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UBC Theses and Dissertations

Problems to illustrate versus problems to initiate the study of calculus Brown, John William 1972

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PROBLEMS TO ILLUSTRATE. VERSUS PROBLEMS TO INITIATE THE STUDY OF. CALCULUS by JOHN WILLIAM BROWN B. Sc., University of B r i t i s h Columbia, 1965 A THESIS. SUBMITTED IN PARTIAL FULFILLMENT OF MASTER OF ARTS i n the Department of Education We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA MARCH, 1972 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT An analysis of the pertinent l i t e r a t u r e showed that there were two.commonly used i n s t r u c t i o n a l strategies for teaching calculus to engineering technology students. Theory-problem i n s t r u c t i o n a l strategy i s being used when an instructor f i r s t develops-the new calculus theory and then " i l l u s t r a t e s " t h i s theory with an applied problem. Problem-theory instruc-t i o n a l strategy i s being used when an i n s t r u c t o r . f i r s t " i n i t i a t e s " the new calculus theory with an applied problem and then develops the new calculus theory The purpose of this study i s to investigate which of the above i n s t r u c t i o n a l strategies i s best for teaching calculus to engineering technology students. The evaluation was done by comparing the achievement of the theory-problem group and the problem-theory group on two te s t s . For the length of the study the two groups were taught i d e n t i c a l content by the same ins t r u c t o r . The only difference was the order i n which the applied problems were presented. There was no s i g n i f i c a n t difference i n achievement of the two groups on the test designed to measure understanding of techniques, p r i n c i p l e s and concepts of calculus. There was no s i g n i f i c a n t difference i n the achievement of the two groups on the test designed to measure success at solving applied problems. i i i The results of t h i s study indicate that students w i l l do as well i f they are taught by an i n s t r u c t i o n a l strategy which uses problems to i l l u s t r a t e calculus theory as they w i l l i f they are taught by an.instructional strategy which uses problems to i n i t i a t e the study of calculus theory. TABLE OF CONTENTS Chapter Page I. THE PROBLEM . . . . . . . . 1 INTRODUCTION . . . .-. ... . . . .. . . . . . . 1 APPLICATIONS AND MATHEMATICS . . . . . . . . . 3 ILLUSTRATE OR INITIATE? . . . . . . . . ... . - . 6 STATEMENT OF THE PROBLEM . . . . . . . . . . . 7 II . REVIEW OF THE LITERATURE . . . . . . . . . . . . 9 INTRODUCTION . . . . . . .•. . . . . . . . . . 9 SURVEY OF CALCULUS TEXTS . . . . . - . . . . . . 9 CONDITIONS OF LEARNING . . . . . , -.. . . . . . 15 MEANINGFUL RECEPTION LEARNING . . .•. . . . . ., 18 ANALYSIS OF RELATED STUDIES . . . . . . . . . . 22 ASSUMPTIONS UNDERLYING THE HYPOTHESES . . . . . 25 THE HYPOTHESES . . . .•. . . . . . . . . . . . 26 II I . EXPERIMENTAL DESIGN . . . . . . . . . . . . . . . 27 INTRODUCTION . . . . . . . . . . . . . . . . . . 27 FORMATION OF GROUPS . . . . . . . . . . . . . . 28 TEACHING SCHEDULE . . . . . . . . . . . . . . . 28 INSTRUCTIONAL PROCEDURES . . . . . . , . . . ..• 30 DEVELOPMENT OF MATERIAL . . . . . . . . . . . . . . 31 DATA GATHERING INSTRUMENTS . . 3 2 V Chapter Page STATISTICAL DESIGN . . . • .33 STATEMENT OF STATISTICAL HYPOTHESES . . . . . . 35 IV. ANALYSIS OF RESULTS . . . . . . .37 THE UNDERSTANDING TEST ... . . .37 TEST OF HYPOTHESIS Hi . . . . . .38 THE PROBLEM TEST . . . . . . . . . . . . . . . . 39 TEST OF HYPOTHESIS H2 . . . . . . . . . . . . . 40 INTERPRETATION OF RESULTS . . . . . . . . . . .41 ADDITIONAL OBSERVATIONS . . . . . . 42 V. CONCLUSIONS . . . . . . . . . . . .43 SUMMARY . . . . . . . . . . . . . . . . . . . . 43 DISCUSSION . . . . . . . . . . . . . . . . . . . 43 THE HAWTHORNE EFFECTS . . . • . . 44 LIMITATIONS OF THE STUDY . . . . .-. . . .-. . .45 SUGGESTIONS FOR FURTHER RESEARCH . . . . . . . .46 BIBLIOGRAPHY . . . . . . . . . . . . . . . . 48 APPENDICES . . . . . . . 54 A. LECTURE TOPICS 54 B. SAMPLE LECTURE . . . . . . . . . . 56 C. SAMPLE PROBLEM SHEET . . . . . . . . . .61 D. TESTING INSTRUMENTS . . . .63 E. SAMPLE OF SCORING ON PROBLEM TEST . . . . . . . . 75 F. EXPERIMENTAL DATA . . . 79 vi LIST OF TABLES Table Page I. PURPOSE OF PROBLEMS IN TEXTBOOKS . . . . . . . .11 II. ANALYSIS OF UNDERSTANDING TEST SCORES . . . . . 38 III. ANALYSIS OF PROBLEM TEST SCORES . . . . . . . . 40 IV. THEORY-PROBLEM SCORES ON-MULTIPLE CHOICE TEST. . 80 V. PROBLEM-THEORY SCORES ON MULTIPLE CHOICE TEST. . 81 VI. THEORY-PROBLEM SCORES ON PROBLEM TEST . . . . . 82 VII. PROBLEM-THEORY SCORES ON PROBLEM TEST . . . . . 83 vii LIST OF FIGURES Figure Page 1. TEACHING SCHEDULE 29 2. DESIGN OF EXPERIMENT 34 CHAPTER I THE PROBLEM INTRODUCTION The purpose of thi s study i s to investigate the effectiveness, i n terms of student achievement, of two instruc-i t i o n a l strategies for teaching calculus to engineering technology students. The Sub-Committee on Mathematics for Institutes of Technology i n Canada (50:3) states the functions of a technical mathematics course as follows: 1. to develop the student's appreciation of the applica-b i l i t y of mathematics i n technology. 1 2. to integrate mathematics with the student's technical programme. 3. to demonstrate that mathematics i s the es s e n t i a l language of technology. 4. to develop the student's a b i l i t y , confidence, and v e r s a t i l i t y i n dealing with physical problems and t h e i r mathema-t i c a l solution. 5. to ensure maximum p a r t i c i p a t i o n of the student i n solving problems involving the application of mathematics within his f i e l d of i n t e r e s t . The Sub-Committee on Mathematics for Institutes of Technology i n Canada also reports; It i s the experience of teachers of mathematics at i n s t i -tutes of technology that the mathematics topics have, at a l l times, to be introduced and related to the students [s p a r t i c u l a r knowledges and interests i n technology and that to expect students to develop t h e i r own applied problems from a course i n pure mathematics i s unreasonable. (50:7) With the report of the Sub-Committee i n mind, instruc-t i o n i n calculus for engineering technology students should be i n t u i t i v e and centered on suitable applied problems that relate as c l o s e l y as possible to the student's own technology and the work the student i s doing i n his other courses. There are two i n s t r u c t i o n a l strategies which are often used to at t a i n the above goal. The f i r s t i n s t r u c t i o n a l strategy w i l l be referred to as theory-problem strategy. When using the theoryrproblem strategy the instructor f i r s t develops i n t u i t i v e l y a technique p r i n c i p l e or concept (henceforth abbreviated TPC) of calculus. The TPC i s then i l l u s t r a t e d by solving an applied problem or example, preferably one which relates to the technology being studied by the student. Presumably theory-problem strategy w i l l motivate the student to learn and understand the new TPC by i l l u s t r a t i n g i t with a problem s i t u a t i o n which the student sees as being useful. Writers of calculus texts w i l l sometimes develop more than one TPC before they present the applied problems or physical s i t u a t i o n s . Thus while using problems to i l l u s t r a t e mathematics they are not s t r i c t l y following the theory-problem d e f i n i t i o n of the investigator.' 3 The second i n s t r u c t i o n a l strategy w i l l be referred to as problem-theory strategy. When problem-theory strategy i s used, the instructor f i r s t looks at a physical s i t u a t i o n , or applied problem which requires the new TPC for i t s solution.' In most cases where an instructor uses problem-theory strategy he and the students together attempt to solve or explain the problem or physical s i t u a t i o n using e x i s t i n g TPC 1s. The instructor.then solves the problem using the new TPC and presents the th e o r e t i c a l development of the new TPC. Presumably the student gains some knowledge about the problem or physical s i t u a t i o n and becomes, aware of the fact that the TPC's he has available, can at best produce only a p a r t i a l explanation or solution. The instructor attempts to get the student interested i n the problem s i t u a t i o n . Subsequently the student.discovers what the new TPC i s intended to accomplish and then understands the development of the relevant TPC. Henceforth, the investigator w i l l use the term theory-problem strategy i f and only i f a problem or physical s i t u a t i o n i s being used to i l l u s t r a t e a TPC of calculus. S i m i l a r l y , the investigator w i l l use the term problem-theory strategy i f and only i f a problem.or physical s i t u a t i o n i s being used to i n i t i a t e the study of a TPC of calculus. APPLICATIONS AND MATHEMATICS The use of problems and physical situations' to i l l u s t r a t e or i n i t i a t e the study of mathematics has been subject to considerable discussion. In the introduction, to his text, Calculus., Morris Kline, j u s t i f i e s , the, use of applications' to motivate the study of calculus. Kline says; The present trend to separate mathematics from science i s ' t r a g i c . . . . the calculus, divorced from applications i s meaningless. We should also keep i n mind that most of the students taking calculus, w i l l be s c i e n t i s t s and engineers [also technicians], and that these students must learn how to use mathematics . . .-. The gap between mathematics and science i n s t r u c t i o n must be f i l l e d , and we'can do so to our own advantage - because thereby we give meaning and motivation to the calculus. (27:vii) Pollak (37:319) supports the use of problems and'physi cal situations to i l l u s t r a t e the use of mathematics i n the classroom.- Pollak (37:319) i n s i s t s that applied problems which claim to be applications to other d i s c i p l i n e s must be "honest" applied mathematics., By "honest", Pollak means that the r e l a t i o n s h i p between the mathematical model and the outsid world must be f u l l y explained to and understood by the student In the opinion of. the i n v e s t i g a t o r . t h i s places an unnecessary r e s t r i c t i o n of the use of. applied problems and physical s i t u a -tions to i n i t i a t e and i l l u s t r a t e much technical mathematics. In some cases the development of the appropriate mathematical model could take'considerably more time than the development of the relevant mathematics and,consume valuable mathematics i n s t r u c t i o n time. If. the applied problem i s chosen so that i t relates to the student's technology the appropriate model may have-been developed i n one. of the student'.s other courses. The investigator believes that. model bu i l d i n g i s an important part of mathematics i n s t r u c t i o n but that i t reaches a point of diminishing returns i n r e l a t i o n s h i p to i n s t r u c t i o n time. . 5 Pollak (37 :325) says "7Anyone who thinks, that mouthing words from some other d i s c i p l i n e and then p u l l i n g a formula out of a hat w i l l motivate students i s probably misleading himself." Willoughby (55:273) discusses some problems i n applying mathematics.. Willoughby (55:274) stresses that applications are important because the ultimate goal of mathematics educa-ti o n for most students i s to make mathematics useful to them. Kline (28:v) says that "Physical problems that c a l l for the creation of mathematics set the stage for.discovery." Looking, toward the future, the Report'of the Cambridge Conference (39) presented a strong case for the c o r r e l a t i o n of elementary school science and mathematics. . The C.U.P.M.'s report e n t i t l e d A Curriculum i n Applied Mathematics (15) says that students should possess a firm knowledge of the techniques of calculus and the a b i l i t y to use them to formulate, solve, and in t e r p r e t problems i n areas of application. I t i s concluded from the above presentation that relevant applied problems are considered important i n the teaching of mathematics. Both the theory-problem and the problem-theory i n s t r u c t i o n a l strategies make use of relevant applied problems. According to the Sub-Committee on Mathematics for Institutes of Technology, one of the functions of a technical mathematics course i s to.correlate mathematics and applications to the student's other subjects and his technology. The student studies mathematics because i t i s required to understand engi-neering theory and solve technical problems. I t i s conceded by 6 the investigator that c o r r e l a t i n g mathematics with other subjects i s somewhat easier i n a technical mathematics course. Howeveras pointed out above, the trend i s toward introducing applications to i n i t i a t e and i l l u s t r a t e the study.of mathematics. The investigator believes that school mathematics teachers and curriculum designers might well learn something from t h e i r technical education counterparts who have been using t h i s approach for many years. ILLUSTRATE OR INITIATE? The Sub-Committee on Mathematics for Institutes of Technology in.Canada indicates that i t i s standard practice to use applications i n teaching a technical calculus course. It i s seen from the above that application problems -or physical situations can be used i n the teaching of calculus TPC's i n two ways. Applied problems and physical situations can. be used to. i l l u s t r a t e TPC's of calculus as i s done by the theory-problem i n s t r u c t i o n a l strategy. An analysis of the way calculus texts use applied problems appears i n Chapter I I . This analysis w i l l provide evidence that some authors of calculus texts use the approach of the theory-problem i n s t r u c t i o n a l strategy i n t h e i r writing. I f one accepts the premise that some instruc-tors w i l l use the same approach as the textbook when teaching calculus then i t can be concluded that some instructors use the theory-problem i n s t r u c t i o n a l strategy. 7 Applied problems and physical situations, can also be used to i n i t i a t e the study.of techniques, p r i n c i p l e s , and concepts of calculus as i s done by the problem-theory instruc-t i o n a l strategy. The Sub-Committee on Mathematics for Institutes of Technology reports; The idea of introducing a new mathematics topic by means of p r a c t i c a l examples r e l a t i n g to the technology, as far as t h i s i s possible, was considered. Such an applied approach to the subject matter could be followed by a somewhat more formal discussion of the theory being dealt with, and f i n a l l y a return to further work on suitable physical examples. This method of applying the mathematics as i t i s studied tends to stimulate and hold the student's i n t e r e s t , and i n s t i l s [sic] confidence when the student sttempts [sic] his own solution of a complete mathematical problem. Further, such an approach.helps to increase the student's retention of the material presented . ... when i t i s seen that the mathematics i s indeed needed i n the other technical subjects. (50:10) STATEMENT OF THE PROBLEM Both of the proposed strategies should motivate the students since they see calculus used to solve problems which are i n t e r e s t i n g and useful to t h e i r technology.' I t . i s conjec-tured by the investigator that students taught by the problem-theory strategy may have a better understanding of calculus and i t s applications than students taught by the theory-problem strategy. It i s the purpose of t h i s study to attempt to answer the following two questions: 1. Do students taught calculus by the problem-theory strategy achieve higher scores than students taught by the theory-problem strategy on a test designed to measure under-standing of techniques, p r i n c i p l e s , and concepts of the calculus? 8 2. Do students taught calculus by the problem-theory, strategy.achieve higher scores than, students taught by.the theory-problem strategy on a.test designed to measure success at solving applied physical problems? A formal statement of the hypotheses can be found at the end of Chapter I I . CHAPTER II REVIEW OF THE LITERATURE INTRODUCTION The following review of the l i t e r a t u r e summarizes some of the current practice, theory and research that i s applicable to the proposed problem. This Chapter w i l l consist of a survey of technical calculus texts, a discussion of the pertinent areas of classroom learning theory and an analysis of si m i l a r studies. SURVEY OF CALCULUS TEXTS The purpose, of th i s survey of technical calculus texts i s to look for instances of the proposed i n s t r u c t i o n a l s t r a t e -gies. The two i n s t r u c t i o n a l strategies defined by the i n v e s t i -gator i n Chapter I are somewhat i d e a l i s t i c . In-actual practice an instructor would alternate between the two strategies using the one he thought more convenient or e f f e c t i v e for the p a r t i c u l a r mathematics topic being developed. S i m i l a r l y , writers of text^-books w i l l develop a. mathematical TPC using the approach, they consider, most e f f e c t i v e . It.would not be unusual to f i n d that a text uses both, of the proposed i n s t r u c t i o n a l strategies. 10 Table 1 on page' 11 summarizes the findings of the investigator's survey of a number of calculus textbooks often considered for use i n technical calculus courses. The texts were analyzed on the basis of how often they used applied problems or physical, situations' to i n i t i a t e and i l l u s t r a t e the development of mathematical TPC's. A text was s.aid to use problems to i n i t i a t e / i l l u s t r a t e mathematics "rarely" i f the problem use occurred on the average less than 20% of the time per chapter or topic (some, texts would devote an entire chapter to the theory of a topic and devote the following chapter to applied problems). A text was said to use problems to i n i t i a t e / i l l u s t r a t e mathematics "often" i f the problem use.occurred on the average between 20% and 80% of the time per chapter or topic. A text was said to use problems to i n i t i a t e / i l l u s t r a t e mathematics "very often" i f the problem use occurred on the average more than 80% of the time per chapter or topic. The text Calculus for Engineering Technology by Blakeley (12) uses an i n t u i t i v e approach to establish the TPC's of calculus which apply to the majority of physical problems. I l l u s t r a t i v e examples and problems are used as the t h e o r e t i c a l development of calculus progresses. The emphasis i s on e l e c t r i -c a l applications so Blakeley's text would not be suitable for y use with most engineering technologies. 11 Table 1 Purpose of Problems i n Textbooks Writer Problems are used to IJ . l u s t r a t e Initiate Rarely Often Very Rarely Often Very Often Often Blakeley / / Douglas & Zeldin V ' / J u s z l i / Kline - / Rice . & Knight Richmond . / . Washington . V . Rarely - on the average less than 20% of the time per chapter or topic. Often - on the average between 20% and 80% of the time per chapter or topic. Very Often - on the average more than 80% of the time per chapter or topic. Blakeley's text uses problems to i l l u s t r a t e mathematics topics very often but rarely uses problems to i n i t i a t e , t h e study of mathematics topics., An i n s t r u c t o r using the approach of Blakeley's text would be using the theory-problem strategy. The text Calculus.and Its Applications by Douglas and Zeldin (17<) was written for use by university engineering undergraduates. Theoretical developments of the TPC's of calculus are f a i r l y rigorous. The authors sometime i n i t i a t e and i l l u s t r a t e the development of a TPC by using an applied problem or physical situation,, but at other times TPC's of calculus are developed without providing a single i l l u s t r a t i v e 12 applied problem. Douglas and Zeldin's text often uses problems to i l l u s t r a t e , mathematics but rar e l y uses problems to i n i t i a t e the study of mathematics. An - instructor using the approach of t h i s text would be alternately using problem-theory and theory-problem strategies part of the time and the rest of the time he would be presenting TPC's without any i n i t i a t i n g or i l l u s t r a t i n g a pplication problems. Analytic Geometry and Calculus by J u s z l i (25) gives a non-rigorous treatment of calculus. The text uses some applied problems to i l l u s t r a t e the TPC 1s of calculus. This text often uses problems to i l l u s t r a t e mathematics but never uses problems to i n i t i a t e the study of mathematics. An ins t r u c -tor using the approach of J u s z l i 1 s . t e x t would be using the theory^problem strategy part of the time. The text Calculus by Kline (27) employs geometrical and physical arguments and generalizations from concrete cases to j u s t i f y the development of the TPC's of calculus. While the t h e o r e t i c a l development i s i n t u i t i v e , Kline i s careful not to overlook a single d e t a i l i n the mathematical development. For example, the rule for d i f f e r e n t i a t i n g the function f(x) = x n i s j u s t i f i e d separately and i n complete d e t a i l for the cases where n i s a po s i t i v e integer, n i s a negative integer, n i s a r a t i o n a l number, and f i n a l l y where n i s any r e a l number. Most technical calculus texts gloss ; over the t h e o r e t i c a l problems involved i n these d i s t i n c t i o n s . Most of the problems which are used by Kline come from the 13 natural sciences with emphasis on physics. This text very often uses problems both to i l l u s t r a t e and i n i t i a t e mathematics topics. Kline's text would not be suitable for use at a techni-c a l i n s t i t u t e because of the laborious attention to d e t a i l and lack of engineering applications. However, i f an instructor used the approach of Kline's text he would be using the problem-theory i n s t r u c t i o n a l strategy. Technical Calculus and Analysis by Rice and Knight (42) presents calculus as a. p r a c t i c a l technique for solving problems actually encountered i n industry. Basic TPC's of calculus are developed by, using i n t u i t i v e geometrical and graphical arguments. There i s no th e o r e t i c a l development of basic TPC's of calculus i n th i s text. The text has many excellent i n d u s t r i a l and engineering applications which are used to i l l u s t r a t e the TPC's of' calculus. Rice and Knight's text very often uses problems to i l l u s t r a t e mathematics but rar e l y uses problems to i n i t i a t e the study of mathematics. An instruc-tor using the approach of t h i s text would not be using either of the i n s t r u c t i o n a l strategies as defined by the investigator because the t h e o r e t i c a l development i s so weak. The text Calculus for Electronics by Richmond (43) shows the theory and methods of calculus which are of the most d i r e c t use i n the study of e l e c t r o n i c s . The text uses problems to i l l u s t r a t e mathematics very often but ra r e l y uses problems to i n i t i a t e the study of mathematics. Richmond's text uses the theory-problem i n s t r u c t i o n a l strategy,, but i s not suitable for most engineering technologies because of the heavy emphasis, on electrical applications; Technical. Calculus with - Analytic Geometry by Washington (53) places emphasis on the more elementary topics in calculus.- The TPC's of calculus are developed in a non-rigorous and intuitive manner, with stress being placed on the interpretation and applications of the mathematics. Numerous applications from industry and technology are included to show the wide application of calculus. Applications' invari-ably follow the theoretical development of the TPC's in. Washington's text. This text very often uses problems to illustrate mathematics but.rarely uses problems to initiate the mathematics. An instructor using the same approach'as Washington's text would be using the theory-problem instruc-tional strategy as defined by the investigator. It should be noted, however, that Washington sometimes develops' a number of principles or concepts before applications are presented.• There are many other texts available, but most are quite rigorous and have too few applied problems. A glance at Table 1 on page'11 shows that there are a number of texts which regularly use problems to illustrate the TPC's of calculus. An instructor using the approach of one of these . texts would be using the theory-problem strategy in the general sense of the investigator's definition. The text.by Kline is the only one found by the investigator to be using the problem-theory strategy. The text, however, is not suitabl for use i n a. technical, calculus course because of the preoccu-pation with d e t a i l and lack of applied engineering problems. CONDITIONS. OF LEARNING In Robert Gagne's The Conditions of Learning, second edition (21), descriptions of eight types of learning are given together with conditions within the learner and condi-tions i n the learning s i t u a t i o n which promote the occurrence of learning. The two higher order types of learning are types of behavior with which formal i n s t r u c t i o n most t y p i c a l l y concerns i t s e l f . These two higher order types of learning are learning of defined concepts and r u l e s , and problem solving^ The type of learning which i s most.likely to occur during the formal i n s t r u c t i o n of calculus i s the learning of defined concepts and rules. According to Gagne; A rule . ., . i s an inferred..capability that enables the  i n d i v i d u a l to.respond to.a class of stimulus situations  with a class of performances, the l a t t e r being predictably related to the former by a class of r e l a t i o n s . (21:191). The learning of the TPC's of calculus during formal i n s t r u c t i o n would be very often i n the form of rule learning. The learning of rules i s understood to include the learning of defined concepts. The conditions within the learning s i t u a t i o n which must be provided by the i n s t r u c t o r to promote the learning of defined concepts and rules are given by Gagne" as follows: 16 1. The conditions of rule learning often.begin with a statement of the general nature of the performance to be expected when learning i s complete . . . . 2. Verbal instructions continue by invoking r e c a l l  of the component concepts . . . . 3. Verbal cues are next given for the rule as a whole. . . . these verbal cues to the rule need not be an exact verb a l i z a t i o n of the entire rule; . . . 4. F i n a l l y , a.verbal question asks the student to demonstrate the rule . . . . The exact form i s not of great importance as long as i t t r u l y requires the student to demonstrate the rule i n i t s f u l l sense . . . . [5. Ask] the student to state the rule verbally, . . . (21:200). Gagne took his conditions within the learning s i t u a t i o n required to promote rule learning and incorporated them into the i n s t r u c t i o n a l sequence given below: Step 1: Inform the learner about the form of the performance to be expected when the.learning i s completed. Step 2: Question the learner i n a way that requires the reinstatement (recall) of the previously learned, concepts that make up the r u l e . Step 3: Use verbal statements (cues) that w i l l lead the learner to put the rule together, as a chain of concepts, i n the proper order. Step 4: By means of a question, ask the learner to "demonstrate" one of [sic] more concrete instances of the r u l e . Step 5: (Optional, but useful for l a t e r i n s t r u c t i o n ) : By a suitable question, require the learner to make.a verbal statement of the r u l e . (21:203). In the problem-theory i n s t r u c t i o n a l strategy, the applied problem or physical s i t u a t i o n i s used to accomplish step 1 and step 2 of Gagne's suggested i n s t r u c t i o n a l sequence. The use of an applied problem or physical s i t u a t i o n can be: used to e f f e c t i v e l y inform the student of the nature of the performance expected when learning i s complete .(50:10)- and 17 to r e c a l l previously learned concepts which make up the r u l e . An examination of the texts which use the theory-problem strategy provides - evidence for the f a c t that theory-problem neglects step 1. The nature of the performance expected from the student i s not made clear when using theory-problem strategy u n t i l exercises are worked-after the theory i s developed. From these texts one also sees that i n the theory^ problem strategy step 2 i s just a statement of previously learned concepts'which make up the r u l e . In the problem-theory strategy the students may actually use the previously learned concepts to formulate and attempt to solve the i n i t i -ating problem. Since the theory-problem strategy does not e f f e c t i v e l y encompass, steps 1 and 2 a s i g n i f i c a n t difference between the two i n s t r u c t i o n a l strategies i n terms of the students.' post test scores might be observed. Gagne"'s eighth type of learning i s c a l l e d problem solving. According to Gagn^, problem solving i s characterized by: 1. The use of rules to achieve some goal. 2. The development of a higher order rule which changes the individual's c a p a b i l i t y . (21:216). The conditions i n the learning s i t u a t i o n required to promote problem solving are: 1. There i s contiguity of the rules that are to be "put together" to achieve solution, and the stimulus s i t u a t i o n that sets.the problem . . . . 18 2. The required contiguity.may be made more highly probable by recent recall of relevant rules. One function of verbal instructions is to ask the questions that stimu-late such recall . . . . 3. Verbal instructions that are externally provided, may "guide" or "channel'! thinking in certain directions. (Such guidance may, of course, be provided by the learner himself in self-instructions.) . . . . (20:222). Gagne* does not incorporate the above learning conditions into an instructional sequence as he did with rule learning. When the problem-theory strategy is used to teach rules, some students may be problem solving. The examples, questions and instructions that are provided by the instructor may result in some of the students arriving at the higher order rule on their own, before it is stated by the instructor. Problem solving by students would be less likely to occur when the theory-problem strategy is used simply because the applied problem would appear to be an exercise after the relevant theory had been developed. Some instructors might, consider, this to be the ideal situation and prefer to use the theory-problem instructional strategy for this reason. MEANINGFUL RECEPTION LEARNING Ausubel (7) argues the case that of the four current approaches to learning theory (neobehavioristic, meaningful verbal reception, Gestalt field theory, and information theory) only meaningful verbal reception learning is important for the acquisition of subject matter knowledge. Meaningful verbal reception learning depends on the relatability of 19 p o t e n t i a l l y meaningful material, vsuch as new mathematical TPC's (Gagne's r u l e s ) , to the learner's e x i s t i n g cognitive structure (7:21). This means that the learner's e x i s t i n g knowledge, i t s organization, s t a b i l i t y and c l a r i t y , w i l l be an important variable influencing meaningful verbal reception learning and retention. D i s c r i m i n a b i l i t y of new learning material from previously learned p r i n c i p l e s has been i d e n t i f i e d as a major cognitive structure variable (7:23) i n meaningful verbal reception learning. Thus a teaching strategy which indicates how a new rule i s related to and d i f f e r e n t from ex i s t i n g rules and concepts i n the cognitive structure w i l l be superior to one which does not. The problem-theory i n s t r u c t i o n a l strategy promotes d i s c r i m i n a b i l i t y when exis t i n g TPC's are used to attempt the problem solution. When the solution cannot be obtained by using these ex i s t i n g TPC's the new required TPC w i l l be i d e n t i f i a b l y d i f f e r e n t , and incorporation into e x i s t i n g cognitive structure w i l l be made easier and more permanent. The theory^-problem strategy w i l l also promote discrim-i n a b i l i t y i f the way i n which e x i s t i n g TPC's are related to and d i f f e r e n t from the new TPC i s made cle a r . This, however,, i s not always done by textbooks using the theory-problem strategy. A teaching strategy which makes a serious e f f o r t to 20 e x p l i c i t l y explore relationships between new and e x i s t i n g p r i n c i p l e s , that points out s i g n i f i c a n t s i m i l a r i t i e s and d i f f erences,.-and attempts to reconcile r e a l or apparent inconsistencies i s said to be employing the p r i n c i p l e . o f "integrative r e c o n c i l i a t i o n " (6:155). The problem-theory i n s t r u c t i o n a l strategy does th i s by using a problem to explore relationships between exis t i n g TPC's and the new TPC which i s required for the solution of the.problem. The theory-problem i n s t r u c t i o n a l strategy does not always employ this p r i n c i p l e . In some of the texts referred to e a r l i e r which use.the theory-problem i n s t r u c t i o n a l strategy, relationships between topics are not f u l l y explored. This, however, need not be the case. A c a r e f u l development of the mathematical theory should take into account a l l relevant relationships and differences between the new TPC and those TPC's which already e x i s t within the cognitive structure. A strategy suggested by Ausubel for promoting i n t e -grative r e c o n c i l i a t i o n i s the use of relevant introductory materials or organizers. An organizer i s introduced i n advance of the learning task and i s at a higher l e v e l of abstraction and generality than the learning task. In the context used here the learning task i s the new mathematics TPC. The main purpose of an organizer i s to bridge the gap between exis t i n g knowledge and that which i s required to perform the task .at hand (6:148). 21 The problem or physical s i t u a t i o n i n the problem-theory strategy may be acting as an.organizer even though the problem i t s e l f may lack generality. This i s because discussion of the problem re s u l t s i n the r e c a l l of exis t i n g TPC's related to the learning task. These e x i s t i n g TPC's w i l l be r e c a l l e d at a higher l e v e l of generality than the actual learning task. When using the theory-problem strategy the relevant e x i s t i n g TPC's related to the learning task could be arranged and presented i n the form of an organizer. In fact, t h i s sometimes, happens by chance when an author i s writing a textbook or an instr u c t o r i s presenting a lesson. The introductory theory, or.problem.is probably most e f f e c t i v e l y used as a comparative organizer. A comparative organizer i s one which i s designed to further the p r i n c i p l e of integrative r e c o n c i l i a t i o n . Such an organizer points out ways i n which previously learned and re l a t e d . p r i n c i p l e s are d i f f e r e n t from the ones i n the learning task. (7:27). It should be made clear that most of the supporting studies for organizers.use written verbal material. Conclu-sive evidence i s not available for organizers presented by another medium such as a classroom i n s t r u c t o r . Some, studies which use organizer theory, with another medium w i l l follow l a t e r i n t h i s chapter. It i s not the intention of the investigator to convince the reader that both i n s t r u c t i o n a l strategies use organizers i n the s t r i c t sense of Ausubel. What should be cle a r , however,. 22 i s that a difference between the two i n s t r u c t i o n a l strategies may i n part be due to the fact that the organizer e f f e c t i s stronger i n one than, the other. ANALYSIS OF RELATED STUDIES In a study by N i e t l i n g (34) problems were used to i n i t i a t e the study of certain topics i n mathematics. Problems on the new topic were presented to students for homework at least,two days p r i o r to the date classroom discussion was to take place. The goal of the problem presentation was to stimulate a student search, and exploration for the solution. It was not intended that, the majority of the students should be able to solve a l l the problems. I t was the b e l i e f of N i e t l i n g that the search and exploration promoted by the problem would help i n learning the related mathematics i n the classroom. Although the mean score of the experimental group exceeded the control group i n both the p i l o t study and the main study, the differences.were not s i g n i f i c a n t at the .05 alpha l e v e l . In the study proposed by the investigator the problems are presented i n the classroom and the students' search i s instructor guided. An opinion survey taken by N i e t l i n g suggested that students i n the experimental group devoted more time to the study of mathematics than the control group. The difference i n study time could account for the higher but not s i g n i f i c a n t l y d i f f e r e n t mean scores obtained by the experimental group. In the study proposed by the 23 investigator, both groups should.spend approximately the same amount of time on mathematics and cover exactly the same content in the form of both theory and problems. "Which should come f i r s t : theory or practice i n teaching problem solving s k i l l s ? " This problem was i n v e s t i -gated by Plants and Venable (35). In t h e i r study, an engi-neering mechanics-dynamics course was presented i n two ways. The "demonstration f i r s t " group received demonstration prob-lems before the theory necessary to explain the problem had been covered i n c l a s s . The "theory f i r s t " group.received the demonstration problems afte r the appropriate theory had been covered i n c l a s s . The "demonstration f i r s t " group scored s i g n i f i c a n t l y higher than the "theory f i r s t " group on a test designed to measure students' a b i l i t y to respond to unusual and unexpected problems. There was no s i g n i f i c a n t difference on tests which measured the technical content of the course. The Plants and Venable study indicates that the problem-theory i n s t r u c t i o n a l strategy of the investigator's proposed study may develop better applied problem solving s k i l l s . Students of calculus experienced the " t h r i l l " of discovery and possessed "superior" knowledge of the theory of calculus when taught by an experience-discovery approach., This, i s the claim made i n a study by Cummins (16). Cummins taught the experimental group and compared, them to students i n a control group taught by another instructor using the " t r a d i t i o n a l " method. The only thing the two groups had i n common was a textbook which the experimental group used only f o r reference. A l l students were given two tests at the end of the study. One, test was prepared by the i n s t r u c t o r of.the experi-mental group and the other was prepared by the i n s t r u c t o r of the t r a d i t i o n a l group. There was no s i g n i f i c a n t difference i n the mean scores of the two groups on the test prepared by the i n -structor of the t r a d i t i o n a l group. There was a s i g n i f i c a n t difference at the .01 alpha l e v e l i n the mean scores of the two groups on the test constructed by the instructor of the experi-mental group.. Thus while the experimental students did score better, i t could have been due to the content or i n s t r u c t o r differences. In the study proposed by the investigator both groups would get i d e n t i c a l content and the same inst r u c t o r . As previously mentioned, i n both i n s t r u c t i o n a l s t r a t -egies, the introductory material, whether i t be problem or theory-, may be acting as an organizer i n the sense of Ausubel. Most of the supporting evidence for organizer theory comes from studies using written verbal material. Following are the r e s u l t s of some studies which used organizers. Grotelueschen and Sjogren (22) investigated the general problem of the e f f e c t of d i f f e r e n t i a l l y structured introductory materials and learning tasks on learning and transfer. They point out that the complexity of the learning task, which varies according to the a b i l i t y of the learner, i s an important variable to consider when ascertaining the effectiveness of the i n t r o -ductory material. Grotelueschen and Sjogren controlled t h i s 25 variable by using adults of superior i n t e l l i g e n c e as subjects. The results of the study strongly indicate that introductory material i s most e f f e c t i v e when i t i s of a general nature and the learning task i s only " p a r t i a l l y ordered." Weisberg (54) compared verbal and v i s u a l organizers i n the learning of earth science concepts and found a s i g n i f i c a n t difference between these organizers. Townsend (52) found that the use of an advance organizer produced a posit i v e s i g n i f i c a n t e f f e c t under programmed i n s t r u c t i o n whereas there was no s i g n i f i -cant e f f e c t under teacher i n s t r u c t i o n . In the l i g h t of these studies i t would probably be d i f f i c u l t to prove that either of the i n s t r u c t i o n a l strategies has an organizer e f f e c t , e specially since they are presented under teacher i n s t r u c t i o n and the learning task i s highly ordered. I t i s not the purpose of this, study to demonstrate that the introductory material of the investigator's study acts as an organizer. I t i s only intended here to point out that some of. the reasons' for using an introduction are the same as those for using an organizer, and that i f a s i g n i f i c a n t difference i s observed on the post test scores i t may i n part be due to the superior organizing e f f e c t of one of the i n s t r u c t i o n a l s t r a t -egies . ASSUMPTIONS UNDERLYING THE HYPOTHESES The following assumptions are based on classroom learning theory and an, analysis of s i m i l a r studies. 26 1. An applied problem or physical s i t u a t i o n can stimulate and motivate students to learn mathematics. 2. Exploration and search for a problem solution develops applied problem solving a b i l i t y . 3. Exploration and search promotes the p r i n c i p l e of in t e -grative r e c o n c i l i a t i o n and has an organizing e f f e c t . 4. A strategy for teaching techniques, p r i n c i p l e s and concepts of calculus should provide the necessary conditions i n the learning s i t u a t i o n . THE HYPOTHESES On the basis of the l i t e r a t u r e reviewed and the assump-tions, the investigator expects the following hypotheses to be true. 1. Students who are taught by the problem-theory strategy w i l l not have any higher scores on the post test designed to measure understanding of techniques, p r i n c i p l e s and concepts of calculus than the students taught by the theory-problem strategy. 2. Students who are taught by the problem-theory i n s t r u c -t i o n a l strategy w i l l have higher scores on the post test designed to measure success at solving applied calculus problems than the students taught by the theory-problem i n s t r u c t i o n a l strategy. A statement of the hypotheses i n s t a t i s t i c a l form w i l l appear at the end of Chapter. I I I . 1 CHAPTER III EXPERIMENTAL DESIGN INTRODUCTION The study took place at the B r i t i s h Columbia I n s t i t u t e of Technology using students from the f i r s t - y e a r Instrumentation and Control Technology program and the material of the i n t r o -ductory calculus course. Both the theory-problem and problem-theory i n s t r u c t i o n a l strategies were i d e n t i c a l except for order. The Sub-Committee on Mathematics for Institutes of Technology indicated that whenever possible.the study of mathematics topics should be i n i t i a t e d with applied problems. A survey of calculus texts showed that most writers use applied problems mainly for i l l u s t r a t i n g the TPC's of calculus. Classroom learning theory did not. indicate that, either i n s t r u c t i o n a l strategy should be vastly superior than the other. I t , i s the purpose, of t h i s study to test the hypotheses stated at the end of Chapter I I . 28 FORMATION OF GROUPS Subjects for the experiment were the 40 f i r s t year Instrumentation and Control students at the B r i t i s h Columbia Insti t u t e of Technology. I t was assumed that the 40 students were representative of t y p i c a l Instrumentation and Control Tech-nology students. The students were assigned to one of two groups i n November 1971 using a table of random numbers. A coin was tossed by another i n s t r u c t o r to decide which group would receive the problem-theory i n s t r u c t i o n a l strategy. The students of the Instrumentation and Control Technology had a d i v e r s i t y of back-grounds. There were a few students whose l a s t mathematics course was taken over three years.ago. There were a few students whose native language was not English. Previous'mathematics background ranged from B r i t i s h Columbia Mathematics 12 to some students who had completed mathematics courses at the second year univ e r s i t y l e v e l . No attempt was made to balance the groups on the basis of these differences since the students were randomly assigned to the groups. TEACHING SCHEDULE The' new timetables for the experimental groups were arranged i n November so the students could be n o t i f i e d i n early December as to t h e i r changed timetable when they returned to classes i n January. I n i t i a l l y both groups had twenty students, but because of drop-outs and f a i l u r e s at Christmas the problem-theory group, had 19 students, and the theory-problem group 14 students when the study began i n January 1972. 29 Each group had three mathematics lectures and a common t u t o r i a l hour each week of the experiment. The times the classes met are shown i n Figure 1 below. Period Monday Tuesday Wednesday. . Thursday, 10 :30-11:20 * T u t o r i a l 11:30-12:20 P-T T-P 12:30- 1:20 P-T T-P P-T 1:30- 2:20 T-P P-T = problem-theory group. T-P =.theory-problem group. *both groups meet together for the t u t o r i a l . Figure 1 Teaching Schedule The. timetable was arranged so that while one group was having lunch the other would be i n class receiving t h e i r mathe-matics lecture. The group to receive i n s t r u c t i o n f i r s t each day was alternated as much as possible. This was done so that stu-dents would not have a l l i n s t r u c t i o n either before or a f t e r lunch. ; I t also meant that one group would not receive better i n s t r u c t i o n simply because the i n s t r u c t o r had had practice with the material. 30 INSTRUCTIONAL PROCEDURES The study took place during the f i r s t eight weeks of 1972. In November 1971 the students concerned were t o l d that in January- 1972 the mathematics class would be s p l i t into two sections i n order to increase student-instructor contact. They were given t h e i r changed timetable i n early December but. were not t o l d they were taking part i n an experiment. The same instructor taught both groups., In order to keep both theory and problem content i d e n t i c a l , the written material for each class was prepared ahead of time and placed on overhead projector transparency r o l l s . I n i t i a l l y the lectures were prepared for the theory-problem group and placed on the overhead r o l l s . The lecture was then recopied placing the problem f i r s t keeping the wording and content for both groups i d e n t i c a l . Half way through the study the investigator decided that instead of recopying the entire lecture for the problem-theory group the applied problem section of the theory-problem lecture r o l l would be cut out.and placed at the beginning to form the problem-theory lecture r o l l . . This change was made i n the time available between classes. When teaching the problem-theory group the instr u c t o r s o l i c i t e d ideas and help from the students i n setting up and attempting to solve the i n i t i a t i n g problem. When discussion had exhausted e x i s t i n g TPC's or a correct method of attack had been suggested, the problem would be. solved i f possible or put aside and the relevant theory developed i n exactly the same manner as for the theory-problem group. When development of the theory was finished the problem solution was completed. When teaching the theory-problem group the theory was f i r s t presented i n lecture s t y l e using the previously prepared overhead r o l l s . The problem was then displayed, and the students were asked how the new theory could be' used to obtain the solution. The problem was then solved. In thi s way both groups received i d e n t i c a l problems and theory. At the end of each.lecture the students of both groups were given i d e n t i c a l problem sheets. The investigator c o l l e c t e d a l l problem sheets and kept a record of the work done by the students of both groups. The same instr u c t o r taught both groups with the same enthusiasm and lecture s t y l e . Both groups were asked questions during the problem solving part of the lecture. The only d i f -ference was the presentation order. DEVELOPMENT OF MATERIAL The subject matter for the course was introductory calculus up to d i f f e r e n t i a t i o n and integration of polynomial functions. A complete outline of the topics covered appears i n Appendix A. Samples of lecture material presented by the d i f -ferent strategies are i n Appendix B. As previously mentioned,. the same physical, problems were used for both groups.. The problems were chosen for their, s u i t a b i l i t y for i n i t i a t i n g and i l l u s t r a t i n g the theory, and whenever, possible for th e i r r e l a t -a b i l i t y to the student's own technology. The problems were developed with the lecture so that they would work'equally well with both strategies. Identical homework problems were assigned to both groups. The f i r s t questions on each problem sheet were designed to exercise the TPC's of calculus that had just,been covered i n the lecture. The l a s t questions were applied problems that sometimes related d i r e c t l y to the student's own. technology. The students had the text Basic Technical Mathematics  with Calculus (second edition) by A. J. Washington available for reference. L i t t l e . r e f e r e n c e to the text was made by the investigator during the course of the experiment. DATA GATHERING. INSTRUMENTS Two tests were designed, constructed and t h e i r r e l i a -b i l i t y determined before the experiment was performed. The r e l i a b i l i t y of the tests was-determined by administering the tests to students of a night school calculus class which covered sim i l a r material. The f i r s t instrument was a twenty question multiple choice test designed to, measure the student's understanding and knowledge of the TPC's of calculus. Some questions 1 on thi s t e s t 33 were sim i l a r to the problems used i n the lectures and on the homework problem sheets. The r e l i a b i l i t y of the. te s t was computed by the University of B r i t i s h Columbia Test and Item Analysis computer program (TIA). A copy of the test appears i n Appendix D. The second instrument was a s i x item subjective test designed to measure the student's success at obtaining solutions to applied problems. Three of the items were applied problems that were related to the ones used i n the lectures and on the homework problem sheets. The other three items were applied problems that presented new situations-. The solution of the problems on th i s test required the student to apply e x i s t i n g TPC's of calculus. The r e l i a b i l i t y of this test was computed by using the "Cronbach a" formula. Each question on the test was scored out of 5'points. A careful grading system was worked out for each question and closely followed when marking the papers. A copy of the te s t i s i n Appendix D and a sample of the scoring appears i n Appendix E. STATISTICAL DESIGN A completely randomized single factor design was used. Since there were only two treatments i t was appropriate to use the t - s t a t i s t i c for the test of s i g n i f i c a n c e . Figure 2 on page 34 shows a block diagram of the design and the symbols used i n the s t a t i s t i c a l analysis. The diagram i s adapted from the one given by Winer (56:28). 34 Basic population of interest, Instrumentation and Control students Random sample of n + n^ subjects, a b J n subjects a J Treatment A Theory-problem C r i t e r i o n measures X. understanding score l a ^ Y^ a problem solving score-Summary s t a t i s t i c s X Y a a xa ya Parameters of population after treatment y xa xa y. ya n^ subjects Treatment B Problem-theory I C r i t e r i o n measures X i b u n ( ^ e r s t a n d i n g score Y ^ problem solving score Summary s t a t i s t i c s X, Y, b b xb yb Parameters of population after treatment y xb y Figure 2 Design of experiment The basic population of in t e r e s t was f i r s t year Instru-mentation and Control students. The actual students used i n the study were assumed to be a random sample from the above population. The random sample of n a + n^ subjects were assigned to either treatment A or treatment B. Treatment A was the theory-problem i n s t r u c t i o n a l strategy, and treatment B was the problem-theory i n s t r u c t i o n a l strategy. At - the s t a r t of the experiment n a and n^ were not equal. Two c r i t e r i o n measures were taken on each subject. The c r i t e r i o n measures were the understanding score and the problem solving score. The summary s t a t i s t i c s computed for each c r i t e r i o n measure were the mean and variance. These summary.statistics were used to perform the t - t e s t and compare the parameters of the population aft e r treatment. The s t r u c t u r a l model for the experiment requires the following basic assumptions: 1. The treatment e f f e c t i s s t a t i s t i c a l l y independent from the experimental error. 2. There i s homogeneity of experimental error. A test for.homogeneity of error variance was performed (see Chapter IV) to check that the data were consistent with the underlying s t r u c t u r a l model, being used. STATEMENT OF STATISTICAL HYPOTHESES The research hypotheses as formulated at the end of Chapter II can be evaluated by means of s t a t i s t i c a l tests of the following n u l l hypotheses: 36 To test hypothesis 1; H l 0 : ^xa = ^xb H 1 1 : yxa ^ yxb Level of si g n i f i c a n c e a = .05 To test hypothesis 2; HV V = yyb H 2 1 : V < ^yb Level of significance a =..05 Decisions concerning the s t a t i s t i c a l n u l l hypotheses w i l l be based on the standard decision rules for the t - s t a t i s t i c . CHAPTER IV ANALYSIS OF RESULTS THE UNDERSTANDING TEST The twenty question multiple choice test designed to measure understanding of the TPC's of calculus was f i r s t used with a group of 20 night school students taking a calculus course that covered s i m i l a r material. The "K-R 20" r e l i a b i l i t y as computed by the University of B r i t i s h Columbia's (U.B.C.) TIA computer program was 0.48. The test had four "bad" items which i f deleted gave a "K-R 20" r e l i a b i l i t y of 0.61. The "bad" items were rewritten and when the test was used i n the study with a t o t a l of 33 students the U.B.C. TIA computer program gave a "K-R 20" r e l i a b i l i t y of 0.77 with a l l twenty items included. In order to check that the experimental data were con-si s t e n t with the underlying s t r u c t u r a l model used i n the s t a t i s -t i c a l design the following hypothesis was tested using the data obtained on the understanding t e s t . The data required for the test are summarized i n Table II on page 38. 38 The test for homogeneity of. variance appears below. H n: cr2 = a 2 0 xa xb H-, : a1 ^ a 2 , 1 xa ' xb Level of sig n i f i c a n c e a =0.10. F , , = s 2 / s 2 =8.39/16.41 = 0.511. observed xa' xb ' C r i t i c a l , value = F 0 5 ( ^ 3 jgj =:.0.402. Since F , > 0.40 2. the c r i t i c a l value, we do not rej e c t obs HQ and conclude the underlying s t r u c t u r a l model i s satisfactory.. TEST OF HYPOTHESIS HI A summary of the analysis of the understanding te s t scores i s - found i n Table II.below. Table II Analysis of Understanding Test Scores Treatment Sample Size Sample Variance . . C r i t e r i o n t A =. T-P n, = 14. a ' s2 = 8.39 xa X = 12.57 a 1.32 B = P-T n b = 19 s 2 = 16.41 xb X, =10.89 b Since the observed t value of 1 . 3 2 was less than the c r i t i c a l value t 9 - 7 5 ( 3 1 ) = 2 . 0 4 the hypothesis H1 Q was not rejected. There was no s i g n i f i c a n t difference i n achievement on the understanding te s t between those students who were taught by the theory-problem i n s t r u c t i o n a l strategy (treatment A) and those taught by the problem-theory i n s t r u c t i o n a l strategy (treat-ment B) . 39 THE PROBLEM TEST The s i x item problem te s t was designed to measure success at solving applied problems. Each problem on t h i s test was scored out of 5'points. The problem test was developed and tested by using two other groups of students who were not i n the study.. The scoring system was developed.by marking the papers of these students which were not i n the study. The scoring system adhered to the following r u l e s : 1. One. point was awarded for the i n i t i a l problem set up. This included i d e n t i f y i n g required formulas and the data given i n the problem. 2. The application of calculus TPC's. required to ar r i v e at the solution was s p l i t into three parts (depending on the method of solution) and one point awarded for including each step co r r e c t l y . 3. One point was awarded f o r performing a l l calculations c o r r e c t l y and making a statement of the problem's solution. Appendix E contains a complete solution of one of the problems and a sample of the scoring. The s i x item problem t e s t was f i r s t used with a group of twenty night school students that covered s i m i l a r material. The "Cronbach a" r e l i a b i l i t y was computed using a ca l c u l a t o r and found to be 0.71. At this stage the .wording of a few questions was changed.. When the test was used i n the study with a t o t a l of 33 students the "Cronbach a" r e l i a b i l i t y was. 0.79. In order- to check that the data were consistent with the underlying s t r u c t u r a l model used in. the s t a t i s t i c a l design the following hypothesis was tested using the data obtained on the problem solving t e s t . The data required f o r the test i s summa-ri z e d i n Table III'below. The t e s t for homogeneity of variance appears below. H n : a 2 =o- 2 , 0 ya yb : a2 yi o2, 1 ya ' yb Level of s i g n i f i c a n c e a = 0.10. F . , = s 2 / s 2 . =• 65.45/84.94 •=. .771. observed ya yb ' C r i t i c a l value = F . 0 5 a 3 f l 8 ) = -402.: Since F , > 0.40 2, the c r i t i c a l value, we do not-reject obs ' ' J H n and conclude the underlying s t r u c t u r a l model i s s a t i s f a c t o r y . TEST OF HYPOTHESIS H2 A summary of the analysis of the problem te s t scores i s found i n Table III below. Table III Analysis of Problem Test Scores Treatment Sample Size Sample Variance C r i t e r i o n t A =- T-P n = 14 a s2 = 65.45 ya Y = 18.2 6 a 0.757 B = P-T n b = 19 s 2 = 8 4.9 4. yb Y, = 15.95 b Since the observed t value of .757 was greater than the c r i t i c a l value t Q5(31) = -1.70 the hypothesis H2Q was not rejected. This indicates that there was not s t a t i s t i c a l l y 41 s i g n i f i c a n t difference i n achievement on the problem test between those students who were taught by the theory-problem instruc-t i o n a l strategy (treatment A) and those taught by the problem-theory i n s t r u c t i o n a l strategy (treatment B)• INTERPRETATION OF RESULTS The f i r s t experimental hypothesis as stated at the end of Chapter II. was not rejected. Students who were taught by the problem-theory i n s t r u c t i o n a l strategy did not have a n y . s t a t i s t i -c a l l y s i g n i f i c a n t higher scores on the post t e s t designed to measure understanding of techniques, p r i n c i p l e s and concepts of calculus than the students taught by the theory-problem i n s t r u c t i o n a l strategy. I t i s i n t e r e s t i n g to note that students of the theory-problem strategy scored higher on the test of understanding although t h i s difference 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 . The second experimental hypothesis as stated at the end of. Chapter II was rejected. Students who were taught by the problem-theory i n s t r u c t i o n a l strategy did not have higher scores on the post t e s t designed to measure success at solving applied calculus problems than the students taught by the theory-problem strategy. It i s i n t e r e s t i n g to note that the students taught by the theory-problem strategy did better on the test designed to measure success at solving applied problems although t h i s d i f -ference was not s i g n i f i c a n t . ADDITIONAL OBSERVATIONS 42 When the study began, there were a few students who got their, timetables mixed up and t r i e d to come to the wrong lecture section. The investigator just refused admittance to the cl a s s -room and within two weeks a l l students came to the correct lectures. The investigator kept a record of each students' lecture attendance. There were only four students that missed more than 4 lectures. The investigator noticed that the problem-theory group seemed to, take more lecture notes than the theory-problem group. Some students i n the theory-problem group just sat and l i s t e n e d to the lecture. The problem-theory lectures always seemed to take a few minutes longer than the theory-problem lectures, probably because of more discussion about, the problem and a student who often complained that the lecture r o l l was turned too fast . The.investigator gave problem sheets after each lecture and kept a record of those completed by the students. There were six of the 19 problem-theory students that completed less than 50% of. the problem sheets. There were two of the 14 theory-problem students that completed less than 50% of the problem sheets. There were 18 of the t o t a l of 33 students who completed a l l of the problem sheets. CHAPTER V CONCLUSIONS SUMMARY This study was designed to determine i f using problems to i n i t i a t e the study of mathematics versus using problems to i l l u s t r a t e the study of mathematics makes any difference i n terms of student achievement on tests designed to measure understanding of TPC's and success at solving applied problems. There was no s i g n i f i c a n t difference between the two i n s t r u c t i o n a l strategies on either of the tests. The results of this study indicate that students w i l l do as well i f problems, are used to i n i t i a t e the study.of mathematics.as they would i f problems are used to i l l u s -t rate the study, of mathematics. The theory-problem group had a higher mean score on both of the tests,.but t h i s difference 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 . DISCUSSION The investigator noted that students of the problem-theory group were more active i n the taking of notes. The problem placed at the beginning of the lecture seemed to arpuse the students' i n t e r e s t . The students of the theory-problem group did i 44 not appear to be as interested at the beginning of the lectures when the theory was introduced. However, their interest seemed to increase during the problem part of the lecture. The result of no significant difference on .the test designed to measure success at solving applied problems is con-trary to the results that Plants and Venable (35) found when investigating a similar problem in an engineering mechanics'-dynamics course. The reason for the difference may have been because Plants and Venables' problem test contained unusual and unexpected problems. In the investigator's problem test the problems were of a more' routine nature, although three of the six were new to the students. THE HAWTHORNE EFFECTS The "attention" Hawthorne effect was controlled by telling the students that two groups were being formed to increase instructor student contact. To the best of the investigator's knowledge, none of the students knew that they were taking part in an experiment. The investigator was never confronted by a student asking if he was in an experiment. The "novelty" Hawthorne effect was controlled in two ways. Both of the experimental groups had the same instructor. While a new physical device was used,.the overhead projector, it was used with both of the groups in the same way. 45 The " i d e n t i f i c a t i o n " Hawthorne e f f e c t was not working d u r i n g t h i s study. Of the two groups i n the study the i n v e s -t i g a t o r expected the problem-theory group to do b e t t e r . In f a c t the theory-problem group d i d b e t t e r , but not s i g n i f i c a n t l y b e t t e r . LIMITATIONS OF THE STUDY The students i n the study were chosen from a s i n g l e technology. The experiment was designed to keep the groups separated d u r i n g l e c t u r e p e r i o d s and lunch hours but o u t s i d e of these times t h e r e was probably some degree of i n t e r a c t i o n . Students o f the d i f f e r e n t groups may have s t u d i e d t o g e t h e r . Students who missed l e c t u r e s may have c o p i e d missed notes from the wrong group. A f t e r each l e c t u r e , problem sheets on the l e c t u r e m a t e r i a l were gi v e n to the s t u d e n t s . The f i r s t q u e s t i o n s on these sheets e x e r c i s e d the TPC's of the l e c t u r e . The l a t e r q uestions were a p p l i e d problems. The purpose df the problem sheets was to keep a l l students doing the same amount o f work. However, the f a c t t h a t the t h e o r y q u e s t i o n s were f i r s t may have r e i n f o r c e d the theory-problem group. T h i s might e x p l a i n why the theory-problem group d i d b e t t e r , but not s i g n i f i c a n t l y b e t t e r , on both of the t e s t s . 46 When the lecture r o l l s were prepared, the theory-problem lecture r o l l was always constructed f i r s t . . While i t was kept i n mind that the order would be changed for the problem-theory .group, the theory-problem lectures might have had better continuity. The fact that the content.of both types of lectures was kept i d e n t i c a l , while desirable from, the design point of view, may have put too much constraint on the two i n s t r u c t i o n a l strategies. I t would be more natural to keep overall, content i n terms of theory.and problems fixed while l e t t i n g the content of i n d i v i d u a l lectures vary. The experiment was run for only eight weeks. Considering the nature of the experiment a longer period of time would have been preferred but this was not f e a s i b l e . SUGGESTIONS FOR FURTHER RESEARCH A study s i m i l a r to t h i s one should be conducted using two d i f f e r e n t technologies., This, would cut down.the i n t e r -action between the two groups. A pretest could be designed and used as a covariate i n analyzing the post test scores. A study s i m i l a r to t h i s one should be conducted with problem sheets also constructed i n the s t y l e of the.lecture. Another study could be conducted using a t h i r d group as a control. This group would be taught equally with both of the i n s t r u c t i o n a l strategies. 47 A study related to this one could be designed to see i f . i t makes any difference i f the applied problems used are not directly related to the student's. own.technology. This could be easily done by.lecturing to two technologies simultaneously and only giving applications to one of the technologies. It is recommended that any study similar to this one be constructed over a longer period of time.. It would also be an advantage to involve other interested instructors. A. study similar to this one should also be conducted in which the problem-theory instructional strategy does not complete the problem solution until after the necessary theory is developed. BIBLIOGRAPHY 49 1. Amidon, E. and Hunter, E, Improving Teaching: The Analysis of. Classroom Verbal, Interaction... 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"Effects of D i f f e r e n t i a l l y Structured Introductory Materials and Learning Tasks on Learning and Transfer," American Educational Research Journal, 5:191-201, March 1968. 23. Heimer, R. T. "Conditions of Learning in.Mathematics: Sequence Theory Development," Review of.Educational. Research, 39:493-508, October 1969. 24. Jerrolds, B. W. "The Effects of Advance Organizers i n Reading for the Retention of S p e c i f i c Facts," Dissertation Abstracts, 28:4532-A, May 1968. 25. J u s z l i , F. L. Analytic Geometry and Calculus. Englewood C l i f f s : P r e n tice-Hall, Inc., 1961. 26. K i l p a t r i c k , J. "Problem Solving i n Mathematics," Review of_Educational Research, 39:523-33, October 1969. 27. Klausmeier, H. J. and Harris, C. W. Analyses of Concept Learning. New York: Academic Press, 1966. 51 28. Kline, M. Calculus: An I n t u i t i v e and.Physical Approach. Pt. One, New York: John Wiley and Sons, Inc., 1967. 29. Livingstone, S. A. "Simulation Games as Advance Organizers i n the Learning of Social Science Materials," ERIC ED 039 156, A p r i l 1970. 30. McKeachie, W. J, (ed.). The.Appraisal of Teaching i n . Large U n i v e r s i t i e s . Ann Arbor: University of Michigan,.1959. 31. Medley, D. M. and Mitzel H. E. . "Some Behavioral Correlates of Teacher Effectiveness," Journal of Educational- Psychology, 50 :239-246 , December 19 59. 32. National Council of Teachers of Mathematics. The Learning of.Mathematics, Its Theory, arid-Practice. Twenty-first Yearbook. Washington: N. C. T. M.,1953, 33. Neisworth,.J. T. "Use of' Advance Organizers with the Educable Mentally Retarded'," Dissertation Abstracts, 28:4539-A, May 1968. 34. N i e t l i n g , L. C. "Using Problems to I n i t i a t e the Study of-Certain Topics i n Mathematics," Dissertation  Abstracts, 29:1372-A, November 1968. 35. Plants, H. L. and Venable, W. S. "Teaching.Problem Solving S k i l l s : Theory or Practice F i r s t , " Engineering.Education, .60 :732-4 , March 1970. or ERIC ED 025 447,,December 1969. 36; Pollak, H. 0. "Problems, of Teaching Applications of Mathematics," Educational. Studies i n Mathematics, 1:1-30 , May 1968. 37. Pollak, H. 0. "Applications of Mathematics," Mathematics Education, 311-334, Chicago: N. C. T. M., 1970. 38. Rector, R. E. "The Relative Effectiveness of Four Strategies for Teaching Mathematical Concepts," Dissertation Abstracts, 29 :520~A, August 1968. 39. Report of the Cambridge Conference on School Mathematics. Goals for School Mathematics. Boston: Houghton M i f f l i n Company, 1963. 40. Report of the Cambridge Conference on the Correlation of Science and Mathematics i n the Schools. Goals for  the Correlation of Elementary Science and Mathema-r  t i c s . Boston: Houghton M i f f l i n Company, 1969. 52 41. Riban, D. M. "An Investigation of the Relationship of Gagn^'s Hie r a r c h i c a l Sequence Model i n Mathematics to the Learning of High School physics," Dissertation Abstracts, 30:4845-A, May 1970, 42. Rice, H. S. and Knight, R. M.. • Technical Calculus and Analysis. New York: McGraw-Hill, Inc.,.1959. 43. Richmond, A. E. Calculus, for Electronics., New York: McGraw-Hill, Inc., 1958. 44. Rosskopf, M. F. "Strategies for Concept Attainment i n Mathematics," Journal of Experimental Education, 37 :78-86, F a l l 1968. 45. Scandura, J . M. • "Prior Learning, Presentation Order and Prerequisite Practice i n Problem Solving," Journal of Experimental Education, 34:12-18, Summer 1966. 46. Scandura,.J. M. and Wells, J. N. "Advance Organizers i n Learning Abstract Mathematics," American Education  Research Journal, 4:295-301, May 1967. 47. Scandura, J. M. "Algorithm Learning and Problem Solving," Journal of. Experimental Education, 34:1-6, Summer 1966. 48. Scandura, J. M. "Problem Solving and Prior Learning," Journal of Experimental Education, 34:7-11, Summer 1966. 49. Shulman, L. S. and Keislar,.E. R. (eds.). Learning by Discovery: A C r i t i c a l Appraisal. Chicago: Rand McNally and Company, 1966. 50. Sub-Committee on Mathematics for I n s t i t u t e of Technology i n Canada.- The... S e c o nd . I n te r im Re po r t. Burnaby : D i v i s i o n of Technical and Vocational Curriculum -Technical Branch B r i t i s h Columbia Department of Education, 1966. 51. Sub-Committee on Mathematics for Institutes df Technology i n Canada. The Third Report. Burnaby: D i v i s i o n of Technical and Vocational Curriculum - Technical. Branch B r i t i s h Columbia Department of Education, 1967. 52. Townsend, R. D. "The Ef f e c t s of an Advance Organizer on Learning to Graphically Analyze Straight Line Kinematics by Classroom.Instruction or Programed Instruction," Dissertation Abstracts, 29:1760-A, December 196 8. 53 53. Washington, A. J. Basic Technical Mathematics With  Calculus. Reading; Addison-Wesley Publishing Company Jnc.,1964. 54i Weisberg, J. S. "The Use of V i s u a l Advance Organizers for Learning Earth Science Concepts," Dissertation  Abstracts,, 30 :3867^A, March 1970. 55. Willoughby, S. S. "Issues i n the Teaching of Mathe^ matics," Mathematics Education,. Chicago: N. S. S. E. 1970 . 56. Winer, B. J . S t a t i s t i c a l P r i n c i p l e s i n Experimental Design. New York: McGraw-Hill, Inc., 1962. 57. Woodward, E. L. "A Comparative Study of Teaching Strategies Involving Advance Organizers and Post Organizers and Discovery.and Non-discovery Techniques Where Instruction i s Mediated by Computer," Dissertation Abstracts, 27:3787-A, May 1967. APPENDIX A LECTURE TOPICS 55 LECTURE TOPICS 1. Functions and functional notation. 2. The l i m i t of a function by the numerical method. 3. The l i m i t of a function by the algebraic delta process. 4. Rates of change at a fixed point by the graphical method, the numerical method, and the delta process. 5. D e f i n i t i o n of the derivative (rate of change at an a r b i t r a r y p oint). 6. Derivatives of polynomial functions by using rules. 7. Derivatives of products and quotients of functions. 8. The power rule and the chain r u l e . 9. Implicit d i f f e r e n t i a t i o n . -10. Curve sketching (polynomials only). 11. Applied maxima-minima problems. 12. D i f f e r e n t i a l s and approximations. 13. The a n t i - d i f f e r e n t i a l . 14. The i n d e f i n i t e i n t e g r a l . 15. The d e f i n i t e i n t e g r a l . 16. The area under and between curves. 17. Area as the l i m i t of a sum and average values of functions, 18. Volumes of rotation by s h e l l s and disks. 19. Centroids. APPENDIX B SAMPLE LECTURE LECTURE 8 THE POWER RULE AND THE CHAIN RULE THEORY-PROBLEM LECTURE Suppose that y = f(x) = u n , where u = u(x) i s some other function of x. I f for example y = f(x) = (10 - x 2 ) ^ then n =5,. u = u(x) = 10 - x 2. We want a rule for d i f f e r e n t i a t i n g functions raised to a power. To find the rul e we use the d e f i n i t i o n of the derivative and the de l t a process. l i m i t f(x + Ax) - f(x) f'(x) = Ax-*0 Ax . l i m i t [u(x +.Ax) ] n - [ u ( x ) ] n  ( ' Ax+0 Ax f , , v _ l i m i t [u + Au] - u K ' Ax+0 Ax T • • J _ n . n-1. , , . > n n f , , . l i m i t u + nu Au + ..• + (Au) - u W Ax+0 Ax i=' ( \ l i m i t [nu"" 1 + . . . + (Au) n ~ 1 ] . Au r l X } Ax-*0 1 Ax f'(x) = nu n - 1u"(x) n T „ „ , n. . n-1 , d(u ) n-1 du RULE SIX (u ) 1 = nu u' or ^ x ' = nu. Example: y = f (x) = (10 - x 2 ) 6 f i n d f ( x ) . Think u = (10 - x 2) n =6. then f'(x) = [(10 - x 2) 6] ' =.6(10 - x 2 ) 5 ( 1 0 - x 2) ' f'(x) = 6(10 - x 2) 5(-2x) = -12x(10 - x 2 ) 5 (x 2 - 2) 3Example: y = f(x) = ( 1 _ x £ ) 2 (The working of th i s example was included i n the actual lecture but w i l l not be included here.) 5 8 Consider the following problem. If z = z(y) = (1 + y 2) 2 j, dz and y = y(x) = x 2 + 1 f i n d or z' (x) . The obvious method would be to compute z(x) = z[y(x)] = [1 + (x 2 + 1) ] 2, and d i f f e r e n -t i a t e t h i s d i r e c t l y . The chain r u l e or function of a function rule w i l l make this job easier. I t can be derived exactly i n the manner of the power rule above, so the derivation w i l l not be given. RULE SEVEN z'(x) = z'(y) y'(x) or = dz. dy_. Q r dx dy dx D z = D z D y. x y x-* The three versions are of course equivalent.- Only the notation i s d i f f e r e n t . Example: Consider the problem posed above. We had z(y) = (1 + y 2 ) h y(x) = xh + 1 z'(y) =-,h(l + y2)~h(l + y 2 ) ' = h{l + y 2 ) " % ( 2 y ) z' (y) =.y(l + Y2)~k y'(x) = hx'h so z'(x) = z'(y) y'(x) = y ( l + y 2 ) ~ ^ x - 3 5 (in the actual lecture the above expression was simplified.) This i s the end of the theory portion of the lecture. On the next page i s the problem portion of the lecture., If i t i s inserted i n front of the above theory the problem-theory lecture i s obtained. 59 Application of the power rule ( u n ) ' = nu n-1 u' Problem: The l e v e l 0 i n feet of l i q u i d i n a cone shaped processing tank i s given, by r* R 9(t) = 0.1t(10 - t ) . Find the rate of outflow q = ,q(t). Solution: Volume of l i q u i d i n a cone shaped tank i s given by V = . j f r r 2 e r 8 J . J.\ by similar traingles — = — so V = V (0 ) = -=r TT£2 K tl J ti by substituting i n the formula for 0 = 0(t) we obtain v = v e t ) = '°3°llR2 t 3 ( i o - t ) 3 using the p r i n c i p l e (outflow) = (-rate of change of volume) we obtain by applying the power rule (actual working was part of the lecture but i s not shown here) q = q(t) = -.OOITT.R^ T 2 ( 1 Q _ T ) 2 ( 1 Q ^ 2 T ) H: Application of the chain rule z •' (x) = z' (y) y 1 (x) . We apply the chain rule to the problem above. We have V = V ( 8 ) and 0 =. 0(t) and wish to f i n d q (t) = . So we j u s t f i n d V ' ( 0 ) and 0'(t) and multiply. (The actual working.of the problem formed part of the lecture. When the answer was obtained i t was com-pared with the one above.) Solution of problems involving related rates. Any two variables which vary-with respect to time and between which a r e l a t i o n i s known to e x i s t , can have the time rate of change of one expressed i n terms of the time rate of change of the other. 6.0 One rate of change i s expressed i n terms of the other by using the chain rule. (These ideas were related to the problem above, the discussion i s not shown here.) Problem: In a certain tank the volume V of l i q u i d at any time t i s related to the l i q u i d l e v e l 0 at any time t by the function V = V(0) = 0.75 8 2.. Find the rate at which the volume i s i n -creasing when, the l e v e l 0 i s increasing at a rate of .5 ft/min. At what rate i s the volume increasing when the l e v e l i s 8 f t . (Actual solution formed part of the lecture but i s not shown here.) This i s the end of the problem portion of the lecture. If pages 58 through 61 are read consecutively you have a t y p i c a l lecture of the theory-problem i n s t r u c t i o n a l strategy. If the pages are read i n the order 60, 61, 58, 59 you have a t y p i c a l lecture of the problem-theory i n s t r u c t i o n a l strategy. APPENDIX C SAMPLE PROBLEM SHEET Mathematics 3 2.223 Number 8 January 1972 48 ABC1 1. Use the power rule to d i f f e r e n t i a t e the following: (a) y = [1 - x 2 ] 3 5 (b) p(v) = ( 5 J2y2) (c) s(t) =: ( t 3 - t ) 1 ^ . (d) H(r) = [rZ+K(L/2) V 2 2. Use the chain rule to f i n d the rates of change indicated. (a) z(y) = y ^; y(x) = x 2 +1 f i n d ^ when x = /3. (b) z = y 2 - 2y; ^ = 4 fin d | | when y = 2. (c) v = ^ e 3 ; | | =-2 ft/min f i n d when 0 = 5 . ( d ) q = p 1- P10 ; i t = " 0 ' 8 f i n d It when p = 15. 3. When a i r expands so that there i s no change i n heat (adia b a t i c change) the r e l a t i o n between pressure and volume i s pv 1' 4 = k, where k i s a constant.. At a ce r t a i n instant the pressure i s 3 atm. and the volume i s 10 i n 3 . The volume i s increasing at a rate of 2 i n 3 / s e c . What i s the time rate of change of pressure at this instant? 4. The voltage E i n vol t s of a certa i n thermocouple as a function of the temperature T i n degrees centigrage i s given E = E(T) = 2.8T + 0.006T2. Both E and T also vary with time i n minutes. If the temperature i s increasing at the rate of l°C/min, how f a s t i s the voltage increasing when T = 100°C? APPENDIX D TESTING INSTRUMENTS 6.4 BRITISH. COLUMBIA INSTITUTE OF TECHNOLOGY COURSE and NUMBER:. MATHEMATICS (INTRO. CALCULUS) 32.223 (PART I). DATE: FEBRUARY 1972 STUDENT NAME: .  TIME ALLOTTED: 1 h. hours STUDENT NUMBER: EXAMINER: TECH: SET: TOTAL PAGES OF EXAM .  SPECIAL INSTRUCTIONS: 1. The student w i l l use his own tables and s l i d e r u l e . 2. Textbooks and notebooks are not allowed. 3. Attempt a l l questions 1 even i f i t means guessing. 4. Indicate your answer by c i r c l i n g the appropriate l e t t e r on the ANSWER SHEET. 1. A student using the method of increments to determine the instantaneous speed of an object obtained the results l i s t e d below: At As . As/.At . 1.0 12.0 .1 1.241 .01 0.129401 .001 0.012994001 .0001 0.001299940001 What i s the instantaneous speed of the object? a) 12.0 b) 13.0 c) 12.9994001 d) .0013 e) not defined The following graph of distance s, i n feet, versus the time t, i n seconds was obtained experimentally, and the equation for the distance function i s not known. Find the instantaneous v e l o c i t y when t i s 10 seconds by using the graphical method of d i f f e r e n t i a t i o n . a) +5.5 ft/sec b) -.18 ft/s e c c) +4.5 ft/sec d) -4.5 ft/sec e) +3.5 ft/sec 50 45 D I 40 S T 35 A N 30 C E 25 I 20 N 15 F E 10 E T 5 2 4 6 8 10 12 14 16 TIME IN SECONDS When finding the derivative of Y = 2 x 2 + 1 by the delta process AY/Ax i s found to be: a) 4x b) 4 xAx + 2 Ax 2 c) 4 x + 2 Ax d) 4 x + Ax e) 2 x + 2 Ax 6 6 4. The distance i n feet t r a v e l l e d by a f a l l i n g object i s given by:. s(t) = +12 + 16t - 16t 2 where t i s time i n seconds. How far has the object t r a v e l l e d when i t s v e l o c i t y i s 0? a) 24 feet b) 0 feet c) 8 feet d) 16 feet e) can't say because the v e l o c i t y i s never 0. 5. If y (x) = x then y' (x) i s given by: / l - 4x a) 1 - .5x b) -2x . c) 1 - .75x / l - 4x / l - 4x / l - 4x d) 1 - 2x e) 1 - 2x 1 - 4x- (1 - 4x) 3 / 2 ds 6. If s(t) = 2t/ 1 - 2t then. ^  i s given by: a) 2 - 3t b) 1 - 4t c) -4t / l - 2t / l - 2t / l - 2t d) 2 - 6t e) 4t - 1 / l - 2t / l -• 2t 7. The equation of the l i n e normal to the curve y(x) = at x = 2 i s given by: a) y = - x + 3 b) y = x + 3 c) y = x - 1 d) y = 1 e) y = -x + 1 x 2 67 An equation for the l i n e tangent to the curve y(x) = 2/ x at the point where x = 1 i s given by: a) y = x - 1 b) y = x + 3 c) y = -x + 1 d) y = -x +.3 e) y = x + 1 4 9. If f (x) = —-—Y a n c^ g ( x) = 2x then what i s f (g(.5)) equal to? a) undefined b) 8 c) -16 d) 0 e) 1 10. An object moving in. the x-y. plane has the following parametric equations describing i t s motion, x = 3 t 2 y = 10 - 4 t 2 If x and y are i n feet and t i s i n seconds, f i n d the magnitude and angle of the v e l o c i t y when t = 1 second, a) 10/53°8' b) 2/36°52' c) 14/30° d) I O Z - S S ^ ' e) e ^ / e s ^ e ' 68 11. If y = y(x) and y 1 (x)>0 for a l l values of x and y"(x)<0 for a l l values of x then which of the following could be part of the graph of y = y(x)? a) -1/6 b) -1/4 c) 1/4 d) does not ex i s t e) cannot be determined by th i s method 13. If S = x 2 then AS - dS i s equal to: a) 0 b) 2xAx c) (Ax) 2 d) 2xAx + (Ax) 2 e) x 2 + (Ax) 2 14. For what value of the constant K w i l l f(x) = x + — x have a r e l a t i v e maximum at x = -21 a) K = -4 b) K = -2 c) K = 2 d) K = 4 e) none of these 15. The graph of y, = f(x) = 5x 3 - 3x 5 has a r e l a t i v e maximum at: a) (0,0).only b) (-1,-2) only c) (1,2) only d) (0,0) and (-1,-2) e) (0,0) and (1,2) 16. If a = s"(t) = 32 f t / s e c 2 , and v(0) = :S' (0) = 10 f t / s e c , and s(0) = 100 f i n d the equation for s ( t ) . a) s(t) = 100+ lOt + 16t 2 b) s(t) = 100 + lOt + 32t 2 c) s(t) = 100 - 16t 2 d) s(t) = 16t 2 + lOt e) s(t) = 32t + 10 17. If f 1 ( x ) = x 3 + x and f ( l ) = 0 then f(x) i s given by: a) f(x) = .25x" + .5x2 + .75. b) f(x) = 4x" + 2x 2 - 6 c) f (x) = 3x 2 + 1 d) f(x) = .25X* - .5x + .5 e) f(x) = xh ^ x 2 3 70 r J i 18. Evaluate: | x/ x dx fix a) 2(5) 5/. 3 - 2/5 b) 2 (5) 5 / 2 - 2/5 c) 2 ( 5 ) 3 / 2 - 2/5 d) /S (5) 2 1 2 " 2 e) none of these 19. The area i n quadrant I under the parabola.y = 9 - x 2 i s equal to: a) 18 b) 36 c) 0 d) 9 e) 27 20. Integrate: ^ ( 1 - 2x) 6 dx a) (1 - 2x) 7 + C b) (I - 2x) 5 + C 7 5 c) (1 - 2x) 7 + C d) -2 (1 - 2x) 7 + C -14 7 e) none of these MATHEMATICS 32.223 CIRCLE YOUR CHOICE NAME SECTION 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. a a a a a a a a a a a a a a a a b b b b b b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c c c c c d d d d d d d d d d d d d d d d d d d d e e e e e e e e e e e e e e e 72 BRITISH COLUMBIA INSTITUTE OF TECHNOLOGY COURSE and NUMBER: MATHEMATICS (INTRO. CALCULUS) 32.223 (PART II) DATE: March 197 2 STUDENT NAME: TIME ALLOTTED: 1% hours STUDENT NUMBER: EXAMINER: J . W. Brown TECH: SET: TOTAL PAGES OF EXAM SPECIAL INSTRUCTIONS: Exam. Booklets Required. INSTRUCTIONS: 1. The student w i l l use his own tables and-slide r u l e . 2. Textbooks and notebooks are.not allowed. 3. Marks w i l l not be awarded to answers where work i s not shown. 4. Examination booklets supplied. 1. A transmission l i n e hangs i n an approximate parabolic shape:as shown below. The distance between the towers i s 200 feet and the sag midway between the towers i s 20 feet. Y + Choose the coordinate system shown above and fi n d the angle between the transmission l i n e and the tower on the rig h t . 73 2. The radius of a certa i n steam engine cylinder i s three inches. What i s the speed of the piston i f steam i s entering the cylinder at a constant rate of 2 f t 3 / s e c ? Assume no compression of steam at the moment. 3. A rectangular processing tank with square bottom and no l i d i s to have a volume of 8 f t 3 . The tank i s to Find the dimensions of the tank i f the length of the welded j o i n t s i s to be a minimum. 4. A steel b a l l of radius 1.0 cm i s given a rubber coating .05 cm thick. Approximately what volume of rubber i s used i n coating the b a l l ? (Use d i f f e r e n t i a l s to obtain your s o l u t i o n ) . 4 i NOTE: Volume of a sphere i s given by ^ TT r . be made out of metal and a l l seams must be welded. 74 5. The v e l o c i t y d i s t r i b u t i o n i n ft/sec of f l u i d p a r t i c l e s moving through a pipe of diameter D i n f t i s parabolic as shown i n the diagram below. If D = 1 f t and V = 25 ft/sec f i n d the equation for max ^ V(x) and estimate the average v e l o c i t y V i n ft/sec of 1 p D / 2 a V f l u i d flow by using V = r V(x) dx. a v D J - D / 2 6. The work done when a gas expands or i s compressed i s J v 2 p(v) dv. If the gas undergoes an V i adiabatic change then p and v are related by the equation pv 1" 5 = constant. Suppose that p = 20 p s i when v = 225 i n 3 and that the gas i s then compressed to 36 i n 3 . . How much work i s required? ( APPENDIX E SAMPLE OF SCORING ON PROBLEM TEST SCORING PROBLEM THREE The following i s the correct solution of problem three and the points earned for each step. Step 1. I n i t i a l problem set up -. 1 point. (1) Volume = V = 8 f t 3 (2) V = hs 2 (3) 8 = hs 2 (4) Length of weld = L = 4s + 4h Step 2. Find function to be minimized - 1 point. The length of weld i s to be made a minimum so we must.express L i n terms of a single independent variable Solve (3) for h and substitute i n (4) to obtain: (5) L = L:(s) = 4s + s Step 3. Look, for horizontal tangents - 1 point. -3 (6) Slope function = L'(s) =4 - 64s Set (6) equal to zero and solve (7) s 3 = 16 so x =2.52' Step 4. Test for maximum or minimum - 1 point. (Student could use f i r s t or second derivative test or the absolute maximum minimum prin c i p l e ) (8) Second derivative = L"(s) =_+192s"4 L"(2.52) = +ve Thus we have a minimum at s =2.52' Step 5. Correct solution and answer to problem - 1 poin (This point was l o s t for an arithmetical error i n any 77 stage of the problem solution.) We want the dimensions of the tank. From (3) we obtain: (9) h = 8/s 2 substituting s = 2.52' into (9) h = 1.26' The dimensions of the tank are: h = 1.261 and s = 2.52' This i s the end of the correct.solution. A STUDENT'S SOLUTION The following student solution scored 3 points. I t was chosen so the reader could see how a student l o s t points. The typed student solution i s arranged as cl o s e l y as possible, to that of the o r i g i n a l student working. The investigator's comments appear i n [square] brackets. 3. Welding to be. minimum 4s + 4h to be minimum V =. s 2h = 8 [at thi s stage step one i s Sub. complete so student gets 1 point] (4/B~7h + 4h) ' = 0 [step two complete so another point (4 (8/h) 3 s + 4h) ' = (4 (2.623)h"J2 + 4h) ' = 11.3*-%h + 4 = 0 [the student has made the more gained] 0 = -5.65h + 4 d i f f i c u l t s ubstitution but completes -5.65xh 4/5.65 = 7.08 step three] = h " 3 / 2 = 1 h ^ 2 [point awarded for step 3] 78 (student solution continued) 1 = /TTW =.2.66 h h 3 =• 1/2.66 h = 0.72 h . = 3 /276*6 V = s 2 (0.72) = 8 h = 1/1.39 =0.72 s 2•= 8/.72 = 11.1 s- = / l l . 1 = 3.32 h = 0.72 f t + s = 3.32 f t This i s the end of the student's solution. The student has made an arithmetical error and thus loses a point for step f i v e . The student does not include any mention of a test for maximum or minimum and thus loses a point for step four. The student's t o t a l score for t h i s problem i s 3/5. APPENDIX F EXPERIMENTAL DATA Table IV Theory-Problem Scores on Multiple Choice Test Student Number 1 2 3 4 Question 5 6 7 8 9 Number 11 13 15 17 20 Total Score 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 18 2 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 16 3 1 1 1 1 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 15 4 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 14 5 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 0 14 6 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 1 1 1 1 1 14 7 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1, 0 0 1 13 8 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 13 9 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 1 1 1 13 10 1 1 0 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 11 11 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 0 0 0 1 1 10 12 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 1 1 1 1 9 13 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 8 14 0 0 0 0 1 0 0 1 1 0 1 0 0 0 0 1. 1 0 .1. 1 8 Table V Problem-Theory. Scores on Multiple- Choice Test Student Number 1 2 3 4 Question 5 6 7 8 9 Number 11 13 15 17 20 Total Score. 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 19 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 17 3 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 15 4 1 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 1 0 14 5 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 14 6 1 1 0 1 1 0 1 1 1 1 0 0 0 0 0 1 1 1 1 1 13 7 1 1 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 1 1 13 8 1 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 12 9 1 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 12 10 1 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 12 11 1 1 0 1 0 0 1 0 1 1 0 0 0 1 0 1 1 1 0 1 11 12 1 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 1 0 1 1 9 13 1 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 9 14 1 1 0 0 0 0 0 0, 1 0 1 0 0 0 0 1 1 1 1 0 8 15 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 8 16 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 . 7 17 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 . 6 18 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 . 5 19 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 3 Table VI 'Theory-problem Scores on the Problem Test Student Number 1 2 Question 3 4 5 6 Total Score 3 4 5 5 5 5 4 28 1 5 3 5 4 5 4 26 4 5 3 4 4 5 5 26 6 5 3 5 3 5 3 24 8 5 3 4 4 5 3 24 7 5 5- 4 1 4 4 23 10 3 3 3 5 5 4 23 2 1 3 4 4 5 ' 5 . 22 14 0 3 3 4 4 4 18 9 1 3 3 0 2 3 12 13 1 0 1 4 1 3 10 5 1 1 3 1 0 1 7 12 0 0 1 2 3 1 7 11 2 1 -'1 0 1 1 6 Table VII Problem-theory Scores on the Problem Test Student Number 1 2 Question 3 4 5 6 Total Score 5 5 5 5 5 5 4 29 1 5 5 4 5 5 5 29 2 5 5 5 5 5 4 29 10 5 5 4 5 5 3 27 7 5 4 5 4. 5 3 26 9 5 3 5 1 5 4 23 11 4 3 4 1 4 4 20 6 5 2 2 1 5 4 19 3 1 3 4 4 1 4 17 4 1 3 4 2 4 3 17 12 0 2 3 4 2 2 13 8 1 0 2 3 1 3 10 15 o- 1 3 1 2 2 9 16 0 3 0 1 1 3 8 14 1 1 1 1 2 1 7 17 2 1 1- 1 1 1 6 18 0 1 1 2 1 0 5 19 0 3 0 2 0 0-' 5 12 4 0 0 0 0 0 4 

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