WW CASE FFFFFFS 0F FFGFF swam _ S‘E‘UBENTS FSFFG FFFSFCAF MFFELSF;j-Lift F0 FFFFF MAFFFFFFFFFAF 5mm ' MFCFFFFF STATE Gerald Clayt, Tar—5‘9 ‘ LIBRARY Michigan State University E :5 e - i _‘ * , , gq This is to certify that the ‘51:; ' - . ¢ ' - ' thesis entitled ' "g, CASE STUDIES OF HIGH SCHOOL STUDENTS USING PHYSICAL MODELS TO STUDY MATHEMATICAL SYSTEMS presented by Gerald Clayton Burke has been accepted towards fulfillment of the requirements for Ph.D. degnmin Secondary Education ”4/, 2/4 ”F- agamm Major professor \ Date 8—6—73 0—7639 ABSTRACT CASE STUDIES OF HIGH SCHOOL STUDENTS USING PHYSICAL MODELS TO STUDY MATHEMATICAL SYSTEMS by Gerald Clayton Burke Purpose The purpose of this study was to examine in detail the cognitive outcomes of high school students using physical models to study the structure and nature of mathematical systems. By mathematizing some physical representation of a phenomena and studying that model, the study was designed to explore how well the students could deal effec- tively with mathematical models, understand the nature of an axiomatic system and the process of logically deducing propositions for investi- gations. Procedure The students chosen for this study were selected from inter— mediate algebra classes taught at Suncoast High School, Riviera Beach, Florida. During the first semester of the 1972—73 school term, stu— dents were introduced to the operations and properties of the real numbers in an algebraic setting with emphasis on the postulational procedure which included operations and properties of a group. At the beginning of the second semester, ten highly motivated and above Gerald Clayton Burke average students were selected to complete a ten—week schedule of activities instead of their regular class work. All sessions were recorded on audio tapes as the students worked in small groups or individually. Ten activity—based exercises, most of which were adapted from Laboratory Manual for Elementary Mathematics by Fitz- gerald, et al., were the source of techniques and procedures. The results of the study were reported using the case study procedure. Findings The results of the research demonstrates that within the con— straints of normal classroom conditions, high school students can achieve a higher level of understanding the entire nature of: (1) model building both physical and abstract, (2) the axiomatic process, and (3) the process by which propositions are logically deduced from other assumptions and proved. n31 .qllsnbivlhnr -s:k§ 7d any; bazaars ..-s:'." ..='*-'i--bsoo'r-n bar Fe:-:p':c:r':as:‘- '10 3:): ya ”.53 ‘-'.r*-‘. ..L_-. _5;_ ,‘.'. :. .Z'b. .,_..p a" -". "."-= :." . ." :'.I'_"I"' .' . , .' ._, CASE STUDIES OF HIGH SCHOOL STUDENTS USING PHYSICAL MODELS TO STUDY MATHEMATICAL SYSTEMS By Gerald Clayton Burke A THESIS Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Secondary Education 1973 [} ACKNOWLEDGEMENTS I am deeply indebted to my wife Myrtis, and our children: Gerunda, Michael, Patricia, Marcus, and Tanya, whose support and understanding during the last four years has meant everything to me in this undertaking. They provided an unyielding catalytic impetus to sustain me. A great debt of gratitude is due the members of my doctoral committee: Dr. William M. Fitzgerald, chairman, Dr. John Wagner, Dr. Joe Byers and Dr. Glen Anderson. Each contributed tremendously in his own manner along the way to the results realized in this report. I owe a special debt of gratitude to generous and audacious students at Suncoast High School who participated in this study over the two year period. .Isadarfl .sbauzsa an o) antdjvxsvs spasm ass erasv ruoi 323! 9d: anixub qnlbunsasnbnu ”ms-c353 13112123353 ..'|'Il'l'-!: 'r'.-'rm fiL‘ bob_'r"r.-'.'; '::"."r" .:"J'--!r...-‘.'-.'.!)."F' :‘-i" :.i TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . CHAPTER I. INTRODUCTION. . . . . . . . . . . . . Rationale for the Study . . . . . . . Pilot Study . . . . . . . . . . . . . Hypothesis. . . . . . . . . . . . Organization of Final Report. . . . . II. A REVIEW OF THE LITERATURE. . . . . . Introduction. . . . . . . . . . . . Review of the Literature. . . . . . Summary . . . . . . . . . . . . . III. PROCEDURE . . . . . . . . . . . . . . Introduction. . . . . . . . . . . . Description of Activities . . . . . . Description of Students . . . . . . . IV. CASE STUDIES RESULTS. . . . . . . Introduction. . . . . . . . . . . . Activities Circle: Turns . . . . . . . Clock Arithmetic: Modulus 8 Equate the Height. . . . . . Zigzag . . . . . . . . . . . Relations. . . . . . . . . . Rectangle: Flips and Turns. Beans and Brussel Sprouts. Tower Puzzle . . . . . . . 10 Instant Insanity . . . . . . Summary . . . . . . . . . . . . . . . \OWNONU'I-L‘LDNH V. SUMMARY AND CONCLUSIONS Summary . . . . . . . . . . . . . . Discussion. . . . Recommendations for Further Research. Conclusions . . . . . . . . . . . . . BIBLIOGRAPHY. . . . . . Equilateral Triangles: Flips and Turns. PAGE iv H \I\IO\N 32 32 32 36 41 45 52 52 54 57 61 .62 63 63 64 65 65 66 . ZHJMT 10 TBIJ HHT'IAHD “-3.3 ”4.1 4.2 4.3 4.4 4.5 4.6 .. 7‘ “ ¥.U.‘V" - 2-1'8538 u I a n I "5%..” gsfixl 3%” 1d". ‘: Equivalence Relation . . . Summary of Test Results . . . . . . . . . . Activity Summary: Equilateral Triangle Activity Summary: Circle Activity Summary: Modulus 8 (:> . . . . . . Activity Summary: Modulus 8‘:> Zigzag . Activity Summary: Rectangle . iv 31 33 36 41 43 49 53 AS . . . . . . . . . . . . . . . . . . . . noleIsfl annalsvlupfl §.£ ifi . . . . . . . . . . . . . . . . . . aJIuaafl ”EDT in grsmmu’ :.£ Lu.!'--ns_=.i‘.rT JI..'.'.3:!.--‘!'.iw-._i '_'-' mm" ;2__.-_ "'1’ .' .‘9: u. I i . I I I I I I l l l I . I -r ' . l " l _ I u - .I .: “- -' l I I 'Iuv...’ ‘ ' GENERI INTRODUCTION Ideas in mathematics once thought to be too difficult for high school students to study are now commonplace in many secondary school curricula. Many of the ideas fundamental to understanding the nature of mathematical structure are no longer reserved for advanced courses or specialized enrichment programs. On matrix theory, Davis makes the following observation: A generation ago, the subject was taught as an intermediate or first-year graduate course in college and was taken by majors in mathematics and theoretical physics. The School Mathematics Study Group (SMSC) has been instrumental in introducing matrix theory into the secondary curriculum. Fuller found analytic geometry suitable for above average students when he wrote: Inasmuch as this particular course is taught almost exclu- sively in colleges, this publication is designed for college freshmen. This book finds ready application in high school which provide for such a study for their mathematically in— clined students. Brumfiel implies the postulation method to correct logical deficiencies in high school geometry when he cited: It is common belief that plane geometry was completed by Euclid 2000 years ago and that nothing has been added to it or 1Philip J. Davis, The Mathematics of Matrices: A First Book of Matrix Theory and Linear Algebra (Boston, Mass.: Ginn and Company, 1965) pp. vi-vii. 2Gordon Fuller, Analytic Geometry (Reading, Mass.: Addison- Wesley Publishing Company, Inc., 1964) p. v. l 2 taken from it since. This is simply not true. That these are logical gaps in Euclid's presentation has been known for a long time. Means to remedy these deficiencies have been known for about sixty years, but strangely enough a mathematically ade- quate and yet elementary treatment of plane geometry in the spirit of Euclid has not appeared in print. This text repre- sents an earnest effort to do this. The current interest in improvement of the secondary school curriculum makes this an appropriate time for such an attempt. These ideas have been incorporated into the high school curricu- lum as a result of the efforts put forth by mathematicians, mathematics educators, classroom teachers, administrators, national curricula study groups and committees. This change in the curriculum has given rise to a challenge in which experimentation and exploration with ideas and ac- tivities can be realized. Those ideas and activities when properly im- plemented, can lead the high school student to a greater understanding of the nature of mathematical structure and the deductive process there- in. Mathematics should be presented in a manner which is conducive to developing critical and creative thinking at all levels. Rationale for the Study The purpose of this study is to observe and report in detail the cognitive attainments of high school students as they study the mathe- matical properties which are embodied in physical models. This writer believes that such an examination will reveal how well individual high school students can in fact deal with the ideas of mathematical systems. By mathematizing some physical presentation of a phenomena then construct- ing a mathematical model of the phenomena and studying that model. This study will explore the students' ability to deal effectively with mathe- 3Charles F, Brumfiel, Robert E. Eicholz, and Merrill E. Shanks, Geometry (Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1960) p. ix. 31H». ..l - n..-' -.-r l n- I '| J I " - a. 'G" 'f'I J""‘ (1' ["5 l..'I I I - ‘ - - .II.' I ... .l II J . I.- c' . x l . E"" '_ I I II E, . u. " 1 p .u-: f" fI. f) r I! ( _ -l I.I.L . . ..-.- .I --. .¢.I ' I "I I I .. n. I I . J'..a.' I ' ‘ 4' - - ' ' -‘ I. J .. .J. ‘ ' ' ' . I I" ' I ' ' J ’.I 3 matical models, the nature of an axiomatic system, and the process of logically deducing propositions for investigation. Blattner makes the following observation concerning models to support this notion: In the beginning of an axiomatic study of a mathematical sys- tem, it is instructive to examine models of the situation, for models have an unexcelled power of clarifying concepts and sug- gest proper questions for investigations. The high school student can play the ”mathematical game" of setting cer- tain rules and understanding certain assumptions as they are related to the idealization of an axiomatic system by means of a physical model. Mathematical systems are characterized by certain operations and properties; this study makes full use of physical models in the real world to represent those Operations and properties so that in the final process, abstract properties of the mathematical system can be under- stood. This notion is supported by the Cambridge Conference which re- ported that: Every application of mathematics depends on a model, and the value of the deduction is more an attribute of the model than it is of the mathematics. We believe that students can be made aware of the distinction between the real world and its various mathematical models. Concrete operations can be executed by the student with a physi- cal model and related to an analogous operation in a mathematical system. Physical models can be used to formulate properties which can be related to analogous properties in a mathematical system. This process of mathe- matizing a physical phenomena leads to a fuller understanding of the 4John W. Blattner, Projective Plane Geometry (San Francisco: Holden-Day, Inc., 1968) p. 17. 5The Report of the Cambridge Conference on School Mathematics, The Goals for School Mathematics (Boston, Mass.: Houghton Mifflin Com- pany, 1963) p. 47. Joodaa dgld sdT o: are god: a; anoiquuaas ninn1so gaibnsjaruh~u bar asluu "is: ,. 5;»rm Isaiavdq a lo Fnuew rd mujags Llhrmlixk -L EH 1ni2¥33_ u ~fl" 'flavaflw ulanR; no -u:11a'n u-n- "w.:- _ '- F‘ I _| J. '11-“; L n - r- '| I ' a I n. . _ 4 relationship between physical models and mathematical systems as models was expressed by Meserve when he wrote: In recent years the emphasis throughout mathematics has shifted to thinking of the mathematical system as a model of the physical situation. The interrelation between a physical model (representation, application) of the mathematical system and the mathematical model (system, representation, abstraction) are be— ing recognized as a basic aspect of mathematics. The trend toward thinking of the mathematical system as a model is based, at least in part, on the use of a variety of mathematical models to represent different aspects of and different approximations of the same-physical situation. The process of model building is used at all levels and in all branches of mathematics. The simplicity employed in formulating the operations and pro- perties of physical models enhances the high school student's ability to transfer a few basic operations and properties that can characterize ab- stract mathematical systems. Although the models will vary with physi- cal representation, an abstract pattern can be formulated which will identify many different and revealing consequences which are familiar to the student from his previous experiences in the study of mathematics. The importance of this simplicity is recognized by Anderson who made the following observation: One striking characteristic of a mathematical model is its simplicity. In designing a mathematical model, we try to focus our attention on the important ideas and ignore the irrelevent ones. Since the operations and properties of the physical models are basic and simple, they can be changed at will and a variety of physical representations can be employed. The basic operations and properties of mathematical systems can likewise be altered. By making changes in 6Bruce E. Meserve, "An Improved Year of Geometry," The Mathe- matics Teacher, LXV (February, 1972) pp. 177—78. 7Richard Anderson, Jack Garon, and Joseph Gremillian, School Mathematics Geometry (Boston, Mass.: Houghton Mifflin Company, 1966) Chapter 12. :;._-.: fut». Hrun-Jf-Eur't. .>:=_; ,::j-_...'l.-.-._ ..i -. ..-'r..v.-:5- -- _ _:. .-';' 5 certain postulates of a mathematical model, new systems can be derived. Lick points out the following: Mathematics as a study of deductive systems allows great, even unlimited flexibility of individual innovations, inven- tions, and creativity. In no other discipline is this true. By slightly altering some of the axioms, or definitions, one can create whole new systems. For example, changing a few of the basic axioms of high school geometry, can lead to new and different geometries (i.e. Euclidean versus non-Euclidean). It is the opinion of this writer that lateral transfers of con- crete ideas of operations and properties acquired during the study of physical models can be made to study the abstract structure of mathe- matical systems. The ultimate realization is one in which students use such acquired knowledge and understanding to reveal the structure and nature of the deductive mathematical systems which are analogs of the physical models. The Cambridge Conference reported the following: I! It is only when the model is fully formulated that the purely deductive methods of mathematics takes over."9 Lick summarizes the challenge proposed in this study when he stated: Mathematics is abstract, mathematics is not nature. How- ever, the key to the study of nature and natural phenomena is the concept of the mathematical model. That is, a mathemati- cal system can be chosen that its terms and assumptions have some meaningful relation to the physical world and so may be a model for a physical situation. This is another beautiful as- pect of mathematics; in one instance it may be used as a tool with application to models representing physical phenomena, and in another it may be an abstract discipline in and of it- self . . . whether pure (i.e. completely abstract) or applied (i.e. having application to physical phenomena), the creation of a mathematical system or model, the study of this entity, and the derivation of consequences and results from it are 8Dale Lick, ”Why Not Mathematics," The Mathematics Teacher, LXIV (January, 1971) p. 85. 9The Report of the Cambridge Conference on School Mathematics, The Goals for School Mathematics (Boston, Mass.: Houghton Mifflin Company, 1963) p. 47. C? ghnfia 3n: uTnSau“:h ..j ."I' rrjniqu in: ”i 31 6 steps in a process than can be extremely challenging and stimu— lating (as well as frustrating at times). This process can and should take place at every level of mathematical training and study.10 Pilot Study In order to realize some of the aforementioned suggestions deal- ing with constructing a mathematical model of a physical situation this investigator conducted a pilot study during the 1971—72 school year. The student population consisted of five classes of first year geometry at Suncoast High School, Riviera Beach, Florida. One significant feature of geometry is that it can be character— ized as a mathematical model of physical space. The pilot study was de- signed to expose the students to some activities that developed a mean- ingful relation between the terms (undefined, defined) and assumptions (postulates, theorems) of a mathematical model and physical objects in the real world; and the nature of an axiomatic system in which the pro- cess of logically deducing propositions was applied. The study included topics in finite geometries based on a mathematical model of a finite number of points and the process of deducing propositions from other as- sumptions of the model. To emphasize the importance of the postulation- al method, a non-Euclidean geometry was introduced and studied. This non-Euclidean geometry was characterized basically by a simple altera— tion of a postulate and some pertinent definitions of a "kind” of geome- try studied earlier. The results obtained from the pilot study served as a guide in assisting this investigator to organize and complete the study for this report. 10Dale Lick, ”Why Not Mathematics,” The Mathematics Teacher, LXIV (January, l97l) p. 85. 7 Finally, the most important significance of this study lies in the fact that it observes a situation which affords students almost un- limited opportunities to do some critical and creative thinking in terms of relating that which is concrete (operations and properties) and derived from the physical models to that which is abstract (opera- tions and properties) and applied to the mathematical system. To assess the significance of this study, the following nonstatistical hy- pothesis is stated. Hypothesis I "' When selected, bright high school students are placed in a circumstance where they can study mathematical models which are embodied in physical models, they will demon— strate their abilities to deal effectively with mathe- matical models; the nature of an axiomatic system, and the process of logically deducing propositions for in- vestigations. The case study procedure is the technique used to determine how well the . I students were able to realize the expectations stated in the hypothesis. Organization of the Final Report Chapter I includes an introduction, rationale for the study, a description of the pilot study, and the assessment hypothesis. Chapter II includes an introduction, a review of the literature and a summary. Chapter III contains the procedure used in the study. An introduction is included; a description of the activities, a description of the stu- dents, and a summary of selected test results of students participating in the study. Chapter IV includes an introduction, a case study report of each activity as listed in Chapter III of this report and a summary. Chapter V includes the summary, discussion recommendations for further research and the conclusions. CHAPTER II A REVIEW OF THE LITERATURE Introduction To deal effectively with the kind of report investigated in this study, an out of the ordinary type of assessment was used to evalu- ate how well the individuals involved in this study performed. The assessment procedure used to evaluate the results obtained from this investigation is the case study approach. A close examination of individual, and/or small group perfor- mance is of utmost importance to the appropriate evaluation of the re— sults obtained, and it was felt that the case study approach would serve the best purpose in that from the point of view of research, this approach uses an intensive investigation of the activity,individual and/or group under observation. Good reports that the most important step in the case study is to identify the unit for investigation in the form of some aspect of an observed behavior or recorded activity.1 This study deals with the activities of students investigating speci- fic activity—based concepts which are observable and recorded. The literature however, did not contain any major studies at the secondary level similar in nature to the study in this report. ICarter V. Good, ”The Individual and Case Study: Diagnosis and Therapy,” Essentials of Educational Research (New York: Appleton-Crofts 1966) p. 313. 9 A major problem at the time of this writing is the limitations imposed upon this investigator due to the almost complete absence of similar studies. Shaughnessy makes mention of the work advanced by Zoltan Dienes who advocates the use of physical materials and games in a manipulative manner before moving gradually to formal mathematical symbols and abstract systems.2 Though most of Dienes work has been done on the primary and intermediate levels, his results have implica— tions for all levels of mathematics. Review of the Literature As was pointed out earlier in this report, research on activity— based mathematical studies at the secondary level are virtually non— existent; and none have been found that reported their results using the case study method. This, however, does not mean that there are no studies dealing with the basic ideas of mathematical systems or models; the nature of axiomatic systems or the process of logically deducing propositions from assumptions for investigations. The pilot study (Chapter I) for this report dealt specifically with the process of characterizing geometry as a mathematical model of physical space by relating physical objects of the real world to components of a mathematical model, and deducing propositions from certain assumptions in the model for investigation. Adler makes the following point con- cerning the nature of geometry as a mathematical model: Students who understand the nature of deductive reasoning and of inductive reasoning can be led to understand what is 2J. Michael Shaughnessy, ”Research in Laboratory Approaches to Mathematics at the Secondary and College Levels" (unpublished paper pre- sented at Michigan State University, 1973). -nallqm1 ejjuan: HJI janzcaads bur. alodmva i-i.e '-{'I-‘.ru.'.i"r. :_-.l_" ..t' ‘Jnub | ..., I i." - '. l.‘..' lO meant when we say Euclidean geometry is a mathematical model of physical space because it has these four characteristics: 1. Each undefined term and each defined term in Eucli- dean geometry is associated with some physical ob- ject. 2. The axioms of Euclidean geometry express certain as— sumed relationships of these undefined terms. 3. Theorems of Euclidean geometry are deductions from these axioms that express other relationships among the undefined terms and/or the defined terms. 4. If the axioms and theorems of Euclidean geometry are interpreted to be assertions about the physical ob- jects associated with its terms, then to the extent that these assertions have been tested by experiment they have been found to be approximately true. The four characteristics mentioned above are readily adaptable to the three major aspects under investigation in this study which are: (l) the study of physical models with concrete operations and properties and are adaptable to analogous abstract operations and properties of mathematical systems; (2) the nature of axiomatic systems derived from the study of the models; and (3) the process involved when propositions are logically deduced from assumptions of the systems and investiga- tions. The combined realization of these aspects should provide oppor- tunities for secondary school students to do some critical and creative thinking. Several studies reviewed in the literature were similar in na- ture to this study in that they dealt with one or more of the above mentioned aspects or some combination of them. Lewis demonstrated that teaching a course in high school geometry based on, or in part on, the components of a mathematical model showed evidence to support the fact that students developed in reflective thinking in non-mathematical areas far greater than in either the traditional course in the subject or no 3Irvin Adler, "What Shall We Teach in High School Geometry?”, The Mathematics Teacher, LXI (March, 1968) pp. 227—28. 11 exposure to the subject at all/I When geometry is organized for the specific purpose to further the ability to think critically, when mate- rials are developed to focus upon this aim, when the teaching method was directed to this end, then the students would come out of the course competent in their ability to do plane geometry and more compe- tent in their analysis of non-mathematical issues than if they had been exposed to the traditional c0urse. The most influential and widely distributed study dealing with the nature of a mathematical system and the inference of proofs was conducted by Fawcett,S In the study, students developed through class discussion a set of undefined terms, definitions and assumptions as the foundation on which they erected their geometric edifice. During the investigation, it was assumed that a student understood the nature of a deductive proof when the following was accomplished: (1) the place and significance of undefined concepts in proving any conclusion, (2) the necessity for clearly defined terms and their effect on the conclusion, (3) the necessity for assumptions or unproved propositions, (4) that no demonstration proves anything that is not implied by the assumptions. The investigator concluded in the report that: Mathematical methods illustrated by a small number of theo- rems yield a control of the subject matter of geometry at least equal to that obtained from the usual formal course by follow- ing (appropriate) procedure . . . it is possible to improve the reflective thinking of secondary school pupils . . . This im— provement is general in character and transfers to a variety of situations. 4Harry Lewis, ”An Experiment in Developing Critical Thinking Through the Teaching of Plane Demonstrative Geometry,” (unpublished doctoral thesis, New York University, 1950). 5Harold P. Fawcett, The Nature of Proof, Thirteenth Yearbook of the National Council of Teachers of Mathematics (New York: Bureau of Publications, Teachers College, Columbia University, 1938). 6Ibid, of Juan .I-r. ark: (.J Edam”... “Ev . Lqua Lojucfirjuib {Isbiu IHL . -'- . . . .- ..-.. r. .. , .JJ-L .' L " ‘J'r..I( I :-I.- =1!.I.-h|:'-d{. , l" i ‘3. I . .. . . _,:|_.r_ t . I} 12 The investigation conducted in this report relies greatly on the reflective thinking of high school students and their ability to transfer knowledge acquired to a variety of familiar as well as unfa- miliar situations. Ulmer conducted a major study in which he pointed out that it is possible to cultivate reflective thinking under normal classroom conditions without sacrificing an understanding of geometric relationships; and that students at all levels are capable of profit— ing from a method when definite provisions are made to study methods of thinking as an important end in itself.7 Another study that investi— gated the role of logical proofs and critical thinking was conducted by Platt.8 In the process of the study, an evaluation was made of the ef- fect of the use of mathematical logic in high school geometry on: (1) the achievement of students in high school geometry, (2) achievement in reasoning in geometry, (3) critical thinking of students, and (4) at- titude of students toward logic, deduction, and proofs in mathematics. The investigator's analysis of the results appeared to support the fol— lowing conclusions: (1) mathematical logic is an appropriate area of study well within the capability of successful high school students, (2) there is no loss of achievement in geometry caused by devoting time to the study of mathematical logic even in the traditional course, (3) in- cluding instruction in mathematical logic appears to produce a more ef- fective treatment of high school geometry with high achieving students in its effect upon student achievement in reasoning in geometry. 7Gilbert Ulmer, ”Teaching Geometry to Cultivate Reflecting Think— ing: An EXperimental Study with 1239 High School Pupils,” Journal of Experimental Education (September, 1939). 8John L. Platt, ”The Effect of the Use of Mathematical Logic in High School Geometry: An Experimental Study” (unpublished doctoral the- sis, Colorado State College, 1967). 13 The investigator reported, however, that the inclusion of mathematical logic in high school geometry does not result in a course which is significantly superior to traditional courses in its over-all effect upon student achievement in reasoning in geometry, deductive thinking and proof in mathematics. The results of this report deal with a variety of physical mo— dels that were used to derive unfamiliar mathematical systems and the procedures used to deduce some theorems (propositions) from the assump- tions of systems and prove them. The pilot study dealt with unfamiliar mathematical systems by studying a non—Euclidean and finite geometries as mathematical systems and using the components (i.e. undefined terms, definitions, assumptions, theorems and laws of logic) of the system to deduce propositions and prove them. Two studies reviewed in the literature that dealt with the un- familiar approach to studying high school geometry as axiomatic systems were formulated by Keezer9 and Beard.10 Keezer formulated an axiomatic system based on three primitive notions related to the following: (1) "point,” (2) a four-termed interior relation among points, and (3) a six-termed equi—distance relation among points. The axioms of the sys- tem were divided into five groups which were concerned with: (1) set relations, (2) interior relation or betweenness, (3) the equi-distance relation (congruence), (4) continuity and parallelism. According to the author, the axioms characterizing the interior relation, logically 98r. J. M. Keezer, ”An Axiom System for Plane Euclidean Geometry" (unpublished doctoral thesis, St. Louis University, 1965). 10Earl M. Beard, ”An Axiom System for High School Geometry” (un— published doctoral thesis, The University of Wisconsin, 1968). Ell-H'J. " . I I. I'_|_' III. 'J'.‘ 14 implied theorems analogous to the betweenness property of point on a Euclidean line. The six-termed equi-distance relation among points is used to define congruence relation on certain point sets; this develop- ment of congruence relation is then used to define equality between line segments. Theorems are proved analogous to congruence properties of line segments and triangles in Euclidean geometry. The investigator assumed a two-dimensional Cartesian space based on the undefined set of points and reported that the set of axioms proved to be consistent in a Cartesian model for the system. Finally, the author demonstrated that the system characterized plane Euclidean geometry by proving that all the plane axioms of a known categorical system for Euclidean geometry are logical consequences of the given system. Beard formulated an axiomatic system for high school geometry based on the principles of isometries. The purpose of the study was to seek answers to the following questions: (1) Is it feasible to develop a course beginning with fundamental ideas suitable for use in the high school that used transformations as the basis for the development of plane geometry? (2) If so, what could be a sequence of fundamental theorems? (3) What comparisons could be made between such a course and the standing or existing geometry course? (4) What special charac— teristics might such a course have that would be useful to other facets of the mathematics curriculum? The content comprised the usual mate- rials found in the first year course with some modifications to accommo- date the major premises of the thesis; such as: the geometry of the triangle, similar figures, circles and parallels. The author reported that the concept of area theory was omitted since a transformational 15 approach does not simplify the topic. The study developed a geometry from fundamentals which meant that no background in Euclidean geometry was necessary. A suggested sequence of theorems for standard topics in high school geometry is given along with some proofs involving transfor- mations for comparative purposes. In the final analysis, the author made a comparison of the proposed and existing geometry courses with respect to: degree of rigor and intuition possible, types of proofs available and concepts of congruence. m The studies reviewed in the literature lends support to the basic ideas expressed in the hypothesis of this study in that high school students can understand the nature and structure of mathematics based on models. That implied relationships between objects found in the physical world and components of an abstract mathematical system can be determined. The studies also imply that is is well within the capabilities of high school students to deal effectively with familiar and unfamiliar mathematical systems which they can study and logically deduce propositions from assumptions in these systems and prove them, thus developing reflective, critical and creative thinking in secondary school mathematics. l4 5 .L 4, 71,24. 4., _'_.,___7 _' . A >. LA‘AA__..- .....L.‘. .CA-c CHAPTER III PROCEDURE Introduction The primary source of techniques, ideas, and methodology for individual and group activities for this study were adapted from Labora— tory Manuel for Elementary Mathematics by Fitzgerald, et al.1 The activities described in the procedure were chosen from: Unit 3 Rela- tions (reflexivity, symmetry, and transitivity); Unit 4 Functions (Tower Puzzle); Unit 8 Mathematical Systems (clock arithmetic, equate heights, zigzag, rectangle: flips and turns, and equilateral triangle: flips and turns); Unit 11 Topology (Beans and Brussel Sprouts). Two other sources included were: Circle2 (counterclockwise rotations) and Instant Insanity.3 These activities provided a situation in which physical mo— dels could be studied and mathematical systems could be developed. The resulting mathematized systems can then be studied, and in fact, be extended to finite and infinite abstract systems having properties and operations embedded in those physical models. 1William M. Fitzgerald, et al., Laboratory Manuel for Elementary Mathematics, 2nd ed. (Boston, Mass.: Prindle Weber, and Schmidt, Inc., 1973). 2Mary P. Dolciani, et al., Modern School Mathematics Algebra and Trigonometry (Boston, Mass.: Houghton—Mifflin Company, 1968) p. 249. 3Commercial Purchase (Salem, Mass.: Parker Bros., 1967). 16 17 Descriptigg of Activities During the school term 1972-73, this investigator taught geome- try and intermediate algebra classes at Suncoast High School, Riviera Beach, Florida. The students chosen for this study were selected from the intermediate algebra classes. During the first semester, students in the algebra classes were introduced to the operations and properties of the real number system in an algebraic setting with emphasis on the postulational system. Operations and properties of a group were stu— died as well. Before planning activities using physical models, students were instructed on the nature and importance of the properties and operations of a mathematical system. At the beginning of the second semester, this writer planned a ten—week schedule of activities for ten students who were above average performers and highly motivated. The students were directed to complete the scheduled activities instead of regular class work. Each activity was planned to last a period of one week. All ses— sions were recorded on audio tapes as the students worked through the weekly activity. Written records were kept of each activity and re- tained in a folder prepared for each student. During the course of the study, students worked in groups and as individuals in their regularly scheduled class periods. There were two students in one class, five in a second class and three in another. The activities were scheduled and completed in the following sequence: 1. Equilateral triangle: Flips and Turns 2. Circle: Turns 3. Clock Arithmetic: Modulus 8 18 4. Equate the Height 5. Zigzag 6. Relations 7. Rectangle: Flips and Turns 8. Beans and Brussel Sprouts 9. Tower Puzzle 10. Instant Insanity 1. Equilateral Triangle: Flips and Turns The students were given an equilateral triangle model prepared from cardboard and a background sheet of paper labeled as in figure 3.1. Stud Figure 3.1 The equi—lateral triangle was attached to the sheet using a thumbtack at the center 0 of the triangle. The students were instructed to consider as elements of the group turns (rotations) and flips of the triangle into itself from the start position as follows: 19 I1 = rotation about 0 counterclockwise 120o r2 = rotation about 0 counterclockwise 2400 f1 = reflection about Kz 180o (pick the triangle up and turn it over) £2 = reflection about My 1800 f3 = reflection about LX 1800 e = the identity element (no turn at all) The symbol 9 meant "followed by" which denoted r1 9 r1 = r2 indi- cated that the triangle was first rotated (r1) counterclockwise 1200 about the center 0 from the start position, then "followed by" r1 or a second counterclockwise rotation (r1) of 120°. This resulted in 2400 which was r2. Other elements of the group were demonstrated based on the conditions for rotations or flips. Tables were constructed for first operations using e1, r1, r2, f1, f2, and f3 "followed by” second operations e1, r1, r2, fl, f2, and £3. The students used the results in the table to investigate five basic properties: closure, identity, inverses, associativity and commutativity which are the pro- perties of an Abelian group. Other properties of the equilateral tri— angle investigated and identified were the order of the group, the generator of the group and whether or not the group was cyclic. Sub— groups of the group were investigated for all of the above mentioned properties of the group. 2. Circle: Turns To investigate the circle, each student was given a circle model prepared from cardboard and a background sheet of paper labeled as shown in figure 3.2. 20 Start K 70 I30 Figure 3.2 The circle was attached to the sheet using a thumbtack which al— lowed the circle to rotate about its center. The students were instruct— ed to consider three counterclockwise rotations about the center of the circle. Those rotations were designated from the start position as follows: a = rotation about the center counterclockwise 900 b = rotation about the center counterclockwise 1800 c = rotation about the center counterclockwise 2700 e = rotation about the center counterclockwise 3600 or no rotation at all As in the first exercise, <:> represented the operation ”followed by." Using the elements of the model and the operation, the five basic pro- perties of an Abelian group were investigated as well as the properties of the order of the group, the generator of the group and Whether or not the group was cyclic. Students were instructed to investigate the 21 physical model for a subgroup and all properties of the subgroup anal- ogous to the group. 3. Clock Arithmetic: Modulus 8 The physical model used in this activity was a circular piece of cardboard with the numerals 0, 1, 2, 3, 4, 5, 6 and 7 painted on the face of the circle and a pointer which could be rotated about the center of the cardboard circle as shown in figure 3.3 Figure 3.3 To build a mathematical system, the numerals on the face of the clock were considered elements, an operation addition was denoted by <:>. The elements and the operation were related as follows: x <:> y = q, where q was found by starting the pointer at 0 then moving it clock- wise x hours, followed by moving it clockwise y hours. Q would repre- sent the "sum" of x and y. Using the numerals 0, l, 2, 3, 4, 5, 6 and 7 as left and right addends, a table was constructed for the operation G; . The system was investigated for the following properties: clo- sure, an identity element, inverses of elements, associativity and commutativity. Multiplication for the system was denoted by (:> where 22 x (:> y = k and k was the number obtained by starting the pointer at 0 and moving it clockwise y hours x times. Using the procedure described above, a table was constructed for <:> and the following properties were investigated: closure, commutativity, associativity, identity ele- ment, an inverse element and the distributive property of (:> over c:). 4. Equate the Height The physical model mathematized and studied in this activity consisted of colored rods and several square pieces of paper which had the same area as one of the square surfaces of any of the colored rods. Each square piece of paper was identified by writing the word "plane” on each piece. The colored rods varied in heights and were used as ele- ments of the set for this system. The following names were noted: orange, blue, brown, black, dark green, yellow, purple, light green, red, white and plane. An operation ”circle—times” denoted by 69 on the set resulted in the following: place the first mentioned rod on its square surface, place a second mentioned rod on its square surface along side of the first rod. The result was the element (rod) which was placed upon the shorter of the two rods to make them the same height. See figure 3.4 for an example of brown (:> red = dark green. 1" 1'7 a! Figure 3.4 23 Using the elements plane, white, red, light green, purple, yel— low, dark green, black, brown, blue and orange as left factors and right factors, a table was constructed for (:J. The system was investigated to determine which of the following properties were exhibited by <:): closure, commutativity, associativity, identity element, and inverses of each element. After the system was investigated, its properties were compared with the system of natural numbers under addition, and the set of integers under the operation of subtraction. 5. Zigzag This system did not consist of a physical model, but was adapted from a series of ”Table System" exercises. The operation "zigzag” was defined on members of set R.= :0, l, 2,13} by the table below: TABLE 3.1 ZIGZAG The system was investigated for the following properties: closure, com- mutativity, associativity, an identity element and inverses of each element. 6. Relations This activity was designed to use a physical model to investi- gate the abstract notion of a relation using colored rods. Due to the nature of the objectives of the activity, definitions were provided for IIIIII’II'“ 3“” Jud .196011: 103;. LIIIA'II‘ LA :0 312:1,7103 30” hi h ”rang-0. alvl'r 24 the following: 1. Cartesian cross product 2. Relation R on a Cartesian cross product 3. Reflexivity, symmetry, and transitivity of a relation R 4. An equivalence relation 5. An equivalence class Considering x and y as elements of a relation R, a table was constructed to investigate the properties of reflexivity, symmetry and transitivity with respect to eight possible combinations based on the following: (x, y) (R if and only if: 1. x ”has the same length as" y 2. x ”is shorter than” y 3. x ”has a different length than” y 4. the length of x exceeds the length of y by an amount equal to the length of the white rod 5. neither x nor y are purple 6. the difference between the lengths of x and y is less than the length of the light green rod 7. x ”is shorter than, or has the same length as" y 8. x and y have the same color, or if the length of x exceeds the length of y by an amount less than the length of the yellow rod TABLE 3.2 EQUIVALENCE RELATION 25 The equivalence relations as defined on the set of colored rods was in- vestigated to determine the following: (x, y) e R if and only if: 1. the length of x differs from the length of y by an amount equal to a multiple of the length of the red rod the names for the colors of x and y begin with the same letter Finally seven non-physical abstract relations were investigated to de— termine which property (ies) was exhibited by each relation, and whether any of the relations was an equivalence relation; where any relation was an equivalence relation, its equivalence class was listed 1. 2. R = {(1, 2), (2, 1)}, as defined on {1, 2 .1. R = {(1, 1), (2, 2), (2, 1)} , as defined on {1, 2}. R = {(1, 1), (2, 2), (l, 2), (2, l) , as defined on 1, 2}. R = i(1,2), (2,3), (3, 4)} , as defined 01121, 2, 3, 4}. R = )(1, 2), (2,3), (1,3)} , as defined on $1, 2, 3}. R = (0, 0), (1, 1), (2, 2), (1, 2), (2, 1), (0, 1), (1, O) ‘, as defined on t 0, 1, 2 (x, y) e R if and only if the last name of x and y begin with the same letter, as defined on the set of students in Suncoast High School. 7. Rectangle: Flips and Turns The physical model used in this activity was prepared from a piece of cardboard, called a frame, by cutting a rectangular hole in it and fitting a rectangular piece of cardboard, with an asterisk placed in the upper left corner, in the hole as shown in figure 3.5. 26 * Ruionqlb frrxw~\¢. Figure 3.5 The rectangle with the asterisk was moved in such a way that the aste- risk was in a different place as the rectangle was fitted into the hole of the frame. Four ways were determined in which the rectangle could fit by starting with the asterisk in the upper corner of the rectangle. See figures 3.5a, 3.5b, 3.5c, and 3.5d. / C 3 - Figure 3.5a Figure 3.5b I Q9 # I , t | d--- -—------o n ' G l I I Figure 3.5c Figure 3.5d 27 3.5a Removed from the frame and replaced in the same way 3.5b Rotate the rectangle 180O (asterisk in lower right corner) 3.5c Flip the rectangle about its vertical axis (asterisk in the upper right corner) 3.5d Flip the rectangle about its horizontal axis (asterisk in the lower left corner) These flips and turns resulted in a set of four transformations of the rectangle denoted by Ea, b, c, dIl. An operation was defined on the set by performing one of the transformations "and then" another. The operation "and then” was denoted by G;- A table was constructed as usual and the system was investigated to determine the following pro- perties: closure, commutativity, associativity, an identity element and inverses for each element, paring elements and their inverses and listing those elements having no inverses; and finally a comparative study of the properties of Rectangle with the Clock Arithmetic System. 8. Beans and Brussel Sprouts Beans and Brussel Sprouts was a topological game for two players. To start the game, an arbitrary number of dots were marked on a piece of paper; players took turns connecting any two dots or connecting a dot to itself. When arcs were drawn, new dots were marked on the arc; no arc was crossed with a connecting line, and no dot was the endpoint of more than three arcs. The following questions were considered by players: 1. Does the game always end? 2. Is there always a winner? 3. Beginning with two dots, what is the maximum number of arcs that can be drawn before the game ends? What is the minimum number of arcs that can be drawn to end the game? Can you predict who will be the winner? 28 4. Apply questions 1, 2, and 3 for a game beginning with three dots; four dots; and five dots. 9. Tower Puzzle The tower puzzle pieces for this study were made from construc- tion paper using the following: 4 squares were cut from a piece of con— struction paper of one color with the following dimensions — 1” x l", 2" x 2", 3” x 3” and 4” x 4”. Using a different color of construction paper, 3 squares were cut with the following dimensions - 1%” x 1%", 2%" x 2%", and 3%” x 3%”. These seven squares represented the seven discs. A piece of construction paper of dimensions 6” x 18” with three circles drawn five inches apart along the length of the paper was used instead of spindles. The circles represented the spindles for this game. The object of the game was to start with a pile of squares in descending size, the largest square on the bottom and the smallest square on the top, at one spindle and move the pile to another spindle. Moves were made according to the following conditions: 1. Only one square could be moved at a time 2. A larger square could never be placed on top of a smaller square Students had to determine the least number of moves necessary to trans— fer the entire pile of squares from one spindle to another using 2 squares, 3 squares, 4 squares, 5 squares, 6 squares and 7 squares; this information was recorded in a table with the number of squares-in one column and the number of moves in another. Students were asked to generalize their finding to a pile of n square based on the apparent results in the table. The following questions were posed for study by the students: 29 1. Why do you think the squares are of alternating colors? 2. Suppose that the squares were all the same color. Could you develop a strategy for determining which square should be moved at each turn? 10. Instant Insanity This game contained four cubes with surfaces painted in the fol- lowing colors: red, white, blue and green. The purpose of the game was to note the arrangement of the colors on the lateral sides of the cubes, before the package was opened: white, blue, green and red on one side and green, white, red and blue on the other and similar arrangements on the other two sides. The cubes were mixed very thoroughly and ten re- assembled in a manner in which they were first observed. Students were asked to formulate any generalization in the arrangements which were successful. Students The population of this study consisted of ten students selected from three intermediate algebra classes taught by this investigator. All ten students were considered to be above average or superior in their class performances. Seven of the students had been part of the student population of the pilot study conducted during the previous school year. Two of the students in the population were seniors, seven were juniors and one a sophomore. A review and study was made of the mathematical performances (i.e. total scores, percentile rank, etc.) of students in the popula- tion on the following tests: Florida State-Wide Ninth Grade Test, the mathematics section of this test is a balance between traditional and 1:2“. .1: 09:! Tulsa m; s ..‘k' . 7‘, __:fi-'I‘3; 9TB 30 9110qu «MIT. .oésitg Ems suld $316236“: S Tang} ‘4' ‘i‘paI 3de io whiz: Isa-33:2! ad: on moles v2.1: to Manoeuvres 55).“: Ibbfa one .10 7:2: hm; '33:, ,-. . x z ch 9.2:: scu'mq ed! 61;} ' ,nuot-q bnf 1;«_"o 3d) 30 contemporary topics. The more contemporary materials include such topics as: number sentences, inequalities, primes, absolute values, clock and remainder arithmetic, set notation, closure, multiplicative and additive inverses, the identity elements for addition and mulitpli- cation, and the commutative, associative and distributive principles. The test does not emphasize the abstract symbolism of logic and sets; School and College Ability Tests (SCAT, Tenth grade), Series II was designed to provide estimates of basic verbal and mathematical ability; Differential Aptitude Test (DAT, Eleventh grade) using diagrams, the abstract reasoning tests measure how easily and clearly students can reason when problems are presented in terms of size, or shape, or posi- tion, or quantity or other non-verbal, non—numerical forms; Preliminary Scholastic Aptitude Test (PSAT, Tenth or Eleventh grade) and Scholastic Aptitude Test (SAT, Eleventh or Twelfth grade); in form and content, the PSAT and the SAT are parallel. The mathematics sections measure the ability to reason with numbers and other mathematical symbols; the sections also contain various kinds of problems to be solved, stress reasoning ability rather than knowledge of specific college preparatory course in mathematics. The Florida State—Wide Twelfth Grade Test, the mathematics sec- tion covers both traditional and modern topics. It includes materials on the number system, set theory, coordinate geometry, data interpreta- tion, algebra and geometry. The table below is a summary of the tests and student's reports by percentiles, and total scores on the PSAT and SAT. . --’. - ., . . . ‘ . - . -. . . - r . .. ...‘ILjIl‘Ihf: Iii—{é W" Y*{ :"'.I , . .> _ v . ‘ .I: . I _ ' .. J ‘ - ‘7, . :7? I: ~If'sbinam-edaam but: 'I‘Sdiov stem! 30 afifldflhg’ “HIM" - LVWIIL‘ 1‘; ed: ,ame-zgelh mdau (abs-1:,- ”flair-1111.1 ,TAQ) :aa'l' obuéijqk Inlanewim' 31 cm as I- .1 Cam oam 0mm :1 own 1. Cam 0mm no I- do me am Ho .1 I- u- an mm .1 mm 09 00 mm mq mm On mm mm om mm ow mm mm mm mm Ms so 00 mm mm mm mm mm Hm om I- so Haum oth warm HHIU Harm Harm Nana Hero Helm Hav< mHADmmm HmMH mo wM , (i.e. rl + r2 means rotation rl "followed by" r2) and the elements of the group (e, r1, r2, fl’ f2, f3) investigate the five basic properties: (1) closure, (2) identity, (3) inverses, (4) associativity and (5) commutativity. Consi- der the subgroup G'(e, r1, r2) and investigate it using the same proper- ties. Determine the generator(s) of the subgroup, the order of the sub— group and whether or not the subgroup is cyclic. TABLE 4.1 ACTIVITY SUMMARY: EQUILATERAL TRIANGLE Students 1. The equilateral triangle characterized the properties of a group. 34 2. The equilateral triangle did not characterize an Abelian group. 3. The subgroup G' characterized the properties of an Abelian group. 4. Determined that r and r l were generators of the 2 subgroup. 5. Determined that the order of the subgroup was 3 . 6. Demonstrated that the subgroup was cyclic. 7. Supplementary activities. Student A(**) This student formulated the following conjecture: Using any of the flip elements (fl’f £3) with the opera— 2) tion O defined as in the activity, if one of the flips was used once it results in the inverse of that element; if the element was used twice, it results in the identity element. An even number of f's resulted in the identity element because we showed that used twice, it results in the identity and the definition of an even number is 2 used as a factor n times; using fl twice results in the identity and e G e any number of times is equal to e Student B(**) This student studied a series of rotations and flips and formulated the following for selected subsets: even: fn e fn = e odd: fno fn e fn = fn Rotations even: r1 9 r1 = r2 35 Student C(**) This student reported that the set of integers under addition was found to be an Abelian group. This investigation could possibly account for the fact that no report on the subgroup was submitted. Student D§**) This student formulated two theorems without proofs: Theorem 1: Flips , Using any flip element, if one of the flips is used once (or odd number of times) it results in the inverse of that element, if used twice (or an even number of times), then the result is the identity element of the set. Theorem 2: Turns Using the elements r1 or r2, if the sums of these were used three or a multiple of three times, the result would be the identity. Students E, F, and G (-) There were no records to indicate that these students had inves- tigated the subgroup for any properties. Student H(**) This student concluded that the properties of the subgroup and the integers under addition satisfied the properties of a group. 36 Activity 2 Circle: Turns Using the circle with center 0, let a, b, c and e represent counterclockwise rotations about the center with the following nota- tions: a = rotation of 900 c = rotation of 2700 b = rotation of 1800 e = no rotation or 3600 Using the operation ”followed by” denoted by (:> and the elements a, b, c, and e as a set, show the following: 1. That S constitutes a group. 2. That S is Abelian. 3. That a is of order 4 in S. 4. That S is cyclic. 5. That (b, e) is a subgroup of S. 6. That the subgroup is cyclic and its generator is b and is of order 2. 7. Supplementary activities. TABLE 4.2 ACTIVITY SUMMARY: CIRCLE Students 37 Student A(<* In addition to reporting that c was a generator of S also with order 4, this student submitted two theorems based on the following con- dition: ”If each element in the set is divided by 10, e = 36, a = 9, b = 18 and c = 27; and given that e and b are even elements; a and c are odd elements.” Theorem 1 , Th? inverse of the even element is that element, i.e. eI = e, b' = b; the inverse of the odd element is the other odd element, a" = c, and cI = a. Theorem 2 1 e'I 1:) cI = b'1 <:) a"1 (corollary to D's theorem) Student B§**) This student submitted three theorems with proofs. Theorem 1 In a circular plane choose elements moving counterclock- wise and using using the operation ”followed by” with the clockwise operation of the element's successor is always C. a (:> 13-1 = c b (E) CI1 = c c C) eI1 = c e (E) aII = c Theorem 2 If you take one operation and "followed by” to the inverse of the operation before the original operation, then you get a. 38 Proof a 63 eII1 = a -l 1) ED a = a <2 G; b-1 = a eec‘1 = a Theorem 3 The operation QB produces a generator in S if an ele- ment and its clockwise or counterclockwise inverse is used. Proof Theorem 1 counterclockwise, c Theorem 2 clockwise, a Student D(**) This student submitted the following theorem to which Student A formulated a corallary. Theorem Given S = (e, a, b, c) with a as a generator and the operation a closed, then e e a = b c. Proof Statements Reasons ll 1. e, a, b, c e S, gen, 9 closed 2. e (:> a = a 2. Identity (table) 3. b a c = (a e a) a 3. Gen. "a" (a £9 a £9 a) a”, 1. By hypothesis 4. b + c = (a e a) e c 4. Gen. ”a", subst. prin. 5. b 9 c = a e (a e c) 5. Associative prop. 6. a a; c = e 6. Gen. ”a”, table 7. a (E) c = a <:) e 7. Step 5, subst. prin. 10. e a e 8: 9 a 39 a 8. Subst. and identity a 9. Steps 2, 4 a b (3) c 10. Transitive prop. This student concluded that c was a generator of the set S. Student G(**) This student submitted the following theorems: Theorem 1 Given the circular plane, if you go in a counterclockwise direction taking one element a) the next element's inverse, its "sum” is always c. l. b 2. a 3. c 4. e e a: e 1:1 e a-1 e a-1 Corollary 1.1 If you take an element and its consecutive clockwise element with (B, then the result is point a. 1. a 2. b 3. c 4. e Theorem 2 e e-1 e e b'1 e 61 An element is a generator "if and only if” it can be generated by adding the next (counterclockwise, clockwise) in the sequence. Proof 1_ Theorem 1 2. Corollary 1.1 40 Student J(**) This student reported that the element ”e" could exist as a sub- group of the set S. In addition to that statement, this student submit- ted two theorems and a corollary for every operation "followed by", <:>. Theorem 1 In a circular plane, if you go in a counterclockwise direction using any element ”followed by” its successor's inverse, then the result is c. Proof 1. a (:> b'1 = c —1 2. b (B c = c —1 3. c <:) e = c 4. e = a = c Corollary 1.1 In a circular plane, by taking the inverse of the element of the next clockwise element, the result is a. l. b (:3 aI1 = a 2. c (E) b.1 = a 3. e (E) cI1 = a 4. a (D eI1 = a Theorem 2 An element is a generator ”if and only if” it can be obtained by using "followed by” or an element next to it. 1. a 2. b @999 41 Activity 3 Clock Arithmetic: Modulus 8 This system involved using a circular piece of cardboard with eight numerals from zero to seven painted on the face of the circle. Attached to the center of the circle was a movable pointer. To build a mathematical system, the numerals on the face of the clock i O, 1, 2, 3, 4, 5, 6, 7} were considered elements and addition denoted by G was defined as: x <:> y = q where q was determined by starting the pointers at O and moving it x hours clockwise followed by y hours clock- wise. To investigate the system for <:) the students responded posi— tively to the following questions: 1. Is the system closed? 2. Is the system commutative? 3. Is the system associative? 4. Is there an identity element? 5. Does each element have an additive inverse? 6. Supplementary activities. Table 4.3 contains the results for <:); Table 4.4 contains the results for (:> TABLE 4.3 ACTIVITY SUMMARY: MODULUS 8 6 Students 42 Student D ** This student reported that the clock arithmetic system under (:> was an Abelian group. Student F(**) This student submitted the following statement of relations: V aab=x5a+b£7(realnumbers); a (:I b = a + b To investigate the system for multiplication denoted by <:) and defined as: x (:> y = k, where k was determined by starting the pointer at 0 and moving clockwise y hours x times. The students responded posi- tively to the following questions: 1. Is the system closed? 2. Is the system commutative? 3. Is the system associative? 4. Is there an identity element? 5. Does each element have a multiplicative inverse? 6. Is there only one multiplicative inverse? 7. Does the distributive law for multiplication over addition hold true in the system? 8. Supplementary activities. TABLE 4.4 ACTIVITY SUMMARY: MODULUS 8 0 Students Student A(**) This student reported that the system was commutative for both G; and C) because the elements in the tables were symmetric about the upper left, lower right diagonal row of elements; and that this system as well as the clock arithmetic system are Abelian groups with respect to 9 and 0 Students B and J(**)# These two students working together concluded that the clock arithmetic system was an Abelian group with respect to (:> and <:> ; the identity elements for (:> and <:) were distinct, and the distri— butive property of (:> over <:> was true, therefore the clock arithmetic system characterized a field. Student C(**) This student submitted the following conclusive report: This system holds all of the properties of a group, even com— mutativity, therefore it is an Abelian group. For multiplication, if you use an even number an odd number of times, you will come up "* ’1) ‘ air-Jung 44 with an even number; if you use an odd number an odd number of times, you will get an odd number. This is a finite set. We found that when multiplying 4 by 4, if you use 4 an even number of times, you will get 0; if you use 4 an odd number of times, you will get 4. Student D(**) This student prepared an elaborate Table of Remainders for <:) and <:) based on the following statement: "The real system's + and x corresponds directly to the mod 8 system for a (E) b = c; c { 8 and a c:, b = c; c E 8.” The student claimed that the tables of re- mainders corresponded directly to the results in the table prepared during the investigation of the system using (:) and (:> . The follow- ing theorem was submitted with a proof: Theorem For the elem nts of the set {0, 1, 2, 3, 4, 5, 6, 7}, i 0, 2, 4, 6 are even elements. If the first even elements of 0, 2, 4, 61 is added to the second even element, then the sum is equal to the third even element added to the fourth even element. Statements Reasons 1. 0 e 2 = 2 1. Definition ofehypothe- sis 2. 4 9 6 = 2 2. Definition of e 3. 0 e 2 = 4 e 2 3. Substitution principle 4. 2 = 2 4. Q. E. D. Finally this student submitted the following conclusive report: For the operation a: 3 G 6 = 2, definition of 9; but if we apply some properties of the real number system, we can see that 4 units and 6 units equals 10 units, simple addition. If we use still another property we can divide 10 by the modu- lus, which in the other system is 8, 10/8 = 1 with a remainder of 2. But look, the remainder in this system (real number) is equal to the result of 4 G) 6 or 2. Two units in the real 45 system is equal to two units in the modulus system. For the operation : 3 6 = 2, definition of . Again by applying the propert1es of the real number system we can say that 3 units times 6 units equals 18 units and 18 units divided by the modulus of the system in which we are working . 18/8 = 2 with a remainder of 2 which is simple division. Again we see that the results are the same 2 units (mod 8) are equal to the units as we define them in the real number system. Student E(**) This student and her associates formulated several theorems in the system for (E) and (:> . This student submitted the following theorem: For every a and every b, when a - b :?7 (real numbers), aC)b equals the remainder of (a - b)/8. The students in this group concluded that their theorems could be used for any modulus. Student H(**) This student submitted the following conclusive report: The system is a finite set which has the properties of an Abelian group. For multiplication, the product of any element and an even number, the product is an even element. Student K **) This student submitted the following theorem: For every a and every b: a - b 5E7 (real numbers) a (:) b = a - b. Activity 4 Eguate the Height In order to investigate this system, a set of colored rods were used including a square piece of paper which had the same area as one of k. 0 O ' .411;- amszosm Iarsvsa bsaeImnx'oi 2938130888 19:? has 3:155:19; 1131'}; _ o o0 . Q u aniwollol 5d: beinimdua jhsbui-a alrtfl’ . A baa a to! 09314.1 .c " 46 the square surfaces of any of the colored rods, and was called "plane." The colored rods including the square pieces of paper were considered elements of the set. Each element, with the piece of paper, was de- signated by color which were: orange, blue, brown, black, dark green, yellow, purple, light green, red, white and ”plane." An operation ”circle times” denoted by (:) was defined as follows: Place the first mentioned element on its square surface, and place the second mentioned element on its square surface immediately to the right of the first. The result will be the element which should be placed upon the shorter of the two to make them the same height. The system was investigated to determine which properties the operation exhibited using the follow— ing questions and statements as a guide: 1. Is 6b closed on the set? 2. Is (3) a commutative operation? 3. Is ® associative? 4. Does this system have an identity element? 5. Does each element have an inverse? 6. Discuss the difference between this system and the natural numbers under ordinary addition. 7. How does this system differ from the set of integers under the operation subtraction? Due to the nature of the questions asked and the small group reporting procedure, this activity is not submitted in table form. The reports show how many students are included in the groups. Students A and D These students performed the operation (:> with the elements in b 3‘s" '2';:'I’-.-;. : . ‘_ 7 .' ‘ :I dbwg' . I 5 - . . I - q. A . sum: 6110393 an: 9:19.qu has ,5'3lzrrua urcupa sumo " “1,," - ' I ' zlh‘lii art: 'ro 3.1.3.11: BID 0: fill-3.11.41.24.12: 213511113 sienna 831 no I iujznflu 1:! menu buorit JO blucu: 1'1 .H, 'fj adj 90 Ill- 33339! 7.57.,“ 1'." 0‘." 3113 lo 47 the set and answered the questions as follows: (1) yes, the set was closed because every element (:> another element results in an element of the set; (2) yes, (2, was a commutative operation. The table showed that the results were symmetric about the upper left to lower right diagonal of "plane." (3) No, 6 was not associative because of the following example: (yellow Q red) ® white 76 yellow Q (red ® white). (4) Yes, the system had an identity element which was "plane,” and (5) the inverse of each element under 65 was the element itself. These two students reported that the natural numbers have no identity, but this system does; the natural numbers are associative under ordinary addition and this system isn't; and that under subtraction, the integers are not commutative. Students B, G and J These students prepared a table using the operation 65 and the elements of the set. The results in the table provided the following answers: (1) yes, the operation was closed, when &a was used with all elements in the set, no new elements were formed; (2) yes, (:> was commu- tative: blue Q yellow = yellow ® blue; (3) no, ® was not associa— tive because (W Q W) ® light green 94 W ® (W ® light green); (4) yes, the system had an identity represented by "plane” and (5) yes, elements had inverses because each element was its own inverse. These students reported that the natural numbers are associative for addition and is an infinite set whereas this system is not associative for addi- tion and is a finite set; the natural numbers have no identity element for addition and this system does. The students concluded that under subtraction, this system and the integers are the same if the absolute 48 value is used for the integers. Students C and H These students constructed a table for (2’ and the elements in the set. The following answers were provided based on the results found in the table: (1) yes, the operation was closed on the set; (2) yes, ® was commutative, verified by blue ® light green = (light green ® blue; (3) no, the system was not associative because (white Q red) ® purple # white £9 (red 69 purple); (4) yes, the system had an identity which was ”plane”; and (5) each element had an inverse which was found to be the element itself. These students reported that this system did not have the pro- perties of a group; the difference between this system and the natural numbers under addition was found to be that the system had an identity element and the natural numbers did not. They reported that the dif- ference between this system and the integers under subtraction was that the system was commutative and the integers were not. Students E, F, and K These students performed the operation 69 with the elements in the set, prepared a table which provided answers to the questions as follows: (1) yes, the set was closed, by the table, a C) b : c where a, b, c are elements in the set; (2) yes, fig vms commutative as the table will show by ”mirror images” of elements; (3) no, (2) was not as- sociative, verified by (blue ® red) ® yellow # blue a (red @yel- low); (4) yes, the identity element was ”plane”; and (5) each element was its own inverse, 49 The students reported that this system was not associative, the natural numbers are; this system had an identity element, the natural numbers do not have one under addition. The difference between the integers and this set was that this system demonstrated commutativity under subtraction and the integers did not. Activity 5 Zigzag This mathematical system was defined by simply presenting a set and a complete operation table instead of having the students to mathe- matize some physical model. This system allowed the student to use those ideas which he had learned from the study of previous physical sys- tems (concrete situations) and apply them to analogous abstract systems. The abstract system of zigzag consisted of the operation é; (zigzag) de- fined on the elements of set R = t 0,]fl 2, 3? by the table: TABLE 4.5 ZIGZAG and the following statements or questions concerning the system: 1. Complete each of the following to make true statements 01,3: 390: 193: 3’ =3 2. For every a of R and for every b or R, is it always true that 3. 5. 50 (a 6 b) is an element of R? a ‘.t) = b ’7 a? Is 6 an associative operation? Is there an identity element? Explain. If so, does every element have an inverse? Students performed this activity in small groups and responded to the questions as a unit. Students A and D These students reported the following results based on the table: 3. 4. 093:2 360:2 193=o 391:3 a 6 b in R was always true. a 9 b = b a was not always true. The operation 6 was not associative. There was an identity element as long as one condition was satisfied and that was using 2 as a left factor with zigzag. Every ele- ment had an inverse as long as a left factor was used with zigzag. Students B, G, and J These students reported the following results based on the table: 350: =2 =0 351:3 HO [\3 w» an» LOU.) b was an element in R. b = b ‘y a wasn't always true. 9 was not an associative Operation. There was an identity element if the first (left factor) 51 was 2 because: 2 9 a = a but a 61 2 # a where a was an element in the set; and every element had an inverse. Students C and H These students reported the following results based on the re- sults of the table: 1. 0 5' 3 = 2 3 l, 0 2 19' 3 = 0 I3 ’ l 3 2. a9 bw an element in R. a, b = bf a was not always true. 3. The operation i was not associative. 4. There was no constant identity element; each element had a different identity; and every element had a different inverse. Students E, F, and K These students submitted the following report based on their study of the results in the table: 1.093=2 3 173:0 3 3 $0 = 2 ’1 = 2. Yes, the system was closed, no, the system was not always commutative. 3. The operation 6 was not associative: (l 5 2) f 3 aé 1 f (2 f 3) 4. There was a number 2 such that: 2 b 0 = 0, 2 9 l = 1, 2 f2 = 2 and 2 i 3 = 3 and each element had an inverse. 52 Activity 6 Relations This activity was designed to use a physical model to investi— gate the abstract notion of a relation using colored rods. Due to the nature of the objectives of the activity, definitions were provided for the following: (1) cartesian cross product; (2) a relation R on a cartesian cross product; (3) reflexivity, symmetry, and transivity of a relation R; (4) equivalence relation; and (5) an equivalence class. The prOperties, equivalence relations and evaluations are listed in Chapter III of this investigation. For some reason or reasons yet un- known, only two students, A/D, completed part of this activity. Other students reported repeatedly that the exercise did not make sense to them and they couldn't understand it. The only question answered con- sistently by most students was: How many relations are there on the set of colored rods? Activity 7 Rectangle: Flips and Turns To investigate flips and turns of the rectangle as a mathemati- cal system, the physical model was prepared as described in Chapter III of this report. A set of four transformations involving flips and turns was de- termined and denoted by {eg b, c, d 3 . An operation was defined on the set by performing one of the transformations ”and then” another. The operation "and then” was denoted by (a? and beginning each time at 53 the starting position. One operation was performed, b ”and then” c which was noted to be the same transformation as d, and placed in the table. Students were to complete the remainder of the table using left transformations "and then” right transformations. The elements of the system is“ b, c, d} and the operation ”and then” were investigated to determine the following properties for the model: (1) closure; (2) commutativity; (3) associativity; (4) an iden- tity element; (5) inverses; and (6) how this system compared with the clock arithmetic system (Activity 3). TABLE 4.6 ACTIVITY SUMMARY: RECTANGLE Students The students responded positively to the first five properties listed to be investigated for this activity, however, the responses to statement six was divided into two major groups: Students A, D, E, and K reported that the rectangle system and the clock arithmetic system satisfied the properties of a group, however, the other students, B, C, F, G, H, and J reported that both systems were Albelian groups. 54 Activity 8 Beans and Brussel Sprouts Beans and Brussel Sprouts was a topological game for two players. To start the game, an arbitrary number of dots were marked on a piece of paper. Two players alternated turns drawing arcs connecting any two dots or connecting one dot to itself. When an arc was drawn, the player had to mark a new dot on the arc, No arc could be crossed and no dot could be the end point of more than three arcs, The winner was the last player able to draw an arc. The following questions were considered: l. Does the game always end? 2. Is there always a winner? 3. Beginning with two dots, what is the maximum number of arcs that can be drawn before the game ends? What is the minimum number of arcs that can be drawn to end the game? Can you predict who will be the winner? 4. Answer Questions 1, 2 and 3 for a game beginning with three dots; four dots; five dots. Students completed this activity with two players performing at a time; the results summarized represents their findings. Students A and D These two students completed an elaborate network of activities indicating the number of dots and connective conditions; who started; who won and the number of arcs at the end of each game. 55 Starting Number Who Who Number Conditions of Dots Started Won of Arcs _ 1. No connection of a dot to itself 2 D D 10 2. One connecting itself 2 A A 10 3. One connecting itself 2 D D 10 4. Two connecting themselves 2 A A 10 5. None connecting a dot to itself 2 D D 10 6. None connecting a dot to itself 3 A D 12 7. One connecting 3 D D 14 8. Two connecting 3 A A 8 9. Three connecting 3 D D 14 10. One connecting 3 A D 16 ll. No connecting 4 D A 20 12. One connecting 4 A D 20 13. Three connecting 4 D A 20 l4. Two connecting 4 A D 20 The students submitted the following summary: a. Game always ends b. Some always win c. No conclusions Students B, G, and J These three students completed the game by playing two at a time and summarized their finding accordingly: 56 Started Finished Winner 1. 7 25 Second player 2. ll 40 Second player 3. 4 l5 First player 4. 6 20 First player The students submitted the following conclusions: 1. The first player won when there was an even number of dots at the start. 2. The second player won when there was an odd number of dots at the start. 3. The number of dots at the finish was a multiple of five. The group responded to the questions concerning the exercise accordingly: 1. Yes, the game always ends. 2. Yes, there is a winner. 3. Starting with two dots, five arcs was the maximum number of arcs that could be drawn before the game ends; five arcs was the mini— mum number of arcs that could be drawn to end a game. The winner can be predicted. Students C and H These two students submitted a summary to the questions asked about the activities. l. No, the game does not end because for every two dots, there is an infinite number of two dots that can be connected to each other. 2. There can be no winner because there are an infinite num- ber of turns that can be taken. 57 3. Beginning with two dots, there is no way of determining the maximum number of arcs; there is no way to determine the minimum. A winner cannot be determined because there is no winner. Students E, F, and K These three students completed the game by playing two at a time and submitted these results: 1. Yes, the game will end. 2. Yes, there will be a winner. 3. No. of Dots Max. Min. Connected Arcs Arcs 2 10 7 3 l4 l4 4 22 18 5 28 22 6 34 __ Activity 9 Tower Puzzle The tower puzzle consisted of a piece of construction paper 6” x 18” with circles drawn 5 inches apart which served as spindles, and squares of appropriate sizes. The object of the game was to move a pile of squares from one ”spindle” to another maintaining the rela- tive positions of the squares, which was the smallest square on the top, largest square on the bottom. Moves were made according to the following conditions: 1. Only one square could be moved at a time. 58 2. A larger square could never be placed on top of a smaller square. Determine the least number of moves necessary to transfer an entire pile of squares from one ”spindle” to another. Try playing the game with 2 squares, 3 squares, 4 squares, etc. Once the least number of moves required to transfer the entire pile of squares has been de— termined, record the information in a table and try to generalize that information to finding a pile of n squares; then determine the least number of moves needed to transfer any pile. Provide answers to the following questions: 1. Why do you think the squares are of alternating color? 2. Suppose that the squares are all the same color. Can you develop a strategy for determining which square should be moved at each turn? The students worked in their groups for this activity, the re- sults represent their combined efforts. Students A and D These two students completed the exercise after many trials and submitted their report which was brief: By working with the tower, it can be found that in order to have the right number of movements, you must not place one colored element on another element of the same color. Using different numbers of squares, they worked for the formula 2r+l where r was the previous number of movements. Students B, G and J These three students completed the moves according to the con- ditions; results were recorded in a table and conclusions drawn {:I'I-I" If“! Ell ..1'r _ . “ii: 1': 59 accordingly: Number of Number of Squares Turns ”Generalization” 0 0 1 1 2(0) + 1 2 3 2(1) + 1 3 7 2(3) + 1 4 15 2(7) + 1 5 31 2(15) + 1 6 63 2(31) + 1 7 127 2(63) + 1 Let y equal the number of consecutive number of squares, the number of turns increase according to the formula 2x + l where x is the previous turn for increasing y's. The least number of turns possible was obtained by always placing a square on another of a different color. Students C and H These students performed the activities, recorded their results and provided the following: Number of Number of Squares Moves 2 3 3 7 4 15 60 Number of Number of Squares Moves 6 63 7 127 n 2x + l When the number of squares was increased by l, the number of moves was increased by twice the preceeding number of moves plus 1. The squares are alternating in color because it's easier to distinguish between them. If all squares were the same color use the smaller of each advancing square. Students E, F, and K These students completed the activity as a team; after many trials, the compiled their results in a table with the following conclusions: The least number of moves to transfer one pile to another spindle Number of Number of Squares Moves 2 3 3 7 4 15 5 3l 6 63 7 127 n 2n + l Conjecture: ” 2n + 1 is the number of moves where n is the preceding number of moves (for the preceding number of squares.) By demonstra- tion we proved it, therefore it is a theorem. The squares are of alternating colors because they are a guide in working the puzzle. The squares must go together in alternating colors.” “4 7 H . " ' _;- L JflJ build 1‘ 9.,. ’ ‘e' 11 550.556 31-2105 .33 adjjnn'xsjhs 93w»: ... _- _ adj 929v askeupa [Is 31.:nan3 n-newied dal v- 1.1 6! If gang» an ionsvbs (JUL) -0 J11 hum an: am: talcum 3 Sch .1 ,3 annsbuia 61 Activity 10 Instant Insanity This game was played with four cubes with their faces colored red, blue, white and green. The purpose of the game was to take note of the arrangement of the colors before the package was opened. At the start of the game, the cubes were mixed very thoroughly and then rear- ranged in a manner similar to the arrangement before the package was opened. Only one student submitted a report on this activity. Student G reported: "The easiest way to complete the problem is to use color combination, first same color next to each other, or any other combina- tion.’ Now a diagram of his solution: Row 1 Row 2 Row 3 Row 4 Side 1 I blue I green I white I red I Side 2 I green blue I red I white I Side 3 I white I green I blue red Side 4 I red I white I green blue Row 1 - blue, green, white, red Row 2 - green, blue, green, white Row 3 - white, red, blue, green Row 4 - red, white, red, blue ”There is no sure way to get a solution for it but the informa- tion given here might be able to generalize something; block positions may be switched.” All of the students had played the game before and their interest in the game was not encouraging as they chose not to 62 try to generalize a solution. Summary The statements, quotes, theorems corollaries and conjectures made by the students and included in this report are exactly as they were made or written. No attempt was made on the part of this inves- tigator to edit or rearrange their thoughts as they perceived them. In several instances, students reported results which were not necessarily valid for generalized cases. These conclusions should be considered conjectures and are marked with a #. The students participating in this investigation were above average and highly motivated. All were con- sidered excellent candidates for further studies in the area of math— ematics; they demonstrated time after time what can be accomplished with this kind of high school student. The students, themselves, pointed out many times that what was accomplished could be part of a normal class situation with any student enrolled in an ordinary algebra class. CHAPTER V SUMMARY AND CONCLUSION Summary The purpose of this study was to examine in detail the cognitive outcomes of high school students using physical models to study the structure and nature of mathematical systems. The study was conducted under normal classroom conditions at Suncoast High School in Riviera Beach, Florida using for the population selected high school students who were considered to be above average and excellent candidates for further studies in mathematics. The hypothesis which stated: When selected bright high school students are placed in a circumstance where they can study mathematical models, they will demonstrate their abilities to deal effectively with mathematical models; the nature of an axiomatic system, and the process of logically deducing propositions for investi— gations was realized in more than a few instances. The physical model being concrete and familiar in nature, as well as manipulative, provided an excellent means by which the student could develop an understanding of the nature and properties of a mathematical system. The students demon- strated time after time that it was possible for them to do critical and creative thinking in terms of relating that which was concrete and derived from physical models to that which was abstract and applied to a mathematical model. One activity proved to be either too abstract or too unfamiliar for the students and no results were obtained. 63 64 The students were familiar with the reflexive, symmetric and transitive properties and the Cartesian cross product concept as they were related to their study of the real number system in a algebraic setting. It is the opinion of this investigator that since this particu— lar activity was not representative of a physical model to be mathema— tized the students were unable to determine the significance of the characteristics of an equivalence relation as they relate to physical models. As was pointed out earlier in the report, the only question answered consistently by most students dealt with the Cartesian cross product of A x A of colored rods. Discussion The results of this research demonstrates that within the con- straints of normal classroom conditions, high school students can achieve a higher level‘of understanding the nature of: (1) model build— ing both physical and abstract, (2) the axiomatic process, and (3) the truth value of inferential propositions. What was accomplished here is not typical, nearly all courses in high school mathematics at some time or another make mention of physical models of mathematical entities and the properties of mathematical models; but all too often, these topics are treated in isolation. The significance of the model, physical or mathematical, should be emphasized throughout the course as the need arises. The content of the intermediate algebra class, from which the students were selected, was taught based on the axiomatic approach which probably contributed to the results obtained. 65 Recommendations for Further Research The following recommendations are made as a direct consequence of this investigator's participation in the pilot and final studies as a classroom teacher. 1. There should be more use of the case study procedure in the elementary and middle school to investigate the content in mathematics using physical models and participatory games to characterize mathematical systems. 2. The physical models mathematized in this study should be used in an attempt to teach high school students about other mathematical systems. 3. That research be done on the effect of the use of these teaching strategies on a wider sample of high school students. Conclusions In conclusion, it can be stated that the major aspects as stated in the hypothesis are reasonable and proper goals to work for and attain with high school students. As was pointed out earlier in this report, models have an unex— celled power to clarify concepts and provide an invaluable means by which difficult and abstract ideas can be made simple and understandable. The students participating in this study submitted fifteen theorems, two corollaries, four conjectures and four generalizations as a testament to that fact. BIBLIOGRAPHY BIBLIOGRAPHY A. Books Anderson, Richard D.; Garon, Jack W.; and Gremillion, Joseph G. School Mathematics Geometry. Boston, Mass.: Houghton Mifflin Company, 1966. Blattner, John. Projective Plane Geometry. San Francisco: Holden-Day, Inc., 1968. Brumfiel, Charles F.; Eicholz, Robert E.; and Shanks, Merrill E. Geometry. Reading, Mass.: Addison—Wesley Publishing Company, Inc., 1960. Davis, Philip J. The Mathematics of Matrices: A First Book of Matrix Theory and Linear Algebra. Boston, Mass.: Ginn and Company, 1965. Dolciani, Mary P.; Beckenbach, Edwin F.; Sharron, Sindey; and Wooton, William. Modern School Mathematics Algebra 2 and Trigonometry. Boston, Mass.: Houghton Mifflin Company, 1968. Fawcett, Harold P. The Nature of Proof. Thirteenth Yearbook of the National Council of Teachers of Mathematics. New York: Bureau of Publications, Teachers College, Columbia University, 1938. Fuller, Gordon. Analytic Geometry. Reading, Mass.: Addison—Wesley Publishing Company, Inc., 1964. Good, Carter V. "The Individual and Case Study: Diagnosis and Therapy," Essentials of Educational Research. New York: Appleton-Century— Crofts, 1966. B. Reports The Report of the Cambridge Conference on School Mathematics. The Goal for School Mathematics. Boston, Mass.: Houghton Mifflin Company, 1963. 67 68 C. Manuels Fitzgerald, William M.; Bellamy, David P.; Boonstra, Paul H.; Greene, Carole E.; Jones, John W.; and Oosse, William J. Laboratory Manuel for Elementary Mathematics. 2nd ed. Boston, Mass.: Prindle, Weber and Schmidt, Inc., 1973. D. Periodicals Adler, Irvin. "What Shall We Teach in High School Geometry?" The Mathe- matics Teacher. LXI. March, 1968. Lick, Dale. "Why Not Mathematics.” The Mathematics Teacher. LXIV. January, 1971. Meserve, Bruce E. ”An Improved Year of Geometry.” The Mathematics Teacher. LXV. February, 1972. Ulmer, Gilbert. "Teaching Geometry to Cultivate Reflective Thinking: An Experimental Study with 1239 High School Pupils.” Journal of Experimental Education. September, 1939. E. Unpublished Materials Beard, Earl M. ”An Axiom System for High School Geometry." Unpublished doctoral dissertation, University of Wisconsin, 1968. Keezer, Sr., J. M. "An Axiom System for Plane Euclidean Geometry.” Un- published doctoral dissertation, St. Louis University, 1965. Lewis, Harry. "An Experiment in Developing Critical Thinking Through the Teaching of Plane Demonstrative Geometry.” Unpublished doctoral dissertation, New York University, 1950. Platt, John L. ”The Effect of the Use of Mathematical Logic in High School Geometry: An Experimental Study." Unpublished doctoral dissertation, Colorado State College, 1967. Shaughnessy, J. Michael. ”Research in Laboratory Approaches to Mathe- matics at the Secondary and College Level.” Unpublished paper presented at Michigan State University, 1973. 2.3;}..5 rs.» .. 31:11:05.. . L .....ntltwrn. ... . ‘1'...» ‘1 at... ...hod.~1~«0o.v (r (it... 3...... .4. .5}. 15:53....- «Ill... (.1 . .vv .rqi. . 5. .ln... 4 :1. (I((.1 .. .1. . ‘36:. J7.1‘.\4kix1l..ao‘owl inr4_.~w . , . l tt....c.2~n.nc¢. 1.}.1111 . vixinnysvvwkirl«o.t.¢ 52...).5. titiwqutet ...t ..x...civuv‘o..2. ictu(\:xv14-x 1....4ai....ul.}. 'afislvca aw.....$4. <5... evun:.¢.x-1:=( ct...!l.£ ((5.... ...... .1412}: 2:51... 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