 ) / (80-$ _ ALMOST RIGHT.YOU'VE CALCULATED THE CHANGE I N DISTANCE AND THE CHANGE 7 IB TIME CORRECTLY , EUT YOU 'VE~ DI VICED" T H EM - INCORRECTLY. 8 THE SLOPE IS THE (CHANGE IN DISTANCE)/(CHANGE IN TIME). TBI AGAIB. $EBD 9 80$ 80 / 8 $ 10$ 4$ _ _ 0 lOU'RE ON THE RIGHT TRACK S I N C E YOU'RE DIVIDING DISTANCE BY TIME 1 BOT YOU'VE ONLY LOOKED AT ONE END OF THE INTERVAL FROM T=4 TO T=8. 2 LOOK AT BOTHENDp OF THE INTERVAL AND DIVIDE THE CHANGE IS DISTANCE •3 BT'T'HE CHANGE I * TIM E . $EBC •4 6 0 / 6 $ 2 0 / 2 $ 5 TOO'RE PARTLT RIGHT SIKCE YOU'RE DIVIDING DISTANCE BY TIME,BUT YOU'VE « ONLY LOOKED AT ONE TOINT ON THE LINE.LOOK AT CHANGE IH"DISIANCE AHD • 7 D I V I D E EI i'HE CHANGE IN TIME FROH T=4 TO 1=8. SEHC 8 1 9 4 1 1 3 2 0 SHEET 2 ILLUSTRATES THE HOTION OF A SECOND SOAT •1 COMPUTE THE AVERAGE SfE::D CF THtBOAT IN THE INTERVAL FROH T = 2 TO T=6 BBS. SEHC '2 PROCEED EXACTLY AS BEFORE. " " " — ' " - $ M i *3 2S 8/4S (9 - 1 ) / ( 6 - S 9 - 1 / 6 - 2 $ _4 T H E AVERAGE SrESr- I S SIMPLY THE CHANGE IH DISTANCE DIVIDED t i T ti £ . '5 CHANGE" IK TIME. T H I S i s ' G I V E N BY •6 V- DEL (S» /DEL ,T) = ( 9 - 1 ) / (6-2) = 8/1 = 2 MILES'HCUR SEHl |7 9/6$ 1/2 f 1.5$ .5$ •8 ALMOST HlGhT S I N C E YG'J HAVE DIVIDED" DI S T A N C E BY TIML EUT'YOtT" HAVE' •9 CilLY CONSIDERED OKI; t'»D OF Th2 INVEHVAI FROM T = 2 TO T=6. «0 C A L C U L A T E THE CHANGE I H DI S T A N C E AND D I V I D E BY T H E CHANGE I H T I K E 196 11 TO GET AVERAGE SPEED. SEN. 12 9$ I S 6$ 13 HO. THIS IS THE VALUE Of A COORDIHATE AT OHE EHD OF T H E INTERVAL. m T H E AVERAGE SPEED IS GIVEN BY CHANGE IH DISTAHCE, 15 DEL (S) , DIVIDED BY CHANGE IH TIME, DEL (T) .FROM P TO Q.CALCULATE Ijj SHIS FR0_K_THE_ JJHAP_H. SENI 17 8$ 4 S 8 HBOHG. YOU'VE CALCULATED THE CHANGE OH OHLY OHE AXIS OF T H E GRAPH.THE i _ AVERAGE SPEED 1 3 THE CHANGE IH DISTANCE DIVIDED BY THE CHARGE IH TIME 0 FROM T=2 TO T « 6 . CALCULATE THIS FBOM THE GRAPH... SEHt 1 5 2 o ii a 3 2 • 3 LOOK AT SHEET 3,WHICH IS JUST THE GRAPH I H SHEET 2 KITH TH.EE POIHTS 4 P,Q, AHD R IHDICATED OH IT. 5 WHAT I S THE SLOPE OF THE LINE SEGMENT JOIHIHG THE POIHTS P AHD Q? SENG 6 TOO NEED TO KNOW THE CHANGE IH DISTANCE AND THE CHANGE IN TIHE BETWEEH 7 P AHD C I " ORDER TO CALCULATE THE SLOPE OF THE LINE SEGHEHT. GO AHEAD... SEHC 8 2$ 8_*$ ) /__{(>_I9-1/6-2$ 9 T H E SLOPE OP THE LINE SEGHEHT JOINING P AHD Q IS GIVEN BY CHANGE IN 0 DISTAHCE DIVIDED BY CHARGE IN TIME, WHICH IS DEL (S)/DEL (T) =8/1 = 2 H/HR. 1 THIS IS_THE SAME AS TF1E AVERAGE SPEED OF THE BOAT BETWEEH P AHD Q. _ SEND 2 9/6$ " 1/2S 1.5J .5$ 3 ALMOST RIGHT SINCE YOU'VE DIVIDED DISTANCE BY TIME.BUT YOU HAVE OHLI jl __HSI0ERED_ONE__ENp_OF_rHE_ INTERVAL_FROK_ P_TO____CALCDLATJ_ TJ_E_ CHAKGE 5 IN DISTAHCE AHD DIVIDE BY THE CHANGE IN TIME FROM P TO Q TO G E T ' T H E SLOPE. SEHD 6 9S 1S 6$ 7 RO. THIS IS THE VALUE OF A COORDINATE AT OVt END OF THE INTERVAL. 8 THE SLOPE OF THE LINE JOINING F AND Q IS GIVEN BY CHANGE IH DISTAXCE, 9 DEL (S) .DIVIDED BY CHANGE IH TIME, DEL (T) .FROM P TO ..CALCULATE O THIS FROM THE GRAPH. SEHC 1 8 $ US 2 HO. TOO'VE CALCULATED THE CHARGE OH OHLI ONE A.IS OF THE GRAPH.THE SLOPE 3 OF THE LINE JOINING P AHD Q IS GIVEN BY CHANGE IN DISTANCE,DEL(S), _ 1 DIVIDED BT CHARGE IN TIME, DEL (T) , FROM P TO Q. CALCULATE THIS FROH 5 T E E GRAPH... SEHD 6 6 ; ; 7 4 8 5 2 1 8 L E T ' S TAKE A SMALLER SIZE INTERVAL THAH BEFORE OH THE GRAPH I H SHEET 3. 9 WHAT IS THE AVERAGE SPEED OF THE BOAT IH T _ £ INTERVAL T«2 TO T=4 HOURS? SEND 0 PROCEED EXACTLY AS BEFORE. " " $EHD 1 1.5$ 3/2$ 1*1/2$ 11/2$ 1 1/2$ 4-1/4-2$ (4-1)/(4-$H1L$ 2 TH_E_AVER-GE_SPEED IS jr=DEL (S ) / J 3 E L (T) = (4- 1) / (4-2) = 1. 5 H/HR. 3 THE SLOPE OP THE LINE SEGMENT JOINING P AND 5 IS Trt THE SAME AS 4 T H E AVERAGE ?PEED OVER THIS INTERVAL. 5 NOTICE THAT THE AVERAGE SPEED IS NO LONGER COHSTAHT! I 6 IF THE GRAt-_ IS HOT A STRAIGHT LINE.THE SLOi.'E IS NO LOHGEP COHSTAHT. SEND 7 I S V 4 $ 1/2$ .5$ 1H1S 8 ALMOST RIGHT SIKCF. TOU HAVE DTVIDED DISTANCJBY TIM E. BUT_IOJ_ _AVE 9 OHLY CONSIDEBEJ ONE END OF THE "INTERVAL. " CALCULATE" THE CHANGE IN 0 DISTAHCE AHD DIVIDE BT THE CHASGE IN TIKE FtiOB T = 2 TO T=H. SEBD i 3$ 2$ 2 HROHG. YOU'VE CALCULATED THE ChAHGE ON OHLY CHE AXIS OP TKE GRAPH. 3 THE SLOPE OF TH? LI!_ JOINING P AND R IS GIVEN BY CHANGE "J DISTANCE, 4 DEL (S) ,DIVIDED_3Y CHANGE I!_ TIME. DEL (T) FROM r__0_R. • 5 CALCULATE THIS FROft THE GRAPH... " - SEHD 6 4$ 7 BO. THIS tS ?BI VALUE OF A COORDINATE AT ONE EN ^ O" THE INTERVAL.THE SLOPE 8 OF THE LINE JOINING P AND S IS GIVEN BY CHANGK IN DISTAHC?,bEL (S>, 9 OIVIDED BY CtiANGE IN TIME, DEL (T) , FROM P TO R. CALCULATE ThI3 FROM 0 THE GRAPH... SEHD 197 t 7 2 4 3 2 3 5 3 WHAT HAPPENS TO THE SPEED AS WE HAKE THE TIHE INTERVAL SBALLER AHD » SMALLER.LET'S CONTINUE WHAT WE DID WITH SHEET 3 BT CONSTRUCTING A TABLE. 5 LENGTH OF INTERVAL(HRS)•* 4 2 1 .5 . 2 5 0 6 SLOPE__OF__LINE (BILES/HR).** 2 1.5 1.25 1. 12 1.06 ? 7 AVERAGE SPEED (BILES/HR) *• 2 1.5 1.25 1. 12 1.06 ? 8 WHAT IS THE L I M I T OF AVERAGE SPEED AS THE INTERVAL SHRINKS TO SIZE 0 7 . SEND J| STUDT_THE TABLE CAREFULLY AND NOTE WHAT HAPPENS TO SLOPE AND AVERAGE 0 SPEED AS THE LENGTH OF THE TIHE INTERVAL SHRINKS FROH DEL(T)=4,DOWN TO 0. SEND 1 1S 1.00$ m i s 2 THE LIHI_T__OF_AVERAGE_.SP.EED _AS _T_HE_INTERVAL APP_R0ACHES_ZER0_IS 1 H/HR. ' 3 HE SEE THAT AS THE~ INTERVAL KEEPS GETTING SMALLER, I T SHRINKS TO A 4 POIHT AT T=2,AS SHOWS OS SHEET 3.THE SPEED AT T=2 IS JUST THE SLOPE OF S> THE LIRE L WHICH TOUCHES THE CURVE AT ONLT ONE POIHT P. THIS LINE IS_ _ 6 CALLED THE TANGENT TO THE CURVE AT POIHT P. SEBD 7 2S TWOS 8 BO. THIS IS THE 7 A L U E OF AVERAGE SPEED FROM T = 2 TO T°4. 9 LENGTH »* .5 .25 . 1 5 . 1 0 0 0 SLOPE • » 1.12 1.06 1.04 1.02 ? J| AVERAGE SPEED** 1.12 1.06 1.04 1.02 7 _ 2 WHAT IS THE LIBIT OF AVERAGE SPEED AS THE INTERVAL SHBIRKS TO SIZE 0 ? ~ SEBD 3 OS OS ZEROS «| HO.THE LEHGTH OF THE IBTERVAL TENDS TO 0 BUT AVERAGE SPEED DOES ROT. 5 LEHGTfl ** .5 .25 .15 . 1 0 0 6 SLOPS *• 1.12 1.06 1.04 1.02 7 7 AVER/iGE SPEED** 1.12 1.06 1.04 1.02 7 8 BHAT IS* THE LIBIT OF AVERAGE SPEED AS THE INTERVAL SHBIKKS TO ~SIZ_ 0 7 ' SEBD 9 1.03$ 1.01$ DEFIHEDS HO LIHITS HOLIBITS 0 HO.J-ET'S TAKE A CLOSER LOOK AT THE INTERVAL AS WE SHRINK IT TO SIZE 0. 1 LENGTH *• .5 .25 .15 . 1 0 0 2 SLOPE *• 1.12 1.06 1.04 1.02 7 3. AVERAGE SPEED** _ 1. 12 1.06 1.04 1.02 7 _ 4 WHAT IS"THE LIBIT OF AVERAGE SPEED AS THE INTERVAL SHRI-kS TO SIZE 0 ? SEND 5 8 6 4 4 2 3 5 7 WHAT IS THE INSTANTAHEOUS SPEED OF THE BOAT AT T H E TIHE,T=2 HOURS? SEHD 8 THIS IS THE SPEED AT THE INSTANT OF TIHE T=2 HRS. STU.I THE TABLE ABOVE 9 *__J____ *_ S H E E T 3._ THEN TRY TO ANSWER... _1_"L_ o" 1 $ i.oos 'OSES'" I B I S 1 • THE SPEED AT TI!.S T=2 HOURS IS DEFINED AS THE LIBIT OF THE AVERAGE 2 SPEED_ AS THE T I B E INTERVAL ABOUT T=2 SHRINKS TO ZERO.THE SHORTER THE 3 TIHE INTERVAL USED.'THE CLOSER T H E AVERAGE S PE EC IS TO THE ACTUAL SP~EED 4 AT TDAT INSTANT. T H E ACTUAL SPEED AT THAT INSTAuT I S JUST THE SLOPE 5 OF THE TANGENT TO THE CURVE. SEHC .6 2"$ ' " " rwo$ 7 WRONG.THIS I S THE VALUE OF AVERAGE SPEED F50B T=2 TO T=6. 8 THE_I N b TAN TANEOUS_ JSP E_D_I S D E F I » E 0 AS THE VJ LU_ 0 F_ A VE KAGE SPE E D W _ E H 9 THE TIBE INTERVAL HAS SHRUNK TO ZERO. LOOK AT THE TABLE ARC TRY AGAIN... $EHI 0 OS 0$ ?_ROS 1 BRONG. THE LENGTH OF THE_ TIB- INTERVAL STARTING AT T=2 IS O.BUT THE SPEED__ •2 IS HOT EQUAL TO 0." " 3 THE INSTANTANEOUS S P E E O IS DEFINED AS THE VALUE 01' AVERAGE SPEED WHEN i4 THE TIHE INTERVAL :'AS SHRUNK TO ZERO. LOOK AT THE TABLE AND TRY AGAIN... SEND '5 1.03$ 1.01$ SO L I B I T $ N 0 LIB IT $ UNCEY-NED'S' '6 THE INSTANTANEOUS SPEED IS DEFINED Ab THE VAL!:E OF .'.VERAGE SPEED WHEH 17 T J I E T I H J E JENTERVAL KAS_SHRU__; TO ZERO. LOOK AT T 3 E T A B L E AND TRY AGAIN.., SEHI 18 9 ~~ ' " •9 4 6 2 7 3 10 LOOK AT SHEET 4 VHICH IS THE GRAPH OF THE HOTIOH OF A FEATHER DROPPED 1.98 • 1 - BOfl A TOWER. S REPRESENTS T H E DISTANCE OF T H E FEATHER FROH THE GROUND. 12 • RHAT I S T H E SPEED OF T H E FEATHER AT T=2 S E C ? SZBl •3 REHEHBER T H E R E L A T I O N S H I P BETWEEN SPEED AND SLO P E ARD THEN FIND THE •4 INSTANTANEOUS SPEED AT T=2 S E C . FROH T H E GRAPH. SEHt 15 -0 S -FOUR $ - FOURS -4MIS I S - 4 S — 4 S l _ THE__AC_TUAL _.S PEED _AT_T=2_ S EC. I S -4 F E E T / S E C . T H I S I S T H E S L O P E OF •7 THE L I N E WHICH TOUCHES T H E CURVE AT ONLY ONE POINT P. T H E S L O P E I S •8 NE G A T I V E S I N C E THE DISTANCE S I S G E T T I N G SMALLER AS T H E TIHE T I H C R E A S E S . '9 THIS_HEANS THAT DEL (S) K I L L BE NEGATIVE. _ . _ . _ SEHC 0 0 S FOURS 1 TOU'RE ALMOST RIGHT . T H E CHANGE IH DI S T A H C E D E L ( S ) I S GIVER BT 2 ( F I R A L_J) 1ST A NCE)j__(TNITI AL_ DISTANCE S I N C E PI S TANCE I S GE T T I H G S H A L L E B 3 AS TIHE GETS BIGGER , DEL (S) WILL BE NE G A T I V E . TRY AGAIN. " SEHI 4 6S 2S 3$ TWOS IS 3.5S -3S 5 H0,_THE S L O P E OF T H E L I N E -RICH TOUCHES T H E CURVE AT POINT P REPRESENTS _ 6 THE IHSTAHTANEOUS S P E E D OF T H E FEATHER AT T=2. 7 THINK ABOUT I T AND TRT AGAIN OR P L E A S E T Y P E REVIEW 8 IF TOU T T P E . ..REVIEW.WE WILL REPEAT T H E L A S T S E C T I O H AGAIH FROH SHEET 2. SEBD 9 REVS BOS DORS 0 GOTO 4 j io ; 2 2 7 3 3 T H I S ACTUAL SPEED I S C A L L E D T H E INSTANTANEOUS SPEED OF THE FEATHER 4 AT T=2 SEC. T H E IHSTAHTANEOUS SPEED IS T H E SLO P E OF THE S T R A I G H T L I N E 5 .HICH TOUCHES THS CUHVE'IN SHEET 4 AT THE POINT" P. T H I S L I N E I S C A L L E D 6 T H E TAHGENT TO THE CURVE AT POINT P. DO TOU UNDERSTAND? SEND 7 . WE'VE DEVELOPED T H E IDEA OF SLOPE R E P R E S E N T I N G SPEED. THE S L O P E OF A ' 1 LIRE J O I N I N G TWO POINTS Oi! THE CUR'/E I S AVERAGE SPEED.THE S L O P E OF A •9 LIRE TOUCHING T H E CURVE AT ONE POINT I S INSTANTANEOUS S P E E D . I S I T C L E A R ? SEBD 0 T E S S CKS B I T S L I T T L E S S_DF.ES THINK S GUESSS 1 THE INSTANTANEOUS SPEED' I S WRITTEN AS V= L I M I T (DEL (SJ / DEL (T) ) AS 2 DEL(T) APPROACHES 0. T H I S I S ABBREVIATED BY WRITING 3 ; V = DS/DT = LIM I T (DEL (S) / DEL (T) ) AS DEL.T) TENDS TO 0. 4 DS/DT I S C A L L E D T H E D E R I V A T I V E S F S WITH RESPECT TO T AND REPRESENTS 5 THE S L O P E OF A L I R E TOUCHING T H E CURVE OF S VS. T AT OHLY ORE PO I N T . SEBD 6 BOS DO NS HOTS 7 THEN" BEAD i f " AGAIH AHD ASK FOR HELP. SEND 8 11 , 9. 4 J l 4 4 4 0 ROW L E T ' S F I N D THE" INSTANTANEOUS" SPEED" BT USING THE EQUATION" OF HPTIO H . 1 S H E E 1 5 ILLUSTR'TES T H E GRA.H OF THE EQUATION OF MOTION, S » 3 T » 2 2! F I R S T LOOK A T T H E T I M E INTERVAL BEGINNING AT T=1 SEC. .3 WHAT I S THF, D STANCE S A T I HE B E G~I H H I MG OF T H I S T I R E I N T E R V A L ? SEHD 4 OSE TB_ EOUATI.N OF MOTION. SEHD 5 3__S T H R E E S S = 33 3 ( 1 ) * 2 $ 6 "THE DI-TA-.E S I S E A S I L Y OBTAINED FROH THE EQUATIOH OF HOTION." 7 S=3T*2=3 ( 1 ) *2=3. 8 BE ARE SIHPLY CJk L C UJ. A TIdG THE_DISTAHCE AT 01 g POINT. SEND 9 SS D I S T S " " 3 T * 2 S S=3T*2S 0 RO. THIS REPRESENTS T H E DISTANCE AT ANT T I K E T, WE WISH TO C A L C U L A T E 1 THE D I S T A H C E AT A SISG1.S POINT T - 1 . 2 OSE THE EQUATION OF MOTION S = 3T*2 TO EVALUATE S AT THE PCIfcT T = 1 . . . S E N D 3 TS 1$ C - L ( T ) S 1 * D E L ( T ) $ 4 WRONG. YOU'RE LOOKING^ AT T H E T I H E INSTEAD OF C A I C U L - T I - - Th- Dl-TAHCE AT A . 5 PARTICOIAR T I M E I H S . A U T . 6 OSE TEE EQUATTOS OF MOTION S=3T«2 TO EVALUATE S AT THE POI H T T=1... SEHD 7 0$ OS _ IN FIN I T Y $ DNDEFIHEDS C OSE T H E EQUATION OF iiOTION S=3T*2 TO EVALUATE S AT THE POI H T 1=1... SEHD 9 12 0 4 1 1 1 2 1 9 9 1 THE LENGTH OF THE TIME INTERVAL INDICATED ON THE GRAPH IS DEL (T). 2 WHICH OF THE FOLLOWING REPRESENTS THE DISTANCE AT THE END OF THE 3 TIHE INTERVAL,THAT IS, AT TIHE T=1 »DEL (T) ? PLEASE ANSWER A,B,C,D,OR E. * A 3(DEL(T))»2 B 3 (1 + DEL (I) ) *2 5 C 3(1»DEL(T)) D 3T*2 (j E NONE OF THE ABOVE SEHD 7 OSE THE EQUATION OF ROTIOS. SEND 8 BS 9. THE TIHE AT THE END OF THE TIHE INTERVAL IS T=1*DEL(T). THEN THE . . 0 DISTANCE AT THIS VALUE OF TIHE IS GIVEN BI S = 3T»2= 3 (1 *DEL (T) ) *2 1 WHICH IS ALSO EQUAL TO 3*6DEL (T) • 3DEL (T) »2 SEBD 2 u : 3 BBONG. THIS IS THE DISTANCE FROH THE ORIGIN AT A POINT IN TIHE T=DEL(T). • IT IS HOT THE DISTANCE AT THE EHD OF THE INTERVAL WHICH STABTS AT T=1. _ TBI AGAIN. send 6 CS 7 BBOHG. THE DISTANCE IS GIVEN BI S=3T*2. IOU HAVE CHOSEN THE DISTANCE AT 8 THE END OF THE INTERVAL FOR AH EQUATION OF HOTION S = 3T=3(1*DEL <T)) . 9 TBI AGAIN. SEHD 0 DS ES J BBONG. THE DISTANCE IS GIVEB BI S=3T*2 AND TBE TIHE AT THE EBP OF THE 2 TIHE INTEBVAL IS T=1*DEL(T). TBI AGAIN. SEBD 3 13 « 4 8 3 3 4 5 HE WOULD LIKE TO FIND THE AVERAGE SPEED IN THIS TIHE INTERVAL FHOS T=1 6 TO T=1•DEL(T).THE CHANGE IN DISTANCE,DEL(S),IS GIVEN BY 7 (DISTANCE AT END OF INTERVAL)-(DISTANCE AT START OF INTERVAL). _ 8 THEM " DEL (S) (3 + 6 DEL (T) + 3DEL (T) *2) -3 9 • WHAT IS THE SIHPLIFIEd FOHH 0 i)EL (T) DEL (T) )-1 OF DEL (S)/DEL (T) ? SEND 1 DO SOKE ALGEDRA TO SIMPLIFY THIS EXPRESSION. SEND 2 SODELS 6 • 3DELS 6« 3DELS 6 ODELS 3(2»DELS 3 (2 • DES 3DEL (T) »6$3 (DEL (T) •$ 3 DEL(S) /DEL (T) = (5DEJ. (T) OCEL (T) »2)/DEL (T)--60DEL (TI SINCE DEL (T) CANCELS 4 IN BOTH NUMERATOR AND DENOHINATOR. THIS IS TBE AVERAGE SPEED 5 I I THE INTERVAL. SEND 6 /DELS DEL (TJ $ DELT$ 7 IOT QUITE. SIMPLIFY THE NUMERATOR AND THEH CANCEL DEL (T) FROM BOTH 8 NUMERATOR AND DENOHINATOR. GO AHEAD... SEBD 9. 3*6DET$_ 3»6S 3 • 6$ _ ' 0 BO. THE •3'" IN~THE "FIRST PART OF THE HUHERATOP CANCELS WITH THE"" •3"' 1 IB THE SECOND PART OF THE DENOMINATOR.THEN YOU CAB CANCEL DEL (T) FROM 2 BOTH NUHERATOR AND DENOMINATOR. TRY AGAIN... SEND 3 6»DEL$ 3VDELS 3+3DELS 3(1*DEL$ 4 BOT QUITE. YOU SLIPPED UP WHEN YOU EXPANDED 3 (1 *DEL(T))*2.THE NUMERATOR •5 SHOULD BE 3 ( l*2DEL (T) «DEL(T) *2)-3 = 6* DEL (T j OPEL (T) » 2 . 6 THEN SiHPLTFITHE iXPRESSION. GO AHEAD.".". " S~EHT 7 14 '8 3 2 7 3 •9 RECALLING THAT INSTANTANEOUS SPEED IS THE LIMIT OF AVERAGE SP2ED A- TBE 0 TIHE INTERVAL SHRINKS TO ZERO,THAT IS,AS DEL(T) APPROACHES ZERO. 1 WHAT IS THE INSTANTANEOUS SPEED AT TIME T= 1 S E C ? SEHC 2 FIND "THE LIMIT OP DEL (SJ/DEL (r> "«S "DEL (T) APPROACHES 01 ' SEBD 3 6S SIXS _4 GOT018 15 3$ 0$ n$' 2$ 95 DELS" 16 THE LIMIT AS DEL (T) TENDS TO 0 IS THE VALUE OF b* DEI- (T) WHEN DEL(T)=0. 17 YOU SECM TO BE UNCLEAR ABOUT WORKING OUT LIMITS.YOL' SHOULD CO_SOME WORK OH 18 LIMITS FOB A FEW MINUTTS BEFORE GOING ON WITH THE MAIN LESSON. ." 19 PLEASE E::T-R TKH BOF.D.-. LIMIT ,OR TBI AGAIN IF .CU !(ISU. SEBI SO LIMITS SOS DONS 200 >1 G0T015 !2 15 •3 2 M »4 LET S=3T«7. WHEH T=2.IT IS EAST TO SEE THAT S=13. BOT HOB DOES SS S BEHAVE WHEN T IS CLOSE TO 2? EXAMINE THE TABLE GIVEN BELOW. Ui I 2.S 2.7 2.J 2.01 2.001 2._000j •7 S=3T*7 14.5 13.6 13.3 13.03 13.003 13.0003 !B I S S CLOSE TO 13 WHEN T IS CLOSE TO 2? SEBD JS D.OES._S..SEEM TO BE GETTING CLOSER TO 13 AS T GETS CLOSES TO 27TES OB 80? SEBC SO TES$ OK$ THINKS SURES i i BE SAT THAT IF S=3T+7,THSN S APPROACHES 13 AS T APPROACHES 2 AND WRITE •2 LIMIT (3T*7) AS T APPROACHES 2 .1S EQUAL TO 13. SEBD 3 BOS DONS HOTS NOS 4 LOOK AT THE TABLE CLOSELT AND ANSWER AGAIN OB ASK FOB HELP. SEBD 5 16 ; , 6 4 4 3 3 1 7 THE LIMIT OF S AS T TENDS TO A PARTICOLAR VALUE SEEMS TO BE SIMPLT 8 THE VALUE OF S AT _THAT__VALUE OF T. THIS IS GENERALLY TRUE EXCEPT ONLT 9 ORDER CERTAIN CON DITIONS, WHICH YO'J WILL LEARN ABOUT LATER IN YOUR 0 CALCULUS COURSE. WHAT IS THE LIMIT OF S=20-6T AS T APPROACHES 3? SEHD 1 FIND TBE VALUE OF S AT THE POINT T=3. SEBD 2 2 $ ~" TWOS ' 20-6 (3) S 20-18$ - - - -3 TBE LIMIT OF 20-6T AS T APPROACHES 3 IS 2. AS WE KAK2 THE VALUE OF T 4 CLOSER AND CLOSER TO 3,TH:~ V.1LUE OF S GETS CLOSER AND CLOSER TO 2, SEBD 5 20$ TWENTYS 20$ 6 BO. THIS IS THE LIMIT OF 20-6T WHEN T=0 SINCE 20-6(0)=20. 7 FIF0_T_HE LIMIT WHEN T=3. GO AHEAD... _ SEBD. 8 6 $ 6(3) 19$ * • 9 BO. YOU'VE ONLY LOOKED AT THE SECOND PART OF THE EXPRESSIOE 20-6T. 0 SUBTRACT THE VALUE OF 6? FROM 20 TO GET THE ANSWER GO AHEAD... SEBD 1 14$ 2 WRONG.THIS IS THE LIMIT OF 20-6T WHEB T= 1 SINCE 2U-6(1) = 14. 3 FIBD THE LIMIT OF 20-6T WHEH _T = 3. SEBD 4 17 5 4 4 2 1 2 6 FIHD T H E_L I MIT 0 F_DE L (S ) /DEL (T)= 6 • 3 D EL (T) AS DEL (T) APPROACHES ZERO. $EHD 7 FIBD THE VALUE OF 6*3DEL(T) AT DEL(T)=0. SEHD 8 6 $ SIXS 6.S 6*3 (0) S / 9 AS DEL (T) GETS CLOSER AND CLOSER TO 0,THE 7ALUE OP 6«3DEL(T) GETS CLOSER 0 ABD CLOSER TO 6.THE LIMIT OF 6*3DEL(T) AS DEL (T) APPROACHES 0 IS 6. SEND 1 3$ THREFS 2 BO. THIS IS THE LIHIT OF 6*DEL (T) AS DEL (T) APPROACHES -1, 3 SINCE 6*3(-1)=3. FIND THE LIMIT AS DEL (T) TENDS TO 0. GO AHEAD... $EBD 4 6*3i 5 WRONG. YOU SHOULD SIMPLIFY THE EXPRESSIOH. 6 PIBP THE VALUE OF 6 + 3DEL(T) WHEN DEL(T)=0.~ GO AHEAD... "' " $ E i ^ 7 9S SINE$ 8 BO. THIS IS THE LIMIT OF 643DEL (T) AS DEL (?) TESDS TO 1. 9 FIHD THE VALUE OF~6*3DEL (T) WHEN DEL(T)=6. CO AHEAD... SEBD . e 18 1 2_2 8 2 LET'S TAKE A SHORT BTEAK "ROM 1 HE LESSON.I'D LIKE TO KNOW HOS TOU FEEL... 3 ..BIGHT NOW. W H I C H O" J-H" CATEGORIES BELOW DESCRIBE BEST TOUR REACTION..HOB.. 4 TO TH E ST A TEME N V. _.. . - ._. . . I AH TENS E. 5 A HOT AT ALL C HODER AT FLY SO 6 B SOHEWHAT D VERY RUCR SO 1 PLEASE ANSWI.R A.B.C.OR D. ' SEHD 8 ANSWER" A. B.C OR D TO DESCRIBE YOL'P !;EAC:iOB" ?ICHT NOW TO THE ST AT EM EN T i i . . 9 I AH TENSE. SEBD 0 BOS YESS 201 1 G0T018 2 AS BS CS DS ROTS SORES BOOEBS VESTS 3 G0T019 » 19 5 2 2 8 6 WHICH CATEGORT BELOW..A.B.C.OB P..BEST DESCRIBES TOOB BEACTIOH TO THE STATEMENT. 7 I FEEL AT EASE. 8 A BOT AT ALL C HODEBATELT SO 9. B_SOMEWHAT D.VEBT HUCH SO _ AHSWER A,B.C.OB D. SEHD 0 PLEASE A,B,C.OB D. SEBD 1 BOS TESS 2 G0TO19 : 3 AS BS CS DS ROTS SOBES HODEBS VESTS « GOT020 5 25 ; 6 2 2 8 7 I AM BBLAZED. ft A BOT AT ALL C BODERATELT SO 9 B SOMEWHAT D VERT MUCH SO 0 PLEASE AHSWER A.B.C.OR D. SEBD 1 ABSWEB A, B.C.OB D...TO THE STATEMENT. ... I AB RELAIED. _ SEBD 2 BOS IESS 3 GOT020 _ IS BS CS DS BOTS SOBES HODEBS TEBTS 5 G0T021 6 21 7 2 2 8 8 I FEEL CALB. 9 A HOT AT ALL C HODEBATELT SO 0 B SOMEWHAT D VERT MUCH SO 1 AHSWER A,B.C. OR D. SEHD 2 ARSWEB A,C.C.OB D TO THE STATEBEHT... I TE'_L CALB. SEBD 3 BOS TESS 4 G0T021 5 AS BS CS DS ROTS SOBES BODEBS VERTS 6 G0TO22 7 2 2 8 2 2 8 . 9 I IE JITTERT. :0 A HO. AT "ALL C BODF^ATELT SO :1 B SOMEWHAT D VERV MUCH SO 2 ___!___ A.B.C.OR ?. SEHD :3 ARSWEB -,_,C.J_ D TO THE STATEMEHT...I AB JITTE_T. SEHD :4 IESS HOi •5 GOT022 ; :6 AS BS CS DS ROTS SOBES ROD'S VESTS !7 GOT023 :'4 23 :9 4 1 2 2 4 10 LEt'S.THT ONCE BORE TO FIND THE IHSTAHTANEOUS SPEED DSIHG THE 3QUA7I0N ! t _ OF fiOTIOH, S = T*2. FIRST, WHILH OF THE FOLLOWING IS THE AVERAGE SPEE" BETWEEH 12 " T=2 AND " T=2 + EEL (T) ? ANSWER A.B.C.D.OR E. " 13 A <2*DEL <T, > - i B (2* DEL (T) ) *2-2*2 14 C_(2*'_ ELJT) ) * 2 - 2 * 2 / (2*DEL(T))-2 D (2* DEL (T) ) * ? . - V - L l\, *2 / DEL (T) 15 E HONE CP THE"ABOVE * . . . - SEHC 16 DO SOBE AI-GE3F.A TO GET DgL(S) AHD DEL (T) BEFOBL FIHDliJG AVERAGE SPEED IR 17 THISIH1ERVAL. SCHC 18 CS 19 THE AVERAGE SPEED IS SiMPLT DEL (S)/DEL (T) IH THE IHTERVAL,WHICH IS 10 GIVEH BT .'2 + DEL {T) ) »2 -2«2 / (2 + DEL CH ) - 2 . SEN! 202 AS BS •0. THIS REPRESENTS DISTANCE.THE DISTANCE AT T=2 IS S=T»2=2»2=« AND THE DISTANCE AT T= 2*DEL (T) IS S= (2*DEL (T) ) *2. YOO HOST DIVIDE DEL (S) BI DEL(T). YOO SHOULD REPEAT THE HATERIAL DEALING WITH SHEET 5 TO BECOHE CLEAR ABOUT THIS. EITH EB TYPE ...REVIEW .OR ANSWER A GAIN. SEBD DS ES BO. THE AVERAGE SPEED IS DEL (S)/DEL (T) IN THE INTERVAL. DEL(S)=FINAL DISTANCE-INITIAL DISTANCE^ (2* DEL (T) ) «2-2*2 AHD _ DEL (T) =FIHAL TIMS-INITIAL TIB E= {2* DEL (T) ) - 2. SIHPLIFY DEL (S) AND DEL (T) AND DIVIDE THE."!.YOU SHOULD REPEAT THE HATERIAL DEALING WITH SHEET 5 TO BE CLEAR ABOUT THIS. TYPE...REVIEW .OH TRT AGAIH. SEHD BEVS HOS DORS OKS GOTOII ; H 3 5 3 2 BHAT IS THE IHSTANTAHEOUS SPEED AT T=»2? SEHD BEBEHBEH THE RELATIOHSHIP BETWEEN AVERAGE SPEED AHD INSTANTANEOUS SPEED •j AND THEN GO AHEAD... SEBD •I a s <t.S FOURS J THE AVERAGE SPEED IS DEL (S)/DEL (T) « (2 + DEL (T) ) *2-2»2 / 2*DEL(T)-2 _ _ •I SIMPLIFYING THE NUMERATOR iND DENOMINATOR GIVES •'; DEL (S) /DEL (T) = 4 DEL (T) + DEL (T) *2 / DEL (T) NCW CANCEL DEL (T) FROM HUHERATOR AND DENOMINATOR A_ D_W_E_E N D_U P WITH U*OSL (T). THE LIBIT OF •»• DEL (T) AS DEL (T) TEHUS TO ZERO IS U. WHICH IS THE IHST. SPEED SEHD 0/0$ 0 $ ZEROS OS 0/0$ JTOO DIDB'T SIMPLIFT THE NUMERATOR AND- DENOMINATOR FIRST IH EEL (S)/DEL (T) BEFORE FINDING THE LIMIT. DO THI- AHD THEN CANCEL DEL (T) FROM BOTIi NUMERATOR AHD DENOMINATOR BEhORE FIHDING THE LI3IT OF DEL(S)/DEL(T). GO AHEAD AND TRY AGAIN. $EBD 2 $ 1 $ DEL$ TOU SEEH TO HAVE SLIPPED UP IN YOUR CALCULATIONS. SIHPLIFTIhG THE HUBEBATOR OF DEL (S)/DEL (T) GIVES (4 • 1 DEL (T) • DEL (T) *2)-« = <i EEL (T) *DEL (T) »2 SIHPLIFYING THE DENOMINATOR GIVES DEL(T).CANCEL CEL(T) FROH NUBERATOR AHD DEHOBINATOB. THEN FIND THE LIMIT OF THE R EBA INING EIPRESSIOR. SEHD _/_$ • $ BOT QUITE. YOU RAVEH'T REDUCED THE EXPRESSION TO SIMPLEST FORM. TOO SHOULD SIMPLIFY THE NUMERATOR AND DENOMINATOR AND CANCEL _DEL_[T_ FROK_BOTH_0F_ THESE. THEN FIND THE LIMIT OF THE REMAINING EXPRESSION SEND 25 " " 2 2 2 CONGRATULATIONS! TOU HAVE COMPLETED A CAI LESSON. I HOPE THAT I ' . » 0 ENJOYED OUR CONVERSATION AS MUCH AS I DID AND THAT Y07 LEARNED SC"-THING TOO. DID TOU EM JOT THIS METHOD OF LEARNING;' ANSWER YES OR HO. SBHE ASSWES YES OR NO. SEHD TESS HCS IF TCI WISH TO HAKE ART COMMENTS ABOUT THE LESSOS,ASK TH. INSTRUCTOR TO PROVIDE YOU WITH A COMMENT 5H-_T. GOOD-BYE FOE HOW... SEHD BATBES CONS PLEASE AHSWER YES OB HO. SEND FILE T2 » 8 2 2 3 lET'S STUD: THE KOTION OF A BOAT AS IT LEAVES A DO SUPPOSE THAT IT'S HOVIHG AT A COHSTAHT SPEED OF 10 BILES/HOUR. 2 0 3 WHAT I S THE DISTANCE S OF THE BOAT FBOH THE DOCK AT ANT TIHE T? REHEHBER THAT DISTANCE 3 (SPEED) (TIHE).BBITE AN EQUATION FOB S IN TEBHS OF T. SEND SEND 10TS 10*T$ 10 TIHES T H E EQUATION WHICH DESCRIBES T H I S I S SIMPLY THE FAMILIAR S P E E D V BEING CONSTANT AT 10 l"0*S/T S T=10/S $ • 0. DISTANCE 3 (SPEED) ( T I H E ) . _IS__T, WBITE AN EQUATION FOB 10 S T $ • 0. DISTANCE 3 (SPEED) (TIME) . I S T . WRITE AN EQUATION FOR_ S S DISS 10SS 1 0 . DISTANCE 3 (SPEED) (TIME). _ I S _ T . WBITE AN EQUATION FOB 2 4 4 8 4 2 LOOK AT SHEET S10XTS S 3 1 0 T $ S=T10S T10S THE HOTION OF THE BOAT IS S=10T. DISTANCE- (SPEED) (TIHE) EQUATION, WITH HILES/HOUR. T » 1 0 S SEHD THE SPEED IS 10 RILES PEB HOUB AHD THE TIHE S IH TERHS OF T...GO AHEAD SEHD THE SPEED IS 10 RILES PEB HOUB AID THE TIHE S IN TERHS OF T. . . GO AHEAD SEHD THE SPEED IS 10 RILES PEB HOUB AND T H E T I H E S IN TERHS OF T...GO AHEAD SEID 1 THAT WAS GIVEN TO TOU.THE GRAPH INDICATES BOH FAR THIS BOAT I S FROM THE DOCK AT ANT TIME. HEAT I S THE SLOPE OF THIS STRAIGHT LIKE REPRESENTING THE BOAT«S BOTIOH IN A TIME INTERVAL T=2 TO T=6 HOURS? ~X"OD "REED TO OBTAIN THE CHANGE IN DISTANCE AND THE CHANGE IH TIRE BETWEEN THESE TWO P0I2TS IN ORDER TO CALCULATE THE SLOPE. GO AHEAD... _10$ __<! $ 0-20) / (6 - J 6 0-2 O/bj-2 $ THE SLOPE IS THE CHANGE IN~DISTANCE DIVIDED BI THE CHANGE II TIME. S E I D S E I D BE B I I T Z THIS AS SLOPE= DEL (S)/DEL (T) _ ( 6 0 - ? 0 ) /J.6-2) = 10_HILES/H00a. _ 1/10$ .1$ 4/40S 2/20S HBOBG. THE SLOPE OF THE STRAIGHT LINE I S THE CHANGE IN BT THE CHANGE IN TIME BETWEEN THESE TWO POINTS. NOW, WHICH I S EQUAL TO SEBD 8/80S " 6 - 2 / 6 C - 2 C S 6 - 2 ) / ( 6 0 - $ DISTANCE DIVIDED 6 / 6 0 S GO AHEAD AND SORK OUT THE SLOPE... SEBD 60S 60/63 201 20/2S _WBOHG._THE SLOPE OF TBE STRAIGHT LINE I S TBE CHAHGE I H D I S T A H ^ E D I V I D E D B I THE CHANGE IN TIME BETWEEN THESE TWO POINTS. HOW, GO AHEAD AND WORK OUT THE SLOPE... SEBD 8 0 / 8 $ S/T S BBOHG. THE SLOPE OF THE STRAIGHT LINE I S THE CHANGE IS DISTANCE DIVIDED BT T H E CHANGE IN TIME BETWEEl' THESE TWO POINTS. NOW, GO AHEAD AND WORK OUT TBE SLOPE... SEHD 3 4 5 8 4 2 BHAT I S THE SLOPE OF THE LINE IN THE TIHE INTERVAL T-4 TO T=8 HOURS? TOU NEED TO OBTft"iH THE CHANGE IN DISTANCE AND THE CHANGE IN TIME BETWEEN THESE TWO POINTS I * ORDER TO CALCULATE THE SLOPE. GO AHEAD... _10$ 40/4$ 0-40) / (8-$80-40/8-4 SCONSTANTS THE SLOPE IS CONSTANT AT 10 HILES/HOUR.THIS IS ALWAYS THE CASE WHEN-$EHD $EBD THE GE".PH OF THE MOTION IS PICTURE OF THE EQUATION OF A STRAIGHT LINE.THE GRAPH IS JUST A MOT10H S=10T,AND SO THE SLOFE OF THE LI?'-I S THE SPEED OF THE BOAT. 1/10$ . 1 $ 4/<4CS THAT ISINCORRECT.SLOPE IS BETWEEN THESE TWO POINTS. 80$ 80/81 40$ SEND 2/20$ 6/60$ 8/80$ 8-4/80-40S8-4)/ .80-$ CHANGE IN DISTANCE DIVIDED 3Y CHANGE IH VIME NOW,GO ABEAD AND CALCULATE THIS SLOPE... SEBD 4$ THAT I S INC0RRECT_» '"T.0v_K IS CHANGE IH DISTANCE DIVIDED BY CHANGE IH TIHE BETWEEN THESE TWO POINTS. NOW, GO AHEAD AN D~ CALCULATE THIS SLOPE... 6 0 / 6 $ 20/2$ THAT IS INCORRECT.SLOPE IS CHANGE IH DISTANCE DIVIDED BI CHANGE IH_TI5E THEST TWO POINTS. ISZTWEEN 4 4 4 4 3 2 NOW,GO AHEAD AND C A L C L A T E TEIS SLOPE... TEND SEHD 204 SHEET 2 ILLUSTRATES THE flOTIOH OF A SECOND BOAT COMPUTE THE AVERAGE SPEED OF THE BOAT IN THE INTERVAL FBOS T-2 TO T»6 BBS. PROCEED EXACTLY AS BEFORE. 2$ 8/4$ (9-1)/(6-S9-1/6-2S TBE AVERAGE SPEED I S SIMPLY THE CHANGE IN DISTANCE DIVIDED Bt THE CHANGE IN TIME.THIS I S GIVEN BY V = DEL (S)/DEL (T) = (9-1)/(6-2)--8/4 = 2 HILES/HOUB 9/6S JB.RONG. 1/2$ 1.5$ .5$ .THE AVERAGE SPEED IS THE CHANGE IN DISTANCE DIVIDED Bt THE CHANGE III TIHE FROM T=2 TO T=6. CALCULATE THIS FROM THE GRAPH... 9$ 1$ 6$ BBONG. THE AVERAGE SPEED IS THE CHANGE IN DISTAHdE PIVIDED BY THE CBABGE IH TIHE FBOH T=2 TO T=6. CALCULATE THIS FROH THE GRAPH... es is WRONG. THE AVERAGE SPEED IS THE CHANGE IH DISTANCE DIVIDED BI THE CHAHGE IB TIHE FBOH T=2 TO T=6. CALCULATE THIS FBOH THE GBAPH... SEHD SEBD SEHD SEBD SEBD SEHD 1 4 4 3 2 LOOK AT SHEET 3.WHICH IS JUST THE GRAPH IH SHEET 2 WITH THBEE POINTS P,Q, AND R INDICATED ON IT. _BHAT I S THE SLOPE OF THE LINE SEGMENT JOINING THE POIETS P AND C? YOU HEED TO KNOW THE CHANGE IN DISTANCE AND THE CHANGE IN TIME BETWEEH t ABD 0 IN ORDER TO CALCULATE THE SLOPE OF THE LINE SEGMENT. GO AHEAD.. _2_S 8/4 $ (9-1) / . 6 - _ - _ / _ ? > THE SLOPE OP THE LINE SEGMENT JOINING P AND Q I S GIVEN 3T CHAHGE IH WROBG. THE SLOPE OF THE STRAIGHT LINE JOINING P AND Q I S GIVEH Bi' CHANGE IB DISTANCE,DEL(S),DIVIDED BY CHANGE IN TIME,DLL(T),FROH P TO Q. JCALCJUL ATE_. THIS FP.OM THE GRAPH... 6 ' " " "• '"" """ 4 8 5 2 1 LET'S TAKE A SMALLER SIZE INTERVAL THAN BEFORE OS T..E GRAPH IN SHEET 3. SEBD SEBD DISTANCE DIVIDED BY CHANGE IN TIME, WHICH IS DEL (S)/DEL (T) = 8/1 = 2 .1/HB. JTHIS J S THE SAME AS THE A V E R S E SPEED OF THE BOAT BETWEEN P AND Q. SEBD 9/6$ 1/2$ 1.5$ .5$ HBOBG. THE SLOPE OF THE STRAIGHT LINE JOINING P AND Q I S GIVEN B I CHANGE "CH DISTANCE, DELJJJ_,Q1JIDED BY CHANGE IN TIME, ftEL (T) , FBOH P TO Q. CALCULATE THIS FROM THE GRAPH... SEBD 9$ 1$ 6$ BBONG. THE SLOPE OF THE STRAIGHT LINE JOINING P AND Q I S GIVEH B t CHAHGE IB DISTANCE,DEL (S) .DIVIDED BY CHAHGE IB TIHE, DEL (T) ,FBOH P TO Q. CALCULATE THIS FBOH THE GRAPH... SEBD 8$ 1$ SEBD WHAT I S THE AV EnAGE SPEED OF-THE BOAT IN THE INTERVAL 1 = 2 TO T=4 HOURS? PBOCEED EXACTLY AS BEFORE. 1.5$ 3/2$ 1*1/2$ 11/2$ 1 1/2$ 4-1/1-2$ (1-1)/(1-$HAL$ THE AVERAGE SPEED IS V = DSL (S)/DEL (T) = (4-1) / .4-2) = 1. 5 M/HR. TBE SLOt E OF THE LINE SEGMENT .JOINING P AND a I S AGAIN THE SASE AS T H E AVERAGE SPEED OVER THIS INTERVAL. SEHD $EBD liOTICE THAT THE AVERA2K SPEED IS NO LONGER CONSTANT!! i r THE GRAPH I S NOT A S T R A I G H T L I N E , T H E S L O P E I S NO LONGER COBSTAfiT. J $ 1/4S 1/2$ .5$ 1HI$ HO. T B E ' A V E R A G E S P E E D I<- T H E CHANGE I N D I S T A N C E D I V I D E D BY T H E CHANGE IB TIHE FROH T=2 TO T=4. C A L C U L A T E T H I S FROM T H E GPAPH. .. 3$ 2$ $EBD SEBD BO. THE AVERAGE SPEED IS THE CHANGE IN DIST AN "t" DIVIDED BY THE CHANGE IH TIHE FROH T=2 TO T=4. CALCULATE THIS FROM THE GRAPH... _«$ BO. THE AVERAGE SPEED I S THE CHANGE IH DISTANCE TIVIDED BY THE CHANGE IB TIHE FROM T=2 TO T=4. CALCULATE THIS FROM THE GRAPH... 7 SEBD SEBD 205 • 3 2 3 5 BBAT HAPPEHS TO THE SPEED AS BE HAKE THE TIHE INTERVAL SHALLER AND SHALLER.LET'S CONTINUE WHAT HE DID WITH SHEET 3 BI CONSTBUCTING A TABLE. LENGTH OF INTERVAL (HRS)»* 0 2 1 SLOPE OF L I N E ( B I L E S / H R ) * * 2 1.5 1.25 AVER AG E__S PEED (BILES/HR) «» 2 1.5 1.25 .5 .25 0 1.12 1.06 1 1.12 1.06 ? BHAT I S THE L I B I T OF AVERAGE SPEED AS THE INTERVAL SHRINKS TO S I Z E 0? SEHD STUDY THE TABLE CAREFULLY AND ROTE WHAT HAPPENS TO SLOPE AND AVERAGE SPEED_AS THE LENGTH OF THE TIHE INTERVAL SHRINKS FROH DEL (T) = 1.DOWN TO 6. SEND IS I.OOS 1HIJ THE L I B I T OF AVERAGE SPEED AS THE INTERVAL APPROACHES ZEBO IS 1 H/HR. BE SEE TH AT. AS TH E...INTERVA L K EEP S_G ETTIN G SB ALL E R_I T_S H RIN KS_ TO_A • POINT AT T=2,AS SHOWN ON SHEET 3.THE SPEED AT T = 2" IS JOST THE SLOPE OF THE LINE L WHICH TOUCHES THE CURVE AT ONLY ONE POIHT P. THIS LIRE I S _CALLED THE TANGENT TO THE CURVE AT_POIHT P. SEBD 2S " " TWOS •0.LET'S TAKE A CLOSES LOOK AT THE IKTERVAL AS HE SHRIHK IT TO S I Z E 0. LEHGTH *« .5 .25 . 15 . 10 0 SLOPE ** 1.12 1.06 1.00 1.02 ? AVERAGE SPEED** 1.12 1.06 1.00 1.02 T _HHAT_ I S THE L I B I T OF AVERAGE SPEED AS THE IHTEBVAL SHRINKS TO S I Z E 0? _ _ SEND OS OS ZEROS lO.LET'S TAKE A CLOSER LOOK AT THE IHTERVAL AS WE SHBIRK IT TO _IZE 0 . LES G_TH ** .5 .25 . 15 . 10 0 SLOPE *• 1.12 1.06 1.00 1.02 ? AVEB1GE SPEED** 1.12 1.06 1.00 1.02 ? JTHAT I S THE L I B I T OF AVERAGE SPEED AS THE IHTERVAL SHBIHKS TO S I Z E 0? 1.03$ 1.01$ DEFINED$ HO L I B I T S NOLIHITS HO.LET'S TAKE A CLOSER LOOK AT THE INTERVAL AS WE SHRINK I T TO SIZE 0 . LEHGTH »» .5 .25 .15 .10 0 SLOPE ** 1.12 1.06 1.00 1.02 ? AVERAGE SPEED*• 1.12 1.06 1.00 1.02 ? BBAT I S THE L I BIT OF A VERAG E SP EEP AS T H E I NT EBV A L S H S I H K S TO SI Z E_ 07 8 1 9 2 3 5 BHAT I S THE IHSTAHTANEOUS SPEED OF THE BOAT AT THE TIHE,T=2 HOURS? THIS I S THE SPEED AT THE INSTANT OF TIBE T=2 HRS. AS WELL AS SHEET 3. THEN TRY TO ANSWER... 1 $ 1.00$ ONES IB I S STUDY THE TABLE ABOVE THE SPEED AT TIBE T=2 HOURS I S DEFINED AS THE L I B I T 01 THE AVERAGE SPEED AS THE TIBE INTERVAL ABOUT T= 2 SHRINKS TO ZERO.-iiS SHORTER THE TIHE INTERVAL USED,THE CLOSER THE AVERAGE SPEED IS TO THE ACTUAL SPEED AT THAT INSTANT. THE ACTUAL SPEED AT THAT IHSTAHT IS JUST THE SLOPE OF TFic TAKGE..T TO THE CURVE. _2$ _TWOS_ _ _ _ _ _ _ INCORRECT. THE SPEED AT T=2 I S THE L I B I T OT'tHE AVERAGE SPEED THAT «E OBTAIiED AS WE SHRANK THE INTERVAL STARTIHG AT 1 = 2 DOWN TO SIZE ZERO WHAT IS THIS VALUE OF SPEED? 0$ 0$ ZEROS INCORRECT. THE SPEED A r T=2 I S THE LIMIT OF THE AVERAGE SPEED THAT VE _OBTAIHED AS WE SHRANK THE INTERVAL STARTING AT T=2 DOWN TO SIZE ZERO. WHAT I S THIS ' VALUE OF SPEED? ~ ~ " * 1.03$ 1.01$ S.0 LIrt.TS HOLIBIT$ UNDEFINED:. IHCORRFCT. THE SPa?D a t 7=2 IS THE L I B I T OF THE AVERAGE SPEED THAT WE OBTAIH-D AS WE SHRANK THE IHTERVAL STARTING AT T=2" DOWN TO SIZE ZERO. BHAT IS THIS VALUE OF SPEED? 9 0 6 2 7 3 LOOK AT SHEET 0 WHICH IS THE GRAPH OF THE HOTIOH OF A FEATHER DROPPED FROH A TOWER. S P.EPRESENTS THE DISTANCE OF THE FEATHER FROM THE GROUND. SEHD SEHD SEBD SEND SEND SEND SEND SEND 206 WHAT I S THE SPEED OF THE FEATHER AT T=2 S E C ? SEND BEHEHBER THE RELATIONSHIP BETWEEN SPEED AND SLOPE AND THEM FIND THE INSTANTANEOUS SPEED AT T=2 S E C FROH THE GRAPH. SEBD -1 S -FOURS - FOURS -IBIS IS-U S =-« S THE ACTUAL SPEED AT T=2 SEC. IS -« F E E T / S E C THIS IS THE SLOPE OF THE LI NE W_H I CH TOUCHES _TH E _CUR.V_E_A.T_ON L Y_0 N E POINT P.THE S LOPE IS BEGAT-VE SINCE THE DISTANCE S IS GETTING SHALLER I S THE TIBE T INCREASES . TBIS BEANS THAT DEL(S) WILL BE NEGATIVE. SEND _ J $ . FOURS HO. FIND THE INSTANTANEOUS SPEED OF THE FEATHER AT T=2 SEC. BT FINDING TBE ' SLOPE OF THE LINE WHICH TOUCHES THE CURVE AT THE POINT P. GO AHEAD... SEBD 6$ 2.S 3$ TWO$_ 1$ __ 3.5$ _3S_ SEHD 10. FIND THE IHSTAHTANEOUS f PEED OP THE PEATHER AT T=2 SEC. BT FINDING THE I SLOPE OF THE L I K E WHICH TOUCHES THE CURVE AT THE POIHT P. GO ABEAD... • REVS HOS. DOHS I GOTO a I 10 ! 2 7 3 TBIS ACTUAL SPEED I S CALLED THE INSTANTANEOUS SPEED OF THE FEATHER AT T=2 SEC. THE INSTANTANEOUS SPEED IS THE SLOPE OF THE STRAIGHT LINE WHICH TOUCHES THE CURVE IN SHEET M AT THE POINT P. THIS LINE IS CALLED "TBE 'TANGENT TO THE CURVE AT POINT P. DO YOU UNDERSTAND BE * VE DEVELOPED THE IDEA OF SLOPE REPRESENTING SPEED.TH? SLOPE CF A LINE JOINING TWO POINTS ON THE CUIiVE IS AVERAGE SPEED.THE SLOPE OF A SEBD LINE TOUCHING THE CURVE AT ONE POINT IS INSTANTANEOUS SPEED.IS IT CLEAR ? SEHD TESS 0K$ BITS LITTLES SURES THINKS GUESSS _THE INSTANTANEOUS SPEED TS SRITTEH AS V = LI BIT (DEL (S)/DEL (T ) ) AS DEL(T) APPROACHES 0. THIS I S ABBREVIATED BY WRITING V=DS/DT = L I B I T (DEL (S)/DEL (T) ) AS DEL (T) TENDS TO 0. DS/DT I S CALLED THE DERIVATIVE SF S WITH RESPECT TO T AND REPRESEHTS THE SLOPE OF A LINE TOUCHING THE CURVE OF S VS.' T AT ONLY ONE POIHT. SEHP BOS DONS HOTS TBEH_REap_IT AGAIH AND ASK FOR HELP. SEND ~ i i ""' " - - . M U M HOW LET'S FIND THE INSTANTANEOUS SPEED BY USIHG__ THE EQUATION OF BOTIOH. SHEET 5 ILLUSTRATES THE GRAPH OF THE" EQUATION OF BOTl"OH„ S=3T»2 FIRST LOOK AT THE TIBE INTERVAL BEGINNING AT T=* 1 SEC. WHAT IS THE DISTAHCE S AT "!:E BEGINNING OF THIS TIHE IHTERVAL? OSE THE EQUATION OF BOTIOH. 3 $ THREES S=3$ 3(1)«2S THE DIST.'HC" S TS_EASI\Y OBTAINED FROH THE EQUATIOH OF BOTIOH. S-3T*2=3"("ljV2-3. BE ABB S I R P L J CALCULATING THE DISTAHCE AT OSE PCIHT. _SS _ DISTS 3T*2$ S=3T*2$ NO. USE THE EQUATION OF BOTION, S=3T*2 AND EVALUATE S AT THE POINT T»1 IH OBDT;R TO OBTAIN THE DISTANCE S AT T=1. GO AHEAD... TS 1 J_ DEL (T) $ 1*DEL$ SEHD SEHD SEBD SEBD POINT ?=• 10. USE THE -QUATIO. OF BOTJON, S = 3 T " AND EVALUATE S JT TH-IS ORDER TO OBTAIN THE DISTANCE S AT T=1. GO AHEAD... _0$ 0$ INFINITY I UNDEFINED* HO. USE THE EQUATION OF BOTION, S=3T*2 AND EVALOATF S "7 T n t POVHT T=1 IH ORDER TO OBTAIN TH- DISTANCE S AT T=1. GO AHEAD., , 12 • 1 1 1 1 2 THE LENGTH OF THE T I B ? TuTEiiVAL INDICATED ON ThE GRAt'H IS DEL (T) . WHICH OF THE FOLLOWING REPRESENTS THF. DISTANCE AT THE YHO It THE TIBF. INTERVAL,THAT IS , AT TIBE T=1«D2L(T)? F-EAS- ANSWER A , B, C, D, OR E. A 3 ( D E L ( T ) ) * 2 * 3 (1 • DEL (T) ) »2 C 3(1*DEL(T)» 1 3T»2 SEBD SEND 207 r r » i t » E BONE OF THE ABOVE OSB THE EQUATION OF H0TI0N. BS TBE TIHE AT THE END OF THE TIHE INTERVAL IS DISTANCE AT THIS VALUE OF TIHE IS GIVEN BI WHICH IS ALSO EQUAL TO 3*6DEI IS BBONG.THE EQUATION OF MOTION IS S=3T»2. THE .BEGINS.AT T=1. . FIND THE VALUE OF T AT THE SUBSTITUTE THIS VALUE INTO THE EQUATION OF T=1*CEL (T). TBEB TBE S=3T»2=3 (1 • DEL (T))»2 (T) *3CEL (T) »2 SEND SEBD SEBD cs BBONG.THE EQUATION OF HOTION I S S=3T«2. THE BEGINS AT T = l . FIND THE VALUE OF T AT THE SUBSTITUTE THIS VALUE INTO THE EQUATION OF _D$ ES HBOBG.THE EQUATIOH OF HOTION I S S=3T»2. THE BEGINS AT T=1. FIND THE VALUE OF T AT THE SOBSTITUTE THIS VALUE INTO THE EQUATION OF TIHE INTERVAL OF LENGTH DEL(T) END OF THE INTERVAL AND HOTION TO GET S... TIHE INTERVAL OP LENGTH DEL (T) END OF TEE INTERVAL AND HOTION TO GET S... SEBD TIHE INTERVAL OF LENGTH DEL (T) END OF TEE INTERVAL AND HOTION TO GET S... SEHD SEBD 13 4 8 3 3 4 _SE_WOULD L I K E TO FIND THE AVERAGE SPEED I N THIS TIHE INTERVAL FBOH T* TO T=1*DEL(T).THE CHANGE IN DISTANCE,DEL(S),IS GIVEN BY (DISTANCE AT END OF INTERVAL)-(DISTANCE AT START Of INTERVAL). TBEB DEL (S) (3 + 6DEL (T) + 3 DEL jT) »2)-3 I _ WHAT IS TEE S I H P L I F I E D FORH • DEL (T) (1 • DEL (T) ) — 1 OF DEL (S)/DEL (T) ? SEND . D0_S0BE ALGEBRA TO SIHPLIFY THIS EXPRESSION. SEND f 6 * 3 i > E L S ~ 6 • 3DELS 6* 3DELS 6 *3DELS 3{2*DELS 3 (2 • DES 3DEL (T) *6S3 (DEL (T) •$ 1 DEL (S) /DEL (T) - (6DEL (T) * 3 DEL (T' *2) /DEL (T) = 6* 3D EL (T) SINCE DEL (T) CANCELS , IB BOTH NUMERATOR AND DENOHINATOR. THIS IS THE AVERAGE SPEED IB THE INTERVAL. SEBD /DELS DEL(T)S DELTS _W80NG. SIHPLIFY THE NUMERATOR BI REMOVING THE BRACKETS AND COLLECTING •L I K E * TERMS.THEN CANCEL DEL (T) FROM BOTH NUMERATOR AND DENOHINATOR. CHECK YOUR ALGEBRA AND TRY AGAIN... SEBD 3*6DET$ 3*6$ 3 • 6$ WRONG. SIMPLIFY THE NUMERATOR BI REMOVING THE BRACKETS AND COLLECTING • L I K E ' TERHS.THEN CANCEL DEL (T) FROH BOTH NUMERATOR AND CENOHIKATCK. CHECK YOUR ALGEBRA AND TRY AGAIN... 6*DELS" 3+DFLS 3+3DELS 3(1*DEL$ WRONG. SIMPLIFY TKiJ NUMERATOR BY REMOVING THE BRACKETS Ai»D COLLECTIF" ' L I K E * TERMS.THEN CANCEL DEL (T) FROM BOTH NUMERATOR AND DENOMINATOR. SEND / CHECK YOUS ALGEBRA AND TRY AGAIN... i 14 I 3 2.7 3 _ _ I RECALLING THAT INSTANTANEOUS" SPEED" IS"" THE LIMIT CF AVERAGE SPEED AS TBE I TIHE INTERVAL SHRINKS TO ZERO,THAT IS,AS DEL(T) APPROACHES ZEEO. t WHAT I S THE INSTANTAll EOIJS SPEET TIME T=1 SE-". ? SEND I FIBD THE LIMIT OF DEL (S) /DLL (T) AS DEL (T) APPROACHES 0. t 6S SIXS i G0"0'"3 • . 3$ OS 0$ 2S 9$ DELS * THE LIMIT A3 DEL (T, TENDS TO 0 IS THE VALUE OF 6*DEL(T) THEN DEL(T) = 0. I IOU SEEM TO BE UNCLEAR ABOUT WORKING OUT LIMITS.10U SHOULD SOSE WORK OH SEHD " S E N D LIMITS FOR A FEW MINUTES BEFORE GOING ON WITH TH1: MAIN LESSON. PLEASE EIITE.' THE "OB D.. . L I M IT ,0R TRY AGAIN I F YOU WISH. JLIMITS _ 0 J DONS GOT015" 15 2 M SEHD 208 2.001 1 3 . 0 0 3 BOT HOH DOES BELOW. 2.0001 13.0003 L E T S=3T*7. WHEN T = 2 , I T I S EAST TO S E E THAT S=13. S DEBATE WHEN T I S CLOSE TO 27 EXAMINE T H E T A B L E GIVEN T 2.5 2.2 2.1 2.01 S-3T*7 11.5 13.6 13.3 13.03 I S S C L O S E TO 13 WHEN T I S C L O S E TO 2? DOES S SEEK TO BE GETTING_.C_L0SER TO 13 AS T GETS CLOSER TO 2 7 I E S OR NO? I E S S OK$ THINKS SUHES I E S A I THAT I F S=3T*7,THEN S APPROACHES 13 AS T APPROACHES 2 AHD WHITE L I B I T ( 3 T * 7 ) AS T APPROACHES 2 , I S EQUAL TO 13. BOS DONS HOTS NOJ LOOK AT THE T A B L E C L O S E L Y AND AHSWER AGAIH OR ASK FOR H E L P . 16 SEBD SEHD SEBD SEBD ' 4 4 3 3 1 I T H E L I B I T OF S AS T TENDS TO A PARTICULAR VALUE SEEHS TO BE S I H P L T »_ THE_ VALUE OF S AT THAT VALUE OF T. T H I S I S GENERALLY TRUE E X C E P T OHLI I ORDER C E R T A I H COHDITIONS,WHICH YOO WILL LEARH ABOUT LATER IN YOUR I C A L C U L U S COURSE. WHAT I S T H E L I B I T OF S=20-6T AS T APPROACHES 3 ? » F I N D T H E VALUE OF S AT THE__PPINT T=3. I 2 S TWOS 2 0 - 6 ( 3 ) $ 2 0 - 1 8 $ » T H E L I B I T OF 20-6T AS T APPROACHES 3 I S 2. AS WE BAKE T H E V A L U E OF T | CLOS_ER_AHD CLOSER TO 3,THE VALUE OF S GETS CLOSER AHD CLOSER TO 2. i 20$ TWENTY$ 20$ I BO. F I B D THE VALUE OP S AT T H E POIHT T » 3 . . . I 6 $ 6 (3) 18$ SEHD $EHD SEBD SEBD » BO. F I B D THE VALUE 0 ? S AT T H E POIHT T O . . . I 1«S I BO. F I N D THE VALUE_OF_S_ AT T H E POIHT T « 3 . „ . _ _ 5 17 > 4 4 2 1 2 1 TJ HP T H E L I B I T_0 F DEL ( S\/P E L (T) = 6 * 3 D E L ( T ) AS S E L ( T ) APPROACHES ZERO. SEHD SEHD SEND F I N D T HE VALUE OF 6 + 3 P E L (T) AT D E L ( T ) = 0 . SEND 6 S S I X $ 6.S 6*3 (0) S _AS D E L ( T ) GETS CLOSER AHD CLOSER TO O.THE VALUE OF 6*3DSL(T) GETS CLOSER AHD CLOSER TO 6.THE L I B I T OF 6 » 3 D E L (T) AS DEL (T) APPROACHES 0 I S 6. $EHD 3$ T HREE$ HO. F I N D THE VALUE OF 6 * 3 P E L (T) AT D E I ( T ) = 0 . . . $END 6*3$ HO. F I B D T H E VALUE OF 6 + 3 D E L (T) AT D E L ( T ) = 0 . . . ? $ NINES "80. F I S S T H E VALUE 6F~6*3D-L(T) AT~DEL(T)=0.". . 18 2 2 8 SEBD SEND L E T ' S TAKE A SHORT BREAK FROil THE L E S S O N . I ' D L I K E TO KNOW HOW YOU F E E L . . . . .BIGHT ! n V . WHICH OF THE C A T E G O R I E S BELOW DESCRIBE BEST YOUR BEACTIOH..NOV.. TO THE _ T / . T E i : : : r . . . . . . . . i AH T E N S E . _ _ _ _ _ _ A NOT AT A L L " ' C HODERATELY SO " " : B SOMEWHAT D VERY BOCH SO P L E A S E ANSWER A.B,C,OR P. SEHD ABSWE'I A.B.J.OH D TO D E S C R I B E -OUR REACTION RIGHT HOW TO THE S T A T E H E H T . . . I AH T E R S E . SEHD _EOS JESV GOT018 ' ~ " AS DS CS DS HOTS SOBES HOD ESS V E S T S GOT019 19 2 2 8 WBICB CATEGORY iSPLOW.. A, B.C. OR P.. BEST D E S C R I B E S TOUR R E A C T I O B TO THE S T A T E B E 5 T . I" F E E L AT E A S E . A HOT AT A L L C BODERATELT SO B SOMEWHAT P VERY HUCH SO AHSWER A,B.C.OR D. SEND 209 PLEASE A,B.C.OS 0. BOS IESS GOTO19 At BS GOTO20 20 ; SEHD CS OS •OTS SOBES BODIES VESTS 2 2 8 I A8 BELAZED. A_HOT AT ALL C MODERATELY SO . B SOMEWHAT D VEST BUCH SO PLEASE ANSWER A,B.C.OB D. ABS8EB A.B.C.OR D.... TO TBE STATEMENT....! AB BELAIEE. SEBD SEHD BOS GOT020 as IESS BS CS D$ • OTS SOBES BODEBS TEBTS GOT021 21 2 2 8 I FEEL CALB. A HOT AT ALL C MODERATELY SO B SOMEWHAT D VERT HOCH SO " ANSWER" A,B.C. OR J . ANSWER A,B.C.OB D TO THE STATEHEHT.. • OS TESS I FEEL CALM. SEBD SEBD G0T021 AS G0TO22 22 2 2 8 BS CS DS • OTS SOBES BODEBS ?EBTJ I AB J I T T E R T . BOT AT ALL SOMEWHAT MODERATELY SO VERT MUCH 30 I ANSWER A.B.C.OB i TESS BOS > G0TO22 _ABSWER A,B.C.OR D. D TO THE STATEMENT. .1 AM JITTERT. SEHD SEND AS GOT023 23 BS CS OS HOTS SOBES BODS VESTS «t 1 2 2 1 LET'S TRT ONCE MORE TO FIND THE INSTAHTANEOOS SPEED USING THE EQUATION OF HOTION, S = T*2. FIRST,WHICH OF THE FOLLOWING TS THE AVER? GE StZIO BETWEES. 1*2 AND T=-'~ D * L ( T ) ? ANSWER A-B.C,D.OR E.~ A ( 2 * D E L . T . ) * 2 B ( 2 * EEL (T) ) *2-2*2 C__(2 + DEL(T) ) *2-2*2 /___(2_3EL ,T) )-2 D J 2 - DEL (T) ) «2-CSi. (T) *2 / DSL (T) E HONE o r THE" ABOVE - - - - - - --DO SOME ALGEBRA TO GET DEL (S. .,.HD DEL (T) BEF'.RE FI3DING AVERAGE SPEED I B _TBIS INTEBVAL. CS TE',2 AVERAGE SPEED I S SIMPLY DEI (S)/DEL (T) IN THE INTERVAL,WHICH IS _G I VE H_B I_ (2 • D E L_( T) ) •2-2*2 /_(2«DEL (T) ) - 2. SEBD AS " BS BBOHG. LOOK AT AH INTERVAL STARTING AT T=2 AND ENDING AT T=2*DBL.T.. THE AVERAGE SPEED I S GIVEN BY DEL (S)/DEL(T) IN THIS INTERVAL SEND SEBD YOU SHOULD REPEAT THE MATERIAL DE is LING 'i-ITS SHEET 5 TO BECOBE CLEAR AEOUT THIS. _ E I T H E R TYPE ...REVIEW .OR ANSWER AG AIW. "DS " ES " " " "'""" " KRONG. LOOK AT AN INTERVAL STARTING AT T=2 AND ENDING AT T=2*DEL(T). TBE AVERAGE SPEED I S GIVEU BY DEL(S)/DEL(T) IN THIS INTERVAL SEHD 210 TOO SHOULD REPEAT THE HATERIAL DEALING NITH SHEET 5 TO BECOHE CLEAR ABOUT THIS. EITHER TTPE ...REVIER ,0R ANSWER AGAIN. SEBD REVS ROS DORS OKS • GOTO11 __J 4 3 5 3 2 BHAT IS THE INSTANTANEOUS SPEED AT T=2? SEBD Jt.EHEBBER THE RELATIONSHIP BETWEEN AVERAGE SPEED AND INSTANTANEOUS SPEED ABD THEN GO AHEAD... SEBD 4 S 4.$' FOURS THE AVERAGE SPEED I S DEL (S )/DEL (T) = (2»DEL ( T H »2- 2»2 / 2 + D E H T 1 - 2 SIMPLIFYING THE NUMERATOR AHD DENOMINATOR GIVES DEL(S)/D£L (T) = 4D£L (T) * t _ L (T) *2 / DEL (T) NOW CANCEL DEL (T) _FBOn BDMEBATOR AND DENOMINATOR AND WE END DP WITH 4*DEL(T). THE L I M I T OF 4 • DEL (T) AS DEL (T) TENDS TO ZERO IS 4, WHICH I S THE INST. SPEED SEBD 0/OS OS ZEROS 0$ 0/OS BO.SIMPLIFY__THE_BUH_ERATOR AND_DENOMINATOR^F__pELJS1/_DEL (T1.THEN CANCEL DEL (T) FROM NUMERATOR AND DENOMINATOR. THEN FINE THE LIMIT OF THE BEHAINING EXPRESSION AS DEL(T) TENDS TO 0...GO ABEAD AND BE CAREFUL... SEBD _2_S 1 $ DELS _ BO.SIMPLIFY THE NUMERATOR AND DENOMINATOR OF DEL (S) /DEL (T) . THEN CASCEL DEL (T) FROM NUMERATOR AND DENOMINATOR. THEN FIND THE LIMIT OF Tiig BEHAINING EXPRESSION AS DEL (T) TENDS TO 0...GO AHEAD AND BE CAREFUL... SEBD /$ •* BO.SIMPLIFY THE NUMERATOR AND DENOMINATOR OF DEL (S)/DEL (T).THEN CANCEL D_L(T) FROM NUMERATOR AND DENOMINATOR. THEN FIND THE LIMIT OF THE BERAlaING EXPRESSION AS DEL(T) TENDS TO 0...GO AHEAD AND BE CAREFUL... SEHD 25 2 2 2 _ CONGRATULATIONS! TOU HAVE COMPLETED A CAI LESSON.I UOPE THAT TOU ENJOYED OUR CONVERSATION AS MUCH AS I DID AND THAT YOU LEAPNED SOMETHING TOO. DID YOU ENJOY T!IIS METHOD OF_LEABHING? AHSWER YES OR NO. - — SEHD AHSWER TES OR NO. SEHD TESS NOS I F TOO WISH TO HAKE ANY COMMENTS ABOUT THE LESSON,ASK THE IHSTBDCTOR TO PROVIDE TOU WITH A COMMENT SHEET. GOOD-BTE FOR HOW... SEBD BATSES DOHS PLEASE AHSWER TES OR HO. SEHD FILE t_3 . • • 1 ' 4 8 2 2 3 LET'S ST"DT TEE MOTION OF A BOAT AS IT LEAVES A DOCK. SUPPOSE THAT IT'S MOVING AT A CONSTANT SPEED CF 10 MILES/HODS. BHAT I S THE DISTANCE S OF THE BOAT FRO_ THE DOCK AT ANY TIME T? S2HD 8 EH EMBER THAT DISTANCE= (SPEED) (TIME).WRITE AN EQUATION FOR S IN T-RBS OF T. SEND 10TS " 10*TS '.0 TIMES S I C . T J S=10TS S=T10S T10» T*10S THE EQUATION WHICH DESCRIBES THE MOTION OF THE BOAT IS 3=10T. THIS I S SIMPLY Trig H . i i i i . r . DISTANCE^ (SPEED) (TIBE) EQUATION, WITH SPEED V l E I N G "CONSTANT A .'"lb "MILES/HOUR. SEBD 10=S/T $ T=10/S _ TOoP_RES. ONSE I S HOT THE CORRECT ORE. TRT AGAIN... ..SEND fO S T $ TOUR ABSWER I S INCORRECT. THINK ABOUT THE PROBLEB AHD TBI AGAIB... ?*.ND S S DISS 10SS 211 THINK ABOUT T H E PROBLEM AGAIN AND T B I TO F I N D T H E RIGBT ANSWER. GO AHEAD.. 2 I M ) 2 LOOK AT SH E E T 1 THAT BAS G I V E N TO IOU.THE GHAPH I N D I C A T E S HOB FAR T H I S BOAT I S FROH T H E DOCK AT AN. T I H E . BHAT I S TH E_S LO P E_ 0 F._T HIS . ST B AIG H T _ L I N E_R EP RESENTING THE BOAT'S SEHD BOTIOH IH A T I H E INTERVAL T = 2 TO T=6 HOURS? SEBD IOO NEED TO OBTAIN T H E CHANGE IN DISTANCE AND T H E CHANGE IH T I H E B E T V E E B _ T H E S E TWO POINTS I N ORDER TO C A L C U L A T E T H E S L O P E . GO AHEAD... SEHD 10S 4 0 / 4 S 0 - 2 0 ) / ( 6 - J 6 0 - 2 0 / 6 - 2 S T H E S L O P E I S T H E CHANGE I N DISTANCE D I V I D E D B I T H E CHANGE I B T I H E . BE WRITE T H I S AS SLOPE=DSL IS)/DEL(T1 WHICH I S EQUAL TO ' ( 6 0 - 2 0 ) / ( 6 - 2 ) =10 HILES/HOUB. ' SEHD 1/10S . 1 $ U/UOS 2/20S 6/60S 8/80S 6 - 2 / 6 0 - 2 0 S 6 - 2 ) / ( 6 0 - S _THAT'_S HOT T H E ANSWER I'H LOOKING FOB. 6 0 S 6 0 / 6 S 2 0 $ 2 0 / 2 $ HO. THINK ABOUT I T AHD T B I AGAIH... 8 0 / 8 $ S/T$ THINK ABOOT IT AHD TBI AGAIH... SEND SEBD 1 0 . 3 4 5 CHECK IOUR WORK AHD TRY AGAIH. 8 4 2 SEHD SEHD SEHD BBAT I S THE S L O P E OF T H E L I N E IN T H E TIHE INTERVAL T»4 TO T=8 HOURS? TOO HEED TO OBTAIN T H E CHANGE IN DISTANCE AND T H E CSASGE IH TIHE BETWEEB T H E S E TWO POINTS IN ORDER T0_CALCULATE__THE S L 0 P E . GO AHEAD... 10$ 4 0 / U S 0-40) / (8-$"80~-40/8-4$C6HS~TAHTS THE S L O P E IS CONSTANT AT 10 HILES/HOUR. T H I S I S ALWAYS T H E C A S E WHEN JtHE GRAIH OF T H E BOTION I S A S T R H G H T L I N E . T H E GRAPH IS J U S T A P I C T U R E OP T H E EQUATION OF H.YIOH S=10T,AHD SO THE SLO P E OF T H E LINE I S THE S P E E D OF T H E BOAT. SEND 1/10S _J$ 4/10$ 2/20S 6/60$ 8/80$ 3- 4 / 8 0 - 4 0 $ 8 - 4 ) / (8 0 - $_ THAT'S HOT T H E ANSWER I'H LOOKING FOB. T H I N ? ABOUT IT AHD T E Y AGAIN... SEND 8 0 S 8 0 / 8 $ 4 0 $ 4S JtO. THATJS INCORRECT... TRT AGAIH... 2 0 / 2 $ SEHD 6 0 / 6 $ TOOB RESPONSE IS INCORRECT. - 4 TRT TO ANSWER AGAIH... SEND 4 4 4 3 2 SHEET 2 ILLUSTRATES THE HOT ION OF A SECOND BOAT _CGrtPUTE THE AVERAGE SPEED OF THE BOAT IH THE IUTERVAL FBOH T=2 TO T=6 BBS. PROCEED EXACTLY AS BEFORE. 2$ 8/4S ( 9 - 1 ) / ( 6 - $ 9 - 1 / 6 - 2 $ SEHD SEND THE AVERAGE S P E E D I S SIMPLY T H E CHANGE IH DISTAHCE D I V I D E D BT T B E CHANGE IN T I B E . T H I S I S GI V E N BY V = --_ ( J ) / D E L (T) = ( 9 - 1 ) / (6-2) =8/4 = 2 MLSS/HCUu 9/6S 1/2$ 1.5$ .5$ BO. CHECK YOUR WORK AND TRT AGAIH... 9 $ 1$ 6S TOOB BE.POHSF. I S INCORRECT. T"Y TO ANSWER AGAIN... 8$ 4$" VOUB AKSWEB I S INCORRECT. iHIHK ABOUT THE PROBLEB AND TBI AGAIN... $EHt> SEHD SEND SEHD 4 4 4 3 2 LOOK AT S B E E T 3.WHICH I S J U S T T H E GRAPH I H SHEET 2 3 1 - - T H - E E POIHTS P,Q, AND R I N D I C A T E D ON I T . BHAT IT- THE SLOPE OF THE LINE SEGMENT JOINING THE POINTS P AND Q? , .00 NEED TO ».NO'J THE CHANGE IN DISTANCE AND THE CHANGE IN TIBS BETWEEN _P A N D Q I N ORDER TO CALCULATE TH. SLOPE OF T l i t LIN P.5EGBENT GO AHEAD.. 2$ ' " "f'.'4$ ' "'" ' » 9 - l ) / . 6 - « 9 - 1 / 6 - 2 S " " """ T H E SLOPE OF THE LINE SEGMENT JOINING P AND Q I S GIVEN BT CHANGE IH DISTAHCE DIVIDED BY CHARGE IH TIME. WHICH I S DEL (S)/DEL (T) =3/4= 2 H/HB. SEND SEND 212 THIS I S THE SAME IS THE AVERAGE SPEED OP THE BOAT BETWEEN P ABD Q. SEBO 9/6$ V 2 $ 1.5$ .5$ THAT'S BOT THE ANSWER I ' f l LOOKING FOB. TBIBK ABOUT IT ABD T B I AGAIN... SEBD 9S IS 6S BO. T B I AGAIB SEBD _8 S 4$_ TOUB ABSWER I S INCORRECT. THINK ABOOT THE PROBLEM ABD TRT AGAIN... SEBD 6 __4 . 8 3. 2 .1 . .. LET'S TAKE A SMALLER SIZE INTERVAL THAN BEFORE ON THE GRAPH IH SHEET 3. BBAT I S THE AVERAGE SPEED OF THE BOAT IN THE INTERVAL T=2 TO T»4 HOURS? SEHD PBOCEED EXACTLI_AS_BEFORE._. SEND 1.5S 3/2$ 1»l/2$ 11/2$ 1 1/2S 4-1/4-2$ (4-1) / (U-SHALS THE AVERAGE SPEED I S V = DEL ( S ) / D E L (T) » («-1) / (1-2) = 1. 5 S/HR. _THE_.SLOPE OF T H E LINE SEGMENT JOINING P AND R I S AGAIN T H E SAME AS THE AVERAGE SPEED OVER THIS INTERVAL. HOTICE THAT T H E AVERAGE SPEED IS NO LONGER COHSTANTM I F THE GRAPH I S HOT A STRAIGHT LINE.THE SLOPS I S NO LONGER COHSTAHT. SEND 1 S 4/4$ 1/2S .5$ 1MIS BBONG. T B I AGAIH... SEHD _3S 2$ _ _ • _ THINK ABOUT THE PROBLEM AGAIN AND TBI TO FIHD THE EIGHT ARSWEB. GO AHEAD.. SEHD «s TOUB BESPOHSE I S IH COB SECT. TRI TO ANSWER AGAIH... SEBD 7 4 3 2 3 5 JRBAT HAPPENS TO T H E SPEED AS WE BAKE THE TIME IBTEBVAL SMALLER ABD _ SMALLER.LET'S CONTINUE WHAT WE DID WITH SHEET 3 BI CONSTRUCTING A TABLE. LENGTH OF INTERVAL ( H R S ) » • 4 2 1 .5 .25 0 SLOPE OF L I N K(MILES/HRJ «« 2 1_. 5 1.25 '.12 1 _06 7 AVERAGE SPEED (M1LES/KR) ** 2 1.5 1.25 '.12 1.06 ? HEAT I S THE LIMIT OF AVERAGE SPEED AS THE INTERVAL SHRINKS TO S I Z E 0? SEHD _STUDI THE TABLE CAREFULLY AND NOTE WHAT HAPPENS TO SLOPE AND AVERAGE SPEED AS TriE LENGTH OP THE TIME INTERVAL SHRINKS FROM DEL (T 1 =4 , DOWl! TO 0. SEBD IS I.OOS 1HI$ THE L I B I T OF AVERAGE SPEED AS THE INTERVAL APPROACHES ZERO IS 1 H/HR. HE SEE THAT AS T H E INTERVAL KEEPS G E T T I N G S M A L L E R . I T SHRINKS TO A POINT AT T=2.AS SHOWN ON S H E E T 3.THE S P E E D AT T=2 I S JUST T H E S L O P E OF _THE LIHE L WHICH TOUCHES THE CURVE AT ONLY ONE POINT P. T H I S L I B E I S C1LLZD T H E TANGENT TO T H E CURVE AT POINT P. SEBD 2$ TWOS TOUR ANSWE'J I S INCORRECT. TV.INK ABOUT T H E PROBLEM AND TRY AGAIH... SEBD OS OS "ZEHOS THAT'S HOT T I E ANSWER I'M LOOKING FOR. THINK ABOUT IT ARD TRY AGAIH... SEHD 1.03$ DEFINEDS NO LIMITS NOLIMITS IOUB~ BEPLY I S NOT COBKKCT. RECONSIDER YOUR AaSBzR AND T B I AGAIH... SEHD 8 4 4 2 3 5 BRAT I S '.HE INSTANTANEOUS SPEED OF THE BOAT «T THE TIHE,T=2 HOURS? SEND THIS I S VHE SPEED AT THE IHSTAIiT OF TIME T=2 HRS. STUDY THE TABLE .'.BOVE AS CELL I S SHEET 3. THEN TRY TO ANS3PR... _ _ • SEBD 1 $ 1.00$ CNE$ 1 M $ " THE SPEED AT TIME T=2 HOURS IS DEFINED AS THE LIMIT OF THE AVERAGE SPEED AS THE TIHE IttTTPVAL ABOUT T=2 SHRINKS TO ZERO.THE SHCRTEP THE TIRE INTERVAL USED,THE CLOSER THE AVERAGE SPEED IS TO THF H-ITUM. SPEED AT THAT INSTANT. Tii£ ACTUAL SPEED AT THAT INSTANT IS J U M THE SLOPE JOF THE TAI.GENT l O THE CURVE. SEBD 2$" " " T V i S 80- THINK ABOUT I T AND VRY AGAIN... SEBD os os zer.^s 213 TOUB BESPONSE IS INCORRECT. TRT TO ANSWER AGAIN... SEND 1.03S 1.01$ NO LIMITS NOLIHITS UHDEFINEDS BBONG. TBI AGAIN SEBD 9 4 6 2 7 3 IQOK AT SHEET 4 WHICH IS THE GRAPH OP THE HOTION OP A FEATHER CROPPED rBOB A TOWER. S REPRESENTS THE DISTANCE OF THE FEATHER FROH THE GBOOHD. HHAT I S THE SPEED OF THE FEATHER AT T=2 SEC.? SEBD _BEHEHBEB THE RELATIONSHIP BETWEEN SPEED AND SLOPE ABD THEN PIBD THE INSTANTANEOUS - SPEED AT T=2 SEC. FROH THE GRAPH. SEBD -4 S -FOURS - POURS -4HIS IS-4 S =-4 $ TBE ACTUA L__SFEED_AT_T= 2__S E C . IS -4 FE ET/SEC._THI S IS T HE _SLOPE _OP -THE L I N E WHICH TOUCHES THE CURVE AT ONLY ONE POINT P.THE SLOPE IS BEGATIVE SINCE THE DISTANCE S IS GETTING SHALLER AS THE TIHE T INCREASES . _TBIS._HEANS THAT DEL (S) WILL BE NEGATIVE. SEND 4 S FOURS TOUR ANSWER IS INCORRECT. THINK ABOUT THE PR03LEH AND TRT AGAIB... SEBD 6$ 2$ 3S TWO$_ 1$ 3.5$ -3$ THINK ABOUT IT AND TRY AGAIN OR PLEASE TYPE REVIEW I P TOO TIPE...REVIEW.WE B I L L REPEAT THE LAST SECTION AGAIB FBOH SHEET 2. SEBD _BET$_ NOS DOBS __ _ _ _ GOTO 4 ~ " . 10 2 7 3 . THIS ACTUAL SPEED I S CALLED THE INSTANTANEOUS SPEED OF THE FEATHEB IT T«2 SEC. THE INSTANTANEOUS SPEED IS THE SLOPE OF THE STRAIGHT LINE BBICH TOUCHES THE CURVE IN SHEET 4 AT THE POINT P. THIS LINE IS CALLED "THE" TANGENT TO THE CURVE AT POINT P. DO YOU UNDERSTAND? SEND BE'VE DEVELOPED THE IDEA OF SLOPE REPRESENTING SPEED.TIE SLOPE OF A LIBE JOINING TWO POINTS ON THE CURVE IS AVERAGE SPEED.THE SLOJE OP A LINE TOUCHING THE CURVE AT ONE POINT IS INSTANTANEOUS SPEED.IS IT CLEAR ? SEND TESS OK$ BITS LITTLES SURES THINK $ GUESSS _THE__INSTANTANEOUS SPEED IS WRITTEN AS V=LIHIT (DEL (S)/DEL (T) ) AS DEL (T) "APPROACHES 0. THIS IS ABBREVIATED BY WRITING V=DS/DT = L I H I T (DSL (S)/DEL (T\) AS CEL(T) TENDS TO 0. DS/DT IS CALLED THE DERIVATIVE SF S WITH RESPECT TO T AND REPRESENTS THE SLOPE OF A LINE TOUCHING THE CURVE OF S VS. T AT ONLY ONE POINT. SEND •OS DONS NOTS THEN READ IT AGAIN AND ASK FOR HELP. SEBD 11 4 4 4 4 4 BOW LET'S FIND THE INSTANTANEOUS SPEED BY DSING THE EQUATION OF HOTION. SHEET 5 ILLUSTRATES THE GRAPH OF THE EQUATION OF HOxICN, £> = 3I-»2 FIRST LOCK AT THE TIHE INTERVAL BEGINNING AT T- I S3C. WHAT_IS THE DISTANCE S AT THE BEGINNING OF THIS TIHE INTERVAL? StND_ OSE THE EQUATION OF HOTION. .- $£HD 3 S THREES S = 3$ 3 (1) «2S THE DISTANCE S IS EASILY OBTAIHLD FROH THE ECUATION OF HOTIOH. S=3T*2 = 3 (1) »2 = 3. BE ARE SIHPLY CALCULATING THE DISTAilCH AT OKE POINT. SEBD _SS DISTS 3T*2$ S=3T*2$ _ IO0B REPLY I S NOT CORRECT. RECONSIDER YOUR ANSWER AND TKT A G a i H . . . SEND TS 1$ t>Et.(Tl' 1»DEL(T)$ TBAT'S NOT THE A N E V R I'H LOOKING FOR. THINK ABOUT IT AND TRT AGAIN... SEHD 0$ OS IN F I N I T Y S UNDEFINEDS BO. VRI AGAIN SEBD 12 4 1 1 1 2 •» THE LEHGTH OF THE TIHE INTERVAL INDICATED ON THE GRAPH IS D c L ( T ) . •HICH OP THE FOLLOWING REPRESENTS THE DISTANCE AT THE EBD OF THE 214 T I B E IHTERVAL.THAT IS,AT TIBE T=1»DEL(T)7 PLEASE ANSWER A.B.C.D.OR B. A 3 ( D E L ( T ) ) * 2 B 3 (1+DEL (T) ) »2 C 3 ( 1 * D E L ( T ) ) D 3T*2 E BOBE OF THE ABOVE OSE T B E EQOATIOH OF BOTIOH. _ _ i SEHD SEHD TBE TIHE IT THE END OF THE TIBE INTERVAL IS T=1*DEL(T). THEH THE DISTAHCE AT THIS VALUE OF T I B E IS GIVEN BY S = 3T«2=3 (1 *DEL (T) ) *2 WHICH IS ALSO EQUAL TO 3+6DEL (T) • 3DEL (T) *2 AS TOOB ABSBEB I S IHCOBBECT. THINK ABOUT THE PROBLEB AND TBI AGAIB. CS SEBD SEBD THINK ABOUT THE PROBLEB AGAIN AND TBI TO FIBD TH- BIGHT ANSWER. GO AHEAD.. SEND DS ES IO0B _HEPLT_IS _HOT COBBECT. BECOHSIDEB IOUR AHSWEB AHD THI AGAIH... SEBD 13 4 8 3 3 1: BE WOOL D LIKE T O F l N D_T HE AVER A G_E_ SPE E D IH THIS T I HE INTERVAL FBOH T=1 TO T*1+DEL(T).THE CHANGE IH DISTANCE, DEL (S) , IS GIVEN BY (DISTANCE AT END OF INTERVAL) - (DISTANCE AT START OF INTERVAL). THEN DEL (S_ (3*6DEL (T) + 3 DEL (T) * 2 ) - 3 _ WHAT IS THE SI M P L I F I E D FORH DEL (T) (1 + D E L ( T ) ) - 1 OF DEL (S)/DEL (T) ? DO SOB- ALGEBRA TO SIHPLIFY THIS EXPRESSION. 6*3DELS 6 • 3DELS 6+ 3 DELS 6 O D E L S 3 ( 2 * D I L S 3 (2 • DES 3DEL (T) *6$3 (CEL DEL (S) /DEL (T) = (6 DEL (T) O D E L (T) *2)/DEL (T)=6*3DEL(T) SIHCE DEL (T) CANCELS IK BOTH KOHERATOR AND DE-OHIBATOR. THIS I S TBE AVERAGE SPEED ~IH THE INTERVAL. ""' /DELS DEL (T) S DELTS »Q. CHECK YOUR WORK AND TRT AGAIN 3*6DET$ 3*6$ 3 • 6S BO. THAT I S INCORRECT... TRT AGAIN... _6»DELS 3*DEL$ _*3DEL$ 3(1*DEL$ HO. CHECK TOUR WORK AND TRT AGAIN... 14 3 2 7 3 SEHD _SZND_ (T) • $ SEND SEHD SEHD SEBD RECALLING THAT INSTANTANEOUS SPEED IS THE L I B I T OF AVERAGE SPEED AS THE TIRE INTERVAL SHRINKS TO ZERO,THAT IS,AS DEL(T) APPROACHES ZERO. S THE INSTANTANEOUS SPEED AT TIBE T=1 S E C ? DEL (T) APPROACHES 0. _R_AT FIND THE L I B I T OF DEL (S)/DEL (T) AS 6 S SIXS GOT018 SEHD SEND 3$ 0$ OS 2S 9S DELS 1 TOO S.-."! TO 2?. UNCLEAR ABOUT WORKING OUT LIHIT.S.iCu SHOULD DO SOME WORK OB _LIHITS FOR A FEW BTNUTES BEFORE GOING ON WITH THE BAIN LESSOR. PLEASE EHTER THE WORD. . . L I B I T ,OR TRY AGAIH I F 100 WISH. LIMITS -OS DOHS GOTO15 SEBD 15 7 4 4 LET S=3T*7. WHEN T=2.IT IS FAST TO SEE THAT S=13. BUT HOW DOES S BEHAVE WHEN T I S CLOSE TO 2? EXAfiIHE THE TABLE GIVEN BELOW. T 2.5 2.2 " . I 2.01 2.C01 _.00G\ S-3T + 7 14.5 I'.o i J . 3 13.03 13.003 13.0003 I S S CIO-E TO 13 WHEN t i S CLOSE TO 2? DOES S SEEN TO BS GETTING CLOSER TO 1j AS T GETS CLOMPS TO 2?TES OR HO? _T_3$ OKS THINKS SURF.S t _ SAI THAT I F S= 7, THEN S APPROACHES 13 A S T .-."PSOACHES 2 AND WHITE " LIM I T ( 3 T * 7 ) AS T APPROACHES 2,IS EQUAL TO 13. BOS DOHS HOTS HO S SEND SEND SEBD 215 LOOK AT THE TABLE CLOSELY AND ANSHER AGAIH OR ASK FOR HELP. 16 4 4 3 3 1 T H E L I B I T OF S AS T TENDS TO A PARTICtJLAH VALUE SEEHS TO BE S I H P L I T H E VALUE OF S AT THAT VALUE OF T. THIS IS GERERALLY TRUE EICEPT ONLI ORDER CERTAIN CONDITIONS,- HICH YOU. 8 ILL_L_EARN _AB.OUT_LAT.ER_I H_TO_UR CALCULUS COURSE. WHAT IS THE LIMIT OF S=20-6T AS T APPROACHES 3? F I N D THE VALUE OF S AT THE POINT T=3. _2..$ __TWO$ 2 0 - 6 ( 3 ) $ 20-18$ T H E L I B I T OP 20-6T AS T APPROACHES 3 IS 2. AS HE BAKE THE VALOE OP T CLOSER AND CLOSER TO 3,THE VALUE OF S GETS CLOSER AND CLOSER TO 2. 2 0 $ TW.S N T TS 20$ „ ; $EHD $EHD SEHD SEHD • 0 . THINK ABOUT I T AND TRY AGAIN... 6 S 6 (3) 18S IO0B_BEPLT IS_HOT CORRECT._ RECOBSIDEB TOUB AHSWE8 AHD T B I AGAIH. 14S TOOB BESPOHSE I S IHCOBRECT. 17 TRT TO AHSWER AGAIH... SEHD SEHD SEBD 4 4 2 1 2 FIBD TBE L I B I T OF DEL (S)/DEL (TJ =6• 3DEL (T) AS DEL (T) APPROACHES ZEBO. SEBD _F_IBD THE VALUE OF 6*3DEL(T) AT DEL(T)=0. _ SEBD 6 S' " S I I $ 6.$ 6 + 3 (0) $ AS DEL(T) GETS CLOSER AND CLOSER TO 0,THE VALUE OF 6+3DEL(T) GETS CLOSEB ABD CLOSER TO 6.THE L I B I T OF 6»3DEL(T) AS DEL (T) APPROACHES 0 I S 6. SEHD 3$ THREES SOUR AHSRER I S IHCORRECT. THINK ABOUT THE PROBLEB AHD TBI AGAIH... J6*3$ __ -BONG. TBI AGAIN. 9$ HIRES •O. THINK ABOUT I T AHD TRT AGAIH. s e i r SEBD SEHD 16 2 2 6 L E T ' S TAKE A SHORT BREAK FROH THE LESSOR.I'D L I K E TO KNOW HOW TOU FEEL... • •BIGHT HOW'. WHICH OF THE CATEGORIES BELOW DESCRIBE BEST TOUR BEACTIOH. .HOW. . TO THE STATEBENT I AH TENSE. A HOT AT ALL C BODERATELT SO VERY HDCH SO B SOMEWHAT PLEASE AHSWER A,B.C.OB D. AHSWER A,B.C.OR D TO DESCRIBE TOUB REACTION 3IGHT HOW TO TBE STAT^flS-T... I AH TENSE. •OS TESS G 0 T 0 1 8 SEND "SEND" AS i: cs DS HOTS S0^»$ nODERS VERTS G 0 T 0 1 9 19 2 2 8 WHICH C.'TEGORT EELOW..A, B.C .OR P..BEST DESCRIBES TC.OR HEACTI03 TO THE STATEBENT. I FEEL AT EASE. A NOT AT ALL C -ODERATELY _C B SOBEWHAT D VERT MUCH SO AHSWER A,B.C.OP D. SEND PLEASE A.B.C.OR D. SEHD •OS IESS GOTO19 AS BS c : DS HOTS SOBES HODEBS VERTS GOT020 2 0 2 2 8 I A- RELAXED. A HOT AT ALL C MODERATELY 30 B SOBEWHAT D VEST MUCH SO 216 PLEASE ANSWER A,B,C,OB D. AHSBEB A,B,C.OB D...TO THE STATEHEHT... . I AH RELAXED. •OS IESS 6OTO20 AS BS CS DS »OTS SOBES GQTQ21 SEBD SEBD BODEBS VESTS 21 2 2 8 I PEEL CALB. 1 HOT AT ALL C BODEBATELT SO B SOBEHHAT D VERT HUCH SO ANSWER A.B.C. OB D. ABSHEK A,B.C.OB D TO THE STATEMENT.. I P E E L CAEH. SEBD SEBD •os _G OT02J_ AS GOT022 _22_ IESS BS cs os • OTS SOBES BODEBS VESTS 2 2 8 I AH J I T T E B T . A HOT AT ALL _ C_ BODEBATELT SO B SOBEWHAT D TEST BUCH SO ANSWER A,B.C.OR D. ANSWER A.B.C.OR D TO THE STATEHEHT...I AB JI T T E R T . TESS BOS G0TO22 AS BS CS _DS :_»OTS SOBES GOT023 23 8 1 2 2 0 SEHD JtEBD BODS "EBIS LET'S TRT ONCE BORE TO FIND TEE INSTANTANEOUS SPEED USING THE EQUATION OF HOTION, S=T*2. FIRST,WHICH OF THE FOLLOWING I S THE AVERAGE SPEED BETHEES T°2 ABD T-""2*DEL (T) ? ANSWEB A,B,C,D,OR E. A ( 2 * D E L ( T ) ) * 2 " B (2 + DEL (T) ) *2-2*2 " " " " C ( 2 * D E L ( T ) ) * 2 - 2 * 2 / t 2 * D E L ( T ) ) - 2 D (2• DEL (T) ) *2-DEL (T) »2 / DEL (T) E BOHE OF THE ABOVE SEBD DO SOBE ALGEBRA TO GET DEL(S) AND DEL (T) BEFORE FINDING AVERAGE SPEED IB THIS IBTERVAL. SEBD _ c s _ TBE AVERAGE SPEED I S SIHPLT DHL (S)/DEL(T) IN THE INTERVAL,WHICH I S GIVEH BT (2*DEL (T) ) *2-2*2 / (? + DEL (T) ) - 2 . SEHD AS BS WRONG... TOO SBCfLD BFPEAT T B E H A T E B I A L C i i A L i - G -'ITH S H E E T 5 TO BECOBE CLEAR ABOUT T H I S . _ _ _ _ _ " " E I T H E B T Y P E ...BEVIEW ,0B AHS.WEB AGAIH. DS. ES WRONG SEBD 100 SHOULD REPEAT THE HATERIAL DEALING BITB SHEET 5 TO BECOBE CLEAR ABOUT THIS. EITHER TYPE ...REVIEW ,0R iNS«EB ACilH. . SEHD REVS BOS DOSS OKS ~ ~ ~ " G0T011 21 • * 3 5 3 "2 BHAT I S THE INSTANTANEOUS SPEED AT T=2? SEHD _6EHEHBER THE RELATIONS "'IP BETWEEN AVERAGE SPEED ABD IBSTABTANEOUS SPE'D • AliD THEN GO A"EAD... " SEHD « S «.$ FOURS THE AVERAGE SPEED IS ""EL (S)/DEL (T) = (2*DEL (T) ) »2-2*2 / 2->0EL(T)-2 217 S I H P L I F T I N G THE NUMERATOR AND DENOHINATOR GIVES DEL (S) /DEL (T) = UDEL (T) + DEL (T) »2 / DEL (T) NOW CANCEL DEL (T) FBOH NUMERATOR AND DENOHINATOR AND WE END UP WITH 1*DEL ( T ) . THE L I B I T OF <**DEL(T) AS DEL (T) TENDS TO ZERO IS a,WHICH I S THE U S T . SPEED 0 / 0 $ 0$ ZEROS OS O/OS BO. CHEC K_ Y0UR WORK AND___HY_AGAIN 2 S I S DELS THAT'S HOT THE ANSWER I'H LOOKING FOB. THINK ABOUT IT AHD T B I AGAIH... 1 \ SEHD SEBD vs_ •s IOUB ANSWER I S INCOBRECT. 25 2 2 2 THINK ABOOT THE PROBLEM AHD TBI AGAIH... SEBD SEBD COBGBATOLATIONS! YOU HAVE COMPLETED A CAI LESSOH.I HOPE THAT YOU ENJOYED OUR CONVERSATION AS MUCH AS I DID AND THAT TOD LEABBED SOHETHIBG TOO. _ DID YOU ENJOY THIS METHOD OF_LEABNING? . ANSWER YES OR NO. ANSWER IES OR HO. TESS HQS SEHD SEBD IF TOO WISH TO HAKE ANY COHHENTS ABOUT THE LESSON,ASK THE IHSTBUCTOB TO PBOVIDE TOU WITH A COHHENT SHEET. GOOD-BYE FOB BOW... SEBD J_ATBE$ _ DONS . ... PLEASE ANSWER YES OB HO. SEHD FILE 218 APPENDIX E L i s t i n g for One CAI Student Session (T ) «source pre #$RUN C A I P R E . 0 4 - 0 A T A P R E U A S T + 1 ) S - P R E T E S 9-»MS0URCE* fEXECUTION tSEGINS P L E A S E ENTER YOUR F I R S T AND LAST NAME • •• H I . I'M YOUR PERSONAL TUTOR FOR TODAY.L E T'S START BY DOING A LESSON ON SOME OF T H E THINGS THAT YOU SHOULD KNOW BEFORE DOING THE MAIN L E S S O N LOOK AT T H E GRAPH SHOWN IN SHEET A. WHAT IS T H E VALUE OF Y AT THE POINT X - 1 7 % 10 OK SUPPOSE THAT YOU ARE TOLD THAT S IS A FUNCTION OF T. WHAT IS T H E VALUE OF S WHEN T « 3 , I F S AND T ARE RELATED BY T H E EQUATION S » 2 W ,READ T H I S AS TWO TIMES ( T SQUARED). 18 GOOD. SUPPOSE THAT A CAR I S T R A V E L L I N G ALONG A HIGHWAY AT A S P E E O OF 60 M I L E S PER HOUR. HOW FAR WILL THE CAR T R A V E L IN 2 HOURS? 12 0 E X C E L L E N T : A CAR P A S S E S T H E 100 M I L E POST ON A HIGHWAY AT 12 NOON.THREE HOURS L".TER, T H E CAR P A S S E S T H E 250 M I L E POST.WE U S E THE SYMBOL D E L ( S ) TO RCTRORD THE CH AN L- E IN DL STANCE, THAT I S , D E L ( S ) » { F I N A L D I S T A N C E ) - ( I N I T I A L D I S T A N C E ) WHAT IS D T L ( S ) IN T H I S CASE? 150 • . 220 EXCELLENT! K E E P UP T H E GOOD WORK. — SUPPOSE THAT A CAR T R A V E L S FROH MONTREAL TO TORONTO,A D I S T A N C E OF J 5 0 M I L E S , A i J D TriE D K I V t K STOPS S E V E R A L TIMES FOR FOOD AND G A S . WHAT S P E E D MUST THE CAR AVERAGE IN ORDER TO MAKE THE T R I P IN 7 HOURS? 50 E X C E L L E N T ! K E E P UP THE GOOD WORK. S U P P O S E THAT A N J T H t R CAR T R A V E L L I N G FROM MONTREAL TO TORONTOC350 M I L E S ) MOVES AT 70 f-lI LES / r lOUR FOR THE F I R S T 5 HOURS AND AT CO M/HR FOR THE NEXT <t HOURS. WHAT IS THE S P E E D OF THIS CAR EXACTLY aO MINUTES A F T E R L E A V I N G MONTREAL? THE SPEED AT A PARTICULAR TIME IS C A L L E D THE INSTANTANEOUS S P E E D . 70 t X L E L L t N T ! K E E P UP THE GOOD WORK. LET'S T A K E A SHORT BREaK FROM THE L E S S O N . I ' D L I K E TO KNOW HOW YOU F E E L . . . . . R I G H T NOW. WHICH OF THE CATEGORIES BELOW DESCRIBE BEST YOUR R E A C T I O N . . N O W . . TO THE STATEMENT I AM T E N S E . A NOT AT A L L C MODERATELY SO B SOMEWHAT 0 VERY MUCH SO PLEASE ANSWER A , B , C , O R D. c WHICH CATEGORY B E L O W . . A , B , C ,OR D. .BEST D E S C R I B E S YOUR R E A C T I O N TO THE STATEMENT I F E E L AT E A S E . A NOT AT A L L C MODERATELY SO B SOMEWHAT D VERY MUCH SO ANSWER A,B,C,OR D. a I AM R E L A X E D . A NOT AT A L L C MODERATELY SO B SOMEWHAT D VERY MUCH SO - - . - . P L E A S E ANSWER A , 8 , C , 0 R D. d I FEEL C A L M . A NOT AT A L L C MODERATELY SO B SOMEWHAT D VERY MUCH SO " " ANSWER A , B , C , OR D. d I AH J I T T E R Y . A NOT AT A L L C MODERATELY SO B SOMEWHAT D VERY MUCH SO " " -ANSWER A,B , C , O R 0. a NOW,LLT'S GET BACK TO THE L E S S O N . . . LOOK AT THE GRAPH OF S V S . T SHOWN IN S H E E T 8 , WHAT If. T H E S L O P E OF THE L I N E INDICATED :iN THE GRAPH? k E X C E L L E N T I K E E P UP Tr i t GOOO WORK. LET'S REVIEW SOME B A S I C A L G E B R A . SAY THAT YOU ARE GIVEN VHE E X P R E S S I O N } ( l + X ) - 2 - i ( l ) * 2 / ( 1 * X ) - 1 WHAT IS THE S I M P L I F I E D FORM OF THIS E X P R E S S I O N ? " --3x NO. THE NUMERATOR BECOMES 3 < 1 + X? « 2 - 3 C1) * 2 » 3 C 1 + 2 X + X - < i )-? « c X + 3 X « 2 « X ( 6 - 3 X ) THE DFNUMIMATOR BECOMES ( 1 * X ) - 1 = X T H E N , T H E EXPRESSION EOUALS X ( 6 + 3 X ) / X » ? " - - - - -WHAT IS THE F I N A L FORM OF THE E X P R E S S I O N ? J(2*X> OK 221 NOW YOU ARE GIVEN AN EXPRESSION ( _ * X ) « 2 - 2 « 2 / (2*X)-2 WHAT IS THE SIMPLIFIED FORM OF THIS EXPRESSION? X*k GOOD. LET'S DO THE SAME EXAMPLE AS BEFORE,BUT NOW WE'LL USE THE SYMBOLS THAT WILL BE USED IN THE MAIN LESSON. GIVEN THE EXPRESSION 3 ( 1 * D E L ( T ) ) » 2 - 3 ( 1 ) * 2 / (1*DEL(T) ) -1 WHAT IS THE SIMPLIFIED FORM OF THIS EXPRESSION? } ( 2 * d e l ( t ) ) EXCELltNTI YOU SEEM TO UNDERSTAND THE CONCEPTS NEEDED TO TAKE THE MAIN LESSON. YOU MAY TAKE A SHORT BREAK OR YOU CAN START THE MAIN LESSON RIGHT AWAY. DO YOU WANT TO TAKE A BREAK? no EXCELLENT I KEEP UP THE GOOD WORK. THEN TYPE...SSOURCE LESSON STOP U •EXECUTION TERMINATED ' s o u r c e l esson #$RUN CAI.O U=DATAMA!N< LAST+1) S-TX . - •MSOURCE* ~ 'EXECUTION BEGINS PLEASE ENTER YOUR FIRST AND LAST NAME LET'S STUDY THE MOTION OF A BOAT AS IT LEAVES A DOCK. " SUPPOSE THAT IT'S MOVING. AT A CONSTANT SPEED OF 10 MILES/Hf",R. WHAT IS THE DISTANCE S OF 7H_ BOAT FROM THE DOCK AT ANY TIM5. T? t U U ) I DON T RECOGNIZE YOUR RESPONSE. TRY AGAIN l O t OK THE EQUATION WHiCH DESCRIBES THE MOTION OF THE BOAT is S-10T. THIS IS SIMPLY THt FAMILIAR DISTMNCE-(SPEEDHTIME) EQUAT I ON, in I TH SPEEO V BEING CONSTANT AT 10 MILES/HOUR. LOOK AT SHEET 1 THAT WAS GIVEN TO YOU.THE GRAPH INDICATES HOW FAR THIS BOAT iS FROM THE DOCK AT ANY 71.!E. WHAT IS ' H E SLOPE OF THIS STRAIGHT L I N . REfP.ESEi;." i r,G THE BOAT'S MOTION IN A TIME IMTERVAL T-2 TO T-6 HOURS? 10 .GOOD. •"• THE SLOPE IS THE CHANGE IN DISTANCE DIVIDED BY THE CHANGE IN TIME. WE WRITE THIS AS SLOf _»UE_(S >/. SLO' ) WHICH IS EQUAL TO <60-20) / (6 -2) -10 MILES/HOUR. WHAT IS THE SLOPE OF THE LINE IN THE TIME INTERVAL l°k TJ T-8 HOURS? "10 ' EXCELLENT I THE SLOPE IS CONSTANT AT 10 MI LES/HOUR.THIS l_ A'.WAYS TH*.: CASE WHEN THE GRAPli CF THE MOTION IS A STPAI Gi l ' l LINE.THE C-RAPH !S JUST A PICTURE Or THE EQUATION OF MOTION S»10T,AND SO THE SLOPE OF THE LINE IS THE SPEED OF Trie BOAT. SHEET 2 ILLUSTRATES THE MOTION OF A SECOND BOAT COMPUTE THE AVEKAOc SPEED OF THE BOAT IN THE INTERVAL PROM T=2 TO T»6 HRS. 2 222 EXCELLENT I KEEP UP THt GOOD WORK. THE AVERAGE SPEED IS SIMPLY THE CHANGE IN DISTANCE DIVIDED BY THE CHANGE IN TIME.THIS IS GIVEN BY V « D E L ( S ) / D E L C T ) - ( 9 - l ) / ( 6 - 2 ) « 8 / 4 - 2 MILES/HOUR LOOK AT SHEET 3,WHICH IS JUST THE GRAPH IN SHEET 2 WITH THREE POINTS P , Q , AND R INDICATED ON IT. WHAT IS THE SLOPE OF THE LINE SEGMENT JOINING THE POINTS P AND 0.? 2 EXCELLENT! KEEP UP THE GOOO WORK. THE SLOPE OF THE LINE SEGMENT JOINING P AND Q IS GIVEN BY CHANGE IN 01 STANCE DIVIDEO BY CHANGE IN TIME,WHICH IS D E L . S ) / D E L ( T ) ' & / 4 = 2 M/HR. THIS IS THE SAME AS THE AVERAGE SPEED OF THE BOAT BETWEEN P AND Q. L E T ' S TAKE A SMALLER SIZE INTERVAL THAN BEFORE ON THE GRAPH IN SHEET. 3. WHAT IS THE AVERAGE SPEED OF THE BOAT IN THE INTERVAL T-2 TO T-4 HOURS? 3/2 EXCELLENT! KEEP UP THE GOOD WORK. THE AVERAGE SPEED IS V=DELCS)/DEL(T) « ( 4 - 1 ) / ( 4 - 2 ) - l . 5 M/HR. THE SLOPE OF THE LINE SEGMENT JOINING P AND R IS AGAIN THE SAME AS THE AVERAGE SPEED OVER THIS INTERVAL. NOTICE THAT THE AVERAGE SPEED IS NO LONGER CONSTANT 11 If THE GRAPH IS NOT A STRAIGHT LINE,THE SLOPE IS NO LONGER CONSTANT. WHAT HAPPENS TC THE SPEED AS WE MAKE THE TIME INTERVAL SMALLER AND SMALLER. LET 'S CONTINUE WHAT WE DID WITH SHEET 3 BY CONSTRUCTING A T/>bLE. LENGTH OF 1 NT E R V A L ( H R S ) * * 4 2 1 .5 .25 0 SLOPE Or L I N E ( M I L E S / H R ) « « 2 1.5 1.25 1.12 1.06 AVERAGE S P E E D(MILES/HR)** 2 1.5 1.25 1.12 1.06 ? WHAT IS THE LIMIT OF AVERAGE SPEED AS THE INTERVAL SHRINKS TO SIZE 0? 0 NO.THE LENGTH OF THE INTERVAL TENDS TO 0 BUT AVERAGE SPEED DOES NOT. LENGTH .5 .25 .15 .10 0 SLOPE * • 1.12 1.06 1.04 1.02 ? AVERAGE SPEED** 1.12 1.06 1.04 1.02 ? WHAT IS THE LIMIT OF AVERAGE SPEED AS THE INTERVAL SHRINKS TO SIZE 0? 1 OK THE LIMIT OF AVERAGE SPEED AS THE INTERVAL APPROACHES ZERO IS 1 M/HR. WE SEE THAT AS T H E IMTtKVAL KEEPS GETTING SMALLER.IT SHRINKS TO A POINT AT T-=2,.*S SH.~>WN ON SHEET 3 . T H E SPEED AT T = 2 iS JUST T H E SLOPE OF THE LINE L WHICH TORCHES THE CURVE AT ONLY ONE POINT P. THIS LINE IS CALLED THE TANGENT 10 THE CURVE AT POINT P. WHAT IS THE INSTANTANEOUS SPEED OF THE BOAT MT THE TIME,T-2 HOURS? 1 -GOOD. THE SPEED AT TIME T=2 H0U.7S IS DEFINED AS THE LIMIT OF THE AVERAGE SPEED AS THE TI Mt INTS-RV/A'. ABOUT T=>2 SHRINKS TO ZERO.THE SHORTER THE TIME INTERV/AI USEI'. T H t CLOSER THE AVERAGE f.PEED IS TO THE ACTUAL SPEED A T THAT INSTANT. YHE ACTUAL SPEED AT THAT INSTANT IS JUST THE SLOPE OF THE TANGENT TO TriE CURVE. LOOK AT SHEET 4 WHICH IS THE GRAPH OF THE MOTION 0? A FEAThER DROPPED FrfOM A TOWER, i RtPREb 1NTS THE D iSTANCE OF THE FEATHER FROM THE GROUND. WHAT IS THE SPEED OF THE FEATHER AT T-2 S E C ? * YOU'RE .'.'.MOST RIGHT.T!'E CHANGE IN DISTANCE D L L ( S ) IS GIVEN BY (FINAL DiSTAiJCt)-< INITIAL DISTANCE). SINCE DISTANCE IS GETTING SMALLER AS TIMt -tTS BIGGtR,DtL(S) WILL BE NEGATIVE. TRY AGAIN. -4 OK THE ACTUAL SPtED AT T-2 SEC. IS -4 FEET/SEC. THIS IS THE SLOPE OF THE LINE WHICH TUUCHtS THE CURVE AT ONLY ONE POINT P.THE SLOPE IS NEGATIVE SINCE THE DISTANCE S IS GETTING SMALLER AS THE TIME T INCREASES THIS MEANS THAT D E L ( S ) WILL BE NE-ATIVE. THIS ACTUAL SPEED IS CALLED THE INSTANTANEOUS SPEED OF THE FEATHER A T T-- SEC. THE INSTANTANEOUS SPEtD IS THE SLOPE OF THE STRAIGHT LINE WHICH TOUCHES TriE CURVE IN SHEET 0 «T THE POINT P. THIS LINE IS CALLED THE TANGENT TO TriE CURVE AT POINT P. OO YOU UNDERSTAND? yes GOOD. TriE INSTANTANEOUS SPEED IS WRITTEN AS V - L I M I T ( O E L C S ) / D E L ( T ) ) AS D E L ( T ) APPROACHES 0. THIS IS ABBREVIATED BY WRITING V«OS/DT-LIMIT(DEL(S)/DEL(T)) AS DEL(T) TENDS TO 0. DS/DT IS CALLED THE DERIVATIVE S F S WITH RESPECT TO T AND REPRESENTS THE SLOPE OF A LINE TOUCHING THE CURVE OF S VS. T AT ONLY ONE POINT. NOW L E T ' S FIND THE INSTANTANEOUS -SPEED BY USING THE EQUATION OF MOTION. SHEET 5 ILLUSTRATES THE GRAPH OF THE EQUATION OF MOTION, S-3T*2 FIRST L O O K AT THE TIME INTERVAL BEGINNING AT T - l SEC. WHAT IS THE DISTANCE S AT THE BEGINNING OF THIS TIME INTERVAL? 3 EXCELLENTI THE DISTANCE S IS EASILY OBTAINED FROM THE EQUATION OF MOTION. S»JT»2«3C1)«2»3. WE A R E SIMPLY CALCULATING THE DISTANCE AT ONE POINT. THE LENGTH O F THE TIME INTERVAL INDICATED ON THE GRAPH IS D E L ( T ) . WHICH O F THE FOLLOWING REPRESENTS THE DISTANCE AT THE END OF THE TIME INTERVAL,THAT IS,AT TIME T-1+DELCT)? PLEASE ANSWER A,B,C,0,OR E. A 3 ( D E L ( T ) ) * 2 B 3(1+DEL(T))«2 C 3(1+DEL(T)> D 3T*2 E NONE OF THE ABOVE 3 I DON T RECOGNIZE YOUR RESPONSE. TRY AGAIN b OK THE T I M L AT THE END O F THE TIME INTERVAL IS T»1+DEL(T). THEN THE OlSTANCt AT T r i i S VALUF OF TIME IS GIVEN EV S-3T«2 C?(1+DEL(T))«2 WHICH IS ALSO EQUAL TO 3+6DEL(T)•30EL(T>*2 WE WOULD L I K E TO FIND THE AVERAGE SPEED I N THIS TIME INTERVAL FROM T - l tO T=:-.*DEL(Y).THE CHANliE IN DISTANCE,DEL(Sj, I S GIVEN BY (DISTANCE AT END O F INTERVAL)-(DI STANCE AT START OF INTERVAL). THEN D E L ( S ; (3+6DEL(T>-JDELCT)*2}-3 . WHAT IS THE SIM P L I F I E D FORM DEL(T) (1+DEL(T))-1 OF D E L ( S ) / D E L ( T ) ? 3<2+de1(C)) GOOD. DEL(S)/DEL<T)=<6DEL(T)*30EL(T)*2)/DEL(T)-6*J0EL<TJ SINCE DEL!T) CANCEL-IN BOTH NUMERATOR >ND DENOMINATOR. THIS IS THE AVERAGE SPEED IN THE !NTEPVAL. RECALLING THAT INSTANTANEOUS SPEFD IS THE LIMIT OF AVERAGE SPEED AS THE TIME INTERVAL SHRINKS TO ZERO,THAT IS,AS DEL(T) APPROACHES ZERO. WHAT iS THE INSTANTANEOUS SPEED AT TIME T»i S E C ? 6 EXCELLENT ! 224 LET'S TAKE A SHORT BREAK FROM THE LESSON.I'D LIKE TO KNOW HOW YOU FEEL... ..RIGHT NOW. WHICH OF THE CATEGORIES BELOW DESCRIBE BEST YOUR REACTION..NOW.. TO THE STATEMENT I AM TENSE. A NOT AT ALL C MODERATELY SO 8 SOMEWHAT D VERY MUCH SO PLEASE ANSWER A,B,C,OR 0. •> WHICH CATEGORY dELOW..A,B,C,OR D..BEST DESCRIBES YOUR REACTION TO THE STATEMENT. I FEEL AT EASE. A NOT AT ALL C MODERATELY SO B SOMEWHAT D VERY MUCH SO ANSWER A,B,C,OR 0. C • . I AM RELAXED. A NOT AT ALL C MODERATELY SO B SOMEWHAT 0 VERY MUCH SO PLEASE ANSWER A,B,C,OR 0. c I FEEL CALM. NOT AT ALL C MODERATELY SO SOMEWHAT D VERY MUCH SO ANSWER A, B,C, OR D. I AM JITTERY. A NOT AT ALL C MODERATE:Y SO • ' B SOMEWHAT 0 VERY MUCH SO ANSWER A,B,C,OR D. • LET'S TRY ONCE MORE TO FIND THE INSTANTANEOUS SPEEO USING THE EQUATION OF MOTION, S=T*2. FIRST,WHICH OF THE FOLLOWING IS THE AVERAGE SPEED BETWEEN T-2 AND T»2*DEL(T)? ANSWER A,B,C,D,OR E. A <2*DEL(T))*2 S (2+DEL(T))*2-2»2 C <2*DEL(T>)*2-2«2 / (2+DEL(T))-2 D (2+DEL(T))»2-DEL(T)*2 / DEL(T) E NONE OF THE ABOVE • NO. THIS REPRESENTS DISTANCE.THE DISTANCE AT T-2 IS S*T»2 = 2*2-k AND THE DISTANCE AT 1-=2*D EL C T ) IS S-<2+DEL(T))*2. YO'J MUST DIVIDE D E L C i ^ BY OEL(T). YOU SHOULD REPEAT THE MATERIAL DEALING WITH SHEET 5 TO BECOME CLEAR ABOUT THIS. EITHER |-.'Pt ...REVIEW ,0R ANSWER AGAIN. e NO. THE AVERAGE SPEED IS DEL(S2,'DEL(T) IN THE INTERVAL. DEL( S )"F INAL 01 STAN CE- I NI T i AL D I ST ANCE= ( 2+L'ELCT) W - 2 * 2 AND ' OEL(T) = FINAL TI ME-INITIAL TIME=(2+DEL(T))-2. S I M P L I F Y D E L ( S ) AND DEL(T) AND DIVIDC THLM.YOU SHOULD REPEAT THE MATERIAL OtAHNG WITH SHEET i TO EE '-LEAP. AEOUT THIS. TYPE.. .REV I EW .OR TRY AGAIN, review NOW LET'S FIND THE INSTANTANEOUS SPEED BY USING THE £QUATION OF MOTION. SHEET 5 ILLUSTRATES THE GRAPH OF THE EQUATION 0'- MOTION, S-=3T»2 FlkST L00£ AT THE TIME INTERVAL BEGINNING AT J"X SEC. WHAT IS THE DISTANCE S AT THE BEGINNING OF THIS TIME INTERVAL? 3 OK THE DISTANCE S IS EASILV OBTM fit, 1) FRO" THE EQUATION OF MOTION. S«;T«2=i(l)«2-3. WE ARE S l h P L V CALCULATING THE DISTANCE AT ONE POINT. T H E LENGTH OF THE TIME INTERVAL I NO ICATED ON THE GRAPH IS D E L ( T ) . WHICH OF THE FOLLOWING REPRESENTS THE DISTANCE AT THE END OF THE T I M E INTERVAL,THAT I S , A T TIME T-1+DEL<T)? PLEASE ANSWER A , B , C , D , O R E . A 3 ( D E L ( T ) ) * - B 3{l+DELCT))*1 C 3 U * D E L ( T ) ) D 3T*2 E NONE OF THE ABOVE c WRONG. THE DISTANCE IS GIVEN BY S - 3 T * _ . YOU HAVE CHOSEN THE DISTANCE AT T H E END OF THE INTERVAL FOR AN EQUATION OF MOTION S - 3 T « 3 { l + D E L C T ) ) . TRY A G A I N , b OK. T H E T I M E AT THE ENO OF THE TIME INTERVAL IS T-1+DELlT) . THEN THE D I S T A N C E AT THIS VALUE OF TIME IS GIVEN BY S« 3 T » _ » 3 ( 1 + D E L ( T ) ) WHICH IS ALSO EQUAL TO .3+6DELCT)*3DEL(T)*2 WE WOULD LIKE TO FIND THE AVERAGE SPEED IN THIS TIME INTERVAL FROM T » l TO T - 1 + D E L ( T ) . T H £ CHANGE IN DI STAN CE, DEL( S ) , I S GIVEN BY (DISTANCE AT END OF IMTERVAL)-(0 I STANCE AT START OF INTERVAL). THEN DEL ( S ) (3+6DELCT)+3DELCT)*2)-3 . WHAT IS THE SIMPLIFIED FORM O E L ( T ) U * 0 E L ( T ) ) - 1 OF DEL(S ) / D E L ( T > ? 3 ( 2 * d e i ( t ) ) GOOD. . . . . . . . . . . . . . - • . OEL(S ) / D E L ( T > » ( 6 D E L ( T ) * 3 D E L C T ) * 2 ) / D E L ( T ) « 6 + 5 D E L ( T ) SINCE DEL(T) CANCELS I N BOTH NUMERATOR AND DENOMINATOR. THIS IS THE AVERAGE SPEED I N T H E INTERVAL. R E C A L L I N G THAT INSTANTANEOUS SPEED IS THE LIMIT OF AVERAGE SPEED AS T H E TIME INTERVAL SHRINKS TO ZERO,THAT IS,AS DEL(T) APPROACHES ZERO. WHAT IS THE INSTANTANEOUS SPEED AT TIME T - l SEC.? 6 E X C E L L E N T 1 L E T ' S TAKE A SHORT BREAK FROM THE LESSON.I 'D LIKE TO KNOW HOW YOU F E E L . . . . .RIGHT NOW. WHICH OF THE CATEGORIES BELOW DESCRI3E BEST YOUR REACT ION..NOW.. TO T H E STATEMENT I AM TENSE. A NOT AT ALL C MODERATELY SO B SOMEWHAT D VERY MUCH SO P L E A S E ANSWER A , B , C , O R D. i> WHICH CA7--0RY . - L G W . A , B , C , OR D. .BEST DESCRIBES "CUR REACTION TO THE STATEMENT I FEEL AT EASE. A NOT AT ALL C MODERATELY SO B SOMEWHAT D VERY MUCH SO ANSWER A , B , C , O R 0. C . - •-I AM RELAXED. A NOT AT ALL C MODERATELY sn . • B SOMEWHAT 0 VERY MUCH SO PLF„',F. ANSWER A ,B ,C ,OR D. C . . • I FEEL CALM. A NOT AT ALL C MODERATELY SO 8 SOMEWHAT D VERY MUCH SO ANSWER A , B , C , OR D. c . . . . . A B I AH JITTERY. NOT AT ALL C MODERATELY SO SOMEWHAT D VERY MUCH SO 226 ANSWER A,B,C,OR 0. • LET'S TRY ONCE MORE TO FIND THE INSTANTANEOUS SPEED USING THE EQUATION OF MOTION, S»T*2. FIRST,WHICH OF THE FOLLOWING IS THE AVERAGE SPEEO BETWEEN T«2 AND T»2 + DEL(T )? ANSWER A,B,C,D,OR E. A ( 2 * D E L ( T ) W B ( 2 * D E L ( T ) ) «2 -2«2 C <2*DEL(T))*2-2«2 / ( 2*DEL(T ) ) - 2 D ( 2*DEL(T ) ) « 2-DEL(T ) » 2 / DEL(T) E NONE OF THE ABOVE C EXCELLENT! KEEP UP THE GOOD WORK. THE AVERAGE SPEED IS SIMPLY DEL(S)/DELCT) IN THE INTERVAL,WHICH IS GIVEN BY ( 2 * D E L ( T ) W - 2 « 2 / (2+DEL(T ) > - 2 . WHAT IS THE INSTANTANEOUS SPEED AT T -2? ft EXCELLENT 1 KEEP UP THE GOOD WORK. THE AVERAGE SPEED IS D E L ( S ) / D E L ( T ) = ( 2 + D E L ( T ) ) « 2 - 2 * 2 / 2+OEL(T)-2 S I M P L I F Y I N G THE NUMERATOR AND DENOMINATOR GIVES DEL(S)/DEL(T)=ltDEL( T)+DEL(T)*2 / OEL(T) NOW CANCEL DEL(T) FROM NUMERATOR AND DENOMINATOR AND WE END UP WITH i»*DEL(T). THE LIMIT OF <**OEL(T) AS DEL(T) TENDS TO ZERO IS U,WHICH IS THE INST. SPEED CONGRATULATIONS! YOU HAVE COMPLETED A CAI LESSON.I HOPE THAT YOU ENJOYED OUR CONVERSATION AS MUCH AS I DIO AND THAT YOU LEARNED SOMETHING TOO. DID YOU ENJOY THIS METHOD OF LEARNING? ANSWER YES OR NO. yes EXCELLENT! KEEP UP THE GOOD WORK. IF YOU WISH TO MAKE ANY COMMENTS ABOUT THE LESSON,ASK THE INSTRUCTOR TO PROVIDE YOU WITH A COMMENT SHEET. GOOD-BYE FOR NOW... STOP 0 •EXECUTION TERMINATED *slg s . -'OFF AT l i t : 32: 39 SUN MAR 18/73 #E 60.580 $3.02 / 'C 20.13 $1.00 | 'C 15,55 $.26 fW 06.006 $13.92 • D 2188 . .. . « $26.02 tP." . ' $ 2 5 2 . « 6 227 APPENDIX F Post test 228 F I N A L T E S T INSTRUCTIONS: Please answer a l l questions. Enter your choice i n the space provided on the answer sheet. Use the scrap paper provided for your c a l c u l a t i o n s . 229 What i s the name for the slope of a straight l i n e segment j o i n i n g two points on the graph of distance vs. time? a tangent b deri v a t i v e c average speed d speed e instantaneous speed What i s the name for the slope of a straight l i n e segment which i s tangent to the curve of distance vs. time? a average speed b instantaneous speed c tangent d change e speed What i s the name f o r the l i m i t of average speed i n an i n t e r v a l as the size of that i n t e r v a l approaches zero? a derivative b average speed c instantaneous speed d tangent e speed What i s the name for the slope of a straight l i n e segment which i s tangent to the curve of S vs. T and i s written as DS/DT? a deriv a t i v e b slope c tangent d change e speed 230 Q) rH •H CO w < H CO M Q 50 40 30 20 10 0 > 0 10 15 20 25 TIME T (hours) Figure 1 5. Refer to the graph i n figure 1. Calculate the average speed i n the i n t e r v a l between T = 5 and T = 20 hours. a b c d e 15 20 2 1 231 OJ t-i •H 1 0 w. CO 14 12 10 8 Graph of S vs. T ~ ? J •• y • 7 0 4 - 5 TIME T (hours) Figure 2 6. Refer to the graph shown i n Figure 2. Compute the average speed i n the i n t e r v a l from T = 1 to T = 3 hours. a 0.2 b 1.0 c 3.3 d 4.0 e 4.5 7. Refer to the graph shown i n Figure 2. Compute the instantaneous speed at T = 2 hours. a 1 b 2 c 3 d 4 e 5 8. Refer to the graph shewn i n Figure 2. Compute the derivative of distance with respect to time, that i s DS/DT at the point T = 2 a 1 b 2 c 3 d 4 e 5 232 9. What i s the value of the expression given below? LIMIT OF (3T+4DEL(T)-6) AS DEL(T) TENDS TO O. a 3T b 3T+4 c 3T-6 d -6 e 3 10. What i s the value of the expression given below? LIMIT OF (4TDEL(T)*2+DEL(T)+5) AS DEL(T) TENDS TO O. a 5 b 4 c 2 d 4T e 4T+5 11. Define the instantaneous speed V, at any time T, as a function of distance(T) and time (T). a V = S/T b V = DEL(S)/DEL(T) c V = LIMIT (S/T) AS T TENDS TO O. d V = LIMIT (PEL(S)/DEL(T)) AS DEL(T) TENDS TO O. , e V = LIMIT (3/T) AS DEL(T ) TENDS TO O. 12. Given that S = 2T represents the distance S of a boat at any time T, compute the average speed of the boat between T = 1 and T = 1+DSL(T). a 0 b 2 c 4 d 6 e 8 13. For . the boat in' problem 12, compute the instantaneous spaed at T~3. a 0 b 2 c -3 d t e h 233 14. Given that S=T*2+8 represents the distance S of a boat at any time T, compute the average speed of the boat between T=l and T=1+DEL(T). a (1+DEL(T))*2+8 b (1+DEL(T))*2-l c (1+DEL (T) )*2-l DEL(T) d (1+DEL(T) )*2+8 DEL(T) e (1+DEL (T) ) - l DEL(T) 15. For the boat i n problem 14, compute the instantaneous speed at T = 1. a (1+DEL(T))*2 b 2+DEL(T) c 8 d 1 e 2 16. Given an equation S = 6T, ca l c u l a t e the derivative of S with respect to T, DS/DT, at the point T = 2. a 0 b 2 c 6 d 12 e 36 17. Given an equation S=T*2, calcu l a t e DEL(S)/DEL(T) a 2T b (T+DEL(T))*2 c T+D£L(T) d 2T+DEL(T) e DEL(T)*2 234 Given an equation S=T*2, calcu l a t e the deriva t i v e of S with respect to T, DS/DT, at the. point T = 4, using the l i m i t d e f i n i t i o n . a 2T b T*2 c 2 d 4 e 8 NAME: 235 ANSWER SHEET 1. 2. 3. 4. 5. 6. 7. 8. 9. r 10. 11. 12. 13. 14. 15. 16. 17. 18. 236 KEY 1. C 2. B 3. C 4. A 5. D 6. E 7. D 8. D 9. C 10. A 11. D 12. B 13. B 14. C 15. E 16. C 17. D 18. E 237 APPENDIX G State-Trait Anxiety Inventory SELF-EVALUATION QUESTIONNAIRE Developed by C. D. Spielberger, R. L. Gorsuch and R. Lushene 238 STAI FORM X-1 NAME : '. DATE DIRECTIONS: A number of statements which people have used to describe themselves are given below. Read each state-ment and then blacken in the appropriate circle to the right of the statement to indicate how you feel right now, that is, at t h i s moment. There are no right or wrong answers. Do not spend too much time on any one statement but give the answer which seems to describe your present feelings best. 1. I feel calm © © ® © 2. I feel s e c u i ^ ^ ^ v -- :- © © © © 3. I am tyense '. , © © ® © 4. I am legretful . ^ ^ s ^ . © © © © 5. I feel at ea&r^ ./.. »• © © © © 6. I feel upset L ...^1. . . ^ r ^ © © © © 7. I am presently worrying over"possible misfortunes/:. ..A © © © © 8. I feel rested y / . ^ T S v ^ ^ . \.. © ® ® © 9. J feel anxious . 1 ©^,® © © 10. I feel comfortable jL. © ® ® © 11. I feel self-confident : /. © © ® © 12. I feel nervous © © ® © 13. I am jittery * © © © © 14. I feel "high strung" ® ® ® © 15. I a:n relaxed '. © © ® © 16. I feel content © © ® © 17. I am worried © © © © 13. I feel over-excited and rattled © © rv © 19. I feel joyful © © @ © 20. I feel pleasant Q ® © ® imjFki CONSULTING PSYCHOLOGISTS PRESS ' y ^ i ^ j j 577 Colieye Avenuo, Palo Alto, California 94306 o > f r S m a 3 o a n s > H M «; 3 e o s SELF-EVALUATION QUESTIONNAIRE STAI FORM X-2 239 NAME : : : : '. DATE DIRECTIONS: A number of statements which people have used to describe themselves are given below. Read each state-ment and then blacken in the appropriate circle to the right of the statement to indicate how you generally feel. There are no right or wrong answers. Do not spend too much time on any one statement but give the answer which seems to describe how you generally feel. 21. I feel pleasant © ® ® ' © 22. I tire q u i c l d j ^ ^ © © ® © 23. I feel like crying ; © ® ® © 24. I wish I could be as happy asc_th^ rs seem to be © ® ® © 25. I am losing out on things/because I can't make up my mind soon enough .... © ® ® © 26. I feel rested L^......^Z ...J... -?^ >v. © © ® © 27. I am '-'calm, cool, and collected" J^^*^^S.. 1 © ® © © 28. I feel that difficulties a^ e piling up so that I cannot overcome \ner^r^.... G © ® © 2i*. I worry too much over something that really doesn't matteeT. © ® ® ® 30. I am happy © ® ® © 31. I am inclined to take things hard © @ ® © 32. f lack self-confidence © ® ® © 33. I teel secure : © © ® © 34. I try to avoid facing a crisis or difficulty © ® ® © 36. I f~el blue © © ® ^ 36. 1 ara content : ;- © © ® © 37. Some unimportant thought runs through my mil-d and bothers mc © ® ® © 38. I take disappointments so keenly that I can't put them out of my mind .... © ® , © © SO. I am a steady person © ^ © ® © 40. I become tense ai:d upset when I think about my present concerns © © ® © 0 1 O 3 3 S w m > o r > CO Copyright (_"> I96S by Charles D. Spiclb'T^rr. [{"production of thin test or any portion thcrci,' by nny prwess without written p.-rinisvirr. of t'nr Publisher is prohibited. 240 APPENDIX H Tables of Correlation C o e f f i c i e n t s 241 ft". T A B L E 17 I N T E R C O R R E L A T I O N MATRIX FOR COMBINE'D GROUPS I :i y< v.. V a t i a b i e 1 2 3 ' 5 6 7 8 9 10 11 14 15 16 17 18 19 20 21 i I f ' / . ' , t j „ ( I 1. P o s t t e s t 1.00 !.o) ^ 2. T o t a l E r r o r s - Main Lesson -.42* 1.00 'V' 3. T o t a l Responses - Main Lesson -.33* . 96* -162!' ,i*2Q* Li ' it ,\ 4. P r o p o r t i o n of E r r o r s - Main Lesson -.42* . 95* 5. T o t a l C o r r e c t - Main Lesson 6. Do you understand? - Main Lesson .46* .13 -.78* -.26* 1.00 . 38* 1.00 t, V '<>• i re 7. Time to Answer (7) Above - Main Lesson -.12 .08 -.23 -.56* 1. 00 3. Had Limit S s c t i o n - Main Lesson -.24 .26* 5P'„ ri, • - . 3 2 * -.10 .20 1.00 9. Average F i r s t Latency - Main Lesson -.12 .00 -.06 f / i'mk i, V ..3.5* -.06 -.01 .23 . 10 1.00 \\>';<;; 10. Average Total Latency - Main Lesson - . l b .33* .3.1*U -.21 .12 .09 . 09 .81* 1.00 11. Enjoyment - Mam Lesson .21 -.17 .10 -.04 .13 - . 06 1 > .02 1.C0 *' L > . . 12. p r e l e s s o n score .27* -.34* -,Z6*"r . 4 8 * .26* - .10 - . 1 7 .06 .07 .24 1;00 ' 13. p r e l e s s o n C o r r e c t on F i t s t Try .21 -.31* -.26*| . 4 2 * .23 -.03 - . 18 .14 .16 .2 6* .'87* 1. 00 14. Average F i r s t '.atvncy - Pr e l e s s o n - .2? .05 -.02 f - . 1 9 - . 05 t — -.04 . 53* .49* .09 '.[oq 4.01 1.00 15. Average T o t a l Latency - Prelesson. - . 3 0 * . 17 .11 '< it - . 2 8 * .01 .03 -.04 . 49* .50* .00 'i>..ii . 94* 1.00 16. Math A b i l i t y - F i r s t T e s t i n g .41* -.27* -.21 t p -|y3>i*.'! . 30* .20 - . 0 7 - .21 - . 32* - . 3 9 * .19 -.38* -.47* 1.00 17. F u l l A - S i a t e - f i r s t T e s t i n g .08 -.10 -.14 f* - . 04 ' -.02 .13 - .20 - . 0 4 -.07 .16 .00<' • : .b4 ! -.03 -.04 • .05 1.00 18. A - T r a i t - F i r s t T e s t i n g . 1 3 -.08 -.10 fj .03 -.06- - . 04 -.10 .00 -.07 .03 . ool- " * ..C6 1 .00 ~. 02 -.01 -.05 1.00 • 19. Short /.-State - F i r s t T e s t i n g .01 -.07 -.13-i')1 -.0.8 .02 -.08 -'• IV .,04 .03 - , ; i i ' • , -. ;0l • .09 .11 .07 .09 . 33 * 1.00 20. A- State - Pr e l e s s o n .07 -.13 - . : . o ; ' .17 .03 -.06 -.05 •' -;.2C* j -. 17 . 03> -.11 -. 08 •12 -.05 -.20 -.08 1.00 1 21. A-State - Main Lesson -.29* .31* .'34* -.28* -.13 • 1 9 .09 .01 .10. v . <.. | -.35* V 02 .00 . 06 J8 . 11 .41" .31'* -.12 ; 1.00-_. .. . ^ .'./.•fv-'k * j r | .25 i s s i g n i f i c a n t & i. the o( = .05 lev- j l • / v l . TA3LE 18 COMPARISON OF SELECTED CORRELATION COEFFICIENTS FOR T, T T Correlation Between Variables Pooled T l T2 1 Aobs Posttest and Proportion of Errors -.42 -.48 -.50 -.03 3.87 Posttest and Total Correct-main .46 .64 .56 -.07 7.22* Posttest and Prelesson Scc?:e .27 .39 .30 .09 . 97 Posttest and Prelesson Latency -.30 -.51 -.30 .19 5. 32 Posttest and Math A b i l i t y . 41 .71 .26 .36 4. 05 Posttest and A-State-main -.29 -.14 -.53 -.12 2.54 Proportion of Errors and Response to "Do you understand?" -.29 -.40 -.27 -.37 .23 Proportion of Errors and Limit Section .33 .03 .55 .12 3.62 Proportion of Errors and Total Latency-main .35 .30 .38 .39 .13 Proportion of Errors and Prelesson Score -.37 - .29 -.44 -.49 .59 Proportion of Errors and Math A b i l i t y -.31 - .35 -.39 -.27 .18 Proportion of Errors and A-State .34- .32 .33 .36 .03 PraJesson Score and Response • to "Do you understand?" .26 . 12 .62 .16 4. 11 Total Lacency-main and Total Laten';y-prelesson . 50 .41 .41 .62 1. 14 Moth A b i l i t y and Total Latency-mai.-. -.39 ' - .24 -.45 -.44 .68 A-Stace Prelesson and Total. Latency-main -.28 -.34 -.50 . 05 3.37 Enjoyuuent and Prelesson Score .24 .49 .28 -.05 3 . 1.2 Enjoyment and A-State-main -.35 -.45 -.01 -.57 3. 97 A-State-main and Math A b i l i t y -.18 -.11 -.12 -.34 .68 A-Trait and A-State-main .41 .24 .46 .4.3 .89 A-Stete Short (Day 1) and A-State-main ' .31 1 .35 .09 .40 • 1.1J + "y~ v /-,\ = 5.99 i s s i g n i f i c a n t at the / - .05 level ^ obs > A. (2 ) T n i c w a i n c <; c A i n u l a t e d r s i n a a Fortran computer pro.grair written by the author.
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A study of computer-assisted instructional strategies and learner characteristics Kaufman, David M. 1973
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Title | A study of computer-assisted instructional strategies and learner characteristics |
Creator |
Kaufman, David M. |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | This study was undertaken to investigate the use of computer-assisted instruction as an instructional laboratory. The concept of an instructional logic was defined as an algorithm followed by the computer program for each instructional unit. This step-by-step logic was repeated for each instructional unit but with different content. This procedure permitted the controlled manipulation of the variable of correctional feedback. Three forms of correctional feedback were defined by varying the information content of the feedback. These were response-sensitive correctional feedback, response-insensitive correctional feedback and no correctional feedback (only that the answer was incorrect). The interaction of correctional feedback with selected learner characteristics was examined as well, These learner traits were mathematical ability, prerequisite knowledge and state anxiety. The effect of correctional feedback and its interaction with these variables was examined. Subjects of the study were a representative sample of sixty-three preservice elementary school teachers from five sections of a mathematics course given in a large education faculty. These subjects were randomly assigned to the three treatment conditions, although they selected the CAI experimental periods in which they would participate. The test of mathematical ability used was the Cooperative Sequential Test of Educational Progress, Mathematics Form 2A (STEP). The state anxiety instrument used was the State-Trait Anxiety Inventory (STAI) and the five item short form used by O'Neil (1972) was administered twice. The eighteen item posttest was constructed by the experimenter and the measure of prerequisite knowledge used was a nine item prelesson with a possible mark of 0, 1 or 2 on each item. The mathematics lesson was a topic in introductory calculus dealing with the concept of derivative. The topic was treated from a physical point of view, using concepts of distance, speed and time to illustrate the mathematical concepts. The main objectives were to show that the derivative is a limit and to show how to use this limit definition to calculate the derivative of a function at a point. The CAI lesson was programmed using an author language developed by the experimenter as a vehicle for implementing the instructional logic and varying the correctional feedback. The language is limited in use but has the advantage of requiring essentially no computer experience of an instructional designer. The main limitation of the language as implemented at the University of British Columbia is the cost, which limited the sample size in this experiment. The results of the study were generally in the expected direction but the effects were not as pronounced as had been hypothesized. The most important finding was the significant difference in proportion of errors on the main lesson between the response-insensitive (T₂) and no correctional feedback (T₃) groups. The importance of this finding was then increased by the significant relationship found between immediate learning and proportion of errors with the effect of learner traits and treatment effects statistically removed. The most effective variable in predicting performance in the experiment was mathematical ability and its effect was statistically controlled when testing the effect of the other variables. prerequisite knowledge was also important in predicting performance and was statistically controlled as well. State anxiety was significant in predicting proportion of errors but not response latency. However, significant treatment-by-A-State interactions were observed for posttest and for response latency. The three treatment groups differed in the expected direction on most of the important variables but the differences were not statistically significant. In particular, the A-State levels for the three groups were ordered as expected, but the differences were not large enough to cause the hypothesized interactions. The results of the study partially supported the hypothesis of the important role of correctional feedback in instruction and its interaction with individual traits of the learner. Finally, the variable of state anxiety was examined and it was found that higher levels of state anxiety led to longer response latencies. Also, state anxiety increased when no correctional feedback was provided to the students as well as when the content became more difficult. This finding confirmed the expected relationship between state anxiety and task difficulty. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0093081 |
URI | http://hdl.handle.net/2429/32409 |
Degree |
Doctor of Education - EdD |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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