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An investigation to determine the effectiveness of pictorial exposition versus symbolic exposition of… Weinstein, Gerald P. 1971

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AN INVESTIGATION TO DETERMINE THE EFFECTIVENESS OF PICTORIAL EXPOSITION VERSUS SYMBOLIC EXPOSITION OF TENTH-GRADE INCIDENCE GEOMETRY by Gerald P. Weinstein B.A., Adelphi University, 1968 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Mathematics Education We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA June, 1971 In presenting th i s thes is in p a r t i a l f u l f i lmen t of the requirements for an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s f o r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t i on o f th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date 5 ' * 7 ' Abstract The purpose of this investigation was to evaluate the effect of two modes of exposition of tenth-grade incidence geometry on logi-cally evaluated problem solving ability. To achieve this purpose two classes of tenth-grade geometry students were chosen to be the experi-mental and control groups. The two treatments, which were of nine class hours duration per group, and were both taught by the investi-gator, involved the use of a set theoretic symbolic-nonrepresentational mode for the experimental group, and a pictorial-representational mode for the control group. The content of the treatments was Eucli-dean incidence geometry-. At the termination of the treatment a criterion test was administered to both groups. The criterion test was composed of two types of problems- Type NR problems, which were believed to be most successfully solved by a symbolic-nonpictorial analysis, and Type R problems, which were believed to be most success-fully solved by a pictorial analysis. Two hypotheses, of null form, were considered: that the mean scores of both groups on Type NR problems would be equal and that the mean scores of both groups on Type R problems would be equal. Both hypotheses were tested by means of an appropriate t-statistic at the .05 level of significance. Analysis of the data indicated that both null hypotheses were not to be rejected. A difference in means on Test NR of the control over experimental group was observed at the .20 level of significance. The implication of the analysis of the data and the restric-tions imposed by the limitations of the study is that the pictorial-representational exposition was as effective as the experimental symbolic-nonrepresentational exposition for Type NR problems and for Type R problems. Since the pictorial-representational mode of expo-sition is generally considered standard practice in the teaching of tenth grade geometry i t should be continued for the present. i TABLE OF CONTENTS Page LIST OF TABLES i i i Chapter 1 . INTRODUCTION 1 Background 1 The Problem 4 Definition of Terms 5 Conjectures Relating to the Problem 11 Hypotheses 12 Statistical Form of Hypotheses 12 2 . REVIEW OF RELATED RESEARCH AND LITERATURE 13 3 . DESIGN OF THE STUDY 18 Method 18 Subjects 20 Procedure 21 Statistical Procedures 22 4 . RESULTS OF THE STUDY 23 Introduction • 23 Tests of the Hypotheses 23 5 . SUMMARY 28 Summary of study 23 Conclusions and Implications 29 Recommendations for Instruction 32 Recommendations for Further Research 33 BIBLIOGRAPHY 35 i i Page APPENDICES 37 Appendix I: Criterion Test 37 Appendix II: Examples of Problem Types and Selec-tion Process and Criteria 43 Appendix III: Outline of Treatment Schedule , 49 Appendix IV: Examples of Exposition Techniques 72 Appendix V: Examples of Scoring Techniques 78 Appendix VI: Training Test 86 Appendix VII: Pilot Study Tests 91 Appendix VIII: Data on Class Composition 97 i i i LIST OF TABLES Table Page 1. Statistics Concerning Hypothesis I 23 2. Results of Statistics Concerning Hypothesis I 2k 3. Statistics Concerning Hypothesis II 25 k. Results of Statistics Concerning Hypothesis II 25 5. Correlations between Individual Test Scores and IQ 26 Chapter 1 INTRODUCTION Baekgr ound Many mathematics educators accept, as does Albert Blank (7:14), the hypothesis that geometry is perhaps the most fertile part of mathematics for the development of both inductive and deductive thinking. Therefore these educators should be concerned to examine critically what is being taught in school geometry courses and how i t is being taught. And i f mathematics educators believe that geometry is an intellectual game which, to be played, draws on and develops both the player's spatial faculties and reasoning powers, then they should be concerned to determine whether students in school are actually taught to play this game or indeed are capable of playing i t in any meaningful way, that is to say, that they have the capa-bility of solving a wide range of problems which require the under-standing of geometry concepts. Geometry is concerned with a direct interplay between the world of physical-spatial experience and the world of abstract concepts. Albert Blank has called Euclidean geometry "applied mathematics" (?:15). However, in accord with Henkin (7*-50), the writer believes Blank's interpretation is extreme, although i t may not be an exaggerated view when pertaining to tenth-year geometry courses. Blank further states, For geometry there is a setting in which the student can 2 experiment and formulate conjectures. Once he has begun to make his own conjectures, he is well motivated to test them logically against known propositions and other conjectures. In geometry, at least, we have not yet completely hidden the mode of thought which makes his subject so exciting to a research mathematician. Geometry has the advantage of being sufficiently close to common experience that long specialized training is not needed to manipulate the concepts.(7:15) There are two aspects of geometry, one spatial-intuitive and the other abstract logical-deductive. That geometry should partake of both aspects, particularly in the secondary school, is sanctioned by many researchers and by proponents of curricular reform. Bruner (4:6ff.) suggests that mathematics learning should include ikonic or image manipulation in problem solving situations, and both Biggs (2:6) and Dienes (8:21) c a l l for the need to foster concrete spatial-intuitive development of conceptualization. Employment of a formal or informal, but not wholly rigorous, axiomatic approach to mathe-matics may well begin in geometry since the student is capable of empirically deriving a "rich" collection of "interesting" theorems, many of whose proofs are within the scope of his deductive ability (Buck, 7:23)* The student can investigate the significant underlying structures of geometry at a closer range than he can the structure of algebra (Buck, 7-23). That is to say, the student can comprehend and apply basic concepts of geometry by means of symmetries, or transformations, or vectors, for example, more easily than comparable basic concepts of algebra which, as Buck claims, "... must wait upon the development of the ring of polynomials over a field."(7*23) In geometry, the axiomatic structure can be a means to explore and make conjectures by using the restrictions imposed by the axioms to suggest consequences of the interaction of the geometric elements 3 of a problem. This is a us& of the axiomatic structure as a component of a heuristic technique (Blank. 7:1?). The role of axiomatics in geometry is stated by Buck as, "It is a valuable experience to learn to reason where intuition is not a guide." (6:470) Any axiomatic development at this time should be "naive" or descriptive since the value of such an approach is to elucidate the structure (meaning) of geometry and not to treat abstract axiomatics for its own sake (Buck, 7:20). That the axiomatic method is only a part of mathematics is implicit in two statements, one by Hadamard, "Logical proofs merely sanction the conquests of intuition," (7:58) and one by Morris Kline, "Rigor will not refine intuition which is not free." (7:59) Suppes claims a need for the axiomatic method in heuristic approaches to problem solving, a use of axiomatics which he finds woefully lacking in the secondary school (7:70), and further he stresses that axioms are important tools in the process of dis-covery. (7:71)* On the other hand a spatial-intuitive approach may be used to clarify issues concerning axioms. For example, in Euclid's Elements. and many modern textbooks which derive from i t , there are logical deficiencies in dealing with order of points on a line. The remedy, at the secondary school level, is not the a priori intro-duction of suitable axioms, but the candid admission that the ques-tion is to be handled by inspection of various figures, drawn on a blackboard, which illustrate the meaning of the axioms (7:46). The proposed import of the above discussion for the present study is that for tenth-grade geometry students, a distinction exists, relevant to learning concepts of axiomatic incidence geometry, between a pictorial representation of the consequences of axioms (a spatial-intuitive aspect of geometry) and a symbolic representation by set 4 notation of the consequences of the same axioms (an abstract logical-deductive aspect of geometry). If the above distinction is shown to have a significant influence on the ability of students to solve incidence problems, then the teaching of axiomatic geometry should be re-evaluated. The above issue is of particular importance with regard to many incidence problems whose pictorial representations would, at certain stages of the solution, involve situations not existing in real world spatial experience. An example of such a situation is the "picture" of a line crossing three sides of a triangle without passing through a vertex. This situation arises in the proof that such a line does not exist. The Problem The question arises as to whether the f u l l range of a student's potential geometry problem-solving ability can be developed in a standard tenth year geometry course, one in which the exposition stresses pictorial representation of the content as opposed to a more indirect symbolic exposition. This study will consider two varie-ties of tenth—year Euclidean incidence problems. Both of the vari-eties involve problems which require logical-deductive reasoning, even for informal solutions. One variety involves solutions that, for tenth-year students, are, in the opinion of the writer, not directly accessible by means of "visual manipulation" whereas the other variety involves solutions that are directly accessible by this means. The varieties of problems will be called Type NR and Type R respec-tively. By a "standard course" is meant a one or two semester course at grade-ten level, taught by a qualified instructor, whose mathe-5 matics training is, in general, limited to undergraduate courses, and a course content which utilizes essentially the content of Geometry by Moise and Downs (17), but which typically excludes or does not stress the concepts of indirect proof, the more difficult aspects of line and plane incidence, or axiomatic systems. The Moise and Downs Geometry was chosen as the textbook for both groups because, in the opinion of the writer, i t more clearly demonstrates the axiomatic nature of geometry and the properties of incidence than any of the other commonly used texts and because i t is widely used in British Columbia. Definition of Terms Although i t is doubtful that one can always categorically place such geometry problems as are likely to be considered in tenth-grade mathematics into classes that are defined by the specific means that students employ to solve them, the writer believes that the majority of problems can be characterized as to membership in one of two varieties according to the involvement of visual manipulation by the student in his solution. By a student's solution is meant his final considered analysis and reasoning composed to answer the question posed. Typically absent from solutions are abandoned approaches and indications of i n i t i a l methods utilized to solve the problem. The f i r s t variety (Type R) of problems is characterized by having the usual solution of the student developed by means of what will be called the "visual mode". "Visual mode" means the employment of visual-analytic techniques, that i s , those techniques by which the 6 student modifies a figure, a drawing which represents the geometric elements associated with the problem, in order to obtain a pictorial representation of a situation which will suggest an analysis of the problem. The above modifications of the figure, collectively called the visual manipulations. are generally accomplished by either ac-tually drawing or mentally visualizing ("picturing") auxiliary lines, standard constructions, extensions to more inclusive figures, re-drawn transformations, or .related supplementary figures. The analyses of, and relationships within the representation of the problem suggested by any i f the above types of modification, are tested and verified by the student using visual observation. That is to say, he determines whether or not the figure behaves as i t should according to previously learned behavior of chalk or ink lines, wire or book-shelves, steel girders or the many common notions of geometry, true or false, to which most children are exposed before they reach grade-ten. The visual observation may suggest to the student theorems and postulates which seem applicable to the problem. It is for Type R problems that figures can readily be constructed which obey the re-strictions and conform to the limitations and requirements imposed by the problem statement, and, in general, unambiguously point to an analysis. Examples involving the visual mode are the following. 1 . Given A ABC, with B-M-C, BM = MC, and m<BAM = m < CAM, prove that AABC is isosceles. For this problem, the technique employed by five of the eight 7 persons who correctly solved the problem (twelve persons were questioned), was, after drawing the figure stated in the question, to draw auxiliary lines. Specifically lines AM^ and rays AB^, AC^and line p, perpen-i—* dicular to AM through M, were constructed. Then the solvers explored the resulting triangle congruences to find more information. The combined use of construction and extension to an inclu- sive figure is illustrated by the proof of the following. 2. In ABC, i f *AD*bisects Lk and B-D-C then show BD/CD = BA/CA. A common technique observed by the writer was to extend A ABC so that i t was included in A BBC with CE parallel to Wand B-A-E. Type R problems are characterized as being those for which solutions are initiated and continued to completion by visual mode inspection and manipulation of the figure. Other examples of Type R problems and their solutions are to be found in Appendix II. Type R problems are generally taught to grade—ten students by tactics which themselves employ techniques of the visual mode. This is a satisfactory approach for teaching the content related to these problems since they are generally analyzed and solved by stu-dents using visual mode techniques. However, the same tactics of teaching, utilizing techniques of the visual mode, which will be termed pictorial-representational mode, are frequently used in the teaching of geometry where the problems, for a student,- may not be readily accessible to analysis using visual-analytic techniques. Such is often the case in the teaching of Type NR problems. The role of visual techniques in the pictorial-representational mode is elucidated by Moise who states, 8 It is customary, in elementary texts, for the reader to be assured that 'the proofs do not depend on the figure 1, but these promises are almost never kept (Whether such promises ought to be kept, in an elementary course, is another question, and the answer should probably be 'no'.). (16:66) The second variety of problems (Type NR) is characterized by having associated with the solution of the problem a figure which cannot be drawn with standard representations of segments, lines, rays, planes, etc. Such figures generally arise from the premise-and-not-conclusion statement of an indirect proof. And such figures as can be drawn do not readily provide information or relationships useful for an analysis of the problem when visual manipulations of the sort used for Type R are employed. The student cannot readily make use of relationships learned from the physical environment to solve Type NR problems. For these problems, the writer contends, i t is not true, as the statement from a teacher's manual claims, that In geometry, the intuitive spatial-visual background is a part of common experience: we see, every day, objects which look like segments, rectangles, and circles; and mass production of consumer goods has made congruence a very familiar idea ... For-mal geometry builds on these perceptions and extends them. (Moise and Downs, 18:13) In the context of the present study, problem solving ability shall mean the ability to obtain a score, a number from zero to ten, when the solution attempts are logically evaluated by the techniques outlined in Appendix V. By logical evaluation of a problem solution is meant the analysis and scoring of the pattern of logical inference which composes the solution. That is to say, the syntax composed of phrases and diagrams which are stated, together with connectives, 9 to form implications or chains of implications is sought for in the solution protocol. These implications are then evaluated as to rele-vance of premise statement and validity of argument or parts of argument by the criteria outlined in Appendix V. It is conjectured by the writer that use of techniques of the visual mode by the student, while appropriate for Type R problems, are inappropriate for the solution of Type NR problems and thus the pictorial-representational mode should not be used to teach Type NR problem solving. The above conjecture was derived from observation by the writer of grade-ten students, the majority of whom devised pictorial models for Type NR problems. These models were such that the real world physical consequences of the visually perceived structure of the model did not logically lead the student in his reasoning to a valid solu-tion of the problem. That is to say, for example, a student may have produced a pictorial model of a "line" crossing three sides of a triangle, avoiding the vertices, to prove the impossibility of the situation; but then the student deduced conclusions (correct or not) that were not valid consequences of the pictorial model used. In the opinion of the writer, this inability to form mental "pictures" occurs because the subject's experience of perceiving his spatial surroundings supplies him with insufficient visual referents, from whose interaction he can draw logical consequences. For example, for most tenth-grade students, there is no visual referent on which to base the concept of several parallels to a given line through a given 10 point not on th© line. On the other hand, there are many visual referents for the geometric concept of three planes whose intersection in space may produce 0 , 1 , 2 , or 3 lines or a single point. Most tenth«»grade geometry problems that f a l l into the category of Type NR are incidence theorems, many of which either because a direct proof is more d i f f i -cult or is lacking require an indirect proof. The figures associated with indirect proof incidence problems are not amenable to easy visual manipulation because the figures must generally represent a state of affairs which is not only a geometric contradiction but is also con-trary to ordinary physical possibility. As a specific example of this lack of facility, consider the following problems. 1. Given convex quadrilateral ABCD with AC bisecting ED, prove area A ABC = area AADC. 2 . Given that ray AB*intersects ray PQ*in segment AP, which of the following may be true: (a) Q-A-P-B, (b) A-B-Q-P, (c) A-Q-B-P, (d) A-P-Q-B, (e) P-Q-A-B ? 3 . Given that lines m and n are distinct, then prove that they cannot intersect in two points. It was observed by the writer during a pilot study that approximately 60 percent of the students of two tenth-year classes which participated could, at the close of the 1969-1970 year's work, solve problems 1 and 2 , whereas only two out of 50 could partially prove or informally justify a solution to problem 3 . even though a discussion of the appropriate axioms had taken place. 11 The set of techniques to which Type NR problems are most sus-ceptible will be called the nonvisual mode. Usually, the techniques of this mode employ various set theoretical interpretations of the problem and employ the logical rules of set theory and associated models, e.g. Venn and Euler diagrams and truth tables, to analyze the problem and to devise a proof. Examples of Type NR problems and their solutions are to be found in Appendix II. Presentation of geometric content in a manner which utilizes techniques of the nonvisual mode, i.e. interpretation of geometric entities and relations in terms of set notation and set operations, will be called the symbolic-nonrepresentational mode of exposition. Conjectures Relating to the Problem After surveying appropriate literature and interpreting the results and after observing,while teaching, the actions of two geo-metry classes, i t is the contention of the writer that: I. The present program of high school geometry: 1. stresses the development of visual-analytic approaches (visual mode) to the exclusion of other techniques, 2. treats the concepts of logic and axiomatics as unrelated topics which are present in the curriculum at this point only because the ideas of geometry easily illustrate them, 3. gives l i t t l e stress to heuristics, strategies, and techniques of problem solving and proof construction beyond implementing the obvious consequences of diagrams and pictures. II. From past experience, the student possesses a repertoire of 12 heuristics and strategies that are almost exclusively of the visual-analytic form, and this: 1. impedes him in the solution of Type NR problems by producing a rigidity of thinking, or Einstellung, in his solution strate-gy, and 2. inhibits the acquisition of nonvisual techniques of problem solving, Hypotheses (null form) It is hypothesized that i f one group of students at the grade-ten level in secondary school is exposed to a treatment (the experi-mental treatment) during which incidence geometry content is taught via a symbolic-nonrepresentational exposition, and another group of grade-ten students is exposed to a treatment (control group treatment) during which the same content is taught via a pictorial-representational exposition, then: H-|: Students of the experimental group treatment will obtain scores on a criterion test which do not differ significantly from scores cf control group treatment students on the same test when the con-tent of the test is Type NR geometry problems. Ez: Students of the experimental group treatment will obtain scores on a criterion test which do not differ significantly from scores of control group treatment students on the same test when the content of the test is Type R geometry problems. Statistical Form of Hypotheses H : The mean scores for the experimental group and control group will 1 not differ significantly («< =.05) on a test of Type NR problems. IL: The mean scores for the experimental group and the control group will not differ significantly (<**=.05) on a test of Type R problems. 13 Chapter 2 REVIEW OF RELATED RESEARCH AND LITERATURE There is good reason to believe that the visual mode is a natural consequent of a child's development. For Piaget, this deve-lopment is a psychological genesis of perceptions of the spatial-visual environment from topological (in which only gross character-istics are preserved by transformation) to projective (in which form, shape, outline, and other finer qualitative properties are preserved) and finally to Euclidean (in which quantitative and metric properties are preserved). Each stage involves experiential clarification of the restrictions and the conservation properties of the previous category (Furth, 10 :23 f f . ) . This is to say that the young child learns progressively which qualities of his physical environment are preserved and which qualities are altered by his manipulation of the environment. The implication for the present study is that the grade ten students will confront a geometry problem as though i t were a physical problem, analyzing i t in terms of its physical counterpart. As Lovell claims, for the child, geometry is essentially a system of internalized physical operations, since mathematical concepts are derived from manipulation of real world objects ( 1 5 : 2 1 6 ) . It is not the use of the visual mode of analyzing questions which the writer believes inhibits problem solving, for Polya and others state that i t is a valid and valuable heuristic for the solution of geometry prob-lems (19*59), but the sole reliance on visual intuition and on the 14 limited realm of one's experience of visual causality. The relationship of visual experience to problem solving is expressed by VanDeGeer who states that i f a subject has worked with Euclidean geometry, a visually oriented study, the content of the topic becomes his field, his reservoir of information, his repertoire of associated techniques, but i f he attempts a problem in non-Euclidean geometry, for example, he is out of his field. "The familiar field brings:a high transparency in the situations which belong to the field. But its visual self-evidence may become a fixation hampering the solution of problems in some cases." (21:143) Bruner and Kenney believe that young children are strongly guided by the perceptual nature of tasks and that they attempt them by analyzing one visual feature of them at a time. Although older children have greater problem solving ability than do younger ones, they are equally oriented toward use of visual aspects of problems, but they analyze several visual facets simultaneously (5S163)» The implication for the pre-sent study is that the form of representation of a problem will be an important factor which determines successful solution by students trained in the "usual techniques" or by common experience. This conjecture is exemplified in a study by J. Sherrill in which i t was found that in solving problems which could be solved with or without reference to a figure, a group of students performed significantly better when an accurate diagram was presented to accompany the prob-lem.- than did a comparable group for whom no diagram was given with the problem. Sherrill found, too, that the latter group performed sig-nificantly better than a group which was given the problem with a 1 5 misleading, distorted, or ambiguous diagram. This study was based on earlier conjectures by Trimble and Brownell which are referred to in Sherrill's work (20:31 f f . ) . Becker and MacLeod found that when abstract models were used, i f the instructor emphasized criti c a l reasoning rather than algorith-mic techniques, transfer of problem solving ability to related tasks increased ( 1 : 1 0 3 ) . Based on consideration of the above study i t is inferred that the nonrepresentational mode treatment of the experi-mental group on Type NR problems will result in increased ability to solve Type R problems. Also of interest is a study by Rugg in which a descriptive geometry program induced increased ability in different aspects of geometry ( 1 2 : 1 5 ) . Sandiford in his article, "Reciprocal Improvement in Learning", says, when quoting Judd, It is not far from the truth tb assert that any subject taught with a view to training pupils in methods of generalization is highly useful as a source of mental training, and that any subject which emphasizes particular items of knowledge and does not stimulate generalization is educationally barren. ( 12 :21 ) Thus, i t may be expected that the experimental group will score higher than the control group on both types of problems by virtue of the former's exposure to an abstract, logical, and generalizable mode of exposition. Of concern at this point is a study by Hall, in which he examined the effects of the training of logical proof on the geometric ability of grade ten students ( 1 3 : 2 2 ) , His experimental group was taught a program of logic followed by a program on geometric simi-larity. Both groups were then given an achievement test in geometry. He found no significant difference between the groups and concluded 16 that "such teaching does not seem to increase the ability to reason deductively in geometry as measured by performance on tests of geo-metric proof." (1306) However, the unit in logic and deduction that Hall taught is alien to the context of geometry, in that very few of his examples are in a geometric framework and the unit appears to f a l l into what Shanks calls a "ritualized and memorized exercise with no understanding of meaning." (7*63) The present study will try to develop deductive ability in both treatment groups by incorporating the logical structure intrinsic to the representational mode and non-representational mode expositions. It is not contended that geometric problem solving ability as construed for the present study is solely dependent on the mode of exposition; the role of a visual factor is s t i l l an open question (Werdelin, 23:38)* However, the writer attempted to ensure that the present study was restricted to one class of problems so that the influence of mode of exposition would become more evident. The evi-dence concerning Einstellung is also ambiguous. Hudgins suggests that drill-oriented teaching induces Einstellung and the presentation of problems with multiple approaches of solutions lessens i t (14:36-40). However in a study very similar to the classic one of Luchins, order of presentation of problem types had no significant effect on prob-lem solving (9:138)* The consequence for the present study was that d r i l l was avoided in the treatment for both groups, and, except for ques-tion I which was chosen for its simplicity, the questions on the criterion test were in random order. That visual representation of geometric concepts is relevant to problem solving is stated by Brian as," ... much of ... problem solving behavior is associated with geometric and visual represen-tation of problems," and chosen in his study of problem solving behavior was "a standard measure of spatial relations ability as the criterion instrument." (3:1202-A) 18 Chapter 3 DESIGN OF THE STUDY Method In Chapter 1 i t was proposed that an experiment be conducted to compare the effectiveness of the representational mode of expo-sition with the nonrepresentational mode of exposition. To this purpose, two intact classes of 28 and 31 tenth-year geometry students in a public secondary school were chosen as the control and experi-mental groups respectively. The instruction for the experimental group (group E) consisted of nine class hours of symbolic-nonrepre-sentational mode exposition of incidence geometry concepts and the instruction for the control group (group C) consisted of nine class hours of the same geometry content presented via the pictorial-representational mode exposition. The instruction for both groups was initiated at the beginning of the semester for geometry. For a detailed outline of the treatment material for both modes of expo-sition, see Appendix III. At the conclusion of the treatment period (approximately two weeks) a criterion test was administered to both groups. The intent of the test was to measure the ability of the students to solve prob-lems whose content was based on material presented during the treat-ment period. The test was composed of four Type R and five Type NR problems, as indicated in Appendices I and II. In order to minimize subjective bias in the oategorization of problems as to type, a panel of four doctoral candidates of the Department of Mathematics Education 19 at the University of British Columbia was chosen to categorize sample items. Based on criteria supplied by the writer, the panel unani-mously categorized the problems selected for the test. Except for question I, which was chosen for its-simplicity, a l l other questions were placed on the test in random order. The reason for utilizing geometric problems involving inci-dence properties rather than metric properties was to avoid, at least in the i n i t i a l stages of solution, involvement of algebraic and computational abilities. If the problems were accessible wholly from a metric approach then i t could not be determined to what extent the student utilized spatial-visual mechanisms; neither could i t be determined whether difficulties arose in translating elements of the geometric problems into algebraic form in the i n i t i a l stages of the problem solution. In order to avoid the above difficulties the prob-lems used in test and treatment were so restricted that they involved algebra of, at most, grade-nine level. Furthermore, problems considered were almost exclusively those for which the involvement of algebra, i f present, occurred after the i n i t i a l analysis had begun. The pool of problem items and the presentation form on the criterion test of the problems selected were determined by an analysis of two pilot studies on different groups of grade—ten students, both conducted by the experimenter. The results of the pilot studies, which consisted of two to four hours of treatment followed by a test (see Appendix VII for.tests), were interpreted for determination of a feasible level of difficulty for questions and treatment material. Completed questions were analyzed to familiarize the writer with the style, vocabulary, 20 and syntax common to grade-ten student test responses. The problem solutions from the criterion test were evaluated on a ten point scale per item. Two aspects of the solution of a problem were considered, the use of legitimate information, that is relevant rules (axioms), definitions, representations (pictures) or notation (set language), and, whether relevant or not, the use of logical argument on the information. Since a l l relevant information, rules, definitions and examples, was included in the test, and since the students had access to their notes, the presence of legitimate in-formation in a response was considered less important than the logical argument employed. Credit was given to logical use of incorrect information to a maximum of four points, and the presence of correct information with no logical argument was given a maximum of two points. For examples of scoring technique, see Appendix V. Subjects Two classes of grade-ten mathematics students were selected from a public secondary school (in a rural, middle to low socio-economic community in the Greater Vancouver area). Because of absence, the registered class sizes of 33 and 30 varied by two or three each day. In neither group was any student absent for more than 2 days, with the exception of one individual who did attend class but did not write the criterion test. The class sizes for the criterion test were: group E, 31 and group C, 28. The students of both groups were hetero-geneous with regard to mathematics background; see Appendix VIII for grades for the previous two years. 21 Procedure Both groups were taught by the experimenter for a total of nine class hours of 60 minutes each, although the effective time per class was approximately 50 minutes. The f i r s t five class periods consisted of exposition, the sixth period consisted of a training test which was intended to familiarize the students with the form of the questions which they would receive on the criterion test, the seventh to ninth periods consisted of exposition and the tenth period contained the criterion test. The experiment was begun one week after the start of the semes-ter, and the instructor normally assigned to both classes reported that no geometry had been discussed within that semester. It may be assumed, therefore, that both groups had equal knowledge of geometry-specifically only the content of informal and common experience or the informal and peripheral content of previous mathematics courses. The method of instruction for the control group was imple-mentation of standard practice in which visual representation of prob-lems were used by the instructor to explain the analysis of a problem and to justify the use of the statements in a proof for the problem. The actual drawings and discussion employed were derived from two sources, (i) observation of Type NR and R material comparable to the material included in the present study, when taught by instructors of grade ten, and (ii) from a reading of standard texts and the teachers' manuals which accompany them. For illustration of the different expo-sition techniques applied to specific problems, consult Appendix IV. Statistical Procedures The data for an individual consisted of two scores, one the sum of points credited for Type NR problems and one the sum of points credited for Type R. The above scores were denoted Test NR score and Test R score respectively. The mean Test NR score from group E was compared for significant difference from equality with the mean Test NR score from group C with a t-statistic. The mean Test R score from group E was compared for significant difference from equality with the mean Test R score from group C by use of a t-statistic. Both compari-sons were made at the .05 level of significance. For each group of subjects, the Pearson product-moment correlation of IQ and test score was computed on both tests. This yielded four correlations. The significance of the difference of each of the correlations from zero was tested at the .10 level. The level of .10 was chosen since the concern of the writer was whether any positive correlation existed; thus an oC as "coarse" as .10 seemed appropriate as i t is the maximum suggested by Glass and Stanley in Statistical Methods in Education and  Psychology (11:282). Comparisons of correlations using the Fisher z-transformation were tested at the .10 level. 23 Chapter 4 RESULTS OF THE STUDY Introduction The results of the s t a t i s t i c a l analysis of the data are presented to support acceptance or rejection of the two hypotheses and to aid i n determination of implications of the study. Tests of the Hypotheses Hypothesis I: The mean scores for group C and group E w i l l not diff e r significantly ( ° < - = . 0 5 ) on Test NR. Table 1 Statistics Concerning Hypothesis I N Mean S.D. Group C 28 13.429 9 . 6 9 7 Group E 31 10.548 6 . 0 7 3 24 Table 2 Results of Statistics Concerning Hypothesis I Results for t-statistic Results for t 1-statistic s pooled 7.981 7.981 d.f. 57 43 t or t' oL/Z, = . 0 5 2.000(approx) 2 . 0 2 0 t or t 1 observed 1.384* 1.354* level of significance of observed t or t 1 .20 .20 * not significant at pC=.05 Table 2 indicates that the observed value of t is 1 . 3 8 4 . This value of t is such that - 2 . 0 0 0 t obs 2 . 0 0 0 for t cK/2 = 2 . 0 0 0 at ° < . = . 0 5 . Therefore hypothesis I is accepted and i t can be concluded that there is no significant difference between the mean scores of group C and group E on Test NR. It should be noted that the standard deviation of scores for the two groups on Test NR was different, 6 . 0 versus 9 . 7 . An F-test confirmed the significance of the difference at the . 05 level. There-fore the data was subjected to the t'-test in accord with the recommen-dation of Introduction to Statistics, by Walpole ( 2 2 : 2 3 0 ) . The result of the t'-test was that t 1 observed is 1 . 3 5 4 . This value of t 1 is such that - 2 . 0 2 0 4 t « obs 4 2 . 0 2 0 for t' o</2 = 2 . 0 2 0 at < * = . 0 5 . Therefore hypothesis I is s t i l l accepted. Hypothesis II: The mean scores for group C and group E will not differ signi-ficantly (c<=.05) on Test R. Table 3 Statistics Concerning Hypothesis II N Mean S.D. Group C 28 I8.96O 7.804 Group E 31 20.840 8.280 Table 4 Results of Statistics Concerning Hypothesis II Results for t-statistic s pooled 8.0586 d.f. 57 t 0C/2, =.05 2.000 t observed O.892 level of significance .40 of observed t * not significant at £X =.05 The observed value of t in Table 4 is 0.892. This value is such that -2.000 <C t obs < 2.000 for t = 2.000 at d = .05. 26 Therefore hypothesis II is accepted. Thus i t is concluded that there is no significant difference between the mean scores of group C and group E on Test R. Correlations between test score and IQ were computed to deter-mine i f there existed a positive relationship. For each group of students, the Pearson product-moment correlation was computed between both Test R and IQ and Test NR and IQ. The IQ scores for each indi-vidual were the most recent entered on the individual's record, and consisted, in the main, of Otis (Form C) and Otis Higher (Form C) tests. The results of the correlational study appear in Table 5« Table 5 Correlations Between Individual Test Scores and IQ group correlation of: correlation, critical r obs. value of r for <X=.10 group E A:Test R and IQ . 4 9 * .301 (mean IQ B:Test NR and IQ 1 1 0 . 2 ) . 3 5 * .301 group C C:Test R and IQ .097 .317 (mean IQ D:Test NR and IQ 107-5 ) . 3 6 * .317 * significant at <X = .10 Each of the observed correlation values, r, was tested for significant difference from the value of zero at the tX =.10 level, 27 1 2 1" using the t-statistic: t Q b s = r/((l-r )/(n - 2)) with d.f. = n - 2. As is indicated in Table 5. correlations A, B, and D are significantly-positive at O6=.10, and correlation C is not significantly different from zero. It was noted that correlation C appeared considerably less than correlation A, the corresponding correlation between individual IQ of a group and Test R score. Therefore the hypothesis that correlation C did not did not differ from correlation A against the alternative hypothesis that correlation A exceeded correlation C was tested u t i l i -zing the Fischer z-transformation at the .10 level. It was found that the former hypothesis would be rejected at a significance level of C< = .076. Therefore i t is concluded that correlation A was greater than correlation C. 28 Chapter 5 SUMMARY Summary of study The purpose of this study was to investigate the effects of the pictorial-representational mode of exposition versus the symbolic-nonrepresentational mode of exposition of grade-ten incidence geometry upon student ability in solving two varieties of geometry problems. To accomplish this purpose two grade-ten classes were selected, one as experimental group (E) and one as control group (C). At the termi-nation of the treatment period of nine class hours which consisted of identical content presented via a symbolic-nonrepresentational mode to group E and a pictorial-representational mode to group C, a c r i -terion test was presented to both groups. The test was composed of two varieties of incidence geometry problems. They were chosen from a pool of problems similar to and including those attempted or solved by students in pilot studies. Problems from the pool were submitted to a panel who classified them as to type according to criteria supplied by the writer (see Appendix II). In the opinion of the writer, an opinion based on analysis of problem solutions by students in pilot studies, the f i r s t variety of problem, classified as Type R,would be more readily solvable by visual-analytic techniques than the other problems in the pool. The second variety of problem, classified as being Type NR, were chara-terized as problems for which a visual-analytic technique of solution appeared inappropriate. For the latter type of problem, i t was believed that symbolic-nonrepresentational techniques of solution would prove more successful than techniques of the visual-analytic approach. Data consisting of two scores per subject were obtained ( a Test NR score and a Test R score, derived from Type NR and Type R problems respectively). The means on Test NR from group C and group E were compared by an appropriate t-statistic as were the means on Test R from groups C and group E. The above statistics were tested at the .05 level of significance. Conclusions and Implications Results of the statistical analysis confirmed both null hy-potheses: that the mean on Test R for group C was not significantly different from the mean on Test R for group E and that the mean on Test NR for group C was not significantly different from the mean on Test NR for group E at the .05 level. Correlation of subjects' IQ and test scores was computed for each group on Tests NR and R to ascertain whether IQ was a highly related factor in determining test score. As Table 5 (see page 26) indicates, IQ was significantly positively correlated with test score at OC=.10 except for the case of IQ for group C and Test R scores where no significant relationship was found. It is inferred from the results of the present study that a pictorial-representational mode of exposition, as is generally stan-dard practice in grade—ten, and a symbolic-nonrepresentational mode produced virtually equivalent problem solving ability as measured by Test R test items. The mean of group C did not significantly exceed 30 that of group E on Test R, although the mode of exposition used in the treatment of group C employed visual mode techniques similar to those which both groups used to solve Type R problems. It is possible that group E learned and firmly established visual mode techniques prior to their participation in this study, and that the treatment for group C did not increase their ability to solve Type R problems beyond that of group E. It is probable that a similarity between the treatment material presented to group C and Type R problems (see Appendix III), together with past familiarity with visual geometric concepts overshadowed any positive TQ-Test R score correlation since the correlation observed was not significantly different from zero at <X=.10. It is probable that for group E, the dissimilarity between the style of presentation and Type R problems made IQ a more potent factor for Test R as indi-cated by the significant correlation (at <X=.10) (see Table 5) • Even though both null hypotheses were accepted i t should be noted to what extent they f e l l short of being rejected. The signifi-cance levels for which Hypotheses I and II would have been rejected were .2 and .4, respectively. It was subjectively judged by the experimenter that the structure of logical inference in attempted solutions of Type NR problems was essentially the same for both groups. By structure of logical inference is meant the pattern by which phrases or diagrams are worded to state implications, or chains of* implications, or various combinations of statements with logical connectives. Representational diagrams were used by both groups, although more fre-quently by group C, and the notation accompanying them reflected that 31 used in the different expositions respectively. Thus i t appears that the content was learned in the context presented and was utilized in that manner. It must be emphasized here that Type NR problems were scored primarily for consistent use of a valid logical argument, not for the formal completeness of a symbolized statement. Thus an informal pictorial illustration of elimination-of-cases was given a higher score than a correct set theoretic statement which was not utilized logically. It is also to be noted that although group E received the geometric content via a set theoretic exposition, the use of "symbolic logic" was avoided; thus both groups had at their disposal the same mechanisms for logical manipulation of information. Since the content and logi-cal form of material in the treatments was essentially the same, i t appears that the relevant difference between the groups was the form, symbolic or pictorial, by which the geometric information of a problem was interpreted by the student. This implies that a visual-pictorial interpretation is equally susceptible to the logical techniques of a grade-ten student as is a symbolic-nonrepresentational interpretation. It was observed that group E students learned the geometric concepts and terminology taught, and that they did so by a symbolic-nonrepre-sentational interpretation as demonstrated by their ability to produce many correct or partially correct proofs involving content novel to them, but utilizing symbolism peculiar to group E exposition. There-fore the symbolic-nonrepresentational interpretation of Type NR items by group E subjects appears to have made these items as capable, of solution as was measured by this study as the pictorial interpretation 32 of group C. In view of the results of the correlations of test score and IQ and the manner by which items were scored, i t is probable that for either group there was a strong reliance on general intelligence (in the manner that IQ indicates this) for Type NR problems. The role of general intelligence and IQ in the scores of Type R problems is unclear. Type R items did not require inference as elaborate or abstract as for Type NR items and the significance of the difference in IQ-test score correlation of group E over group C, significant at a .10 level, pre-sents no obvious relationship. These conclusions must be viewed in light of the limitations of the study. Apart from the possession of a cursory set of common notions, both groups were naive with respect to incidence geometry and with respect to logically precise inference via implication and indirect argument. Both groups, although of heterogeneous academic background, had a knowledge of set notation and set operations. There-fore its use in the treatment of group E did not represent an un-balanced introduction of new concepts of notation. It is possible that the duration of treatment, nine class hours per group, although suffi-cient to convey the concepts and terminology of the geometry, was insufficient to convey the concept of logical proof- the vehicle by which much of the material was presented, and, as a result, may have masked any long term inferior or superior effectiveness of the symbolic mode of exposition. Recommendations for Instructions As a consequence of this study i t is recommended that the 33 standard practice of teaching incidence geometry concepts to grade-ten students by the pictorial-representational exposition should be continued. Furthermore the inferential structure of the geometry content should also be developed in this manner. Until further research clarifies its role in problem solving the symbolic-nonrepresentational mode, as employed in the present study, should be minimally stressed. This should apply even tb<those situations arising from indirect argument where a pictorial representation does not provide, from its own structure, an adequate basis for deducing correct logical consequences. It is suggested that a symbolic-nonrepresentational mode of exposition involving geometric concepts be deferred until more sophistication concerning logical inference and proof is attained. Perhaps the symbolic-nonrepresentational mode should be introduced in a context which does not admit the strongly entrenched visual approaches— in courses such as abstract algebra or number theory. Recommendations for Further Research It is recommended that further research be conducted to clarify issues arising from this study. A f u l l year or halfjear comparison of the treatments of this study might resolve the issue of whether subjects are more capable of utilizing geometric information symboli-cally interpreted after becoming acquainted with and proficient in the techniques of logical proof as are visually trained subjects. It is suggested that comparisons of the treatments be conducted with groups of a more homogeneous composition than those of the present study. This might clarify the observed difference in correlation of IQ with Test R scores for groups E and C, and might isolate other factors which influence test scores. It is also recommended that a systematic and quantitative scheme be constructed in order to classify and rank 34 the logical inference structure of problem responses with more pre-cision than was possible in the present study. 35 BIBLIOGRAPHY 1 . Becker, J., and G. MacLeod. "Teaching, Discovery, and the Problem of Transfer of Training in Mathematics," Research in Mathematics  Education. 1 9 6 7 , pp. 9 3 - 1 2 4 . 2 . Biggs, Edith. Mathematics in Primary Schools• London: Her Majesty's Stationery Office, 1 9 6 6 . 3 . Brian, Richard Boring. "Processes of Mathematics: A Definitional Development and an Experimental Investigation of Their Rela-tionship to Mathematical Problem Solving Behavior,11 Disserta-tion Abstracts, LXVII, (1966), 1 2 0 2-A. 4 . Bruner, Jerome S. Toward a Theory of Instruction. Cambridge: Harvard University Press, 1966. 5. » and others. Studies in Cognitive Growth. New York:- John Wiley and Sons, 1 9 6 6 . 6 . Buck, Charles. "What Should High School Geometry Be?," The Mathe-matics Teacher. LXI, (May, 1968), 4 6 6 - 4 7 0 . 7 . Conference Board of the Mathematical Sciences. The Role of Axioma-tics and Problem Solving in Mathematics. Washington,D.C.: Ginn and Co., 1966. 8. Dienes, Z.P. "Some Basic Processes Involved in Mathematics Learning," Research in Mathematics Education, 1967, pp. 2 1 - 3 4 . 9 . Faltheim, A. Learning, Problem Solving, and After-Effects. Uppsala: Appelbergs Boktryekeri, 195^ » 1 0 . Furth, H. Piaget and Knowledge. Englewood Cliffs: Prentice-Hall, 1 9 6 9 . 1 1 . Glass, Gene V., and Julian C. Stanley. Statistical Methods in Education and Psychology. Englewood Cliffs : Prentice-Hall, 1 9 7 0 . 1 2 . Grose, and Birney (eds.). Transfer of Learning. Princeton: D. VanNostrand Inc., 1963. 1 3 . Hall, William E. "An Investigation to Determine the Effects of Teaching Elementary Logic to Tenth Grade Geometry Students," Unpublished master's thesis, University of British Columbia, 1 9 6 8 . 1 4 . Hudgins, B. Problem Solving in the Classroom. New York: MacMillan and Company, 1 9 6 6 T 36 15. Klausmeier, H., and C. Harris. Analyses of Concept Learning. New York: Academic Press, 1966. 16. Moise, Edwin. Elementary Geometry from an Advanced Standpoint. Reading: Addison-Wesley, 1963. 17. , and Floyd Downs,Jr. Geometry. Reading: Addison-Wesley, 1964 . 18. . Geometry: Teachers' Manual. Reading: Addison-Wesley, 1964. 19. Polya, G. Mathematical Discovery, Vol. II. New York: John Wiley and Sons, 1965. 20. Sherrill, J. "The Effects of Differing Presentation of Mathemati-cal Word Problems upon Student Achievement," Research Reporting Sections-National Council of Teachers of Mathematics 48th  Annual Meeting. Columbus: Science and Math Education Informa-tion Analysis Center, 1970. 21. VanDeGeer, J.P. A Psychological Study of Problem Solving. Haarlem: Uitgeverij de Toorts, 1957. 22. Walpole, Ronald E. Introduction to Statistics. New York: MacMillan Company, 1968. 23. Werdelin, I. Geometric Ability and the Space Factors in Boys and Girls. Copenhagen: Ejnar Monksgaard, 1961. APPENDIX I CRITERION TEST QUIZ (2) Name: Justify a l l answers with diagrams or set notation. I. If ray AB* intersect ray PQ i s AP, state the betweenness r e l a -tionship of A, B, P, and Q. I I . Prove that an angle,<ABC, together with i t s interior i s a convex set. III. Prove that 2 distinct lines which cross are contained i n exactly one plane. Note: Xou can use a l l of the rules except the note i n Definition I I . Hint: Prove this directly. IV. If A, B, C, and D are 4 distinct points i n a plane, what i s the maximum number of lines i n the plane I can use i f each line i s to contain exactly one pair of points? How are the points arranged? V. Note: Three points i n space whioh are not collinear determine exactly one plane, the unique plane which contains them. 1. Prove that i f the least number of planes determined by the distinct points A, B, C, and D i s 4, then A, B, and C cannot  be collinear. 2. Can the least number of planes determined by A, B, C, and D be Ji les ( ) , No ( ). 3< If the least number of planes determined by A, B, C, and D is one, must any 3 of A, 8, C, and D be collinear? les ( ) , No ( ). VI. Assume that k, m, and n are 3 distinct lines i n plane E such that the lines cross i n 2 points. Show that the following statement i s false: "No pair of the lines i s parallel . " VII. A, B, C, and D are 4 distinct points on line m, with C-A-D and B-A-C. 1. Where must B be i f AD intersect BD i s BD? -> -» — 2. Where must B be i f AD intersect BD i s AB? 3. State the betweenness relationship of A, B, C, and D i f BA intersect BB i s B. VIII. If a, b, and c are 3 distinct lines i n plane E with a parallel to c and b parallel to c, then prove a i s parallel to b. IX. Given triangle ^ ABC and line m, both i n the same plane such that m does not contain A, B, or C (a vertex of the triangle), prove that m cannot cross a l l 3 sides of the triangle. 40 RULES: Rule I: If P and Q are any 2 points, then there i s EXACTLY one line m which contains them. Rule II: 1. Any plane contains at least 3 points which are not on a l i n e . 2 . SPACE has at least 4 points not on a plane. Rule III: If P and Q are any 2 points i n plane E, then the line m which contains P and Q l i e s completely i n plane E. Rule IV: 1. If P, Q, R are §ny_ 3 points (in SPACE) then there i s at least one plane which contains P,Q,R. 2 . If P, Q, R are any 3 points (in SPACE) which do not l i e on some line, then there i s EXACTLY one plane E which contains P.Q.R. Rule V: If P i s ANY point not on line m then there i s EXACTLY one line t which contains P and i s parallel to m. DEFINITIONS: I. Points are collinear i f there i s at least one line which contains a l l of them. Note: 2 points are ALWAYS collinear. 3 points MAY be collinear. I I . Points are coplanar i f there i s at least one plane which contains them. Note: 2 points are ALWAYS coplanar. 3 points are ALWAYS coplanar. 4 points MAY be coplanar. A line and a point not on i t are always coplanar. 2 lines which cross are always coplanar. III. 2 lines m, n are parallel, m n, i f 1. m and n are different lines, 2. m does not intersect (cross) n, 3. mi and n li e in the same plane. IV. A set of points is CONVEX i f , for ANY choice of 2 points P, Q in the set, the segment PQ lies completely in the set. Note: Any line is convex. Any ray is convex. Any segment is convex. A circle is not convex. A circle with its 'interior' is convex. V. Note: Lines have an infinite number of points. VI. Note: If line PQ lies in plane E then so does segment PQ. VII. The interior of angle x£BAC formed by rays i l and AC* is the intersection of half planes formed by AB on side C and AC on side B. 42 VII. Note: P-Q-R-S means as a diagram: , • I f : f P Q R S or « a i • APPENDIX II EXAMPLES OF PROBLEM TYPES AND SELECTION PROCESS AND CRITERIA A pool of potential criterion test items was obtained from questions similar to those actually used i n pilo t study treatments and tests. The problems i n the pool were either solved completely or substantially by at least 1/3 of the pilo t study groups or the problems were similar to those generally discussed i n grade-ten geometry. Therefore, i t was believed that any of the problems which appeared on the criterion test, whether solved at the Atime or not, was within the range of d i f f i c u l t y of problems to which grade-ten students are normally exposed. Sixteen problems of the pool were chosen such that, i n the opinion of the writer, eight of the problems clearly exemplified the characteristics of Type R and eight the characteristics of Type NR. These sixteen problems were submitted i n random order to a panel for independent categorization as to type according to the following c r i t e r i a . Categorize the item as Type R i f you believe that i t i s l i k e l y that an average grade—ten student would u t i l i z e techniques of the visual-analytic mode to produce a solution. Categorize the item as Type NR i f you believe: 1. that an average grade—ten student would not generally employ visual mode techniques to solve the problem, or 2. that i f the student did employ visual mode techniques his solution would probably not be a valid l o g i c a l argument. A description of the terms used above was given to each mem-ber of the panel as were several examples. Those items for which there was unanimous agreement among the panel and writer as to type were chosen to compose the criterion test. Examples of problem types Examples of both Type NR and Type R problems are to be found in the criterion test, training test and pilot tests in the following order. Criterion test (Appendix I) Type NR: Questions II, III, VI, VIII, IX Type R: Questions I, IV, V, VII Training test (Appendix VI) Type NR: Questions III, IV, V, VI Type R: Questions I, II, VII Pilot test (A) (Appendix VII) Type NR: Questions II, III, V, VI, VII Type R: Questions I, IV, VIII Pilot test (B) (Appendix VII) Type NR: VI Type R: I, II, III, IV, V, VII, VIII, IX, X Further examples of problems of both types are the following which were used in the treatment exposition and pilot study training programs• Examples of Type R problems: 1 . Prove that in a plane the perpendicular bisector of a segment AB is the set of a l l points P such that PA = PB. 2. Prove that the line segment joining the midpoints of adjaoent sides of a triangle is parallel to the third side and one-half its length. 3. Prove that i f a point P is not on a line m then there exists at least one perpendicular from P to m. 4. If two parallel lines are cut by a transversal then prove that the bisectors of any pair of corresponding angles are parallel. 5. In a plane i f a line m intersects a parallelogram dividing its interior into two regions of equal area, then prove that m inter-sects the diagonals of the parallelogram at their point of inter-section. 6. In a plane, i f two circles C-j and C 2 are externally tangent at D and the circles are congruent, with AB and ND diameters of C1 and C 2 respectively, and line NC tangent to C.j at C and intersecting C 2 at^E, then prove m(AC) = m(DC) + m(DE). 7. Given a line BC and a point A not on BC* such that triangle ABC has AB>AC, and P any point on ^  such that P-B-C, prove AP>AB. 3. If A, B, C, D are four different points in space, how many lines can pass through pairs of them i f : (i) A, B, and C are collinear? (i i ) no three are collinear? ( i i i ) the points are noncoplanar? (iv) A, B, C, and D are coplanar? 9. Given triangle AABC with B-T-C and AT is the bisector of <BAC and AT is the median, prove that triangle AABC is isosceles. 10. Point D is in the interior of triangle AABC. Prove that angle 4. ADB is greater than angle 4. ACB, i.e., i f m fcADB)>m^CBj then ADB >^ACB. Examples of Type NR problems: 1. Show that a half plane contains at least three noncollinear points. 2. Show that i f m is parallel to n and m is parallel to p, where m, n and p are three distinct lines in space, then n is parallel to p, without considering perpendiculars. 3. Prove that i f a,b,t, and m are different lines a l l in £lane E such that t is parallel to a, t\ is parallel to b, and m intersects b and t, then m A a f . 4. Given a taingle ABC and a line m in the same plane, prove that i f m contains no vertex of the triangle, then m cannot intersect a l l three sides. 5. If A-M-C on line m, then prove that A and M are on the same side of any line n different from ra which contains point C. 6. If B-M-C and A, a point not on BC, then prove that M is in the interior of angle 4 BAC. 7. Show that a half plane H contains at least two distinct points. 8 . Given triangle AABC and m, a line, both in the same plane, then prove that i f m intersects AB between A and B, then m must inter-sect either AC or BC. 9. Given a plane E and a line m in E which forms two half-planes G and H, prove that G ^ ^ a n d H ^ ^ . 10. If a, b, and t are different lines in a plane E, and t is para-l l e l to a, and t is parallel to b, then prove that a is parallel to b. 11. Prove that given a line m in space and a point P not on ra, then there is exactly one plane containing both the point P and the line m. 12. If m and n are two different lines in space which intersect, then prove that their intersection contains exactly one point. 13. Prove that every ray is convex. 14. If m and n are two different lines which intersect, prove that there is exactly one plane E which contains them. APPENDIX III OUTLINE OF TREATMENT SCHEDULE 50 Session 1, Group C 1. Discussion of what a proof accomplishes with regard to making a general statement. 2 . Example, "6< average of 6 and 8<8." Do you think that the average of two numbers is always between the smaller and larger value? 3 . Class asked i f above property of average is true for a l l positive numbers, one positive and one negative number, or for two negative numbers. 4. Discussion of whether testing some possibilities would guarantee that the statement about average is always true. 5. Statement was symbolized: AC(A+B)/2^B, when A,BS are any two numbers with A<*B. 6. Proof of A <(A+B)/2 < B, with A < B given.. i.e. 1. A^B , was assumed. 2 . A+A^A+B , by a rule of algebra. 3 . Zk< A+B therefore A<(A+B)/2 , by another rule of algebra. 7. It is not that the two rules of algebra used above made reference to numbers that was important, but that they made reference to a whole class of numbers that was important 8. Discussion of idea that a proof will make a statement about a large set of possible situations. 0 51 9. Discussion and proof of other examples. 1. The sum of two odd numbers is always even. 2. The product of two odd numbers is always odd. End of session Session 1, Group E Same content as Session 1, Group C. Session 2, Group C 1. Geometry discussed as a game related to the real world in certain ways. 2. Discussion that the objects in geometry have many of the properties of common objects; for example, many objects have edges which one could stretch a string (line-like) along and many objects have surfaces like the floor or black-board (plane-like). 3. Geometry has objects which resemble stretched string— c a l l them lines and things which resemble blackboard— call them planes. 52 k. The class was asked what a line in geometry "looked" like. 5 . The discussion concluded with the idea that lines could not be seen but only representations of them could be seen. 6. The class was asked to define a line. 7. The discussion concluded that any definition made use of "line" or "straight" or "curve" whose definitions need the idea of line to begin with. 8. It was concluded that a line could not be defined without involving other concepts which could not be defined like points. 9. It was suggested that while lines could not be seen or defined they could be described by talking about how they behaved, that i s , i f one had a l i s t of rules that specified what line could "do" and not "do", one could get a good idea of what lines were. 1 0 . It was suggested that the rules should, i f possible, make lines and planes behave like the objects which suggested them. 1 1 . It was suggested that, in order to talk about lines we should have a picture or caricature of a line, something which has some of the properties of a line. 1 2 . Fig. 1 , 2 , 3 were drawn on the blackboard of two lines crossing. Fig. 1 Fig. 2 Fig. 3 13. The issue of what picture would best represent the behavior of two lines crossing was discussed. 14. Opinions were obtained that where lines cross they should cross in one point. 15- It was decided that the picture in Fig. 1 would be a good way to represent the situation of two lines crossing. 16. The idea of how a "point" should be pictured was discussed and i t was concluded that a chalk dot no larger in diameter than the width of a chalk mark for a line would be a good idea. End of Session Session 2, Group E 1. Geometry discussed as a game related to the real world in certain ways. 2. Discussion that objects in geometry have many of the pro-perties of common objects; for example, many objects have edges along which one could stretch a string (line-like) and many objects have surfaces like floors or blackboards (plane-like). 3. Geometry has objects which resemble stretched strings— c a l l them lines and things which resemble blackboards— 54 c a l l them planes. 4 . The class was asked what a line in geometry "looked" like. 5 . The discussion concluded with the idea that lines could not be seen but only representations of them could be seen. 6. The class was asked to define a line. 7. The discussion concluded that any definition made use of "line" or "straight" or "curve" whose definitions needed the idea of line to begin with. 8. It was concluded that a line could not be defined without involving other concepts which could not be defined, like point. 9. It was suggested that, while lines could not be seen or defined, they could be described by talking about how they behaved, that is i f one had a l i s t of rules that specified what lines could "do" and not "do", one could get a good idea of what lines were. 10/i It was suggested that these rules should, i f possible, make lines and planes behave like the objects which suggested them. 11. It was suggested that, in order to talk about lines, we should have a means of describing some of their properties in a convenient fashion. 12. The idea of a line or a set of points was suggested,and i t was claimed that set intersection was, for example, a convenient way to explore the situation of two lines crossing. 13* The notation to describe this was introduced as pf\q_, whioh was to be the set of points where p crossed q. 14. The issue of what pf|q would be could be described by-having pflq = |" or or £A,B^ - . 15. It was decided that, for example, i f pflq represented what two lines which crossed had in common at a place where they crossed then pflq = • a set of one point. End of Session Session 3. Group C 1. It was suggested that we should formally state some of the basic rules of geometry. 2. Geometry would be played with things called points, sets of points called lines and sets of points called planes, the totality of points being called space. 3. The f i r s t rule was stated: Rule I: If P and Q are two distinct points then there is exactly one line, m, which contains P and Q. 4. The phrase "exactly one" was explained to mean that there was a line which contained P and Q and there was only one line. 5. To illustrate the idea of "exactly one" the solution of o equations was considered; We say that "x =a" has a solu-tion, in fact i t has two, but 3x+5=23 has a solution and has only one solution. 56 6. To illustrate Rule I consider Figs. 1,2. Fig. 1 Fig. 2 7. In Fig. 1, i t is shown that line m contains P and Q (We can't have two points such that no line may contain the.), Fig. 2 shows "two lines" containing P and Q, but Rule I tells us that these two lines must be the same. 8. It was agreed that capital letters would name points and lower case letters would name lines and script letters would name planes. 9. It was agreed that the following conventions would be used to depict lines and planes, see Fi.g 3.4. 10. The conventions were described as "caricatures" of points, lines and planes, not the objects themselves or even their "pictures" i f such things existed. 11. A line could be named by one symbol m or as ^ LB^ from any two points on i t . 12. Note that the caricatures "behave" in a fashion similar to what we expect for lines and planes. End of Session Fig. 3 Fig. k 57 Session 3> Group E 1. It was suggested that we should formally state some of the basic rules of geometry. 2. Geometry would be played with things called points, sets of points called lines and sets of points called planes, the totality of points is called space. 3. The f i r s t rule was stated: Rule I: If P and Q are two distinct points then there is exactly one line, m, which contains P and Q . 4. The phrase "exactly one" was explained to mean that there was a line which contained P and Q and only one such line. 5. To illustrate the idea of "exactly one" the solution of equations was considered: We say that "x -a" has a solution, in fact i t has two, but 3x+5=23 has a solution and only one solution. 6. To symbolize Rule I consider: Rule I: If P and Q are distinct points, i.e. ( P ^  Q), then there is exactly one line m, such that ^ P . Q ^ m , i.e. P e ra, Q e- m. 7. Thus. Rule I states: If P , Q are distinct points then we always have a line m, such that £p»Qi ^ m a n < * ifcfp»Q3 c-m» and 1P,Q3 n, then m - n. 8. It was agreed that the following convention would be used to stand for points, lines and planes. 9. A line could be named by one symbol as "m" or by "AB" where A and B are any two points on the line. 10. Note that then we can express the idea of a point on a line by P & m or a line in a plane as 1 C £ . End of Session Session 4, Group C 1. The following definition of "collinear" was given; A set of points is collinear i f there is some line, m, which contains them. 2. In Fig. 1, A,B,D are collinear, but A,B,C are not. A e ° 4. -.--> c • Fig. 1 3. Note that 3 points may or may not be collinear, but 2 points always are collinear. 4. Discussion of what "straightness" of a line meant. 5. The idea that Rule I contributed to what straightness would mean was discussed. 6. It was noted that the representation for a line satisfied the restriction of Rule I. 7. A definition of "coplanar" was given as a set of points (points and lines or a,;set of lines) is coplanar i f there is some plane which contains them. 59 8. In Figs. 2,3,4 note that A,B,C,D are not coplanar, but A,B,C,E are coplanar. 9. Note that in Fig. 3i A,B,M are coplanar and in Fig. 4, m and n are coplanar. ,10. Note that 3 points seem to always be coplanar, but 4 or more points may or may not be coplanar. 11. A rule was proposed to make planes "bigger" than lines and space "bigger" than planes. 12. Discussion lead up to a statement of Rule II. Rule II: 1. Every plane contains at least 3 points which are not collinear. 2. Space contains at least 4 points which are not coplanar. 13* A definition of parallel lines was given; Two lines m,n are parallel, m/J n i f 1. iru and n are distinct, and 2. m and n l i e in the same plane,and 3. m does not cross n. 14. This idea was illustrated in Figs. 5»6»7» Fig. 2 Fig. 3 Fig. 4 L Fig. 5 Fig. 6 Fig. 7 60 15. The possibilities of 2 lines m and n being parallel (Fig. 5). both in the same plane but intersecting (Fig. 6), and skew lines (Fig. 7) were discussed. 16. Fig. 7 was used to il l u s t r a t e the need for condition 2 in the definition of parallel lines. 17* The idea of what would make a plane f l a t was discussed. 13. To express "flatness" Rule III was stated. Rule III: If P and Q are 2 points i n plane then the line which contains P and Q l i e s completely in plane Fig. 8 Fig. 9 19. Fig. 8 was drawn to show how Rule III prevented the s i t u -ation depicted. 20. Fig. 9 was drawn to show how Rule III made a plane " f l a t " i n a l l directions unlike a cylinder which i s " f l a t " in only one direction. 21. Rule IV was stated to express the idea that planes are "thin." Rule IV: 1. Given any ^  points P.Q.ft there i s at least one plane which contains them. 2. Given any 3 points P,Q,R which are not collinear then there i s exactly one plane which contains them. 22. Thus, as Fig. 10 shows, planes are "thin." ^ "2 c_ £ £p H Fig. 10 End of Session 61 Session 4, Group E 1. The following definition of "collinear" was given: A set of points is collinear i f there is some line m which con-tains them. 2. Note we can say P,Q,R are collinear by £P,Q,R}- <^ m. 3. Two points are always collinear, but 3 points may or may not be collinear. 4. Discussion of what "straightness" of a line meant. 5. The idea that Rule I contributes to what straightness would mean was discussed. 6. It was noted that our notation £p,Q^cm for every P,Q and £p,o}cm and £P ,Q^cn implies that m = n, obeyed the wording of Rule I. 7. A definition of "coplanar" was given: A set of points (points and lines or lines and other lines) is coplanar i f there is some plane which contains them. 8. Our notation for this situation i s : If P,Q,m are coplanar then we write Pe£_tQz£_,m c£_ • 9. Note that 3 points seem to always be coplanar, but 4 or more points may or may not be coplanar. 10. A rule was proposed to make planes "bigger" than lines and space "bigger" than planes. 62 1 1 . Rule II was stated. Rule I I : 1 . Every plane contains at least three noncollinear points. 2. Space contains at least four noncoplanar points. 12. A definition of parallel lines was given: Two lines m and n are parallel, written m(|n i f 1. m and n are distinct (m^n), and 2. m and n are i n the same plane (mc£*,n^ ), and 3. m/ln = j> . 13> Consider the three possibilities for two lines. 1 . m|/ n 2. mOn + <f> , and *n £ , < = - £ 3. mdn = p and mUn for any plane; the two lines i n 3 are said to be skew. 14. The idea of what would make a plane f l a t was discussed. 1 5 * To express this idea, Rule III was stated. Rule III: If P and Q are two points in plane £l then the line which contains P and Q, line m, li e s completely i n plane . 16. This rule was expressed as follows: I f £p,Q^c£lthen **»CH» when |p,Q^cm. 17. I t was noted that Rule III made planes f l a t i n a l l direc-tions, unlike a cylinder which was f l a t i n only one direc-tion. 18. Rule IV was stated to express the idea of "thin-ness" of 63 a plane. Rule I V : 1. Given any 3 points P,Q,R there is at least one plane which contains them. 2. Given any 3 noncollinear points P,Q,R there is exactly one plane which contains them. Thus, i f P,Q,R are noncollinear then one cannot have £ P , Q , R } C . ^ , and £p TQ,R^<=^- with ZLt End of Session Session 5t Group C. 1. The idea of betweenness was discussed and defined as: A-B-C means A,B,C are collinear and B is between A and C, i.e. the distance of A to B plus the distance of B to C is equal to the distance of A to C. 2. Rays and segments were defined in terms of betweenness as: AB means segment AB is defined as A,B,and a l l the points between A and B. 3. Fig. 1 illustrated the idea of betweenness. y 0 . _ .—> Fig. 1 k. We write A-B-C or C-B-A to symbolize the arrangement of points in Fig. 1, 5. A ray was defined as follows: In Fig. 2, the segment AB and a l l points X such that A-B-X — > make ray AB; A is called the vertex. <e — — > Fig. 2 6. The idea of convexity was defined as: A set of points is convex i f whenever P and Q are in the set PQ is in the set. 7 . Examples were given in Fig. 3 | 4 , 5 . «y e Fig. 3 Fig. k Fig. 5 8. Lines, rays and segments are convex as illustrated by Fig. 3 , 4 , 5 . 9. The concept of indirect proof was introduced and discussed,; 1 0 . The technique of considering the implication as (premise) implies (statement) and then assuming (premise) and (oppo-site of statement) and deriving a contradiction was discussed. End of Session Session 5# Group E 1. The idea of betweenness was discussed and defined as: A-B-C means A,B,C are collinear and B is between A and C, i.e. 6 5 the distance of A to B plus the distance of B to C is equal to the distance of A to C. 2. Rays and segments were defined in terms of betweenness: AB means segment AB is defined as A,B, and a l l the points between A and B. — * — 3 . A ray was defined as follows: Ray AB is AB U X, where A-B-X; A is called the vertex. 4. The idea of convexity was defined as: A set of points is convex i f whenever P and Q are in the set, PQ is in the set. 5 . This was stated as: T is convex if{p,o3 C~H implies PQ c l . 6 . It was noted that rays, lines, and segments were convex. 7. Examples of convex and nonconvex sets were mentioned. 8. The concept of indirect proof was introduced and discussed. 9. The technique of considering the implication as (premise) implies (statement) and then assuming (premise) and (oppo-site of statement) and deriving a contradiction was discussed. End of Session Session 6 , Groups C and E Training test. End of Session 66 Session 7i Group C 1. Problems from the training test were done in class. 2. Prove that a plane is a convex set. See Fig. 1. Fig. 1 3. As in Fig. 1, to show plane f £ - is a convex set we need to choose ANY two points P,Q and ask i f PQ is in - — <—? 4. By Rule III, PQ is in C_ , so PQ which is part of PQ is in «^  ; thus <E- is convex. 5. Prove that two distinct lines cannot intersect in two points. 6 . Fig. 2 shows the "opposite statement." Fig. 2 7. As in Fig. 2, m and n intersect in P and Q, two different points, but from Rule I we know exactly one line contains P and Q^Q, so m = n = *PQT this is a contradiction since we were given m ^ n. 8. Prove that i f a line intersects a plane ^ , and does not li e completely in cE then the intersection of the line and the plane is only one point. 9. Fig. 3 shows the "opposite statement." Fig. 3 67 10. As in Fig. 3. line m crosses plane ct in at least two points. 11. But by Rule III, the line which contains any two of these points must l i e completely in ^ , this contradicts the assumption that m does not l i e in c£_ . End of Session Session 7, Group E 1. Problems from the training test were proved. 2 . Prove that a plane is a convex set. 3. To show plane ^ convex, we choose any two points in , P and Q, and are asked to show PQ^ . 4. We know by Rule III, that i f £ P , Q ^ £. then^bT* and PQ c: <PQ> so PQ C r£ , and is convex. 5. Prove that two distinct lines cannot intersect in two points. 6. The opposite statement is ra^n, and mHn = £P tQ^ • P^ Q. 7. By Rule I, we know that there is exactly one line, m, which contains ^P,Q^ . 8. This contradicts m^ n. 68 9. Prove that i f a line p intersects a plane £ , and does not li e completely in _^ then the intersection of p and <1 is only one point. 10. The opposite statement is p f) cZ_ - ^P^Pg.... J 11. We are given p 0 ^ , aid p ^  <: < * r ^ ? I r-12. By Rule III, P | P 2 c t , but this line P 1P 2 = p Cf. . End of Session Session 8, Group C 1. The plane separation postulate was stated as... See Fig. 1. If line m lies in plane £ then: 1. The plane is divided into 3 convex regions m,H,K, where H,K are called half-planes, and 2. If P lies in H and Q lies in K then PQ crosses m. Fig. 1 2. The class was asked to guide the instructor in the proof of the following theorem: 3. If m lies in E_ and ray AB^lies in <I and A is on m, then ray A^ ^ (except for A) lies completely on one side of m. 4. The following figure was drawn, Fig. 2, which illustrated possibilities for the opposite of the statement. H 7) H O ) I * K 5 . It was noted that i f any part of ray AB was in K, call that point C, then CB would cross m at A, as is seen in the figure since a ray can cross a line in only one point. 6 . Then we have Fig. 2(c) as our opposite statement. 7. Then, i t was noted that we would have C-A-B, but this is impossible since A is the vertex of the ray. End of Session Session 8, Group E 1. The plane separation postulate was stated as: If a line m lies in plane £ then: 1. The plane is divided into 3 convex regions m,H,K where H,K are called half-planes, and 2. If P lies in H and Q lies in K then PQ crosses m. 2. That is, £. =HUKvm and i f P£H and Q t K, then PQ n m f <j> . 3 . The class was asked to guide the instructor in the proof of the following theorem: k. If m lies in £ and ray AB lies in 1L And A is on m, then ray AB (except for A) lies completely on one side of m. 5 . That is A £ m, show AB*n H = <j> or Alffl K = 0 <—9 6 . If not, some C £. AB in K and B E H. 70 7. Then CBq m ± (f) , in fact CBcAB'so CBOm = A since^AB^m = A. 8. But then C-A-B, which is impossible since A is the vertex. End of Session Session 9t Group C 1. The class was asked to guide the instructor in the proof of the following theorem: 2. Line m, and £± ABC l i e in plane £[ . Show that i f m crosses AB between A and B, that m must cross BC or AC. 3. Fig. 1,2 show possibilities for the opposite statement and Fig. 3 shows the desired state of affairs. Fig. 1 Fig. 2 Fig. 3 4. The situation in Fig. 1 was shown to be impossible since m cannot cross AB more than once. 5. Fig. 2 was shown to be impossible since, i f i t were true, then A and B would be on opposite sides of m by the plane separation postulate and since C is not on m, i t must be on the "A" side or "B" side of m; i f i t is on the "A" side then B and C are on opposite sides of m, thus BC crosses m, a contradiction. 6. The same argument for C on the "B" side of m. End of Session 71 Session 9» Group E 1. The class was asked to guide the instructor in the proof of the following theorem: 2. Line m, and /S. ABC l i e in plane . Show that i f m crosses AB between A and B, that ra must cross BC or AC. 3. It was noted that i f raHAB + j> then ml AB = mOAB = dPJ . 4. Then, by the plane separation postulate, A and B are on opposite sides of m. 5. We know c/m, so 0 ^  H or C a , when H is the "A" half plane and K is the "B" half plane, i.e. A € H, B £ K. 6. If C L H, then CB^m # ^  and i f C £K then CA^ m j (f> . 7. Both of these situations are impossible. End of Session APPENDIX IV EXAMPLES OF EXPOSITION TECHNIQUES 73 In order to illustrate the use of the pictorial-representa-tional mode of exposition, consider the following treatment of inci-dence chosen from a geometry text (17)* The student is informed early in the text that he already knows certain "facts about geometry", for example, that "Two straight lines cannot cross each other in more than one point"( 17 )• The student is also informed that "Pos-tulates describe fundamental properties of space" and "... the idea of point, line and plane are suggested by physical objects" and "When we use the term line, we shall always have in mind the idea of a straight line. A straight line extends infinitely far in both directions. Usually we shall indicate this in ... illustration by putting arrow-heads at the ends of the part of a line we draw..."( 17 )• Even though the student is warned that these statements are not definitions, the idea that geometry is a formalization of the physical world is implied. The standard teaching tactic (pictorial-representational) approaches incidence theorems, as the two that follow, in the manner outlined below. Th. 1- If a line m intersects a plane E and does not l i e in E then prove that m intersects E in exactly one point. Th. 2- Given triangle A ABC and a line m in the same plane, prove that i f m contains a point between A and B then m must intersect one of the other sides, AC or BC. 74 Fig. 4 Fig. 5 Fig. 6 The negation-of-conclusion-statement for an indirect proof of Theorem 1 is often depicted as in Fig. 4 or Fig. 5« Through inspection of these figures a contradiction of previous theorems or postulates is sought. Specifically, for the depiction in Fig. 4, the contradiction sought is that line m containing the two points of intersection, A and B, with plane E does not l i e in plane E and yet is equal to line H Note that in Fig. 5, which also is a depiction of the negation-of-the-con-clusion of theorem 1, is less suitable for consideration of contra-dictions, since, as drawn, the presence of lines AB and BC may tend to confuse the direction of reasoning. Also note that Fig. 6, a depiction of the "desired state of affairs", does not clearly suggest a course of action for analysis of the problem, and thus would not be utilized in the visual mode analysis of the problem except as a preliminary statement of what is to be proved. For theorem 2 a similar but more complicated situation exists. The depiction in Fig. 7 represents the negation-of-the^conclusion of theorem 2 and is a possible diagram to be used in the visual mode program as follows: (Note that Fig. 8 represents the "desired state of affairs.") Fig. 7 Fig. 8 A pictorial-representational mode exposition of the proof of theorem 2 could be outlined as follows: 1. As can be seen in the diagram in Fig. 7, i f line m does not intersect AC* or BC, then 2. by the plane separation postulate, A,B, and C are a l l on the same side of line m. 3. But A and B are on opposite sides of m. Therefore we have a contradiction. 4. Thus we must reject the assumption that m does not cross AC or BC. The same problems, theorems 1 and 2, are taught via the sym-bolic-nonrepresentational mode of exposition by set theoretical ideas. It should be noted that the use of Venn diagrams to illustrate concepts 76 of set theory is included in the symbolic-nonrepresentational mode since a Venn diagram is not derived from visual manipulation of the geometric elements of the problem involved. Theorem 1 is stated in the same form as for the visual mode: i f a line m intersects a plane E and does not l i e in E then m intersects E in exactly one point, and is then translated into set language: i f mfl E f <j> and m^fE then 1 . assume m f \ E ^  ^ i.e. m/\E = { p 1 f P2, ... \ or mAE = <f> . 2. We are given mr\E ^  ^ . 3. We are given m<j^ E . 4. Statement 2 rules out mAE = ^ . 5. P1 a n d ^ l i e in E implies that the line that contains them, P1P2. lies completely in E .(This postulate had been studied in class.) 6. Therefore, m = RJ~P2 E. 7. Statement 3 rules out mcE, 8 . therefore statement 1 must be rejected. 9 . Thus we have mAE = { P } . The symbolic-nonrepresentational mode for teaching the proof of theorem 2 employs a set theoretic statement of the plane separation postulate. This form of the postulate was taught together with the postulate statement to the experimental group. The control group was taught a pictorial form of the postulate statement. A symbolic-nonrepresentational proof of theorem 2 is outlined as follows. 1 . If a line m and AABC l i e in the same plane and m intersects AB between A and B, then prove that m must intersect BC or AC. 2. m intersects AB at T, A-T-B. 3. Assume m does not intersect BC or AC. Then m/^ BC = <f) and m A AC = <f> . 4. By the plane separation postulate, A and B are on opposite sides of m, i.e. A(m)B. 5. By the plane separation postulate, i f m/vAC = (ft then A and C are on the same side of m, i.e. A,C(m). 6. If A(m)B and A,C(m), then B(m)C. 7. Also by the plane separation postulate, B(m)C implies that mrtBC ± 0. 8. Thus we have a contradiction to the hypothesis in 3* 9. Conclusion: m intersects AC or BC. APPENDIX V EXAMPLES OF SCORING TECHNIQUES The responses from the subjects to the problems of the c r i -terion test were scored by the writer in the following manner. A response, which consists of sentences and phrases, diagrams, or set theoretic symbols and connections such as arrows, was read by the writer. The response was analyzed for logical structure. That is to say, the response was interpreted to consist of statements and implications between the statements and finally a chain of implica-tions forming a logical argument. A response which was interpreted to consist of statements containing only those rules (axioms) and definitions relevant to the problem and having a valid logical argu-ment leading to the desired conclusion was given f u l l credit (10 points). The presence of irrelevant information caused loss of 1 point from the final score. Relevant information with no valid argu-ment present was given a maximum score of 2 points and a logical argument using completely irrelevant information was given a maximum score of 4 points. In order to illustrate the scoring for responses, the follow-ing are a range of solution possibilities. The solution possi-bilities are stated in the logical form against which the logically interpreted response of the students were compared. The problem chosen for the illustration is of question VIII on the criterion test. If a,b, and c are 3 distinct lines in plane E with a parallel to c and b parallel to c, then prove a is parallel to b. Example 1_. (diagram: A) diagram A 80 1. assume opposite of the statement, i.e. (a is not parallel to b.) 2. Then a and b cross, at P. 3. a is parallel to c, therefore P is not on c. 4. b is parallel to c, therefore a and b are parallel to c through P. 5. Rules state that there is exactly one parallel to o through P, therefore statement 1 is false. 6. Therefore a is parallel to b. Score: 10 points. Comment: logic is valid throughout, diagram used for reference only, only relevant information used Example 2. ft. P ^" 1. If a is not parallel to b, then a and b cross at P. 2. Rule V states that there can be only one line parallel to c through P. 3. Therefore (by 1 and 2) a does not cross b and is then parallel to b. Score: (8-10) points. Comment: no reason stated that P is not on c, only relevant information used, diagram used for reference only, indirect proof implicit, not stated Example 2* (no diagram used) 1. c II a, c II b is given (diagram: A) c diagram A 81 2. Assume a^b. 3. Then by 2 aAb = ^P.\ k. a Ho Implies P c. 5* Rule 7 states that there exists exactly one line parallel to c through P. 6. Therefore, statement 2 is false, and a||b. Score: 10 points Comment: logic valid throughout, only relevant information used Example 4. (no diagram used) 2. Therefore both a and b are parallel to c at P. 3. But Rule V states that there exists exactly one line parallel to c at P. 4. Therefore a l l b. Score: (9-10) points Comment: argument does not account for P<^  c. 1. There are 2 possibilities for a \\ c and b |l c (A or B). Example 5_« (diagrams A,B,C,D) 2. If (A) is false and a "H b then we have (C), which is impossible by the plane separation postulate. 3* If (B) is false and &\b then we have D which is false by Rule V. Score: (6-8) Comment: diagram makes implicit use of plane separation postulate and convexity, but this is not stated Example 6. (no diagram used) 1. If a\b, then ar\b $ (j) . 2« I£ a l l °» then b\c since Rule V states there exists exactly one parallel at a point P. 3. Therefore a lib. Score: (6-8) Comment: argument from 1 to 2 not stated, assumption of 2 is not needed Example 7_« (diagram: A) a. diagram A 1. If a^b and a c then bl^c, by. Rule V. 2. Therefore a|( b. Soore: (6-8) Comment: argument from 1 to 2 not stated, diagram indirectly used. Example 8. (diagram: A) diagram A 1. If a crosses b, then (A). 2. But by Rule V there is only one parallel to c through P. 3. Therefore aj| b. Score: (4-6) Comment: argument from 1 and (A) to 3 not stated. 1 • I£ a U D then a crosses b. 2. Therefore a or b crosses c by (A). 3. But a II c and b H c. 4. Therefore a|| b. Score: (2-4) Comment: argument from 1 and (A) to 4 not stated. Example 10. (no diagram used) 1. If a^b then a crosses b at P. 2. Therefore a,b are both parallel to c at P. 3. Rule V states that there is exactly one parallel to c at P (call Example £• (diagram:A) diagram A i t m). 4. Therefore there cannot be 3 parallels (a,b,m). 5. Therefore a lib. Score: (4-6) Comment: argument from 1 to 2 not stated, argument 2 to 4 not stated. Example 11. (no diagram used) 1. If al{b, then a Ab = . 2. But a H c. 3. Therefore a ^ b. Score: (2-4) Comment: argument from 1,2 to 3 not stated. 1. If &\b then (A). 2. Therefore i f b is extended in (A), b will cross c. 3. Therefore b\c. 4. But b ll c, 5. a lib. Example 12. (diagram:A) diagram A Score: (2-4) Comment: argument from 1 ,,(A) to 2 not stated. Example 13_. (diagram:A) 1. There is no other way to draw A. 2. a Hb Score: (0-2) Comment: argument from (A) to 1 not stated. The remainder of the Type NR problems, II, III, VI, IX, are evaluated in the same fashion as problem VIII. Question I is given 10 points for a correct statement of betweenness or lacking this, 0-3 points for a plausible diagram of the situation. Question IV is given 10 points i f answer "6" is given; i f not,then 0-8 points for a plausible diagram. Question V is given 10 points- 6 points for #1, 2 points for #2, and 2 points for #3. Question VII is given 10 points- 2 to #1, 2 to #2, and 6 to #3. diagram A APPENDIX VI TRAINING TEST 87 Quiz (1) Name: I. Given A, B, C, and D are 4 distinct points on a line, and given that B-C-D, and A-C-D yes no (1) must you conclude A-B-C? ( ) X~ ) (2) is B^ A-D possible? ( ) ( ) (3) is B-A-C possible? ( ) ( ) (4) is A-C-B possible? ( ) ( ) II. Given A, B, C, D, and E are 5 distinct points on a line and given that B-C-E, D-B-C, and A-B-D yes no (1) must you conclude B-A-C? ( ) T~ ) (2) must you conclude D-B-E? ( ) ( ) (3) is A-B-C possible? ( ) ( ) (4) is B-C-A possible? ( ) ( ) III. Prove that a plane is a convex set. IV. Prove that two distinct line p and q cannot intersect in two points. Hint: Prove that the opposite statement is false. V. Prove that i f a line p intersects a plane E and p does not com-pletely l i e in E, then the intersection of p and E is only one point. Hint: Prove that the opposite statement is false. VI. Prove: If p, q, t are 3 distinct lines in a plane and p is parallel to q, and t intersects p, then t must intersect q. Hint: Prove that the opposite statement is false. VII. A, B, and C are 3 distinct collinear points. C is on ray AB. If ray AB^is intersected with ray CB? which of the following (one or more) describe the possibilities for the intersection of AB* and CB? (1) AB ( ) t» CB ( ) (2) AC ( ) (5) AB ( ) (3) B! ( ) ( 6 ) f e ( ) 89 RULES: Rule I: If P and Q are any. 3_ points then there is EXACTLY one line m which contains them. Rule II: 1. Any plane contains at least 3 points which are not on a line. 2. SPACE has at least 4 points not on a plane. Rule III: If P and Q are any 2 points in plane E, then the line m which contains P and Q lies completely in plane E. Rule IV: 1. If P, Q, R are any. 3 points (in SPACE) then there is at least one plane which contains P, Q, R. 2. If P, Q, R are any 3 points (in SPACE) which do not l i e on some line, then there is EXACTLY one plane E which contains P, Qt H. Rule V: If P is ANY point not on line m then there is EXACTLY one line m which contains P, and is parallel to m. DEFINITIONS: I. Points are collinear i f there is at least one line which contains them. Note: 2 points are ALWAYS collinear. 3 points MAY be collinear. II. Points are coplanar i f there is at least one plane which contains them. Note: 2 points are ALWAYS coplanar. 3 points are ALWAYS coplanar. 4 points MAY be coplanar. A line and a point not on i t are always coplanar. 2 lines which cross are always coplanar. III. 2 lines m, n are parallel, m)|n,if 1. m and n are different lines, 2. m does not intersect (cross) n, 3. ra and n l i e in the same plane. IV. A set of points is CONVEX i f , for anv_ choice of 2 points P,Q in the set, the segment PQ lies completely in the set. Note: Any line is convex. Any segment is convex. Any ray is convex. A circle is not convex. A circle with its 'interior* is convex. V. Note: Lines have an infinite number of points. ^ y ——. VI. Note: If line PQ lies in plane E then so does segment PQ. APPENDIX VII PILOT STUDY TESTS Test A Name: INSTRUCTIONS: Answer as many questions as you can on the answer sheet. Show a l l work. When doing a proof, you may informally give statement and reasons; you may use the numbers of the postulates on the attached sheet and need not write out their names in f u l l . I. Given A, B, C, and D as four distinct points on a line, and given that B-C-D and A-C-D, yes no (1) must you conclude A-B-C? \ ) 7/" ) (2) is B-A-D possible? ( ) ( ) (3) is B-A-C possible? ( ) ( ) (4) is A-C-B possible? ( ) ( ) II. Prove that a plane is a convex set. Write out the proof on the paper provided. III. Given A, B, C, and D as four distinct points in space, (1) Prove that the least number of planes needed to contain A, B, C, and D is either four or one; what w i l l determine whether i t is four or one? (2) Prove that i f the least number of planes needed to contain A, B, C, and D is four, then A, B, and C cannot be collinear. IV. Given A, B, C, D, and E as five distinct points on a line and that B-C-E, A-B-D, and D-B-C, yes no (1) must you conclude B-A-C? \ ) ( ) (2) must you conclude D-B-E? ( ) ( ) (3) is A-B-C possible? (4) is B-C-A possible? ( ) ( ) ( ) ( ) V. Prove that a half-plane is a convex set. VI. Prove that two distinct lines p and q cannot intersect in two or more points. VII. Prove that i f a line p intersects a plane E and p does not li e in E, then the intersection of p and E is only one point. VIII. A, B, and C are three distinct collinear points and C is on ray AB. If you intersect AB with CB, which of the following (one or more) describe the possibilities for the intersection (1) AB (2) BC ( ( ) ) (5) AC* (6) A l ( ( ) ) (3) AC W cS ( ( ) ) (7) BC (8) & ( ( ) ) Postulate List for Test A 94 Abbreviation Point, Line, or Space Postulate  L1 For every two points in space, there is exactly one line which contains both points. L2 Every plane contains at least three non-collinear points. L3 Space contains at least four noncoplanar pojrts. L4 If two points l i e in a plane then the line which contains them lies in that plane. L5 Any three points l i e in at least one plane. L6 Any three noncollinear points l i e in exactly one plane. L7 If two planes intersect, then their inter-section is a line. Separation Postulate  S1 Given a line and a plane containing i t , the points of the plane that do not l i e on the line form two sets such that (1) each of the sets is convex, and (2) i f P is in one of the sets and Q is in the other, then the segment PQ intersects the line. Recalling the definition of two lines being parallel as: two lines are parallel (p|| q) i f (1) p and q are different lines, (2) p and q l i e in the same plane, and (3) p and q do not cross, we have the para-l l e l postulate. Parallel Postulate P1 Through a given external point there is only one parallel to a given line. Test B 95 Name: ____________________________ INSTRUCTIONS: For each of the problems 1-6, draw a line through the number of the correct answer. I. If you are given ray AB* which contains point C (different from A or B), then ray A& intersected with CA* is (1) the line AB (4) the segment AC (2) the point A , (5) the segment CB (3) the point B II. If you intersect ray AB* with cl, you get - > - > — > CO either AB or BC (not both) (4) either AB or CB (2) either AC or c l (5) either BC or CB (3) either AC* or c f III. If the answer to AB intersected with CB is CB, then (1) B is between A and C (2) C is between A and B (3) A is between B and C (4) cannot determine; not enough information IV. Which of the following (one or more) is true? (1) Three points may l i e on the same line but not l i e in one plane. (2) Three points may not necessarily l i e on one line but s t i l l can l i e in one plane. (3) Three points must always l i e on two or more lines. (4) You can always find three lines to contain four points. V. Which of the following are true? (1) It can happen that of four points, three can l i e on a line and a l l four can l i e in one plane. (2) There may be no plane which holds a l l four, and yet any three of them l i e in one plane. (3) A circle may be found which passes through any three points i f you cannot find a line to pass through them. (4) Of any five points chosen in a plane, six lines in that plane are always sufficient to contain them and join every pair of points. VI. Given a AABC and a line m which intersects AB (not A or B, however), then (1) m must intersect C (2) m must intersect AC (but not at C) (3) m can intersect AC and BC (not in C) (if) m must intersect BC i f BC>AC. A, B, C, D, E a l l l i e in one plane.(I shall write A-C-E to mean C is between A and E.) Answer true or false for each of the following by circling. VII. A-C-E and B-C-D must yield A-C-D or A-C-B. T F VIII. A-C-B and C-B-D must yield A-B-D. T F IX. A-C-B and G-B-D and E-C-D must yield A-E-D. T F X. A-B-C and B-C-D must yield A-B-D and A-C-D. T F APPENDIX VIII DATA ON CLASS CLASS COMPOSITION 98 Table 6 Past Math Achievement of Control Group Subject number Math marks past two years 1 67- 68 G.M.10-C 68- 69 Math 11-C 2 * 68- 69 Math 9-C 69- 70 Math 10-F 3 68- 69 Math 8-P 69- 70 Math 9-P 4 68- 69 Math 8-A 69- 70 Math 9-B 5 68- 69 Math 8-C 69- 70 Math 9-C 6 68- 69 Math 8-C 69- 70 Math 9-B 7 68- 69 Math 8-C 69- 70 Math 9-C 8 68- 69 Math 8-P 69- 70 Math 9-P 9 68- 69 Math 8-C 69- 70 Math 9-B 10** 69- 70 Math 9-DS 70- 71 Math 9-C 11* 68- 69 Math 9-C 69- 70 Math 10-F 12** 69- 70 Math 9-DS 70- 71 Math 9-C 13 68- 69 Math 8-C 69- 70 Math 9-C 14* 68- 69 Math 9-P 69- 70 Math 10-F * indicates subject had taken grade ** indicates a deferred standing gr, Subject number Math marks past two rears 15* 68- 69 69- 70 Math Math 9- P 10-F 16 68- 69 69- 70 Math Math 8- B 9- C 17 68- 69 69- 70 Math Math 8- C 9- B 18 68- 69 69- 70 Math Math 8- C 9- C 19 68- 69 69- 70 Math Math 8- B 9- C 20 68- 69 69- 70 Math Math 8- A 9-A 21** 68- 69 69- 70 Math Math 8- F 9- DS 22 68- 69 69- 70 G.M.9-C Math 9-C 23** 68- 69 69- 70 Math Math 9-F 10- DS 24 68- 69 69- 70 Math Math 8- C 9- C 25* 68- 69 69- 70 Math Math 9- P 10- F 26 68- 69 69- 70 Math Math 8- C 9- B 27 69-70 69-70 Math Math 8-A 9- B 28 69-70 69-70 Math Math 8-A 9- B geometry previously 99 Table 7 Past Math Achievement of Experimental Group Subject number 1 2** 3* 4* 5 6 7 8 9 10 11** 12 13 14* Math marks past two years 68- 69 Math 8-A 69- 70 Math 9-A 69- 70 Math 9-P 70- 71 Math 10-DS 69- 70 G.M.9-C 70- 71 Math 10-C 69- 70 Math 9-P 70- 71 Math 10-DS 69- 70 Math 8-B 70- 71 Math 9-A 69- 70 Math88-A Math 9-B 70- 71 Math 9-A 68- 69 Math 8-P 69- 70 Math 9-P 68- 69 Math 8-C 69- 70 Math 9-P 68- 69 Math 8-B 69- 70 Math 9-B 68- 69 Math 8-B 69- 70 Math 9-B 69- 70 Math 9-DS 70- 71 Math 9-C 68- 69 Math 8-B 69- 70 Math 9-C 69- 70 Math 9-F 70- 71 Math 9-C 69- 70 Math 10-F 70- 71 Math 9-F Subject number 17 18** 19 20 21 22** 23 24* 25** 26 27 28 29 30** Math marks past two yars 68- 69 Math 8-B 69- 70 Math 9-B 68- 69 Math 8-P 69- 70 Math 9-DS 68- 69 Math 7-B 69- 70 Math 8-B 68- 69 Math 8-B 69- 70 Math 9-B 68- 69 Math 8-C 69- 70 Math 9-C 69- 70 Math 9-DS 70- 71 Math 9-C 68- 69 Math 8-C 69- 70 Math 9-C 69- 70 Math 9-C 70- 71 Math 10-F 69- 70 Math49-DS 70- 71 Math 9-P 68- 69 Math 8-C 69- 70 Math 9-P 68- 69 Math 8-B 69- 70 Math 9-P 68- 69 Math 8-C 69- 70 Math 9-C 68- 69 Math 8-C 69- 70 Math 9-P 69- 70 Math 9-C 70- 71 Math 10-DS Table 7 (continued) 100 Subject number Math marks Subject number Math marks past two years past two sears 15* 69-70 Math 9-C 31 68-69 Math 8-C 70-71 Math 10-F 69-70 Math 9-C 16 68-69 Math 8-A 69-70 Math 9-B * indicates subject had taken grade ten geometry previously ** indicates a deferred standing grade 

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