AN APPROACH TO TEACHING MATHEMAMAL iNDUCTION w AEOLESCEs’é-I BOYS Thesis for {he Degree of Ph. D. MICHEGAN STATE UNEVERSITY FRAME HGWARf} HéLDEBRAND 1968 -_AL LIBRARY Michigan State University TH E80. This is to certify that the thesis entitled AN APPROACH TO TEACHING MATHEMATICAL INDUCTION TO ADOLESCENT BOYS presented by Francis Howard Hildebrand has been accepted towards fulfillment of the requirements for W/fiflm/ Major profess; Date I [7 0-169 ABSTRACT AN APPROACH TO TEACHING-MATHEMATICAL INDUCTION TO ADOLESCENT BOYS By Francis Howard Hildebrand This study was primarily concerned with developing an approach and materials to promote the achievement of adolescent boys in mathematical induction. A second objective was to determine the relationship between selected response patterns of fifty seventh grade boys and their achievement in mathematical induction. The design of the study was a one group, pretest-- post test type. The two most important facets of this study were the sequence of problems in mathematical induction listed as variable x(l) and the scales of response patterns listed as variable x(i) , i = 2, 3, h, 5, 6, 7, 8 . The statistical design for the study was to answer basic questions of correlation and analysis of variance. A.model for student evaluation was used in an effort to standardize scoring procedures. Partial credit for each induction.problem was awarded on the basis of this model. The materials were designed for use in a relatively unstructured school environment covering 960 minutes of treat- ment. In such an environment scores of high achieving students were found to correlate significantly with three scales measuring reSponse patterns. Correlations with an .05 con- fidence level were found on two scales: One scale x(2) , measured the enthusiastic, easygoing, individual who partici- pates in group activities. A.second scale x(5) , measured the assertive, independent and aggressive individual who was spontaneous. .A correlation with an .01 confidence level which measured self confidence was found with a third scale x(7) . Such an individual has an adequate self image, and a high regard for his own worth. The materials developed included three basic orientations, algebra, geometry and approximation. In drawing conclusions many unanswered questions involving the indiscriminate use of physical models as a means of pro- moting conceptualization and problem solving in mathematical induction occurred. Students who were dependent on models, those who did not move rapidly from.the problem statement to representational forms or otherwise translate the problem into their own mode of expression, did not solve the problems. It was also clear that students need to be trained to collect and organize information as it is generated in the problem solving process. Those prdblems which were solved by very few students shared the quality of requiring extensive organization of information. AN APPROACH TO TEACHING MATHEMATICAL INDUCTION TO ADOLESCENT BOYS By Francis Howard Hildebrand A THESIS Submitted to Nfichigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Education 1968 G urfi'alal‘i . 0‘ L ~ ‘ t J {2" .ACKNOWLEDGMENTS For their help in carrying out the research on which this report is based, and for their numerous suggestions and wise counsel, I express my appreciation to my advisor, Professor Calhoun C. Collier and to Professors Harold Anderson, Jehn Mason, and JOhn wagner, the other members of my committee. To Mr. Russell Wblfe and his staff at Laingsburg Community School and to Mr. Paul Souder and his staff at Owosso Junior High School, I give Special thanks for their assistance in providing subjects and for cooperating with this research. Finally, I am directly indebted to my wife, Nancy, whose help, encouragement and patience have contributed greatly to my success on this study. ii TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS......................................... ii LIST OF TABLES.......................................... v LIST OF F‘IGImSOOOOO0.0.00000000000000000000000000000000 Vi CHAPTER I. INTRODUCTION.................................. 1 Need for the Study.......................... 1 Definition of Terms......................... 3 PhilOSOphy and Assumptions of the Study..... 5 Limitations of the Study.................... 9 Organization'of the Study 10 II. PATTERNS, PROBLEMS AND RELATED RESEARCH....... ll Statement of the Prdblem......... ..... ...... ll Pertinent Research.......................... ll Reaponse Pattern Scales..................... 18 Induction Material.......................... 21 Summary.........L........................... 26 III. PURPOSE AND PROCEDURE......................... 27 Selection of Subjects....................... 27 Design of the Study......................... 30 Daily Routine............................... 31 An Analysis of PrOblem 6.... ..... ........... 3h An Analysis of Prdblem 7.................... 36 An Analysis of An Unsolved Prdblem.......... 37 iii CHAPTER PAGE Summary..... ..... .... ............ ........... #0 IV. ANALYSIS OF DATA,............................. #2 Discussion.................................. 59 Summary..................................... 60 V. CONCLUSIONS AND IMPLICATIONS.................. 61 Conclusions................................. 61 ExplicationSOO0.00.00.00.00...OOOOOOOOOOO... 63 APPENDIX A. RESPONSE PATTERNS............................. 66 B. TESTS......................................... 86 C. PROBLEMS AND DAYFBY-DAY ROUTINE............... 92 D. SOLUTION INDEX TO ALL MATHEMATICAL MMTERIAL... 11% E. SUBJECTS SCORES SCALE-BY-SCALE................ 120 F. BELIOGRAPHYOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO... 123 iv LIST OF TABLES Scale Scores ...... .............................. A Scatter Plot.................................. Analysis of Variance............................ Simple Correlation Matrix....................... Unmatched F Statistics Between Means............ Coefficients for Least Square Fit xi on xj...... Coefficients for Least Square Fit xj on xi...... Standard Error of Estimate...................... Standard Error of Least Square Beta............. F Value for Least Square Beta................... PAGE 45 A7 51 ' 52 53 51+ 56 57 58 LIST OF FIGURES FIGURE _ PAGE 1. Approxj—mation Of AreaOOOOO0.00000000000000000000 1"" 2. mil-y SChedllleOO0......IO...OOOOOOOOOOOOOOOOOOOO 33 vi CHAPTER I INTRODUCTION Need for the Study Organization of the mathematics curriculum is a current problem faced by public schools. Some schools group children homogeneously according to their mathematical achievement. Dif- ferent curricula provide for very able, average and less able groups. Other schools emphasize flexible intraclass grouping. Materials to handle diverse groups vary from those devised by the University of Illinois Committee on School Mathematics (UICSM) employing the discovery approach1 to the more formal 'definition-- theorem--proof’ techniques provided by the School Mathematics Study Group (SMSG).2 In the past decade the United States government, through the National Science Foundation has demonstrated its concern with the state of mathematical learning; and in particular has lUniversity of Illinois Committee on School Mathematics, High School mathematics, (Revised edition; Urbana: University of Illinois Press, 1962). 2School Mathematics Study Group, Mathematics for Junior High School Volume I, Parts I and II (New Haven: Yale University Press, 19 l . contributed to projects developing text materials, to academic year and summer institutes, and to in-service programs for improvement of mathematics backgrounds for elementary, secondary, and college teachers. An identifying characteristic of projects such as UICSM and SMSG was that each was subject matter oriented. The primary concern of these professional projects was improving the subject matter of school mathematics. While mathematics educators and the United States government have shown concern for the organizational aspects of the school, the mathematical background of the teacher, and the develOpment of neW'materials, they have displayed little concern about the effects of new content or its organization upon the children involved. In short, it appears that school mathematics programs have developed in terms of adult mathemati- cian's goals. The researcher designed this study to prepare materials to teach induction and to examine student responses to the induction material. Further, motivation for the study grew out of the researcher's Observations of students in elementary mathematics who failed as defined by test scores. These students were observed to share certain characteristics or similar ways of responding. One such characteristic was the students” apparent lack of confidence in their efforts in mathematical study. A second characteristic was the students' apparent inclination to withdraw from class participating-~they participated less as their test records deteriorated. These characteristics were the result of personal observations; no supporting studies were found. Definition 9£_Terms Specific terms have Special meanings in this study as follows: Autonomy is a display of confidence and is implied by the student's positive reSponse to a question such as: Can you work well, without making mistakes, when you are watched? Independence involves the student's perceptions of con- straints arising in his interpersonal relations and his interaction with the environment. Independence is implied by the student’s positive response to a question such as: Do you completely understand what you read? Mathematical Induction is a study of sequences of particular patterns or cases of symbols which suggest a continuation from which a conjecture arises and may be generalized to a theorem. Mathematical induction as defined here is distinct from other forms of induction. For example, Baconian Induction as a logical method is a process of attaining general statements on the basis of observations, comparisons and experiments through intermediate generalizations and with regard for negative as well as positive 3 instances. 3W'e‘bsters Third New International Dictionary, unabridged, (Chicago: G and C Merriam Co., 1966), p. 160. Another example is scientific induction which is a method used in systematic pursuit of intersubjectively accessible know- ledge and involving as necessary conditions 1) the recognition and formulation of a problem, 2) the collection of data through observation and if possible experiment, 3) formulation of hypotheses, A) testing and confirmation of the hypotheses formulated.h Persistence is a display of sustained attention on the part of the student. It is implied by a student's positive response to a question such as: If someone turns on noisy music while you are trying to work, do you continue working? Pl§y_is the simulation of the student’s internalized visions. Play is the opportunity for the student to indulge in spontaneous activity with objects, ideas or peOple. If he deems an activity play, the student is highly responsive. Response Patterns are constellations of responses where a response is a change in the student associated with or correlated with a stimulus or set of stimuli. Self Image is a demonstration of the student's con- ceptualization of himself. This conceptualization is implied by examining the individual subject's response to a question such as: When you write about your personal thoughts, do you enjoy telling about yourself? hwebsters Third New International Dictionary, unabridged, (Chicago: G and C Merriam 00., 1966), p. 2033. Preference in_Problem Structure refers to the perceptual structure a student prefers in prdblem analysis._ An example of such preference is the student's free selection of one of three distinct structures for solving a given problem. The student may choose a geometric, algebraic or verbal approach to problem solution. Philosophy and Assumptions 9£_the Study Funded projects by the United States government such as SMSG and UICSM has shown lively interest in school mathematics curricula during the last decade. A predominating interest of most projects was the improvement of text materials and of teachers' competency in subject matter. Though this researcher agrees that subject matter is important, his first assumption is that organization of the materials and the response patterns of the individual students are important contributing factors in determining the level of student success in school mathematics. Important controlling factors which support the learning of mathematics remain essentially unknown, because the methods by which any type of mathematical learning takes place are relatively obscure. Henderson stated: The teacher can not depend upon any special type of lesson, such as "supervised study," to guarantee success in teaching and learning . . . . There is no decisive proof that any particular philosophy of teaching . . . will guarantee better results than any other method . . . . 5 In discussing the National Longitudinal Study of Mathematical Abilities, Dessart noted that: . . . Only through attempts to find interrela- tions among the many variables interacting in a learning situation can findings become more th cross sections of a learning situation . . . . A review of the literature indicates that few of the variables in learning mathematics have been detected, and neither interrelations nor significance of Observed phenomena have been sufficiently explored. This suggests that exploration is necessary in terms of identifying fundamental cognitive characteristics and ability expectations in the field of mathematics. .A second assumption of this study is that the child is basically an integrated unit in terms of his emotional structure and his physiology. Further, he tends toward individual autonomy in his cognitive development only if social interaction exists. The tendency toward autonomy in many children is assumed to be constrained by an existing set of reSponse patterns. If the response patterns are adequate for the potential intellectual outcomes of the organism, successful action is applied toward 5Kenneth B. Henderson, "Research on Teaching Secondary School Mathematics," Handbook of Research on Teaching, ed. N. L. Gage (Chicago: Rand McNally and Co., 15657? p. 1025. 6Donald J. Dessart, "Mathematics in Secondary Schools," Review 93 Educational Research, XXXIV (June, 1961;), 307. individual autonomy, particularly in intellectual matters. A third assumption of this study is that action is necessary for success in mathematics. Mathematics is something children do. Children are not passive about mathematics but rather become involved both intellectually and affectively. Another basic assumption is that achievement in induction implies both the existence and successful application of a model of intellectual activity. Failure to succeed is no assurance that the use of the model failed since the difficulty may be a matter of motivation. A final assumption is that mathematics learning is less isolated than many suspect. Such an assumption suggests that learning mathematics is not a simple matter of having a desire to sit down and do something. Desire may be a necessary but far from a sufficient condition for achievement. .A statement about the relationship between the researcher and his subjects seems in order. The experimenter most accurately describes the interaction between himself and the participating children as a form of play. The materials in mathematical in- duction extend to the youngsters an qpportunity to enter into spontaneous activity in abstraction. They are encouraged to explore the material by whatever route their intuition suggests. This approach opens the way for the student to become involved affectively as well as cognitively. Thus, the induction prdblems given the subjects, were, within limits, a form of play. The mathematical problem play involved activity and was limited only by the youngsters' imagination. Some students found such play extremely motivating. They seemed to want the same feeling of power over mathematics that they had over their possessions as children. Those students who fell captive to the prdblem form of play, lost track of time and no longer classed their efforts as work, since their activity supported developing affective components which generated satisfaction. Play is highly contagious to most children and con- sequently when participation does not occur there must be inhibiting factors. Some of the inhibiting factors might be described as a pathology and relate directly to response patterns not adequate for the child's organization of develop- ment. Seven response patterns were identified as part of this research. Their specific characteristics are listed in Chapter Two. From this phiIOSOphical orientation comes the under- lying purpose of this study. This study was primarily concerned with developing an approach and materials to promote achievement in mathematical induction by adolescent boys. A second objective was to determine the relationship between selected response patterns of fifty seventh-grade boys and their achievement in mathematical induction. For a highly complex intellectual activity, such as success in mathematical induction, appropriate affective components appear to exist in terms of specific response patterns. Limitations 9;: the Study This study was limited in several ways. No taxonomy of problem difficulty was available for sequencing materials in mathematical induction. There was no computer program available employing such tools as step-down F statistics for the item determination of the scales. One investigator can effectively observe only a small number of students in a class- room. The sample used for the study was not ideal. The selection of the sample was from schools which were willing to cooperate in a research study, rather than selection by a random sampling. The sample was limited to twenty-five male students in each of two classes. Preparatory learning, critical in any study, is still largely unknown. This area is in need of research.7 No informa- tion was available on the matter of individual preferences for prdblem structure and what role such a preference plays in strategy formation. The lack of information relating reSponse patterns and preference affinity constitutes a significant limitation of this research. 7Fred T. Tyler, "Issues Related to Readiness to Learn," The Sixty-third Yearbook gf_the National Society for the Stud gf_Education (Chicago: university of Chicago Press, I§6hl, p. 23 . 8The Research and Development Center at Stanford University is currently conducting an investigation in this area. 10 Organization gf_the Study This research study is organized into five chapters. Chapter One, the Introduction, cites the need for the study, definition of terms, and the philosophy and assumptions on which the study was based. Limitations of the study and organization of the research report are also included. Chapter Two, Patterns, Prdblems and Related Research, contains a careful delineation of the hypotheses and a survey of other research related to the study. The structure of the induction materials and descriptive characteristics of the response patterns are also a part of this chapter. Chapter Three, Purpose and Procedures, reports the design of the study and how the subjects were selected, ex- plains the daily routine and gives student reactions to selected prdblems. Chapter Four, Analyses and Results, describes the results of the given sequence of induction material, correla— tion ratios and related statistics between individual and collective reSponse patterns and achievement in induction. The results of the analyses of data complete this chapter. The last'chapter, Conclusions and Implications, includes the summary and suggestions for further research. CHAPTER II PATTERNS, PROBLEMS AND RELATED RESEARCH Statement of the Prdblem This study was primarily concerned with developing an approach and materials to promote the achievement of adolescent boys in mathematical induction. ,A second objective was to determine the relationship between selected response patterns of fifty seventh-grade boys and their achievement in mathematical induction. The hypotheses to be tested is whether or not sig- nificant differences exist for correlation between achievement scales and selected response pattern scales. Specifically the null hypotheses is: Between scale x(l) and each scale x(i) , i = 2, 3, h, 5, 6, 7, 8 the correlation is not significant. Pertinent Research The research literature provided data pertinent to the mathematical aspects, identification of responses, and selection of the sample used in this study. Hadamard con- sidered questions of the affective nature of students and their success in the field of mathematics. In a personal reference, Hadamard stated that he knew that powerful emotions ll 12 favored different kinds of mental creation.l Such kinds of creation are in large measure related through emotion to selective attention. Hadamard seemed to recognize emotion as potential influence in invention in mathematics but he suggested no measures or materials to determine Specific affective re- sponses. Piaget seemed to agree with Hadamard's position about affective influence when he stated that: . . . mathematical concepts are not derived from the materials themselves, but from an appreciation of the significance of the Operations performed with the materials.2 3 Poincare saw success in invention and problem solving as the result of prior unconscious work that was influenced by affective characteristics. He hypothesized that individuals involved in invention are so emotional in their commitments that fragments remain in their subconscious even when conscious effort is suspended. In his study "Personality Factors and Success in Mathematics." KOchnower" found that his sample leaned toward 1Jacques Hadamard, The Psychology gf_Invention in_the Mathematical Field (Princeton: Princeton University Press, 19HS), p. 10. 2K. Lovell, The Growth of Basic Mathematical and Scientific Conce ts in Children-(London: University of London Press, Ltd., 156E}, p. 3. 3Hadamard, op, cit., p. 19. MW. KOchnower, "Personality Factors and Success in Mathematics," ig Points, XLIII (April, 1961), 21. 13 the Hamlet factor.5 Kechnower's Subjects were highly self- sufficient and individualistic, tended to be loners, displayed obsessive and compulsive behavior and a tendency to react emotionally and excitably. In Kbchnower’s study the achievers had personalities which rejected imposed cultural demands. KOchnower concluded that a potential profile of a successful mathematics student shows that such a student is sensitive, insecure, introspective, and tends to avoid group activities. Certain response patterns and spatial ability seem to be inextricably related to success in mathematics. Both Piaget and Lovell found that the child's thinking is largely defined by his perceptions, particularly those formed in the very early stages of school.6 Lovell also argued that there is a taxonomy for analogies relative to understanding. The prOblem of defining such a taxonomy includes distinguishing between perceptual space and representational space. Discrimination between a circle and a triangle is an example of a perceptual space. An example of ,representational space is the mental representation of a circle and a triangle without the aid of a drawing or model. Intuitive transformations according to Lovell are critical to the formation of representational Space from perceptual space. 5The Hamlet factor is characterized by obstructive, reflective, and individualistic behavior. 1h Consider the following problem: \J Approximation g£_Area Figure l The student is asked to find the area of the above bounded region. Lovell described the necessity of a transforma- tion in these words: If the pupil has not sufficient mental manoeuvrability to be able to deal with the 'bits and pieces of squares' in his mind . . . there is little one can do to get him to understand. Related to the need for intuitive transformations and perceptual orientation is the difficulty of converting a given perception into symbolic form for effective mathematical analysis and production. Common physical models and words must eventually be ruled out. The descriptive forms most often successfully substituted are vague internalized images. Such images symbolize the ideas without imposing irrelevant conditions. Quoting from Les Definitions dans l‘Enseignement in Science §t_Methode, Poincare stated that: 7Ibid., p. 119. 15 Almost all . . . wish to know not merely whether all the syllogisms of a demonstration are correct, but why they link together . . . they are not conscious of what they crave, and if they do not get satisfagtion they vaguely feel that something is lacking. Hadamard asserted that every mathematical research compelled him to build a schema necessarily of a vague character so that it was not deceptive.9 Engle seemed to support Hadamard’s views on vagueness in the individual's production of new ideas. She declared that: . . . limiting scientific language to terms that have achieved definition may well cut off a rich source of future knowings. Though these and other researchers commented on vague internalized models, no support for the use of physical models or their impor- tance in determining student success in the perception of necessary mathematical transformations was found. In his doctoral dissertation, Schunert indicated that differential assignments were significant in mathematics achievement. He defined achievement from an analysis of the youngsters responses over a given totality of work rather than 8Hadamard, _p, cit., pp. th-S. 9Ibid., p. 77. 10Mary Engel, "Thesis--Antithesis," American Psychologist, XXI (August, 1966), 786. 16 an analysis of a very restricted range of problems. His study indicated a need for information on related factors of differ- ential assignments and, by implication, aims specifically at factors for learning.11 Another factor that influences success in mathematics is the problem of sex differences. Sex difference in problem solving appears early in development and under different guises. Miltonl2 found that males adOpt an analytic attitude toward natural events and toward formal prOblems. MicDavid,l3 and Crandall and RObsonl)+ found that boys appear to be more analytic, more independent and more persistent than girls in problem situations. This difference increases with time, and llJim.Schunert, "The Association of Mathematical Achievement with Certain Factors Resident in the Teacher, in the Teaching, in the Pupil and in the School," Journal g§_ Emerimental Educationg, XIX (1951), p. 237. 12G. A, Milton, "Five Studies of the Relation Between Sex Role Identification and Achievement in PrOblem Solving," cited by Martin L. and Lois W} Hoffman, eds., Review'2£_Child Development Research (New Ybrk: Russell Sage Foundation, 156M), P- 157- l3J. W; McDavid, "Imitative Behavior in Preschool Children," cited by Martin L. and Lois W. Hoffman, eds., Review of Child Development Research (New Ybrk: Russell Sage Foundation, 196A), p. 157. ill-V. J. Crandall and A. Robson, "Children’s Repetition Choices in an Intellectual Achievement Situation Following Success and Failure," cited by Martin L. and Lois W. Hoffman, eds., Review of Child Development Research (New YOrk: Russell Sage FbundatiOH, 196A), p. 162. 17 by late adolescence and adulthood, the typical female feels inadequate in problems requiring analytical reasoning. Sweeney'sls results confirmed this when he found that differences in prOblem solving can be demonstrated for groups of men and women matched with respect to intelligence. ‘Witkinl6 also found a marked difference between boys and girls in modes of perception, with girls being the more dependent on visual framework. To take into account known factors supported by others' research, the subjects for this study of relationships between response patterns and mathematical achievement were deliberately limited to a sample of boys. Though certain response patterns such as persistence, independence, autonomy and self image were linked to mathematical success by various researchers, no available evaluative instru- ments, in toto, were deemed suitable for measuring the desired response patterns. Therefore, modifications of existing instru- 17 ments were made with the aid of a testing collaborator. Jerome Bruner has examined the question of induction and one point in 15Edward J. Sweeney, "Sex Differences in PrOblem Solving" (unpublished Ph.D. dissertation, Department of Psychology, Stanford University, 1965), p. 57. l6H..A. Witkin, "The Nature and Importance of Individual Differences in Perception," The JOurnal 9; Personality, XVIII (December, 1916), 162. 17Donna Palonen, Department of Psychology, Michigan State University. 18 particular is important for this study. The child is postulated as passing through three stages of development. The earliest stage is the enactive level where the child manipulates objects. The second stage is the iconic l§y§l_where the child deals with mental representation of the physical objects. The third stage is the symbolic level where the child is dealing in mental images.18 ReSponse Pattern Scales Personality scales which have been published since 1955, and which were commercially available from western Psychological Service, Beverly Hills, California, were examined. ane of the tests seemed to identify the Specific response patterns sought, but the Junior Senior High School Personality Questionnaire (HSPQ)19 was satisfactory in some respects. HSPQ scales were purchased and then modified, both in Specific wording and in scoring, to more closely identify the desired response patterns. The resulting scales are identified as x(i) , i = 2, 3, A, 5, 6 .20 The choice of symbols to identify these specific scales was decided upon since, at this time, no precise words were appropriate 18"Improving Mathematics Education for Elementary Teachers," W. R. HOusten, ed., Michigan State University, sponsored by Science and Mathematics Teaching Center, p. 26. l9HSPQ,Form A, Second Edition (1963) was purchased from The Institute for Personality and Ability Testing, Champaign, Illinois. 2OIndividual scales identifying reSponse patterns can be found in appendix A. 19 to identify the variable concerned. All interpretations or labels relating to items on the scales used were prepared by the re- searcher. The author assumes scale reliability and validity for purposes of this study. The following descriptive terms used to help the reader characterize response patterns are those of the author of this research. The items for scale x(2) were chosen to identify a reSponse pattern which was characterized by a happy, enthusiastic, easygoing, warmhearted extrovert, who enjoys participating in group activities. Items in scale x(3) were chosen to identify a response pattern characterized by a calm, secure, self-assured, emotionally stable individual, who faces reality. The twenty items in scale x(h) were chosen to identify a response pattern characterized by a decisive, resourceful, self-sufficient individual, who is relaxed and tolerant rather than aggressive and competitive. Scale x(S) was characterized by an assertive, independent, aggressive, venturesome individual, who is uninhibited and reacts spontaneously. The x(6) scale was characterized by an individual who disregards rules, and follows his own urges. He tends to be un- dependable and has a casual, careless attitude. This person would tend to be impulsive. Unlike scales x(2) to x(6), scale x(T) was a selection of incomplete sentences designed to identify the 20 individual who tended to be confident, who had a positive self image, and who had a high regard for his own worth. The scale and scoring for response x(7) were modified from The Forer Structured Sentence Completion Test.21 Scale x(8), the DraWeA-Person Test without modifica- tions, was used to test whether or not perceptual and motor factors of coordination were related to the individual's achievement in induction. The scales x(2) through x(6) were administered to the twenty-five subjects in each section, as a group. The subjects responded to scales x(7) and x(8) at the second sitting, on an individual basis. No time limits were set so that the subjects felt no overt pressure to hurry. Each of the five scales x(2) through x(6) consisted of twenty items, with each item having three possible responses. The students' responses to each item were scored 0, l or 2 according to the numerical weight assigned that response.22 Scale x(7) contained 15 items for completion scored 0, l or 2 . Scale x(8), identified by the Draw-A—Person Test, which was not modified, was scored according to Bodwin and 23 Bruck's procedures. 21Bertram R. Fbrer, The Forer Structured Sentence Comple- tion Test, (Beverly Hills: western Psychological Services, 1957). 22See Appendix A for the unique integer assigned each scale item by this researcher. 23R. F. Bodwin and M. Bruck, "The Adaptation and Validation of the Draw—A-Figure Test as a Measure of Self Concept," (Mimeo- graphed). 21 Each subject's score on a given scale was the sum of the integers resulting from his selected responses. This range, from O to A0 for each of the six scales x(2) - x(7) , scattered the individuals responses on a continuum. The higher scores indicated the more consistent patterns. Induction Material A wide variety of mathematical induction problems was developed for assignments. For purposes of this study, mathematical induction was restricted to three primary areas of conceptualization. One conceptualization was basically algebraic and consequently did not fit a simple physical model. PrOblem.h, given to the students at the fourth meeting, was of this type. It should be added that the subjects who became involved in the induction process rapidly oriented themselves to perceive what was intended in each new problem. After prOblems were handed out, the only help provided either as to approach or procedure was in answer to questions or in discussion after all those who wished to, had completed or given up on the problem. Though a casual reader may ponder the intent of such prOblems as number A, given below, the participating students were apparently seldom confused about the intent of the problem. 22 Problem .LL 1 = O + 1 2 + 3 + A = 1 + 8 5 + 6 + 7 + 8 + 9 = 8 + 27 10 + 11 + 12 + 13 + 1H + 15 + 16 = 27 + 6h If this process is continued what will the two numbers be on the right of the equal Sign? Can you find a way to write the solution for any given line? A second conceptualization was basically geometric and was interpretable through physical models, although the orienta- tion did not demand that physical models be used. Some students applied representational Space in the form of pencil and paper configurations as opposed to using either physical models or computational forms for orientation to geometric concepts. Problem 7, handed out at the fifth class meeting, was of such a type. Problem I_ Ybu are to assume that the cube shown on the following page has been immersed in paint and is completely covered. If the cube is then cut through each plane, as indicated by the dashed lines, the question is: In each case how'many of the resulting small cubes will be painted on how many faces? For the general case fill in the table below: 23 (a) Given n cuts there are cubes with 6 painted faces? (b) Given n cuts there are cubes with 5 painted faces? (c) Given n cuts there are cubes with A painted faces? (d) Given n cuts there are cubes with 3 painted faces? (e) Given n cuts there are cubes with 2 painted faces? (f) Given n cuts there are cubes with 1 painted face? (g) Given n cuts there are cubes with O painted faces? Paysical Model for PrOblem Z. -—-D’——— l l t, A r— . l _;1:;_:-. I ‘ ' ‘I' ' I unn——-l-——- —|— — -aVl/ I l i II; I | .l’ ._-_.L___I--_./I’l ' I I ' I __, l I — — C )— I' t i : i |':. I F __j-—P—r—-VH l I ' lrl. .I I -' V' .l- -l- r -- ILA I ' ' ,"H --J-4--r--l l I g I I I 2h All students who successfully completed Problem 7 worked out specific cases for n = l, 2, 3 and A before they solved the more general case or the nth case. A third conceptualization appealed to a dimension distinct from either algebra or geometry. This conceptualiza- tion involved application of the principle of approximation and extensions of approximation. Experience with induction materials which require methods of approximation are one approach to creating a need for mathematical proof. A student, confronted with the problem 1/20 + 1/21 + 1/22 + 1/23 + . . . + 1/2n = who, through analysis of a number line model . 315.2533. 0' 1 7* '2 collects successively the partial sums: 0 s1 = 1/2 s2 = 1/20 + 1/21 53 = 1/20 + 1/21 + 1/22 and extends the procedure until he believes intuitively that 1/20 + 1/21 + 1/22 + 1/23 + . . . + 1/2n s; 2 has established a real need for mathematical proof. An immediate physical model which fits this question is displayed by the following problem. 25 Problem _l_O Part (a) A ball is dropped from an original height of 100 feet. Each time it hits the ground it bounces back one half the previous fall. How far does the ball fall? In order to determine the students' backgrounds for dealing with inductive material involving the three previously discussed conceptualizations, an eight item Pre-test was pre- pared and administered by this researcher.2h The subjects had no time limit in which to complete this test. They received one point for each correct answer. Over a period of six weeks, 25 the researcher presented a variety of induction problems for the subjects to study and solve. At the end of the second and fourth weeks, the subjects responded to forty-minute quizzes.26 Quiz I, containing four items, had a total possible score of [four points. Quiz scores were purposely weighted to fit cumulative points on other material. Partial credit was awarded. Quiz II, also contained four items but had a possible score of six points. These quizzes sampled the induction ideas covered in the previous two weeks' work and supplied additional data on the subjects' gain in inductive skills. 2”The Pre-test is included in Appendix B. 25See PrOblems in Appendix C. 26See Quiz I and Quiz II in Appendix B. 26 The Post Test was designed to evaluate the gain Shown on problems parallel to, if not identical to, items on the Pre- test. The gain was 61 points as indicated in Table 1, page A7, of Chapter IV. It should be noted that no item was missed by everyone on the Post Test yet no prOblem was solved by more than 12 students. Summapy In Chapter II, literature pertinent to the study was presented and discussed. Both psychological and mathematical aspects involved in learning induction were included. The experimental studies included were chosen in an effort to clarify some obstacles to learning. Conceptualizations of the induction material were given with examples from the prOblems. The complete stock of induction problems given to the students are included in Appendix C. ,A description of the administration and scoring of tests and quizzes were also presented in this chapter. Though the scales used to identify response patterns and the format for scoring student responses were briefly described in this chapter, the complete set of these scales are included in Appendix A. CHAPTER III PURPOSE AND PROCEDURE Selection pf Subjects The selection of subjects for this study came from two schools. One school was the Laingsburg Community School, the other, the Owosso Junior High School. The schools made modifica- tions to meet the research needs in terms of changing rooms, releasing time for students and disseminating information to parents and other interested individuals in the community. Originally both schools released their total seventh grade pOpulation for this study. The total number of male students in the seventh grade in Lainngurg was 29. Because of anticipated attrition, all the seventh grade boys from the Laingsburg School system were invited to participate. In Owosso all the available seventh graders, 78 boys and girls, were invited to participate. From these, twenty-five males were randomly selected to be the subjects for the study. The subjects were told that the experimental study would not affect their permanent records as no grades would be recorded. Parental consent was obtained for each participant. From the original fifty-four males selected, 16 in Laingsburg and 2A in Owosso completed the study. Various 27 28 logistical problems contributed to the elimination of some subjects from the experimental program. For example, some boys missed a psychological examination because of sickness. If they missed such a test and also the makeup test, the students were dropped from the study because of missing data. Certain other students simply failed to come to class. The Laingsburg students were generally from two types of homes. Some lived on subsistence-type farms. Others came from homes where their fathers were skilled or semi-Skilled workers of median income in the Lansing industrial complex. In general, the Laingsburg boys displayed poor orienta- tion in operating within an evaluation model. A.typical Laingsburg boy had not developed a level of s0phistication sufficient to generate complex conceptualizations. The average subject in this category found little excitement in prOblems involving mathematical induction and unhesitatingly remarked, upon inquiry, that everything at school was boring and worthless. To them intellectual matters were of little concern. When further questioned about current aspects of life they found exciting, the subjects were not uniform in their feelings. Even in discussions of mechanical concepts such as might be employed in repairing or designing automobiles, airplanes, missiles and Space flights, most of them did not seem particularly enthusiastic. Few of the students could recall facts about current space activity. Individual boys could not recall names of astronauts except JOhn Glenn and none were able to associate details of 29 achievement more Specific than "space" with his name. As a group, the Laingsburg boys had little perception of details in life about them. The overwhelming majority of this group considered reading a chore. School for them was a place of drudgery and work-~of listening and doing meaningless things. Observation of the youngsters indicated they had average attention spans of ten minutes for mental activities. During problem.sessions, the subjects who chose not to work used the time for social interaction with other youngsters. One could infer that these subjects could not get enough time just to be "listened to". Later, in their own evaluations of this workless period of time the subjects stated that they had been involved in intensely difficult effort. The majority of students from.the Owosso School were primarily from low or medium income families that lived in city dwellings with parents and several siblings. Though they came from dissimilar backgrounds, a majority of the subjects from Owosso responded to intellectual activities in much the same fashion as did the boys from Laingsburg. The question of what the Owosso boys considered play was also interesting. .A large, disinterested group did not involve themselves in inductive type play. They did not consider reading a pleasurable activity. Topics in mathematical induction did not interest the majority of this group. Each subject who showed disinterest was also noted to be very weak in steps one and two according to the 30 model used for evaluating achievement.1 The larger interested group was at Owosso. Desigi 93 the Study The basic design of this study is a one group, pre- test-~post test type. The two most important facets of this study were the sequence of problems in mathematical induction listed as variable x(l) and the scales of response patterns listed as variable x(i), i = 2, 3, II, 5, 6, 7, 8. The problems selected for the mathematical portion of the study were based on prOblems previously used with students of various abilities and at various school levels. These problems were selected with the knowledge that a study of induction is one area of mathematical learning in which a student succeeds according to his own efforts alone. The mathematical induction prOblems were framed in terms of anticipated seventh graders' levels of SOphistication. The programming was carried out around a total of 960 minutes of contact between the experimenter and the subjects. The basic dayéby-day mechanics for distributing and retrieving materials are indicated in Table 1. The schedule for scaling response patterns related to this research was organized by the school counselor for each school. The test 1See the evaluation model on page ME. 31 collaborator, Miss Donna Palonen, a Ph.D. candidate in psychology, scaled the response patterns of the students during six weeks of the period set aside for mathematics instruction, on days when no instruction in induction was scheduled. Scales x(2) through x(6) were administered to groups, while scales x(7) and x(8) were administered to individuals. Each scale was hand scored, in one Sitting. The subjects' final reSponse scores on each scale were not submitted to the experimenter until all the subjects' achieve- ment scores in induction had been tallied at the end of the study. All data were then punched on a data deck with a card for each subject and submitted for an analysis of variance routine at the computation center. Daily Routine The basic daily research schedule for each school was a function of the necessary demands imposed by the school's routine. Laingsburg was on a class schedule which readily allowed the experimenter to meet class three mornings a week for fifty minutes, or one-hundred fifty minutes per week, during a period of six weeks and two days. (One meeting was lost due to Easter holidays.) The Owosso school employed an alternating schedule of A and B days in such a fashion that when Monday was an A day this week it would be a B day the following week. Hence, when the experimental group met on Monday, wednesday, Friday one week, 32 they met on Tuesday and Thursday the following week. In addition a three day week (a Monday, wednesday, Friday week) was lost at Easter. The class in Owosso met twenty contact class periods covering nine weeks. In a letter to the subjects‘ parents, the school adminis- tration invited the subjects to participate in an experiment in mathematics learning. The initial class meeting consisted of orientation and collection of the permission slips which had been signed by the parents. At this same meeting, the experimenter informed the subjects about the anticipated routine which included the sequence of events listed in Figure 2. The experimenter was the only adult involved with the subjects during the instruction in induction. A typical class period consisted of the following activities: During the first portion of the period students turned in written work from the previous meeting and asked questions about Specific problems. Those boys who had no material to hand in or who were without questions were urged to re-examine portions of the problems they had considered previously. While the boys handed in all the previously introduced material, the experimenter distributed a new set of materials. .A discussion followed each distribution of new materials and the subjects' questions centered on patterns and structures pertinent to the new problem. Questions were handled in two ways. In some instances, the questioning involved an individual and in other instances, small groups of subjects. Questions concerning problem solutions 33 Daily Schedule Figure 2 mOH>Hom anooam our mnOHpmosd Pme PmOnH HH ease H name mm-sm omumm sm-mm mm Hm ONImH wHINHImHImH :HImH NH HHIOH mnwub 6A-: mImIH meoapoam pmopnmhm Soapwpoowao weapon: 3.4 56789 lO 13 1h 15 16 17 18 19 2o 3A were not answered directly. subjects were led to consider either the formation of specific and finite patterns, as in Problem A cited on page 22, from which they could organize a structured solution of strategies for testing their intuitive ideas for correctness. As time passed, the work-question sessions involving groups took up most of the class period. An additional aspect of the study reported here is best identified by analyzing problems. It appears that problem solution is related to cognitive preference in some cases, and training as well as preference in others. Three problems are analyzed in some detail here to exemplify such styles. PrOblem 6 may be viewed as basically algebraic--its solution depends on recognition and manipulation of symbolism. PrOblem 7 may be viewed as basically geometric and depends primarily on Spatial understanding. PrOblem 13 may be viewed as a multi-step transformation problem. .A solution requires transformations of words into appropriate symbolic form. If this is prOperly completed, the solution requires a student to collect sufficient detailed information. Ap_Apalysis 9£_PrOb1em 6_ Consider Problem 6: 2 + h + 6 + . . . + 999,996 + 999,998 + 1,000,000 = ? This prOblem was particularly interesting for several reasons. One reason was that the subjects found it difficult to relate it to perceptual space in a meaningful fashion. The students 35 various approaches using representational space were not sufficiently discriminating to evolve a solution. Mbst of the subjects did not understand why there would be only 250,000 terms left if they added the first term to the last, the second to the next to the last, the third to the third from the last and continued to collapse the terms in this manner until the matchings were complete.2 The computational aSpects of handling series seemed to dominate the subjects” thinking, so that many of them summed only the six obvious terms, forgetting the meaning of the ellipses for the moment. When the subjects were asked the meaning of the three dots in PrOblem 6 they answered that these dots represented "missing numerals". When asked to Show how they had considered the missing numerals in their computation, most would point to the six given numerals and then, at last, recognize that they had omitted the numerals denoted by the dots. Even then some subjects did not recognize that they had omitted most of the numbers in the series. Those subjects who failed to solve problems such as PrOblem 6 were extremely weak in terms of recall which is pre- requisite to analyzing problems. In other words, for them, the necessary components of step one of the evaluation model related in Chapter IV were missing. A typical subject in this 2J. J. Gibson and E. J. Gibson, "Perceptual Learning," The Psychological Review, LXII (January, 1955), 1+0. 36 category could not compute accurately using two-digit multipli- cation. Such subjects did not seem to have a particular aversion for multiplication as a process, they simply did not recognize that the principles of multiplication were valuable tools to be used in prOblem situations. The students who failed to solve this series problem could not extend the intuitive suggestions imposed by the representational symbols. In a sense, a conceptualization such as an intuitive imposition required a second order abstraction, since the representational objective, in this instance, was a sequence of names for ideas. The three dots were a symbol, representing a sequence of other missing symbols, themselves representative of ideas. A_I_1_ Analysis 93 Problem 7 Problem 7, simply stated was this: .A cube is completely covered with paint and cut by orthogonal planes in standard position, parallel to the faces, thus reducing the given cube into eight smaller, regular cubes. Each of the smaller cubes will have paint on some of the surfaces. The problem is to identify how many faces of each of the small cubes are painted. Extend the procedure to answer the same question for two planes regularly spaced through each dimension, then three and finally for 2. planes. (See the model on page 23 for an illustration of this problem. ) 37 Observation of the subjects' cognitive preference in attacking PrOblem 7 hints at how profound the initial approach may be in mathematical learning. In this study, subjects who played too long with a wooden model apparently penalized them- selves. All those who built more than one distinct example of the larger cube by using small wooden cubes, failed to solve the problem, while the successful students neither played with the wooden model more than a few minutes nor reconstructed a larger cube more than once. All successful students drew pictures that showed fewer than five cuts, and then spent considerable time studying their drawings. Those students who could not construct a three-dimensional perspective of the cube on a piece of paper failed to solve the problem. Each student who solved some portion of the prOblem, constructed at least one perspective drawing. Successful sub- jects usually controlled computational demands but depended heavily on intuitive direction to complete their problems. Ap_Ana1ysiS f A2 Unsolved PrOblem PrOblem l3. PrOblem l3, given out at the eighth class meeting went unsolved by both subject groups. The original prOblem follows: Agent 007 has infiltrated the Red Chinese Department of State as a spy for British intelligence. Mao has decided it is time for a purge of his Department of State. Mao will handle the purge in the following way. He will have all members of the Department of State brought into 38 a room. There will be p_ of them and they will be told to sit in a circle. Then Mao's executioner called Chopper will come through and eliminate every other one around the circle until only one remains. Mao wishes to preserve the brightest member and assumes this will determine the brightest. Suppose you are 007 and you would know what the number p_ would be just ten minutes before you entered the room. Where should you Sit in order to be the lone survivor? Try some samples if you wish. The following sketch may be helpful. In this prOblem the subjects had to determine a scheme whereby they could select the position in a ring or circle of Objects that would remain unchanged if someone began at a given initial point and proceeded to eliminate alternate objects continuously around the circle or ring until only one object remained. Only one subject earned partial credit for this problem and his credit was for an incomplete solution. A detailed inspection of subject responses suggested that a contributing reason for failure rested on the subjects’ lack of formal training in data classification and analysis. The subjects gave the additional impression that success on PrOblem l3 depended on the use of powerful cues from the environment, accommodated by their intuition. If they generated incomplete 39 patterns, either their intuition failed to function or they failed to recognize a critical step in the analysis. An examination of the partial solution submitted indicated that the subject compiled data to n mm mmmandm mo 25m m.aa moamoa o.oa ommmOm o.o memmm H.m mamw m.m Hmam m.m Hmmma o.m oars. m.m mmmwa a .m momma m.m ommm onamHsmn mamaoom shaman scam muz_ho mHmNA.o NO.H mm.o mm.o 00.0 m: .0 $0 0m .0 0m.0 NH.0 00.0 m mum.m 0H mnm c>h4n> CDFJRJ Ol—‘m 71 continued: 7. a. b. 10. a. b. 12. a. b. 13. a. b. Do you often make big plans and get excited only to find they won’t work? yes, sometimes, no. Are you satisfied that you come up to what others expect of you? yes: usually, no. Do you sometimes feel happy and sometimes feel depressed without real reason? yes, uncertain, no. If someone asks you to do a new and difficult job, do you: feel glad and show them how well you do, in between, feel you will make a mess of the job? Can you always tell what your real feelings are, for instance whether or not you are tired or just bored? yes, perhaps, no. Do you sometimes feel you are not much good, and that you never do anything worthwhile? yes, perhaps, no. Do you feel that you are getting along well, and that you do everything that is expected of you? yes: perhaps, no. 72 x(3) continued: SCORE lh. Do you find yourself humming tunes if someone else started them? 0 a. yes, 1 b. perhaps, 2 c. no. 15. Which of these changes in school would you rather vote for: 2 a. putting sl w people in classes together, 1 b. uncertain, 0 c. doing away with punishment? 16. When things are going wonderfully, do you: O a. actually almost "jump for joy", 1 b. uncertain, 2 c. feel good inside, but remain calm? 17. When you wait in line, do you often: 2 a. wait patiently, l b. undertain, O c. fidget and think of leaving. 18. Do you wish you could learn to be more carefree and light-hearted about school work? 0 a. yes, 1 b. perhaps, 2 e. no. 19. Do you find it easy to go up and introduce yourself to an important person? 2 a. yes, 1 b. perhaps, 0 c. no. 20. Do your feelings get so bottled up that you feel you could'burst? O a. often, 1 b. sometimes, 2 c. seldom. 73 Response Pattern xghz The x(h) variable was determined by scoring the student's responses to the following questionnaire in the fashion indicated. Forty points were the maximum on this questionnaire. SCORE l. [DI—'0 0‘ OI—‘N ?' [DI—'0 0" Ol—‘m O" a. b. c. Oi—‘m 6. Do you think there is a fair chance that you will be a well-known, pOpular figure when you grow up? yes, perhaps, no. When you are given higher grades than you usually make, do you feel that the teacher might have made a mistake? yes: perhaps, no. In first grade, did you always go to school without your mother's having to force you? yBS, perhaps, no. Do you tend to be quiet when out with a group? yes, sometimes, no. When a new fad starts, say in dress or manner of speaking, do you: start early and follow it, uncertain, wait and watch before following? Wbuld you rather be: a builder of bridges, uncertain, a member of a traveling circus? X(1+) SCORE [DI—’0 Ol-‘m “DI-'0 OI-‘I'D OHM 71+ continued: 7. Are you, like a lot of peOple, slightly afraid of lightning? a. yes, b. perhaps, c. no. 8. Do you think that the average committee of your a. b. c. 10. a. b. 0. 11. a. b. 12. a. b. 13. a. b. classmates takes too much time and does poorer work than you alone could do? yES, perhaps, no. Which kind of friends do you like-~those who: "horseplay", uncertain, are serious all the time? If you were not a human being, would you rather be: an eagle, uncertain, a seal in a colony by the sea? Do you think that life has been happier and more satisfying for you than for others? yes, perhaps, no. Do you have trouble remembering someone’s joke in order to repeat it? yes, sometimes, no. If someone puts on noisy music while you are trying to work, do you continue your work? yes, perhaps, no. X(1+) 75 continued: SCORE 1h. 0 l 2 [DI-‘0 Ol—‘m own) Oi—‘m [DI-’0 a. b. c. 15. a. b. c. 16. a. b. c. 17. a. b. 18. a. b. 19. a. b. c. 20. a. b. WOuld you rather Spend time and money on: a popular dance, uncertain, a book on earning more money. Do you feel that most of your wants are reasonably well satisfied? yes: perhaps, no. When you read an adventure story, do you: get bothered about whether it is going to end happily, uncertain, just enjoy the story as it goes along? When you do badly in an important game, do you: say-~its just a game, uncertain, get angry? When you are walking in a quiet street in the dark, do you often get the feeling you are being followed? yes: perhaps, no. When someone is disagreeing with you, do you: let him have his way, uncertain, tend to interrupt before he finishes? Do small troubles, sometimes which are really unimportant, get on your nerves? yes: perhaps, no. 76 ReSponse Pattern xfi5) The x(5) variable was determined by scoring the student's responses to the following questionnaire in the fashion indicated. Forty points were the maximum on this questionnaire. SCORE l. a. b. c. I'Df-‘O Ol—‘m P’ a. b. c. [\DI-‘O a. b. c. OHM If a friend's ideas differ from yours, do you keep still to maintain good feelings? yes, sometimes, no. Do you laugh with your friends more in class than other peOple do? yes, perhaps, no. When you finish school, will you prefer to: do something that will make people like you, uncertain, make a lot of money? Have you told your parents that some teachers are too old-fashioned to understand you and your friends? yeS, perhaps, no. Do you prefer having teachers tell you how things should be done? yes, perhaps, no. In a trip with naturalists, would you find it more fun to: catch birds and preserve them, uncertain, make artistic photos and paintings? lll.|. Iqlu. . x(5) SCORE NI-‘O NI—‘O [\Dl-‘O OHM [DI-'0 77 continued: 7. a. b. c. a. b. c. a. C. 12. a. b. C. l3. a. b. c. If you accidentally say something odd in public, do you remain uncomfortable long and find it difficult to forget? yes: perhaps, no. Are you known among your friends for going "for broke" for things you like? yes, perhaps, no. In school would you rather be: a librarian, uncertain, an athletic director. On your birthday, do you prefer: to be asked in advance about what you would prefer for a gift, uncertain, to have the fun of getting a complete surprise? Do you sometimes feel, before a big party, that you are not interested in going? yes, perhaps, no. Can you talk to a group of strangers without being self-conscious? yes, perhaps, no. Are your feelings easily hurt? yES, perhaps, no. x(5) 78 continued: SCORE 1h. NI-‘O 0|—-m NI—‘O or-am Ol-'l\) Ol—‘m a. b. 20. a. b. Can you work just as well, without making more mistakes, when you are watched? yes, perhaps, no. Do you sometimes feel unwilling to try something, though you know it isn't difficult? yes, perhaps, no. Eb you stand up before class without looking nervous? Yes: perhaps, no. How would you rate yourself? inclined to be moody, in between, not at all moody. In school, do you feel your teachers: approve of you, uncertain, hardly know you are present? Are you so afraid of consequences that you avoid making decisions one way or the other? often, sometimes, never. Do you have periods of feeling just "run down"? seldom, sometimes, often. 79 Response Pattern xS62 The x(6) variable was determined by scoring the student's responses to the following questionnaire in the fashion indicated. Forty points were the maximum on this questionnaire. SCORE 1. Which of these says better what you are like? 0 a. a dependable leader, 1 b. in between, 2 c. charming, good looking. 2. Do you like to tell peOple to follOW’prOper rules and regulations? 0 a. yes, 1 b. sometimes, 2 c. no. 3. Are you usually patient with people who Speak very fast or very slowly? O a. yes, 1 b. sometimes, 2 c. no. A. If you found another pupil doing a job you had been told to do, would you: 2 a. ask him to turn the job over to you, 1 b. uncertain, O c. let the teacher decide? 5. Are you steady and sure in what you do? 2 a. seldom, l b. sometimes, 0 c. always. 6. With people who take a long time to answer a question, do you: O a. give them all the time they want, 1 b. in between, 2 c. try to hurry them, getting angry if they are slow? x(6) SCORE Ol—‘l’D own) MI—‘O Ol—‘m 80 continued: 7. Do you spend most of your allowance each week for fun? a. yes, b. perhaps, c. no. 8. Do other peOple often get in your way? a. yes, b. perhaps, o. no. 9. If you were working with groups in class, would 11. a. b. Co 12. a. b. 13. a. b. you rather: walk around to carry things from one person to another, uncertain, specialize in showing peOple what to do? Do you take trouble to be sure you are right before you say anything in class? always, generally, not usually. would you rather be: a traveling TV actor, uncertain, a medical doctor? Do peOple say that you are a person who can always be counted on to do things exactly and methodically? yes) perhaps, no. would you like to be extremely good-looking, so that peOple would notice you? yeS, perhaps, no. 81 x(6) continued: SCORE 1h. When something is bothering you, do you think it's better to: 0 a. withhold action until you become calm, l b. uncertain, 2 c. explode. 15. Do you sometimes say silly things, just to see what people will say? 2 a. yes, 1 b. perhaps, 0 e. no. 16. Do you ever suggest to the teacher a new subject for the class to discuss? O a. yes, 1 b. perhaps, 2 c. no. 17. would you rather Spend a break between morning and afternoon classes in: 2 a. a card game, 1 b. uncertain, 0 c. catching up on homework? 18. Do you usually: 0 a. follow your own ideas, 1 b. uncertain, 2 c. do the same as other peOple? 19. Do you sometimes go on and do something you very much want to do, even though you feel a bit ashamed of yourself? 2 a. yes, 1 b. perhaps, 0 c. no. 20. Do you think that to be polite you must learn to control your feelings? 0 a. yes, 1 b. perhaps, 2 c. no. 82 Response Pattern x§72 The x(7) variable was determined by scoring the student's responses to the following questionnaire in the fashion indicated: SCORE PROCEDURE 1. When he was completely on his own, he success 2 neutral 1 fail O 2. It looked impossible, so he try 2 neutral 1 quit 0 3. My first reaction to him was positive 2 neutral 1 negative 0 A. When others made fun of him, he stand up 2 neutral, 1 other 0 5. When he met his principal, he act appropriate 2 neutral 1 scared O 6. If I think the class is too hard for me, I try 2 neutral 1 quit O 7. When I have to make a decision, I make it 2 think 1 avoid O 8. When they looked at me, I positive 2 neutral 1 negative 0 x(7) continued: SCORE PROCEDURE cope 2 quit 0 handle 2 crumble 0 positive neutral negative C>FJRD cry recover 09C) give neutral no CDEJIU take over 2 neutral 1 avoid 0 spoke nervous C30) 10. 11. 12. 13. 1H. 15. 83 If I can't get what I want, I When I am criticized, I People seem to think that I After they knocked him down, he When they asked my opinion, I When they put me in charge, I When his turn came to Speak, he ReSponse Pattern xg82 The x(8) variable was determined by scoring the results of the following directions: Draw a picture of a person. What is (his, her) age? Draw a picture of a person of the Opposite sex. What is (his her) age? What is the best trait of the first person? What is the worst trait of the first person? What is the best trait of the second person? What is the worst trait of the second person? What adjective best describes the first person? What adjective best describes the second person? What are the wishes of the first person? What are the wishes of the second person? The completed responses were scored according to Bodwin and Bruck’s procedure: 1. Shading: Light, dim, subtle, and uncertain lines which furtively accent particular parts of the figure. Patterned or stylized Shading. Table of Shading (0-20%) (21-ho%) ’ (Al-60%) (61—80%) (81-100%) 5 h 3 2 1 Markedly Markedly absent present 2. Detail in Figure: Unessential features or details added to the figure or background. This analysis follows the table for shading. x(8) 3. 85 continued: Asymmetry: Imbalanced and lopsided arrangement of the body parts in respect to size, shape, or position on the opposite sides of the center. This analysis follows the table of Shading. Mixed age: Disparity in the physiological maturation of various body parts such as breasts emphasized in an other- wise childish body. This analysis follows the table for shading. Immaturity: Drawing is marked by elaborate treatment of the mid-line such as Adam's apple, tie, buttons, buckle, and fly on trousers. There is emphasis on mouth and/or breasts. This analysis follows the table on shading. APPENDIX B TESTS 87 Pretest Compute the following sum: l+2+3+...+1+98+h99+500= Compute the following sum: 2 + 2-2 + 2-22 + + 2-2-2-2°2-2°2'2-2-2 = What is the name of the following number? 1,000,000,000,000,000,000 Find a set of elements with exactly the same number of elements in it as there are in the set of even counting numbers. {2, A, 6, 8, ..., 2-n, ...} If one has 6h feet of fence Show which rectangular shape encloses the greatest area. In Alpha Land {the land of pure imagination} the addition and multiplication tables for digits are: —F Z3 V7 C3 ' ll ‘7' C] [3 [1 V7 E] AS [3 [3» ZS. V7 Y7 C21 VOA <7 [3 ‘7' C] C] C] VZAsV7VC C] [X C] ‘“V Complete the following Alpha addition and multiplication: (a) '(b) DCJV VGA ‘VCI DEV VD-D +VAD Ell—VA v__|:|.__. V_V_D 88 What is the approximate area of the bounded region indicated below? Four lines, extended indefinitely far in each of the lines two directions, are placed in a plane in any position you choose. How many regions apoost will the plane be cut into? 89 Quiz I Find the sum of the series listed below: 1 + 2 + 3 + ... + 697 + 698 + 699 + 700 Find the sum of the series listed below: 2 + h + 6 + ... + 69% + 696 + 698 + 700 Part (a) What number is represented by 25 Part (b) What number is represented by 210 Part (c) 20 What number is represented by 2 If n is termed a variable and if you are told that n.e.N+ what do you know about n ? Quiz II Find the sum of the following series: 2 + h + 6 + ... + h96 + #98 + 500 = If a completely painted cube is cut 26 times through each dimension how many resulting cubes will be painted on exactly one side? If a geometrical solid is convex and has 15 faces as well as 10 vertices, how many edges must it have? If there are 370 grains of wheat in a cubic inch, how many grains are in a cubic yard? 90 Post Test 1. Find the sum: 1 + 2 + 3 + ... + 1,000,000 2. Find the sum: 2 + h + 6 + ... + l,OO0,000 = 3. Given the system: A, V, C], O, 04. VA, (7?, VD, V0, (700, ... fill in the blanks (a) [:1 UN (b) C3 C] M '00 (>13? VCI—CJ god VA_.<>A +V vvflva ‘7‘“:— h. If a cube is cut five times each way find: (a) How many small cubes result ? (b) How many small cubes are painted on three Sides ? (c) How many small cubes are painted on no Sides ? 5. What is the approximate sum of? 21 + 22 + 23 + . . . 10 91 6. What is the approximate area of: o; f5 \n 7. Fill in the blanks, assuming unique answers are intended: {Just guess what is wanted}. F) T) fi; J-) TE) ——»T-__{) fl’qk’ _— . 8. If a solid object is convex and has 12 vertices with 7 faces it will have edges? APPENDIX C PROBLEMS AND DAY-BY-DAY ROUTINE 93 SECOND MEETING Preliminary material introduced to and discussed with the student prior to his initial attempts at problem solving included the following: §_E_r§_ N+ = {1, 2, 3, ..., n, ...} VARIABLES n is a member (a number) of some set and is denoted by n€N+. OPERATIONS If a€N+ and b€N+ then [a+b]€ N+. If aeN+ and b€N+ then [a-b]€N+. SOME SHORI'HAND NOTATION 2+2+2 3-2 2+2+2+2=h~2 2+2+2+ooo+2=n’2 n 2'2 = 22 2°2°2 = 23 n 2.2.2. 0.0 .211 = 2 QUESTION If as N+ and naN+ write down what you think an COUNT ING Counting means pairing sets l-l . means . 9h QUESTION Count the even numbers in N+ . If you have a set of numbers all of which fit the form 2-n and you know neN+ what do you know about the set of numbers? THINK Guess what a related set of numbers looks like in a form Similar to the form of the set just discussed. THIRD MEETING Add the Series: 1 + 2 + 3 + h = PROBLEM 1. Try to find the sum of the following series without actually adding each term to the next: 1 + 2 + 3 + h + 5 + 6 + 7 + 8 + 9 + 10 = PROBLEM.g' If you found a way to add the last series try this one: 1 + 2 + 3 + ... + 98 + 99 + 100 = PROBLEM‘3 If you were successful on PrOblem 2 try this one: l+2+3+...+[n-2]+[n-l]+n= 95 FOURTH MEETING PROBLEMI&_ l = O + 1 2 + 3 + h = l + 8 5 + 6 + 7 + 8 + 9 = 8 + 27 10 + 11 + 12 + 13 + 1h + 15 + 16 = 27 + 6h If this process is continued what will the two numbers be on the right of the equal Sign? Can you find a way to write the solution for any given line? PROBLEM‘E l + 2 + 3 + ... + 999,998 + 999,999 + 1,000,000 PROBLEMI§_ 2 + h + 6 + ... + 999,996 + 999,998 + 1,000,000 = FTPTH.MEETING PROBLEMIK Ybu are to assume that the cube shown on the following page has been immersed in paint and is completely covered. If the cube is then cut through each plane as indicated by the dashed lines the question is: (In each case how many of the resulting small cubes will be painted on how many faces?) (a) Given n cuts there are cubes with 6 painted faces? (b) Given n cuts there are cubes with 5 painted faces? (c) Given n cuts there are cubes with h painted faces? (d) Given n cuts there are cubes with 3 painted faces? 96 cubes with 2 cuts there are (e) Given n painted faces? cubes with l cuts there are (f) Given n painted face? cubes with 0 (g) Given n cuts there are painted faces? cubes? (h) Given n cuts there are 4’ ------fl-'-"- ___ _ "|"""|"|"|'l P-------- ,l- -.. #3:.-- l I I Il"||' -*- -- l —-- A a . . --.ak dull-I'L _--- 97 PROBLEM g Once upon a time many centuries ago a mathematician in Egypt invented a game played on a board with Sixty-four squares. There were eight squares along each edge. The king was so pleased with the game he offered the mathemati- cian any reward he might wish. The mathematician asked to have one grain of wheat placed on the first square, two on the second, four on the third and thus double the quantity on each successive square of the board. The king thought this request trivial and ordered it be carried out. Long before the jOb was completed the king learned the error he had made and promptly had the mathematician beheaded. It would have taken Egypt thousands of years to have grown all the necessary wheat. If there are 370 grains of wheat in a cubic inch how large would the bin have to have been to hold the wheat required to satisfy the mathematician’s request? If it turns out to be important 265 is approxi- mately 37,000,000,000,000,000,000 . IS this number of any use? 98 PROBLEM 9 A.cube may be dissected into 8 subcubes with ease. May a cube be dissected into 9 subcubes? May it be cut into 10, ll, ... ? Consider only those counting numbers less than SM and identify the number of cubes into which a given cube may be dissected. Note the example--how many subcubes are there? \\\\ \\\\ \\\: //// 99 SIXTH MEETING PROBLEMIIQ. Part (a) A ball is dr0pped from an original height of 100 feet. Each time it hits the ground it bounces back 50% of or one half the previous fall. How far does the ball fall? Part (b) How far does the ball travel ? PROBLEM E Explain why the shaded area constitutes one half the total area indicated within the bounds of the region below. lOO SEVENTH MEETING PROBLEM lg Given an n-gon ; how many 'non-intersecting' lines cut it into how many triangles? / / / / / / l / / SIDES LINES TRIANGLES I. l 2 5 2 3 6 3 u lOl EIGHTH MEETING PROBLEM ii Agent 007 has infiltrated the Red Chinese department of state as a Spy for British intelligence. Mao has decided it is time for a purge of his department of state. Mao will handle the purge in the following way. He will have all members of the department of state brought into a room. There will be n of them and they will be told to Sit in a circle. Then Mao's executioner called chOpper will come through and eliminate every other one around the circle until only one remains. Mao wishes to preserve the brightest member and assumes this will determine the brightest. Suppose you are 007 and you would know what the number n was to be just ten minutes before you entered the room. Where Should you sit in order to be the lone survivor? Try some samples if you wish. The following sketch may be helpful. 102 PROBLEM 2% Complete the following patterns in at least five different ways and find reasons for your solutions. (a) 2) LL: 6: 8: _) .9 1 (b) 9:01910: 2 2 ) ' (C) IT; TF3?) _: _1 .v The following patterns are so structured that if you read my mind you will find a unique answer for each blank. Read my mind and fill in the blanks. This suggestion Should not surprise you. Much of mathematics is one form of mind reading or another. (d) A, V, C], VA, (7V7, (7g,._,____’__, 0 1311—4»: T1, TT, __.., _._.,__. (f) #6, fl), N’G, N~,N**, —""") 103 NINTH MEETING ‘PROBLEM 12 FACES VERTICES EDGES CUBE i I l l 9..-- ____ 5 5‘ 8 SQUARE PYRAMID OCTAHEDRON 8 6 12 I I I ' TRUNCATED : CUBE 7 10 15 L ...... / Consider the various convex solids above. Can you find a relationship among the faces, vertices and edges? 10h PROBLEM is 2 + h + 6 + ... + [2-n - 2] + 2-n = PROBLEMIIZ. 2 + h + 6 + ... + 2(n - l) + 2-n = we Part (a) 22 + #2 + 62 + ... + [2(n - 1)]2 + [2-n]2 = Part (b) l2 + 32 + 52 + ... + (2-n - l)2 = Part (c) l + h + 7 + ... + (3-n e 2) = TENTH MEETING PROBLEM 19 Part (a) Show how -§LE- and 2-n - 2-n2 + 2-n3 - 2-nu ... n + l are related and show the form of the set builder.. Part (b) Show what 8'“ has for an uiv lent form s m eq 8* a a series and Show the set builder. Part (c) ShOW‘What §§%—E£%- has for an equivalent form as a series according to n odd or even and Show the set builder. 105 PROBLEM g9 So construct the four sided figure in the half circle below that one Side is the diameter and that the area of the figure is largest. \ Suppose instead of four sides as above you are asked to fill in the following table: SIDES BEST PROCEDURE 2 2 sides equal 3 [note--this is the above case} ? 1* '2 ELEVENTH MEETING In the following problem count each and every line in each circle exactly once and fill 1 in the accompanying table where indicated: 1 ‘1 g;;;;EEEff//N‘SIESHIIIIP 1 l 4 \ PROBLEM a ' .2 7' POINTS LINES 1 O 7 3 ‘3 2 l 3 3 6 4- . 8 28 108 THIRTEENTH MEETING PROBLEMI23 In the figure below determine if: C1 = C2 + C3 and D1 = Dé + D3 . PROBLEM 23 In the figure below determine if Fl = F2 + F3 . Can you make a conjecture you believe at this point? Think about it and draw some pictures. 109 FOURTEENTH MEETING PROBLEM 25. .2 LINES GREATEST NUMBER OF REGIONS l 2 J Plane -LINES GREATEST NUMBER OF REGIONS 2 h LINES GREATEST NUMBER ' OF REGIONS 3 7 Plane -Suppose you had n lines. How many regions could you . form 22 most? 110 FIFTEENTH MEETING PROBLEM‘gé Plane ,CIRCLES 2 l Plane CIRCLES h 2 Plane 1 2 h 3 5 . CIRCLES 8 6 ., 3 GREATEST NUMBER OF REGIONS 2 GREATEST NUMBER OF REGIONS h GREATEST NUMBER OF REGIONS 8 If n circles were placed in a plane how many regions could be formed ap_most? 111 FTITEENTHIMEETING PROBLEM g; If you consider the region bounded by the two straight lines and the curve how might you determine its area? N h 5 \‘é F575" 3" ' / WOuld this give you a better way of determining the region area? How could you do a better job? a ' 114 ' L\ . a B 4 A // A \\\“' 1.. R) w 4:- \n 112 PROBLEM g§_ Suppose you wanted to determine the area of the circle below and did not know anything about 1T' or did not recall the formula. What could you do? N N\ WOuld this give a better result? Can you extend this idea? \\ \ N / // ///// /// 113 Make the best estimation of the area of the bounded region below. APPENDIX D SOLUTION INDEX TO ALL MATHEMATICAL MATERIAL 115 SOLUTIONS FOR PRETEST: 1. 125,250 2. Acceptable solutions: :3! 2,0h8 with the exact answer being 2,068 . 3. One quintillion. A. [1, 3, 5, 7, ... (2n - 1), ...} or (1, 2, 3, ... n, ...} infinitely many solutions exist. 5. A square of 16' on a Side. 6. a) line 1, v b) C], A . line 2, C] line 3, V2, C7 7. Acceptable solutions: 2 E> <> b) line 1, a) 125 b) 8 c) M3 11 Acceptable solutions: lOm+1fi. 8. 6.65 miles on a side. 9. 1-8-15-20-22—27—29—3u—36-38-39-u1—u3-u5-u6—u8-u9- 50-51-52-53- 10. ll. l2. 13. 00040me 10 ll 12 13 1h 15 l6 17 117 a) 200 feet b) 300 feet Split the Shaded triangle forming two rectangles. n sides, n-3 lines, n-2 triangles. The students were expected to gain solution by forming some classification scheme such as the following finite matrix: Chair to be selected 1 3 5 7 9 ll 13 15 -X- -X- *- -X- *- 9(- 9(- * * *- *- * * 9(- Where the pattern now emerges for testing. The pattern to be tested may be examined in the following way: (1) Only odd numbers occur and the pattern of the solution set fits an eXponential sequence. (2) For 3 peOple the solution is a one digit sequence or Simply 3 . (3) For n peOple where 34 10.5 20 27 15 2h 10 12 19 SS x(l) 1.7 5. 1. 1. 00.5 00. 9‘ 9’ YD x(2) 29 16 16 13 19 12 18 1h 20 10 13 20 30 10 18 17 X(3) 22 X(h) 15 10 11 7 16 12 13 1h 1h 15 18 17 l2 l5 X(5) 2A 19 17 20 20 1A 21 18 22 13 15 16 22 22 17 2O x(6) Ox [3 tn 01 O\ \o tn 01 t3 on O\ O\\OO\I: x(7) 7 10 11 18 l7 6 13 L 13 l9 l3 x(8) 23 25 29 28 33 22 2h 17 26 20 32 33 25 33 27 21 BIBLIOGRAPHY 12h BOOKS Bell, E. T. Men of Mathematics. New YOrk: Simon and Schuster, 1937- Berlyne, D. E. Soviet Research 9p_Intellectual Processes in Children. Society for Research in Child DevelopmeHE: Monographs, Vol. XXVIII, No. 2. Chicago: University of Chicago Press, 1963. Bloom, Benjamin S. (ed. ). Taxonomy 9; Educational Objectives. Handbook I: CO itive Domain. New YOrk: David McKay Company, Inc., 1956. Bruner, J. S. The Process of Education. Cambridge: Harvard University Press, 1960. Bruner, J. S., and Rose R. Olver. Develgpment gf_Eguivalence Transformations ip Children. Society for Research in Child Development: IMonographs, Vol. XXVIII, No. 2. Chicago: University of Chicago Press, 1963. Bruner, J. S. and others. .A Study_ offl fling. New YOrk: John Wiley and Sons, Inc., 1965.1 Carroll, J. B. Language and Thougpt. Englewood Cliffs: Prentice Hall, 1961? Chomsky, Noam. Syntactic Structures. The Hague: Mouton and Company, 1965. Dienes, Z. P. An Experimental Study of Mathematics Learning. London: —Leicester University Press, 1963. Duncker, Karl. On PrOblem-Solving. Trans. Dynne S. Lees. Psychological MOnographs, Vol. LVIII, No. 5. washington, D. C.: The American Psychological Association, Inc., 19h5. Flavell, JOhn H. The Developmental Psychology gf_Jean Piaget. New YOrk: D. Van Nostrand Company, 1963. Gage, N. L. Handbook of Research on Teaching. Chicago: Rand McNally and Company, 1965.— Gagne, Robert M; The Conditions of Learning. New YOrk: Holt, Rinehart and Winston, 1967.?— 125 Hadamard, Jacques. The Psychology_ of Invention in the Mathematical Field. Princeton: Princeton University Press, 19h5. Hunt, J. McV. Intelligence and Experience. New YOrk: The Ronald Press, 1961. JOhnson, Donald. The Psychology Lf Thought and Judgment. New YOrk: Harper and Brothers, 1955. Kaplan, A. The Conduct g£_In ui . San Francisco: Chandler Publishing 00., 1963. Kennedy, wallace A. A Multidimensional Studerf Mathematically Gifted Adolescents. Society for Research in Child Development: Monographs, Vol. XXV, No. 31. Chicago: University of Chicago Press, 1960. Lovell, K. The Growth Lf Basic Mathematical and Scientific Concepts in Children. London: University of London Press, 1967:. Mendelson, Elliott. Introduction to Mathematical Logic. New YOrk: D.'Van Nostrand Company, 196R. Miller, G. A., and others. Plans and The Structure Lf Behavior. New YOrk: Henry Holt and Company, 1960. Morrisett, L. N., and J.'Vinsonha1er (eds.). Mathematical Learning. Society for Research in Child Development: Monographs, Vol. XXX, No. 1. Chicago: University of Chicago Press, 1965. NMrphy, Gardner. Personality, A.Biosocia1 Appm mach to Ori ins and Structure. New York: Harper and Brothers, 1967. Piaget, Jean. Psychology g£_Intelligence. Paterson: Littlefield, Adams and Company, 1960. . The Langgage and Thougpt pf The Child. New YOrk: Meridian Books, 1962. . The Child's Conception of Geomet . New York: Harper and Row Publishers, 1960. Piaget, Jean, and Barbel Inhelder. The Growth Lf Logical Thinking From.Childhood to Adolescence. —New York: Basic Books, 1958. IYDI..I..A 126 Polya, G. Patterns of Plausible Inference. Princeton: Princeton University Press, 1951+. . Induction and Analog in Mathematics. Princeton: Princeton University Press, 1951+. Sexton, P. C. Education and Income. New York: The Viking Press, 1961. School Mathematics Study Group. Mathematics for Junior Hijh School. Vol. I, Parts I and II. New Haven: Yale University Press, 1961. Spiker, Charles C. Verbal Factors E the Discriminati_on_ Learning of Children. Society for Research in Child Development: Monographs, Vol. XXVIII, No. 2. Chicago: University of Chicago Press, 1963. University of IELlinois Committee on School Mathematics. High School Mathematics. Revised edition. Urbana: University of Illinois Press, 1962. Vygo‘bslqr,j.L. S. Thought and Langlgge. Cambridge: MIT Press, 1962. Websters Third New International Dictionary. Chicago: G and C Merriam Co. , 1966. Wertheimer, Max. Productive Thinking. New York: Harper and Row, 1959. ARTICLES AND PERIODICALS Aiken, Lewis R. "Personality Correlates of Attitude Toward Mathematics ," Journal ,1: Educational Research, LVI (1963), 1:781:80. Callicutt, Wade. "Problems of Predicting Success in Algebra," National Association 93 Secondary School Principals Bulletin, XLV (November, 1961), 107-111. Cattell, R. B. "Personality Traits Associated with Abilities-II With Verbal and Mathematical Abilities," Journal 9: Educational Psycholog, XXXVI (1916), h75-E86. 127 Crandall, V. J. and A. Robson. "Children's Repetition Choices in an Intellectual Achievement Situation Following Success and Failure," Review of Child Development Research, Martin L. and Lois Hoffman, editors. New York: Russell Sage Foundation, 1961+, p. 162. Cronback, Lee J. "Learning Research and Curriculum Development ," Journal 93 Research in Science Teaching, II, (1961+), 20E-70 Dessart, Donald J. "Mathematics in The Secondary School," Review 9; Educational Research, XXXIV (196A), 298-312. Duckworth, Eleanor. "Piaget Rediscovered, " Journal o__f_ Research in Science Teaching, II, (196M), 172* 5. Engel, Mary. "'I'hesis--Antithesis," American Psychologifl, XXI (August, 1966), 786. Gibson, J. J. and E. F. Gibson. "Perceptual learning," The Psychological R_e__,view LXII (January, 1955) , 1+0. Henderson, Kenneth B. "Research on Teaching Secondary School Mathematics," Handbook 9_f Research on Teachin N. L. Gage, editor. Rand McNally and Company, 1965, pp. 1007-1030. Hilgard, E. R. "Creativity and Problem-Solving," Creativity and Its Cultivation, Harold H. Anderson, editor. New York: Harper and Row Publishers, 1959, pp. 162-180. Hills, John R. "Can We Find The Latent Scientist?" School _a_n__d Society, LXXXV (1957), 30608. Kochnower, W. "Personality Factors and Success in Mathematics," Hi 5g Points, XLIII (April, 1961), 65-72. Lambert, P. "Mathematical Ability and Masculinity," Arithmetic Teacher, VII (January, 1960), l9-21. Lee, D. M. "A Study of Specific Ability and Attainment in Mathematics," British Journal of Educational Psychology, XXV (NW member, 19557} 178-89. McDavid, J. W. "Imitative Behavior in Preschool Children," Review of Ch__i_ld Development Research Martin L. and Lois W. Hoffman, editors. New York: Russell Sage Foundation, 1961, p. 157. 128 Milton, G. A. "Five Studies of the Relation Between Sex Role Identification and Achievement in Problem Solving," vaiew of Child Development Research, Martin L. and Lois W. Hoffman, editors. New York: Russell Sage Foundation, l96h, p. 157. Mink, Oscar G. "Experience and Cognitive Structure," Journal 9f Research i§_Science Teaching, II (l96h), 196-203. Mnsatti, C. L. "Les Caracteres Perceptifs des Objects et la Theory Mathematique des Groupes," Acta Psychology, XIV (1958), 41-53 Plank, Emma and Robert. "Emotional Components on Arithmetic Learning as Seen Through Autobiography," The Psychoanalytic Study of the Child IX (195 H), 273- 293- Poffenberger, Thomas and others. "Factors in the Formation of Attitudes Toward Mathematics," Journal of Educational Research, LII (1959), 171- 76. Riffenburgh, R. H. "Personality Traits Associated with Mathematical Ability: A Plea for Research," Journal of Experimental Education XXIX (December, i§557"18§¥92. Schunert, Jim. "The Association of Mathematical Achievement with Certain Factors Resident in the Teacher, in the Teaching, in the Pupil and in the School," J0urnal of Experimental Education, XIX (1957), 237. Sears, P. S. and E. R. Hilgard. "The Teacher's Role in The Motivation of the Learner," Theories 2: Learning gg§_ Instruction, E. R. Hilgard, editor. Chicago: National Society for the Study of Education, 196A, pp. 182-209. Shulman, L. "Improving Mathematics Education for Elementary Teachers," W. R. Houston, editor. Muchigan State University, sponsored by The Science and Mathematics Teaching Center, p. 26. Smedslund, Jan. "Internal Necessity and Contradiction in Children’ s Thinking," J0urnal of Research in Science Swift, Joan W. "Effects of Early Group Experience: The Nursery School and Day Nursery," Review of Child Development Research, Martin L. and Lois W. Hoffman, editors. New York: Russell Sage Foundation, l96h, pp. 210-228. 129 Tyler, Fred T. "Issues Related to Readiness To Learn," Theories of Learning and Instruction, E. R. Hilgard, editor. Chicago: National Society for the Study of Education, l96h, pp. 210-239. Whitkin, H. A. "The Nature and Importance of Individual Differences in Perception," The JOurnal of Personality, XVIII (December, 1919), 162. UNPUBLISHED MATERIALS Hildebrand, F. H. "Some Cognitive Components Involved in Mathematical Induction," Unpublished Paper in Education 982, Michigan State University, 1966. (mineographed.) Knight, Sarah. "Curiosity." Unpublished Paper in Education 982, Michigan State University, 1966. (Mimeographed.) Rosenfield, I. J. "Mathematical Ability As a Function of Perceptual Field-Dependency and Certain Personality Variables." Unpublished Ph.D. dissertation, University of Oklahoma, l958. Sweeney, Edward J. "Sex Differences in PrOblem Solving." Unpublished Ph.D. dissertation, Department of Psychology, Stanford University, 1965. "'TITIWLUITILEJEILHEIflflfnjrjfifliflllflyWNW'ES