THE EFFECTIVENESS OF SIMPLE ENUMERATION AS A STRATEGY FOR DISCOVERY by ISABEL CAMPBELL LEASK B.Ed., University of British Columbia, I 9 6 I A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Education We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1968 Iri p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by hits r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada i i ABSTRACT Leask,'-I.C. The e f f e c t i v e n e s s of simple enumeration as a s t r a t e g y f o r d i s c o v e r y . Problem This study i s r e l a t e d to the controversy.surrounding the r e l a t i v e m erits of teaching by di s c o v e r y and e x p o s i t o r y methods. S p e c i f i c a l l y , i t i n v e s t i g a t e d the e f f e c t i v e n e s s of treatment w i t h simple enumeration as a s t r a t e g y f o r d i s c o v e r y compared to treatment u s i n g an e x p o s i t o r y method. I t was hypothesized t h a t the two treatments would y i e l d the same mathematical achievement, but the simple enumeration t r e a t -ment would y i e l d more mathematical and non-mathematical t r a n s -f e r e f f e c t than the e x p o s i t o r y treatment. Procedure The subjects comprised s i x c l a s s e s i n Mathematics 12. They had been randomly assigned to c l a s s e s at the beginning of the school year and three c l a s s e s were assigned to each treatment group. A l l classes were taught a u n i t on a r i t h -metic and geometric progressions by the experimenter. Equivalence of the groups was e s t a b l i s h e d i n terms of the co v a r i a t e s I. Q. and previous term mark. The measuring instruments c o n s i s t e d of the Lorge-Thorndike I n t e l l i g e n c e Test, Form 1, L e v e l H of the 1961). Multi-Level Edition; a mathematical content test; and a mathematical transfer test. In addition, the Nonverbal Battery of Form 1 of the Lorge-Thorndike Test was used as a pretest to measure the ability of students to generalize and discover principles from examples. Form 2 was used as a posttest measure to determine whether any improvement in ability to generalize had occurred as a result of the experience with the unit on progressions. The generalized t-test was used to compare means of achievement on a l l tests. A l l results were analysed at the University of British Columbia Computing Centre. Conclusions On the basis of results on the tests, the following conclusions were reached: 1. Treatment with simple enumeration yielded the same level of mathematical achievement as treatment with an expository method. 2. Treatment with simple enumeration yielded significantly greater effect on a mathematical transfer test than treatment nith an expository method. An examination of I. Q. levels showed that the superiority in performance was largely located at the medium I. Q.. level. 3. Treatment with simple enumeration was no more effective than treatment with an expository method when the criterion measured general transfer. Both groups showed significant improvement in ability to generalize after studying the unit on arithmetic and geometric progressions. The improvement was mainly located at the medium and low I. Q. levels and was independent of teaching method. The implication of this study is that i f concern is centred on acquisition of facts, simple enumeration is no more effective than an expository teaching method. However, if there is concern for pupil participation and for training students to advance independently to related but more difficult material, then discovery-orientated lessons are advantageous. V ACKNOWLEDGMENT The writer wishes to express her appreciation to the members of her committee: Dr. T.A. Howitz (Chairman), Dr. R. Ya3ui, and Dr. S. S. Lee. V i TABLE OP CONTENTS CHAPTER PAGE I. THE PROBLEM 1 Background of the Problem • • • 1 The Problem $ Definition of Terms 6 Discovery 6 Expository Method 7 Strategy . . . . . 7 Simple Enumeration . . . . . . • 8 Hypotheses to be Tested 8 Justification for the Study, . . 9 II. A REVIEW OP THE LITERATURE 10 Studies Having General Relevance to the Problem 10 Studies Having Specific Relevance to the Problem \\\ Literature Concerning Opinions on Discovery 23 Programs Based on the Discovery Method . . . 25 Summary 26 III. PROCEDURE AND DESIGN 27 Design 27 Subjects 29 Control 30 v i i CHAPTER PAGE Instructional Material. . 3l Measuring Instruments . . . . . . . . . . . . 31 Lorge-Thorndike Intelligence Test 31 Mathematical Content Test 3lj. Mathematical Transfer Test . . . . . . . . 3U> Statistical Procedures 35 Limitations of the Study 35 Summary • 37 IV. RESULTS OP THE STUDY 38 Analysis of Data 38 Equivalence of the Two Treatment Groups on the Basis of I. Q,. and Previous Term Marks 39 Hypotheses Testing l\.0 Summary of Results. I4.8 V. SUMMARY AND CONCLUSIONS 50 The Problem 50 Procedures $0 Findings $1 Conclusions 52 Suggestions for Future Research . . . . . . . 5^ BIBLIOGRAPHY . 56 APPENDICES 59 v i i i LIST OP TABLES TABLE PAGE I. Means of I. Q. Scores and of Previous Term Marks by Treatments and I. Q,. Levels I4.X II. Means of Mathematics Achievement Scores by Treatments and I. Q,. Levels . . . . . . . . . . I|2 III. Means of Mathematics Transfer Test Scores by Treatments and I. Q,. Levels IV. Means of Posttest and Pretest Scores of Nonverbal Battery by Treatments and I. Q. Levels. . . . . lj.6 V. Means of Posttest Scores by Treatments and I. Q. Levels Ii-7 CHAPTER I THE PROBLEM --Background of the Problem During the past three decades much attention has been -directed towards the various ramifications of teaching by discovery as opposed to those of teaching by expository methods. Indicative of the importance attached to discovery in the teaching of mathematics is the fact that it has been widely accepted in the basic philosophy of textbook writers. Evidence of this is seen in such statements as "this text-book guides the student in discovering mathematical prin-ciples and furnishes him with extensive exercise material and applied problems to strengthen his comprehension of 1 these principles and of their usefulness." The fact that this is part of the avowed philosophy in many textbooks does not necessarily mean that it is always present in the con-tent material. A l l too frequently the textbooks follow a definition, illustration, and practice format. It seems clear from the literature that discovery is not just a method but that it embodies many methods and strategies. Hendrix cites four methods of discovery which Mary P. Dolciani, Simon L. Berman, and William Wooton, Modern Algebra and Trigonometry Structure and Method. Book 2> (Boston: Houghton Mifflin Company, 1963), p. 2. she labels inductive, nonverbal awareness, incidental and 2 deductive. Henderson has identified the following seven strategies for discovery: analogy, simple enumeration, agreement, difference, difference and agreement, concomitant 3 variation, and independent action. Schaaf classifies methods of generalization as empirical and rational pro-cedures. He states that simple enumeration, analogy, continuity of form, and statistical procedures belong to the empirical category, while deduction, variation, formal k analogy, and inverse deduction are rational procedures. Among the arguments advanced in support of discovery are that it Is the essence of mathematical thinking; i t can be applied to other fields; and i t creates greater involve-ment and interest among students. Beberman, for example$ states that "the discovery method develops interest in mathematics and power in mathematical thinking. Because of the students' independence of rote rules and routines, i t also develops versatility in applying mathematics." Bruner 2 Gertrude Hendrix, "A New Clue to Transfer of Training, Elementary School Journal, XLVIII (December, 191+7), p. 197. •^ Kenneth B. Henderson, "Strategies for Teaching by the Discovery Method," Updating Mathematics, I (November, 195>8), pp. 57-60, and I (April, 1959), pp. 61^61+. h Oscar Schaaf, "Student Discovery of Algebraic Prin-ciples as a Means of Developing Ability to Generalise," The Mathematics Teacher, XLVIII (May, 1955), Pp. 32J+-27. Max Beberman, An Emerging Program of Secondary School Mathematics (Cambridge: Harvard University Press, 1958T, P. 03* ~ l i s t s the benefits which accrue from discovery as increase in intellectual potency, a shift from extrinsic to intrinsic rewards, a provision for learning the heuristics of discovery 6 and an aid in the conservation of memory. Suchman, who has experimented in the field of inquiry training, postulates that "some have been prompted to reformulate their methods to capitalize on the intense motivation and deep insight that seem to accrue from the 'discovery' approach to concept 7 attainment." Kersh claims that students acquire what psychologists call a "learning set" or strategy for discovery 8 which assists them in the solution of new problems. Others who advocate the use of the discovery method concur with these claims and consistently expound the superiority of discovery~orientated learning. Ausubel is one of the educators who question the extensive claims made for discovery. He concedes that it can be valuable in the early stages of learning and in the teaching of the scientific approach to problem solving. Nevertheless, he feels that the crucial issue is not whether Jerome S„ Bruner, Essays for the Left Hand (Cambridge Harvard University Press, 1963), p. 83. 7 J. Richard Suchman, "Inquiry Training: Building Skills for Autonomous Discovery," Merrill-Palmer Quarterly of Behavior and Development, VII (April,~~1961), p. lJj.fi Q Bert Y. Kersh, "Learning by Discovery: Instructional Strategies," The Arithmetic Teacher, XII (October, 1965), pp. l+ll|-17. k learning by discovery enhances learning, retention, and transferability, but whether i t does so sufficiently for those who are capable of learning meaningfully without it to warrant the time spent. In addition, he questions the feasibility of discovery as a technique for transmitting content to students who have mastered the rudiments and vocabulary of a subject. It is his contention that such students can accomplish as much, in as proficient a manner, 9 and in less time, by means of good expository teaching. The obvious lack of unanimity of opinion regarding the relative merits of discovery and expository methods is also prevalent in the results of empirical research. This is substantiated in the review of the literature in the second chapter. Wittrock has cited several reasons for the equivocal nature of the results of studies based on discovery. The first of these is the semantic inconsistency in labeling different treatments. Some of the researchers were concerned with the amount and kind of external guidance, some with the role of verbalization, and soma with the rate of presentation. Wittrock points out that it is necessary to identify more accurately the relevant teaching-learning variables and to direct research based on interactions of the methods with different types of teachers, pupils, and subject matter. David P. Au3ubel, "Learning by Discovery: Rationale and Mystique," Bulletin of the Rational Association of Secondary School Principals, XLV TDecember, 1961), pp. 18-56. Perhaps the most salient factor in the contradictory evidence is the differing specifications of discovery, guided dis-covery, and exposition. These terms have not been reduced 10 to uniformly operational definitions by the experts. If one accepts the premise that various strategies for discovery do exist, and that they constitute unique approaches to the discovery process, then i t should be possible to investigate whether these strategies can be taught to students, whether some of the strategies are more applicable to the teaching of mathematics than others, and whether certain topics are more amenable to specific strategies than others. It may be possible to analyse such instructional procedures and techniques and to identify the behavior elicited by each. Such analysis could lead to more accurate prediction of learning outcomes and to more per-ceptive discernment of the comparative effects of different strategies. The Problem The present study is concerned with the use of the strategy of simple enumeration as a technique for discovery in two ways. The first concern is with the effectiveness of this strategy in learning specific mathematical material. M.C. Wittrock, '"Verbal Stimuli in Concept Formation: Learning by Discovery," JournaJ. oj? Educational Psychology, LIV, No. i+, (I963), pp. TFjrgo. The second concern is whether students who have used this strategy will display marked superiority on transfer tasks which may or may not be related to the mathematical material. Specifically, the present study will attempt to answer the following questions: Can students be taught to discover generalizations using the strategy of simple enumeration? Are students who have acquired this strategy superior to those who have been taught by an expository method when mastery of subject matter is measured? Is there any significant difference in the ability of the two groups to transfer their knowledge to similar but unfamiliar mathematical materials? Does the group which has used the strategy of simple enumeration achieve at a significantly higher level than the group which has been taught by an expository method when the task requires generalization from examples which are not directly related to mathematical materials? Definition of Terms Discovery. Bruner describes discovery as "a matter of rearranging or transforming evidence in such a way that one is enabled to go beyond the evidence so reassembled to nev; 11 Insights." The crux of discovery in the present study is the recognition and understanding of relationships among Bruner, ojp. cit., p. 8 2 . concrete examples, the application of this recognition, and the operation of putting i t into a compact rule. The students are presented with an ordered, structured series of examples which are designed to maximize the opportunity for discovery of a generalization associated with the examples. By studying examples and answering questions, the students are expected to discover the underlying principle and formulate a symbolic generalization. Although students are expected to use the generalization in the solution of practice problems, they are not expected to verbalize the principle before using i t . Expository method. The i n i t i a l step in instruction by the expository method is the derivation and presentation of the rule by the instructor. A complete explanation of the rule is given both verbally and symbolically. This is followed by the working of several examples illustrating the principle. To minimize rote memorization, emphasis is placed on the relation of the examples to the principle involved. Strategy. This is a plan of action designed to lead the students to knowledge, skills, and attitudes. The measure ment of the degree to which the strategy has been acquired is in terms of increase in mean scores between a pretest and a posttest involving generalization from examples. 8 Simple enumeration. This consists of the presentation of many instances of the generalization to be discovered. The students form hypotheses based on the examples and test these to determine which is correct. One counter-example is sufficient to warrant rejection of a hypothesis. Hypotheses to be Tested In general, it is assumed that there can be sufficient distinction made in teaching methods so that i t is possible to compare the methods in terms of student performance on mathematical and transfer tasks. To acquire experimental evidence for this study, the investigator spent three weeks during the month of May, 1968 teaching six Grade XII classes at Delbrook Senior Secondary School in S. D. No. (North Vancouver). The following hypotheses were tested in the study. 1. Treatment with simple enumeration as a strategy for discovery will yield the same mathematical achievement.effect as treatment by an expository method. 2' Treatment with simple enumeration will yield greater mathematical transfer effect to unfamiliar mathematical materials than treatment with an expository method. 3. Treatment with simple enumeration will yield more transfer effect to non-mathematical materials than treatment with an expository method. Justification for the Study Since there are entire programs being developed in the fields of science and mathematics which are based on the philosophy of discovery, i t would soem expedient to investigate specific facets of this approach. One of the crucial related problems or questions which is classified as "unansx>reredM by researchers in mathematics is, "What are the optimum methods 12 for inducing and utilizing discovery methods?1' - The present study deals with one aspect of the discovery process and is an effort to determine whether the use of simple enumeration to lead to generalizations Is an effective procedure. If students who have been exposed to this strategy for discovery during the study of a specific unit in mathematics show evidence of superiority in dealing with a transfer task, then it can be interpreted as evidence that they have acquired a "learning set" or strategy for discovering generalizations which is superior to that of students who have been exposed to an expository approach. Since the study is conducted in the classroom environment, using materials from the curriculum of Grade XII mathematics, it should be of interest to teachers involved in teaching at this level, as well as to textbook writers and curriculum consultants. Kenneth E. Brown and Theodore L. Abell, Analysis of Research in the Teaching of Mathematics, U.S. Department of Health, Education and Welfare (Washington: 1965), P« 19. CHAPTER II A REVIEW OP THE LITERATURE The purpose of this study is to investigate the effectiveness of simple enumeration as a strategy for dis-covery. Much has been written concerning the discovery method and many studies have been conducted in an effort to establish empirical verification for the claims made for i t . This chapter constitutes a review of some of the pertinent materials related to the discovery method. For organ-izational purposes the literature is classified according to studies having general relevance to the problem, studies having specific relevance to the problem, literature related to opinions regarding discovery methods, and a brief review of mathematical programs based on the discovery principle. Studies Having General Relevance to the Problem Katona conducted intensive studies involving the learning of principles for problem solving. His investigations were directed to the solution of card-trick and match-stick pattern problems and demonstrated the relative ineffectiveness of memorizing verbal principles compared to understanding. He considered that learning which involves meaningful wholes favors transfer to problem solving situations. The subjects who had the greatest success in solving card-trick problems were those who listened to an explanation of the basic problem and watched a step-by-step demonstration. This group was followed in order by the group which learned verbal principles, the group which memorized steps, and the group which had no training.* As a follow-up to Katona's studies Hilgard, Edgren, and Irvine investigated the cause of errors made by the most successful students who apparently understood the work. They designed five methods of training for understanding with a view to determining which method would best eliminate errors in transfer problems. Overall differences in methods were slight. The prevalence of careless errors suggested that it is important in teaching for understanding to devise methods that are open to review or checking. The authors concluded that attitudes of caution or carelessness are more important in determining error level than lack of under-standing. The best results were obtained with the method 2 which was easiest to check. Ray conducted a study comparing directed discovery and "tell-and-do" methods for learning micrometer skill s . 1 G. Katona, Organizing and Memorizing: Studies in Psychology of Learning and Teaching (New York: Columbia University Press, 194/0.) 2 Ernest R. Hilgard, Robert D. Edgren, Robert P. Irvine, "Errors in Transfer Following Learning with Under-standing: Further Studies with Katona's Card-Trick Exper-iments, w jlpurnal jof Ejjperijne^t^ Pjycholog^, XLVII, No. 6, (1954), PpTTjTFoV While there was no significant difference in manipulative performance based on knowledge of the micrometer or in ability to solve problems, there was significant difference in retention and effective application after one week and six weeks. This difference favored the directed-discovery method,-^ Haslerud and Myers were concerned with discovery learning as applied to decoding sentences from concrete instances, as opposed to using specific instructions for decoding. The directed procedure was better for original learning but there was no significant difference on the k transfer task. On the basis of recorded results the conclusions of the experimenters supporting the contention that derived principles transfer more readily than given principles seems somewhat questionable. Cronbach is highly critical of this study, maintaining that it suffers from prejudice in the analysis of the data and stating that inferences were made on differences in test scores without 5 comparing each test at the discovery-non-discovery level. • / " i , 3 Willis, E. Ray, "Pupil Discovery vs, Direct Instruction," Journal of Experimental Education, XXVI (March, 196l), pp. 271-'B?0. *~~ ' k G. Haslerud, Shirley Myers, "The Transfer Value of Given and Individually Derived Principles," Journal of Educational Psychology, XLIX (December, 19567TpP^293"97. Lee Cronbach, "The Logic of Experiments in Dis-covery," in L.S. Shulman & E.H. Keislar (Ed.) Learning by Discovery: A Critical Approach (Chicago? Rand McNally, 196"oT pp. 76-927""""™ Craig attempted to determine the effect of directing learners' discovery of established relations upon retention and ability to discover new relations. Groups were given different amounts of direction during discovery of the bases determining solution of multiple-choice verbal items. The group receiving greater direction learned more relations on three trials. Both of the groups wore discovery groups. Craig interpreted his results as evidence that experimenters should be liberal with information designed to assist learners in the discovery of principles. Large amounts of external direction insure that the learner will have more 6 knowledge to direct future discovery. Wittrock experimented with deciphering sentences using groups in which the rule and answer were both given, the rule was given v/ithout the answer, the answer was given without the rule, and neither answer nor rule was given. The superior groups were those given the rule. The group which had neither rule nor answer required more time to learn and showed higher retention scores than learning scores. 7 The converse was true of the other groups. 6 R.C. Craig, "Directed versus Independent Discovery of Established Relations," Journal of Educational Psychology, XLVII (April, 1956), pp. 223ZW» 7 M.C. Wittrock, "Verbal Stimuli in Concept Formation: Learning by Discovery," Journal of Educational Psychology, I, IV, No. h,, (1963), pp. 133-907" Kittell employed a word task with sixth-grade subjects. The three groups were designated as having minimum, inter-mediate, and maximum direction. The groups received varying amounts of practice with the principles involved. Because the task was difficult, the discovery group averaged less than three out of fifteen principles discovered. The inter-mediate group had l i t t l e discovery experience but had practice in application of principles. The third group had l i t t l e practice time and no opportunity for discovery. The Inter-mediate group was superior in applying principles and in discovering new principles from examples. It was thought that the discovery behavior of the first group may have been repressed rather than reinforced because of the extreme 8 difficulty of the task. Studies Having Specific Relevance to the Problem Hendrix investigated to what extent the way in which one learns a generalization affects his ability to recognize an opportunity to use i t . She used the concept that the sum of the first 'n' terms of an odd integer sequence is p *n '. The findings indicated that the group which dis-covered the principle independently and left it unverbalized exceeded those who discovered and verbalized, while both of 8 Jack E. Kittell, "An Experimental Study of the Effect of External Direction During Learning on Transfer and Retention of Principles," Journal of Educational Psychology, XLVIII (November, TWTT7~V^' 391=405. these groups exceeded in transfer those for whom the prin-ciple was stated and illustrated. Her claim that the immediate flash of unverbalized awareness is what actually accounts for transfer power, and her separation of the discovery phenomena from the process of composing sentences which express the discoveries provided a new and startling proposition in learning theory. She stated that the dawning of a generalization on an unverbalized awareness level seemed to be an internal process and the indication that the process has occurred is the organism's new power for self-direction. Her experiments were significant at the .12 level but she admits that part of th© results were invalidated by the control group. There was also great difficulty in designing an appropriate instrument to test achievement of unverbalized awareness and a transfer test which would present varying degrees of remoteness from the 9 original examples used. Cummins used discovery in teaching first year calculus students. In his study each group was given two achievement tests, one of which was designed especially for the discovery group and the other for the traditional. The discovery group had significantly better results on the Gertrude Hendrix, "A New Clue to Transfer of Training," Elementary School Journal, XL VII '(December, 19i|7)» pp. 197-208. f i r s t test but there were no s i g n i f i c a n t differences on the 10 t r a d i t i o n a l t e s t s . Retzer and Henderson conducted a study based on the conjecture that i f a group of students are taught such concepts as variable, open sentence, universal set, generalization, Instances and counter-instances of generalizations, and are given practice i n applying the con-cepts and i n writing generalizations, they w i l l be able to state c o r r e c t l y the re l a t i o n s they discover when taught by the method of guided discovery. The experiment involved the use of a Sentences of Logic unit by the treatment group. This group was able to verbalize universal generalizations involved i n the c r i t e r i o n measure. The authors f e e l that the research suggests that an alternative to delaying verbal-i z a t i o n of discoveries would be to include the teaching of l o g i c a l components of universal generalizations as an e x p l i c i t part of the curriculum. In thi s way i t would be possible to ask f o r immediate v e r b a l i z a t i o n of a discovered generalization and to expect a great deal of p r e c i s i o n on 11 the part of the students. Szabo expresses a similar view-point when he says, "In f a c t , there i s evidence to support 10 Kenneth Cummins, "A Student Experience-Discovery Approach to Teaching Calculus," The Mathematics Teacher, L I I I (March, I960), pp. 162-70. 11 Kenneth A. Retzer, Kenneth B. Henderson, "Effect of Teaching Concepts of Logic on Verbalization of Discovered Mathematical Generalizations," The Mathematics Teacher, LX (November, 1967), pp. 707-100 17 the fact that too-early verbalization of discovered general-izations with mathematically immature children can be damaging to the learning, due mainly to lack of verbal fac i l i t y . When students become more mature, they should be encouraged to give precise verbalization of generalizations after they demon-12 strate an awareness of those generalizations." Gagne and Bassler did a study of retention on non-metric elementary geometry and found that a narrowing of practice had a negative effect on retention. They also asserted that the major concepts which had been discovered •wore well retained but the subordinate knowledge used to 13 develop the concepts was quickly forgotten. Worthen prepared two methods of task presentation at the f i f t h and sixth-grade levels. His was a classroom experiment extending over a period of six weeks. He measured in i t i a l learning, retention, transfer of concepts, and transfer of heuristics. He also conducted an analysis of teacher behavior to ensure adherence to the models in each treatment.. Expository groups were superior in i n i t i a l learning at the .01 level. The discovery group was superior on retention and transfer of heuristics and slightly superior on the transfer task. The superiority of the discovery group 12 Steven Szabo, "Some Remarks on Discovery," The Mathematics Teacher, LX (December, 19&7), p. 839. "^Robert M. Gagne, Otto C. Bassler, "Study of Retention of some Topics of Elementary Non-Metric Geometry," Journal of Educational Psychology, LIV, No. 3, (1963), pp. 123-31-on the majority of inter-treatment comparisons varied from I k the . 0 2 5 to the . 0 8 level. Gagne' and Brown conducted a study on the summation of a series task. The results of their study favored the discovery method. A l l groups followed self-instructional programs designed to give three approaches. The discovery group was required to discover the rules for the problem series and had no practice in application. The guided discovery group carried through a series of steps to lead to formulation of the rules but had no formal practice in application. The directed group was told the rules and given formal practice. The test measured only the learners' ability to discover new rules from different problem series and was not a test of recall or application. Tho results shovjed the best performance by the guided discovery group and the least effective performance by the rule and example 15 group with the discovery group between them. This study is open to criticism on the basis of lack of sound didactic teaching to the non-discovery or rule and example group. The guided discovery group was taught to look for a structural relationship based on the terms of the series. The non-Blaine R. Worthen, "Discovery and Expository Task Presentation in Elementary Mathematics," Journal of Educational Psychology Monograph Supplement, LIX TFebruary, 19oT), No. 1, Part 2. 15 Robert M. Gagne., Larry T. Brown, "Some Factors in the Programming of Conceptual Learning," Journal of Exper-imental Psychology, LXII (October, 1961), pp. 313°2T. discovery group should have been made aware of the existence of the relationship even though they were not required to discover i t . Since the present study is based on one of Schaaf's premises concerning methods of generalizing, a fairly detailed treatment of his original study will be given. He proposed a course in grade nine algebra and conducted a class based on his designed program. The theme chosen as a guide-line for the direction of the course was improvement in ability to generalize, and this determined to a large degree the nature of class procedures. Schaaf hypothesized that students would learn to generalize i f they were given sufficient practice, and if they discovered for themselves as many as possible of the mathematical principles involvedo This required a program which provided guidance and mathematical experiences designed to make students aware of concepts and principles. The tasks which Schaaf set himself in his, study were to analyse different processes of generalizing, determine the characteristics of. a superior generalizer, formulate lessons and procedures to aid students in developing ability to generalize in mathematical and non-mathematical situations, and evaluate in terms of specified criteria of generalizing ability and mathematical achievement. His criteria of a superior generalizer consisted of fifteen items. Methods of generalizing were classified as empirical or rational. Empirical methods included simple enumeration, analogy, continuity of form, and statistical procedures, while rational methods included deduction, variation, formal analogy, and inverse deduction. Various approaches involving -all methods were used in the presentation of the lessons. Evidence from observers' reports, students' notebooks, teachers' notes, and responses on students' reactions indicated that the course did lead students to acquire the desired abilities. Schaaf concluded that his experimental group made significantly greater improvement in ability to draw conclusions, to recognize non-justifiable conclusions, and to interpret graphical data than the status group which was used for comparative purposes. Although less time was spent on the study of algebra than in average classes, the. achievement of the experimental group as measured by the Lankton First Year Algebra Test was significantly greater than'was predicted by the results on the Iowa Algebra 16 Aptitude Test. Kersh was concerned in his studies with the motivating effect of learning by discovery. In his studies he used the odd-numbers rule and the constant-difference rule for series as learning tasks. He accepted the premise that learning by discovery is superior and investigated whether a possible explanation was that discovery made learning more meaningful 16 Oscar Schaaf, "Student Discovery of Algebraic Principles as a Means of Developing Ability to Generalize," The Mathematics Teacher, XLVIII (May, 1955), pp. 32u,-27. 21 in the cognitive sense of understanding or organization. His data suggested the inadequacy of the meaning theory but revealed that students were motivated to continue learning and practising after the formal period of instruction was over. Analyses were made of differences in treatment, differences in test periods, and differences attributed to interaction of treatments and time periods. The guided group was superior to the unguided group in use of rules, retention, and transfer. The rote group was superior to 17 others in a l l respects but was not Included in the results. Tuckman and others recently reported a study, the purpose of which was to induce a search set in individuals through prior experience and to determine the conditions which would allow the search set to transfer. The task was a four by six matrix of two-digit numbers which were to be added but in each case a shortcut method could be used. They performed three experiments to investigate the effects of appropriate and inappropriate practice experiences on students' tendency to search for and find shortcut solutions to problems. The strategy of looking for, and s k i l l in finding a shortcut, were termed a search set. In the first experiment where the criterion problems resembled practice 17 Bert Y. Kersh, "The Adequacy of Meaning as an Explanation for the Superiority of Learning by Directed Discovery," Journal of Educational Psychology, XLIX (October, 1958) PP. 282-92; "The Motivating"Effect of Learning by Discovery," Journal of Educational Psychology. LIII (April, 1962), P P . 65-71. problems, the subjects having search experience were more likely to search for and find correct solutions* In the second and third experiments where the problems were dis-similar to practice problems, the subjects who had search experience were more likely to search for shortcuts but were singularly unsuccessful in finding correct solutions. The researchers concluded that search s k i l l has limited transfer possibilities as compared to search strategy. They recommend a conceptual distinction between searching and finding. Although they started the experiment with the idea of combining these concepts, they concluded that this was not a sound idea and recommended the use of the term "search set" to refer to the strategy of search and perhaps the term "learning set" to the s k i l l of finding a shortcut solution. Another implication of the study was that limited exposure to a problem-solving approach might induce students to adopt a strategy but leave them without the s k i l l to apply i t . On this assumption the researchers suggest that it is necessary to provide a level of s k i l l commensurate with the commitment to the strategy or the latter cannot be used effectively. This implies that It is essential to give extensive practice 18 sequences. 18 Bruce V/. Tuckraan and others, "Induction and Transfer of Search Sets," Journal of Educational Psychology, LIX (April, 1968), pp. 59°6"3\ 23 Literature Concerning Opinions of Discovery As mentioned in the fi r s t chapter, Ausubel is one of the most severe and vociferous critics of the discovery method. In his articles he concedes that, under certain conditions, there is a defensible rationale for discovery. Unfortunately there has been a tendency to use it as a panacea and to attempt to extrapolate its advantages to a l l age levels, to a l l levels of subject sophistication, to a l l educational objectives, and to a l l kinds of learning. He 19 20 refutes claims made by Bruner and Hendrix. Hendrix says that verbalization is not only unnecessary for transfer of ideas and understanding but is harmful i f used for these purposes, and that language only enters the picture because of a need to attach a symbol or label to subverbal insight so that it can be recorded and communicated to others. Ausubel says that the function of language is not just to label, and that verbalization does more than attach a symbol to thought. It constitutes part of the process of abstraction. An individual who is using language to express an idea is engaged In an intellectual process of generating a level of insight which transcends the subverbal awareness stage in every respect. In explaining the apparent success of programs 19 Bruner, loc. cit. 20 Hendrix, loc. cit. based on the discovery method, Ausubel hypothesizes that they succeed because they are highly organized and systematic and because they use discovery judiciously in the early stages and gradually attenuate i t . The courses have been taught by well-trained and enthusiastic teachers and are a 21 testimonial to didactic verbal exposition. Ausubel's criticisms are based on personal opinion and have not been subjected to empirical verification. Taba states that learning by discovery as presently pursued pertains to the cognitive aspects of learning and is limited in content to mathematics and science. She feels that there are two aspects of the transactional process—» assimilation of content and operation of cognitive processes to organize and use the content. She favors a discovery approach because i t is the chief mode of intellectual productivity and autonomy. The individual is better equipped to move into unknown areas, gather data, and abstract con-cepts. The learner develops an attitude of search and a set to learn and becomes freed from extrinsic rewards. While Taba is a proponent of discovery she does state that there should be a balance between receptive and assimilative 21 " David P. Ausubel, "Some Psychological and Educational Limitations of Learning by Discovery," The Arithmetic Teacher, XI (May, 196b,), pp. 2 9 0 - 3 0 2 . "Learning by Discovery: Rationale and Mystique," Bulletin of the National Association of Secondary School Principals, XLV (December, I96I), pp.Toxrs. learning and views the task of the educator and curriculum 22 planner as securing this balance in instruction. Programs Based on the Discovery Method Interest in research has not been confined to individuals. The University of Illinois has prepared materials for teaching secondary mathematics and for training teachers. In discussing the basic philosophy on which the program is based, Beberman says, "We believe that a student will come to understand mathematics when his text-book and teacher use unambiguous language and when h9 is 23 enabled to discover generalizations for himself." The technique of delaying verbalization of important discoveries is a prominent feature of the UICSM program. It is also characterized by careful sequencing and structuring of con-cepts and by pupil involvement in the discovery of these concepts. Davis, with the Madison Project, is also attempting to put the discovery aspect into the mathematics curriculum for a l l grades. . Although the philosophy of the authors is discovery-orientated, the apparent success of the 22 Hilda Taba, "Learning by Discovery, Psychological and Educational Rationale," Elementary School Journal, LXIII (March, 1963), pp. 3O8-15. 23 • Beberman, o_p_. cit., p. 2h Robert B. Davis, "Madison Project of Syracuse University," The Mathematics Teacher, LIII (November, I960) 571-75. programs cannot be interpreted as confirmation of the claims made for discovery. Many factors are operative and no accurate measure of any of the contributing factors has been attempted. Summary The majority of the research would appear to favor the use of some form of discovery. While the superiority of discovery techniques is not always evident in the immediate learning situation, most experimenters concede that these techniques are more effective with reference to transfer and retention. Because of the wide diversity of materials used, the range in the age of subjects, and lack of uniformity in definition of methods, the precise areas in which discovery techniques produce maximum positive effects on learning have not been established. Much of the research is general in nature and l i t t l e attention has been directed to the analysis of the many strategies which are involved in the discovery process. In addition, a great deal of the current literature is based on opinion and has not been subjected to rigidly controlled research. Therefore, it cannot be accepted as factual information concerning the relative merits of discovery and expository methods. CHAPTER III PROCEDURE AND DESIGN This chapter is concerned with the design of the study and the procedures used to implement the design. It contains information pertaining to instructional methods and materials, subjects, tests and measures, statistical procedures, and limitations of the study. Complete details of lesson pro-cedures with written exercises and copies of tests con-structed by the experimenter are contained in Appendices A and B respectively. Design As previously indicated, two treatments were used in this study. One treatment involved a lecture or expository method of presenting specific mathematical material and the other involved the presentation of the same material by the use of simple enumeration as a strategy for discovery. In both treatments seven one-hour periods were devoted to the study of arithmetic and geometric progressions. In the expository treatment each period commenced with a formal presentation by the instructor of pertinent principles and definitions related to a particular phase of the unit. During the development of the lesson, no attempt was made to have the students participate In, or contribute to a discussion. If questions were posed by students, they were immediately answered by the instructor. At the completion of the lecture-type presentation, the students were given written exercises comprised of two basic types of questions— those involving a direct application of the rules which had' been developed in the lesson, and those which were problem-type questions designed to show a practical application of mathematical concepts. Some of the written exercises were completed during class time with the instructor moving about the room and answering any questions which were asked by students. The solutions to the exercises were made available only after the exercises had been completed. Students were expected to check their own work and any difficulties which aroso out of the exercises were dealt with either with the class as a whole or with individuals at an appropriate time. The second treatment made use of the strategy of simple enumeration to lead to the discovery of generalizations. Students were presented with examples of the principles to be discovered, followed by a series of questions structured to maximize the opportunity for discovery. Because there is a tendency for only a few students to be involved in the discovery process when this type of lesson is presented orally, the students were given individual copies of the examples and questions in an effort to ensure that the maximum number of students actually discovered the desired principles. After students had discovered the principle and worked several examples, they proceeded to the written 29 exercises. For those who were unable to discover the prin-ciples from the original examples, further illustrations were provided on the blackboard. The only direct information given to students was concerned with notation or labeling of concepts. It was felt that precise terras were essential for effective communication in the written exercises. As In the case of the previous treatment, the instructor moved about the room but responded to questions in a different manner. Instead of giving direct answers, the instructor asked further questions which were designed to promote under-standing. The solutions were provided in the same manner as for the expository treatment. Sub jects At the outset of the study, 158 twelfth-grade students in 6 classes at Delbrook Senior Secondary School in S. D. No. l±t\. (North Vancouver) made up the sample population for the two treatments. This is an academically-orientated school with a total population of approximately 750 students, and is located in an upper middle class socio-economic area. During the course of the study, thirty-one of the students were eliminated because of absence from a period of instruction or a test. The students had been randomly assigned to classes by computer at the beginning of the school term, and with the exception of one class, a l l had been taught by one teacher throughout the year. For the present study three classes were randomly assigned to each treatment and a l l classes were taught by one experimenter. Control It is conceded that many factors are operating in a classroom situation, and that rigid control of sources of extraneous variation over a period of time in such circum-stances is virtually impossible. Howover, an attempt was made to eliminate as many sources of extraneous variation as possible. The fact that the classes were a l l taught by the experimenter eliminated the teacher variable and any inter-action due to the presence of a different instructor in the classroom was comparable for a l l classes. Since the experimenter had previously taught at Delbrook and was known to the students, the 'Hawthorne effect' would be minimal i f i t existed at a l l . In addition, there was no difference in the time allocated to each group, in the examples, or in written exercises. The only major difference in the conduct of the classes was in the diversification of method of presenting the materials. Because a l l other aspects of the learning situation were treated alike, it should be possible to attribute any major differences in results on criterion tests to the variation in teaching method. 31 Instructional Materials The unit on arithmetic and geometric progressions i s part of the mathematics curriculum at the Grade XII l e v e l i n the Province of B r i t i s h Columbia. Although no textbook was used during the course of the experiment, the lessons were -based on Chapter XIII of the prescribed textbook which i s "Modern Algebra and Trigonometry, Book 2" written by Dolciani, Berman, and Wooton and published by Houghton M i f f l i n Company. This unit was chosen because, i n the opinion of the experimenter, i t i s suitable f o r the use of the strategy of simple enumeration, and because i t i s new material and should be l e s s subject to e f f e c t s created by previous knowledge. Measuring Instruments Lorge-Thorndike Intelligence Test. Before any content material was taught, the Lorge-Thorndike Intelligence Test, Form 1, Level H of the 196h, Multi-Level E d i t i o n was administered to a l l students. The f i r s t part of the test i s composed of one hundred items i n f i v e subtests c l a s s i f i e d as vocabulary, verbal c l a s s i f i c a t i o n , sentence completion, arithmetic reasoning, and verbal analogy. The remainder of the test i s a Nonverbal Battery of eighty items subdivided into three tests which are categorized as p i c t o r i a l c l a s s i f i c a t i o n , p i c t o r i a l analogy, and numerical r e l a t i o n -ships. Separate d i f f e r e n t i a l I. Q. scores are given for Verbal and Nonverbal Batteries but these can be combined to 32 form a composite I. Q. score which is the simple unweighted average of these two scores. On the basis of the composite score, students were assigned to high, medium, and low I. Q. subgroups. The Multi-Level test was standardized on a nation-wide sample of Grade XII students in the f a l l of 19&3 and bears a 196I|. copyright date. The mean is set at 100 with a standard deviation of 16. It is stated that the average college freshman class may have a mean as high as 115 to I3O and the spread of scores about these values will be con-siderably more restricted than in the general population. The statistical information contained in the administration manual is limited because of the recency of the date of revision of the test. It does indicate that preliminary data obtained when the order of presentation of Form 1 and Form 2 was rotated yielded reliability coefficients of .90 for the Verbal Battery and .92 for the Nonverbal Battery. Because the sample used to establish these coefficients was small, the authors state that the results should only be regarded as suggestive. No information is available on the validity of the Multi-Level Edition. This test represents a revision and refinement of the Separate Level Edition published in 1951+. Information with respect to predictive validity of the latter is meager in that a correlation coefficient of .67 at the ninth-grade level is the only statistic quoted. Con-current validity of this edition as measured by correlation 33 w i t h three other well-known group i n t e l l i g e n c e t e s t a was . 7 7 , . 7 9 , and .8I4. f o r the Verbal B a t t e r y and .65 , . 7 1 , and f o r the Nonverbal B a t t e r y . The authors f e e l that the M u l t i -L e v e l E d i t i o n has a higher r e l i a b i l i t y than the Separate L e v e l E d i t i o n and that c o r r e l a t i o n s w i t h other t e s t s would be at l e a s t as h i g h as those of the l a t t e r . There i s general concurrence of o p i n i o n among the reviewers of the Lorge-Thorndike t e s t s that they are among the best of the group i n t e l l i g e n c e t e s t s . They are w e l l -designed and constructed, and they provide r e l i a b l e measures of v e r b a l and nonverbal reasoning. The only c r i t i c i s m p r o f f e r e d i s that there i s a l a c k of adequate data on p r e d i c t i v e v a l i d i t y . ^ * The Nonverbal B a t t e r y of Form 1 was used as a p r e t e s t score to measure the a b i l i t y of the students t o see r e l a t i o n -ships among examples, discover a p r i n c i p l e , and: apply the p r i n c i p l e . At the end of the i n s t r u c t i o n p e r i o d the Non-v e r b a l B a t t e r y of Form 2 was used as a _posttest. These pre-t e s t and p o s t t e s t scores were used to determine whether thero was any s i g n i f i c a n t growth i n a b i l i t y to g e n e r a l i z e w i t h i n the groups, and a l s o to compare any d i f f e r e n c e s between groups which might be a t t r i b u t e d to the v a r i a t i o n i n treatment. 1 Oscar K. Buros, (ed.), The F i f t h Mental Measurements Yearbook, (Highland Park, New Jersey! The Gryphon Press, 19F9T, PP. h,78-8I|. Mathematical Content Test. This test consisted of twenty-five multiple choice items constructed by the experimenter and administered during a period of forty minutes. The items were designed to measure the degree to which the students had acquired a knowledge of the principles involved in the unit on arithmetic and geometric progressions. The content validity of each item was reviewed by the regular classroom teacher and the experimenter. Upon completion of data collection, the internal consistency of each item as an index of item reliability was examined by means of point biserial correlation coefficient between the total score and responses to each item. The correlation coefficients ranged from ,0l| to .lj.9, with a mean correlation of .30. Three items had correlation coefficients less than .20. Although it is apparent that some of the items were non-discriminatory, a l l were included in the analysis of results. Mathematics Transfer Test. This instrument was also constructed by the experimenter. It was a twenty-item test based on mathematical material somewhat related to.sequences but requiring a transfer and extension' of the knowledge acquired. The items viere comprised of examples from which students were required to generalize. Point biserial correlation coefficients ranged from .12 to ,5>lj. with a mean correlation value of .39. Only one item of the test had a correlation coefficient less than .20. The time allocated for the test was thirty minutes. Statistical Procedures The processing of data was done at the University of British Columbia Computing Centre using the generalized t-test. The basic computations performed by this test are the calculation of means and standard deviations of sets of variables, and the calculation of 't' values for specified combinations of these variables. Limitations of the Study A clear and unmistakable limitation of this study lies in the experimenter-constructed measuring instruments. From the information concerning the point biserial correlation coefficients, i t is obvious that the tests require refinement. In the case of the Mathematics Content Test, the experimenter feels that i t could be improved by increasing the number of items with a corresponding increase in the time allotted. The fact that there were many omissions in the solutions of the Mathematics Transfer Test would indicate that insufficient time was given for this test. It is not suggested that a l l items could or should have been completed by a l l students but the task appears to have been more difficult than anticipated for the time assigned to i t . A second limitation of the study is one which is commonly held to be true of a l l classroom experiments. McDonald states that "even with the best intentions on the 36 part of school personnel, ordinary school and class conditions are not highly suitable for experimentation." He suggests that task and method variables ought to be tested under controlled conditions and that it is better to do a small study in which a few well-defined variables may be manipulated 2 efficiently. It could, of course, be argued that an exper-iment conducted within the time limits and learning conditions representative of typical school behavior and curriculum might well be generalized to classroom situations with a greater degree of confidence than could be assumed from a short-term laboratory experiment. A further limitation of the study is that it is based on a comparatively short unit of instruction and Is done with what might be considered a select group of students. The experimenter feels that thi3 group is representative of the population taking the Grade XII mathematics course in the lower mainland or any other urban area of British Columbia. It is doubtful that the results are applicable to populations in outlying districts because students in these schools may be taking mathematics because of a lack of choice of options, whereas, in more populated areas, only students who are on the academic-technical program and want mathematics for a major are likely to enrol in the course. Frederick J. McDonald, "Meaningful Learning and Retention: Task and Method Variables," Review of Educational Research, XXXIV (I96J4J, p. 5^2. 37 This study i s l i m i t e d to the use of only one of a possible eight strategies which have been suggested f o r the purpose of discovery. It i s obvious that most students w i l l have had previous experience with t h i s strategy but i t i s hoped that the newness of the material may help to counteract the previous experience. Summary In t h i s study, six Grade X I I mathematics classes were assigned to two treatment groups. During the course of seven class periods the groups were taught a unit on arithmetic and geometric progressions. An expository method was used i n teaching one group while the strategy of simple enumeration to lead to discovery was used i n the second group. On the basis of r e s u l t s from c r i t e r i o n measures, comparisons were made with respect to a c q u i s i t i o n of knowledge and trans-f e r a b i l i t y by using the t - t e s t . The major l i m i t a t i o n s of the study are inherent i n the experimenter-constructed tests and i n the classroom environment i n which the study was conducted. CHAPTER IV RESULTS OP THE STUDY The purpose of this chapter is to present and inter-pret the data obtained in the study, and in so doing, to test the hypotheses enumerated in the f i r s t chapter. Analysis of the Data Using criteria based on mathematical content, mathematical transfer, and general transfer, the results of teaching by simple enumeration were compared with those of teaching by expository method. The specific hypotheses were tested at the .05 level by the use of the t-test. The basic assumptions associated with the. use of this statistic are normal distribution, homogeneity of variance, and independent observation within each sample and between groups. Some researchers have suggested that the importance of the f i r s t two assumptions may be overrated. Lindquist says that unless variances are so heterogeneous as to be readily apparent, the effect on the test will be negligible.^" Boneau contends that in a large number of research situations the probability statements resulting from the use of ' t' tests, even when the 1 E. Lindquist, Design and. Analysis of Experiments, (Boston: Houghton Mifflin, 19^3), pp. Ttf-Bo*. 39 f i r s t two assumptions are violated, will be highly accurate.2 Under the conditions of the present study the t-test was deemed to be an adequate and suitable statistic. Equivalence of the Two Treatment Groups on the Basis of I. ft. and Previous Term Marks Prior to testing the hypotheses, the groups were compared with respect to I. ft, and previous terra marks. Since these criteria are considered to be predictors of achievement, any major differences in these areas would have a marked effect on results and would require adjustment by means of analysis of covariance. ° The Lorge-Thorndike Intelligence Test, Form 1, Level H of the I96I4. Multi-Level Edition was administered to a l l students. Students were then arbitrarily classified into high, medium, and low I. Q. subgroups within each treatment. Those having differential I. ft. scores of 125 or more were in the high category, those with scores from HI4. to I2J4. were in the medium category, and those less than lib. were in the low I. ft. category. This resulted in a distribution of 16 high, 33 medium, and 15 low for the discovery group and 9 high, 33 medium, and 21 low for the expository group. The reason for the classification was to enable the experimenter to determine whether any significant differences in treatment C. Boneau, "The Effects of Violations of Assumptions Underlying the t Test," Psychological Bulletin, LVII ( i 9 6 0 ) , pp. 1{.9~61{.. ko groups ranged through a l l I. Q. levels or whether they could be attributed to a specific level. The results of the comparisons for equivalence of the groups are shown in Table I. It indicates that there are no significant differences in the total group with respect to either criterion. Examination of the subgroups indicates that the minor difference in I. Q.. is located in the low level category. These results signify that the treatment groups are equivalent on the basis of I. Q. and previous terra marks, and that analysis of covariance to adjust for i n i t i a l differences is unnecessary. Hypotheses Testing There are three major hypotheses to be tested in this study. Essentially they are concerned with the effects of two different treatments in teaching a unit on arithmetic and geometric progressions at the Grade XII level. The hypotheses are first examined according to the treatment effect on the total sample in each group. Then the effects on subgroups classified according to I. Q. level are analysed where applicable. The hypotheses that treatment with simple enumeration will yield the same level of mathematical achievement as treatment with an expository method was accepted. The results for treatment groups and I. Q. levels as shown in Table II show no significant differences between groups or at any level. TABLE I MEANS OP I. Q. SCORES AND OP PREVIOUS TERM MARKS BY TREATMENTS AND I. Q. LEVELS I. Q,. TERM MARK Discovery Expository df t Discovery Expository df t Total 119.63 ( 8 4 1 ) 116.70 (8.98) 125 1.895 6 7 4 1 (13.68) 65.91*. (IO.6I4,) 125 0.675 High 130.88 ( 5 4 3 ) 130.14 (3.61) 23 0.212 73.63 (12.91) 76.33 (llf.Olj.) 23 -O.l4.88 Medium 118.73 (3.37) 119.36 (2.79) 6)4 -O.836 6I4..6I (13.514,) 66.79 ( 8 4 7 ) 6I4 KJ.785 Low 109.60 (2.61|) 106.62 (5.38) 3k 1.977 66.93 (1345) 60.ll). ( 8 4 8 ) 3k 1.859 Note: Figure in parentheses is standard deviation for each corresponding mean. Examination of standard deviations for treatments shows homogeneity of variances involved. h2 TABLE II -MEANS OF MATHEMATICS ACHIEVEMENT SCORES BY TREATMENTS AND I. Q. LEVELS Discovery Expository d f T o t a l High Medium Low 17.81 (2.91) 19 . 0 6 ( 2. 35) 17.39 (3 . 0 5 ) 17.1+0 ( 2 . 9 0 ) 17.06 (3 . 2 3 ) 19.10+ (3.36) 17.09 (2 .90) 16.00 (3.26) 125 23 61+ 1.375 -0.331+ 0.1+11+ 1.330 Note: Figure in parentheses is standard deviation for each corresponding mean. k3 The inference drawn from these results is that the use of simple enumeration as a strategy for discovery is not any more effective than teaching by an expository method when the criterion is concerned with acquisition of specific knowledge or mastery of subject matter. The information in Table III supports the hypothesis that treatment with simple enumeration will yield more mathematical transfer than treatment with an expository method. On a total sample basis the superior achievement of the simple enumeration treatment group is significant at the .01 level. The breakdown by I. levels shows that, at the high I. Q. level, achievement was approximately equal in both treatments, whereas there was significant superiority for the simple enumeration treatment at the medium level, and substantial, although not significant, superiority at the low level. Since the high I. Q. subgroup in the expository treat-ment achieved on a similar level to the comparable group in simple enumeration, it seems clear that treatment at this level had l i t t l e relevance. The explanation for this lack of change is probably that since these students had shown marked ability to generalize in the pretest, the experience gained' in the mathematical unit was not as effective for them as for other groups. The results at the other levels indicate that ability to generalize from simple enumerations can be improved in a relatively short time with a minimum of practice. TABLE III MEANS OP MATHEMATICS TRANSFER TEST SCORES BY TREATMENTS AND I. Q. LEVELS - Discovery Expository df t Total 8.30 (3.30) 6.76 (2.98) 125 ' 2.752 High 9.1+1+ (3.50) 9-33 (3.50) 23 0.071 Medium 8.15 (3.28) 6.61+ (2.1+3) 61+ 2.131 Low 7-1+0 (2.97) 5.86 (3.ol+) 3l+ 1.516 Note: Figure in parentheses is standard each corresponding mean. deviation for *- Significant at .05 level. Significant at .01 level. The hypothesis that treatment with simple enumeration will yield more transfer effect to non-mathematical materials than treatment with an expository method was rejected. Con-sideration of this hypothesis led to certain pertinent related questions. Did either or both treatments lead to improved ability to generalize from examples? If so, was the improvement significant? Did treatment with simple enumeration lead to greater measurable improvement than treatment with an expository method? To assess the ability of students to generalize prior to any treatment, the score of the Nonverbal Battery of the Lorge-Thorndike Intelligence Test, Form 1 was used as a pretest measure. The subtests involve pictorial classification, pictorial analogy, and numerical relation-ships. According to the authors they measure ability to see relationships and generalize. Following the period of instruction, the Nonverbal Battery of Form 2 was administered as a posttest. Table IV, which shows the pretest and posttest means by treatment groups and I. Q. levels, indicates that both treatment groups shox^ ed significant improvement. The conclusion based on the evidence of this table is that the improvement was not attributable to treatment. Table V, which specifically compares the posttest scores of the two treatments corroborates this conclusion. The implication is that the use of simple enumeration as a strategy for dis-covery is no more effective than teaching by an expository method when the transfer test is non-mathematical. This is TABLE IV MEANS OP POSTTEST AND PRETEST SCORES OP NONVERBAL BATTERY BY TREATMENTS AND'I. Q,. LEVELS DISCOVERY EXPOSITORY Posttest Pretest df t Posttest Pretest df t Total .52.11+ '"'"(7.93) 1+6.67 ll.Jk) 126 "l+. 0-1+9 121L 3.269 High 58.75 (8.81+) 55.63 (5 . 5 8 ) 30 1.196 56.00 (6.1+8) 5 1 . 5 5 (6.58) 16 0.1+69 Medium 51.15 ( 5 . 9 4 ) 61+ \ . 5 o o it 51.97 ( 5 . 5 5 ) 1+7.61 (5 . 0 0 ) 61+ *3.356 Low 1+7.27 . (6.1+0) 39.67 TJ+-89) 28 3.651+ 1+3.05 (6.9i+) 36.19 (6.27) l+o 3-359 *• Significant at .002 level. -»-«- Figure in parentheses is standard deviation for each corresponding mean. -p-47 TABLE V MEANS OF POSTTEST SCORES BY TREATMENTS AND I. Q. LEVELS Discovery Expository d f t Total 52.11]. (7.93) Ij,9.57 (7.77) 125 1.814 High 58.75 (8.8IJ.) 56.00 (6.I4.8) 23 0.815 Medium 51.15 (5.94) 51.97 (5.55) 6li -0.578 Low k7.27 ( 6 4 0 ) 34 1.857 Note: Figure in parentheses is each corresponding mean. standard deviation for 1+8 in contrast to the results obtained when the test involved mathematical transfer. There are at least two possible explanations for these results. The first is that the unit covered in the mathematics classes was concerned with the structural relationships of numbers in sequences and this would tend to make students more aware of patterns when they wrote the posttest. Recognition of patterns is basic to the discovery of generalizations but it would appear that having relationships explained is equally as effective as discovering them for oneself. The second explanation is that there was only a time lapse of three weeks between tests and the students probably felt familiar with the format of the post-test. Summary of Results The findings of this study indicate the use of simple enumeration as a strategy for discovery of general principles is more effective than teaching by an expository method when the criterion is a transfer test which emphasizes mathematical content related to the specific material taught to students. However, when the criterion seeks to measure mathematical knowledge acquired in the study of a specific unit, there is no significant difference attributable to method. The significant gain in general transfer ability as measured by pretest and posttest scores, was comparable for both treatment groups. This gain appeared to be related to an awareness of 1+9 the existence of patterns in sequences of numbers, rather than to any particular treatment. CHAPTER V SUMMARY AND CONCLUSIONS The Problem This study was specifically concerned with the effectiveness of the use of simple enumeration as a strategy for discovery as compared to an expository approach. It was hypothesized that the two treatments would yield the same mathematical achievement, but the simple enumeration treatment would yield more mathematical and non-mathematical transfer than the expository treatment. Procedures The material chosen for the study was arithmetic and geometric progressions as outlined in the Grade XII curriculum for the Province of British Columbia. The subjects were enrolled in six classes at Delbrook Senior Secondary School in S. D. No. Ijlj. (North Vancouver). Three classes were arbitrarily assigned to each treatment, and treatment groups were subdivided on the basis of I. Q. as measured by the Lorge-Thorndike Intelligence Test, Form 1, Level H. The experimenter taught both groups for a period of three weeks. Students who missed a period of instruction or a test were , eliminated from the study when the data were analysed. In the final results, there were 63 students in the expository group and 6I4. in the discovery group. At the outset of the study, the groups were compared to see whether any significant differences existed with respect to I. ft. or previous term marks. The groups were regarded as equivalent with respect to these covariates since there were no significant 1 1 1 values in the comparison of means. Since a l l other important factors were controlled, i t could, be assumed that any major differences in achievement or transfer were attributable to variation in teaching methods. In order to investigate the effectiveness of simple enumeration as a strategy for discovery, the groups were com-pared on the basis of criteria which measured mathematical achievement, mathematical transfer, and non-mathematical transfer. Findings The mean achievement of the groups on the criteria was compared by the use of the t-test with the level of significance set at . 0 5 . Decisions on the three major hypotheses were as follows: 1 . The hypothesis that simple enumeration would yield the same mathematical achievement as treatment by an expository method was accepted. The results for the total treatment samples and for I. 'ft. levels were approximately equal. 2. The experimental hypothesis that treatment with simple enumeration would yield more mathematical transfer than treatment with an expository method was also accepted. Analysis on the basis of I. Q,. levels revealed that the majority of the difference was located at the medium level. 3. The experimental hypothesis that treatment with simple enumeration would yield more non-mathematical transfer effect was rejected. Both treatment groups showed significant improvement in ability to see relationships and generalize but the treatment had no apparent influence on the improvement. Conclusions The fi r s t three conclusions enumerated below are based on the observed results of the study. The remainder, while not substantiated by data, are included as observations of the experimenter and are an attempt at objective analysis of the average classroom situation. 1. There is no clear-cut advantage in the use of simple enumeration for teaching mathematics i f the concern is centred on acquisition of facts. 2« The use of simple enumeration has a mathematical transfer effect which does not appear to accrue when the same material is taught by an expository method. This transfer effect does not extend to S3 non-mathematical material but is only prevalent when the task involves a situation with which students are familiar. 3. The results of the pretest-posttest measures should be a reminder to teachers to exercise extreme restraint in the interpretation of I. Q. test results. The particular test used in this study can be regarded as a measure of verbal and non-verbal reasoning but cannot be regarded as a measure of some intangible factor called mental capacity. It is obvious from the pretest-posttest improvement exhibited by both treatment groups that the scores are largely a function of environment and experience. ij.. There is a need for greater emphasis on a discovery attitude and approach, not only in the classroom, but by textbook writers. An examination of the average textbook shows a "tell-and-do" approach which leads to rote memorization rather than under-standing. It is the opinion of the experimenter that the discovery attitude must start at the earliest elementary level and be developed through-out the grades. It is difficult to change attitudes to any measurable degree at the secondary level. •The stress placed on this aspect of learning in a-philosophy underlying curricula is not evident in practice. £. There is a need to de-emphasize the "mark" approach to learning. 6 . It is not suggested that a l l mathematical material should be taught by a discovery method. A judicious mixture of methods is. probably most effective. Under existing conditions In the average classroom, many students never experience the satisfaction of solving problems Independently and with under-standing, nor do they acquire any ability to analyse problem situations. Suggestions for Future Research 1. The present study could be replicated after revision of the experimenter-constructed tests. It could also be designed at a different grade level with different instructional material. 2. There are many possibilities for enlargement of the scope of the present study to include more strategies, measures of attitude, and measures of retention. 3. It is possible that valuable information could be obtained by designing a study based on individual instruction rather than on a classroom situation. This would facilitate more rigid control and would also provide an opportunity to analyse s o l u t i o n processes more accurately. There i s need f o r research designed to investigate the most suitable methods for teaching the high I. Q.. students. Such methods should not only maximize the i r rate of progress but should provide them with challenging tasks through which they w i l l have an opportunity to develop new s k i l l s and str a t e g i e s . 56 BIBLIOGRAPHY Ausubel, David P. "Learning by Discovery: Rationale and Mystique," Bulletin of the National Association of Secondary School Principals, XLV (December, 1961, pp. 15-56. . "Some Psychological and Educational Limitations of Learning by Discovery," The Arithmetic Teacher, XI. (May, 1961+), pp. 292-302. Beberman, Max. An Emerging Program of Secondary School Mathematics. Cambridge: Harvard University Press, 1958. Boneau, C. "The Effects of Violations of Assumptions Under-- lying the t Test," Psychological Bulletin, LVII (I960) pp. 59-61+. Brown, Kenneth E. and Theodore L. Abell. Analysis of Research in the Teaching of Mathematics. U.S. Department of Health, Education and Welfare (Washington: 1965). Bruner, Jerome S. Essays for the Left Hand. Cambridge: Harvard University Press, 1963. Buros, Oscar K. (ed.). The Fifth Mental Measurements Year-book. Highland Park, New Jersey: The Gryohon Press, 19F9. Craig, R.C. "Directed Versus Independent Discovery of Established Relations," Journal of Educational Psychology, XLVII (April, 1956), pp. 223-3I™ ~~ ™ Cronbach, Lee. "The Logic of Experiments in Discovery," in L.S. Shulman & E.H. Kei3lar (ed.). Learning by Discovery: A Critical Approach. (Chicago: Rand McNally, 1966), pp. 76-92. Cummins, Kenneth. "A Student Experience-Discovery Approach to Teaching Calculus," The Mathematics Teacher, LIII (March, I960), pp. 162-70 Davis, Robert B. "Madison Project of Syracuse University," The Mathematics Teacher, LIII (November, I960), pp. 571-Dolciani, Mary P., Simon L. Berman, and William Wooton. Modern Algebra and Trigonometry Structure and Method, Book 2. Boston: Houghton Mifflin Company, 1963. 57 Gagne, Robert M. and Otto C. Bassler. "Study of Retention of Some Topics of Elementary Non-Metric Geometry," Journal of Educational Psychology, LIV, No. 3, (I963), pp. I23-31. Gagne, Robert M. and Larry T. Brown. "Some Factors in the Programming of Conceptual Learning," Journal of Experimental Psychology, LXII (October, 1961), PP. 313-21. Haslerud, G.M. and Shirley Myers. "The Transfer Value of Given and Individually Derived Principles," Journal of Educational Psychology, XLIX (December, 1958), PP. 293-97. Hays, William L. Statistics for Psychologists. New York: Holt, Rinehart and Winston, 196)4.. Henderson, Kenneth B. "Strategies for Teaching by the Discovery Method," Updating Mathematics I (November, . 1958 and April, 1959), pp. 57-60 and pp. 61-6)4.. Hendrix, Gertrude. "A New Clue to Transfer of Training," Elementary School Journal, XLVIII (December, I9I4.7), pp. 197-208. Hilgard, Ernest R., Robert D. Edgren, and Robert P. Irvine, "Errors in Transfer Following Learning with Under-standing: Further Studies with Katona's Card-Trick Experiments," Journal of Experimental Psychology, XLVII, No. 6, (1951+), PP. kSl^k-Katona, G. Organizing and Memorizing: Studies in the Psychology of Learning and Teaching. New York: Columbia University Press, 1914,0. Kersh, Bert Y. "The Adequacy of Meaning as an Explanation for the Superiority of Learning by Directed Discovery," Journal of Educational Psychology, XLIX (October, 1958), pp. 282-92. . "The Motivating Effect of Learning by Discovery," Journal of Educational Psychology, LIII (April, I962), pp. 65-71. "Learning by Discovery: Instructional Strategies," The Arithmetic Teacher, XII (October, 1965), pp. I4.li4.-i7. Kittell, Jack E. "An Experimental Study of the Effect of External Direction During Learning on Transfer and Retention of Principles," Journal of Educational Psychology, XLVIII (November, 195777 pt>. 39I-4O5. 58 Lindquist, E.P. Design and Analysis of Experiments. Boston: Houghton M i f f l i n , 1953* McDonald, Frederick J. "Meaningful Learning and Retention: Task and Method Variables," Review of Educational Research, XXXIV (I96I+), p. 5I+2T Ray, W i l l i s E. "Pupil Discovery Versus Direct Instruction," Journal of Experimental Education, XXVI (March, I96I), pp. 271-"5o. Retzer, Kenneth A. and Kenneth B. Henderson. "Effect of Teaching Concepts of Logic on V e r b a l i z a t i o n of Discovered Mathematical Generalizations," The Mathematics Teacher, LX-(November, 1967), pp. 707-10. • Schaaf, Oscar P. "Student Discovery of Algebraic P r i n c i p l e s as a Means of Developing A b i l i t y to"Generalize," The Mathematics Teacher, XLVIII (May, 1955), pp. 321+-27. Suchman, J. Richard. "Inquiry Training: Building S k i l l s f o r Autonomous Discovery," Merrill-Palmer Quarterly of Behavior and Development, VII ( A p r i l , 1961), pp. 11+5-69. Szabo, Steven. "Some Remarks on Discovery," The Mathematics Teacher, LX (December, 1967), pp. 839=1+2. Taba, Hilda. "Learning by Discovery, Psychological and Educational Rationale," Elementary School Journal, •LX'III (March, 1963), pp."308-15. " ~ ~ ' Tuckman, Bruce W. and others. "Induction and Transfer of Search Sets," Journal of Educational Psychology, LIX pp. 59-68. Wittrock, M.C. "Verbal Stimuli i n Concept Formation: Learning by Discovery," Journal of Educational Psychology. L r v , No. 1+, (1963), pp. 103-90. ' " Worthen, Blaine R. ^Discovery and Expository Task Presentation i n Elementary Mathematics," Journal of Educational Psychology Monograph Supplement, LIX TFibruary, I960), N o T l , Part 2. APPENDICES 60 APPENDIX A PAGE Teaching Procedures 6l Written Exercises 8 3 61 TEACHING PROCEDURES Introduction The concept of a function was reviewed because i t is on the basis of this concept that the teaching of sequences i s structured. The review was neither long nor detailed because of the students' considerable previous experience with functions. There was no d i f f e r e n t i a t i o n i n treatment In the review because i t was not considered to be part of the subject matter on which the experiment was based. In both groups the review was l a r g e l y conducted by questioning and examples. Materials Concept of a function - a set of ordered p a i r s . The domain and range of a function and the function r u l e . Notation - f(x) = 2x + 1, x 1 (the set of integers) Examples of functions which students have previously had such as; f (x) = x 2 (x£ R) f (x) = cos x (0°< x 0 6 0 ° ) f (x) = x3 (x£ R) Given the tables below, write the rule which determines f ( x ) . X 1 2 3 k 5 6 7 8 f(x) 1 3 5 7 9 11 13 15 X 1 2 3 k 5 6 7 8 f(x) 2 k 8 16 32 61. 128 256 X 1 2 3 h 5 6 f(x) 1/2 l A 1/8 1/16 1/32 l M In the examples above, the terms indicated by f(x) form a sequence. A sequence is related to a function. The domain of the function is the set of counting numbers. f(x) is found according to the function rule and is an ordered set in one-to-one correspondence with the counting numbers. Thus, If f is a function, a sequence could be defined as f ( l ) , f ( 2 ) , f(3) etc. 63 LESSON I Concepts 1. A sequence is a set of numbers in one-to-one correspondence with the set of natural"numbers. 2. An arithmetic sequence has the special character-istic that there is always a common difference between terms. 3. The nth term of an arithmetic sequence is defined to be L = a + (n - l)d, where 'a1 is the first term, 'a' is the common difference, and 'n' is the number of terms. If. The terms sequence and progression are synonymous in describing sets of numbers. Method for Expository Group 1. Definition of arithmetic sequence: A sequence is arithmetic i f and only i f each term after the first can be obtained from the preceding term by adding to i t a fixed number called the common difference. Examples: 1, 2, 3, 5, 6, 1, 3, 5, . 7, 9, 8, 11, 1I4, 17, 20, 7, lu 1, - 2 , - 5 , 1, 3/2, 2, 5/2, 3* Note that the common difference can be positive or negative. It is not difficult to identify an arithmetic sequence or to continue the sequence when the first term and common difference are known. 2. To find a given term of an arithmetic sequence: Example: 7, 10, 13, Suppose that one wished to know the 5>0th term of this sequence. Note the following pattern: Term No. I 10 1 2 n = k + (n - 1)3 This pattern is applicable to a l l arithmetic sequences. In general, i f the first term is represented by 'a', the common difference by 'd', and ' n' is any given term, then the nth term is represented by t„ and is defined by t = a + (n - l)d. n n Examples; Given the A. P. 5 , 9, 13, 17, ••• By inspection a = 5 and d = 1+. Find the 8th term and the 50th term. t 8 = 5 + (8 - 1)LL = 5 + 28 t 5 o = 5 + (50 - 1)1L = 5 + (1I9)1L = 201 If any three of the terms a, d, n, or t^ are given, the other term can be found by application of the formula. Example: In an arithmetic sequence the first terra is 3 and the 24-th term is 72. Find the common difference and write the first three terms of the sequence = 3 + (2k " D d 72 = 3 + 23<i d = 69/23 65 The f i r s t three terms of the sequence are 3, 6, and 9. Example: Find the l 2 t h term of the sequence 5, 3» 1, ~1, • • • By inspection a = 5 , d = - 2 , and n = 12. t 1 2 = 5 + (H ) ( - 2 ) = -17 Method for Discovery Group 1 . E s t a b l i s h the f a c t that sequences can be c l a s s i f i e d according to the method of obtaining consecutive terms. These examples were used to i d e n t i f y the arithmetic sequence. k, 7, 1 0 , 13, 1 , 2 , kt 8,, 3 , 6, 1 2 , 2U-, JL, 0 , -II, - 6 , 6, 3, 1 1 / 2 , 3 A * 1 , 2 , 3» 4» 5, Students were required to write the next three terms of each sequence, to state s p e c i f i c a l l y the operation by which any term was obtained from the previous term, and to c l a s s i f y the sequences into two groups on the basis of d i f f e r e n t methods of obtaining consecutive terms. The instructor then stated that only those sequences i n which the operation of addition was used would be considered. To f i n d an expression f o r the nth term of such a sequence, the following method was used: Examine the sequence 1, II, 7, 10, 13, 16, The f i r s t term i s The second term i s 1 + The t h i r d term is 1 + or 1 + ( ) ( ) The fourth term i s 1 + .... or 1 + ( ) ( ) The tenth term i s 1 + . . . . ( ) ( ) The f i f t i e t h torm i s .... + The hundredth term i s ...... + The nth term i s . . . . . . . + To find the general expression for the nth terra: Examine the sequence x, x + y, x + 2y, What is the ij.th terra? the 10th term? the £0th term? the nth term? Examine this sequence: a, a + d, a + 2d, a + 3d, What is the 8th term? the 100th term? the nth term? On the basis of the last sequence, designate the nth term as t n and write an expression for t n in terms of 'a' and 'd'. What does 'a' represent? What does 'd' represent? The terminology arithmetic sequence or arithmetic progression is used to describe the above type of sequence. What do you think is the main character-istic of an arithmetic sequence? The same examples were used for the discovery group as for the expository group, the difference being that the discovery group worked the examples with-out the instructor's help whereas the instructor worked the examples for the expository group. 67 LESSON II Concepts 1. The sum of an arithmetic progression. 2. The summation notation. Method for Expository Group 1. The series representing the sum of the first one hundred integers was first used, S100 = 1 + 2 + 3 + * + 1 0 0 S 1 Q 0 =100 + 99 + 98 + + 1 2 S100 = 1 0 1 * 1 0 1 + 1 Q 1 + + 1 0 1 2 S 1 0 0 = 100(101) S = l o o ( i o i ) 100 2 2. The general sequence can be represented by a, a + d, a + 2d, .... a + (n - l)d. The sum of n terms of this sequence can be written: S = a + (a + d) + (a + pd) + a + n (n - l ) d For convenience call the nth term the last term and designate it with the letter ' l ' . Then write the sum as S n = a + (a + d) + (a + 2d) + + 1 S = 1 + (1 - d) + (1-- 2d) + + a 2S n = (a+1) + (a+1) • (a+1) + +(a+l) 2S = n (a + 1) n S = n ( a + 1) or S = nfea (n - l)dj (Substitute n 2 n 2 f o r 1 } 68 Examples: 1. Find the sum of the sequence of 30 terms i f the first term is 6 and the 30th term is b%. 2. Find the sum of the first 80 terras of a sequence whose first terra is 3 and whose common difference is 2. 3. Suppose the sura of the fi r s t 30 terms of an arithmetic sequence is IO2O and the fir s t terra is 5» Find the common difference. 3. The Summation Notation As a means of shortening the writing out of the series, the Greek letter £ is used to denote summation. For example, (3i) means the same as 3(1) + 3(2) + 3(3) + 3(I4.) Note that i t is not necessary to write out the complete series. One can calculate the first and last terms and the number of terms and use the formula s _ n(a + 1) + 3(5) or 3 + 6 + 9 + 12 + 15 i=l n 2 or S 2X3 + i S l 2 Examples: (a) 6 (d) 69 Method for Discovery Group 1 . The story of Gauss solving the problem, of the sum of the first 100 positive integers in an incredibly short time was used to introduce the topic of finding the sum of an arithmetic sequence. The ""•students were then asked to try to find the clue given that 1 + 2 + 3 + 4 + + 1 0 0 = 5 0 5 0 . "Subsequently the following additional examples were given: 1 + 2 + 3 + + 1 0 = 5 5 l + 2 + 3 + + 2 0 = 2 1 0 1 + 2 + 3 + + ho = 8 2 0 1 + 2 + 3 + + 5 0 = 1 2 7 5 1 + 2 + 3 + + 1 0 0 0 = 5 0 0 5 0 0 , Note: The majority of the students established the formula S = n (n +• 1 ) ' 1 0 0 2 The examples 1 + 3 + 5 + 7 + . . . . . + L L 9 = 6 2 5 5 +.8 + 11 + . . . . + 6 8 = 8O3 were then given and students were asked to check their formula. The next hint consisted of a chart as illustrated. X \ X X X X X X X X \ X X X X X X X X \ X X X X X X X x \ x X X X X x x x \ x X X X X X X x \ X The portion on the right of the diagonal line is a replica of the left portion turned upside down. By answering the questions: How many rows in the complete diagram? How many crosses in each row? What is the total number of crosses? What is the total number in the left section? How may the total number of crosses in each row be derived? Students were able to establish the fact that the sum was 4 - ( . 6 . . Similar diagrams were used to establish 70 the general case that S n = ^ a + ^\ where a i s the f i r s t terra and 1 i s the l a s t terra of the sequence. 2. The meaning of the summation notation was explained ' "to t h i s group by the in s t r u c t o r . In view of the f a c t that t h i s i s not a p r i n c i p l e to be discovered but merely a notation device, i t was not deemed -necessary to d i f f e r e n t i a t e between groups for th i s part of the lesson. The only difference was that t h i s group was not t o l d that i t i s only necessary to f i n d the f i r s t and l a s t terras of the sequence and use the formula but arrived at t h i s decision by working the examples. 3. The term series was also introduced as the indicated sum of a sequence. 71 LESSON III Concepts 1. An arithmetic mean between two numbers is the average of the two numbers. 2« Terms between any two given terms of an arithmetic sequence are called arithmetic means. Method for Expository Group 1. Define an arithmetic mean between two numbers. Extend the definition to the general case. i.e. The arithmetic mean between x and y is x'+ y. 2 2. Define arithmetic means as stated under concepts. Example: In the arithmetic sequence 1, l\.t 7, 10, the numbers ij. and 7 are said to be arithmetic means between 1 and 10. 3. To find the arithmetic means between any two given terms of an arithmetic sequence, use a diagram. Example: Insert three arithmetic means between 5 and 2 l . 5, , 21 Note that there are 5 terms in the sequence. The first term is 5 and the f i f t h term is 21. t^ = a + lj.d 21 = 5 + lfd d = k Since the common difference is LL, the means to be inserted are 9, 13, and 17. Example: Given that the third term of an arithmetic progression is 7 and the twelfth term is 25. Find the first term and write the first 12 terms. 72 Method f o r Discovery Group 1. A series of examples was given involving the i n s e r t i o n of one term between two given terms with the i n s t r u c t i o n to make an arithmetic progression of three terms. Example: , 16 -The f i n a l example was x, , y 2. The examples were then extended to i n s e r t i o n of two, three, and four arithmetic means. Example: Insert two arithmetic means between 3 and 15. (Note; Most students did a l l examples without d i f f i c u l t y . Some immediately subtracted 3 from 15 i n the above example and then divided by 2 and obtained a difference of 6. They quickly ascertained that t h i s did not give an arithmetic progression and had no d i f f i c u l t y i n a r r i v i n g at the correct solution.) 73 LESSON IV Concepts .1. A geometric sequence i s one i n which the r a t i o between any two consecutive terms i s constant. 2. The nth term of a geometric sequence i s defined to be t n = arn " , where 'a' i s the f i r s t terra, • r 1 i s the common r a t i o , and 'n* represents the number of a given term. Method f o r Expository Group 1. A d e f i n i t i o n of a geometric sequence was given followed by these examples; k* 8* 16, 32, 12, 6, 3, 3/2, 2, -6, 18, -Ski v Ratios were examined between two consecutive terms to i l l u s t r a t e the d e f i n i t i o n and to es t a b l i s h the fact that the r a t i o may be any r a t i o n a l number. 2. The sequence k> 8> 16, 32, was examined as follows: \ = k t 3 = k(2)2 ^ = 4 ( 2 ) 3 n - 1 3# Examples worked by the instructor and cl a s s . (a) Find the I|.th term of a geometric progression whose f i r s t term is 2 and whose common r a t i o i s k> (b) Which term of the geometric progression - 8 l , 27, -3» .... is 1/9? (c) The seventh term of a geometric progression i s 256 and the f i r s t term i s k» What i s the f i f t h term? The f i r s t , t h i r d , and f i f t h examples from the f i r s t lesson were used to introduce the geometric sequence. Students had previously written the next three terms of each and had sp e c i f i e d the "basis on which they had decided on the terms. It was established that t h i s type of sequence i s c a l l e d geometric. The sequence 3, 9, 27, 8l, .... was presented with instructions to f i n d the r a t i o between the second and f i r s t terms, the t h i r d and second terms, and the fourth and t h i r d terms. Ratios were also checked between terms of the preceding examples. This established that the r a t i o between con-secutive terms is a constant. The example 3, 9, 27, 8l, was analysed i n a manner similar to the analysis of an arithmetic progression i n Lesson I. The concept was extended to the sequence x, xy, xy2, ...... f o r which the nth term was required. The f i n a l example was the sequence a, ar, f o r which the nth term was required. On the basis of the l a s t example students were required to write an expression f o r t n i n terms of >a' and ' r ' . -I.e. t 3 3( ) 3( )( ) or 3 ( ) or 3( ) 3( ) 75 LESSON V Concepts 1. The sum of a finite geometric sequence. 2. The summation notation. Method for Expository Group 1. The formula for the geometric series was established as follows: The general geometric progression is represented by a, ar, ar , a r 3 , ar n - 1 The sum of the above progression can be represented by the expression S„ = a + ar + a r 2 + a r 3 + + a r n ~ 2 rS n = ar + ar 2 + a r 3 + n-1 n + ar + ar S - rS_ = a - ar 1 1 n n S n ( l - r) = a - ar 1 1 S = a - a r ? r f 1 n 1 - r If the nth term is considered to be the last term and is represented by the letter ' l 1 , this formula can also be written in the form S n = * • 1 2 . These two examples were worked with the class. (a) Find the sum of the first five terms of the geometric progression with a = 6 , r = l / 2 . (b) Given that S n = - I I L , n = 3 , a = - 2 , find the common ratio and write the fi r s t three terms of the geometric sequence. 3» Two examples were used to. Illustrate the use of the.summation notation for a geometric series. (a) J 5(2) 0-1 (b) 9(1/3) r-1 Method for Discovery Group Study the three examples given below which show how to find the sum of a finite geometric progression without actually adding the terms. In each case the second statement has been obtained from the fir s t by multiplying both sides of the equation by the same number. Identify the number by which both sides of the equation are multiplied and decide how i t is related to the series. Example 1: Given the finite geometric progression 2, 5> 8> 1&> 32. Find the sum of this sequence. - S^ = -62 or S^ = 62 Example 2'-Given the G. P. 8, IL» 2, 1, l / 2 . Find the sum of the S^ = 2 + 5 + 8 + 16 + 32 = 5 + 8 + 16 + 32 + 65 - 6^ terms. s^ - i / 2 s 5 = 8 - 1 / 5 = 8 - 1 / 5 S^ = 2(8 - 1/5) or 15 1/2 Example 3 : Find the sum of the G. P. 2, 6, 18, 5l|, I 6 2 , I4-86. S 6 = 2 + 6 + 18 + Sk + 1^2 + I4.86 336 = 6 + 18 + 5u, + I62 + I4.86 + 4 5 8 s6 - 3S 6 = 2 - 4 5 8 s 6 ( i - 3) = 2 - 4 5 8 After you have studied the above examples use the same process to develop a method for finding the sum below. S = a + ar + ar 2 + ar 3 + ar^ + n -1 n - 1 + ar Note: Examples for practice and for the use of the summation notation were the same as those for the expository group. 78 LESSON VI Concepts 1. The geometric mean or mean proportional. 2. Geometric means. Method f o r Expository Group 1. A single term between two terms i n a geometric sequence is c a l l e d a geometric mean or mean proportional. Two methods of fin d i n g a mean proportional were discussed. (a) Find a mean proportional between II and 9. *1 = k t ^ = lp? 2 or 9 = Ip?2 r = 3/2 or -3/2 The mean proportional i s 6 or -6. (b) Use the idea that the r a t i o between con-secutive terms i s a constant. Let the mean proportional be 'x 1. The sequence i s II, x, 9 Then 2. = 9 or x~ = 36 Ii x x = +6 or x = -6 2» The geometric mean between 'a' and 'b' i s represented byfab. 3. Examples used to demonstrate how to f i n d tx*o or more geometric means. Geometric means are the terms between any designated terms of a geometric sequence. Example: Find two geometric means between 1 and 27. 1> —> _> 27 a = 1 79 ar3 =27 r 3 = 27 or r = 3 The means are 3 and 9 Example: Find three geometric means between 1/525 and 25/21. a = 1/525 and a r 4 = 25/2I rk = (25/21)(525/1) rk = 2 5 2 or r 2 = 25 and r = +5 or -5 The means are t ™ » and t Method for Discovery Group In the same way as i t is possible to find an arithmetic mean between two given numbers, it is possible to find a geometric mean between two terms. The single geometric mean is called' a mean pro-portional. The following exercises were given to the discovery group: 1. (a) Insert one number between II and 9 so that the resulting sequence will be in geometric progression. 5, 9 (b) Insert one geometric mean between 2 and 8. (c) Insert one geometric mean between x and y. (d) Is there a possibility of using a different number than the one you used? If so, why? 2. Find two geometric means, between 1 and 27. 1* 27 3. Find two geometric means between m2 and m^-4. i;. Find three geometric means between l/ 5 2 5 and 25/21. 5 . Find four geometric means between -7 and -22\\.* 80 LESSON VII Concepts 1. The sum of an infinite geometric series. 2» Rewriting a repeating decimal as an infinite geometric series and using this method to write a repeating decimal as a common fraction. Method for Expository Group 1. The sum of a geometric series is given by the formula _ a - ar n n 1 - r Examine the sequence 1, l / 2 , l/k> l / 8 , 1 ~ l ( l / 2 l l 1 - 1/8 s3 = 1 -1/2 o r r^-172 = 1, °7 1 - 1/2 1 " V 2 S - 1 - K l / 2 ) n n *" 1 - 1/2 Note that as 'n1 becomes very large the quantity represented by ( l / 2 ) n becomes very small. In fact, as 'n* Increases without bound this quantity approaches zero, and the sum of the sequence approaches ^ ~ "1/2 ° r *^ ' ^ r n ^ °^ ^ e 3 u m °^ the above sequence is said to be 2» Note that the quantity represented by ar n in the formula can only approach zero i f | r| is less than 1. The sum of an infinite geometric progression whose ratio has an absolute value less than 1 is defined by a S = — r * A repeating decimal can be written in the form of an infinite geometric series. For example 0.555 ... can be written in the following form: 0.5 + .05 + .005 + .005 + The terms of the series form a sequence whose f i r s t term i s 0 .5 and whose common r a t i o i s l / l O or . l e The l i m i t of the series i s 0 . 5 = 5V 1 - .1 9 3 . Practice Examples: Change the following repeating decimals to equivalent common f r a c t i o n s . 0 . 6 5 . . . 0.127 Find the sum of the i n f i n i t e geometric sequence with a = 3 and r = 2/3. Method for Discovery Group 1 . Consider the series 1 + 1/2 + l A + l / 8 + S l =• 1 s 2 = 1 + 1/2 = 1 1/2 S 3 = 1 + 1/2 + 1/IL = 1 3/ii = l + 1/2 + l / l j . + 1/8 = 1 7/8 = 1 + 1/2 + l A + 1/8 + 1/16 = 1 15/16 s 6 = 1 + 1/2 + i A + 1/8 + 1/16 + 1/32 = 1 31/32 On the b a s i s o f the above examples e s t i m a t e the va lue of S, n = 1 + 1/2 + l A + 1/8 + 1/16 + I/32 + l/6k + X U 1/128 + 1/256 + 1/512 Check your guess by using the formula f o r finding the sum of a geometric sequence. s = a - ar n n 1 - r Suppose n = 20 1 1(1/2) "20" Then S n = n 1 - 1/2 82 and i f n - 100 1 s . - - M l / 2 ) 1 0 ° n 1 - 1/2 As 'n 1 gets progressively larger, wr+at i s happening to the term enclosed by the rectangle? As 'n' approaches i n f i n i t y , what value does t h i s term approach? Would t h i s be true f o r any 'r'? If not, what r e s t r i c t i o n would you place on 'r'? What do you think i s the l i m i t of the above series? What do you think would be an appropriate formula for evaluating the l i m i t of the sura of an i n f i n i t e geometric sequence? 2. The repeating decimal 0.555«». can be rewritten i n th i s manner: 0.5 + 0.05 +0.005 + . . . As !n' increases without bound, what i s the l i m i t of S n f o r t h i s series? 3. Practice examples as for the expository group. WRITTEN EXERCISE - LESSON 1 Write the f i r s t six terms of the sequence associated with each of these functions. In each example the domain of ' n' is the set of counting numbers. Find the Indicated term of each sequence. (a) The 30th term of 1/3, 1, 5/3* (b) The 8th term of i , 0 . 8 i , 0 . 6 i , (c) The 35th term of \f2~ + 1, i~2~, - 1, (d) The liOth term of x - y, x, x + y, (e) The 20th terra of -3x 2 , -x2, x2, True or False: (a) In the sequence whose nth term is I i — 1 — i , the 7th term is 25/1+. 1 + n (b) In a sequence whose nth term is n 2 - $n + 6, the fi r s t and fourth terms are the same. (c) Two different arithmetic sequences always have different common differences. (d) It is possible for two different arithmetic sequences to have the same first three terms. The 5th term of an arithmetic progression or sequence is 9. The ll+th term Is 1+5. Write the first three terms. Find the next three terras in each sequence. (a) 87, 7k, 61, (c) -8/3, -35/12, -19/6, (b) 19, 37, 55, 81+ 6. A b a l l which r o l l s o f f a penthouse terrace f a l l s 16 feet i n the f i r s t second, 1+8 feet i n the second, and 80 feet i n the t h i r d second. If i t continues to f a l l i n t h i s manner, how f a r w i l l i t f a l l i n the 7th second? 7» A missi l e f i r e d v e r t i c a l l y upward r i s e s 15,81+0 feet i n the f i r s t second, 15,808 feet i n the following second, and 15,776 feet i n the t h i r d second. How many feet does i t r i s e i n the l+5th second? How many feet and i n what d i r e c t i o n does i t move i n the last second of the ninth minute a f t e r i t i s f i r e d ? 8. The speed of sound i n a i r i s about 332*1 meters per second at 1° C. This increases about .6 meters per second f o r each degree of increase i n the temperature of the a i r . Express t h i s i n an arithmetic sequence. What i s the nth term? WRITTEN EXERCISE - LESSON 2 Find the indicated sums: (a) The sum of the first 17 terms of the sequence whose fi r s t term is 6 and whose common difference is I+. (b) The sum of the sequence with first term of 13, last . term of 89, and difference of 1+. (c) The sum of 20 terms of the sequence 2, -1 , -I+, -7, Find the sums of these arithmetic series. (a) (b) j£ (c) 5 3j . (3k - 1) y~ (2 - 3n) k=T n=l (d) 12 y^ ( 3r - II) r=l Write the first three terms of each of these arithmetic progressions. (a) a = 3 1 = 17 S n = 100 (b) a = 8 n = 17 S n = I83.6 Use summation notation to write each of the following: (a) (2 + 3«1) + (2 + 3'2) + (2 + 3'3) + (2 + 3 *LL) (b) (5*1 + 2) + (5*2 + 2) + (5*3 + 2).+ ( 5 A + 2) (c) (1 - 3.-I2) + (l - 3 * 2 2 ) + (1 - 3 . - 3 2 ) + (1 - 3 A 2 ) (d) 2 + 5 + 8 + 1 1 (e) 1 - 1 - 3 - 5 - 7 True or False: (a) In every arithmetic sequence with a common difference of 5 , s 2 0 = s 1 9 + 5 . 86 (b) In the arithmetic sequence -12, -19, s n = 0. (c) In any arithmetic sequence whose first term is a and whose common difference is d, = + 2a + 7&. 6. On a construction job a laborer is told to carry 20 joists from the lumber pile and place them on the ground at II foot intervals. The closest placement to the lumber pile is 60 feet. Starting at the lumber pile and finishing there as well, i f he carries one joist at a time how far does he have to walk in order to place the 20 joists? 7. If the taxi rate is fOfi for the first mile and ILO^ for each additional mile, x^ hat is the fare from a suburb to the airport which is 12 miles away? 8. Find the sum of the positive integers less than 100 which are divisible by 6. 9« The largest integer in an arithmetic progression of con-secutive even integers is 9 times the smallest. The sum of the progression is 90. Find the largest and smallest integers. 87 WRITTEN EXERCISE - LESSON 3 1. Find: (a) Three arithmetic means between 10 and 16. (b) Five arithmetic means between -7 and 6. (c) Five arithmetic means between -2 and -6. (d) Six arithmetic means between -2 and 12. (e) Nine arithmetic means between -10 and 0. (f) One arithmetic mean between a + bi and a - bi. 2. True or False: (a) One could find 100 arithmetic means between 6 and 7* (b) The arithmetic mean between -11 and -lb. is -I3. (c) For any three numbers x, y, and z, i f z is the average of x and y, then there is an arithmetic progression which begins x, y, z.... (d) If -77' is the arithmetic mean between 2 and x, then x = 2vC - 2. (e) If a, b, c, d, e, are any five numbers in an arithmetic sequence, then 2c = ae. 3. The seven weights in a set for analytic balance are in an arithmetic sequence. The largest is 2p grams and the smallest 1 gram. Find the weights of the other five. 11. A man driving along a road at 60 m.p.h. (88 f t . per sec.) applies the brakes and comes to a complete stop in 22 seconds. If the speeds at which he is travelling in successive seconds form an arithmetic progression, how fast did he travel the 7th second after braking? 5. A young man's salary increased for five years in arithmetic sequence. If his salary the fir s t year was ^ILILOO and the fi f t h year was $6000, what was his salary in each of the other years? 6. The reciprocal of one number is 5 times the reciprocal of another. Seven times their arithmetic mean is li greater than their product. Find a l l such pairs of numbers. 88 7. Write i n summation notation, (a) 2a + lia + 8a + (b) 1*3 + 2-4 + 3*5 + . . . . . . (c) IL - 7 + 10 - 13 + . . . . (d) - 3 - I + I + 3 + 5 + ... + 19. -.8. -Write -the f i r s t four terms of 9 fel 9. Write the tenth term of a/2, 3a/2, 5a/2, . . . . . . . . 10. What i s the arithmetic mean of a ,,. and — f L , _ ? a + b a - b . 89 WRITTEN EXERCISE - LESSON II 1. Write the fi r s t four terras of each of these geometric sequences. (a) a = - 9 , r = 2- (b) a = -3 , r = -1/3 (c.) a = 1/12, r = 11. 2. Find the indicated terms: (a) The Iith term of the G.P. with a = IL and r = II. (b) The 7th term of 5, 10., 20, (c) The 10th term of ~{~J7 f~o~7 -2f"3T (d) The 9th term of 39, 13, IL 1/3, 3« There are two geometric progressions of real numbers with a first term of 7 and a 5th. terra of 112. Find the two values for r which will generate the two series. II. Find the nth term of each sequence in Question 2« (i.e. Write an expression for the nth term) 5. The length of the arc of the first swing of a pendulum is 10 inches. The length of each succeeding swing is l/9 less than the preceding one. How long is the seventh swing? the ninth? Write the expression for computing the answer but do not do the computations. 6. The first term of a G. P. is 27 and the common ratio is 1/3. For what value of n is t n = 1/3? 7. If the value of a car depreciates 20$ the f i r s t year and 5$ each year after the f i r s t , what is the value of a car which is four years old and originally cost $3000? 8. A, jar contains 500 cubic inches of air. Qn its first stroke an air pump removes 20$ of the air- leaving 80$ of 500 cubic inches in the jar. On the second stroke it removes 20$ of the remaining air and so on for the following strokes. How much air is left after the f i f t h stroke? 9. True or False: (a) The terms of a geometric sequence grow constantly smaller or larger but they never fluctuate back and forth. 90 (b) The (n+1) term of the geometric sequence 1/2, l/3> 2 / 9 , . . . i s (l / 2 ) ( 2 / 3)n. (c) The sequence 5, 5, S» .... i s both arithmetic and geometric. "(d) A geometric sequence i s uniquely determined i f you know the f i r s t term and the common r a t i o . """Ce) If the nth terra of a geometric sequence i s (1/2) (5) n " 1 , then the 1+th term i s 32. (f) In any geometric sequence each term i s d i v i s i b l e by a l l preceding terms. (g) I f a i s the f i r s t term and r the common r a t i o , then i n any geometric sequence the product of the fourth and f i f t h terms is always a 2 r ' . (h) I f you multiply each term of a geometric sequence by 3, the r e s u l t i n g sequence w i l l also be geometric. WRITTEN EXERCISE - LESSON 5 Find the sums of the following geometric progressions: (a) a = 1, r = 2, n = 1. (b) a = 12, r = 3/2, n = 11. (c) a = I I , 1 = 32I+, r = 3. (d) 1000, 100, 10, ... when n = 7. Find the sums of these geometric series: ( - 2 / 3 ) U / 2 ) k " 1 r=l k=l In a finite geometric sequence the last term is 8,192 and the ratio is -I}.. If the sum of the sequence is 65£IL, find the first terra. The 5th terra of a geometric sequence is 2I1 and the 10th terra is 768. Find the sequence and the sum of the first 7 terms. The side of a square is 10 inches. The midpoints of the sides are joined to form an inscribed square as shown in the diagram on the next page, with the process continued until there are five squares. Find the sum of the perimeters of the five squares. If the half l i f e of the uranium 23O isotope is 2O.8 days, how much of a given amount of the isotope will be left after lOli days. The sum of the fir s t and second terras of a geometric progression is -3 and the sum of the 5th and 6th terms Is -3/I6. Find the sum of the fi r s t 8 terms. If a youngster decided to put \£ in a toy bank today, 2/ tomorrow, II/ the next day, and so forth for 3I days, how many digits are there in the number of pennies he should put in on the 3lst day? (log 2 = .3OI) Is the sura for the month more than |l0,000,000? 93 WRITTEN EXERCISE - LESSON 6 1. In each case find the mean proportional between the given numbers: (a) 5 and 20 (b) 3 and 6 (c) -7 and -189 (d) a + bi and a - bi 2. Insert the given number of geometric means and write the resulting finite geometric progression. (a) Three between IL / 3 and 27/6I4,. (b) Two between 1 and 27 • (c) Seven between 3 and IL 8 . (d) Three between -15 and ° 1215. 3. The third term of a geometric progression is 5 . The 6th term is 8/Y"lT. Find the terms between these two terms. II. The product of three real numbers which are in geometric progression is -61L. If the f i r s t number is II times the third, what are the numbers? 5 . If -6IL/9 is the 6th term of a geometric progression whose common ratio is -2/3, what is t]_? 6. If ~ = ~ prove that ab + cs is a mean proportional b d between a2 + c 2 and b 2 + d2. 7. Which Is larger-!~the arithmetic mean or the geometric mean between 2 and %l Does the arithmetic mean ever equal the geometric mean? If so, when? 8. Find x, given that 2x - 7 is the geometric mean between x ~ 5 and 2x + 11. 95 WRITTEN EXERCISE - LESSON 7 1. Find the sums of these Infinite geometric sequences: (a) a = 6, r = -1/3. (b) 0 .1 , 0.01, 0.001, . (c) 6, 2, 2/3, (d) 3, 1, 1/3, (e) 1, -1/2, l/k, -1/8, 2. Find the common fraction equivalent of each of these repeating decimals: 3. A pendulum Is brought to rest by air resistance. The firs t arc through which the bob of the pendulum swings is 1L0 cm., and each swing thereafter is .98 as long as the previous arc. Find the total distance the bob has travelled by the time it has come to rest. I I . A ship which is 101 miles from shore sustains damage and takes in water. It starts at once for shore at the rate of 10 m.p.h. but due to the damage the rate each hour decreases and is 9/l0 that of the preceding hour. Will the ship reach the shore safely? 5. A rubber ball dropped I4.O feet rebounds on each bounce 2/5 of the distance from which it f e l l . How far will i t travel before coming to rest? 6. In an unending series of equilateral triangles, the vertices of each triangle after the fi r s t are the mid-points of the sides of the preceding triangle. The sides of the fir s t triangle are each one foot long. Find the sum of the perimeters of a l l the triangles. 7. Find the sum of the areas of a l l the triangles in Question 6, given that the area of an equilateral triangle of side a is o > /1 8. Of the values a, t„, n, r, and Sn, three are given. Find the other two. (a) .I38I38 (b) . 7373- . - . (c) .l59ij.9li.9 • • • a (a) a = 1, r = 2, n = 7. (b) a = 1/3, r = 3, S N = I L O / 3 -(c) a = 30, t n = .003, S N =33-333. 96 APPENDIX B INSTRUMENTS PAGE Mathematics Content Test 97 Mathematics T r a n s f e r Test 101 97 MATHEMATICS CONTENT TEST Multiple Choice Sequences and Series 1. If a-^ , a^, a , ^ s a n arithmetic sequence, then a-^ - a^ + is equal to: A. a n B. a„ C. a D. a + a^ E. -a^ 1 2 3 1 3 2 1. 2. The set of numbers 3, 11, 19, 27, is best described as: A. a sequence B. a series C. an inf i n i t e sequence D. an Infinite series E. none of these 2. 3. An inf i n i t e geometric progression with a ratio of 1/2 has a sum of 12. What is the sum of a geometric progression whose terms are the squares of the original progression? A. 12 B. 36 C. J18 D. 72 E. I L L L L 3. II. The f i r s t number of a sequence is 729, and each succeeding^ number i 3 found by multiplying the preceding terra by I / 3 . Sj^ is equal to; A. 1093 B. 1092 C 1089 D. 1080 E. 29,160 5>. The set of numbers 7, 11, 15>, 19, is an example of: A. a geometric series B. a geometric sequence C. an arithmetic series D. an arithmetic sequence E. an algebraic sequence jp. 6. If a, b, c is an arithmetic progression, which of the following is a true statement? A. 2a = b + c B. 2b = c + a C. 2c = a + b D. b 2 = ca E. c = 2(a + b) 6. 7. If the f i r s t term of an arithmetic sequence is - 3 , and the common difference is 2, then the nth term, t n , i s : 98 A. 5 - 3n B. 6n - 3 C. 2n - 3 D. n(n - 2 ) E. 2n - 5 7. 8. The 18th term of the sequence 5 , 1, ~3» i s : A. 77 b. 73 C. -71 D. -67 E. -63 8. ' 9 . The 9 t h term of the series 3 - 6 + 1 2 - 2 5 + . . • • I s : A. 768 B. -768 C. 27 D. -15 E. -55 9 . 10. The po s i t i v e geometric mean between 3 and 5 5 i s : A. 3 /~2 B. 27 C 9 y 2 D. 6 V 3 E. 9 y/"3 10. :. 11. Two arithmetic means between 11 and 29 are: A. 17 and 23 B. 16 and 25 C. 15 and 25 D. 18 and 22 E. None of the preceding 11. __________ 12. If a, b, c i s a geometric progression with p o s i t i v e terms, which of the following i s an arithmetic progression? A. a - 1, b - 1, c - 1 B. a, b, c C. 10 a, 10 b, 10° D. log a, log b, log c E. ab, be, ca 12. ' 13. The sum of the f i r s t 20 terms of the series (-6) + (-2) + (2) + (6) + ... i s : A. 320 B. 70 C. 1280 D. 35 E. 650 13. ' 15. Two geometric means between 7 and 189 are: A. 68 and 129 B. 28 and 75 0. 2l and 63 D. 67 and 127 E. None of the preceding 15. 15. Which term of the geometric progression l / 8 , -l/5> l/2» .... i s -5? A. the 5 t h B. the 6 t h C. the 7th D. the 8 t h E. -5 i s not a term of thi s G. P. 15. 99 16. As n approaches i n f i n i t y , the limit of S n is S = T - a - i f : 1 - r A. r = 1 B. r is less than 1 C. r is greater than 1 D. | r | > 1 E. | r 1 < 1 16. _ 17. The arithmetic mean between x * a and x " a Is: A. 2 i f a = 0 B. 1 C. 2 D. x E. a/x 17. 18. The repeating decimal .23 is equivalent to the geometric series whose common ratio and f i r s t terra have the values: A. r = 0.1, a = .23 B. r = 0.01, a = 23 C. r = 0.01, a = .23 D. r = .23, a = 0.01 E. r = 0.1, a = 23 18. 19. The sum of the infin i t e geometric series 9 + 6 + li + i s : A. 36 B. 27 C 21 2/3 D. $\ E. 19. . 20. The sum of the geometric series given by 36(-2/3) k" 1 i s : I k = 1 A. 220 B. 7 ^ C -220 D. -Jj^ E. __9_ 20. 21. The value of 9 (2n - 3) i s : n = 1 22. If a, b, and c form an arithmetic progression, which of the following is not necessarily an arithmetic progression? 100 A. c, b, a B. a + 2, b + 2, c + 2 C. 3a, 3b, 3c D. a 2 , b 2 , c 2 E. None of these 22. _____________ 23. If the t h i r d and seventh terms of an A. P. are 5 and 11, then the f i f t e e n t h term i s : A. 21.5 B. 22.5 C. 23.5 D. 23 E. 21}. 23. 21+. The l i m i t i n g sura of the series 1 + (9/l0) + ( 9 / l 0 ) 2 + «••• i s • A. 1 B. 9 C 90 D. 10 E. 10/9 24,. 25. The number . 5 l 5 l 5 l . ». can be written as a f r a c t i o n . When reduced to lowest terras, the sum of the numerator and denominator i 3 : A. 3O B. 50 C. 150 D. 100 E. None of these 25. 101 MATHEMATICS TRANSFER TEST 1. The numbers below are arranged in triangular form. Study the arrangement and answer the questions which follow i t . 1 1 1 1 2 1 1 3 3 1 . 1 l i 6 L L 1 1 5 10 10 5 1 (a) F i l l in the next row according to the pattern which has been established. (b) If the pattern continued, what would the second term in the 1965th row be? -(c) What would the third term in the 22nd row be? (d) What would the sum of the terms in the 11th row be? (e) How many terms would there be in the 1968th row? (f) If you start with a positive sign In a row and alternate positive and negative, signs in each row, what is the sum of the terms in row 769' • 2. Answer the questions following the triangular integer pattern below. 1 1 2 1 1 2 3 2 1 1 2 3 ^ 3 2 1 I 2 3 ^ 5 i i 3 2 l 102 (a) What would the middle term in the 76th row be i f the pattern continued? (b) What would the 6th terra in the 196lst row be? (c) What would the sura of the terms in the 50th row be? (d) How many integers would be in the array up to and including the lpth row? 3. Complete the examples below and provide a formula for the sura of n terms. ( a ) 1 = (b) 1 + 1 = 1*2 1*2 2*3 (c) _____ + _____ + = 1*2 2*3 yk (d) 1*2 2<3 y\\ T T T " + "2^3" + "iV" + ~WT + +n(n i 1) UNDERLINE THE CORRECT ANSWER FOR QUESTIONS 1+, 5, 6, 7 and 8 1+. A harmonic progression is a sequence of numbers such that their reciprocals are in arithmetic progression, i.e. If ii, 1, and 10 are in arithmetic progression, then l/l+, 1/7. and l/lO are in harmonic progression. Let S n represent the sura of the fir s t n terms of a harmonic progression. For example S^ represents the sura of the first three terras. If the first three terms of a harmonic progression are 3, II, and 6,- then: A. S^ = 20 B. S^ = 2$ C S^ = 1+9 D. S 6 = 1+9 E. S 2 = 1/2 S^ 5. The next two terms in the infinite sequence 0, 1, 1, 2, 3, $> 8, 13, 21, are: A. 2V and 3 I 1 B. 3J4. and I1I4. C. 3I4. and 5$ D. 29 and I L L L E. None of these 6. The sura to infinity of 1 + _2 + _1 + _2 7 72 7 3 7 I L 1 2 + — P - + -=r-+ i s : 7 5 7 6 A. 1/5 B. I/2I4. C. 5 A 8 D. 1/16 E. 9 A 8 7. The arithmetic average of the first n positive numbers is: A. _n_ B. nf C. n D. n - l E. n + 1 2 ^ ~ 2 ~ " ~ ~ 8. When simplified, the product. (1 - 1/3) (1 - l A ) (1 - l /5) (1 - l /n) becomes: A. _1__ B. __2_ C 2(n - 1) D. 2 n n n n(n + 1) E. 3 n(n + 1) 9. The sum of the first two positive odd integers is The sum of the first three positive odd integers is The sum of the first four positive odd integers is The sum of the first five positive odd integers is The sum of the first 100 positive odd integers is The sura of the fir s t n positive odd integers is 10. The diagram shows how an object would look as it falls toward the surface of three different planets. Find the pattern which seems to appear and predict how far the object will f a l l in 10 seconds. (t represents time and d represents distance.) Planet R Planet G Planet B © d = 0 © d = 0 §a = o & d = 3 0 d = 5 ® d = 16 '& d = 12 @ d = 20 <9 d = 6IL 0 d = 27 © d = I|5 0 d =ll|ii '© d = # d = 80 ®d =256 © d = — ~ $ d = _ 0 d = — -105 APPENDIX C RAW DATA Interpretation Student Number. The fir s t digit refers to the treat-ment group. The digit "1" designates expository treatment and "2" designates simple enumeration treatment. Sex. Each student is designated UM" for male or "F" for female. Term Mark. The mark assigned by the regular teacher in the term preceding the experimental unit. Pretest. Raw score on Lorge-Thorndike Nonverbal Battery, Form 1, Level H. Posttest. Raw score on Lorge-Thorndike Nonverbal Battery, Form 2, Level H. Mathematics Test. Raw score on Mathematics Content Test. Mathematics Transfer. Raw score on Mathematics Transfer Test. I. Q. A composite score which is the simple unweighted average of the Verbal and Nonverbal Batteries of the Lorge-Thorndike Intelligence Test, Form 1, Level H of the 196ii Multi-Level Edition. RAW DATA Student Term Pre- Post-Number Sex Mark test test Math. Math. Test Transfer I.Q, 15 10 121 16 k 118 17 k IO3 18 127 19 10 120 15 3 111 17 5 Ilk 18 5 115 15 8 122 17 p 121 22 k 111 20 o 109 13 9 119 13 5 111 22 9 122 18 7 120 21 6 122 18 10 111 19 2 112 I36 17 11 17 9 111 15 10 117 17 5 119 21 9 97 22 11 129 16 3 100 20 3 120 17 k IIIL 16 9 112 19 7 135 20 6 122 17 8 117 8 5 101 16 8 119 125 13 k 15 6 112 21 5 133 13 3 98 20 11 129 19 8 120 20 11 122 15 1 105 21 8 119 101 102 IO3 105 105 106 107 108 109 110 111 112 ^ Ilk 115 116 117 118 119 120 121 122 123 12k 125 126 127 128 129 I30 131 132 I33 134 1 3 ? I36 137 I38 139 240 llj.1 li+2 1^ 3 M 63 F 69 F 62 F 76 M 60 M 60 M 71 M 56 F 72 M 69 F 60 F 67 F 59 F *7 F 7k F 67 F 72 M 50 M 58 M 65 F 55 M F 55 F §3 M 81 M 57 F 72 F 62 M 58 F 83 F 79 M 68 M k8 F 58 M i+5 M 65 M 79 F 52 F 79 M 8k M 66 F 58 F 88 50 39 I k2 \ s 65 Student Term Pre- Post- Math* Math. Number Sex Mark test test Test Transfer I.Q 227 228 -229 230 231 232 233 235 235 236 2 3 I 238 2,39 2^0 2 1^ 24-2 iti 2 1 + 6 257 248 259 250 251 252 253 255 255 256 257 258 259 260 261 262 26 26 M P M P M M P P M M F M F M M F F M F M M M M M M M F F F M M M M M M M F M 71 50 63 72 58 85 67 75 3_ 78 85 90 86 75 92 81 85 29 67 90 68 58 58 77 79 72 78 78 55 65 60 65 90 £ 3 57 65 55 2 7 5 6 ? 59 P 39 55 51 51 50 62 ft 55 52 65 36 16 16 16 18 15 21 18 21 15 17 20 23 21 18 19 19 19 19 16 20 25 21 18 16 21 18 20 18 21 17 17 19 12 20 22 19 12 17 12 10 5 7 5 5 6 11 5 i5 9 9 10 10 13 12 7 6 \ 8 8 7 16 8 7 3 17 10 7 2 9 15 11 8 8 7 12} l2o 122 129 110 129 106 120 Il5 125 118 I38 112 126 113 129 I3O 117 n 2 115 130 115 113 111 125 125 X 23 113 139 l l 5 119 109 110 155 109 I36 123 l l 5
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The Effectiveness of simple enumeration as a strategy for discovery Leask, Isabel Campbell 1968
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Title | The Effectiveness of simple enumeration as a strategy for discovery |
Creator |
Leask, Isabel Campbell |
Publisher | University of British Columbia |
Date Issued | 1968 |
Description | Problem: This study is related to the controversy surrounding the relative merits of teaching by discovery and expository methods. Specifically, it investigated the effectiveness of treatment with simple enumeration as a strategy for discovery compared to treatment using an expository method. It was hypothesized that the two treatments would yield the same mathematical achievement, but the simple enumeration treatment would yield more mathematical and non-mathematical transfer effect than the expository treatment. Procedure: The subjects comprised six classes in Mathematics 12. They had been randomly assigned to classes at the beginning of the school year and three classes were assigned to each treatment group. All classes were taught a unit on arithmetic and geometric progressions by the experimenter. Equivalence of the groups was established in terms of the covariates I. Q. and previous term mark. The measuring instruments consisted of the Lorge-Thorndike Intelligence Test, Form 1, Level H of the 1964 Multi-Level Edition; a mathematical content test; and a mathematical transfer test. In addition, the Nonverbal Battery of Form 1 of the Lorge-Thorndike Test was used as a pretest to measure the ability of students to generalize and discover principles from examples. Form 2 was used as a posttest measure to determine whether any improvement in ability to generalize had occurred as a result of the experience with the unit on progressions. The generalized t-test was used to compare means of achievement on all tests. All results were analysed at the University of British Columbia Computing Centre. Conclusions: On the basis of results on the tests, the following conclusions were reached: 1. Treatment with simple enumeration yielded the same level of mathematical achievement as treatment with an expository method. 2. Treatment with simple enumeration yielded significantly greater effect on a mathematical transfer test than treatment with an expository method. An examination of I. Q. levels showed that the superiority in performance was largely located at the medium I. Q. level. 3. Treatment with simple enumeration was no more effective than treatment with an expository method when the criterion measured general transfer. Both groups showed significant improvement in ability to generalize after studying the unit on arithmetic and geometric progressions. The improvement was mainly located at the medium and low I. Q. levels and was independent of teaching method. The implication of this study is that if concern is centred on acquisition of facts, simple enumeration is no more effective than an expository teaching method. However, if there is concern for pupil participation and for training students to advance independently to related but more difficult material, then discovery-orientated lessons are advantageous. |
Subject |
Learning by discovery |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0104492 |
URI | http://hdl.handle.net/2429/36492 |
Degree |
Master of Arts - MA |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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