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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 this  thesis  an advanced degree at  in p a r t i a l  the U n i v e r s i t y of B r i t i s h  the L i b r a r y s h a l l make i t I  f u r t h e r agree  f u l f i l m e n t o f the requirements f o r  freely available  that permission  for  Columbia,  I agree  r e f e r e n c e and  f o r e x t e n s i v e copying o f  this  that  study. thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department o r by h i s of  this  written  representatives. thesis  It  for financial  is understood that c o p y i n g o r p u b l i c a t i o n gain shall  permission.  Department The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  5  ' *  7  Columbia  '  not  be allowed without my  Abstract The purpose of this investigation was to evaluate the effect of two modes of exposition of tenth-grade incidence geometry on l o g i cally evaluated problem solving a b i l i t y .  To achieve this purpose two  classes of tenth-grade geometry students were chosen to be the experimental and control groups.  The two treatments, which were of nine  class hours duration per group, and were both taught by the investigator, 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 successf u l l y 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 i n 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 restrictions imposed by the limitations of the study i s that the pictorialrepresentational 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 exposition i s generally considered standard practice i n the teaching of tenth grade geometry i t should be continued for the present.  i  TABLE OF CONTENTS Page i i i  LIST OF TABLES 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 Introduction  23 •  Tests of the Hypotheses 5 . SUMMARY  23 23 28  Summary of study  23  Conclusions and Implications  29  Recommendations for Instruction  32  Recommendations for Further Research  33  BIBLIOGRAPHY  35  ii Page APPENDICES  37  Appendix I: Criterion Test  37  Appendix II: Examples of Problem Types and Selection 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  iii  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 i s perhaps the most f e r t i l e part of mathematics for the development of both inductive and deductive thinking.  Therefore these educators should be concerned to examine  c r i t i c a l l y what i s being taught i n school geometry courses and how i t i s being taught.  And i f mathematics educators believe that geometry  i s 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 i n school are actually taught to play this game or indeed are capable of playing i t i n any meaningful way, that i s to say, that they have the capab i l i t y of solving a wide range of problems which require the understanding of geometry concepts. Geometry i s 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, i n accord with Henkin (7*-50), the writer believes Blank's interpretation i s extreme, although i t may not be an exaggerated view when pertaining to tenth-year geometry courses.  Blank further states,  For geometry there i s a setting i n which the student can  2  experiment and formulate conjectures. Once he has begun to make his own conjectures, he i s 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 i s 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 i n the secondary school, i s sanctioned by many researchers and by proponents of curricular reform.  Bruner  (4:6ff.) suggests that mathematics learning should include ikonic or image manipulation i n problem solving situations, and both Biggs (2:6) and Dienes (8:21) c a l l for the need to foster concrete spatialintuitive development of conceptualization.  Employment of a formal  or informal, but not wholly rigorous, axiomatic approach to mathematics may well begin i n geometry since the student i s capable of empirically deriving a "rich" collection of "interesting" theorems, many of whose proofs are within the scope of his deductive a b i l i t y (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 i s 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 i s a us& of the axiomatic structure as a component of a heuristic technique (Blank. 7:1?). The role of axiomatics i n geometry i s stated by Buck as, "It i s a valuable experience to learn to reason where intuition i s not a guide." (6:470) Any axiomatic development at this time should be "naive" or descriptive since the value of such an approach i s to elucidate the structure (meaning) of geometry and not to treat abstract axiomatics for i t s own sake (Buck, 7:20). That the axiomatic method i s only a part of mathematics i s implicit i n two statements, one by Hadamard, "Logical proofs merely sanction the conquests of intuition," (7:58) and one by Morris Kline, "Rigor w i l l not refine intuition which i s not free." (7:59) Suppes claims a need for the axiomatic method i n heuristic approaches to problem solving, a use of axiomatics which he finds woefully lacking i n the secondary school (7:70), and further he stresses that axioms are important tools i n the process of discovery. (7:71)* On the other hand a spatial-intuitive approach may be used to c l a r i f y issues concerning axioms. For example, i n Euclid's Elements. and many modern textbooks which derive from i t , there are logical deficiencies i n dealing with order of points on a line.  The  remedy, at the secondary school level, i s not the a priori introduction of suitable axioms, but the candid admission that the question i s 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 i s 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 spatialintuitive aspect of geometry) and a symbolic representation by set  4  notation of the consequences of the same axioms (an abstract logicaldeductive aspect of geometry). I f the above distinction i s shown to have a significant influence on the a b i l i t y of students to solve incidence problems, then the teaching of axiomatic geometry should be re-evaluated.  The above issue i s of particular importance with regard  to many incidence problems whose pictorial representations would, at certain stages of the solution, involve situations not existing i n real world spatial experience.  An example of such a situation i s  the "picture" of a line crossing three sides of a triangle without passing through a vertex.  This situation arises i n 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 a b i l i t y can be developed i n a standard tenth year geometry course, one i n which the exposition stresses pictorial representation of the content as opposed to a more indirect symbolic exposition. This study w i l l consider two varieties of tenth—year Euclidean incidence problems. Both of the v a r i eties involve problems which require logical-deductive reasoning, even for informal solutions.  One variety involves solutions that,  for tenth-year students, are, i n 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 w i l l be called Type NR and Type R respectively.  By a "standard course" i s meant a one or two semester course  at grade-ten level, taught by a qualified instructor, whose mathe-  5 matics training i s , i n 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 d i f f i c u l t aspects of line and plane incidence, or axiomatic systems. The Moise and Downs Geometry was chosen as the textbook for both groups because, i n 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 i s widely used i n British Columbia. Definition of Terms Although i t i s doubtful that one can always categorically place such geometry problems as are likely to be considered i n tenthgrade 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 i n one of two varieties according to the involvement of visual manipulation by the student i n his solution.  By a student's solution i s meant  his f i n a l 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 i s characterized by having the usual solution of the student developed by means of what w i l l 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 w i l l suggest an analysis of the problem. The above modifications of the figure, collectively called the visual manipulations. are generally accomplished by either actually drawing or mentally visualizing ("picturing") auxiliary lines, standard constructions, extensions to more inclusive figures, redrawn 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 i s 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 bookshelves, steel girders or the many common notions of geometry, true or false, to which most children are exposed before they reach gradeten.  The visual observation may suggest to the student theorems and  postulates which seem applicable to the problem. I t i s for Type R problems that figures can readily be constructed which obey the restrictions 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 i s 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 i n 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 inclusive figure i s 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 i n A BBC with CE parallel to W a n d 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 i n Appendix I I . Type R problems are generally taught to grade—ten students by tactics which themselves employ techniques of the visual mode. This i s a satisfactory approach for teaching the content related to these problems since they are generally analyzed and solved by students using visual mode techniques.  However, the same tactics of  teaching, utilizing techniques of the visual mode, which w i l l be termed pictorial-representational mode, are frequently used i n the teaching of geometry where the problems, for a student,- may not be readily accessible to analysis using visual-analytic techniques. Such i s often the case i n the teaching of Type NR problems. The role of visual techniques i n the pictorial-representational mode i s elucidated by Moise who states,  8 It i s customary, i n elementary texts, for the reader to be assured that 'the proofs do not depend on the figure , but these promises are almost never kept (Whether such promises ought to be kept, i n an elementary course, i s another question, and the answer should probably be 'no'.). (16:66) 1  The second variety of problems (Type NR) i s 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-notconclusion 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 i s not true, as the statement from a teacher's manual claims, that In geometry, the intuitive spatial-visual background i s 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 ... Formal geometry builds on these perceptions and extends them. (Moise and Downs, 18:13) In the context of the present study, problem solving a b i l i t y 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 i n Appendix V.  By logical evaluation of a problem solution  i s meant the analysis and scoring of the pattern of logical inference which composes the solution.  That i s to say, the syntax composed of  phrases and diagrams which are stated, together with connectives,  9 to form implications or chains of implications i s sought for i n 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 i n Appendix V. It i s 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 pictorialrepresentational 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 i n his reasoning to a valid solution of the problem. That i s 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 i s 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 i s more d i f f i cult or i s 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 i s not only a geometric contradiction but i s also contrary to ordinary physical possibility. As a specific example of this lack of f a c i l i t y , 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 i n two points. It was observed by the writer during a pilot study that approximately 6 0 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 susceptible w i l l 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 i n Appendix I I . Presentation of geometric content i n a manner which utilizes techniques of the nonvisual mode, i . e . interpretation of geometric entities and relations i n terms of set notation and set operations, w i l l 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 geometry classes, i t i s 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 i n 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 i n the solution of Type NR problems by producing a r i g i d i t y of thinking, or Einstellung, i n his solution strategy, and 2. inhibits the acquisition of nonvisual techniques of problem solving, Hypotheses (null form) It i s hypothesized that i f one group of students at the gradeten level i n secondary school i s exposed to a treatment (the experimental treatment) during which incidence geometry content i s taught via a symbolic-nonrepresentational exposition, and another group of grade-ten students i s exposed to a treatment (control group treatment) during which the same content i s taught via a pictorial-representational exposition, then: H-|: Students of the experimental group treatment w i l l obtain scores on a criterion test which do not differ significantly from scores cf control group treatment students on the same test when the content of the test i s Type NR geometry problems. Ez  :  Students of the experimental group treatment w i l l 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 i s Type R geometry problems.  Statistical Form of Hypotheses H : The mean scores for the experimental group and control group w i l l not differ significantly («< =.05) on a test of Type NR problems. 1  IL: The mean scores for the experimental group and the control group w i l l not differ significantly (<**=.05) on a test of Type R problems.  13  Chapter 2 REVIEW OF RELATED RESEARCH AND LITERATURE There i s good reason to believe that the visual mode i s a natural consequent of a child's development. For Piaget, this development i s a psychological genesis of perceptions of the spatialvisual environment from topological (in which only gross characteri s t i c s are preserved by transformation) to projective (in which form, shape, outline, and other finer qualitative properties are preserved) and f i n a l l y 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, 1 0 : 2 3 f f . ) .  This i s 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 i s that the grade ten students w i l l confront a geometry problem as though i t were a physical problem, analyzing i t i n terms of i t s physical counterpart. As Lovell claims, for the child, geometry i s essentially a system of internalized physical operations, since mathematical concepts are derived from manipulation of real world objects  (15:216).  It i s 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 i s a valid and valuable heuristic for the solution of geometry problems (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 i s 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 f i e l d , his reservoir of information, his repertoire of associated techniques, but i f he attempts a problem i n non-Euclidean geometry, for example, he i s out of his f i e l d .  "The familiar f i e l d  brings:a high transparency i n the situations which belong to the f i e l d . But i t s visual self-evidence may become a fixation hampering the solution of problems i n 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 a b i l i t y 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 i s that the form of representation of a problem w i l l be an important factor which determines successful solution by students trained i n the "usual techniques" or by common experience. This conjecture i s exemplified i n a study by J . S h e r r i l l i n which i t was found that i n 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 problem.- than did a comparable group for whom no diagram was given with the problem. S h e r r i l l found, too, that the latter group performed significantly better than a group which was given the problem with a  1  misleading, distorted, or ambiguous diagram.  5  This study was based on  earlier conjectures by Trimble and Brownell which are referred to in Sherrill's work  (20:31  ff.).  Becker and MacLeod found that when abstract models were used, i f the instructor emphasized c r i t i c a l reasoning rather than algorithmic techniques, transfer of problem solving ability to related tasks increased  (1:103).  Based on consideration of the above study i t i s  inferred that the nonrepresentational mode treatment of the experimental group on Type NR problems w i l l result i n increased ability to solve Type R problems.  Also of interest i s a study by Rugg i n which  a descriptive geometry program induced increased ability i n different aspects of geometry  (12:15).  Sandiford i n his article, "Reciprocal  Improvement i n Learning", says, when quoting Judd, It i s not far from the truth tb assert that any subject taught with a view to training pupils i n methods of generalization i s highly useful as a source of mental training, and that any subject which emphasizes particular items of knowledge and does not stimulate generalization i s educationally barren. ( 1 2 : 2 1 ) Thus, i t may be expected that the experimental group w i l l 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 i s a study by Hall, i n which he examined the effects of the training of logical proof on the geometric ability of grade ten students  (13:22),  His experimental group was  taught a program of logic followed by a program on geometric similarity.  Both groups were then given an achievement test i n geometry.  He found no significant difference between the groups and concluded  16 that "such teaching does not seem to increase the a b i l i t y to reason deductively i n geometry as measured by performance on tests of geometric proof." (1306) However, the unit i n logic and deduction that Hall taught i s alien to the context of geometry, i n that very few of his examples are i n 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 w i l l try to develop deductive ability i n both treatment groups by incorporating the logical structure intrinsic to the representational mode and nonrepresentational mode expositions. It i s not contended that geometric problem solving a b i l i t y as construed for the present study i s solely dependent on the mode of exposition; the role of a visual factor i s 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 evidence concerning Einstellung i s 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 i n a study very similar to the classic one of Luchins, order of presentation of problem types had no significant effect on problem solving (9:138)*  The consequence for the present study was that  d r i l l was avoided i n the treatment for both groups, and, except for question I which was chosen for i t s simplicity, the questions on the criterion test were i n random order. That visual representation of geometric concepts i s relevant  to problem solving i s stated by Brian as," ... much of ... problem solving behavior i s associated with geometric and visual representation of problems," and chosen i n his study of problem solving behavior was "a standard measure of spatial relations a b i l i t y 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 exposition with the nonrepresentational mode of exposition.  To this  purpose, two intact classes of 28 and 31 tenth-year geometry students i n a public secondary school were chosen as the control and experimental groups respectively. The instruction for the experimental group (group E) consisted of nine class hours of symbolic-nonrepresentational 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 pictorialrepresentational 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 exposition, 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 a b i l i t y of the students to solve problems whose content was based on material presented during the treatment period.  The test was composed of four Type R and five Type NR  problems, as indicated i n Appendices I and I I . In order to minimize subjective bias i n 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 i n random order. The reason for u t i l i z i n g geometric problems involving i n c i dence properties rather than metric properties was to avoid, at least i n the i n i t i a l stages of solution, involvement of algebraic and computational a b i l i t i e s .  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 d i f f i c u l t i e s arose i n translating elements of the geometric problems into algebraic form i n the i n i t i a l stages of the problem solution.  In order to avoid the above d i f f i c u l t i e s the prob-  lems used i n 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 d i f f i c u l t y 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 i s 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 i n formation i n 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 socioeconomic community i n 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 heterogeneous 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 semester, 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 geometryspecifically 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 implementation of standard practice in which visual representation of problems 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 i n the present study, when taught by instructors of grade ten, and ( i i ) from a reading of standard texts and the teachers' manuals which accompany them. For illustration of the different exposition 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 - s t a t i s t i c .  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 - s t a t i s t i c . sons were made at the .05 level of significance.  Both compari-  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 i s the maximum suggested by Glass and Stanley i n Statistical Methods i n 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 r e s u l t s of the s t a t i s t i c a l analysis of the data are presented to support acceptance or r e j e c t i o n of the two hypotheses and to a i d i n determination of implications of the study. Tests of the Hypotheses Hypothesis I : The mean scores f o r group C and group E w i l l not d i f f e r s i g n i f i c a n t l y ( ° < - = . 0 5 ) on Test NR. Table 1 S t a t i s t i c s Concerning Hypothesis I  N  Mean  S.D.  Group C  28  13.429  9.697  Group E  31  10.548  6.073  24  Table 2 Results of Statistics Concerning Hypothesis I  Results for t-statistic  Results for t -statistic  s pooled  7.981  7.981  d.f.  57  43  t or t' oL/Z, t or t  1  =.05  observed  level of significance of observed t or t  1  2.000(approx)  2.020  1.384*  1.354*  .20  .20  1  * not significant at pC=.05  Table 2 indicates that the observed value of t i s 1 . 3 8 4 . value of t i s such that - 2 . 0 0 0 °<.=.05.  t obs  This  2 . 0 0 0 for t cK/2 = 2 . 0 0 0 at  Therefore hypothesis I i s accepted and i t can be concluded  that there i s 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 . 0 5 level.  There-  fore the data was subjected to the t'-test i n accord with the recommendation of Introduction to Statistics, by Walpole ( 2 2 : 2 3 0 ) . of the t'-test was that t such that  1  observed i s 1 . 3 5 4 .  - 2 . 0 2 0 4 t « obs 4 2 . 0 2 0  The result  This value of t  1  is  for t' o</2 = 2 . 0 2 0 at < * = . 0 5 .  Therefore hypothesis I i s s t i l l accepted.  Hypothesis I I : The mean scores for group C and group E w i l l not differ significantly (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. t  0C/2,  57 =.05  t observed  level of significance of observed t  2.000 O.892  .40  * not significant at £X =.05 The observed value of t i n Table 4 i s 0.892. This value i s such that -2.000 <C t obs < 2.000 for t  = 2.000 at d = .05.  26  Therefore hypothesis II i s accepted.  Thus i t i s concluded that there  i s 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 determine 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 individual were the most recent entered on theindividual's record, and consisted, i n the main, of Otis (Form C) and Otis Higher (Form C) tests.  The results of the correlational study appear i n Table 5 « Table 5 Correlations Between Individual Test Scores and IQ  group  correlation of:  correlation, r obs.  critical value of r for <X=.10  group E  A:Test R and IQ  .49*  .301  (mean IQ  B:Test NR and IQ  .35*  .301  group C  C:Test R and IQ  .097  .317  (mean IQ  D:Test NR and IQ  .36*  .317  110.2)  107-5)  * significant at  <X = . 1 0  Each of the observed correlation values, r, was tested for significant difference from the value of zero at the tX =.10 level,  27 2 using the t - s t a t i s t i c : t  Q b s  = r / ( ( l - r )/(n - 2))  1  1" with d.f. = n - 2.  As i s indicated i n Table 5. correlations A, B, and D are significantlypositive at  O6=.10,  and correlation C i s 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 zing the Fischer z-transformation at the .10 level.  tested u t i l i -  I t was found that  the former hypothesis would be rejected at a significance level of C<  = .076.  Therefore i t i s 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 symbolicnonrepresentational mode of exposition of grade-ten incidence geometry upon student ability i n 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 termination 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 i n 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 I I ) . In the opinion of the writer, an opinion based on analysis of problem solutions by students i n 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 i n 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 s t a t i s t i c a l analysis confirmed both null hypotheses: 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 i n 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 i s inferred from the results of the present study that a pictorial-representational mode of exposition, as i s generally standard practice i n grade—ten, and a symbolic-nonrepresentational mode produced virtually equivalent problem solving a b i l i t y 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 i n the treatment of group C employed visual mode techniques similar to those which both groups used to solve Type R problems. I t i s possible that group E learned and firmly established visual mode techniques prior to their participation i n this study, and that the treatment for group C did not increase their a b i l i t y to solve Type R problems beyond that of group E. I t i s 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.  I t i s 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 indicated 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 s i g n i f i -  cance levels for which Hypotheses I and II would have been rejected were .2 and .4, respectively.  I t was subjectively judged by the  experimenter that the structure of logical inference i n attempted solutions of Type NR problems was essentially the same for both groups. By structure of logical inference i s 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 f r e quently by group C, and the notation accompanying them  reflected that  31 used i n the different expositions respectively.  Thus i t appears that  the content was learned i n the context presented and was utilized i n 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 i s 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 l o g i cal form of material i n 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 i s equally susceptible to the logical techniques of a grade-ten student as i s 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-nonrepresentational interpretation as demonstrated by their a b i l i t y to produce many correct or partially correct proofs involving content novel to them, but u t i l i z i n g 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 i s 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 i s unclear. Type R items did not require inference as elaborate or abstract as for Type NR items and the significance of the difference i n IQ-test score correlation of group E over group C, significant at a .10 level, presents no obvious relationship. These conclusions must be viewed i n 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 i t s use i n the treatment of group E did not represent an unbalanced introduction of new concepts of notation.  It i s possible that  the duration of treatment, nine class hours per group, although s u f f i 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 i s 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 i n this manner. Until further research c l a r i f i e s i t s role i n problem solving the symbolic-nonrepresentational mode, as employed i n 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 i t s own structure, an adequate basis for deducing correct logical consequences.  I t i s suggested that a symbolic-nonrepresentational  mode of exposition involving geometric concepts be deferred until more sophistication concerning logical inference and proof i s attained. Perhaps the symbolic-nonrepresentational mode should be introduced i n a context which does not admit the strongly entrenched visual approaches— i n courses such as abstract algebra or number theory. Recommendations for Further Research It i s 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 u t i l i z i n g geometric information symbolically interpreted after becoming acquainted with and proficient i n the techniques of logical proof as are visually trained subjects. I t i s suggested that comparisons of the treatments be conducted with groups of a more homogeneous composition than those of the present study.  This might c l a r i f y the observed difference i n correlation of  IQ with Test R scores for groups E and C, and might isolate other factors which influence test scores. I t i s also recommended that a systematic and quantitative scheme be constructed i n order to classify and rank  34 the logical inference structure of problem responses with more precision than was possible i n the present study.  35 BIBLIOGRAPHY 1 . Becker, J., and G. MacLeod. "Teaching, Discovery, and the Problem of Transfer of Training i n Mathematics," Research i n Mathematics Education. 1 9 6 7 , pp. 9 3 - 1 2 4 . 2 . Biggs, Edith. Mathematics i n 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 Relationship to Mathematical Problem Solving Behavior, Dissertation Abstracts, LXVII, (1966), 1 2 0 2 - A . 11  4 . Bruner, Jerome S. Toward a Theory of Instruction. Harvard University Press, 1966. 5.  Cambridge:  » and others. Studies i n Cognitive Growth. New York:- John Wiley and Sons, 1 9 6 6 .  6 . Buck, Charles. "What Should High School Geometry Be?," The Mathematics Teacher. LXI, (May, 1968), 4 6 6 - 4 7 0 . 7 . Conference Board of the Mathematical Sciences. The Role of Axiomatics and Problem Solving in Mathematics. Washington,D.C.: Ginn and Co., 1966. 8. Dienes, Z.P. "Some Basic Processes Involved i n Mathematics Learning," Research i n 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 C l i f f s : Prentice-Hall,  1969.  1 1 . Glass, Gene V., and Julian C. Stanley. Statistical Methods i n Education and Psychology. Englewood C l i f f s : 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, 1968.  1 4 . Hudgins, B. Problem Solving i n the Classroom. New York: MacMillan and Company, 1 9 6 6 T  36  15. Klausmeier, H., and C. Harris. Analyses of Concept Learning. York: Academic Press, 1966.  New  16. Moise, Edwin. Elementary Geometry from an Advanced Standpoint. Reading: Addison-Wesley, 1963. 17.  , and Floyd Downs,Jr. Geometry. Reading: AddisonWesley, 1 9 6 4 .  18.  . Geometry: Teachers' Manual.  Reading: Addison-  Wesley, 1964. 19. Polya, G. Mathematical Discovery, Vol. I I . New York: John Wiley and Sons, 1965. 20. Sherrill, J . "The Effects of Differing Presentation of Mathematical Word Problems upon Student Achievement," Research Reporting Sections-National Council of Teachers of Mathematics 48th Annual Meeting. Columbus: Science and Math Education Information Analysis Center, 1970. 21. VanDeGeer, J.P. A Psychological Study of Problem Solving. Uitgeverij de Toorts, 1957.  Haarlem:  22. Walpole, Ronald E. Introduction to Statistics. New York: MacMillan Company, 1968. 23. Werdelin, I. Geometric Ability and the Space Factors i n Boys and Girls. Copenhagen: Ejnar Monksgaard, 1961.  APPENDIX I  CRITERION TEST  QUIZ (2) Name: J u s t i f y a l l answers with diagrams or set notation. I . I f ray AB* i n t e r s e c t ray PQ i s AP, state the betweenness  rela-  tionship of A, B, P, and Q. I I . Prove that an angle,<ABC, together with i t s i n t e r i o r i s a convex set. I I I . Prove that 2 d i s t i n c t l i n e s which cross are contained i n exactly one plane. Note: Xou can use a l l of the rules except the note i n D e f i n i t i o n I I . Hint: Prove t h i s d i r e c t l y . IV. I f A, B, C, and D are 4 d i s t i n c t points i n a plane, what i s the maximum number of l i n e s i n the plane I can use i f each l i n e i s to contain exactly one pair of points? How are the points arranged? V. Note: Three points i n space whioh are not c o l l i n e a r determine exactly one plane, the unique plane which contains them. 1. Prove that i f the l e a s t number of planes determined by the d i s t i n c t points A, B, C, and D i s 4, then A, B, and C cannot be c o l l i n e a r . 2. Can the l e a s t number of planes determined by A, B, C, and D be  Ji  les (  ) , No (  ).  3< I f the l e a s t number of planes determined by A, B, C, and D i s  one,  must any 3 of A, 8, C, and D be collinear? les (  ) , No (  ).  VI. Assume that k, m, and n are 3 d i s t i n c t l i n e s i n plane E such that the l i n e s cross i n 2 points.  Show that the following statement  i s f a l s e : "No pair of the l i n e s i s p a r a l l e l . " VII. A, B, C, and D are 4 d i s t i n c t points on l i n e m, with C-A-D and B-A-C. 1. Where must B be i f AD intersect  ->  2. Where must B be i f AD intersect  BD i s BD?  -»  —  BD i s AB?  3. State the betweenness r e l a t i o n s h i p of A, B, C, and D i f BA intersect  BB i s B.  V I I I . I f a, b, and c are 3 d i s t i n c t l i n e s i n plane E with a p a r a l l e l to c and b p a r a l l e l to c, then prove a i s p a r a l l e l to b. IX. Given t r i a n g l e ^ ABC and l i n e m, both i n the same plane such that m does not contain A, B, or C (a vertex of the t r i a n g l e ) , that m cannot cross a l l 3 sides of the t r i a n g l e .  prove  40 RULES: Rule I : I f P and Q are any 2 points, then there i s EXACTLY one l i n e m which contains them. Rule I I : 1 . Any plane contains a t l e a s t 3 points which are not on a l i n e . 2 . SPACE has a t l e a s t 4 points not on a plane. Rule I I I : I f P and Q are any 2 points i n plane E, then the l i n e m which contains P and Q l i e s completely i n plane E . Rule IV: 1. I f P, Q, R are §ny_ 3 points ( i n SPACE) then there i s a t least one plane which contains P,Q,R. 2 . I f P, Q, R are any 3 points ( i n SPACE) which do not l i e on some l i n e , then there i s EXACTLY one plane E which contains P.Q.R. Rule V: I f P i s ANY point not on l i n e m then there i s EXACTLY one l i n e t which contains P and i s p a r a l l e l t o m.  DEFINITIONS: I . Points are c o l l i n e a r i f there i s a t l e a s t one l i n e which contains a l l of them. Note: 2 points are ALWAYS c o l l i n e a r . 3 points MAY be c o l l i n e a r . I I . Points are coplanar i f there i s a t l e a s t 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 l i e i n the same plane. IV. A set of points i s CONVEX i f , for ANY choice of 2 points P, Q in the set, the segment PQ l i e s completely i n the set. Note: Any line i s convex. Any ray i s convex. Any segment i s convex. A circle i s not convex. A circle with i t s 'interior' i s convex. V. Note: Lines have an infinite number of points. VI. Note: If line PQ l i e s i n plane E then so does segment PQ. VII. The interior of angle x£BAC formed by rays i l and AC* i s 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: ,  or  •  I  f  P  Q  R  «  a  i  :  f  S  •  APPENDIX II  EXAMPLES OF PROBLEM TYPES AND SELECTION PROCESS AND CRITERIA  A pool of p o t e n t i a l c r i t e r i o n t e s t items was obtained from questions s i m i l a r to those a c t u a l l y used i n p i l o t study treatments and t e s t s .  The problems i n the pool were either solved completely  or s u b s t a n t i a l l y by a t l e a s t 1/3 of the p i l o 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 c r i t e r i o n test, whether solved a t 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 c l e a r l y exemplified the c h a r a c t e r i s t i c s of Type R and eight the c h a r a c t e r i s t i c s of Type NR.  These sixteen problems were submitted i n random order to a  panel f o r 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 v i s u a l - a n a l y t i c mode to produce a s o l u t i o n . Categorize the item as Type NR i f you believe: 1. that an average grade—ten student would not generally employ v i s u a l mode techniques to solve the problem, or 2. that i f the student d i d employ v i s u a l mode techniques his solution would probably not be a v a l i d l o g i c a l argument. A d e s c r i p t i o n of the terms used above was given to each member of the panel as were several examples.  Those items f o r which  there was unanimous agreement among the panel and writer as to type were chosen to compose the c r i t e r i o n t e s t .  Examples of problem types Examples of both Type NR and Type R problems are to be found i n the criterion test, training test and pilot tests i n 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 i n the treatment exposition and pilot study training programs• Examples of Type R problems: 1 . Prove that i n a plane the perpendicular bisector of a segment AB i s 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 i s parallel to the third side and one-half i t s length. 3. Prove that i f a point P i s 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 i t s interior into two regions of equal area, then prove that m intersects the diagonals of the parallelogram at their point of intersection. 6. In a plane, i f two circles C-j and C are externally tangent at D 2  and the circles are congruent, with AB and ND diameters of C and 1  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. I f A, B, C, D are four different points i n 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 i s the bisector of <BAC and AT i s the median, prove that triangle AABC i s isosceles.  10. Point D i s i n the interior of triangle AABC.  Prove that angle 4. ADB  i s 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 i s parallel to n and m i s parallel to p, where m, n and p are three distinct lines i n space, then n i s parallel to p, without considering perpendiculars. 3. Prove that i f a,b,t, and m are different lines a l l i n £lane E such that t i s parallel to a, t \ i s parallel to b, and m intersects b and t, then m A a f 4 . Given a taingle  .  ABC and a line m i n 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 fromrawhich contains point C. 6. I f B-M-C and A, a point not on BC, then prove that M i s i n 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 i n the same plane, then prove that i f m intersects AB between A and B, then m must intersect either AC or BC. 9. Given a plane E and a line m i n 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 i n a plane E, and t i s paral l e l to a, and t i s parallel to b, then prove that a i s parallel to b. 11. Prove that given a line m i n space and a point P not on ra, then there i s exactly one plane containing both the point P and the line m. 12. If m and n are two different lines i n space which intersect, then prove that their intersection contains exactly one point. 13. Prove that every ray i s convex. 14. If m and n are two different lines which intersect, prove that there i s 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 i s always between the smaller and larger value? 3 . Class asked i f above property of average i s 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 i s always true. 5. Statement was symbolized: AC(A+B)/2^B, when A,B  S  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 3 . Zk< A+B  , by a rule of algebra. therefore A<(A+B)/2  , by another rule  of algebra. 7. I t i s 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 w i l l 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 i s always even. 2. The product of two odd numbers i s 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 i n 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 blackboard (plane-like). 3. Geometry has objects which resemble stretched s t r i n g — c a l l them lines and things which resemble blackboard— c a l l them planes.  52  k. The class was asked what a line i n geometry "looked" l i k e . 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, i n 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 i n one point. 15- I t was decided that the picture i n 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 i n 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 i n certain ways. 2. Discussion that objects i n geometry have many of the properties 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 s t r i n g s — 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 i n geometry "looked" l i k e . 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 i s 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. I t was suggested that, i n 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 byhaving pflq = |"  or  or £A,B^- .  15. I t was decided that, for example, i f pflq represented what two lines which crossed had i n common at a place where they crossed then pflq  =  •  a set of one point.  End of Session  Session 3. Group C 1. I t 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 i s 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 solution, i n 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 i s 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 t e l l s 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 . I t was agreed that the following conventions would be used to depict lines and planes, see Fi.g 3 . 4 .  3  Fig. k  10. TheFig. 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" i n a fashion similar to what we expect for lines and planes. End of Session  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 i s called space. 3. The f i r s t rule was stated: Rule I: If P and Q are two distinct points then there i s 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 solution and only one a  solution. 6. To symbolize Rule I consider: Rule I: If P and Q are distinct points, i . e . ( P ^ Q), then there i s exactly one line m, such that ^ P . Q ^ m , i.e.  P  era,Q e- m.  7. Thus. Rule I states: If P , Q are distinct points then we always have a line m, such that and  1P,Q3  n, then  £»Qi p  ^  m  a  n  <  *  cf »Q3 -  if  p  »  c m  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 l i n e . 10. Note that then we can express the idea of a point on a line by P & m or a line i n a plane as 1 C £ . End of Session  Session 4, Group C 1. The following definition of "collinear" was given; A set of points i s collinear i f there i s some line, m, which contains them. 2. In Fig. 1, A,B,D are collinear, but A,B,C are not. 4.  e  A  °  -.-->  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. I t 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) i s 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.  Fig. 2  Fig. 3  Fig. 4  9. Note that i n Fig. 3i A,B,M are coplanar and i n F i g . 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 I I . Rule I I : 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 i n the same plane,and 3. m does not cross n. 14. This idea was illustrated i n Figs. 5»6»7»  L  Fig. 5  Fig. 6  Fig. 7  60 15. The p o s s i b i l i t i e s of 2 l i n e s m and n being p a r a l l e l ( F i g . 5). both i n the same plane but i n t e r s e c t i n g ( F i g . 6), and skew l i n e s ( F i g . 7) were discussed. 16. F i g . 7 was used to i l l u s t r a t e the need f o r condition 2 i n the d e f i n i t i o n of p a r a l l e l l i n e s . 17* The idea of what would make a plane f l a t was discussed. 13. To express "flatness" Rule I I I was stated. Rule I I I : I f P and Q are 2 points i n plane  then the  l i n e which contains P and Q l i e s completely i n plane  Fig. 9  Fig. 8  19. F i g . 8 was drawn to show how Rule I I I prevented the s i t u ation depicted. 20. F i g . 9 was drawn to show how Rule I I I made a plane " f l a t " i n a l l directions unlike a cylinder which i s " f l a t " i n only one d i r e c t i o n . 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 a t l e a s t one plane which contains them. 2. Given any 3 points P,Q,R which are not c o l l i n e a r then there i s exactly one plane which contains them. 22. Thus, as F i g . 10 shows, planes are "thin."  H  F i g . 10  ^  "2 c_ £  £p  61 End of Session  Session 4, Group E 1. The following definition of "collinear" was given: A set of points i s collinear i f there i s some line m which contains 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. I t was noted that our notation £p,Q^cm for every P,Q and  £p,o}cm and £ P , Q ^ c n  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) i s coplanar i f there i s some plane which contains them. 8. Our notation for this situation i s : If P,Q,m are coplanar then we write  Pe£_ Qz£_,m c£_ • t  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 I I was stated. Rule I I : 1 . Every plane contains a t l e a s t three noncollinear points. 2. Space contains a t l e a s t four noncoplanar points. 12. A d e f i n i t i o n of p a r a l l e l l i n e s was given: Two l i n e s m and n are p a r a l l e l , written m(|n i f 1. m and n are d i s t i n c t (m^n), and 2. m and n are i n the same plane ( m c £ * , n ^ ), and 3. m/ln =  j>  .  13> Consider the three p o s s i b i l i t i e s f o r two l i n e s . 1 . m|/ n 2. mOn  + <f> , and  3. mdn = p  *n  and mUn  £  ,  <=-£  f o r any plane; the two l i n e s  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 t h i s idea, Rule I I I was stated. Rule I I I : I f P and Q are two points i n plane £l then the l i n e which contains P and Q, l i n e m, l i e s completely i n plane  .  16. This r u l e was expressed as follows: I f £ p , Q ^ c £ l t h e n  **»H» when C  |p,Q^cm.  17. I t was noted that Rule I I I made planes f l a t i n a l l d i r e c tions, unlike a cylinder which was f l a t i n only one d i r e c 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 i s 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 Q,R^<=^T  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 i s between A and C, i.e. the distance of A to B plus the distance of B to C i s equal to the distance of A to C. 2. Rays and segments were defined i n terms of betweenness as: AB means segment AB i s 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 i n 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 i s called the vertex.  <e  — —  >  Fig. 2 6. The idea of convexity was defined as: A set of points i s convex i f whenever P and Q are i n the set PQ i s in the set. 7.  Examples were given in Fig. 3 | 4 , 5 .  e  «y  Fig. k  Fig. 3  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 (opposite 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: means A,B,C  A-B-C  are collinear and B i s between A and C, i . e .  65  the distance of A to B plus the distance of B to C i s equal to the distance of A to C. 2. Rays and segments were defined i n terms of betweenness: AB means segment AB i s defined as A,B, and a l l the points between A and B. — *  —  3 . A ray was defined as follows: Ray AB i s AB U X, where A-B-X; A i s called the vertex. 4. The idea of convexity was defined as: A set of points is convex i f whenever P and Q are i n the set, PQ i s i n the set. 5.  This was stated as: T i s convex  if{p,o3 C~H  implies PQ c l .  6 . I t 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 (opposite of statement) and deriving a contradiction was discussed. End of Session  Session 6 , Groups C and E Training test. End of Session  66  Session 7 i Group C 1. Problems from the training test were done i n class. 2. Prove that a plane i s a convex set. See Fig. 1.  Fig. 1 3. As i n Fig. 1, to show plane f £ - i s a convex set we need to choose ANY two points P,Q and ask i f PQ i s i n -  4. By Rule III, PQ i s i n  <—?  —  C_ , so PQ which i s part of PQ  ; thus <E- i s convex.  is i n «^  5. Prove that two distinct lines cannot intersect i n two points. 6 . Fig. 2 shows the "opposite statement."  Fig. 2 7. As i n Fig. 2, m and n intersect i n 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 i s a contradiction since  we were given m ^ n. 8. Prove that i f a line intersects a plane ^ l i e completely i n cE  then the intersection of the line  and the plane i s only one point. 9. Fig. 3 shows the "opposite statement."  Fig.  3  , and does not  67 10. As i n Fig. 3. line m crosses plane c t i n at least two points. 11. But by Rule III, the line which contains any two of these points must l i e completely i n ^  , 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 i s a convex set. 3. To show plane ^convex, we choose any two points i n 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  i s convex.  5. Prove that two distinct lines cannot intersect i n two points. 6. The opposite statement i sra^n,and mHn = £ P Q ^ t  • P^Q.  7. By Rule I, we know that there i s 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  l i e completely i n ^_ then the intersection of p and <1 i s only one point. 10. The opposite statement i s 11. We are given p 0 ^ <  *  p f) cZ_ - ^P^Pg.... J , aid p ^ <:  r  12. By Rule III, P | P c t 2  ^  ?  , but this line P P 1  2  I  r-  = p Cf.  .  End of Session  Session 8, Group C 1. The plane separation postulate was stated as... See Fig. 1. If line m l i e s i n plane £  then:  1. The plane i s divided into 3 convex regions m,H,K, where H,K are called half-planes, and 2. If P l i e s i n H and Q l i e s i n K then PQ crosses m.  Fig. 1 2. The class was asked to guide the instructor i n the proof of the following theorem: 3. If m lies i n E_ and ray AB^lies i n <I  and A i s 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) O)  I*  H K  5 . I t was noted that i f any part of ray AB was i n K, c a l l that point C, then CB would cross m at A, as i s seen i n the figure since a ray can cross a line i n 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 i s impossible since A i s 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 i n plane £  then:  1. The plane i s divided into 3 convex regions m,H,K where H,K are called half-planes, and 2. If P lies i n H and Q l i e s i n K then PQ crosses m. 2. That i s , £.  =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 i n the proof of the following theorem: k. I f m lies in £  and ray AB lies i n 1L And A i s on m,  then ray AB (except for A) lies completely on one side of m. 5 . That i s A £ m, show AB*n H = <j> <—9  or Alffl K = 0  6 . If not, some C £. AB i n K and B E H.  70 7. Then CBq m ± (f)  , in fact CB AB'so CBOm = A since^AB^m = A. c  8. But then C-A-B, which i s impossible since A i s 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 i n 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 i n 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 i s not on m, i t must be on the "A" side or "B" side of m; i f i t i s 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 i n the proof of the following theorem: 2. Line m, and /S. ABC l i e i n plane  . Show that i f m  crosses AB between A and B, thatramust cross BC or AC. 3 . I t 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 i s the "A" half  plane and K i s 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-representational mode of exposition, consider the following treatment of i n c i dence chosen from a geometry text (17)* The student i s informed early i n the text that he already knows certain "facts about geometry", for example, that "Two straight lines cannot cross each other i n more than one point"( 17 )• The student i s also informed that "Postulates 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 i n mind the idea of a straight line.  A straight line extends infinitely far i n both directions.  Usually we shall indicate this i n ... illustration by putting arrowheads at the ends of the part of a line we draw..."( 17 )• Even though the student i s warned that these statements are not definitions, the idea that geometry i s a formalization of the physical world i s implied.  The standard teaching tactic (pictorial-representational)  approaches incidence theorems, as the two that follow, i n the manner outlined below. Th. 1- If a line m intersects a plane E and does not l i e i n E then prove that m intersects E i n exactly one point. Th. 2 - Given triangle A ABC and a line m i n 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 i s often depicted as i n Fig. 4 or Fig. 5«  Through inspection of these  figures a contradiction of previous theorems or postulates i s sought. Specifically, for the depiction in Fig. 4, the contradiction sought i s 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 i s equal to line H  Note  that i n Fig. 5, which also i s a depiction of the negation-of-the-conclusion of theorem 1, i s less suitable for consideration of contradictions, 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 i n the visual mode analysis of the problem except as a preliminary statement of what i s to be proved. For theorem 2 a similar but more complicated situation exists. The depiction i n Fig. 7 represents the  negation-of-the^conclusion  of theorem 2 and i s a possible diagram to be used i n 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 i n the diagram i n 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 symbolic-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 i s included i n the symbolic-nonrepresentational  mode  since a Venn diagram i s not derived from visual manipulation of the geometric elements of the problem involved.  Theorem 1 i s 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 i n E then m intersects E i n exactly one point, and i s then translated into set language: i f mfl E f <j> and m^fE then 1 . assume m f \ E ^ ^ m/\E = { p  1 f  i.e.  P , ... \  or mAE = <f> .  2  2 . We are given mr\E ^ ^ . 3. We are given m<j^E . 4. Statement 2 rules out mAE = ^ . 5. P a n d ^ l i e i n E implies that the line that contains 1  them, P 1 P 2 . lies completely i n E .(This postulate had been studied in class.) 6. Therefore, m = RJ~P  2  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 symbolicnonrepresentational proof of theorem 2 i s outlined as follows. 1 . If a line m and AABC l i e i n 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 i n 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 i n 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 f i n a l l y a chain of implications 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 argument 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 f i n a l score. Relevant information with no valid argument 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 following are a range of solution possibilities.  The solution possi-  b i l i t i e s are stated i n the logical form against which the logically interpreted response of the students were compared. The problem chosen for the illustration i s of question VIII on the criterion test. 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. Example 1_. (diagram: A) diagram A  80 1. assume opposite of the statement, i . e . (a i s not parallel to b.) 2. Then a and b cross, at P. 3. a i s parallel to c, therefore P i s not on c. 4. b i s parallel to c, therefore a and b are parallel to c through P. 5. Rules state that there i s exactly one parallel to o through P, therefore statement 1 i s false. 6. Therefore a i s parallel to b. Score: 10 points. Comment: logic i s valid throughout, diagram used for reference only, only relevant information used ft.  Example 2.  P  ^"  (diagram: A)  c diagram A  1. If a i s 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 i s then parallel to b. Score: (8-10) points. Comment: no reason stated that P i s 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  i s given  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 i s 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. Example 5_« (diagrams A,B,C,D)  1. There are 2 possibilities for a \\ c and b |l c (A or B).  2. If (A) i s false and a "H b then we have (C), which i s impossible by the plane separation postulate. 3* If (B) i s false and &\b then we have D which i s false by Rule V. Score: (6-8) Comment: diagram makes implicit use of plane separation postulate and convexity, but this i s not stated Example 6. (no diagram used) 1. I f a\b, then ar\b $ (j) . 2« I£ l l °» then b\c a  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 i s not needed Example 7_«  a.  (diagram: 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 i s only one parallel to c through P. 3. Therefore aj| b. Score: (4-6) Comment: argument from 1 and (A) to 3 not stated. Example £• (diagram:A) diagram A 1 • I£ U a  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 i s exactly one parallel to c at P (call 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. Example 12. (diagram:A) diagram A 1. If &\b then (A). 2. Therefore i f b i s extended i n (A), b w i l l cross c. 3. Therefore b\c. 4. But b ll c, 5. a lib.  Score: (2-4) Comment: argument from 1 ,,(A) to 2 not stated.  Example 13_. (diagram:A) diagram A 1. There i s 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 i n the same fashion as problem VIII. Question I i s given 10 points for a correct statement of betweenness or lacking this, 0-3 points for a plausible diagram of the situation.  Question IV i s  given 10 points i f answer "6" i s given; i f not,then 0-8 points for a plausible diagram. Question V i s given 10 points- 6 points for #1, 2 points for #2, and 2 points for #3. points- 2 to #1, 2 to #2, and 6 to #3.  Question VII i s given 10  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) i s B^A-D possible?  ( )  ( )  (3) i s B-A-C possible?  ( )  ( )  (4) i s 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) i s A-B-C possible?  ( )  ( )  (4) i s B-C-A possible?  ( )  ( )  III. Prove that a plane i s a convex set. IV. Prove that two distinct line p and q cannot intersect i n two points. Hint: Prove that the opposite statement i s false. V. Prove that i f a line p intersects a plane E and p does not completely l i e i n E, then the intersection of p and E i s only one point. Hint: Prove that the opposite statement i s false. VI. Prove: If p, q, t are 3 distinct lines i n a plane and p i s parallel to q, and t intersects p, then t must intersect q. Hint: Prove that the opposite statement i s false.  VII. A, B, and C are 3 distinct collinear points.  C i s 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  (  )  AC  (  )  (5) AB  (  )  (  )  (6)fe  (  )  (2)  (3) B!  89 RULES: Rule I: I f P and Q are any. 3_ points then there i s EXACTLY one line m which contains them. Rule I I : 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 I I I : I f P and Q are any 2 points i n plane E, then the line m which contains P and Q lies completely i n plane E. Rule IV: 1. I f P, Q, R are any. 3 points (in SPACE) then there i s at least one plane which contains P, Q, R. 2. I f 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,  Qt H .  Rule V: I f P i s ANY point not on line m then there i s EXACTLY one line m 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 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,if 1. m and n are different lines, 2. m does not intersect (cross) n, 3.raand n l i e i n the same plane. IV. A set of points i s CONVEX i f , for anv_ choice of 2 points P,Q i n the set, the segment PQ l i e s completely i n the set. Note: Any line i s convex. Any segment i s convex. Any ray i s convex. A circle i s not convex. A circle with i t s 'interior* i s convex. V. Note: Lines have an infinite number of points. ^  y  ——.  VI. Note: I f line PQ l i e s i n 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 i n 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) i s B-A-D possible?  ( )  ( )  (3) i s B-A-C possible?  (  )  ( )  (4) i s A-C-B possible?  (  )  ( )  I I . Prove that a plane i s a convex set. Write out the proof on the paper provided. III. Given A, B, C, and D as four distinct points i n space, (1) Prove that the least number of planes needed to contain A, B, C, and D i s either four or one; what w i l l determine whether i t i s four or one? (2) Prove that i f the least number of planes needed to contain A, B, C, and D i s 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, (1) must you conclude B-A-C?  yes \ )  (2) must you conclude D-B-E?  (  no ( ) )  ( )  (3) i s A-B-C possible?  (  )  (  )  (4) i s B-C-A possible?  (  )  (  )  V. Prove that a half-plane i s a convex set. VI. Prove that two distinct lines p and q cannot intersect i n two or more points. VII. Prove that i f a line p intersects a plane E and p does not l i e i n E, then the intersection of p and E i s only one point. VIII. A, B, and C are three distinct collinear points and C i s on ray AB. If you intersect AB with CB, which of the following (one or more) describe the possibilities for the intersection  (1) AB  (  )  (5) AC*  ( )  (2) BC  (  )  (6) A l  ( )  (3) AC  (  )  (7) BC  (  W cS  ( )  (8)  ( )  &  )  94 Postulate List for Test A Abbreviation L1  Point, Line, or Space Postulate For every two points in space, there is exactly one line which contains both points.  L2  Every plane contains at least three noncollinear 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 i n 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 i s 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 i s convex, and (2) i f P i s in one of the sets and Q i s 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 i n the same plane, and (3) p and q do not cross, we have the paral l e l postulate. Parallel Postulate P1  Through a given external point there i s only one parallel to a given l i n e .  95  Test B 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* i s (1) the line AB  (4) the segment AC  (2) the point A  ,  (5) the segment CB  (3) the point B I I . If you intersect ray AB* with cl, CO  you get  - >  either AB or BC (not both)  (2) either AC or c l  - >  — >  (4) either AB or CB (5) either BC or CB  (3) either AC* or c f III. If the answer to AB intersected with CB i s CB, then (1) B i s between A and C (2) C i s between A and B (3) A i s between B and C (4) cannot determine; not enough information IV. Which of the following (one or more) i s true? (1) Three points may l i e on the same line but not l i e i n one plane. (2) Three points may not necessarily l i e on one line but s t i l l can l i e i n 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) I t can happen that of four points, three can l i e on a line and a l l four can l i e i n one plane. (2) There may be no plane which holds a l l four, and yet any three of them l i e i n one plane. (3) A c i r c l e 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 i n a plane, six lines i n 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 i n C) (if)  m  must intersect BC i f BC>AC.  A, B, C, D, E a l l l i e i n one plane.(I shall write A-C-E to mean C i s between A and E.) Answer true or false for each of the following by c i r c l i n g . 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  Subject number  Math marks past two rears  1  67-68 G.M.10-C 68-69 Math 11-C  15*  68-69 Math 9- P 69-70 Math 10-F  2 *  68-69 Math 9-C 69-70 Math 10-F  16  68-69 Math 8- B 69-70 Math 9-C  3  68-69 Math 8-P 69-70 Math 9-P  17  68-69 Math 8-C 69-70 Math 9- B  4  68-69 Math 8-A 69-70 Math 9-B  18  68-69 Math 8-C 69-70 Math 9-C  5  68-69 Math 8-C 69-70 Math 9-C  19  68-69 Math 8- B 69-70 Math 9-C  6  68-69 Math 8-C 69-70 Math 9-B  20  68-69 Math 8-A 69-70 Math 9-A  7  68-69 Math 8-C 69-70 Math 9-C  21**  68-69 Math 8-F 69-70 Math 9-DS  8  68-69 Math 8-P 69-70 Math 9-P  22  68-69 G.M.9-C 69-70 Math 9-C  9  68-69 Math 8-C 69-70 Math 9-B  23**  68-69 Math 9-F 69-70 Math 10-DS  10**  69-70 Math 9-DS 70-71 Math 9-C  24  68-69 Math 8-C 69-70 Math 9-C  11*  68-69 Math 9-C 69- 70 Math 10-F  25*  68-69 Math 9- P 69-70 Math 10-F  12**  69-70 Math 9-DS 70-71 Math 9-C  26  68-69 Math 8-C 69-70 Math 9- B  13  68-69 Math 8-C 69-70 Math 9-C  27  69-70 Math 8-A 69-70 Math 9-B  14*  68-69 Math 9-P 69- 70 Math 10-F  28  69-70 Math 8-A 69-70 Math 9- B  * indicates subject had taken grade ** indicates a deferred standing gr,  geometry previously  99 Table 7 Past Math Achievement of Experimental Group Math marks past two yars  Subject number  Math marks past two years  Subject number  1  68-69 Math 8-A 69-70 Math 9-A  17  68-69 Math 8-B 69-70 Math 9-B  2**  69-70 Math 9-P 70- 71 Math 10-DS  18**  68-69 Math 8-P 69-70 Math 9-DS  3*  69- 70 G.M.9-C 70- 71 Math 10-C  19  68-69 Math 7-B 69-70 Math 8-B  4*  69-70 Math 9-P 70- 71 Math 10-DS  20  68-69 Math 8-B 69-70 Math 9-B  5  69-70 Math 8-B 70- 71 Math 9-A  21  68-69 Math 8-C 69-70 Math 9-C  6  69-70 Math88-A Math 9-B 70- 71 Math 9-A  22**  69-70 Math 9-DS 70- 71 Math 9-C  7  68-69 Math 8-P 69-70 Math 9-P  23  68-69 Math 8-C 69-70 Math 9-C  8  68-69 Math 8-C 69-70 Math 9-P  24*  69-70 Math 9-C 70- 71 Math 10-F  9  68-69 Math 8-B 69-70 Math 9-B  25**  69-70 Math 9-DS 70-71 Math 9-P  10  68-69 Math 8-B 69-70 Math 9-B  26  68-69 Math 8-C 69-70 Math 9-P  11**  69-70 Math 9-DS 70- 71 Math 9-C  27  68-69 Math 8-B 69-70 Math 9-P  12  68-69 Math 8-B 69-70 Math 9-C  28  68-69 Math 8-C 69-70 Math 9-C  13  69- 70 Math 9-F 70- 71 Math 9-C  29  68-69 Math 8-C 69-70 Math 9-P  14*  69-70 Math 10-F 70- 71 Math 9-F  30**  69-70 Math 9-C 70- 71 Math 10-DS  4  100  Table 7 (continued) Subject number  Math marks past two years  15*  69-70 Math 9-C 70-71 Math 10-F  16  68-69 Math 8-A 69-70 Math 9-B  Subject number  Math marks past two sears  31  68-69 Math 8-C 69-70 Math 9-C  * indicates subject had taken grade ten geometry previously ** indicates a deferred standing grade  

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