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A piagetian analysis of intellectual performance on first-year university physics examinations Hewson, Mariana Gay A'Beckett 1971

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A PIAGETIAN ANALYSIS OF INTELLECTUAL PERFORMANCE ON FIRST-YEAR UNIVERSITY PHYSICS EXAMINATIONS by Mariana Gay A'Beckett Hewson B.Sc, University of the Witwatersrand, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Science Education We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f S u ^ c t . tcc\.vA/L^_V-to«>% The U n i v e r s i t y o f B r i t i s h C o l umbia V a n c o u v e r 8, Canada Date ABSTRACT A PIAGETIAN ANALYSIS OF INTELLECTUAL PERFORMANCE ON FIRST-YEAR UNIVERSITY PHYSICS EXAMINATIONS The Problem This thesis i s addressed to a problem in classroom practice identified as formative evaluation, that i s , the development of a method of u t i l i z i n g Piaget's theory of i n t e l l e c t u a l development to evaluate the i n t e l l e c t u a l performance of students contending with the formal concepts and methods of inquiry presented in a f i r s t year physics course with a view to improving instruction i n the course. The method of studying the problem required a good understanding and analysis of the relevant aspects of Piaget's theory so that i t could be reformulated i n such.a way as to be unable i n identifying formal behaviour of individuals' answers to specially selected examination items. This resulted i n the formulation of a methodology i n the form of an inventory of descriptors, and a method of analysis for identifying behaviour at the f i n a l stage of Piaget's developmental sequence, namely the formal operations stage. Method of Study The inventory of descriptors was used to identify formal operational behaviour of students performance on selected Piaget tasks, providing infor-mation concerning their maximum potential level of i n t e l l e c t u a l development. It was then used to identify physics examination items which required formal operations for their solution and the formal operational behaviour displayed by students in responding to the selected items thereby providing information concerning the actual level of i n t e l l e c t u a l performance displayed by students i n classroom situations. A comparison of id e n t i f i e d i n t e l l e c t u a l behaviours provided information concerning the actual level of intellectual performance displayed by students in classroom situations. A comparison of identified intellectual behaviours provided information concerning the usefulness of such an instrument in education. Results of the Study It was concluded that the inventory of descriptors adequately described and identified intellectual behaviour at the formal operations level, both in student performance on the Piaget tasks, and in student performance on selected items from the physics examination paper. The inventory of descriptors proved to be of potential value to formative evaluation in the classroom situation. ACKN OWLEDGEMEN TS My great respect and grateful thanks go to Dr. Walter Boldt who supervised, advised and en-couraged me, and above a l l , taught me so much. Thanks are due also to Dr. Clifford Anastasiou for his valuable advice, to my committee for agreeing to be there, and to my fellow students who always proved most stimulating. To my baby Fraser John who nearly made this thesis impossible and Peter my husbn.nd who saved i t so often. I. THE PROBLEM AND ITS CONTEXT 1 A. CONTEXT OF THE STUDY 1 B. STATEMENT OF THE PROBLEM 3 General Problem 3 Specific Problem 4 C. METHOD OF STUDY 5 D. SCOPE AND LIMITATIONS 8 E. DESCRIPTION OF TERMS 10 II. THEORETICAL BACKGROUND OF THE STUDY - PIAGET'S THEORY OF INTELLECTUAL DEVELOPMENT .13 A. . THE PIAGETIAN FRAMEWORK 13 Theoretical Constructs 13 Tasks and Stages 16 Piaget Tasks 16 Stages 17 B. PIAGET'S CLINICAL METHOD 18 C. STAGE OF CONCRETE OPERATIONS 20 Intellectual Structures 20 Elementary Groupements of Classes 21 Elementary Groupements of Relations 23 Characteristics of the Intellectual Structures 2k D. PSYCHOLOGICAL'FUNCTIONING AT THE CONCRETE OPERATIONS STAGE. 26 Empirically Oriented Thought * 26 Non-Integrated Thought 27 Non-Generalizable Thought 29 E. STAGE OF FORMAL OPERATIONS .... 29 Intellectual Structures 29 C ombinatorial S ystem . o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 INRC Group of Operations 32 Properties of the INRC Group 32 Propositional Logic . . . . . . . . . . a . . . . • • • « » « . . o . o . . • • • . . . • « 33 Formal Operational Schemes 34 F. PSYCHOLOGICAL FUNCTIONING AT THE FORMAL OPERATIONS STAGE 36 Hypothetical Thought .« 37. Thinking within a Framework of Related Ideas 37 Integrating the Ideas - Control of the Variables 38 Establishing the Relationships Between the Ideas 39 Deductive Thought . . . o o . o e e . o * o . « . . e « o o . 9 . . . o . . » . . . . . . . . . 39 G. SUBSTAGES A AND B OF THE FORMAL OPERATIONS STAGE 39 H. SUMMARY 40 III. PROCEDURES AND DATA USED IN THE STUDY 41 A. DEVELOPMENT OF AU INVENTORY FOR MAKING PIA.GETIAN INTER-PRETATIONS OF INTELLECTUAL PERFORMANCE: FORMAL OPERATIONS STAGE 41 Basis for Development » 4 l Description of the Inventory 42 B. STUDENT PERFORMANCE ON THE PIAGET TASKS 49 Selection of Piaget Tasks 49 Selection of Students 56 Administration of Piaget Tasks 56 Method of Recording Piaget Task Performance 57 Method of Analysing Piaget Task Performance 57 C. STUDENT PERFORMANCE ON SELECTED EXAMINATION ITEMS .... 58 Selection of Examination Items 58 Method of Analysing Student Performance on Selected Examination Items 58 D. COMPARISON OF PIAGET TASK PERFORMANCE WITH EXAMINATION ITEM PERFORMANCE -WD ASSESSMENT OF RESULTS 59 Method of Comparison 59 Method of Assessment . . « . « « . * . . . . « . « . . . « « « « « . o . . . . . . . . 59 IV. APPLICATION OF THE INVENTORY FOR ANALYSING STUDENT PERFOR-BLANCS ft«o*«»o*Oftoo*o«e*e*»»oooo»**o««e«o«o»e«**«*0*»ooo*o« OX j .2i"bi*ocinc"bxon. •©©©©©•©©©©•©©©•©©©©©©©•©••©•©©•»©o»©©©©©»©©© ol. A. THE PIAGET TASKS .... 61 Method of Analysis and Reporting of Results 6 l • Synopses, Analyses and Summaries of Student Performance. B. THE EXAMINATION ITEMS ' 97 Method of Analysis and Reporting of Results 97 Responses, Analyses and Summaries of Student Performance V. DISCUSSION OF RESULTS, SUMMARY AND CONCLUSIONS 128 A. COMPARISON OF STUDENT PERFORMANCE ON PIAGET TASKS AND SELECTED EXAMINATION ITEMS 128 B. IMPLICATIONS. OF COMPARISONS 131 Forma.tive Evaluation 131 Classroom Practice 132 C. CRITIQUE OF USEFULNESS OF THE INVENTORY 133 D. FURTHER RESEARCH 13^ E. . SUMMARY AND CONCLUSIONS 135 Bibliography 13T Appendixes A Transcripts of Student Performance On The Piaget Tasks 138 B Physics 110 Examination Papers 150 C Bases of Selection and Rejection of Examination Items 151* 1 The Dichotomous D i v i s i o n of Classes i n t o Sub-Classes forming the A d d i t i v e Groupment of Classes 22 2 A Two-Way C l a s s i f i c a t i o n of Classes forming the M u l t i p l i c a t i v e Groupment of Classes 23 3 The System of S e r i a l Ordering Forming the Groupment of Relations 2k k Summary of the Operations Involved i n the Groupments of Classes and R e l a t i o n s . (The designation of the l e t t e r s i s given i n Tables 1, 2, 3) 25 5 The l 6 Binary Operations Derived from the Generalized M u l t i p l i c a t i v e C l a s s i f i c a t i o n 30 6, The I n t e r r e l a t i o n s h i p s of the INRC Group, Using the Binary Combinations (pvq) as the I n i t i a l P r o p o s i t i o n 31 7 ^ Example of the I n t e r r e l a t i o n s h i p s of INRC Group of Operations i n the Simple Pendulum Task 33 8 P r o p o s i t i o n a l Statements Based on the 16 Binary Operations 3U 9 Synopsis and A n a l y s i s of Performance on the Angles of Incidence and R e f l e c t i o n Task: Student J.V Sk 10 Synopsis and A n a l y s i s of Performance on the O s c i l l a t i o n of a Pendulum Task: Student J.W 66 11 Synopsis and A n a l y s i s of Performance on the Combination of L i q u i d s Task: Student J.V 68 12 Synopsis and A n a l y s i s of Performance on the Balance Task: Student J.V 70 13 Summary of Performance On Piaget Tasks: Student J.V. 72 Ik Synopsis and Analysis of Performance on the Angles of Incidence and Reflection Task: Student B.H. 73 15 Synopsis and Analysis of Performance on the Oscillation of Pendulum Task: Student B.H. 75 16 Synopsis and Analysis of Performance on the Combination of Liquids Task: Student B.H. .»•... . .••••..•••..©•.•••• 79 "'17 Synopsis and Analysis of Performance on the Balance Task: S tudent B «H. « . o « . . . . . © . « » o . . . « * • . . . » . . • • © • » . . • • . . . . . . . © . . . . . . . 82 18 Summary of Performance on Piaget Tasks: Student B.H 85 19 Synopsis and Analysis of Performance on the Angles of Incidence and Reflection Task: Student L.vJ. 86 20 Synopsis and Analysis of Performance on the Oscillation of Pendulum Task: Student L.W. . . . . a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 21 Synopsis and Analysis of Performance on the Combination of Liquids Task: Student L.W 91 22 Synopsis and Analysis of Performance on the Balance Task: Student L.W. 93 23 ' Summary of Performance on Piaget Tasks: Student L.W 95 2k Overall Summary of Student Performance on a l l Piaget Tasks . . . . 96 25 Expected Response and Analysis of Performance on Selected Examination Item 1 99 26 Expected Response and Analysis of Expected Performance on • Selected Examination Item 2 102 27 Summary of Expected Performance on Selected Examination Items . 106 28 Student Response and Analysis of Performance' on Selected Examination Item'l: Student J.V. 107 29 Student Response and Analysis of Performance on Selected Examination Item 2: Student J.V. 110 30 Summary of Performance on Selected Examination Items: S "bud© 1*1 "t» J »\r 0 « o o » o o o e a « o o a « * « * o o * * e * » « o * * « o » « * e 0 * * w * 9 o » « * * a * « * o H 3 31 Student Response and Analysis of Performance on Selected Examination Item 1: Student B.H. 114 TABLE PAGE 32 Student Response and A n a l y s i s of Performance on Sele c t e d Examination Item 2: Student B.E . . . 0 0 0 117 33 Summary of Performance on Selected Examination Items: 3^4- Student Response and A n a l y s i s of Performance on Selected Examination Item 1: Student L.W • 121 35 Student Response and A n a l y s i s of Performance on Se l e c t e d Examination Item 2: Student L.V. 123 36 Summary of Performance on Selected Examination Items: S uHCLtSl'l't' L O'W « «a«e««o«««»*o««oooo«««eoo««oooo«o*««oe«o*o««o«*e*« 126 37 O v e r a l l Summary of Student Performance on Both Selected 38 Comparison of Student Performance on the P i a g e t Tasks and Selected Examination Items 130 CHAPTER I THE PROBLEM AND ITS CONTEXT The problem being investigated in this study is described in the context of educational evaluation. A discussion of the relevant aspects of educational evaluation is given f i r s t followed by a description of the general and specific problem being considered. The method of study is presented in outline, and the scope and limitations of the study are discussed. Finally,for purposes of clarification, a description is given of the terms used in this study. A. CONTEXT OF THE STUDY The philosopher Urmson (l9o9) defines evaluation as the intention to make a judgment about a phenomenon. According to Smith (1962, p. l ) , a precondition for judging a phenomenon is understanding i t . To understand and judge a pheno-menon such as intellectual development, for example, requires that the phenomenon be observed, analyzed into it's various elements, and the elements classified into a set of descriptive categories to which normative criteria can be applied (Westbury, 1970, p. 25l). Piaget's theory of intellectual development is often seen as constituting such a set of descriptive categories. The role of educational evaluation is twofold, and is described by Scriven (1967) as formative evaluation which provides feedback information for an ongoing educational program such as curriculum development or methods of instruction, and summative evaluation which assesses the worth of a new educational program after its completion. Teachers often need to perform the functions of both kinds of evaluation in the classroom situation. The function of formative evaluation is to provide the teacher with information for the improvement of a course of instruction while the teaching is being done. Summative evaluation serves the function of establishing the merits of a course of instruction upon its com-pletion. Formative evaluation includes, among other things, an evaluation of the level of intellectual performance required of students to perform classroom tasks successfully and an evaluation of the intellectual skills the learner is capable of performing. By careful matching of the intended level of intellectual functioning required to perform a task with observed intellectual capability and performance, the teacher can optimize conditions for learning, predict possible sources of difficulty in performing the task, and identify possible sources of difficulty with the task i t s e l f . The evaluation of intellectual behaviour, according to Piaget, would include a description of the operational mechanisms governing intellectual behaviour, (piaget, 1953, p. xviii) and not simply obtaining a measure of intellectual skills by means of various tests. This approach to evaluating intellectual behaviour appears to have an advantage in that i t enables one to judge the educational value in the learner's interaction with the environ-ment by understanding the intellectual processes by which the learner copes with his environment. Piaget's creative theorizing, however, poses a methodological problem when i t comes to dealing with problems of classroom practice, such as the problem in formative evaluation just described. It is the methodological problem, then, that is the focal point of the present study. In summary, the study has been oriented in a general area of educational evaluation - Scriven's theoretical view of evaluation, and a specific area, formative evaluation involved in classroom practice. It should be added that the study presented here was carried out within the context,of the Physics Education Evaluation Project of the University of British Columbia (1970/1971), B. STATEMENT OF THE PROBLEM General Problem ' The general problem to be investigated in this study is the development of a method for u t i l i z i n g Piaget's theory of intellectual development for evaluating the intellectual performance of students contending with formal concepts and methods of enquiry as presented in a first-year university physics course. In the classroom situation an educational theory is much needed for guiding teachers in handling the problems involved in the evaluation of intellectual performance. Skager and Broadbent (1968) discuss these problems and advocate that evaluative techniques should be based on a theoretical model in order to give generality and validity to the evaluative criteria. Unfortunately, current educational theory provides l i t t l e direct help for teachers in dealing with problems related to the learning of complex subject matter in areas such as the sciences (Easley, 1969). Cognitive psychology has touched on aspects of this problem (Ausubel, 1968), but there is not yet a theory of cognitive development formulated in such a way that i t can be used to meet the problems that arise in classroom practice (Easley, 1969). Of the available theories pertinent to this area of concern, Piaget's theory appears to be the most useful. The theory appears at least potentially useful to the classroom teacher for providing a means of identifying the way in which the intellect of the infant evolves into the intellect of the adult. Easley (1969) points out that Piaget's theory could be used to judge the educational value of a child's interaction with the.environment by providing an understanding of the way in which the child copes with his environment. It could be used to understand the way in which informal conceptual structures and intuitive methods of inquiry used by the child evolve into the more formal structures and modes of inquiry used by adults and that teachers seek to help students attain. Piaget describes developmental changes in the internal intellectual structures which are manifest in the psychological functioning of an indivi-dual performing tasks especially designed by Piaget for this purpose. Analysis of performance on the Piaget tasks provides information concerning the person's maximum intellectual potential for investigating and understanding his world. Unfortunately Piaget's theory lacks sufficiently clear formulation to guide classroom teachers in analyzing and interpreting performance on the Piaget tasks and, more seriously, lacks a methodology for evaluating intellec-tual performance on classroom tasks. The Specific Problem Students in a course in first year Physics at the University of British Columbia have an average chronological age of about 18 years, and have a l l passed either one or two High School physics courses, (a Physics 110 course requirement). Presumably students in this course could be expected, in Piagetian terms, to have reached the final stage of intellectual development, namely the stage of formal operations. This stage involves the ability to use formal, logical, hypothetical-deductive thought in coping with intellec-tual problems. Other factors being equal, students in the Physics 110 course should be able to answer correctly physics examination items which can be shown to require formal operational thought for their solution. Conversely, i t would be expected that students who do not prove capable of using formal operational thought would f a i l to respond correctly to these examination items. The specific problem investigated in this study is the development of a method for utilizing Piaget's theory of intellectual development for evaluating intellectual behaviour of students in the Physics 110 course coping with items on a final examination for the course, and to explore the usefulness of tho. method for rnQotinfi problems in science teaching. C. METHOD OF STUDY The problem of the study was investigated in five stages. Stage one involved an analysis of Piaget's theory of intellectual development with special emphasis on the concrete and formal operations -stages. Descriptions, or elements termed "descriptors" of psychological functioning related to theoretical constructs of processes and structure at these stages of development, were identified. The descriptors together with the related constructs constituted the theoretical basis on which the remainder of the study was developed. The results of Stage One of the study are presented in Chapter II. Stage two of the study was the development of an inventory for making a Piagetian analysis and interpretation of intellectual performance. An important reason for the development of the inventory of descriptors was to. meet the problem of facilitating the analyses of student performance on the Piaget tasks and selected examination items. A measure of consistency would be ensured i f the analyses were made in terms of descriptions of adequate behaviour acceptable for classification into a particular stage of development. Concerning the problems involved in analysing the performance of individuals Piaget writes: The important thing is to find a number of rules of interpretation which will unite the maximum of flexibility with the maximum of strictness, in so far as these two requisites can be reconciled ... we must find out what rules must be followed to avoid problems of premature judgment. (Piaget, 1929, p.23). The rules which Piaget subsequently discusses (Piaget, 1929, introduction) were inadequate for identifying formal operational thought in the performance of an individual. The major concern of this study was therefore to develop an inventory of descriptors for identifying formal operational thought which in effect serve as "rules of interpretation" advocated by Piaget. The inventory of descriptive categories representing different facets of intellectual performance at the formal operations stage was developed, and the descriptors elicited in Stage one were classified into the categories of the inventory. Exemplars of psychological functioning to which the descriptors were thought to apply were obtained from the Piagetian literature (inhelder and Piaget, 1958), and were included in the inventory for reasons of clarity and ease of application in analysis and interpretion of intellectual performance. The development of the inventory is described in the first part of Chapter III. In Stage three, data were collected on the performance of Physics 110 students on Piaget tasks at the formal operations stage and on selected items on the final course examination. Since most students at the first year university level were expected to be at the formal operations stage of intellectual development, application of the inventory to performance data on the Piaget tasks was seen as a way of checking the validity of the inventory. In addition performance data on the Piaget tasks was seen as an indication of maximum intellectual potential of the students performing the tasks. The performance data on the examination items were intended for use in demonstrating the applicability of the inventory to evaluating intellectual performance on a typical classroom task. Selection and descriptions of students and tasks, as well.as procedures for collecting and recording task performance is reported in the second part of Chapter III. A clinical technique was used in the administering of the Piaget tasks as any standardization of administration is a process to which Piaget strongly opposed. If subjects are to be stimulated into displaying their intuitive»competence and weakness in intellectual thought, they must be given the freedom to follow their thoughts to conclusions. This freedon may be curtailed in the administering of standardized tasks, and there is always the danger that such tests tap only the surface of a subject's cognitive s k i l l s , and would "not provide a reliable index of the real quality of his understanding" (Ginsburg and Opper, 1969, p.227). In view of these objections to standardization, a clinical approach was used in the administration of the Piaget tasks which was basically unscheduled (Piaget, 1929, p. 7-23) (See Chapter II. p. 10). The analyses of student performance were, however, standardized in that the descriptors and their exemplars were used as standards against which performance was compared. In Stage four of the study, an attempt was made to illustrate the applicability of the inventory to analyzing and interpreting performance data. Transcripts of videorecordings of task performance were analysed and interpreted .by means of the inventory. The inventory was applied to the examination items by analysing first the responses expected by the instructor and a tutor for the course, and secondly to the actual responses to the items made by the students involved in the study. Details of the analyses and summaries of the results are presented in Chapter IV of the study. Stage five included a comparison of the results of analysis of per-formance on the Piaget tasks with the actual performance on the examination items. Implications of the comparisons for the purposes of formative evaluation and improvement of classroom practice are suggested, and a critique of the usefulness of the inventory for these purposes is presented. The results of Stage five of the study are presented in Chapter V. D. SCOPE AND LIMITATIONS OF THE STUDY The study was confined to a consideration of the formal operations stage of Piaget's theory. The inventory developed, therefore, is applicable only to Piaget tasks, classroom tasks, and students at this level of intellectual behaviour. The inventory developed was used to analyse and interpret performance on tasks that involved both verbal and non-verbal responses. The inventory therefore, can be applied to classroom tasks of both kinds provided that a record of performance is available for analysis. The inventory has been applied by persons both trained and untrained in Piaget theory, although considerable familiarity with the inventory was required before application. The inventory, therefore could be useful to teachers who are willing to thoroughly familiarize themselves with the inventory. Piaget's theory i t s e l f , being an original contribution to epistemoloty, presents difficulties for establishing the construct validity of the inven-tory in that i t is constantly being modified and articulated. Piaget's older volumes use a somewhat different vocabulary from his more recent writings. Furthermore, i t has been drawn to the writer's attention that the English translations of Piaget's theory are sometimes inaccurate. The theory itself, is primarily philosophical in nature and often very difficult for untrained individuals to understand. These difficulties with the theory pose a serious problem in developing a valid inventory which can be easily used in the classroom situation. Even for a trained person, use of the inventory is difficult and time-consuming. Further, while the information attainable through the use of the inventory is detailed and in-depth, i t is not clear at present how a teacher might go about using this information for improving instruction. Without the assistance of a person trained in Piage-tian theory, a training program using both written materials and demon-strations would seem desirable before attempting to use the inventory. The study is exploratory in nature, and attempts to find a way of using Piaget's theory for meeting a problem of classroom practice in evaluation without deviating too far from Piaget's view of retaining as much flexibility in procedural rules as possible. Retaining this flexibility raises difficulties in developing a reliable instrument. Standardization of procedures for administering tasks and applying the inventory, therefore, has not been undertaken. The reliability of the instrument, in terms of consistency of application and results across interviewers, consequently, is much more dependent on a thorough understanding of Piagetian theory than i t is on following procedural rules. E. DESCRIPTION OF TERMS In an attempt to clarify Piaget's theory of intellectual development, definitions of some of the terms used by Piaget introduced in this thesis are provided. ~ (See also Furth, 1969, pp 259-265). 1. Accomodation - The modification of structures of a biological organism by assimilated elements. Likewise the modification of Intellectual structures to deal with assimilated "objects of knowing". 2. Adaptation - In a biological sense involves a balanced state between an organism and i t s environment - in a Piagetian sense i t involves the establishing of a state of equilibrium between assimilation and accomodation. 3. Assimilation - The integration of external elements into evolving or completed structures of an organism. Likewise, an "object of knowing" is incorporated by the existing i n t e l l e c t u a l structures as "knowable". k. l6 Binary Operations - The t o t a l number of propositional statements which are the product of two given propositions - form a commutative group, subject to the laws of the INRC group of operations -constitute a "structures whole". 5. Concrete Operations - Characteristics of the f i r s t stage of operational intelligence - concerned with a limited extension of empirical r e a l i t y . 6. Elementary Groupments - Incomplete l o g i c a l systems governed by the five operations of composition, inversion, identity, associativity, and tautology, of which the operation of tautology is a restrictive con-dition. 7. Epistemology - The theoretical study of the nature of knowledge. 8. Equilibration - In a biological sense involves an internal regulatory process characteristic of most biological systems - in a Piagetian sense i t applies to the process of the development of the intelligence in which a balance is established between assimilation and accomodation. Can also be described as the process by which the individuals environmental interactions are balanced with his autoregulations. 9. Equilibrium - The state of balance between assimilation and accomodation. 10. Formal Operations - Mature, l o g i c a l , hypothetical - deductive thought characteristic of the f i n a l stage of operational intelligence -Develops from concrete operations as a result of the co-ordinating of the elementary groupments into a single system. 11. Formal Operational Schemes - See Operational Schemata. 12. INRC Group of Operation's - The means for mentally transforming data about the real world so that they can be organized and used selectively. A group of four transformations also called operations, i.e. identity operation, negation operation, reciprocity operation and correlation operation, which when applied to particular propositional statements gives use to the formal operatory schemes - Serve as heuristic principles. 13. Intellectual Behaviour - The behaviour manifest in an individual as a result of his stage of i n t e l l e c t u a l development. Ik. Intellectual Structures - The internal organization of interrelated and co-ordinated forms of "knowing actions" or schemes which organize r e a l i t y i n terms of concepts such as object, cause, space and time. 15. Intelligence - The t o t a l number of possible • i n t e l l e c t u a l co-ordinations or transactions that characterize the adaptive behaviour of the individual towards his environment. 16. Logic - In Piagetian sense - formal system which can be used to describe the i n t e l l e c t u a l structures of intelligent behaviour - A system of operations carried out on propositional statements. 17. Logic - According to others - "the laws of thought" (Bode); "the theory of inquiry" (Dewey); a system of syntactical c a l c u l i " (Russell and Whitehead); a theory of proof as "natural deduction" (Vaughan). 18. Logical Operation - An internal transformation of one propositional statement into another - subject to the laws of a perfect mathematical group, i . e . IHRC group of operations. • • '• . • .19. Operation - A reversible, internalized mental action which i s co-ordinated with others i n an integrated structure - analogous to a l o g i c a l operation or an internal transformation. I f used as a structure, should be termed operational scheme. 20. Operational Schemata - (inhelder and Piaget, 1958) Formal Operational Schemes - (Piaget 1969, P« 1^ 0 - ikk) The term formal operational schemes has recently replaced the older . term operational schemata. However, as constant reference i s made to the Inhelder and Piaget (1958) text in this thesis, i t i s important to note that their meanings are the same. (Recently Piaget has used the term Schema to mean simply the figurative aspect or image of an object). A co-ordinated set of higher order schemes which imply the diverse p o s s i b i l i t i e s implicit i n the propositional logic - based on the combinational system - obey the conditions of the INRC" group of operations - considered to be examples of equilibrium between the i n t e l l e c t u a l processes of assimilation and accomodation. 21. Propositional Logic - Operations performed on propositional statements which arise from the commutative group of the l6 Binary Operations, which are subject to the laws of the INRC group of operations. 22. Psychological Functioning - Overt i n t e l l e c t u a l behaviour which results from and reflects the internal i n t e l l e c t u a l structures of intelligence. 23. • Scheme ' - The internal general form of a specific knowing ac t i v i t y . Schemes become co-ordinated into higher order schemes, e.g. formal operational schemes,,, and are collectively included in the term int e l l e c t u a l structures. 2k. Stages - Consecutive identifiable periods of the i n t e l l e c t u a l development of a child.' 25. Structured Whole - See l6 Binary Operations. 2 1 . Propositional Logic - Operations performed on propositional state-ments which arise from the commutative group of the 1 6 Binary Oper-ations, which are subject to the laws of the INRC group of operations. 2 2 . Psychological,Functioning - Overt intellectual behaviour which re-sults from and reflects the internal intellectual structures of in-telligence . 2 3 . Scheme - The internal general form of a specific knowing activity. Schemes become co-ordinated into • higher order schemes, e. .g. formal operational schemes, and are collectively included in the term intellectual structures. 24. Stages - Consecutive, 1 identifiable periods of the intellectual devel-opment of a child. 2 5 . Structured Whole - See 1 6 Binary Operations. . THEORETICAL BACKGROUND OF THE STUDY PIAGET'S THEORY OF INTELLECTUAL DEVELOPMENT , . The focal point of this Chapter is the presentation of aspects of Piaget's theory of intellectual development relevant to the study. The relevant aspects are the intellectual structures and corresponding psy-chological functioning of individuals at the concrete and formal opera-tions stages of development. The f i r s t part of the Chapter presents the general theoretical framework developed by Piaget, and the aspects of major concern in the theory. The remainder of the Chapter presents a description of Piaget's clinical method used in the study, followed by a more detailed description of the salient aspects of structures and func-tioning of the intellect at the concrete and formal operations stages. A. THE PIAGETIAN FRAMEWORK Piaget address himself to' the problem of 'What is Intelligence?' He claims that "every psychological explanation comes sooner or later to lean either on biology or on logic." (Piaget, 1950, p. 3)« It is to both these two sources of knowledge that Piaget turns to ansxrer the prob-lem. Theoretical Constructs Piaget uses biological concepts at the most general level of his theory to describe processes by which one form of intelligence develops into another. At a more specific level he makes use of symbolic logic as an instrument or technique to analyse and describe psychological behaviour and the structure of thought at different levels of intellec-tual developmento Piaget defines intelligence as follows: Intelligence constitutes the state of equilibrium towards which a l l the successive adaptations of a sensorimotor and cognitive nature9 as well as a l l assimilatory and accoma-datory interactions between the organism and the environ-ment (are directed) - (Piaget 9 1950* P» H). Piaget is describing the development of the epigenetic system in which human intelligence evolves through a series of adaptations or unstable equilibria, towards a final stable equilibrium. Each equil-ibrium is considered to be analogous to the adaptation of an organism to its environment,, involving both environmental interactions and auto-regulations of that organisms In other words, the adaptation of intelligence depends "as much on progressive internal co-ordinations as on information acquired through experience„" (Piaget 9 1970 9 p. 703).. The final stable equilibrium results when a real balance exists between the individual 8s environmental interactions and his auto-regulations. Each intermediates unstable equilibrium develops from the one prior to i t s and enables a further equilibrium to be reached. The whole process is called equilibration. Every knowing activity has a general internal intellectual struc-ture which Piaget refers to as a scheme«, It is through the elabora-tion of schemes that the child develops the ability to deal with his environment,. To explain the elaboration of schemes Piaget uses the analogy of accommodation and assimilation of food in an organism or of energy in photosynthesis and explains further his use of the term equilibrium. Assimilation,, from the biological point of view9 is: .o.the integration of external elements into evolving or completed structures of an organism...e.g. the assimila-tion of food consists of a chemical transformation that incorporates i t into the substance of the organism; (pho-tosynthesis i s the) integration of radiation energy i n the metabolic cycle of the plant (p. 706-707), -yJ The incorporating of environmental data i s through the process of assimilation. Just as an organism can only assimilate those mater-i a l s which i t can deal with or u t i l i z e , so the i n t e l l e c t assimilates only those objects, a c t i v i t i e s , experiences or ideas with which the schemes can deal. Analogous to the way i n which the organism confers a quality on the materials assimilated, the assimilating scheme confers  a quality on that which i t assimilates. For example a baby has a very simple scheme, the grasping scheme. I t assimilates objects as "grasp-able" and confers the quality of "graspability" on the "object of knowing." I f , as a result of the interaction the scheme becomes modi-fied to deal with that particular "object of knowing," then that process i s called accomodation. The balance between these two processes des-cribes the adaptation of the scheme to the environment with which i t interacts * As a result of the process of adaptation, a process of differen-t i a t i o n becomes evident. Piaget says that i n t e l l e c t u a l development can be seen as a process i n which the mental organization of a child changes from being undifferentiated through progressive stages of d i f f e r -entiation to clearly and highly differentiated mental processes. (Piaget, 1969, p. 152). As the mental processes become more d i f f e r -entiated, they also become correspondingly more co-ordinated. The processes of equilibration (including assimilation and accomo-dation) and differentiation and co-ordination, constitute the general adaptive patterns of a biological nature, applied by Piaget to the problem of epistemologyo They w i l l not be considered further i n this study. Instead, the processes by which the functioning of the int e l l e c t u a l structures develop w i l l be discussed since this i s of greater immediate importance to the present study. Piaget has attempted to describe the in t e l l e c t u a l structures i n terms of the l o g i c a l operations characteristic of l o g i c a l systems referred to as groups and l a t t i c e s . Piaget explains his position on the use of the l o g i c a l operations to describe the structures of the i n t e l l e c t by taking the view that symbolic logic " i s the mirror of thought and not vice versa." (Piaget, 1950, p. 27) i . e . logic i s the result of man's i n t e l l e c t u a l thought and can therefore be used to describe, i n some measures at least, those i n t e l l e c t u a l structures which i t r e f l e c t s . / As the child develops i n t e l l e c t u a l l y he progresses through a series of identifiable i n t e l l e c t u a l stages. Whether or not a child has reached a particular stage (or substage) of in t e l l e c t u a l develop-ment depends on his behaviour when he i s faced with particular tasks. The nature of these behaviours constitute a necessary l i n k between Piaget's theoretical constructs and r e a l i t y . Piaget describes his theory with the term "axiomatics",; by which he means, "axiomatics (can) replace the inductive science t<rhich forms the essential lin k to r e a l i t y . " (p. 28). The psychological functioning or behaviour of an individual i s considered to be the empirical, observable manifestation of his internal i n t e l l e c t u a l structures. The Piaget Tasks The tasks are designed to produce inductive, experimental evidence which substantiates the deductive aspects of Piaget's theory i . e . the theoretical constructs. They are based on actions involving the manipulation of simple apparatus. The l e v e l of psychological func-tioning of the subjects i s determined from both verbal and overt responses made when attempting a task. Piaget describes the psychological functioning displayed i n performing the tasks as operational„ In the psychological sense, "oper-ations are actions intemalized 8 reversible, and co-ordinated into sys-tems characterized by laws...which apply to the system as a whole." (Piaget, 1953. P« 8)« In the l o g i c a l sense,, operations refer to the symbolic manipulations involved i n l o g i c . Piaget explains psycholog-i c a l functioning which i s the manifestation of the in t e l l e c t u a l structures i n terms of l o g i c a l operations. This w i l l be explained i n greater d e t a i l l a t e r i n this Chapter. The tasks have been specially designed so as to illuminate d i f f -erent and particular psychological functionings of individuals at different stages of development. The Stages The psychological functioning (or behaviour) of individuals has been observed and described by Piaget and his co-workers as consisting of "four main stages.which extend over the period from b i r t h to maturity." (Piaget, 1953. P«9). These stages are the sensory-motor  stage (0-2 years), the pre-operational stage (2-5 years), the concrete operations stage (5-H years), and the formal operations stage (11-15 years and beyond). The ages given are approximate, the important fact being that the order of appearance of the stages i s constant. At about 15 years of age a child should optimally be capable of mature, logical, hypothetical-deductive thought. The manifestation of this sort of psychological functioning is found at' the formal operations stage. Rudimentary, incomplete forms of adult psychological functioning typifies the concrete operations stage. It is possible for the developmental sequence of some individuals to stop at the concrete operations stage, or to show evidence of having made the transition to the final stage without having fully attained the level of psychological functioning characteristic of that stage. Factors such as experience, social environment and neuro-physiological conditions play a role in the rate and extent of development. B. PIAGET1S CLINICAL METHOD Piaget and Inhelder*s investigative technique has been primarily descriptive and analytical. They advocate that a clinical interviewing technique be used in the administration of the tasks, and that the interviewer should attempt to e l i c i t the maximum potential of the subject without unwittingly "teaching" him the answer. The procedure consists of the interviewer showing the apparatus to the subject and then'posing the problem to be solved. The interviewer uses his discretion as to the extent of his questioning and its nature. The questioning technique should, however, be as unscheduled as possible in order to provide freedom for the subject to answer to the best of his ability in his own way. There are a number of drawbacks to the use of the clinical method which have earned Piaget ;a measure of criticism. Firstly, the method is time-consuming, resulting in a limitation to the number of tests performed and number of subjects used. The necessity of allowing the subjects the opportunity for spontaneous questions and answers makes i t difficult to standardize the administration of the Piaget Tasks. It is possible to ensure that the selection of the Piaget Tasks and the conditions are the same, but the questions asked can only be approximately the same in each case. In the introduction to The Child's Conception of the World (1929), Piaget indicates that the subjects' answers should be related to a scale or schedule that serves as a standard of comparison, both qualitative-ly and quantitatively. While this would serve the real need of standardizing the interpretation or analysis of subject performance on the tasks, no such scale or schedule seems to be available, and the major concern of this study is addressed to this need. The development of a means for standardizing the analysis and interpretation of subject performance on the Piaget Tasks may also be applicable, to other situations involving subject performance, for example student performance in educational evaluation. Further drawbacks in the clinical method arise in situations where the subject is reticent, and does not communicate a l l his thoughts due to shyness or feelings of inferiority, or where he feels that certain events are too obvious to be commented upon or even cases of disinterest. It is occasionally difficult, in the case of young children, to distinguish play (or romanticizing) from belief. Confusion in a subject can be the result of previous teaching rather than an indicant of his intellectual structures. While i t is clear that there are problems involved in the clinical method used by Piaget, i t is probably true that this is the most authentic method for observing the maximum potential of an individ-ual's intellectual functioning,, especially when compared with respon-ses to standard tests. C. STAGE OF CONCRETE OPERATIONS An individual who reaches the concrete operations stage of in-tellectual development is able to use operational thoughfwhich was not possible at the earlier stages. Operations are "the means for mentally transforming data about the real world so that they can be organized and used selectively in the solution of problems." (In-helder and Piaget, 1958, p. XIII). Piaget uses the word "operation" in this context to emphasize that the individual performs mental actions which are derived from the "interiorisation" of physical ac-tions. For example, the operation of addition can be performed both physically and mentally, and the mental operation is the result of the interiorisation of the physical operation. Piaget also emphasizes that operations are reversible, and "constitute set-theoretical struc-tures." (Piaget, 1970, p. 705). For example, the addition operation is reversed by the subtraction operation. Particular logical oper-ations, vis. operations of combination, associativity, inversion, identity and closure apply to the logical system of addition and sub-traction as a whole, constituting what Piaget calls the "set-theoret-s i c a l structures." Intellectual Structures When describing the intellectual structures of the concrete oper-ations stage in terms of the logical systems of classes and relations, Piaget noticed certain restrictions. The logical system of classes and relations normally form a perfect mathematical sot described as a lattice and a group, having logical operations of combina-tion, associativity, inversion, identity and closure (see Table k). At tho concrete operations stage, however, the operation of tautology is also present, and i t is this operation which restricts tho operation of closure, resulting in a failure to constitute a formal logic or perfect group and lattice structures. (Piaget, 1953, p. 17 ) . The psychological functioning of individuals at this level is correspondingly limited, e.g. in the lack of ability to generalize and the restriction to the handling of concrete sub-ject matter. Piaget has introduced the term elementary groupement to describe the limited kind of logic used by individuals at this stage. Elementary Groupements of Classes There are two kinds of systems of classes: 1, Additive Classes are typified by dichotomous biological classif-ications (see Table 1) . A class of objects having certain proper-ties, e.g. a class of a l l animals, is designated the letter C. The sub-classes of C are designated the letters B and B', and represent vertebrates and invertebrates respectively. Similarly, the sub-classes of B are designated the letters A and A', which represent mam-mals and non-mammals respectively. The sub-classes of B 1 are not designated, forming what is known as a semi-lattice. The Dichotomous Division of Classes into Sub-Classes forming the Additive Groupoment of Classes. Animals vertebrate's invertebrates mammals non-mammals Example 2, Multiplicative Classes result from the multiplication of two classes designated B-^  and B2» in which a l l the objects in B-^  are contained,in B£ and vice versa. For example, B-^  represents ani-mals divided into sub-classes of A^, vertebrates, and A-^ ', inver-tebrates. Class B2 represents animals divided into sub-classes of A2, aquatic animals, and Ag', terrestrial animals. The pro-duct of B-j^  and B£ consists of four different combinations of ani-mals according to their habitat and their structural type (see Table 2 ) . This system of multiplicative classes conforms to the operations characteristic of a perfect group. A Two-Way Classification of Classes forming the Multiplicative Groupment of Classes. (animals) vertebrates invertebrates A • A„ A l A 2 A 8 A ' 1 A 2 terrestrial B 2 (animals) ^ 1 ^ 2 = v e rtebrates, aquatic. ^ 1 ° ^ 2 B invertebrates, aquatic. A]_ A 2 S = vertebrate, terrestrial. ^ l ' ^ 2 8 = S invertebrate, terrestrial. Elementary Groupment of Relations The operations involved in the elementary groupements of rela-tions also involve the operations of composition, inversion, identity, tautology, and association (see Table k) used in the system of serial ordering. For example, a number of objects such as wooden rods may be arranged in order of increasing or decreasing length. The relation-ships between each two consecutive rods and between each rod and the f i r s t rod constitute the system of relations (see Table 3 ) . The System of Ser i a l Ordering Forming the Groupement- of Relations. a, b, and c = the relation expressing the difference between A. and B, A . and C , A and D. a' and b* = the relation expressing the difference between B and C, and C and D. The details of the operations of composition, inversion ident-i t y , tautology and association at the concrete operations stage w i l l not be discussed i n further d e t a i l as the present study i s more concerned with the stage of formal operation. The reader i s re-ferred to Piaget's Traite de Logique (1949) for further informa-tion. A summary i s provided i n Table 4. Characteristics of the Intellectual Structures For the purposes of this study the three most important charac-t e r i s t i c s of the l o g i c a l structures of the concrete operations stage are: 1. The different operations cannot be integrated with each other. This means that the individual lacks the a b i l i t y to l i n k operations together to give a truly logical interpretation of his experience. TABLE h Summary of the Operations Involved in the Groupments of Classes and Relations. (The designation of the letters is given in Tables 1, 2, and 3) Operation Groupment of Groupment of Additive Classes Multiplicative Classes Groupment of Relations composition any element in a set may be combined with any other element of the same set, and. the logical result is itself an element included in that set A + A' = B B + B' = C B l x B2 A x A 2 + A]_ A2' a + a' + A1*A2 + A^Ag' Inversion for each element in a set, there is only one element which when added to i t effectively' nullifies the original operation _A-A' = -B B^ x B 2 B, B, A = B - A' and (where: B 2 means a + (-a) = 0 eliminating B2) A* = B - A Identity for each element in a set, there is only one element which when added to another in the set leaves i t unchanged A + (0) = A B1 = Z fwhere Z is the most general class) a + (0) = a Tautology when an element in a set is added to itself the result is the same element (i.e. the effect is not cumulative) A + A = A .(BjBg) x (A^A2' ) = A ^ ' a + a = a Associativity elements of a set are put together in different ways to give the same result (A + A') + B - (B XB 2) x (A]_A2') (a+a') +b* A + (A* + B) = (BXB2) x (A ^ ' ) = a + (a' + b*) Closure one element in a set when added to another produces a third element which is included in the set See Table 1 See Table 2 See Table 3 2 e Inversion (or negation) is the most important and the most clearly defined operation. It has the characteristic "of being able to return to (the) original state or starting point...and results from the co-ordination of the actions of combining, dissociating, ordering and the setting up of correspondences." (Piaget, 1953» p. 13). It allows the individual to construct a proposition and then to reverse the direction. This reversibility of thought enables the individual to conserve (or hold invariant under transformation), . weight, volume, distance, etc. Inversion involves an annulment of the composition operation to give the original class, in the case of the groupements of classes. For in the inversion operation in the groupement of relations, however, the result of eliminating the difference is a null difference relationship. This gives rise to a statement of equivalence or symmetry, which is better described as a reciprocal operation. 3. The operation of tautology is restrictive, and results in an incomplete operation of associativity, forming an imperfect group and a semi-lattice. A perfect mathematical group has the property of closure, which is not found here due to the restrictions of the tautological operation. D. PSYCHOLOGICAL FUNCTIONING AT THE CONCRETE OPERATIONS STAGE. Empirically Oriented Thought The individual at this stage of development is capable of solving some problems, but in a rather limited way. Piaget says: "Concrete operations consist of nothing more than a direct organization of immed-iately given data." (Inhelder and Piaget, 1958, p. 2^9). The empirical or immediately given data may be concrete objects or events, e.g. discs of different colours or shapes, or may be graphic representations or even imaginative representations of actual objects. The important point here i s that the objects need to be such that the individual can easily dissociate the objects' properties from his own actions, e.g. different colours or different lengths. These properties "can be objectified more readily" (p. 249) than properties such as weight and density, which cannot be easily represented i n drawings or i n the imagination. In this way i t can be understood why concrete operations can be described as "nothing more than a limited extension of empirical r e a l i t y . " (p.250). At this stage, form such as change i n shape or weight cannot be divorced from the subject matter, because the l o g i c available to the individual i s i n s u f f i c i e n t l y f l e x i b l e . The individual i s constantly using the empirical facts as the starting point i n his reasoning. Non-Integrated Thought The individual has no means by which he can integrate his thoughts. The propositions he makes are not integrated within a system having the effect of li m i t i n g their usefulness. He can handle the l o g i c of classes and relations, but the mental operations he performs "function only with reference to observations or representations regarded as true, and not on the basis of a mere hypothesis." (Piaget and Inhelder, 1969» p. 132). Furthermore, he i s only able to integrate his thoughts by "bringing classes or relations together by a class inclusion or contiguous linkage which moves from one element to the next." (Inhelder and Piaget, 1958, P« 274)» In other words the reasoning process i s constituted by a step by step progression of thoughts or propositions. These may be obtained simply by "decomposing and recomposing the con-tent of propositions", (p.292) or from combinations obtained by simple " t r i a l and error", (p. 311) or even "haphazard" variations of given empirical data (piaget, 1953, p.19). The individual thus collects a l l the information concerning the problem with which he establishes an approximation of "the whole picture" and can at least establish the invariance in the empirical facts under transformation. The lack of integration in concrete operational thought is further seen in the way in which the individual "is only able to introduce or eliminate (a) variable in order to see i f (that variable) itsel f plays an active role, and not as a means of studying the other (variables)." (inhelder and Piaget, 1958, p.285). Moreover, the limited intellectual ability is observed in the way in which he fails truly to separate variables. The limitation is particularly clear "in cases where a factor cannot be physically S e p a r a t e d . " (p.28k). Once again the empirical nature of the individual's thought is apparent. Examples of the limitation of unintegrated operations can be shown firstly in the way in which the individual at this stage attempts to understand the concept of proportionality between the weights and lengths on a balance. He may search for a common denominator of the two relations of weights and lengths and instead of suggesting the •proportion W/W* = L'/L he may think the relation is additive, resulting in an inequality of differences where W - W = L' - L. He interprets the problem in concrete terms, applying the elementary groupments of additive classes to the relationship between weight and length. (W and L represent weight and length on the one side of the.balance and V/' and L' represent weight and length on the other side). Secondly, the individual is able to increase the weight of a pendulum so as to establish the effect of the weight factor. As he lacks the ability to integrate the logical operations he performs, he does not consider the possible effect :of different lengths of the pendula while experimenting on the effect of weight. Non-Generalizable Thought A further limitation of concrete operational thought is "that i t cannot be immediately generalized to a l l physical properties." • (p. 2k9). The generalizations which are made, are simply "hypotheses which do no more than outline plans for possible action." (p. 25l). They are better described as solutions for particular problems or cases which are empirically based, and are an approximation of ' the whole picture'. E. STAGE OF FORMAL OPERATIONS An individual at the formal operations stage is capable of using logical hypothetical-deductive thought. • Intellectual Structures The two main intellectual structures of formal operational thought can be described in terms of operations characteristic of the Combina-tional System and the' INRC Group. The integration of the Combinational System and INRC Group result in what Piaget calls' a structured whole which constitutes a truly formal logic. The individual is now able . to use propositional logic, and specific combinations of propositional logic which form the formal operational schemes. The Combinatorial System The structure of formal operational thought is based on the dual structure of a complete lattice and a perfect group. The operations of the structure of the groupements of classes and relations (previously separated in the semi-lattice and group systems) become integrated, forming the system of the 16 Binary Operations. This is a combinatorial system which is derived from a generalization of multiplicative classification in which both the multiplication of the elements and the resultant products are considered. In addition to the single "n x n" combinations of elements, the products thereby obtained are combined in pairs, triplets, one set of four and a set of zero combinations, giving a "set of a l l subsets." There are a total of 16 Binary Operations obtained in this way. (See Table 5).. TABLE 5. The 16 Binary Operations Derived from the Generalized Multiplicative Classifications. B-| (animals) Al H' vertebrates invertebrates A XA 2 ' (1) 1 A1, A 2 (2) A 2 aquatic AlA 2' (3) | 'Al 'A2 ' | i i A2' terrestrial ~£>2 (animals) A-L+AI* = B ^ = A 2 + A 2 * = B 2 i.e. vertebrates (A-^) + invertebrates ( A - ^ 1 ) constitute the class of animlas ( B ^ ) and aquatic animals ( A 2 ) + terrestrial animals ( A 2 ' ) constitute the class of animals ( B Q ) . The multiplication of B-j_ x B 2 gives l6 possible binary combinations: (using the numerical representations given in Table 5) 1, 2, 3 and 4 = The *n x n' combinations (giving a total of 4) 12, 13, 14, 23, 24, 34, = The 'set of a l l subsets1 123, 134, 124, 234, combinations (giving a 1234, and 0. total of 12). The 16 Binary Operations are related to each other in the sense that each one can be derived from the others by operations conforming to those of the complete lattice and perfect group. (The restrictive operation of tautology does not apply to the l6 Binary Operations). Piaget's books, Traite de Logique (1949) and Logic and Psychology (1953) give further details on this subject. The INRC Group of Operations The combinatorial system, is structures by four operations or transformations. Any one combination can be transformed into any other by means of an operation. The operations are those of identity ( i ) , negation (N), reciprocity(R), and correlation (C). They form a group called the INRC group which has a particular structure and properties i.e. the group is commutative. The interrelationships of the INRC group are shown in Table 6. Properties of the INRC Group Identity Operation (l)leaves the original proposition unchanged, i.e. I(pvq) = (pvq). The identity operator is also the resultant of every operation and its inverse, i.e. N(pvq) = (p.q). N e g a t i o n O p e r a t i o n (N) changes b o t h the s i g n s and the o p e r a t o r o f the o r i g i n a l p r o p o s i t i o n i . e . N(pvq) = ( p . q ) . R e c i p r o c a l O p e r a t i o n (R) changes the s i g n o f each s ta tement i n the p r o p o s i t i o n , b u t keeps the same o p e r a t o r , i . e . , R(pvq) = ( p v q ) . C o r r e l a t e O p e r a t i o n (C) changes the o p e r a t o r o f the p r o p o s i t i o n , i . e . , C(pvq) = ( p . q ) . Key : v = e i t h e r , o r b o t h ( d i s j u n c t i o n ) ; • . = and ( c o n j u n c t i o n ) ~~ = n o t ( n e g a t i o n ) TABLE 6 The I n t e r r e l a t i o n s h i p s o f the INRC G r o u p , u s i n g the B i n a r y Combina t ion (pvq) as the I n i t i a l P r o p o s i t i o n . I n the t a s k r e q u i r i n g the d e t e r m i n a t i o n o f f a c t o r s a f f e c t i n g o s c i l l a t i o n s o f a s imp le pendulum, (pvq) r e p r e s e n t s the s ta tement t h a t e i t h e r w e i g h t (p) o f bob or l e n g t h (q) o f suspending s t r i n g o r b o t h , i n f l u e n c e the r a t e o f o s c i l l a t i o n . The i n t e r r e l a t i o n s h i p s between the INRC t r a n s f o r m a t i o n s i s shown i n T a b l e 7 u s i n g t h i s example An Example of the Interrelationships of INRC Group of Operations in the Simple Pendulum Task. either weight or length or both A •R either weight or Alength but not both together C both weight and length not weight and not length nor both The INRC group of operations allow the individual to perform operations on propositions themselves without regard to the reality of the content. The mind is no longer limited to operating with the logic of classes and relations characteristic of the concrete operations stage. "The propositional operations ... form a single system such that i t is possible to move with accuracy from anyone of its sixteen elements to each of the others." (inhelder and Piaget, 1958, p.303). The INRC group of operations provide the means for doing this. Piaget refers to these operations as 'Second Order Operations' or 'Interpropositional Operations'. Propositional Logic "Propositional logic is a logic of a l l possible combinations, whether these combinations arise in relation to experimental problems or., purely verbal questions" (inhelder and Piaget, 1950, p. 253). Verbal statements are in fact substituted for objects, and are symbolically represented by letters such as p, q, etc. When the symbolic represen-tations of the verbal statements are connected by what is called a combinatorial operation such as conjunction (.), disjunction (v), implication (=>) etc., they become propositional variables, e.g. in the pre-positional statements (p.q), (pvq) and (p=«q). The logic of " a l l possible combinations" for the propositional variables p, p, q and q gives rise to sixteen possible combinations forming the combinatorial system, or l 6 Binary Operations. The derivation of these has been shown in Table 5 . Each propositional statement has particular implications, e.g. ( p3q) = (p.q)v(p.q)v(p.q), which can be represented by a Venn diagram. The propositional statements based on the l 6 Binary Operations are given in Table 8 , including the implications of each. Formal Operational Schemes Formal,operational schemes are sets of propositional statements which relate to a particular concept such as proportionality and are strongly linked to each other within the combinatorial system and the INRC group of operations. They are to be considered as representations of the cognitive process of equilibration between assimilation and accomodation. The links between the operations involved in dealing with the concept of proportions, for instance, are established by the individual as a result of his need to interpret the concept in the course of his experiences. "VJhen the need is felt , he TABLE 8 ' Propositional Statements Based on the 1 6 Binary Operations. Operation Propositional Statement 1. Disjunction = (p .qMp.q)v(p. q) 2. Conjoint Negation (p.q) (p.q) 3. Conjunction (p.q) (p-q) 4 . Incompatibility (p-q) = (p.q)v(p.q)v(p. ci) 5 . Implicat ion (p^q) = (p .qMp.q)v(p. q) 6. Non-Implication (pAq) (p.q) q) 7 . Converse Implication (q^p) (p.q)v(p.q)v(p. 8. Negation of Implication (q«>) = (p.q) 9. Equivalence (p=q.) = (p.q)v(p.q) 10. Reciprocal Exclusion (P=q) = (p.q)v(p.q) 11. Independence P U J (p . q M p.q) 12. Inverse of Independence P [q] = (p.q)v(p.q) 13. Independence q [pl (p.q)v(p.q) 1 4 . Inverse of Independence •q [p] = (p . q M p.q) 1 5 . Tautology p*q = (p.q)v(p.q)v(p. q)v(p.q) 1 6 . Contradiction [o] = nothing true. manages to work them out spontaneously." (p .308) , providing he i s cap-able of using propositional l o g i c . There are eight formal operational schemes described by Piagot and Inhelder (1958). namely, combinatorial operations, proportions, co-ordination of two systems of reference, multiplicative compensation, concept of mechanical equilibrium, notion of probability, correlation, and forms of conservation which cannot be empirically v e r i f i e d , e.g. inertia„ In the example of the formal operational schemes of proportion-a l i t y necessary for understanding the simple balance, the INRC operations are related to each other as shown below: Increase of wt & d i s t . on Increase of wt. or d i s -l e f t arm of balance _ tance on l e f t arm~of balance Increase of wt. & d i s t 0 c n ~~ , Increase of wt. or d i s -right arm of balance tance on right arm of balance (1) I f an increase of weight and distance on the l e f t arm of the balance i s designated by p and q respectively, end an increase of weight and distance on the right arm of the balance i s designated p' and q' respectively, then the proportionality schema can be written with the propositional statements representing each of the statements i n (1) as follows: ( P °Q.) = ( P Vq) 0 0 . 0 0 0 0 . . 0 O 0 . O O 0 . . . 0 . . . . . « . . o . . . . . . . .....(2) ( p ' . q ' ) ( p ' v q « ) Applying the INRC group of operations described on page to the propositional statement (p.q); i t i s shown that each propositional statement i n (2) can be obtained as follows: I(p.q) = (p.q) N(p.q) = (p«vq«) R(p.q) = (p'.q1) C(p.q) = (pvq) Thus, by substituting the INRC operations into (2) and omitting the propositional statements the following relationship is obtained: The proportion involves 'second order operations' (or interpropositional operations) where relations are established between relations, an important feature of operational schemata. F. PSYCHOLOGICAL FUNCTIONING AT THE FORMAL OPERATIONS STAGE Five distinctive features of psychological functioning at the formal operations stage have been selected for the purpose of the pre-sent study. Firstly, there is the ability to reason by hypothesis, i.e. to reason from a proposition that is assumed, or seems to be a likely explanation or theory. Secondly, there is the ability to think within a framework of related possible ideas. Thirdly, there is the ability to integrate ideas. The fourth characteristic is that of establishing the relationships between the ideas, and thereby consoli-dating the hypothesis. Lastly, deductive reasoning involves the ability to apply a general principle or hypothesis to particular cases in order to verify the hypothesis. Thinking at the formal operations stage is often referred to as hypothetical-deductive thought. Each of the aspects will be described in further detail below. Hypothetical Thought There are different aspects involved in hypothetical thinking which are discussed in order of the most general to the most specific. An individual is capable of accepting unproven facts as true, or assuming what seems to be a likely explanation, and which he is able to investigate or think about in a systematic logical way. In order to hypothesize, the individual must be able to consider a l l the logical possibilities including those which may not be physically possible (e.g. in the concept of density i t may be hypothesized that weight and volume are independent of each other. In the physical sense, however, volume and weight cannot be separated). Hypothetical thought also involves the ability to deal with objects by using "verbal elements rather than the objects themselves (Inhelder and Piaget, 1958, p. 252). In other words, an individual uses exclusively hypothetical terms when . verbally formulating information which cannot be imagined. "Verbal statements which are simply substituted for objects are used in the construction of propositional logic." (p. 252). Thinking within a Framework of Related Possible Ideas An hypothesis which "seems to be a likely explanation" of a problem must be based on a limited number of possibilities which are logically related to each other. The individual intuitively appreciates the framework which limits the scope of the possibilities. The individual establishes a particular framework of related possible ideas by f i r s t recognizing the most significant variables (operative factors) involved in the hypothesis. For example, in a problem concerning floating objects, the individual may hypothesize that volume and weight are important. The^operative factors are firstly that the greater the weight of the object the less the ability to float, and secondly, the greater the volume of the object, the greater the ability to float. The individual can use these operative factors as foundations for the whole framework of possible ideas. The limitations to the number of possibilities involved are determined by: (l) The logic of the system. Within a particular system, only a certain number of possible propositions can be made. In a binary combinational system (e.g. increase and decrease in volume) there are 16 possible combinations or propositions which form the "related possible ideas". (2) The relevance of the possibilities. Some of-the possibilities may be neglected because they have been found to be irre-levant and unimportant, or are obviously so. The individual must appreciate the implications of particular ideas in order to check their validity against the observable or given facts. He is then able to select those ideas that are valid and summarize them, in a statement. Integrating the Ideas - Control of the Variables The "related possible ideas" must be integrated by the individual in order to appreciate the implications of each. This integration or co-ordination of operations is essentially the process by which the individual controls the variables involved in the hypothesis. In instances where the variables involved are not physically separable, i t is possible to establish the effect of individual variables by using the concept of " a l l other things being equal". By keeping a l l the variables equal except one, the individual nullifies their effect, and is free to investigate the effect of one variable at a time. Establishing the Relationship Between Ideas The integration of the "related possible ideas" enables tho indi-vidual to identify the relationships that exist between them. This gives coherence and structure to the hypothesis, making i t logically sound. Deductive Thought The hypothesis is applied to particular proven cases or facts which are test"-• cases for its verification. The individual must be able to recognize appropriate facts which are relevant to the hypothesis,' i f i t is correct. G. SUBSTAGES A AND B OF THE FORMAL OPERATIONS STAGE Substage A is considered a transitional stage between concrete and formal operations. The intellectual structures are present, but in a latent form, and are therefore not functioning adequately. The intellectual structures at Substage B are, however, well established and functional,, Piaget maintains that the ability to use the 16 Binary Operations and the INRC transformations develops as a whole. Some aspects of propositional logic and some of tho operational schemata may remain latent simply because the individual has not had the oppor-tunity or experience required to make them functional. The differences in psychological functioning between Substages A and B of the formal operations stage are a matter of degree (Inhelder and Piaget, 1958, p. 120). The individual at Substage A is described as hesitant, inconsistent, relatively unsystematic and uncoordinated, while the individual at Substage B is more certain, consistent, systematic, co-ordinated and capable of sporadic elaborations on what he is doing. H. SUMMARY This Chapter covers those aspects of Piaget's theory of intellectual development which are important for this study. The basic theoretical constructs and their implications in the concrete and formal operations stages are discussed in some detail. A good understanding of this part of the theory is required for understanding the inventory and its formulation which is presented in Chapter III. While the inventory i s , at this stage, concerned only with identifying formal operational behaviour, i t is less meaningful to discuss formal operational behaviour without first discussing concrete operational behaviour which theoretically must always precede i t in the developmental sequence. PROCEDURE AND DATA USED IN THE STUDY A. DEVELOPMENT OF AN INVENTORY FOR MAKING PIAGETIAN INTERPRETATIONS OF INTELLECTUAL PERFORMANCE: FORMAL OPERATIONS STAGE. The inventory is intended to act as a model and guide for interpre-ting intellectual behaviour of individuals at the formal operations stage. In' the discussion of Piaget's clinical method of administering the Piaget tasks (Chapter II, p. 1 8 ) , i t was emphasized that such a schedule was needed, especially as the clinical method of administering the Piaget tasks obviates any standardized procedures. The inventory is intended as an aid to the classroom teacher for classifying intellectual performance of individuals into Piagetian-like categories of intellectual behaviour in order to indentify their level of intellectual development. Basis for Development The salient aspects of Piaget's theory concerned particularly with the stage of formal operations were summarized and analyzed in Chapter 2 , pages . It was found that the psychological functioning of individuals could be divided into five main categories representing aspects of Piaget's concept of formal operational thought. (Chapter II, pp 2 9 ) . The categories are taken to constitute a model of intellectual behaviour characteristically used by an individual at this stage of intellectual development. Each category is delineated by descriptors of the psychological functioning of individuals at this stage. The descriptors are based on material'found in Piaget's writings. Each descriptor is illustrated, for the purposes of clarification, by two exemplars, obtained from Piaget's.descriptions of subjects performing Piaget's tasks. The exemplars depict details of psychological functioning characteristic of the descriptors and, at the same time, illustrating the intellectual structures which underlie and are made manifest by the functions performed. The structures exemplified are the 1 6 Binary . Operations (constituting the combinatorial system) and the INRC group of operations. (See Chapter II, pp. 30-33). Description of the Inventory The inventory presented on page kk, is divided into three parts. Part one (l.O) gives a brief description of two main characteristics of the intellectual structure of formal operational thought termed in Piagetian writings as the 1 6 Binary Operations ( l . l ) and the INRC Group  of Operations (1.2). Part two (2.0) contains five categories of psychological functioning of individuals at the formal operations stage (2.1, 2.2, 2.3, 2.k3 and 2.5). Each category Is delineated by descriptors to be used for interpreting intellectual performance on selected Piaget tasks and class-room examination items (2.11, 2.12, 2.21, 2.22, 2.23 etc.). Part three (3.0) contains six descriptors (3.11, 3.12, 3.13, 3.21, 3.22, and 3.23) for use in distinguishing between substage A (3.l), the transition stage, and substage B (3.2), the well established stage of formal operations. These descriptors differ from those in the second part (2.00) in the extent to which the functions can be performed and are considered to be a further refinement of the descriptors pertaining to the formal operations stage. (See Chapter II, p. 39). The inventory is intended for use in interpreting the psychological functioning involved in intellectual performance. To illustrate the use of the inventory, actual performances were recorded on videotape. Trans-crips of the recordings were made and then condensed into synopses which were subsequently analyzed for behaviours that could be adequately des-cribed, by the descriptors. A summary was made of the actual descriptors used and an attempt made to establish whether or not the individual was using formal operational thought, and i f so, the substage A or B, at which he was functioning. An individual was considered to be using formal oper-ational thought when the descriptors could be used to describe actually observed behaviour. Since a l l the descriptors of the inventory, taken as a whole, describe a model of intellectual performance at the formal opera-tions stage, the extent to which the descriptors match actual performance provides information about what aspects of intellectual functioning were used in a particular task. Utilization of the inventory was. extended to a comparison of the intellectual performance of which an individual was po-tentially capable as indicated by performance on the Piaget tasks with per-formance on examination items. Piaget tasks are specifically aimed at eliciting the maximum poten-t i a l of an individual whereas this is not necessarily so for intellectual tasks often given in the classroom, for example. For this reason, the study f i r s t illustrates application of the inventory to actual performance on selected Piaget tasks at the formal operations stage. The user of the in-ventory can then compare the intellectual behaviours elicited by a non-Piagetian task with intellectual behaviours that a person is maximally able to perform at the formal operations stage of intellectual development. The following categories, descriptors, and examplars constitute the inventory developed and described above. INVENTORY FOR MAKING PIAGETIAN INTERPRETATIONS OF INTELLECTUAL PERFORMANCE: FORMAL OPERATIONS STAGE Part 1.0: Brief Description of the Logical Structures of the Formal  Operations Stage. 1.1 The l6 Binary Operations A l l the possible products of two propositions are compiled and the relationship between the elements form a commutative . • group. (Piaget, 1953, p. 37). 1.2 The INRC Group of Operations " The laws pertaining to the combined structure of the lattice and the group, enabling an individual to transform one proposition into another within the combinatorial system. (See Inhelder and Piaget, 1958, P.13M. Part 2.0: Categories of Psychological Functioning at the Formal Operations  St age. CATEGORY 2.1: HYPOTHETICAL THOUGHT 2.11 ABILITY TO ACCEPT UNPROVEN FACTS AS HYPOTIIETICALLY TRUE, IN ORDER TO DEDUCE THE REAL FROM THE POSSIBLE. (Piaget 1953, p.18, 19. Inhelder & Piaget, 1958, p. 25l). 2.111 I: "How do you know you have to bring the weight toward the centre?" ' S: "The idea just came to me, I wanted to try." (Inhelder and Piaget, 1958, p. 173). . The S has a possible idea or theory in mind which is within the limited framework of the possible interactions of the given variables of increase and decrease of weight on both sides of the balance (designated by p, p, p*, p'), and increase and decrease of distance from the fulcrum on both sides (q_, q, q' and q'). The interaction of these variables constitutes a combinatorial system. 2.112 S: "If I bring i t (weight) in halfway, the value of the weight is cut in half. I know but I can't explain i t . " (p.173). The S intuitively appreciates that his hypothesis involves a transformation of a reciprocal nature. 2.12 ABILITY TO CONSIDER THE LOGICAL POSSIBILITIES INDEPENDENT OF THE CONTENT (Inhelder & Piaget, 1958, 252, 293) 2.121 S explains the balance task by saying: "The distance and the weights-; i t 's a system of compensations." (p. 174). S is con-sidering the combinations involved in a general sense which also holds true for the empirical facts. He is not restricted to actually manipulating the apparatus to obtain combinations. 2.122 The statement: "it 's a system of compensations" (p. .174) in the balance task involves the S in carrying out the INRC transformations on the possible combinations from which he devises a compensatory relationship, I _ C (p.177). or I (p.q) C(pvq) (p. 178) R (P.q)~N(pvq) (where p and p designate increase and decrease of weight on one side, and'q and q designate an increase and decrease of distance on the same side). CATEGORY 2.2: THINKING WITHIN A FRAMEWORK OF RELATED POSSIBLE IDEAS . 2.21 ABILITY TO INTUITIVELY INTEGRATE THOUGHTS WITHIN A SYSTEM OF RELATED POSSIBLE STATEMENTS (Piaget, 1953, p. 39). 2.211 "A.t the same time that the subject combines the (four differ-ent colourless liquids) given in the experimental context, he also combines the propositional statements which express the results of these combinations of facts, and in this way men-tally organizes the system of binary operations consisting in conjunctions, disjunctions, exclusions etc." (Inhelder & Pia-get, 1958, p. 122). For example, i f p and p designate the presence and absence of the colour reaction, and q and q des-ignate the presence and absence of liquid 4,.then a statement may be made that liquid 4 is incompatible with the presence of the colour, i .e. (p/q). Also, i f q and q are changed to designate the presence or absence of liquid 2, a statement can be made that liquidAhas no effect on the colour reaction, and is neutral, i .e. (p*q), which is a tautological statement (p. 118-119). 2.212 The statement that p is incompatible with q, i .e. (p/q) (where liquid 4 is_designated by q), has particular implications, i .e. (p/q) = (p.q) v (p.q) v (p.q). In order to reach "the statement p/q, the S must be able to integrate his thoughts which involves using the INRC transformations, e.g. I (p.q) = R(p.q). 2.22 ABILITY TO FORMULATE THE OPERATIVE FACTORS INVOLVED MD ARRANGE EXPERIMENT OR THOUGHT SEQUENCE ACCORDINGLY (Piaget, 1958, p. 19, Inhelder & Piaget, 1958, p. 250). 2.221 S: "The greater the distance, the smaller the weight should be." (p. 17k). In order to say this, S must f i r s t have con-sidered a l l the possible interactions involved in the com-binatorial system of variables. He then can arrange his thought sequence in order to experiment, e.g. S: "If I re-placed this weight (1 unit) with that one (2 units), i t would only go halfway up." (p. 175). 2.222 The statement S: "The greater the distance, the smaller the weight should be," (p. 175) is a proposition which involves the transformation of reciprocity e.g. I(p.q) = R(p.q)» i.e. an increase in weight (p) together with a decrease in dist-ance (q) has the same effect as a decrease in weight (p) with an increase in distance (q) on the same side of the balance. 2.23 ABILITY TO INFER THE IMPLICATIONS OF THE STATEMENTS (WITHIN THE FRAME-WORK OF IDEAS), AND SELECT THE TRUE STATEMENTS AND DISCARD THE FALSE, AND SYNTHESIZE A STATEMENT OF NECESSARY MD POSSIBLE CONDITIONS. (Piaget, 1953, pp. 19', 39). 2.231 In the liquids task, the implications of the statement (p/q) are (p.q) v (p.q) v (p.q). (See 2.211 and 2.212 for the meaning of the symbols). S realizes that either colour appears in the absence of liquid 4, or colour disappears in the presence of liquid 4, or colour is absent in the absence of liquid 4, or a l l possibilities. He then selects these for their validity with respect to the observable facts and then clearly states a syn-thesis or summary of his reasoning and observations, e.g. S: "Liquid 4 cancels i t a l l (i.e. the colour)." (Inhelder & Piaget, 1958, p. 117). 2.232 In selecting the true statements and discarding the false, the S uses the BIRC transformations, e.g. the proposition; (p/q) = (p.q) v (p.q) v (p.q) (See. 2.211 and 2.212) is checked by carrying out the identity transformation on each statement and comparing i t with the observable facts. Each statement would be selected as true in this case, thus confirming the proposi-tion; (p/q). CATEGORY 2.3: INTEGRATING THE IDEAS - CONTROL OF THE VARIABLES 2.31 ABILITY TO SEPARATE THE VARIABLES BY NEUTRALIZING OR ELIMINATING FAC-TORS WHICH CANNOT BE PHYSICALLY SEPARATED (Inhelder & Piaget, 1958, p. 284). 2.311 S: "It's the length of the string that makes i t go faster or slower, the weight doesn't play any role." (p. 75). In the pendulum task, the variables are; modification or lack of m o d i f i c a t i o n of length (p and p ) ; a mo d i f i c a t i o n or lack of mod i f i c a t i o n of weight (q and q ) , and a mo d i f i c a t i o n or lack of m o d i f i c a t i o n of the frequency of o s c i l l a t i o n i n the pendulum (x and x ) . S: "v a r i e s the length of the s t r i n g with equal weights" (p.76), thereby n e u t r a l i z i n g the e f f e c t of weight while observing the e f f e c t of length. Student also uses d i f f e r e n t weights on equal s t r i n g lengths, thereby n e u t r a l i z i n g the e f f e c t of length. In both cases, weight cannot be p h y s i c a l l y separated from l e n g t h . The t r u t h statement made by student i s composed of four combinations of the v a r i a b l e s . (p.q.x)v(p.q.x)v(p.q.x)v(p.q.x) This means that with, or without, a m o d i f i c a t i o n i n weight, a mod i f i c a t i o n i n length r e s u l t s i n a m o d i f i c a t i o n i n frequency of o s c i l l a t i o n and v i c e versa. This i s b e t t e r w r i t t e n ; p[q]x. In a d d i t i o n , student also holds the other two v a r i a b l e s constant, i . e . m o d i f i c a t i o n i n amplitude (r) and mod i f i c a t i o n i n i n i t i a l force a p p l i e d ( s ) . Thus the whole t r u t h statement i s ; p[q.r.s.]x. There are sixteen true combinations: (a combinatorial system) implied i n t h i s statement (see p. 77). 2.312 In order t o n e u t r a l i z e or eliminate f a c t o r s the student must manipulate the binary operations, i . e . perform the INRC transformations. The d e c i s i o n t o h o l d a l l the v a r i a b l e s constant and then t o vary one at a time so as to e s t a b l i s h i t s r o l e involves the transformation of negation, i . e . student may hypothesize weight influences the frequency of the pendulum. i . e . (using the designation given i n 2.7l), (p^>x) = (p.x)v(p.x)v(p.x) However i n the experimentation student discovers that weight has no e f f e c t on frequency of the pendulum, i . e . (p*'x) = (p.x) v(p.x)v(p.x) v(p.x"). Student r e a l i z e s that negation i s inv o l v e d , i . e . (p=>x) = N (p.x) and therefore discounts h i s hypothesis (p^x) i n favour of the experimental evidence th a t (p'?x) , saying; "Nothing has changed", (p. 76). 2.32 ABILITY TO CONTROL THE VARIABLES BY STUDYING THE ROLE OF ONE FACTOR BY VARYING ANOTHER (Inhelder & Piaget, 1958, p.285). 2.321 Pendulum task: S: "When i t ' s smaller ( i . e . length i s s h o r t e r ) , the weight goes f a s t e r . I t ' s because I didn't put on the same weight (that I didn't prove anything)-. Now I ' l l put on the same weight", (p. 7*0. Student must therefore a c t i v e l y c o n t r o l the weight v a r i a b l e by ensuring there are equal weights on both pendula. As i n 2.311, the 16 Binary Operations must be considered. 2.322 Student experiments with two v a r i a b l e s (weight and length) at once, and may hypothesize (p.q)=»x, i . e . weight and length a f f e c t the frequency of the pendulum. The possible interaction between weight and length produces a cumbersome number of possibilities to be tested. S may therefore change the hypothesis (p.aj^x into ( p v q)oxs i .e. weight or length or both affect, the fre-quency of.pendulum. This involves a correlation transformation i .e. (p.q) = C (pvq) which will alter the subsequent reasoning pattern. CATEGORY 2 . 4 ; ESTABLISHING THE RELATIONSHIP BETWEEN THE IDEAS 2.41 ABILITY TO INTERPOLATE MEANING BETWEEN•THE SUCCESSIVE STATEMENTS, ESTABLISHING RELATIONS BETWEEN RELATIONS (Inhelder & Piaget, 1 9 5 8 , p. 2 5 4 , 2 7 9 ) . 2.411 S initially establishes the relationship between the weights and distances on one side at a time. In order to do this, he works within a combinatorial system, e.g. using p, p, q and "qt q' and q B , (see 2 . 1 1 1 ) , he establishes (p.q) and (p.q) will give a ba-lance with (p 8.q !) and (p'.q'). 2.412 In order to explain the balance system, S has to work out the relationship between these propositions for which he uses INRC group of operations to establish the existence of a compensa-tory relationship, which in this case is; I C. (See Inhelder R = N &. Piaget, 1 9 5 8 , p. 176-181 for further details). CATEGORY 2 . 5 : DEDUCTIVE THOUGHT 2 . 5 1 ABILITY TO PREDICT THE REAL SITUATION IF THE HYPOTHETICAL CONDITION WAS FULFILLED, AND BY OBSERVING THE CONSEQUENCES, VERIFY THE HYPOTHESIS (Inhelder & Piaget, 1 9 5 8 , p. 2 5 1 , 2 7 9 ) . 2 . 5 1 1 Angles of incidence and reflection task; S hypothesizes; "The more the target.approaches the plunger, the more the plunger must (of necessity) also approach the target." (p. 1 1 ) . S then predicts a real situation by saying: "For example, i f there were a (perpendicular) line here, the ball would come back exactly the same x-ray." (p. 1 1 - 1 2 ) . S then verifies the law by putting.the plunger at 4 5 ° , followed by "several angles chosen at random" and demonstrates the law of equality of angles of incidence and reflection. In order to establish the statement that the angle of incidence (p) implies the angle of reflection (q), i .e . (pjq), S has to consider the four possibilities, i .e. (p.q)v(p.q)v(p.q)v(p„q). This involves the combinatorial sys-tem, and S must establish that (p.q) never occurs in order to say; (]»q) = (P«q)v(p.q)v(p.q) Thus S verifies his hypothesis that ( p 3 q ) by demonstrating (p.q) never happens. 2.512 Tho negation transformation of the hypothesized statement (p^q) (as in 2.511) , is (p.q' » i.e. (psq) = N(p.q). In order to make sense of the possible combinations, S thus uses the INRC group of operations, and, in addition, compiles the log-ical consequences of a statement such as (poq). The v e r i f i -cation of such a statement can then be demonstrated empiri-cally by checking the validity of each consequence, which involves using the identity transformation. 3.0 PSYCHOLOGICAL FUNCTIONING AT SUBSTAGES A AND B OF THE FORMAL OPERATIONS STAGE 3.1 SUBSTAGE A 3.11 Approach is hesitant, uncoordinated, unsystematic and uncertain, resulting in non-rigorous proofs and a tendency to jump to con-clusions .TTrihelder and Pia.oet,. 1958, op. 62, 116, 120, 292, 294 310, 311) 3.12 Ability to use concept of " a l l other things being equal" in a rudimentary manner, (p. *+3) " 3.13 Tendency to make proofs and generalizations which are restricted to empirical facts. {p. 11, 58)• ~ 3.2 SUBSTAGE B 3.21 Approach is systematic, integrated, sure and organized, result-ing in exhaustive and rigorous proofs, without jumping to con-elusions, (pp. 53, 1Z1, zyz, 294, 311 ). ; 3.22 Ability to use concept of " a l l other things being equal" in a general sen ieT~(p . 43-44, 277") ' 3*23 Tendency to make l og i ca l proofs and generalizations based on con-cept of "Xogical necessruyT" (p. 11) B. ' STUDENT PERFORMANCE ON THE PIAGET TASKS Selection of Piaget Tasks The tasks used by Piaget and his co-workers are described in Inhelder and Piaget's book, The Growth of Logical Thinking from Childhood to Adoles-cence (1958). The tasks f a l l into two categories; those which require propositional logic for their solution; and those which involve formal  operational schemes. Both propositional logic and formal operational schemes are involved in formal operational thought, and both evolve from the 'structured whole' or unified system of 16 Binary Operations. Four tasks were selected for purposes of illustrating a possible use of the inventory; two from each category identified above. The Pendulum Oscillation task and the Angles of Incidence and Reflection task require propositional logic for their solution, while the Combination of Liquids task and Balance task involve particular formal operational schemes. The restriction to four tasks was due to the limited time available for the interviews. Only three of the eight formal operational schemes described by Piaget are involved in the Combination of Liquids and Balance tasks. Piaget, however, maintains that once an individual has reached the stage of formal operations he is potentially capable of using a l l the formal opera-tional schemes. (Piaget, 1953, Inhelder & Piaget, 1958, p. 303). An individual's lack of experience in specific fields may result in particular formal operational schemes remaining latent. The interviewer, using the clinical method, may be able to e l i c i t the functioning of latent formal operational schemes. The four tasks selected for the purposes of this study are described below in terms of the apparatus involved. The nature of the task, the concept involved, and its relevance to Piaget's theory. Angles of Incidence and Reflection Task angle of rotation ebound wall launcher target area .target positions Diagram 1. Pin-ball Apparatus Apparatus. A square board with a hard plastic-lined rebound wall is used as a type of pin-ball apparatus. Balls are launched with a device con-sisting of a tube and a spring plunger. The launcher can be pivoted in a restricted area around a fixed point, so that the target cannot be hit _ directly. The ball is fired against the rebound wall which causes i t to reflect toward the target area of the board.A target (a second ball) is placed at position 1, and subsequently moved through positions 2, 3 and 4 by the interviewer (I). Nature of the Task. The subject (S) is instructed to aim ball in the launcher at the target in position 1. After hitting the target in pos-ition 1, the investigator moves i t to positions 2, 3 and 4 each time asking S to predict the direction which which the launcher should be aimed, and to provide reasons for his predictions. S_ is asked to hit the target in each position. Concept and Relevance. The apparatus involves the reflection of a ball from a hard surface. Under ideal conditions, the angle of incidence will equal the angle of reflection. If the target is moved to position 2, the launcher must be aimed at a new point on the rebound wall, a l i t t l e to the right of the original point. In this case the angles of incidence and reflection remain equal, but are both increased in size. The same thing happens when the target is moved to position 3« Position 4 lies on the line of reflection of the ball aimed at the target at position 3 , there-fore no change in direction of the launcher is made (See diagram 2 ) . Launcher Diagram 2„ Positions of target on pin-ball apparatus A person at the formal operations stage is capable of the proposi-tional logic required to perform the task successfully, i.e. he can make use of the propositional logic involved in reciprocal implications (Chapter II,p. 21). The_S is able to reason from the generalization of the law of incidence and reflection to a particular case and is able to verbalize the law in so far as he recognized this necessity. The S at the concrete operations stage can also perform the task, but resorts to a t r i a l and error technique without indicating the formal solution to the problem. Pendulum Oscillation Task Diagram 3* Pendulum apparatus. Apparatus. Two pendula consist of two adjustable strings, suspended from' a wooden frame with terminal hooks on which different bobs may be hung. The bobs consist of three pairs of wooden blocks-, a heavy pair ( 0 ) , a medium weight pair (X) and a light pair(C). The blocks differ only in weight, not in size or shape. Mature of the Task. The student is asked to establish which of the four variables (weight of the bob, length of the string, amplitude of swing, effect of i n i t i a l force on bob) affect the frequency of oscillation of the pendulum.* Concept and Relevance. The frequency of oscillation of a pendulum at least to a first approximation, is dependent only on the length of the string. In solving the problem the student has to separate and control the variables in order to establish which of them affects the frequency of oscillation. This involves using the concept of " a l l other things being equal." The variables must then be selected or excluded according to their established effects. This process particularly demonstrates the operation of exclusion, an operation in propositional logic (Chapter II, p. 3 M . A student who is not at the formal operations stage is unable to separate and control the variables by using the concept of " a l l other things being equal." He may vary two- factors at once, e.g. weight of bob and length of string, and falsely conclude from his results that only one of the factors affect the frequency of oscillation of the pendulum. Combination'of Liquids Task Apparatus. Four similar bottles, each containing colourless and odourless * On occasions the student was asked for the period of oscillation of the pendulum, but this does not alter the task. liquids, are labelled 1, 2, 3 and 4 . Bottle 1 contains dilute sulphuric acid; bottle 2 contains water; bottle 3 contains dilute hydrogen peroxide; bottle 4 contains sodium thiosulphate solution. A f i f t h bottle labelled 'g8 is describe^, as an indicator, and contains a solution of potassium iodide. A supply of small beakers is provided in which the S can combine the liquids. The hydrogen peroxide (3) reacts with the potassium iodide (g) in an acid medium (1) to give a yellow solution (i.e. 1 + 3 + g). The water (2) is neutral, and has no effect on the reaction, but the sod-ium thiosulphate solution (4) acts as a bleach, and removes any yellow colour that may be formed. Nature of the Task. The S is shown a beaker containing a 'mystery* colour-less liquid which is a combination of (1 + 3) previously prepared by I. S is told that the mystery liquid is obtained somehow, from the given l i -quids. A few drops of g are added to this liquid, and a deep yellow colour results. S is asked to reproduce the mystery liquid using bottles 1, 2, 3 and 4 as he wishes, and test for i t using the indicator. When S accom-plishes this, he is asked to distinguish between the effects of liquids 2 and 4, and i f possible to guess what they might be. Concept and Relevance. The solution to this task involves combining the four given liquids in different ways. A combinatorial system is involved, giving rise to 16 possible combinations (Chaptern» P- 3^). The_S intui-tively uses relationships of implication, disjunction and exclusion in solving the problem. It is important that I establish that_S is capable of making the systematic combinations rather than using a haphazard, t r i a l and error technique, not characteristic of formal operational thought. I must also ascertain that S is capable of using the formal operation schemes of combinations to establish the role of liquids 2 and 4, which involves using the INRC group of operations (Chapter 2, p. 19). Balance Task Diagram 4. Balance apparatus Apparatus. A uniform rod supported by a central fulcrum has nine hooks, uniformly spaced, on each side on which varying weights (standard type) can be hung. Nature of the Task. S i s asked to set up a variety of balanced systems, using the weights supplied. I f S f a i l s to do more than set up equal weights at equal distances on both sides of the balance, I then semi-structures the situation by setting up one side and asking j> to balance the other side i n a variety of ways. S i s asked to explain the relationship between vreight and distance, for a balance i n equilibrium. Concept and Relevance. I observes S and establishes whether _S i s aware of and capable of using and understanding the relationship W/W• = l'/l» where W and VJ? are the weights and L and L' are the distances of the weights on the fulcrum. The formal operational schemes of proportionality and mechan-i c a l equilibrium are involved i n this task (ChapterII, p. 35-36). A person who i s not at the formal operations stage would not only, be unable to verbalize the law, but would tend to resort to t r i a l and error experimen-tation. He may successfully use the formula W/W = l ' / l without understand-i * i> A b 6 <b <i> & 6 6 6 5 il O i 6 box of weights ing i t . It is essential that I establish the level of -understanding of S. Selection of Students Fifteen student volunteers from the f i r s t year physics course (Phy-sics 110) at the University of British Columbia were invited to take part in the study. They were each asked to do the four Piaget tasks and their performance was recorded on videotape. The performances of three students were then selected from the original fifteen for detailed analysis and interpretation using the inventory. The selection of the three students was based on the success of the interview technique. The elimination of students was due either to incomplete interviews or to unsatisfactory use of the clinical method of interviewing on the part of the interviewer. Administration of Piaget Tasks Each interview lasted approximately two hours, the f i r s t fifteen min-utes of which were spent in informal talk, intended primarily for estab-lishing a friendly, relaxed atmosphere. The interviewer asked questions concerning the student's background in physics, his appreciation of physics, and his plans for his future career. The information was recorded in an informal questionnaire but was not used subsequently, and is therefore not included in the thesis. The clinical method of interviewing was used. The interviewer intro-duced the apparatus, posed the problem to be solved and then assumed a sort of non-directive role in which the student was allowed to follow his own line of reasoning. The interviewer asked for clarifications, explana-tions, justifications and indications of procedure on the part of the student. The Angles of Incidence and Reflection task involved more direction from the interviewer as more specific questions were asked concerning the differ-' ent positions of the target. The Piaget tasks were designed to show the maximum intellectual po-tential of the individuals. In some cases the interviewer found i t necessary to prompt the student to think further, while a t the same time taking cnre to resist the temptation to "teach" rather than "probe." Method of Recording Piaget Task Performance A l l the interviews were recorded on videotape, using the apparatus and facilities of a TV studio. Transcripts were then made from videotapes in which both the comments and actions were noted. The transcripts were tabulated in order to facilitate the analysis and interpretations using the Inventory. Method of Analysing Piaget Task Performance Transcripts of student performance on the Piaget tasks x*ere f i r s t con-densed into synopses which summarized the intellectual performance dis-played. One f u l l transcript is given in Appendix A, which was cross-keyed to the appropriate synopses for checking purposes. The synopses were ana-lyzed for statements and actions which could be adequately interpreted with one or more of the descriptors in the inventory. The appropriate descrip-t o r s) used to interpret the statements and actions were recorded, and the inventory code number provided. A summary was made of the descriptors used to interpret the student's performance in each task, separately, and a l l the tasks taken together. A final summary showed the overall Piaget task performance of a l l three students. C. STUDENT PERFORMANCE ON SELECTED EXAMINATION ITEMS Selection of Examination Items The f i n a l examination i n the f i r s t year Physics 110 course at the University of B r i t i s h Columbia was used as an example to show the a p p l i -c a b i l i t y of the inventory to non-Piagetian task situations. In order to assess whether the examination could potentially e l i c i t i n t e l l e c t u a l per-formance at the formal operations stage the inventory was used to select examination items requiring formal operational thought for their solution. Before the inventory could be used, i t was necessary to eliminate a l l items which could not be c l a s s i f i e d into any of the categories of the inventory. The class instructor and a class tutor were asked to give their own versions of the correct solutions to each item. They were also asked to indicate items involving only r e c a l l of information presented i n the l e c -tures, t u t o r i a l s , or textbook. The solutions to the items involving more than simple r e c a l l were then analyzed for reasoning sequences which could be interpreted using one or more of the descriptors. The descriptors used to interpret the expected i n t e l l e c t u a l performance, were recorded and the code number given. A summary was made of the descriptors used to interpret the expected i n t e l l -ectual performance for each selected item. The complete physics examination and a table showing reasons for selection of individual items i s given i n Appendix B. Method of Analysing Student Performance on Selected Examination Items Student written responses to the selected examination items were re-corded, as wall as the mark credited them by the examiners. The responses were then analysed for statements and reasoning sequences which could be interpreted using one or more of the descriptors. The descriptors used to interpret the in t e l l e c t u a l performance displayed were noted and the code number given. A summary was made of the descriptors used for each student on each item and on a l l the'items taken together. D. COMPARISON OF PIAGET TASK PERFORMANCE WITH EXAMINATION ITEM PERFORMANCE MD ASSESSMENT OF RESULTS Method of Comparison The analyses and interpretations of the students' overall performances on the Piaget tasks and on the selected examination items were compared • for congruency. The appropriate descriptors for each student were summar-ized i n terms of their overall performance on Piaget Tasks and overall performance on the selected examination items. The comparison was consid-ered congruent when adjacent rows either both contained descriptors, or were both empty. Conversely, a description was considered non-congruent when two adjacent rows differed markedly i n the number of descriptors l i s t e d . Method of Assessment The inventory for making Piagetian interpretations of i n t e l l e c t u a l performance at the formal operations stage was developed for analysing individuals' i n t e l l e c t u a l performance on the Piaget tasks as well as on selected classroom examination items. F i r s t l y the extent of agreement (congruency) between the model of formal operational thought as represented by the descriptors of the inventory and actual performance of individuals on the Piaget tasks was examined- for implications regarding the i n t e l l e c t u a l functioning potentially available to the individual. Secondly the extent of agreement (congruency) between the model ( i . e . the descriptors of the inventory) and a c t u a l performance of i n d i v i d u a l s on the s e l e c t e d examination items, was examined f o r i m p l i c a t i o n s regarding the i n t e l l e c t u a l f u n c t i o n i n g demonstrated by i n d i v i d u a l s i n a classroom examination s i t u a t i o n . T h i r d l y , the extent of agreement between the Piaget task performance and s e l e c t e d examination item response f o r each i n d i v i d u a l was examined i n order to compare the i n t e l l e c t u a l f u n c t i o n i n g displayed i n responding t o the examination items with the i n t e l l e c t u a l f u n c t i o n i n g of which the i n d i v i d u a l was p o t e n t i a l l y capable at that time. These three comparisons provide information which allows f o r speculation concerning the i n t e l l e c t u a l demands of the examination items, and the c a l i b r e of the students' i n t e l l e c t u a l responses t o the examination items. The data thus obtained were used to speculate on the a p p l i c a b i l i t y of Piaget's theory of i n t e l l e c t u a l development to science education e v a l u a t i o n , by means of the inventory formulated f o r t h i s purpose. APPLICATION OF THE INVENTORY FOR ANALYSING STUDENT PERFORMANCE ON PIAGET TASKS AND SELECTED EXAMINATION ITEMS Introduction The chapter is divided into two parts. Part A contains synopses of the performances of three students on each of the four Piaget tasks and an analysis of each student performance in terms of formal operation thought using the inventory given in Chapter III. Part B contains f i r s t l y , the expected responses by instructors of the course for two selected course examination items and an analysis of the expected responses in terms of for-mal operational thought. Secondly, this part contains an analysis of the responses of the three students to the two selected examination items in terms of formal operational thought apparently used in responding to the examination items. Summaries of the analyses are given for each student on the Piaget tasks and selected examination items. Overall summaries of the student performance on a l l the Piaget tasks and on both selected examination item's are given at the end of Part A and Part B respectively. A. THE PIAGET TASKS Method of Analysis and Reporting of Results The information synopsized in this part of the chapter was obtained from written transcripts made from videorecordings of actual task perfor-mance. A f u l l transcript of the performance of Student L. W. on a l l the Piaget tasks is given in Appendix A, and is coded for cross-reference to the corresponding synopses given in this section. The reader may use these transcripts for the purpose of checking the accuracy of the synop-ses. Transcripts for the remaining two students have not been included in the Appendix for reasons of brevity. The synopses record a l l the statements and actions of the students considered significant for the task at hand. Repetitions, and asides were omitted, and lengthy explanations were summarized. The synopses include descriptions of student performance that reflect as accurately as possi-ble the mode of thought used by the students when doing the tasks. The analyses were made by carefully reading the synopses and extract-ing and collating from them those sections which could be described by one or more of the descriptors of the inventory. The analysed sections were then placed in a table with the appropriate descriptor listed along-side. For purposes of clarity, some of the descriptors were elaborated to indicate in which way they were found to be appropriate to the analysed statements or actions. The elaborations are indicated by means of paren-theses. The analyses given in the table are cross-keyed for reference to the appropriate line of the synopses. Tables 9-12, 14-17, and 19-22 con-tain the results of the analyses of student performance on the Piaget tasks. Summaries of the analyses of student performance on the four Piaget tasks are presented in table form. The tables contain the descriptors used to describe performance on each task. An overall summary of the des-criptors used to describe student performance on a l l the tasks is also pre-sented. Tables 13, 18 and 23 give the summaries for individual tasks and Table 24 presents the overall summary of student performance across tasks. The summaries are compiled in this way in order to facilitate a com-parison between the students intellectual performance on the Piaget tasks and performance on the selected examination items (Table 30). In some instances of student performance'a descriptor could only partially be applied. In order to indicate a doubt about the applicability of the entire descriptor to such a case, the descriptor code number used to identify the descriptor applied is qualified by means of parentheses. In Table 15 code number (2.1l) is a case in point. When the overall summaries were made the qualified descriptors were not noted when the same descriptors was found adequate in describing other aspects of student performance in the same task. For convenience certain abbreviations and symbols have been adopted in recording the data. The symbols Jl and /_R refer to the angles of incidence and reflection with respect to the normal to the rebound wall, whereas, j \ and Jr_ refer to the angles of incidence and reflection with respect to the reflecting surface or rebound wall.' y r Different liquids in the combination of liquids tasks were referred to by 1, 2, 3, and U, and g (the indicator) while combinations are indicated by expressions such as ( l + 3 + g). ' In the balance task the arrangements of the weights on the hooks on both the left and right side of the fulcrum are written as equations. For example, lOOgm x 8L = 50gm x 4R means that a lOOgram weight was hung on the eighth hook from the fulcrum on the left side of the fulcrum to balance a 50gram weight hung on the fourth hook from the fulcrum on the right side. The dashes (—) indicate hesitations in speech by the student, whereas dots (...) indicate that words have been omitted. TABLE 9 SYNOPSIS MD ANALYSIS OF PERFORMANCE Oil THE ANGLES OF INCIDENCE AND REFLECTION TASK: STUDENT J.V. Synopsis The student (s) explains f i r s t l y that, "you have to keep the angles the same" and then demonstrates with a p e n c i l held perpendicular to the rebound w a l l that a p a r t i c u l a r point on the rebound w a l l must be found so that the b a l l rebounds. Student has some d i f f i c u l t y i n v e r b a l i s i n g e x a c t l y what she. means and r e c a l l s i n Grade 11 she was t o l d to think i n terms of the angles t o the perpendicular. Student attempts to explain again the necessity of the two angles (/_I and /?<.) being the same. 5 Student aims at target 1 and c o r r e c t l y adjusts the launcher t o give a near h i t . Student c o r r e c t l y adjusts launcher t o the r i g h t f o r h i t t i n g target 2, and explains e a s i l y t h a t the t o t a l angle (/I + /_R) i s now greater, the point of rebound must be s h i f t e d t o the r i g h t so that the angles (of incidence and r e f l e c t i o n ) are equal. A f t e r f u r t h e r questioning by the i n v e s t i g a t o r ( i ) , student maintians that the angles of incidence and r e f l e c t i o n are always equal when a h i t i s made. Student i s a l i t t l e confused about the changing s i z e of the t o t a l angle with d i f f e r e n t p o s i t i o n s , but i s capable o f working i t 10 out. Otherwise student c l e a r l y understands the p r o p o r t i o n a l i t y , r e c i p r o c i t y and e q u a l i t y operations involved i n the task. Analysis Aspects of Student Performance Descriptors Used Desc r i p t o r s Code Numbers 1. Student immediately resorts • t o a l . a . Tendency to make l o g i c a l proofs 3.23 g e n e r a l i z a t i o n of the equality of and generalizations based on angles of incidence and r e f l e c t i o n , concept of " l o g i c a l n ecessity", and the r o l e of the perpendicular, b. A b i l i t y t o consider the l o g i c a l 2.12 due p a r t l y to r e c a l l (5). Student p o s s i b i l i t i e s ( o f the e q u a l i t y c o n f i d e n t l y states l a t e r on t h a t ' of the angles) independent of the angles are always equal for the content (before handling the any p o s i t i o n of the launcher apparatus). ( 9 , 10). Aspects of Student Performance 2. Student has some d i f f i c u l t y in ver bally explaining the concept, but is competent in adjusting the launcher direction for different target positions. Student is capable after some thought (6) of explaining the change of size of the angles and change in point of rebound, in order to keep the angles of incidence and reflection equal ( l l ) . 3. Student clearly understands the proportionality, reciprocity and equality operations involved in the task (12).. Table 9 (continued) Descriptors Used Descriptors . . . Code Numbers 2. a. Ability to int u i t i v e l y and 2.21 exp l i c i t l y integrate thought within a system of related possible statements (concerning angle equality, angle size, and . point of rebound.) b. Ability to formulate operative 2.22 factors involved (in concept of the equality of.the angles in order to think through the implications for the angle size and point of rebound). c. Ab i l i t y to infer the implica- 2.23 tions of the statements (of the equality of the angles), and select the true and discard the false statements in order to synthesise a statement of the necessary and possible conditions (i.e. to establish the change in angle size and change in rebound point cor-responding to the change in launches direction). 3. Ab i l i t y to interpolate meaning 2 .U l between the statements (con-cerning equality of angles, change in size, change in re-bound point), establishing relations between relations, (i.e. proportionality, reciprocity and equality.) SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE OSCILLATION OF A PENDULUM TASK: STUDENT J.V. • • •Synopsis S tests for the effect of length on the frequency of o s c i l l a t i o n , and says, "Put the same mass on both and displace them the same amount, I hope." She speculates that "the short one would ... be faster." After experimenting, S concludes that the frequency of o s c i l l a t i o n "varies one over the length somehow." S then tests the effect of different masses, and makes the lengths of the strings equal.. She concludes that mass has no effect on the frequency of o s c i l l a t i o n . V'hen •testing for 5 the effect of impetus, S ensures that amplitude is constant as well as length. At f i r s t 3 cannot decide on the effect of impetus, but after establishing that different amplitudes did not affect the frequency, she repeated the impetus test. S refers to the fact that she has two different masses, but adds that "that doesn't affect i t " . S concluded that "frequency varies inversely to the length." " I t doesn't vary with mass or amplitude" ... and impetus "doesn't seem to make any difference." S 10 thus successfully completed the task and excluded each of the noneffective variables. Final l y I gives S a brass and a wood cylinder and asks . i f the frequency of o s c i l l a t i o n would be affected, ~3 replies, "No, i f you keep a l l the other variables constant," and i s suf f i c i e n t l y convinced that no further testing i s required. Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. _S_ immediately recognizes the problem involved and f i r s t tests the effect of length on the frequency of o s c i l -lation, and e x p l i c i t l y ensures that the mass and amplitude are held constant. ( 1-2). I . a. b. A b i l i t y to use^concept of " a l l other 3.21 things being equal" i n a general sense. A b i l i t y to separate the variables (not 2.31 being tested) by neutralizing their ef-fect (through making them equal on both pendula). A b i l i t y to control the variables by 2.32 studying the role of one factor, (e.g. mass) by varying the others, (such as length, amplitude and impetus). Aspects of Student Perforimnce Descriptor Descriptors Used Code Numbers 2. J3 concludes that the frequency of oscil-lation varies inversely to the length of the pendulum ( 3 )« S_ establishes that neither mass, impetus, nor amplitude have any effect on the fre-quency of oscillation. 3 . .S is sufficiently convinced that mass does not affect the period of oscillation that she indicates that using different masses while testing for the effect of impetus will not make any difference (9 ). Also S does not feel i t necessary to retest the effects of the brass and wood cylinders ( l4) d. Ability to intuitively integrate 2.21 thoughts within a system of related possible statements (concerning the interaction of the four variables). e. Ability to formulate operative fac- 2.22 tors involved (in that the effect of each factor has to be tested inde-pendently to make sense out of exper-imentation) , and arrange experiment accordingly. f. Approach is systematic, integrated, 3.22 sure, resulting in exhaustive and rigorous proofs, without jumping to conclusions. 2. Tendency to make logical generalizations 3.23 based on concept of "logical necessity." 3 . Ability to consider the logical possibil- 2.12 ities (implications) of the non-effect of mass on frequency of oscillation) inde-pendent of the content (ie. the.fact that the masses were actually different). SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE COMBINATION OF LIQUIDS TASK: STUDENT J.V. Synopsis S begins by adding the indicator to liquids 1, 2, 3 and 4 separately. After hesitating because she thought the mystery liquid was said to come from one bottle, she says, "So I •will have to try all possible combinationso" She says she will combine (3 4- 4), (1 + 2), (1 + 3), (2 + 4), (1 + 4), (2 + 3), then (1 + 2 + 3), (4 + 2 + 3), and (1 + 2 + 3 + ^). I advises S to write down the possible.combinations, which she does in an orderly, systematic fashion. S then adds liquid combinations, each time referring to her l ist 5 of possibilities. She takes (1 + g), then (3 + g) and (4 + g), then (2 + 3 + g). S_ thinks, then adds (1 + 3 + g) and gets positive reaction. S_ asks, "Shall I keep trying for more?11 and is told to find out "to her satisfaction" the answer to the problem. S tries a combination of (2 + 4 + g), then (1 + 4 + g), getting no reaction. S then thinks, and de-cides to add (3 + g) and (1 + 2). S comments that the colour should change, because the mixture already 10 contains (1 + 3), "unless 2 stops i t from changing." S then combines (4 + 3.+ 2 + g) and proposes to combine (1 + 2 + 4 + g). .S. is asked how many possi-ble combinations there are. She records 14 possibilities, omitting zero and ( 1 + 2 + 3 + 4 ) , and comments that she can remember doing something similar in Grade 9. S is asked to identify 2, and to differentiate i t from 4. S replies that 2 "doesn't seem to react 15 with anything else" and proposes to add (1 + 3 + 4 + g) to see i f 4 is the same as 2. She combines them, gets no reaction, and concludes that 4 is different from 2, because 2 with (1 + 3 + g) changed colour, and 4 with (1 + 3 + g) stopped the colour change. S refuses to guess what liqtiid 2 might be, and adds that "you could get to know more about chemistry working around here." Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. S. recognizes that a number of liquid combinations are possible. She writes them out systematically and proceeds to test each possibility methodically 1. a. Ability to consider logical possi-bilities, (i.e. combinations), independ-ent of the content, (i.e. independent of the actual combinations). 2.12 Aspects of Student Performance Descriptor Descriptors Used Code Numbers referring to the l i s t (1-5). 2. S_ hypothesises the role of 2 when added to (1 + 3 + g)» commenting that the colour should change, un-less 2 stops i t (from changing).11 (11) . 3o S establishes the difference between 2 and 4 with controlled experimenta-tion, i.e. S f i r s t establishes that ( 1 + 3 + 2 + g) produces the colour reaction, and then compares the effect with that of 2 by adding (1 + 3 + 4. + g). (15-19). b. Ability to intuitively integrate thoughts 2.21 within a system of related statements (i.e. possible combinations of liquids). c. Approach is systematic, integrated and 3.21 sure, testing is rigorous, without jumping to conclusions. 2«,a. Ability to formulate operative factors in- 2.22 volved (in considering the role of 2) and arrange experiment accordingly. b. Ability to infer the implications of the 2.23 statements (concerning the possible role of 2) . c. Tendency to make logical generalizations 3.23 based on concept of "logical necessity." 3.a. Ability to control the variables by study- 2.32 ing the role of one factor (i.e. 2 or 4) by varying another (i.e. in this case us-ing (1 + 3 + g) in both casesc) b. Ability to interpolate meaning between 2,b-1 successive statements, (e.g. concerning the difference between 2 and 4) establish-ing relations between relations. I O N \ 0 SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE BALANCE TASK: STUDENT J. V. Synopsis j3 begins by balancing 1 0 0 gm x &L = 5 0 gm x 4P, but immediately recognizes her mistake and corrects her-se l f with 5 0 gm x &L = 1 0 0 gm x 4R, S writes the formula F_ x D. = x D^ , and explains her balance say-ing "the distance in this case ( l e f t side) should be twice as large as that one (right side). The mass is half as large so they should be equal." I_ asks for further variations, S_ tr i e s 80 gm x IL = 40 gm x 2R and explains saying that the formula always works, I_ puts 3 0 gm x 2L, S thinks and then balances i t with - 5 2 0 gm x 3R» and when I suggests changing round the masses, 3. replies that i t couldn't be done as i t would be heavier on the right side. On being asked for reasons, S says "I don't know," but on further question-ing adds that the formula means "that the further away from the fulcrum that you put a certain weight, the more i t would bring i t dowa.. .the weight of the ruler would also be acting.. .don11 know exactly why." I moves one mass further out and asks ">Ihat am I doing?" She says, "Well you're increasing their 1 0 weight almost, except you're not really,. .You're making the a b i l i t y to bring i t dox-m on that side greater." On being asked to balance 3 0 gm x 4L, S says "that's 1 2 0 gm so that would be 2 0 gm x 6R." I_ adds an extra 1 0 gm to the 3 0 gm that i s balanced ( 3 0 gm x 4L = 2 0 gm x 6R) , S says "You're .just increasing the weight, so I could add to i t . " On further questioning S. shows she understands very well the proportions, multiplicative compensation 1 5 and mechanical equilibrium involved in the balance task. Although 5 could not give exact reasons, in terms of force for example, for explaining the balance, i t was evident that her reasoning was clear and well organized. She worked rapidly and easily. Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1 . S. shows a b i l i t y to accurately bal-ance the apparatus in each example .provided by I, and S explains say-1 . a. A b i l i t y to i n t u i t i v e l y integrate thoughts within a system of related possible state-ments (concerning increase and decrease of 2 , 2 1 o Aspects of Student Performance Descriptors Used "Descriptor Code Number ing: "The distance i n this case (le f t ) should be twice as large as that one (right). The mass i s half •as large, so they should be equal." ( 3) "The further away from the f u l -crum that you put a certain weight, the more i t would bring i t down, you're making the a b i l i t y to bring i t down on that side greater" (11). 2. S_ understands the concepts of pro-portionality, mechanical equilibrium and multiplicative compensation, e.g. in saying that i f the weight i s i n -creased on one side, the weight on the other side can either be increased or moved outwards (3-11). 3. _S refers to the formula x D^ = Fg x Dg» While she cannot verbalize exactly what i t means in terms of force, her reasoning i s nevertheless clear and organized when using weight and dis-tance (2,17). 2. 3. weight and distance on both sides.) A b i l i t y to formulate operative factors i n - 2.22 volved ( i . e . interaction of distance and weight), and arrange experiment accordingly, ( i . e . set up the balances.) A b i l i t y to consider the l o g i c a l p o s s i b i l i t i e s 2.12 (of the interaction of distance and weight) independent of the actual apparatus. Tendency to make l o g i c a l generalizations based 3.23 on the concept of "logical necessity." A b i l i t y to predict the real situation (using 2.51 the general formula), and by observing the consequences (check on i t s v a l i d i t y ) . A b i l i t y to interpolate meaning between state- 2.41 ments concerning the increase and decrease of weights and distances, establishing relations between relations, ( i . e . in formula W/D1 = W'/D.) Approach i s systematic, integrated and sure, 3.2; showing no tendency to jump to conclusions. i -<3 SUMMARY OF PERFORMANCE ON PIAC-ET TASKS: STUDENT J.V. Task DESCRIPTORS USED Angles Pendulum Liquids Balance Combined Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 . 2 3 2 . 3 1 2.3:1 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 + + + + + + + + + + + + + + + + + + + + +. + + + + + + + + + + The table shows Student J.V. i s capable of using formal operational thought at the substage B l e v e l . SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE ANGLES OF INCIDENCE AND REFLECTION TASK: STUDENT B.H. Synopsis J3 begins by explaining that the midpoint between the firing ball and the target ball must be found, and extrapolated to the rebound wall to form the point at which the launcher must be aimed. As explanation, S_ says, "The angle of incidence equals the angle — well, i t rebounds symmetrically- off — well, the re-bounds are equal — er — the angles are equal." The angle of incidence equals the angle of reflection." On being asked to demonstrate these angles. j3 hazily indicates/_I and/R_. As he seems muddled, he is 5 asked again, and the second time he describes the angles made with the rebound wall ( i . e . / i and£r). S_ admits he is recalling his experience with the light experiment and with playing pool. _S is asked to hit the ball at target 1. He sets up the launcher using imaginary angles, fires, misses to the right, S_ explains he is trying to hit the halfway mark. _3 correctly resets the launcher to the le f t , fires and misses. 10 S_ explains his actions saying, "Well i f the angles are supposed to be equal, and i f you make this angle (£) smaller, then you have to have a smaller angle here Qr). Then i t will go off in that direction right. If this angle (Ji) is larger then this angle is larger Qr), and i t (ball) should come further in (to l e f t ) . " I_ moves the target to position 2, S, correctly readjusts the launcher to the right and explains as 15 above. X moves the target to position 3, and S_ says, "I have to move i t (launcher) to the l e f t . . .(thinks) — er — No, not to the left, to the right." On being asked what happens when the launcher is aimed at 90°, S says, " i t should bounce right back." Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. J3_ quotes the general law "The angle of incidence equals the angle of re-flection." ( 4 ) and is able to use the law in subsequent reasoning (11-13). 1. Ability to consider logical possibilities independent of the content (i.e. general use of law incidence and reflection.) 2.12 V.0 Aspects of Student Performance Descriptor Descriptors Used Code Numbers 2. S. aims the launcher in the correct directions for hitting the target ball. (15-17). 3. S explains his actions saying, "Well i f the angles are supposed to be e-qual, and i f you make this angle ((i) smaller, then you have to have a smaller angle here Qr). Then i t will go off in that direction (right). If this angle Q±) Is larger then this angle is larger (r), and it (ball) should come further in (to left" Cll-14). 2. Ability to intuitively integrate thoughts 2.12 within a system of related possible state-ments (concerning sizes of the angles of incidence and reflection, and the direction of the fired ball). 3. a. Ability to formulate, operative factors in- 2.22 volved (i.e. size of angles, direction of ball), and arrange experiment and thought sequence accordingly. b. Ability to infer the implications of the 2.23 statements (concerning size of angles, dir-ection of ball), select the true statements and discard the false, and synthesize a statement of necessary and possible conditions. c. Ability to interpolate meaning between the 2.4l successive statements, establishing relations between relations (reciprocal implication and equality of the angles). d. Ability to predict the real situations i f the 2.51 hypothetical condition was fulfilled (i.e. the equality of the angles), and by observing the consequences verify the hypothesis, e. Tendency to make logical proof and generaliz- 3«23 ation based on concept of "logical necessity" (i.e. equality of the angles,[i and[r). SYNOPSIS A N D ANALYSIS OF PERFORMANCE ON THE OSCILLATION OF PENDULUM TASK: STUDENT B.H. Student begins by announcing that he intends to e s t a b l i s h the e f f e c t of weight on the frequency, and proposes using d i f f e r e n t weights and equal lenths. He pr e d i c t s that the heavier one should de-crease the frequency. As-explanation, student says, "In order to compare — you have t o have some-th i n g — that i s constant." While experimenting, although there i s very l i t t l e difference i n frequencies of the two bobs, student tends to r e l y on his own convictions (hypothesis) rather than the experimental evidence. On being given the wood and brass cylinders student s t i l l maintains the l i g h t one has a higher frequency of o s c i l l a t i o n , but admits that " . . . i t doesn't seem t o be as much as I thought i t would be." Student then shortens both s t r i n g s , t r i e s again and concludes "They're o s c i l l a t i n g at a higher frequency." and l a t e r adds, "Shortening the strings seems t o decrease the accuracy of i t . . . i f you decrease the length, the frequency seems to be c l o s e r together — they swing more i n time." Student admits that he has not j u s t i f i e d h i s hypothesis that weight a f f e c t s the frequency of o s c i l l a t i o n . Student then proposes t o e s t a b l i s h the e f f e c t of length of s t r i n g on the frequency and experiments with one pendulum only, using the brass bob on a long s t r i n g and a short s t r i n g . Student claims he i s simply getting an impression of the e f f e c t . He concludes: "As length decreases frequency increases," and reasons that the d i f f e r e n t e f f e c t of d i f f e r e n t weights would be "harder t o see" when using short s t r i n g s . Student i s asked about the other v a r i a b l e s . He proceeds to t e s t the e f f e c t of impetus, taking care t o ensure the lengths are the same, by using one pendulum, but inadvertently uses very d i f f e r e n t amplitudes, which i s pointed out. Student i s asked how he i s measuring the d i f f e r e n c e , and he r e p l i e s " j u s t by the speed and the distance i t moves." In s t r u c t o r suggests that student use two pendula, and student them makes the bobs the same. After many t r i e s student claims that he can t e l l nothing from h i s experimentation. He reasons that amplitude doesn't seem t o have any e f f e c t , and i f one pendulum i s given "some sort of acceleration — (thinks) — that shouldn't a f f e c t i t e i t h e r — i f the amplitude didn't matter — g i v i n g i t a push i s the same as changing the amplitude. Student i s asked to demonstrate c o n c l u s i v e l y h i s hypothesis that both length and weight a f f e c t the frequency of o s c i l l a t i o n . Student suggests, " I f I put a l i g h t e r mass on a small s t r i n g , and a heavier mass on a long s t r i n g , then i t should amplify the e f f e c t . " Student t e s t s h i s hypothesis, and concludes: "The short s t r i n g and l i g h t mass seem to have a much higher frequency than the large s t r i n g . " Instructor suggests interchanging the bob. Student says the e f f e c t sould be about the same, "depending on how they vary." Student then refutes h i s statement, saying that the short s t r i n g with heavy bob " w i l l s t i l l have a higher frequency," e x p l a i n i n g that "the d i f f e r e n c e i n weight i s n ' t that much, and the difference i n the s t r i n g i s . " Student i s asked to prove con-c l u s i v e l y h i s hypothesis concerning the e f f e c t s of length and mass, He attempts to do t h i s , but i n s i s t s on varying both the length and weight together, and becomes more and more confused. .Student eventually asks "What did I say?" then says "I thought mass didn't have anything to do with i t . " "I was working on an assumption — I was t r y i n g t o go back i n my memory and I guess my memory was wrong". Analysis Aspects of Student Performance Descriptors Used Descriptor Code ITumbers 2. S tends to r e l y on h i s own convictions (hypothesis) rather than the experimental evidence (U). S f i n a l l y claims "I was working on an assumption — I was t r y i n g to back i n my memory, and I guess my memory was wrong." ( 3 l ) . S reasons that i f the amplitude has no e f f e c t on the frequency, then impetus w i l l a lso have no e f f e c t , but does not succeed i n demonstrating h i s point (19-30). 1. a. A b i l i t y to accept unproven (2 .1 l ) facts as h y p o t h e t i c a l l y t r u e , but without deducing. the r e a l from the p o s s i b l e . b. A b i l i t y t o consider the 2.12 l o g i c a l p o s s i b i l i t i e s (of weight and length a f f e c t i n g the frequency of o s c i l l a t i o n ) , independent of the content, ( i . e . the experimental evidence). 2. a. A b i l i t y t o consider l o g i c a l 2.12 p o s s i b i l i t i e s (concerning e f f e c t of amplitude and impetus), independent of the content. . b. A b i l i t y t o i n t u i t i v e l y integrate 2.21 thoughts (concerning e f f e c t of amplitude and impetus) within a system of r e l a t e d p o s s i b l e s t a t e -ment s. c. A b i l i t y to formulate operative f a c - (2.22) tors i n v o l v e d , ( i . e . i n amplitude and impetus) and arrange thought sequence accordingly (but not h i s experiment). Aspects of Student Performance _S_ proposes establishing the effect of weight on frequency, by using different weights and equal lengths ( 1 ), but does not actually do i t . S tests effect of impetus taking care "to ensure the lengths are equal, for-getting to check amplitude ( l5)« attempts to check effects of weight and length on frequency by varying both together (29) Descriptors Used Descriptor Code Numbers d. Ability to infer the implications 2.23 of the statement, (i.e. that am-plitude has no effect on the fre-quency) and select the true state-ments &nd discard the false, and synthesize a statement of necessary and possible conditions (i.e. con-concerning effect of impetus on fre-quency) . e. Ability to interpolate meaning be- 2 ,4 l tween the successive statements, establishing relations between re-lations, (i.e. interconnections be-tween amplitude and impetus and their effect on frequency of oscil-lation) . 3. a. Indicated ability to separate the (2.31)* variables (of length and weight), but neutralising the effect of length (by making lengths equal). b. Ability to control the variables (2.32)* by studying the role of one factor (weight) by varying another (length). c. Inability to actually separate or control variables.* Aspects of Student Performance 4. _S suggests, "If I put a lighter mass on a small string and a heavier mass on a long string, then i t should amplify the effect" (24). 5 . .S shortens both strings while test-ing effect of different weights on frequency ( 7 ) . _3 claims he measures the effect "by simply getting an impression of the effect" (12) and by measuring "just the soeed and the distance i t moves (17X * S concludes, "As the length decreases, the frequency increases" and reasons that the different effect of different weights would be "harder to see" when using short strings (l4) „• S tests effect of impetus by specifically keeping a constant length, but inadvert-ently varying amplitude ( 1 5 ) . 5 concludes; "The short string and light mass seem to have a much higher frequency than the large mass and long string)* (24),. S insists on varying both length and weight together, becoming more and more confused (29) . Deser:.V< cor Descriptors Used Code Numbers 4. a. Ability to interpolate meaning be- 2.4l tween successive statements concern-ing effects of weight and length, es-tablishing relations between relations (even though based on an inaccuracy.) b. Ability to predict the real situation (2 .51) i f the hypothetical condition was ful-filled, but tunable to verify hypothesis from observing consequences, due to basic inaccuracy. 5. a. Approach.is hesitant, uncoordinated, un- 3 .11 systematic and uncertain, resulting in non rigorous proofs and a tendency to jump to conclusions, b. Ability to use concept of "all other 3.12 things being equal" in a rudimentary manner. SYNOPSIS AND ANALYSIS OF PERFORMANCE CM THE COMBINATION OF LIQUIDS TASK: STUDENT B.K. J3 begins by suggesting that he^will try each one and see i f they turn yellow with the indicator. I guess it 's by trial and error. It can be a mixture can it?" S tries (4 + g) and (3 + g) then (2 + g) and (1 + g) and (3 + 4 + g). He decides "It's going to be a mixture. Well I guess I could try a mixture of ail four plus indicator" and mixes (1+2 +3 +4+ g). Then tries (4 + g) + (2 + g) then (2 + g) + (3 + g) + (1 + g) . 5 On being asked i f _S_ can say anything more about the liquids, he replies: "Well, something must happen between those three, so the indicator can show that," S_ is asked i f 1, 2, and 3 are al l important and he decides to test (1 + 2 + 4 + g), commenting: "I'm just trying to think of a wsy-of a l l the possibilities." I_ suggests that he writes them down. He writes 1, 2, 3 4, 1 2 , 13, 14, 23,24, 34, 123, 34.5, 134 and 124, commenting; "I haven't tried them all ." 10 j} then adds 4 to (1 + 2 + 3 + g) and comments "It's not that." He then tries (1 + 2+ 4 + g) and ( 2 + 3 + 4 + g ) , and in passing notes that (1 + 3 + g) give the positive reaction. S is asked about the role of 2. S replies "1, 2 and 3 first showed colour, 1 and 3 showed colour, so 2 can't have any effect." 15 On being asked to distinquish between 2 and 4, S tries (1 + 3 + g), gets the colour, adds 4, and finds colour is removed. He concludes, "2 is different from 4 in that 2 in combination with 1 and 3 gives the yellow colour but 4 in combination with 1 and 3 does not. Therefore they're different." 5 suggests 2 could be water. _S_ is asked how many possibilities there are. He makes guesses such as 10 and suggests 4, but notes 20 that he has 14 on his l is t . S. fails to remember the formula, and also forgets that he actually mixed ( 1 + 2 + 3 + 4 + g) twice but does not have the combination of all four on his l i s t . Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1, After making several combinations, _3_ comments, "I'm just trying to think of a way—of a l l the possibilities" (8 ), l . a . Ability to consider the logical possibilities, (i.e. the total num-ber of possible combinations), 2.12 Aspects of Student Performance and systematically writes down 14 of the 1 6 possibilities (9 ). S^  guesses that the total number of "possibilities may be 4' ( 1 9 ) . 2 . After making several combinations and identifying (1 + 3 + g) as the solution, S is asked about the role of 2 , and also to distinquish be-tween 2 and 4. He comments " 1 , 2 and 3 first showed colour, 1 and 3 ••' showed colour, so 2 can't have any effect." Also "2 is different from 4 in that 2 in combination with 1 end 3 gives the yellow colour, but 4 in combination with 1 and 3 does not. Therefore they're different." ( 1 5 ) . L/'escnptor Descriptors Used Code Numbers independent of the content, (i.e. of actually doing the experiments). b. Ability to intuitively integrate 2..21 thoughts within a system of related possible statements (i.e. combina-tions of liquids). 2.a. Ability to formulate the operative 2.22 factors involved and arrange exper-iment and thought sequence accord-ingly, (i.e. by comparing (1 + 3 + 2 + g) with ( 1 + 3 + 4 + g)). b. Ability to infer the implications of 2 . 2 3 the statements, and select the true statements and discard the false, and synthesize a statement of necessary and possible conditions^concerning the role of 2 and 4). c. Ability to control the variables by 2 . 3 2 studying the role of one factor ( l i -quids 2 and 4 respectively) by varying another, (in this case, keeping ( 1 + 3 + g) as a constant). d. Ability to interpolate meaning between 2.41 successive statements (combinations), establishing relations between relations (i.e. roles of 2 and 4). e. Ability to use concept of "all other 3.22 things being equal" in a general sense. 1 CO o Aspects of Student Performance 3 twice combines a l l k l i q u i d s but f a i l s to l i s t them i n the t o t a l number of p o s s i b i l i t i e s feo). The sequence of performed com-binations i s not systematic. J3_ guesses the t o t a l number of p o s s i b l e combinations as 10, and but s t i c k s to Ik which i s the t o t a l number he l i s t e d ( l 9 ) . Descriptors Used Descriptor Code Numbers 3.a. Approach i s h e s i t a n t , uncoordinated, '3.11 uncertain, ( i . e . when a c t u a l l y com-bi n i n g l i q u i d s ) ... with a tendency to jump to conclusions (hazards guesses as to to t a l number of combinations). b. Tendency to make ge n e r a l i z a t i o n s which 3.13' are r e s t r i c t e d to e m p i r i c a l f a c t s , ( i . e . number of combinations a c t u a l l y l i s t e d ) . TABLE 17 SYNOPSIS AND ANALYSIS OP PERFORMANCE • ON THE BALANCE TASK: STUDENT B.H. S_ begins by balancing the apparatus with 100 x 4L = 100 x 4R, and. explains saying that "the same weights are equidistant from the fulcrum." He then adds 50 gw x 3L>and 50 gm x % to the previous balance system, and on being asked i f i t will balance,Sreplies "No," and rearranges the weight so that they are equidistant from the fulcrum. On being asked for more interesting variations, 3_ puts 50 gm x h\, - 100 gm x 2R. He explains 5 • saying, "You take twice the distance of that side , and half the weight. This side is half the distance of that and twice the weight. Then i t should be balanced." _S balances 30 gm x 21, with 30 gm x 2R, then 60 gm x 1R, suggests 15 gm x HR, and is finally advised to use the 20 x 3R, and gives a muddled explanation saying, "As long as you've got an equal number, say with 20 gm at 3, would be equal to the same thing as 30 gm at 2." S_ is pushed for further explanation ;10 and says "As you change the distance, you're getting, ah—farther away from the fulcrum, - you're getting more force." 5 is asked to explain in terms of force. He says "...the farther you are away from the fulcrum, the less weight you need — to provide the same force — on the other side." On being asked to prove this, S suggests a balance system with 100 gm x IL =10 gm x 10R. S then sets up 30 gm x 2L = 15 10 gm x 6R, and prepares to substitute 10 gm x 6R for 60 gm x IE. He explains saying "If you need less weight as you go out, then i f you go in you will need more weight ." On being asked i f the weight in the balance 30 gm x 2L = 20 gm x 3 L can be interchanged, S. replies "No," and attempts to explain saying, "Viell i t was balanced initially, and now you have taken one weight off one side and put i t on the other, so i t won't be balanced." "You haven't got a constant on both sides. You call i t a 20 constant 'K' I guess, and you multiply the weight times the distance, and now you can't.You have got more weight at the same distance, so i t won't be balanced." ,S is asked to balance 30 gra x 3L. He suggests 15 gm x &R, hesitates and reconsiders, mumbles about 50 gm and 25 gm, and eventually suggests 4-5 gm x 2R, or 10 gm x 9 R . S_ is asked what would happen i f the 10 gm weight on the left side were moved further, inwards, S_ 2-5 replies first that the left side will go down, reconsiders, says that the right side will go down, and explains sayin.g"because you're—going in—uh—-decreasing the distance''so on my side right i t remains constant. Your equivalent weight is going to become smaller." On being asked what he should do to main-tain the balance i f more weight is added to the 30 gm x 31.» S says that he could add an equivalent weight to his side, or he could move his existing weight outwards." 30 Analysis Aspects of Student Performance 1. _S_ explains the balance in terms of the constant 'Kr (implicit reference to the formula W'/D = W/D') , as well as in terms-of force (20-21,14). 2. S is capable of setting up balances in-volving different weights and distances, e.g. 50 gm x 4L = lOO^giP. x 2 R ( 1 ), and 100 gm x IL = 10 gm x 10?. (15). 3. 3 explains the balances with such state-ments as "You take twice the distance of that side , and half the weight. This side is half the distance and twice the weight. Then i t should be balanced," and "As you change the distance, you're geti". ting, ah—farther away from the fulcrum, you're getting more force." Also "... the farther you are away from the fulcrum, the less weight you need to provide the same force—as on the other side,"and, "If you need less weight as you go out, then, i f ; you go in, you vail need more weight." 06). Descriptors Used "'Descriptor Code numbers 1. Ability to consider the logical possibilities independent of the content, (i.e. use of general con-cepts) . 2. a. Ability to intuitively integrate thought within a. system of related possible statements (concerning vari-ables of weight and distance on both sides of the fulcrum) . b. Ability to formulate.operative factors involved, (i.e. the relationship be-tween the variables), and arrange experiment accordingly. 3. a. Ability to infer the implications of the statements (concerning the in-crease and decrease of weights and distances) and select the true state-ments and discard the false, and syn-thesize a statement of the necessary conditions (for balance to occur). b. Ability to interpolate meaning between the successive statements establishing relations between relations, (i.e. W'/D = W/D'). c. Tendency to make logical generalize tions based on concept of "logical necessity" (implied in formula W'/D = W/D'). 2.12 2.2] 2.22 2.23 2.41 3.23 Aspects of Student Performance Descriptors Used Descriptor Code Numbers 4. attempts a balance with 50 (?n x 3L = 50 .gm x 5?- ( 2 )» but recon-siders and corrects himself making a symmetrical arrangement. 3 explains that interchange of weights in 30 gm x 2L = 20 gm x 3ft system won't balance saying "Well, i t was balanced initially, and now you have taken one weight off one side and put i t on the other, so i t won't be bal-anced," (19) . S__ hesitates and has to reconsider when thinking of a system for balancing 30 gm x-3L ( 23). S is confused about the effect on the "Tulcrum of moving a weight inwards, but clarifies later.(25-30). 4.a. Approach is hesitant, uncoordinated unsystematic and uncertain, result-ing in non-rigorous proofs and a tendency to jump to conclusions. 3.11 SUMMARY OF PERFORMANCE ON PIAGET TASKS: STUDENT B.N. Task DESCRIP! ORS USED Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 , 2 3 2 . 3 1 2.j2 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 Angles + . .j. + ' + + + Pendulum '"(+) ( + ) . + (+ ) (+ ) + + + Liquids + + + + + + • - + • + Balance * + + i + + • Combined ( + ) + + + + (+ ) + + + + . + + + + The table shows Student B .H. was capable of using formal operational thought predominantly at substage A l e v e l . SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE ANGLES OF INCIDENCE AND REFLECTION TASK: STUDENT L . V . ' . Synopsis S begins by explaining that i t i s necessary to aim for a point on the rebound w a l l i n such a way that i f the t o t a l angle of rebound i s divided into two equal angles (2) , the l i n e d i v i d i n g the angles w i l l be perpendicular to the rebound wall (4). S i s trying to state the law of the equality of the angles of incidence and r e f l e c t i o n (2,4,6). S. explains further that i f the rebound w a l l were rotated, the p o s i t i o n of the perpendicular would change, the angle of incidence becomes greater as does the angle of r e f l e c t i o n (8). 3 also stipulates that the surface must be f l a t for the law to hold. S devises a technique for proving that the angles are equal using a glass sheet, or carbon paper and paper to trace the path of the f i r e d b a l l (12). 3 i s asked what happens to the angles when the launcher po s i t i o n i s changed, for example to.the r i g h t . She r e p l i e s that "the angle i t h i t s at w i l l be wider, ...but i t w i l l h i t somewhere over here (to rig h t ) ' and you w i l l have to bring your target over here (co r i g h t ) " (16) . 3 adds: " I f you moved i t the other way, the angle i s getting smaller u n t i l you get i t perpendicular with (the rebound wall) in which case i t w i l l go and come r i g h t back again, and your target would have to be even with (the launcher) " (18). ••.•Tape ended before interview was concluded. Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. 3 attempts to verbalize the law.of equality of angles of incidence and r e f l e c t i o n , and seems to understand the basic concept (1-4 ). 1. Tendency to make ...generalizations based on concept of " l o g i c a l necess-i t y , " (concerning the equality of angles of incidence and r e f l e c t i o n ) . 3.23 Note: Synopses of student L.'.T. are cross-referenced to t r a n s c r i p t s i n Appendix A. Aspects of Student Performance S_ discusses the apparatus in terms of the position of the rebound wall and the smoothness of the sur-face rather than the position of the launcher (4-6 ). 2. S_ indicates that i f the launcher is moved to the right, the angles of incidence and reflection will be "wider," and the rebound point will move to the right (lO-ll) • Con-versely, i f the launcher is moved to the left, the angles get smaller until the lines of incidence and reflection coincide with the per-pendicular, when the ball "will go and come right back again." ( ]_4 '). < 19 (continued) ' . , • • . • Descriptor Descriptors Used Code Numbers 2.a. Ability to intuitively integrate thoughts 2.21 within a system of related possible statements (concerning sizes of angles and points of rebound). b. Ability to formulate operative factors 2.22 involved and arrange thought sequence accordingly, (e.g. in explaining effect of different positions of launcher). c. Ability to infer the implications of 2*23 the statements, (concerning size of angles and positions of rebound points) select the true statements and discard the false, and synthesize a statement of necessary and possible conditions. d. Ability to interpolate meaning between 2.41 the successive statements (concerning ' size of angles, and points of rebound), establishing relations between rela-tions (i.e. reciprocal implications). 1 CO -v3 SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE OSCILLATION OF A PENDULUM TASK: STUDENT L.W. Synopsis 3 begins by making the lengths the same i n order to e s t a b l i s h the e f f e c t of d i f f e r e n t weights. She uses the heavy and l i g h t bobs, saying "This shows i t most .obviously." She.also takes care that the bobs are dropped from the same point ( 2 ) . S concludes weight has no e f f e c t on the period of o s c i l l a t i o n (3). " 3. then determines the e f f e c t of force (impetus) on the period of o s c i l l a t i o n . She f i r s t pushes 5 both pendula equally e s t a b l i s h i n g t h a t the period i s the same, i . e . a l l v a r i a b l e s are being held constant (8). .3 then pushes the r i g h t pendulum and drops the l e f t pendulum. She says, i n surprise, "They both turn a t the same t i n e , but I thought force had something to do with i t "(8). S i s asked i f she can j u s t i f y the r e s u l t s , she thinks, then says "I guess so! I f you use F = ma. You had more force, so a c c e l e r a t i o n was greater, masses are the same, so that should mean they are equal i n period (10). 10 S then checks the e f f e c t of amplitude, ensuring t h a t the lengths and weights are equal (14), S. explains that i n doing experiments a con t r o l i s necessary, and the " c o n t r o l should always be constant ...so that you can compare them "(16). Si concludes that amplitude does not a f f e c t the period of o s c i l l -a t i o n (2), and explains the r e s u l t s saying, "V-ell t h i s one ( r i g h t pendulum) has a. greater amplitude,'but i t has a greater speed too, so that i t ' s . . . t h e y both have the same period "(24).. She a l s o explains i n 15 terms of the formula F = ma (30). 3 predicts "the shorter the s t r i n g the greater the period "(31), and " i f t h i s one ( r i g h t pendu-lum); was h a l f that one ( l e f t pendulum) i t should take h a l f as long f o r t h i s ( r i g h t pendulum)' to get back to the poin t as t h i s one does ( l e f t pendulum)"(33) • ,5 swings the pendula to demonstrate what she i s saying. 20 S summarizes, "Force, no d i f f e r e n c e , mass the same, as long as length was the same. Amplitude d i d n ' t make any d i f f e r e n c e . Length of s t r i n g did make a d i f f e r e n c e " (37). 1 CO CO Aspects of Student Performance 1. _S_ attempts to j u s t i f y the r e s u l t s i n terms of the formula F = ma (9, 14 ), 2 . S p r e d i c t s and demonstrates "The shorter the s t r i n g , the greater the period" and " I f t h i s one ( r i g h t pendulum), was h a l f that • one ( l e f t pendulum), i t should take h a l f as long for t h i s ( r i g h t pendulum)j to get back to the p o i n t as t h i s one does ( l e f t pendulum) " ( 1 7 I 2 0 ) . Descriptors Used Descriptor Code Numbers I , A b i l i t y to consider the l o g i c a l p o s s i -b i l i t i e s (concerning the e f f e c t of the v a r i a b l e s ) , independent of the content. 2-aS: A b i l i t y to i n t u i t i v e l y integrate-'thoughts 3 a . within a system of r e l a t e d p o s s i b l e statements (concerning e f f e c t of length on period of o s c i l l a t i o n ) . 2a<3: A b i l i t y 'bo formulate operative f a c t o r s 3 b . (involved with length v a r i a b l e ) and arrange experiment and thought sequence accordingly. 2 & A b i l i t y to i n f e r the i m p l i c a t i o n s of the 3 c statements (concerning length v a r i a b l e ) , and s e l e c t the true statements and d i s -card the f a l s e , and synthesize a s t a t e -ment' of the necessary and p o s s i b l e con-d i t i o n s (concerning the e f f e c t of the y a r i a b l e on the frequency of o s c i l l a t i o n ) , 2 d , A b i l i t y to p r e d i c t the r e a l s i t u a t i o n i f the h y p o t h e t i c a l c o n d i t i o n ( t h a t length a f f e c t s the period of o s c i l l a t i o n ) was f u l f i l l e d , and by observing the conse-quences v e r i f y the hypothesis. 2 . 1 2 2 . 2 1 2 . 2 2 2 . 5 1 1 co Aspects of Student Performance 3 . J3 takes care to vary one v a r i a b l e while keeping a l l the others con-stant, e.g. 3 experiments with ' weight ( 1 ) force (impetus) ( 5 ) > amplitude ( 1 1 ) , then length ( 1 7 ) . • S explains that i n doing e x p e r i -ments a c o n t r o l i s necessary, and the " c o n t r o l should be always con-st a n t . . . so that you can compare r;.-'v- them." ( 1 2 ) , 4. 3_ explains her r e s u l t s f o r impetus v a r i a b l e saying, " I f you use F = ma, you had more force, so a c c e l e r -a t i o n was greater, masses are the same, so that should mean they equal i n period " (8-10). S_ explains her r e s u l t s f o r the am-p l i t u d e v a r i a b l e saying; "Well, t h i s one (.right pendulum) has a greater amplitude, but i t has a greater speed too, so...they both have the same period " ( 1 4 ) . Tjes'eripCor" Descriptors Used Code Numbers 3 d . A b i l i t y to separate the v a r i a b l e s by 2.31 n e u t r a l i z i n g the e f f e c t of f a c t o r s such as weight, length, impetus, and amplitude, which cannot be p h y s i c a l l y separated i n the pendulum. e. A b i l i t y to c o n t r o l the v a r i a b l e s by 2.32 studying the r o l e of one f a c t o r by varying others, ( i . e . keeping them constant i n t h i s case). f . -Approach to task i s systematic, i n - 3 .21 tegrated, and organized, r e s u l t i n g i n exhaustive rigorous proofs, with-out jumping to conclusions, g. A b i l i t y to use the concept of " a l l 3 .22 other things being equal" i n a gen-e r a l sense. 4 . A b i l i t y to i n t e r p o l a t e meaning between 2 . 4 l the successive statements, (concerning force, mass and a c c e l e r a t i o n , and am-p l i t u d e and speed), e s t a b l i s h i n g r e l a -t i o n s between r e l a t i o n s . 1 VO O SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE COrTBINATION OF LIQUIDS TASK : STUDENT L.W. synopsis S begins by adding g to 1, 2 , 3 and k (b). J3 then adds ( 1 + g) + (2 + g) then (1 + g) + (3 + g), ana obtains the p o s i t i v e colour r e a c t i o n ( 8 ) . On being asked i f she i s sure about i t , j3 says, "Well, I could t r y the l a s t one," adds ( 1 + g) + (k + g) and concludes the mystery l i q u i d was (1 + 3 + g) ( 1 2 ) . S_ checks her r e s u l t s by remixing ( 1 -b 3 + g) (14), and also (2 +_ g) -i- (b + g) (16) . On being pushed f o r mors information about the liquidsy^says "Well, keep going on d i f f e r e n t com- 5 binations to see which one works " (19 }„ SL says she has forgotten how to f i g u r e out the t o t a l number of combinations, but writes out "4 s i n g l e s , 6 doubles, b t r i p l e s , 1 a l l four " (25) ^  S i s asked about l i q u i d s 2 and b ( 2 6 ) . 3_ r e p l i e s t h a t neither (4 + g) nor (2 + g) nor (2 + b + g) give a r e a c t i o n , "so there's. nothing of what you're t r y i n g to determine i n any of those two 11 (27). . J. i s asked i f what she has said proves c o n c l u s i v e l y that 2 and b have nothing to do with the r e a c t i o n ( 2 8 ) . 10 S_ r e p l i e s "you might have to do other--. They (1 + 3) might contain something and you'd need something to se t o f f the r e a c t i o n . 2 and k may need another l i q u i d or something to mix i n with i t --. j u s t to set o f f the reaction...You'd have to do other experiments "* (29). S_ i s asked i f she can d i s t i n g u i s h between 2 and b, and although I pushes, S, does not attempt any fur t h e r experimentation, and simply summarizes that the r i g h t combination i s 1 and 3. "Anything with 15 j u s t ( 1 + g) would turn a: t i n y b i t yellow. With j u s t 3 i n i t , or 2 and b, i t would be c l e a r . I think that's a l l " (34). *The JE. missed a valuable clue, and should have asked_S what furth e r experimentation she would consider r e l e v a n t ( 2 9 ) . S lacked motivation to experiment e x t e n s i v e l y . ^ 0 Aspects of Student performance Descriptors Used Descriptor" Code Numbers 1. J3. i s able to account f or 15 of the 16 possible combinations, and writes them as "4 s i n g l e s , 6 doubles, 4 t r i u l e s , and 1 a l l four " (5 )-S_ experiments by systematically adding (1 + g ) / ( 2 + g), (3 + g) (4 + g), (1 + 2 + g), (1 + 3 + g) (1 + 4 + g) and (2 + 4 + g)-. ( ).. _S_ mentions further experimentation; but does not s p e c i f y (1-4,13). l . a . A b i l i t y to i n t u i t i v e l y integrate thoughts w i t h i n a system of r e l a t e d p o s s i b l e statements, (concerning the p o s s i b l e l i q u i d combinations). b. A b i l i t y to formulate operative f a c -t o r s involved and arrange experiment . ?,nd thought sequence accordingly. c. Approach i s systematic, integrated, sure and o r g a n i z e d . - — 2.21 2.22 (3.21) i vO SYNOPSIS AND ANALYSIS OF PERFORMANCE ON THE BALANCE TASK : STUDENT L.W. Synopsis _S_ begins by balancing the apparatus with 2 0 0 gm x 8L = 200 gm x 8R (6). She explains saying, "You've added more weight to one side so you have to add more equal weight to the other side to keep a balance on the fulcrum. It doesn't matter how much weight you add, but you have to add i t the same to both sides to (get a) balance." ( 1 0 ) . "It should be the same weight and distance from the centre." (12), On being asked to give more variations, S maintains her argument about having equal weights 5 and equal distances on both sides of the fulcrum ( 1 6 ) . S is asked again for variations "using differ-ent weights on each side" (1?) , and she responds by setting up 5 0 0 gm x 9L = ( 2 0 0 gm + 2 0 0 gm + 100 gm) x 9Ri and says "Well, you can add i t — you can have 2 0 0 gm + 2 0 0 gm and another 100 gm to make up the 5 0 0 gm, but you can't put them in different places," ( 2 0 \ and, "You can have any combination you want to make up the weight on the opposite side, but they have to be the same distance " ( 2 2 ) and gives a 10 further example, 400 gm x 9L = ( 2 0 0 gm ± 2 0 0 gm) x 9?. (24). ^ is asked i f 1 0 0 gm can be made to balance with 5 ° g™ ( 2 ? ) . J3 replies "Well, i f you put 2 0 0 gm x 9L and 1 0 0 gm x % — No! — the other way round (i.e. 2 0 0 x % = 1 0 0 gm x 9P.) i t should work. The 2 0 0 gm on this side is at the half distance, and this one ( 1 0 0 gm x 9R) has half the weight and twice the distance from the fulcrum. It should s t i l l balance " (28)O I_ asks i f this is a general rule. S_replies: "Yes — i t should be'." ( 3 0 ) „ I releases the apparatus, the lef t side drops, S comments, "No, it's not!" ( 3 2 ) . S. changes the balance to 2 0 0 gm x 4L = 1 0 0 gm x 8R, saying "Well, before this CR) was 9 and this (L) was 5 , so this (i) was more -" ( 3 2 ) . S. is asked to balance 40 gm x % using 5 0 gm- ( 3 3 ) which she places at 4 , i.e. 40 gm x 5L = 5 0 gm x 4R. She explains saying, "You've got them in the same ratio, this is 40 at the 5 t h hook .which 2 0 equals 2 0 0 and 5 0 at the 4 t h hook which equals 2 0 0 as well-" ( 3 8 ) , "Distance and mass should be in a balance-}1 (40), S_correctly balances 3 0 gm x 2L with 2 0 gm x 3R. saying "It's the same ratio, the greater the distance, the smaller the weight you need to balance ." ( 4 4 ) . On being asked i f the two weights could be interchanged ( 4 9 ) , 3 responds by changing the weights and the distances, ' i.e. 3 0 gm x 3L = 3 0 gm x 2R (50).25 1 SO VjO Aspects of Student Performance 1. 3 eventually balances 200 gm x "5L = 100 gm x 8R (17) and 40 gm x 5L = 50 gm x 4R (19) and 30 gra x 2L = 20 gm x 3R (23). In explanation 3 explains the first balance by saying that the 200 gm is at half the distance, •. " of the 100 gm and is twice its weight (14). "You've got them in the same ratio 11 (20). "Distance arid mass should be in balance " (2l)» and finally "It's the same ratio, the greater the distance the smaller the weight you n.~:ed " . to balance (the other side) "(23). 2. initially insists that a balance must be established with equal weights and equal distances, but shows that she is capable of setting up other balances 0--4,12 ). _S suggests firstly that 200 gm x 9L = 100 gm x 5R (12), then changes the weights around to 200 gm x 5L = 100 gm x 9ft (l4) but on seeing that no balance was obtained, changed the dis-tances to 200 gm x 4L = 100 gm x 8R , D e s c r i p t o r Descriptors Used 'Code Numbers 1. a. Ability to intuitively integrate 2.21 thought within a system of related possible statements, (concerning weights and distances on both sides of the fulcrum). b. Ability to formulate operative fac- 2.22 tors involved (i.e. V/'.'/D =W/D') and arrange experiment and thought sequence accordingly, (i.e. setting up balances and reasoning), c. Ability to interpolate meaning between 2.4l the successive statements, establishing relations between relations, (concerning proportionality and mechanical equil-ibrium) . 2. a. Approach is hesitant, uncoordinated and 3.11 uncertain. SOT-MARY OF PERFORMANCE ON PIAGET TASKS: STUDENT L.W. Task DE SCRIPTORS D3ED Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 . 2 3 2 . 3 1 2 . 3 2 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 Angles + + + + + Pendulum + + + + + + + • + ! Liquids + + ( + ) Balance + + + -Combined + + + + + • + + + + + + + The table shows Student L.V. is capable of using formal operational thought predominantly »t substage B level, but with some indication of. using substage A level. Comment: Student L.W. seemed to be capable of adequately performing a l l the tasks. She needed, however, to be pushed and seemed to be self restricted to the easiest answers. • OVERALL SUMMARY OF STUDENT PERFORMANCE ON ALL PIAGET TASKS Student DS SCRIPTORS USED Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 . 2 3 2 . 3 1 2 . 3 2 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 J.V. . .+... + ..+ ... + . + +•' +' + . B.H. ( + ) + + + ! . . . + .(+>. . + + ;_' + + + + + ' L.W. < + + + + + + : + + + + The table shows that a l l students were capable of using formal operations. Student L.W. and J,v seemed to function mainly at the substage B level of formal operations while B.H. tended to' function more at the substage A level. i VO ON B. THE EXAMINATION ITEMS Method of Analysis and Reporting of Results Two course examination items were selected from the Physics 110 examination on the basis that the responses expected require formal oper-ational thought. The selected examination items and responses expected by the course instructor and class tutor to these items are given in Tables 2-5 and 26 . The Tables include the instructor's best guesses about the information to be recalled in responding according to expectation, and the maximum mark obtainable for each item.. The Tables give a detailed analysis of the expected performance, using the inventory, and display the formal operational thought expected for each item. The entire examination is given in Appendix B, and brief reasons for the selection of the individual items given in Appendix C. The actual responses given by the students to the selected examin-ation items were transcribed from their examination papers. The marks credited to each student on each item by the examiners were also trans-cribed . The analyses of the expected responses and actual student responses to the selected test items were made in.the same way as they were for the students performance on the Piaget tasks. The responses were read carefully and those sections which could be described by one or more of the descriptors of the inventory were placed in a table with the approp-riate descriptor(s) listed alongside. (See Tables 28, 29, 31, 32, 34 and 35) Where necessary, the descriptors were elaborated to indicate in which way the descriptors were found to be appropriate to the responses given. The elaborations are indicated by parentheses. As in the ana-lyses of student performance on the Piaget tasks, qualified descriptors could be used i f cases arose where a descriptor was found to be nearly adequate. As in the case of the Piaget task performances, summaries wore com-piled for the expected responses and for the responses of each student on both the selected examination items. (See Tables 27, 30, 33, and 36) The summary of the overall performance across items was made so as to display the descriptors which were found to be appropriate to the students performance, at least once in their responses to both the selected exam-ination items (See Table 37) » ' E X P E C T E D R E S P O N S E A N D A N A L Y S I S O F P E R FOR I " ! A N C E O N S E L E C T E D E X A M I N A T I O N I T E M I Item 1 (maximum marks, 8) Nuclear Energy can be converted into heat by nuclear fusion as well as by nuclear fission. Could one not make the best use of these processes by f i r s t splitting atoms (nuclear fission, heat will be pro-duced), and then re-uniting the parts again (nuclear fusion, heat will be produced)? Repeating this cycle over and over again, one would have an inexhaustible energy source. Explain, in terms of the binding ener-gies of nuclei, why this process is impossible. Expected Response Nuclear fission gains energy by splitting heavy nuclei. The result of this fission is medium sized nuclei. Nuclear fusion gains energy by uniting light nuclei. The result of this fusion is medium sized nuclei. As medium sized nuclei have the maximum binding energy per nucleon, energy has to be pro-vided to either split or unite medium sized nuclei. If one splits heavy nuclei one obtains energy. The same energy has to be invested to reunite the nuclei. Thus the overall gain is zero, as predicted by the law of conservation of energy. Analysis Aspects of Expected Performance Descriptors Used Descriptor Code Numbers 1. S_ is asked to explain why the given hypothesis that a continuous cycle of nuclear fission and nuclear fusion would produce an inexhaustible energy source would be impossible. 1. Ability to accept unproven facts (con-cerning nuclear fusion and nuclear fission) as hypothetically true, in order to deduce the real from the im-possible, (i.e. the actual effects of nuclear fusion and nuclear fission). 2.11 Aspects of Expected Performance 2. must consider the logical possi-bilities involved in nuclear fusion and nuclear fission as separate pro-cesses and as integrated processes, • in terms of the binding energy of the nuclei. 3. has to consider the interaction of factors such as nuclear fusion and fission, heavy, light and medium atoms, the energy produced and the energy gained, and high and low binding energies. 4. S must reason that nuclear fission occurs \im "splitting heavy nuclei, and that nuclear fusion occurs when uniting light nuc-l e i , and that both processes result in energy gained and medium sized nuclei, 5. S must infer the implications of the • Statements concerning nuclear fusion and fission, and reason that as medium sized nuclei have the maximum binding energy per nucleon, energy would have to be pro-vided to either split or unit them, and that the overall gain in energy would be zero. Descriptor Descriptors Used Code Number 2. Ability to consider the logical possi- 2.12 bilities involved, independent of the content (i.e. as a theoretical consid-eration) . 3. Ability to intuitively integrate thoughts 2.21 within a system of related possible'state-ments (concerning interaction of the fac-tors) . 4. Ability to formulate operative factors 2.22 involved (in nuclear fusion and fission) and arrange thought sequence accordingly. 5a. Ability to infer the implications of the 2.23 statements, select the true statements and discard the false, and synthesize a statement of necessary and possible con-ditions (concerning nuclear fusion and fission). b. Ability to interpolate meaning between 2.41 the successive statements (concerning Aspects of Expected Performance Descriptors Used Descriptor Code Numbers c. the gain or loss of energy), estab-lishing relations between relations, (i.e. relations between the nuclear fusion and fission processes), Ability to predict the real situation i f the hypothetical condition (that nuclear fusion and fission could give an inexhaustible supply of energy) was fulf i l l e d , and by observing the consequences (.reject) the hypothesis. 2.51 6. 5 must approach this problem systematically in order to reason and prove the falsehood of the given hypothesis. 6. Approach is systematic, integrated, sure and organized, resulting in ex-haustive and rigorous proofs without jumping to conclusions. 3.21 7. S may eventually rely on or state the law of conservation of energy which effectively falsifies the hypothesis. 7. Tendency to make logical proofs and generalizations based on concept of "logical necessity," (i.e. law of con-servation of energy). 3.23 EXPECTED RESPONSE AND ANALYSIS OF EXPECTED PERFORMANCE ON SELECTED EXAMINATION ITEM 2 Item 2 (maximum marks, 12) Space explorers discover a ring of charged particles orbiting around a mysterious cloud. The ring consists' of positive hydrogen ions and negative oxygen ions, circulating in the same direction. The -speed of the hydrogen ions is 1 km/sao, the speed of the oxygen atoms is 2 km/sec, the radius of orbit is .the same for both kinds of particles. The number of particles per cubic-meter is too small to allow" the ions to combine. For the same reason, no electric or magnetic forces between the ions could account for the motion. The ex-plorers discuss the following explanations to account for the ^ . , ,. circular orbits of the ions. Try to rule out as many of these * explanations as possible. Give your reasons. | \pEEj v °~  x Kr* a. The circular orbits are due to gravitational attraction ^ ^ by a massive star within the cloud: Could be/ /; Cannot <a^ i be r 7 b. The circular orbits are due to a charged object hidden in the cloud: Could be j I Cannot be / / c. The circular orbits are due to a magnetic field at right angles to the plane of the orbits: Could be j /, Cannot be / / Note A certain amount of recall is involved in answering this item: v~ 1. The S_ must know the dynamics of satellite motion, i.e. attractive force = Mga^ . (mass x centripetal acceleration). 2. The properties of the proposed forces (gravitational, coulomb, magnetic) e.g. the magnetic force is given by F = qvB and the relative directions of (qv), 3 and F by the Right Hand Rule. ~c" 3. Like charges repel each other while unlike charges attract each other. Expected Response None of the three hypotheses account for the circular orbits of the ions. a. If the hypothesis were true, the 0" and H + ions would move with the same speed, i.e. G K M . mv therefore G M _ cons-tant r L . = r v2" v? Thus implications of hypothesis are inconsistent with given information; b. If the hypothesis were true, only one of the ions (negative or positive) would be attracted to the centre object, depending on its charge. The other one would not move in orbit, as i t would be repelled by the centre object. 3 9 i2-mr^  Thus implications of hypothesis are inconsistent with given information. If the hypothesis were true, the ions could not move in the same direction due to difference in charge, One kind would move clockwise while the other kind would move counter-clockwise. qvS _ mr^  c r Thus implications of hypothesis are inconsistent with given information. Analysis 'Descriptor Code Numbers Aspects of Expected Performance Descriptors Used 1. S has to accept the three hypothetical explanations of the given problem, and deduce from each its truth or false-hood in terms of the recalled physics information. 1. Ability to accept unproven facts (in-volved in suggested explanations) as hypothetically true in order to deduce the real from the possible. 2.11 1 H O Aspects of Expected Performance 2. §. has to consider variables of the equality or inequality of radius of ions from object, of mass, charge, direction of current flow, and velocity of ions. 3. These variables must be considered "by S in terms of their interactions as noted in the relevant information (formulae and laws) that must be recalled for this item. 4. For each-hypothesis S has to formu-late the appropriate and relevant variables so as to apply the recalled formulae or laws. 5. S must establish the implications of possible statements made in the terms of the recalled formulae or laws so .as to be able to state the conditions arising out of each suggested hypoth-esis. S must decide whether or not the impli-cations of the suggested hypotheses agree with the given facts. S_ thus can verify or reject the suggested hypothesis by checking his deduced impli-cations against the given conditions. — . Descriptor Descriptors Used Code Numbers 2. Ability to consider the logical possi- 2.12 bilities independent of the content (i.e. theoretically). 3. Ability to intuitively integrate thoughts 2.21 ' within a system of related possible state-ments (concerning the variables). 4. Ability to formulate operative factors 2.22 . involved and arrange thought sequence accordingly. 5a. Ability to infer the implications of the 2.23 statements, select the true statements and discard the false, and synthesize a statement of necessary and possible con-ditions. b. Ability to interpolate meaning between the 2.41 successive statements (made in each hy-pothesis) establishing relations between relations (i.e. in judging the agreement). c. Ability to predict the real situation 2.51 (using knowledge of physics formulae and laws) i f the hypothetical condition was fulfilled, and by observing the consequences verify (or reject) the hypothesis. TABLE 26 (continued) Aspects of Expected Performance Descriptor Descriptors Used Code Numbers 6. S, must think through the implications of each hypothesis carefully and sys-tematically, and use recalled physics knowledge as the logical reason for accepting or rejecting each hypothesis. 6a. Approach is systematic, integrated, '3.21 sure and organized, resulting in exhaustive and rigorous proofs, with-out jumping to conclusions, b. Tendency to make logical proofs and 3.23 generalizations based on concept of "logical necessity." I H SUMMARY OF EXPECTED PERFORMANCE ON SELECTED EXAMINATION ITEMS Selected Items DESCRIPTORS USED 1 2 Combined Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 . 2 3 2 . 3 1 2 o ^ 2 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 + + ; + + + + + + + + + + + + + + + + + + + + + + + The table shows that both selected examination items require formal operational thought at the sub-stage B level for the expected responses. Descriptors 2 . 3 1 , 2 . 3 2 and 3 . 2 2 are generally not applicable to written responses to examination items. They would, however, be more applicable to laboratory situa-tions. • -STUDENT RESPONSE AND ANALYSIS OF PERFORMANCE ON SELECTED EXAMINATION ITEM 1: STUDENT J.V. Student Response (marks obtained 8/8) Once a heavy atom has undergone nuclear fission to become a medium-sized atom, and cnce a light atom has undergone nuclear fusion to become a medium-sized atom, they have become medium-sized atoms with the binding energy of the nuclei of these medium-sized atoms so great that the same amount of energy would be needed to extract or produce any energy by either splitting them or reuniting them with.another nucleus so there could be no inexhaustible supply of energy. Analysis Descriptor Code Numbers Aspects of Student Performance Descriptors Used 1. S_ did not accept and reason from the hypothesis that there could be "an inexhaustible energy source." But S_ effectively refuted the hy-pothesis by f i r s t dealing with the implications of nuclear fusion and nuclear fission, then linked the two as suggested in the hypothesis with the resulting contradiction, 2. _S_ considered the logical possibil-ities of nuclear fusion and nuclear fission separately and concluded that "there could be no inexhaustible supply of energy." ( 5 ) * 1. 2. Ability to accept unproven facts as hypothetically true in order to de-duce the real from the possible. Ability to consider the logical possi-bi l i t i e s independent of the content(of nuclear fusion and nuclear fission.) 2.11 2.12 o Aspects of Student Performance 3 . 3 integrated factors, i.e. nuclear fusion and nuclear fission, heavy, medium and light atoms, energy pro-duced or "extracted," and binding energy of the nuclei in order to reach her conclusion. 4. S_arranged her argument as a function of the energy needed and produced. She stated that both splitting a heavy atom and uniting a light atom results in "medium sized atoms."( 2 ) . 5. §_ states "...the binding energy of these medium-sized nuclei (is) so great that the same amount of energy would be needed to extract or produce any energy by either splitting them or reuniting them with another nucleus." ( 3 ) , S simply refutes the hypothesis by say-ing "... so there could be no inexhaus-tible supply of energy" ( 5 ) indicating that she has found the hypothesis to be incompatible with the logical implications of the premises on which the hypothesis is based. ~ Descriptor Descriptors Used Code Number 3 . Ability to intuitively integrate 2.21 thoughts within a system of related •possible statements (concerning im-plications of nuclear fusion and fission). 4. Ability to formulate operative factors 2.22 involved and arrange thought sequences accordingly. 5a. Ability to infer the implications of the 2.23 statements, select the true statements and discard the false, arid synthesize a statement of necessary and possible conditions. b. Ability to interpolate meaning between 2.41 the successive statements (concerning the production (gain) or extraction (loss) of energy) establishing relations between relations. c. Ability to predict the real situation i f 2.51 the hypothetical condition was f u l f i l l e d and by verifying the consequences (reject) the hypothesis. 1 H O CO Aspects of Student Performance Descriptors Used Descriptor Code Numbers 6. S systematically considers the logical implications of the given premises in order to refute the suggested hypothesis. 6. Approach is systematic, integrated, sure and organized, resulting in ex-haustive and rigorous proofs without jumping to conclusions. 3.21 7. 3. proves logically that there "could be no inexhaustible supply of energy" by showing in effect, the law of conservation of energy. 7. Tendency to make logical proofs and generalizations based on concept of "logical necessity." 3.23 I H O STUDENT RESPONSE AND ANALYSIS OF PERFC SELECTED EXAMINATION ITEM 2: STUDENT J.V. RMANCE CN Student Response (marks obtained 8/]_2) (a) "could be." Both oxygen and hydrogen atoms have mass, and so could be attracted by a massive star i f there is one. (b) "cannot be."- If the charged objects hidden in the cloud was of a net positive charge, i t would attract only the 0 " ions and would repel the H + ions, or, i f the object was of a net negative charge i t would attract only the H + ions and would repel the 0 " ionsT However, the cloud seems to attract positive and negative ions. (c) "cannot be." The H ions and 0 ions are circulating in the same direction. Therefore there are two directions of current flow. Using the Right Hand Rule 1 of these ions but not both would exert a fore to the center of the clouds on only one set (0~ or . Note No credit was.given for (a). Analysis Aspects of Student Performance Descriptors Used Descriptor Cede Numbers 1. accepts each hypothesis in turn and deduces the implications of each. The implications of hypothe-1. Ability to accept unproven facts as hypothetically true in order to deduce the real from the possible. 2.11 TABLE Aspects of Student Performance 2. S. considers the variables of mass, charge, attractive and repulsive forces, and directions of current .flow. 3. _S__ considers the variables of charge in terms of the attractive and re-pulsive forces, and the directions ' of current flow in terms of the Right Hand Rule. 4. For hypothesis (b), S correctly form-ulates the variables of charge, attrac-tion and repulsion, and for hypothesis (c), the variables of directions of current flow, magnetic force and Right . Hand Rule. J3 incorrectly formulates variables of mass and attraction and repulsion for hypothesis (a), and reasons from these points. 5. _S considers the implications of the hy-potheses in terms of the recalled form-ulae or laws, and is able to state the conditions for each hypothesis, i.e. Hypothesis (b), S_ states that the cloud would attract either the positive or the negative ions, not both (5 )'» Hy-pothesis (c), S. states that the magnetic field would exert a force on only one set of ions (0~ or H+) but not both.(8,9). 29 (continued) Descriptor Descriptors Used Code Numbs 2. Ability to consider the logical 2.12 possibilities independent of the content. 3. Ability to intuitively integrate thoughts 2.21 within a system of related possible statements. 4. Ability to formulate operative factors in- 2.22 volved and arrange thought sequence accordingly. 5a. Ability to infer the implications of the 2.23 statements, select the true statements and discard the false,, and synthesize a statement of necessary and possible con-ditions. b. Ability to interpolate meaning between 2.41 the successive statements establishing relations between relations (for each hypothesis). Aspects of Student Performance Descriptor Descriptors Used Code Numbers ,S rejects hypotheses (b) and (c) by checking her deduced implica-tions against the given conditions which are that both positive and negative ions are attracted by the cloud, (therefore hypothesis (b) is rejected), and that both the types of ions are travelling in the same direction, (therefore hypothesis (c) is rejected). 6. S reasons systematically and meth-odically through the implications of hypotheses (b) and (c), and uses recalled physics knowledge as the logical basis for rejecting the hypothesis. Reasoning for hypothesis (a) was incomplete, and therefore inaccurate. 5 c Ability to predict the real situation 2.51 i f hypothetical condition was fu l f i l l e d , and by observing the consequences (re-ject) the hypothesis. 6a. Approach is systematic, integrated, sure 3.21 and organized resulting in exhaustive and rigorous proofs, without jumping to conclusions (applicable to hypotheses (b) and (c). b. Tendency to make logical proofs and gen- 3.23 eralizations based on concept of "logical necessity." . I H to SUMMARY OF PERFORMANCE ON SELECTED EXAMINATION ITEMS: STUDENT J.V. Selected Items DESCRIPTORS USED 1 2 Combined Formal Operations Substage A Substage B 2 . 1 1 2 . 1 2 2 . 2 1 2 . 2 2 2 . 2 3 2 . 3 1 2 . ; 2 2.41 2 . 5 1 3 . 1 1 3 . 1 2 3 . 1 3 3 . 2 1 3 . 2 2 3 . 2 3 + + + + . .+.. . + + :; + , + + • + + .. • - • '• ' + + + + ; + + i + + + + ' + + The table shows that Student J.V. used formal operational thought at the substage B level in her responses to the selected examination items. STUDENT RESPONSE AND ANALYSIS OF PERFORMANCE ON SELECTED EXAMINATION ITST' 1: STUDENT B.H. Student Response (Mark obtained 5/8) The binding energy of an atom increases with atomic weight, to a c e r t a i n point, around 80. -Then the binding energy decreases, presuriably due to e l e c t r o s t a t i c r e p u l s i v e f o r c e s . A t 80, binding energies are greatest, s p l i t t i n g apart or u n i t i n g n u c l e i would require great amounts of energy. However, the l i g h t elements and very heavy elements have smaller binding energies, since l e s s force i s keeping them together they are easier to s p l i t arid to u n i t e . But when say i s s p l i t i t produces l i g h t e r n u c l e i of higher binding energies which, require great amounts of energy to reunite. T h i s i s the same as H when i t fusss^ only i t produces heavier n u c l e i with higher binding energies which are very hard to s p l i t . Thus i t would be impossible to 'have a c o n t i n -uous cycle since both reactions go to n u c l e i with high binding energies which cannot be fused or s p l i t ( p r a c t i c a l l y ) . A n a l y s i s Aspects of Student Performance Descriptor Descriptors Used Code Numbers 1. _S_ sets about disproving the given hypothesis concerning nuclear f u -s i o n and f i s s i o n , and concludes h i s argument saying, "Thus i t would be impossible to have a . continuous cycle." ( 8 ) . 1. A b i l i t y to accept unproven f a c t s as 2.11 h y p o t h e t i c a l l y true i n order to de-duce the r e a l from the p o s s i b l e . 2. S__ considers the e f f e c t of high and low binding energies with respect to heavy and l i g h t elements, and f i s s i o n and f u s i o n . 2. A b i l i t y to consider the l o g i c a l p o s s i -b i l i t i e s involved independent of the content, ( i . e . t h e o r e t i c a l l y ) • • • • Aspects of Student Performance 3. 3_ considers the interaction of factors such as the ease of nu-clear fusion, and nuclear fission, heavy, light and medium elements, and high and low binding energies. 4. S_ reasons that when the binding energy of elements is high, i t is very difficult to split or fuse the nuclei as great amounts of energy would be required. The opposite is true of light and heavy elements. .S attempts to explain that the fusion of light elements, e.g. H and the figsion of heavy elements, e.g. U , both result in medium sized nuclei with high binding energies. The argument is implicit rather than explicit, 5. J> reasons that both the fusion of light elements and the fission of heavy elements produce nuclei with high binding energies which require great amounts of energy to unite or split apart, "Thus i t would be impossible to have, a continuous cycle." ( 8 ) . Descriptor Descriptors Used Code Numbers 3. Ability to intuitively integrate ••• 2.21 thoughts within a system of related possible statements (concerning the effects of nuclear fusion and nuclear fission). 4. Ability to formulate operative factors 2.22 involved and arrange thought sequence accordinslv. 5a. Ability to infer the implications of the 2.23 statements, select the true statements and discard the false and synthesize a statement of necessary and possible con-ditions. b. Ability to interpolate meaning between 2.41 the successive statements concerning the gain or loss of energy, establishing re-lations between relations (i.e. between , nuclear fusion and nuclear fission). £ Aspects of Student Performance Descriptors Descriptor-Code llumbers 6. Approach to problem is not very-coherent although accurate. 7. _S_ does not comment on the con-servation of energy aspect, but is more concerned with the im-practicability of splitting nuclei which have a high binding energy. Ability to predict the real situation i f the hypothetical condition was ful-filled, and by observing the conse-quences (.reject) the hypothesis. 6. Approach is uncoordinated and unsys-tematic, resulting in (.seemingly) non-rigorous proof . o 7. Tendency to make proofs and generali-zations which are restricted to empir-ical facts. 2.51 3.11 3.13 STUDENT RESPONSE AND ANALYSIS OF PERFORMANCE ON SELECTED EXAMINATION ITEM 2: STUDENT B.H. Student Response (marks obtained 4/l2) (a) "Cannot be." If i t were a gravitational attraction, then the force exerted on the particles would the hydrogen ions, there would be a greater Fr and hence they would be closer to the cloud than the H + ions. Since they are in a ring then i t couldn't be a gravitational attraction. (b) "Cannot be." If this were the case then those particles that had the same charge as th-it of the 5 object in the cloud would be repelled and would not stay in orbit. Both charges would not be present in the orbit i f i t were a charged object within, the cloud. (c) "Could be." If there were a continuous magnetic field at right angles to the plane orbits, then there would be an electric field at right angles to the magnetic field. The particles would then be kept in a circular orbit and their velocity would depend on their mass. 10 Note Only reasoning in (b) was given credit. Analysis Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. S__accepts each of the three hypotheti-cal explanations and is prepared to • make deductions from each. However accurate reasoning is displayed for hy-pothesis (b) only. 1. Ability to accept unproven facts as hypo-thetically true in order to deduce th9 real from the possible. 2.11 2. considers the variables of mass, dis-tance from cloud, charge, attraction and repulsion, and (direction of current flow), and velocity of the particles. 2. Ability to consider the logical possibili ties independent of the content. 2.12 H Aspects of Student Performance 3. For hypothesis (b) S is able to inte-grate the variables of charge with the variables of attraction and repul-sion of ions. For hypotheses (a) and (c) S i s incorrect i n his reasoning but Ke nevertheless attempts to inte-grate the variables. 4 . For hypothesis (b) S reasons in terms of the attraction and repulsion of differently charged ions. For hypo-thesis (a) S incorrectly reasons in terms of the distance from the centre, which i s not the operative factor i n this case. 5. For hypothesis (b), S infers that "Both charges would not be present i n the or-b i t i f i t .were a charged object within the cloud." ( 6 ) . For hypotheses (a) and (c) _S_ makes cer-tain inferences but they are incorrect. S. rejects hypothesis (b) by checking his deduced implications against the given conditions, i . e . that both positive and negative charged ions are in orbit around the cloud. —. . . — - . . . Descriptor Descriptors Used Code Numbers 3 . A b i l i t y to i n t u i t i v e l y integrate thoughts 2.21 within a system of related possible state-ments . 4 . A b i l i t y to formulate operative factors i n - 2.22 volved and arrange thought accordingly. 5a. A.bility to infer the implications of the statements, select the true statements and discard the false, and synthesize a state-ment of necessary and possible conditions. b. A b i l i t y to interpolate meaning between the successive statements (made in hypothesis (b) establishing relations between relations ( i . e . i n judging the agreement of the hypo-thesis with the given conditions). c. A b i l i t y to predict the r e a l situation i f the hypothetical condition was f u l f i l l e d , 1 H H CO 2.23 2.41 2 . 5 1 Aspects of Student Performance Descriptor Descriptors Used Code Numbers 6 . For hypothesis (b) carefully and methodically deduces the implications and logically rejects the hypothesis. For hypotheses (a) and (c), 3 selects the wrong operative factors or incor-rectly infers the implications of the hypothesis, giving some evidence of uncertain, non-rigorous thinking. and by observing the consequences (reject) the hypothesis. 6 a . Approach is systematic, integrated, 3 . 2 1 sure and organized, resulting in ex-haustive and rigorous proofs, with-out jumping to conclusions, (appli-cable to hypothesis (b) only) . b. Tendency to make logical proofs. . . . 3 . 2 3 based on concept of "logical necessity" (applicable to hypothesis (b) only). c. Approach is . . . uncertain, resulting 3 . 1 1 in non-rigorous proofs and a tendency to jump to conclusions (applicable to hypotheses (a) and (c)). I H H MD SUMMARY OF PERFORMANCE ON SELECTED EXAMINATION ITEMS: • STUDENT B.N. Selected Items DESCRIPTORS USED . Formal Operations Substage A Substage B 2.11 2»D.2 2.21 2.22 2.23 2,31 2 © *>2 .41 2.51 3.11 3.12 3.13 3.21 3.22 3.23 1 + + + + + + • + + 2 + + + + + + + Combined + + + + + + + + + + The table shows Student B.R". used formal operational thought at both the substage A and substage B levels. This indicates that B.H. was.not fully capable of using formal operational thought at a l l times. STUDENT RESPONSE AND ANALYSIS OF PERFORMANCE ON . SELECTED EXAMINATION ITEM 1: STUDENT L.W. Student Response (marks obtained 5 /8 ) T h i s i s impossible because only heavy n u c l e i can go through the process of nuclear f i s s i o n and extremely l i g h t n u c l e i only can go through the process of nuclear f u s i o n . I f i t were p o s s i b l e to get a material whose nucleus had a binding energy a t the meeting point of nuclear f u s i o n and f i s s i o n , i t might be possible to s p l i t t h i s nucleus, and then reunite i t to produce an inexhaustible energy source, but there i s no element as of yet with a nuclear binding energy at e x a c t l y t h i s point. Analysis A.spects of Student Performance Descr i p t o r s Used Descriptor Code Numbers 2. S hypothesizes that "a material whose nucleus had a binding energy at the meeting point of nuclear fusion ana f i s s i o n , " could be s p l i t and reunited "to produce an inexhaustible energy source." (4). This hypothesis i s i n -accurate as such a "material" would have the highest nuclear binding energy, and could not be s p l i t o r . r e -united to form an inexhaustible supply of energy. S ^rejects the suggested hypothesis be-cause "only heavy n u c l e i can go through the process of nuclear fusion and extremely l i g h t n u c l e i only can go through l a . ( A b i l i t y to hypothesize...but i n t h i s case the hypothesis i s inaccurate and therefore does not contribute to the problem). b. A b i l i t y to consider l o g i c a l p o s s i b i l i -t i e s independent of the content. 2. Approach i s h e s i t a n t , uncoordinated, unsystematic and unsure, r e s u l t i n g i n non-rigorous.proofs and a tendency to jump to conclusions. (2.11) 2.12 3.11 Aspects of Student Performance Descriptors Used Descriptor Code Number the process of nuclear fusion." (2 ). I t i s not clear on what grounds S r e j e c t s the hypothesis except that she has a c c u r a t e l y given the conditions f o r nuclear fusion and f i s s i o n . This could be a case of sinple r e c a l l , or of jumping to conclusions. Note S may not have r e c a l l e d information concerning mediuri sized n u c l e i . STUDENT RESPONSE AND ANALYSIS 0? SELECTED EXAMINATION I: STUDENT L.N. Student Response (marks obtained 0/12) (a) ' "Could be." A massive star of no charge one way or the other would not attract the ions. I f there were no forces acting on the ions they would go off into space, but the gravitational force of the large central mass would keep changing the direction of n. of the ions, thus the movement would be or b i t a l . Fc zr-mv2 , Fg = GMm r T " T . (b) "Cannot be." I f the object was positively charged the negative oxygen ions would be attracted, and i f i t were negatively charged the positive hydrogen ions would be attracted. Since however the ions stay in constant orbit the object in the cloud cannot be charged. (c) "Could be," - Using.the Right Hand Rule, as the direction .of the current constantly changes the magnetic f i e l d w i l l turn at-the same rate always remaining at right angles to the plane of the orbits, 10 and thus the force w i l l continually change. Note Response (b) closely approximates the examiner's expected response. However no credit was given. Analysis .' : ' Aspects of Student Performance Descriptors Used Descriptor Code Numbers 1. S_ accepts each of the three hypothe-t i c a l explanations of the given pro-blem, and i s prepared to make deduc-tions from each. However accurate reasoning is displayed only for (b). 1. A b i l i t y to accept unproven facts as hypothetical!y true in order to de-duce the real fron the possible. 2.11 Aspects of Student Performance. 2, _S_ considers the variables of charge, (mass) a t t r a c t i o n (and repulsion), d i r e c t i o n of the current. .3. For hypothesis (b) § i s able to i n t e -grate the variables of positive and negative charge with the variables of a t t r a c t i o n and repulsion of ions. For hypotheses (a) and (c) 3 i s incorrect i n her reasoning, but nevertheless attempts to integrate the variables. 4. For hypothesis (b) S_ reasons i n terms of the a t t r a c t i o n of negative and positive ions to the object, depending on the charge of the object. For hypothesis (c) S reasons i n terms of the d i r e c t i o n of the current and the Right Hand Rule, but i s i n s u f f i c i e n t l y rigorous i n her reasoning. 5. For hypothesis (b), .3 i n f e r s that " i f the object was p o s i t i v e l y charged, the negative oxygen ions would be-.-attracted, and-if i t were negatively charged the p o s i t i v e hydrogen ions would be attracted " (6 ). For hypotheses (a) and ( c ) , S_ attempts certain inferences which are incorrect and not relevant. Descriptor Descriptors Used' Code Numbers 2 . A b i l i t y to consider the l o g i c a l p o s s i - 2 . 1 2 b i l i t i e s independent of the content ( i . e . t h e o r e t i c a l l y ) 3 . A b i l i t y to i n t u i t i v e l y integrate,. - 2 . 2 1 thoughts within a system of related . possible statements. k. A b i l i t y to formulate operative factors " 2 . 2 2 involved and arrange thought sequence accordingly. 5a. .Ability to i n f e r the implications of the 2 . 2 3 statements, select the true statements and discard the f a l s e , and synthesize a statement of the necessary and possible conditions. b. A b i l i t y to interpolate meaning between the 2 .h-l successive statements (made i n hypothesis (b)) establishing r e l a t i o n s between r e l a -tions ( i . e . i n judging the agreement of i H •P-Aspects of Student Performance • .. Descriptor Descriptors Used Code.Numbers .£ rejects hypothesis (b) by checking her deduced implications against the given conditions, and concludes, "Since however the ions stay i n con-, stant orbit, the object i n the cloud cannot be charged." (7 )• 6. For hypothesis (b), S carefully and methodically deduces the implications and l o g i c a l l y rejects the hypothesis. For hypothesis (a), S selects the wrong operative factors, and for hy-pothesis (c) S is unable accurately to deduce the implications, giving evidence of uncertain non-rigorous thinking. the hypothesis with the given condi-tions). c. A b i l i t y to predict the r e a l situation 2.51 i f the hypothetical condition was f u l -f i l l e d and by observing the consequences verify (or reject) the hypothesis. 6a. Approach, for hypothesis (b) i s system- 3.21 atic, integrated, sure and organized, resulting i n exhaustive and rigorous proofs, without jumping to conclusions. b. Tendency to make lo g i c a l proofs...based 3.23 on concept of " l o g i c a l necessity." (hypothesis (b) only) c. Approach i s urisystematic...unsure, re- 3.11 suiting i n non-rigorous proofs and a tendency to jump to conclusions.- (hy-potheses (a) and (c)) -TABLE 36 -SUMMARY OF PERFORMANCE ON SELECTED'EXAMINATION ITEMS: ' STUDENT L.W. / j * * f Selected Items DESCRIPTORS USED Formal Operations Substage A Substage B 2.11 2.12 2.21 2.22 2.23 2.31 2.32 2.41 2.51 3.11 3.12 3.13 3.21 3.22 3.23 1 (+) + - . - i . . . . . . . .1 . .. + j 2 + ' + + + + • " - ; + + + . + Combined + ' + + i + i + + + + + + *u* * T J 6 t a b l e S h C W S s t u d e n t L , ¥ * u s e d f°rmal operational thought at substage A and substage B, indicating that she vas not f u l l y capable of using formal operational thought at a l l times. . . • ON OVERALL SUMMARY OF STUDENT PERFORMANCE ON BOTH SELECTED EXAMINATION ITEMS Student DESCRIPTORS USED Formal Operations • Substage A Substage B 2.11 2.12 2.21 2.22 2.23 2.31 2.32 2.41 2.51 3.11 3.12 3.13 3.21 3.22 3.23 J.V. + + + " + + - - A - - . , . + + .; • + +. B.H. . + + + + + :- - • - - - .-+ + ; + + + L.W. + + + + + + + + + + : The table shows student J .V\ used formal operations at the substage B level, while both students B.H. and L.W. used formal operation thoughts at substages A and B levels indicating that they were not fully capable of using formal operational thought at a l l times. DISCUSSION OF RESULTS, SUMMARY' AND CONCLUSIONS The f i n a l Chapter contains the comparison of the results of the analyses of student performance on the Piaget tasks with the analyses of the Actual student performance on the examination items. The inferences made from the comparisons are presented. The implications of the comparisons are discussed in the context of both formative evaluation and the improvement of classroom practice. A critique of the usefulness of the inventory i s followed by suggestions for further research. The Chapter ends with a summary of the. problem and the conclusions of the study. A. COMPARISON OF STUDENT PERFORMANCE ON PIAGET TASKS AND SELECTED EXAMINATION.ITEMS The overall results of student performance on the Piaget tasks and s e l -ected examination items are displayed i n a table so as to f a c i l i t a t e the com-parison between them (See Table 38) . The comparison was made by looking for congruency between the descriptors used to describe performance on the Piaget tasks and selected examination.items, for each student i n turn. The results were considered congruent when the same, or nearly the same descriptors were identified for a student i n both the Piaget tasks and .selected examination items. Conversely, the results were considered non-congruent when the des-criptors identified for the same student i n the Piaget tasks and selected ex-amination items did not agree. Three comparisons are considered. F i r s t l y , the comparison of the inven-tory of descriptors with student performance on the Piaget tasks provides information which identifies the potential l e v e l of i n t e l l e c t u a l development of the students, and i s a summary of Table 24, Secondly, the comparison of the inventory of descriptors with student performance on selected examination items provides information about the actual l e v e l of i n t e l l e c t u a l development d i s -played i n non-Piagetian conditions, and i s a summary of Table 37. Thirdly, the comparison of the students potential and actual levels of i n t e l l e c t u a l develop-ment, the l a t t e r displayed i n non-Piagetian conditions, i s a summary of informa-tion obtained from Table 38. Fin a l l y , Table 38 i s used to provide information concerning the adequacy of the inventory of descriptors i n describing student performance on Piaget tasks and selected examination items. Summary of Comparisons 1. Inventory of descriptors compared with student performance on Piaget tasks, i . e . potential l e v e l of i n t e l l e c t u a l development. • Student. J.V. displayed i n t e l l e c t u a l functioning at the formal operations, sub-stage B l e v e l . Both students B.H. and L.W. displayed potential i n t e l l e c t u a l fun-ctioning at the formal operations l e v e l , with indications of operating at the substage B l e v e l as well as at the substage A l e v e l , from which i t could be i n -ferred that their formal operational thought was not f u l l y developed (Table 24). 2. Inventory of descriptors compared with student performance on selected exam- ination items, i . e . actual i n t e l l e c t u a l performance displayed i n non-Piagetian  conditions. ' . Student J.V. displayed actual i n t e l l e c t u a l functioning at the formal opera-tions, substage B l e v e l . Both students B.H. and L.W. displayed actual i n t e l l e c -tual functioning that was both at substage A and B levels, from which i t could be inferred that they did not actually use f u l l y developed formal operational thought (Table 37). 3. Student potential l e v e l of i n t e l l e c t u a l development compared with student ' actual i n t e l l e c t u a l performance displayed i n non-Piagetian conditions. A l l three students showed congruency between actual i n t e l l e c t u a l performance and potential l e v e l of i n t e l l e c t u a l development (Table 38). COMPARISON OF STUDENT PERFORMANCE ON THE PIAGET TASKS AND .SELECTED EXAMINATION ITEMS Stu-dent _'~ ' DESCRIPTORS USED .- '. ;^  ; OVERALL • •PERFORMANCE J.V. B.H. L.W. Formal Operations • -•- - - -- Substage A - -Substage B -' 2.11 2.12 2.21 2.22 2.23 2.31 2.32 2.41 2.51 3.11 3.12 3.13 3.21 3.22 -3.23 + + + • + + + - + + + + + . Piaget Tasks + + + + + + - - + Exam. Items (+) + + + +• (+) + + + + + + + + Piaget Tasks + + + + + + + + + Exam. Items + + + + + + • + + +; + + Piaget Tasks + + + + + + + ..+ ... - +; + Exam. Items This table shows the congruency.between intellectual performance observed on the Piaget tasks and i n t e l l -ectual performance observed i n the responses, to the selected examination items for students J.Y., ?,.H., and. L.w. "• " ": 1 ' ' ' I • • H o 4. Adequacy of inventory of descriptors i n describing student performance on Piaget tasks and on selected examination items. i . e . on the examination items. Descriptors 2.31, 2.32, 3.12 and 3.22 were not found useful i n practical experimental situations, i . e . i n the Piaget tasks. The fif t e e n descriptors however, seemed to cover adequately both the-or e t i c a l and practical situations. In conclusion, i t i s apparent: a) that the Inventory of descriptors can be used to identify and select examination items that require formal operational thought for their s o l -ution; b) that these selected examination items can be used as a means of evaluat-ing student performance in examinations i n terms of the l e v e l of i n t e l l -ectual development displayed by the students i n the classroom situation; c) that Piaget's theory of in t e l l e c t u a l development can be further developed i n order to become of practical use i n the classroom. Formative Evaluation The inventory of descriptors i s an Instrument designed to identify i n -t e l l e c t u a l behaviour of individuals at the formal operations stage, and can be applied to both verbal and non-verbal behaviour. Such an instrument could prove valuable i n the formative evaluation involved in curriculum design. The designs of new science curricula: are very often based on one/or. more theories of education. An independent method of evaluating the i n t e l l e c -tual progress of students using the curricula i s advantageous, especially i f Descriptor 2.11 was not found to be useful i n theoretical situations, B. IMPLICATIONS OF THE COMPARISONS the evaluative c r i t e r i a are based on an acceptable theory of in t e l l e c t u a l development. The inventory of descriptors for indenti-fying the formal operations l e v e l of development could he used in the formative evaluation of curriculum development in three ways; 1) evaluation of the s u i t a b i l i t y of subject matter being presented to different age groups i n the terms of the i n t e l l e c t u a l standards required to master the concepts. 2) evaluation of the s u i t a b i l i t y of different methods of instruction for different age groups, and 3) evaluation of the in t e l l e c t u a l demands made by different forms of examinations, e.g. multiple choice, essay questions, prac t i c a l examination?, etc. Classroom Practice The inventory of descriptors could be used by teachers i n the classroom situation to evaluate a course of instruction in progress i n order to improve i t . This could be effected by; 1) evaluating the lev e l of i n t e l l e c t u a l performance required of students in order to carry out classroom tasks and unserstand new concepts, 2) evaluating the potential i n t e l l e c t u a l s k i l l s of the students in order to ascertain the level at which they are able to function, 3) matching the selected classroom task against the ident i f i e d i n t e l l e c t u a l capacity of the students. Teachers could thus optimize conditions for learning by t a i l o r i n g the courses to meet the i n t e l l e c t u a l needs of the students. They could anticipate the source of possible d i f f i c u l t i e s for the students in performing the tasjs and understanding concepts, and could take steps to present d i f f i c u l t subject matter i n such a way as to minimize the d i f f i c u l t i e s . C. CRITIQUE OF THE USEFULNESS OF THE INVENTORY The inventory of descriptors has been shown by this study to identify successfully i n t e l l e c t u a l behaviour of individuals at the formal operational lev e l from both performance on the Piaget tasks as well as performance in non-Piagetian conditions. The inventory can be applied to behaviour that i s both verbal and non-verbal i n nature, provided that i t i s recorded. The advantage of this method of identifying i n t e l l e c t u a l behaviour i s that i t analyses behaviour i n depth, producing more thorough, and meaningful information than might otherwise be the case. The instrument identifies the maximum i n t e l l e c t u a l potential of individuals as well as the actual i n t e l l e c -tual l e v e l displayed by the individual i n different circumstances. The f l e x i b i l i t y provided i n the administration of the Piaget tasks allows for freedom i n handling different personalities i n different conditions. The inventory of descriptors provides a means of standardizing the interpretation of the data i n such a way as to add a measure of r e l i a b i l i t y to the results. The v a l i d i t y of the instrument i s ensured by the fact that i t adheres as closely as possible to the Piaget texts. The format of the Inventory serves both as a means of helping potential users of the inventory to understand the descriptors, as well as to demon-strate their coherence and relationship with Piaget theory both generally and s p e c i f i c a l l y . There are two main disadvantages with this method of identifying i n t e l l -ectual lev e l s . F i r s t l y , the technique involves in-depth analyses of behav-iour. The c l i n i c a l technique of administering the Piaget tasks requires time and patience. In addition, the application of the inventory to the data to be analysed i s time consuming, as fine details and nuances of behaviour must be considered. The complex nature of the inventory compounds the d i f f i c u l t y i n applying i t . The method thus limits the number of individuals whose behaviour can be analysed, and makes i t impossible for mass testing. Moreover, the method does not lend i t s e l f to quantification. The second disadvantage i s found i n the need for users of the inventory to be familiar with Piaget theory and to be well trained in the application of the inventory to the data. This requirement further limits the use of the inventory in the classroom. In summary, i t i s apparent that the methodology developed in this study i s at present directed towards in-depth analysis of the i n t e l l e c t u a l performance of a limited number of individuals. It i s applicable to subject matter such as that found i n a classroom situation as well as to student performance on classroom tasks. D. FURTHER RESEARCH The inventory of descriptors i s designed at present to identify i n t e l l e c t u a l behaviour at the formal operations l e v e l . Further research i s required to refine the instrument i t s e l f . The interrelationships between the theoretical constructs y descriptors and exemplars, should be established with more rigour. The methodology for analysing r e l a t i v e l y instructional and structural data on i n t e l l e c t u a l performance, represented by the inventory, require considerable development i n order to make i t possible for classroom teachers to determine existing and changing modes of thought and the possible reason for these s h i f t s . This would serve to improve the l o g i c a l v a l i d i t y of the inventory, as well as to render i t more usable and reliable and there-fore more useful. An inventory of descriptors of behaviour at the concrete operations stage would make the analyses of behaviour more convincing. I f an individual is shown not to be performing at the formal operations stage, i t is more meaningful to reinforce the results by identifying his behaviour as being at the concrete operations stage. The importance of the methodology developed in this study lies in it's potential usefulness for the classroom. If the inventory could provide a way of systematically analyzing intellectual performance with a view to discriminating between individuals at the formal operations stage and the concrete operations stage, i t could facilitate the improvement of classroom instruction. Similarly i f the pre-operational stage and sensory motor stage could be represented by an inventory of descriptors, the theory of Jean Piaget, instead of being potentially significant for classroom practice, could become of practical importance. At this point i t is opportune to draw the reader's attention to the very recent work of Professor Klaus G. Witz (1971). This work may well represent a major step forward in the direction described above and is recommended to the reader interested in a more rigorous methodology for applying Piagetian theory to problems of classroom practice. E. SUMMARY AND CONCLUSIONS This thesis is addressed to a problem in classroom practice indentified as formative evaluation, that i s , the development of a method of utilizing Piaget's theory of intellectual development for evaluating the intellectual performance of students contending with the formal concepts and methods of inquiry presented in a first year physics course with a view to improving instruction in this course. The method of studying the problem required a good understanding and analysis of the relevant aspects of Piaget's theory so that'it could be re-formulated in such a way to be usable in identifying formal behaviour of individuals in answering specially selected examination items. This resulted in the formulation of a methodology in the form of an inventory of descriptors, and a method of analysis designed for identifying behaviour at the f i n a l stage of Piaget*s developmental sequence, namely,.the formal operations stage. The inventory of descriptors was used to identify formal operations; behaviour of students performance on selected Piaget tasks, providing information concerning their maximum potential l e v e l of i n t e l l e c t u a l development. It was then used to identify physcio examination items which required formal operations for their solution and the formal operational behaviour displayed by students i n responding to the selected items, thereby providing information concerning the actual level of i n t e l l e c t u a l performance displayed by students in classroom conditions. A comparison of iden t i f i e d i n t e l l e c t u a l behaviours provided information concerning the usefulness of the instrument as well as information concerning the potential usefulness of such an instrument i n science education. It was concluded that the inventory of descriptors adequately described and iden t i f i e d i n t e l l e c t u a l behaviour at the formal operations l e v e l , both in student performance on the Piaget tasks, and i n student performance on selected items from the physics examination paper. The inventory of descriptors may well be of potential value to formative evaluation i n the classroom situation. i \ Ausubel, David P., Educational Psychology, A Cognitive View. New York: Holt, Rinehart and Winston, Inc., 196b. Easley, J. A. Jnr., "Concerning Educational Applications of Piaget Theory." (Ur-bana, Illinois: College of Education, University of Illin o i s , mimeographed), 19 1969. Furth, Hans G., Piaget and Knowledge, Theoretical Foundations. Englewood C l i f f s , New Jersey: Prentice-Hall, Inc., 1969. Ginsburg, H., and Opper, S., Piaget's Theory of Intellectual Development: An  Introduction. Englewood C l i f f s , New Jersey: Prentice-Hall, Inc., 1969. Inhelder, B., and Piaget, J., The Growth of Logical Thinking from Childhood to  Adolescence. New York: Basic Books, Inc., iy5tf. Piaget, J., The Child's Conception of the World. Harcourt, Brace and World, 1929. Piaget, J., Traite de Logique. Paris: A. Colin, 1949. Piaget, J., The Psychology of Intelligence. London: Routledge and Kegan Paul, Ltd., 1950. ' Piaget, J., Logic and Psychology. Manchester: Manchester University Press, 1953* Piaget, J., "Piaget 1s Theory," Carmichael's Manual of Child Psychology, Vol. I, - Editor P.H. Mussen. New Yorki John Wiley and Sons, Inc., 1970. pp. 703-732. . Piaget, J., and Inhelder, B., The Psychology of the Child. New York: Basic Books, Inc., 1969. Scriven, M., "The Methodology of Evaluation.. Perspectives of Curriculum Evalua-tion," American Educational Research Association Monograph Series on Curric-ulum Evaluation, Vol. I, Chicago: Rand MlcNally, I967. pp. 39-b3. Skager, R. W., and Broadbent.,. L. A., "Cognitive Structures and Educational Evaluation," CSEIP Occasional Report, No. 5. July 1968, U.C.L.A. Smith, B. 0., M. Meux, et. a l . "A Study of the Logic of Teaching." (Urbana, I l l i n -ois. Bureau of Educational Research, University of Illin o i s , mimeographed) ,1962. Urmson, J. D., "On Grading," "Philosophical Essays on Teaching. (Edited by Ber-thram Bandman and Robert S. Cuttchen.; Philadephia: Lippincott, 1969, pp. 194-217. Westbury, I., "Curriculum Evaluation," Review of Educational Research. Vol. 40, No. 2, April 1970. pp. 239-260. ; : Witz, K. G., "Analysis of a Framework in Young Children." Urbana, Illinois; Mathematics Dept., University of Il l i n o i s , mimeographed). 1971. TRANSCRIPTS OF STUDENT PERFORMANCE ON THE PIAGET TASKS Complete transcripts of the performance of student L.W. on the four Piaget tasks are given. The transcripts are taken l i t e r a l l y from the video tapes made of the interview, and descriptions of student actions are included. The ennumoratlon of the comments enables the reader to make cross-reference to the appropriate synopses given in the text for the purpose of checking their accuracy. TRANSCRIPT OF STUDENT PERFORMANCE ON THE ANGLES OF INCIDENCE AND REFLECTION TASK: STUDENT L . W . • I n t e r v i e w e r 1. (Shows apparatus and poses the problem.) 3« How do you s p l i t the angle? 5. Why d i d you take the p e r -p e n d i c u l a r ? 7. I f t h i s ( rebound w a l l ) was t u r n e d around a l i t t l e , what would happen? 9 . OK. I s t h i s a r u l e w h i c h h o l d s f o r a l l cases? 11. Can you prove i t ? 13. OK, I ge t the i d e a now t h a t t h e y s h o u l d be e q u a l , b u t what happens when you s t a r t chang ing t h e p o s i t i o n o f the l auncher? S tudent 2. W e l l you a im i t so i t h i t s t h e r e ( r e -bound w a l l ) . I f you s p l i t the angle up between t h e r e and t h e r e — so b o t h those ang les a re e q u a l , ( i . e . jl and 4. Y o u b i s e c t the ang le — l i k e t h e angle t h a t t h i s comes i n o n , i t has — uhm — you t ake t h e p e r p e n d i c u l a r t o t h i s ( rebound w a l l ) , and i f i t ( b a l l ) goes i n on t h i s ang le here (zJ)» i t must come out (rebound) a t the same ang le (/_R). 6. Uhm, w e l l you want t o f i n d out what ang le i t comes i n o n , and i f you t ake — . W e l l i f t h i s ( w a l l ) was t u r n e d around a b i t — w e l l , i t a lways has t o be p e r p e n d i c u l a r t o t h e s u r f a c e . 8. Then the p e r p e n d i c u l a r would come out a t an ang le l i k e t h i s ( d e m o n s t r a t e s ) . So then the angle ( / I ) would be g r e a t e r and i t ( /R) would come ou t — g r e a t e r . 10.;Uh — y e s , i f something comes i n on a c e r t a i n a n g l e , p r o v i d i n g the s u r f a c e i s f l a t , i t i s . I f i t h i t s here i n waves , or something ~ i t - w i l l come o f f a t a d i f f e r e n t a n g l e . 12. W e l l , I ' m n o t a v e r y good shot ( l a u g h s ) •uhm — W e l l , i f you were t o — i f t h i s (board) was g l a s s or someth ing , ( t a k e s a p i e c e o f paper) and you p u t t h i s ( p a -per ) underneath and drew a l i n e out p e r p e n d i c u l a r t o i t ( rebound w a l l ) , and you drew a s t r a i g h t l i n e w h i c h would meet a t t h i s p o i n t here ( l i n e o f i n c i d e n c e ) , t h e n get a p i e c e o f carbon paper and p u t i t on t o p o f the paper , and see the p a t h t h a t the b a l l makes on the carbon paper when coming o u t , and you measure those two a n g l e s , and t h e y s h o u l d be e q u a l . 15. And what about the angles, are they s t i l l the same? 17. And the other way? 19. OK, what about the other angles? 14. You have to change the position of your target too. 16. Well the angle that i t h i t s , i t w i l l be wider, — but i t w i l l h i t some-where over here (to right) so you w i l l have to bring your target over here (to r i g h t ) . 18. I f you moved i t the other way, the angle i s getting smaller u n t i l you got i t perpendicular with the (re-bound wall) i n which case i t w i l l go and come ri g h t back again and your target would have to be even with i t ( i . e . the launcher). End of Tape. TRANSCRIPT OF STUDENT PERFORMANCE ON THE OSCILLATION OF THE PENDULUM TASK: • STUDENT L.W. Interviewer Student 1.(Shows apparatus, and poses the problem.) , 4, What did you mean by "C" has a smaller distance to go? 7. You don't need to do i t too high. 9 . Can you j u s t i f y that? Are you happy with it? 2. You put them (the pendula) both the same length (adjusts string) and try to determine i f weight i s a factor. Take a l i g h t and heavy one. This shows i t most obviously. Uh -- you p u l l i t back —"'How sh a l l I determine this? (Uses ruler as starting point). 3. They're both turning at the same time. This one "C" ( l i g h t bob) has a shorter distance to go, but they're s t i l l turn-ing at the same time, so the period i s s t i l l the same. 5. Well, this one (heavy bob) i s s t i l l going further so i t ' s going faster. This one ( l i g h t bob) i s going a smaller distance. 6. I f you want to determine — to see i f the length does anything — now l e t ' s see — (begins shortening) — Well, you could t r y the force, but this wouldn't be very accurate because you don't know i f your forces are the same. (Puts a heavy bob on both pendula). 8. (Pushes pendula equally). That was the same force, and they are travelling at very much the same amplitude — I mean (Drops l e f t pendulum, pushes right pendulum) Let's t r y i t . (Unin-t e l l i g i b l e remark of surprise) They both turn at the same time, but I thought ; that — force had something to do with i t t They're both turning at the same time, so force must be — I didn't put any;force on this one ( l e f t pendulum), and this one I did (right pendulum). 10. Uhm (thinks) No, not r e a l l y when — I guess sot I f you use F = ma — you had more force — so acceleration i s greater. 11. Well, have we finished? 13. (Suggests S use a ruler i f she wishes). 15. How do you know that? 17. Uh hum. 19. Uh hum. 21. Something else? 23.( Indicates that only i n i t i a l swings should be considered.) 25. OK - that makes sense does it? 27. And the force? 29. Can you use the same explanation to j u s t i f y forcet Masses are the same — so that should mean they are equal i n period, l i k e they both travel at the same time. I can't remember any other — . 12. No, we have length and (removes weights) amplitude. 14. (Shortens r i g h t pendulum then lengthens.'' i t to make i t equal the length of the . l e f t pendulum). Measuring, amplitude, you need the same weight. 16. Well, when you're doing experiments, we already said that mass had nothing to do with i t , so you should — well, you see — this i s your control. Your control should be always constant. And this should be the one to vary ( i . e . right pendulum), and. i f you're comparing them and not the aotual weights, — , something else — i n a system, these (lengths) should be the same. So that you can compare them. 18. So, p u l l this (right pendulum) back to a further amplitude. (Swings them and watches). 20. Well, they're turning at the same time so the period must s t i l l be equal. 22. (Swings weights again giving the right • pendulum larger amplitude). This one (right pendulum) i s taking longer now, but you can't t e l l i f you've given i t a tiny b i t of force or not. I said force didn't have anything to do with i t either. Let's see, uhm. (Swings them again). 24. Well this one (right pendulum) has a greater amplitude but i t has a greater speed too so that i t ' s — You can just-i f y i t that way that they both have the same period. 26. Uhm. • 28. What did I say about the force? I t didn't — i t didn't matter about force. 30. Uhm, Well, we used the same mass. I f you use same mass, then mass i s constant — You're applying a greater force to 32. Shorter the string, the greater the period? 34. Uh hum. (Noncommital). 36. OK. So now can you summarize? 38. Was that confusing you? this , so the acceleration w i l l be greater within a certain period of time. I f you use F = ma, and the force i s greater and the a i s greater, m i s constant, then i t should be the same r a t i o . (Shortens right pendulum and puts on both heavy weights. Allows both weights to drop). 31. I'd say, the shorter the string\the greater the period. 33. Uh hum. (Watches) — Well, the shorter the string — Oh, i f t h i s (right pend-ulum) was half that ( l e f t pendulum) i t should take half as long for t h i s (right pendulum) to get back to the point as this one does ( l e f t pendulum). 35. I f they (the pendula) are down here,, with the same length and same weight you get the same period. I f th i s one i s shorter, the faster i t goes. 37. Force, no difference. Mass, the same as long as length was the. same. Amp-litude, didn't make any difference. Length of string did make a difference. I t frustrates me because I can't remem-ber. I know we take this stuff i n Physics to determine what's dependent on one another and when you can't remem-ber i t — I f you're doing that for the f i r s t time I think you'd accept what you saw, but knowing i n doubt before, I was just wondering -39. Yes — uh hum — I think this i s — I think i f you've had i t before but never learned i t thoroughly enough to remember i t , well then, i t confuses you the second time. TRANSCRIPT OF STUDENT PERFORMANCE ON THE COMBINATION OF LIQUIDS TASK: STUDENT L. W. Interviewer 1. (Shows apparatus and poses question). 3. I'm not t e l l i n g you exactly where i t comes from! 5« I t w i l l go pencil yellow, darker than that. 7. Yes. 9. Yes, that's positive. H i ' Are you quite sure about it? 13. Uh hum (noncommital). 8. 10. 12, 14. 15* Is that the only way you could get the color? Is there anything else you could t e l l me? Could you t e l l me i f any of the others are important as well? 16, Student Shall I do it? - You took i t (the liquid) from one of the beakers? (Adds each l i q u i d singly i n beakers. Adds indicator to each. Adds more g to (1 + g). I'd say i t was this one, (1 + g). (Adds more indicator and shakes each). I'm assuming that i t ' s changing — when — Can I use as many of these ...? (Tries ((1 + g) + 2 + g)) and ((1 + g) + (3 + g)). Is that the colour? Then i t ' s a combination of this and this and this and plus indicator ( i . e . ( 1 + 3 + g)). Uh — Well, I could t r y the l a s t one. ((1 + g) + (4 + g)) I'd say those, ( 1 + 3 ) and the g ( i . e . ( 1 + 3 + g ) ) . . I ' l l see i f i t doesn't turn when I mix them up. (Tries ( 1 + 3 + g ) again and gets color). 17. Uh hum — Well, I mixed the other two i n these combinations here (1 + 4 + g)., and (1 + 2 + g) and they didn't work. And here I have the indicators i n by them-selves ((2 + g) and (4 + g)) and they didn't work as well. So i t had to be a combination and i t was this combination that turned i t (1 + 3 + g). 18. Just say for the argument that i t hadn't turned with (1 + 3 + g) , what would you have done then? 19. 20. How many combinations? 21. 22. Could you write them out? 23. You mean on mixing? Well, keep going on different combinations to see which one works. (Thinks) Uhm, forgotten how you figure that Writes: (4 singles, 6 doubles, 4 t r i p l e s , 1 a l l four) So that's 15. 24. 1 5 ? 26. A l l right, good. Well, what about liquids 2 and 4? What can you t e l l me about the p o s s i b i l i t i e s from what you've done a l -ready? 2 5 -28. Well, we haven't any tech-nique for identifying acids or bases. So, l e t ' s see. On what would you base your hypothesis that they both ( 2 and 4) have nothing to do with the reaction? Do you f e e l that what you've said proves i t conclusively? 3 0 . Uh hum. Can you distinguish between 2 and 4? Yea - 4 singles, 6 doubles, 4 t r i p l e s , and 1 a l l four. 27. Well, when this one (4) doesn't react with g, and whatever you were trying for i s not i n this one (2) and you also know that when you mix the 2 together that there i s no reaction shown off by the g. So there's nothing of what you're trying to determine i n any of those two. Also, depending on the g you've got, acid,base, or water. 2 9 . 3 1 . . 3 2 . (Corrects S for she i s naming wrong beakers) 3 3 . Can you make any predictions? You've got 16 possible combin-ations there - could you pre-d i c t for each one what the re-action would be? Yow might have to do other — They ( 1 + 3 ) might contain something and you'd need something to set off the reaction. 2 and 4 may need another l i q u i d or something to mix i n with i t — just to set off the reaction. This doesn't prove anything. You'd have to do other experiments. As they are, no, I don't think so. With another indicator, you might be able to. (4 + g) has turned off clear (cloudy). ( 2 + g) i s completely, clear. This (4 + g) i s not much changed, i t ' s off-color i n ( 2 + 3 + g) but not enough. I suppose you could distinguish them i n that way, but i t ' s not enough for what we want. I t may have been that I put too much indicator i n , or more of 2 or 3 than I should have. But there's not enough change to say any-thing different about them. I would have to do further tests. 34. Well, anything with 1 + 3 + g would turn yellow, and there'd have to be the combination of 1 and 3. Anything with just 1 + g would turn a tiny b i t yellow. With just 3 i n i t , or 2 or 4, i t would be clear. I think that's . a l l . TRANSCRIPT OF STUDENT PERFORMANCE ON THE BALANCE TASK: STUDENT L.W. Interviewer 1. (Shows apparatus and explains problem.) Set up a balanced, system. 3. Just so you can balance both sides. 5. What have you got there? 7. Uh hum - Why did you do that? 9. Yes, i t ' s a crude apparatus . - don't worry about that. Can you explain what's going on here? 11. Does i t matter where you put it? 13. Give me some more variations now. 15. Do the weights change? Student 2. This - uhm - so you can determine - ? 4. Uhm. (Tr i e s ) . 6. (Puts 200gm x 8L = 200gm x 8R and adds a lOgm x 7L). 8. Before, the balance was s l i g h t l y this way (slanted to right) and I was wonder-ing - now i t ' s gone - this side i s heav-i e r but that's because this i s closer to the end. (S concerned about small un-balance i n apparatus). 10. You've added more weight to one side so you have to add more equal weight to the other side to keep a balance on the fulcrum. I t doesn't matter how much weight you add but you have to add i t the same to both sides to balance. 12. No, i f you put i t here on this side (R) you have to put the same distance on this side (L). I t should be same weight and distance from centre. 14. I f you want to prove distance you put this further out. That's more weight - no balance - That's why i t has to be the same distance. 16. No, the weight hasn't changed. There's more weight at end of balance — what's pulling i t down i s — see, you've no longer got the fulcrum in the centre of the system, so i t w i l l be unbalanced. It's as i f you've got more weight on this side. So you've got to put i t to the same distance. Doesn't matter how close you bring i t i n to the fulcrum. I t has s t i l l always got to be the same (dis -tance); i f you had — i t doesn't matter hovr small weights. Size of the weight shouldn't matter i f i t ' s balanced. (Moves weights to various equal positions on ruler, and keeps a balance). 1?. Can you set up something using different weights on each side? 19. Just explain what you're going to do. 21. Uh hum. 23. Uh hum. 25. OK. 27. Is i t possible to use different weights? Can I put 100 gm on my side (L) and 50gm on your side (R)? Could you work out a balance. 29. Is this a general rule? 31. (Releases balance and the le f t side drops). Can you see what's the matter? Can you adjust l e f t side now? Any reason? 33. Fine (Takes off weights) I'm going to use 40 gm, how about that? (Puts on 40gm x 5 L and gives S 5°gI<i to balance on other side)• 18. You can have — (puts $00gm x 9L = ((200gm + 200gm + lOOgm x 9(R)). 20. Well, you can add i t — You can have 200gm + 200 gm and another lOOgm to make up the 500gm, but you can't have the — you can't put them in different places. It has to be the same place to do i t (balance). 22. You can have any combination you want to make up the weight on the opposite side but they have to be the same dis-tance. 24. And i f you wanted a 400gm on this side (L) and make i t equal, you could put the other 200gm there to make i t equal, and i t wouldn't matter, (i.e. 400gm x 9L = (200gm + 200gm) x 9 R ) ) . 26. I f you're going to have a l l the weight on one (L) hook - you have to have i t a l l on one hook here (R) too. 35• Are you.sure? •28. Well, i f you put 200gm here at 9L and lOOgm x 5R — no! — the other way around (i.e. 200gm x 5L = lOOgm x 9R) i t should work. The 200 lbs. on this side (R) is at half the distance and this one (lOOgm x 9R) here has half the weight and twice the distance from the fulcrum. It should s t i l l balance. 30. Yes (thinks) It should be... 32. No, i t not (a general rule)j — uhm (thinks, then changes balance to lOOgm x 8R = 200gm x 4L). Well, before this (R) was 9 and this (L) one was 5 - so this (L) was more. . 34. (Hangs 50gm at 4, i.e. 40gm x 5L = 50gm x 4R). 36. It should be — uhm (Thinks). 37. Any reason? 39. Yes? 41. There i s a formula for thi s , but l e t ' s not worry about formulae. Can you explain to me i n terms of ra t i o s . . . I think you've probably got i t . Let's give you another p o s s i b i l i t y . Put this one there (30gm x 2L), can you balance that without being exactly symmetrical? 43. How do you know? 45. Just for interest, i f I were to start counting from the edges (1, 2, 3, 4, 5, 6.) would that be important. 47. Yes, i s that important? 49. Another thing - could we swap these, my 30gm and your 20gm? Would that be alright? 51. Oh - you have to change the distance? 38. I think so. Well, you've got them i n the same rat i o , this i s 40gm at the 5th hook which equals 200, and 50gm at the 4th hook which equals 200 as well. 40. Distance and mass should be i n bal-ance. 42. That's 30gm ... (Puts 20gm x 3 R ) • 44. You have the same rat i o of weight. This one i s 30gm at 2(L). Here 20gm at 3(R). It's the same r a t i o , l i k e the greater the distance, the smaller the weight that you need to balance. 46. No - i t ' s the distance from the f u l -crum. 48. Distance from fulcrum i s . I t doesn't matter what you've got out here. 50. You'd have to treat them l i k e ... (changes distance so that 20gm x 3L = 30gm x 2R). 52. Yes. 1. a b 2. a b 3. a b 4 . a b PHYSICS 110 EXAMINATION PAPER ' UNIVERSITY OF BRITISH COLUMBIA APRIL 1970 Give a definition of FORCE (Classical Mechanics), as an equation: Explain the symbols used: Give a definition of MOMENTUM (Classical Mechanics), as an equation: Explain the symbols used: Give a definition of WORK (Classical Mechanics), as an equation: Explain the symbols used: Give a definition of ELECTRIC POTENTIAL DIFFERENCE, as an equation: Explain the symbols used: 5. An object moves on a circle with constant speed. a) Give an equation for the force required to keep i t in orbit: b) Explain the symbols used: 6. Give a relationship between the wavelength A» the frequency f, and the speed of propagation v of a wave: 7. When the potential energy of a system increases there w i l l also be an increase in mass. Give an equation governing this mass increase: 8. Somebody claims that telepathic signals are a special kind of waves. T'fhat general kind of experiment would he have to perform to demonstrate that they actually are waves (in the meaning of the word "waves" as used in physics)? 9. Nuclear Energy can be converted into heat by nuclear fusion as well as by nuclear fission. Could one not make the best use of these processes by f i r s t splitting atoms (nuclear fission, heat w i l l be produced), and then re-uniting the parts again (nuclear fusion, heat w i l l be produced)? Repeating this cycle over and over again, one would have an inexhaustible energy source. Explain in terms of the binding energies of nuclei, why this process is impossible. YOU HAVE THE CHOICE TO OMIT QUESTION 10 OR QUESTION 11 10. A radioactive sample explodes in a laboratory. Immediately after the explosion a Geiger counter in the room records $60 counts/sec. One day l a t e r , the same counter records 2U0 counts/sec. Assuming that a safe level of radiation as indicated by the counter would be 1 count/sec, what would be a reasonable estimate of the number of days since the explosion for people to safely re-enter the laboratory? (Give reasons with your answer). YOU HAVE THE CHOICE TO OMIT QUESTION 10 OR QUESTION 11 11. A microphone stands at some distance from a wall. At some greater distance, a loudspeaker emits sound. The frequency of the sound i s steadily increased beginning from zero, while the intensity of the sound i s kept constant. When the frequency i s below 1,500 cycles/ s e c , sound w i l l be picked up by the microphone. At 1,500 cycles/ s e c , no sound i s received. When the frequency i s increased further, the microphone w i l l pick up the sound again. a) Explain this phenomenon and b) predict at which higher frequency there w i l l be the next minimum so that no sound w i l l be received by the microphone. b) ANS: NEXT MINIMUM OCCURS AT: cycles/sec. YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 12, 13 OR lU 12. An indirect way for measuring currents (frequently used to measure strong current pulses) i s to pass the current, Ip, through the primary c o i l of a transformer and to display the secondary emf, s , with an oscillscope. The current begins to flow at time + . The oscilloscope trace ^_ of the secondary emf,s looks like t h i s : After careful consideration, give a qualitative graph of the primary current Ip. YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 12, 13, or Ik 13. A proton moves, with steadily increasing speed, along a straight line (x axis of a coordinate s y s t e m ) ! T h e only forces involved i n this motion are caused by el e c t r i c and/or magnatic f i e l d s . These el e c t r i c and/or magnetic fields are constant in time and homogeneous in space. Which one or more of the fields l i s t e d below has to be, or could be, present? (disregard signs, "along x axis" means: i n direction of + or -x) There has to be an e l e c t r i c f i e l d along the x axis f~J There could be an e l e c t r i c f i e l d along the x axis f~J There has to be an el e c t r i c f i e l d along the y axis /*~7 There could be an electric field along the y axis / / There has to be an electric field along the z axis / / There could be an electric field along the z axis / / There has to be a magnetic field along the x axis / / There could be a magnetic field along the x axis / / There has to be a magnetic field along the y axis / / There could be a magnetic field along the y axis / / There'hasto be a magnetic field along the z axis / / There could be a magnetic field along the z axis / / If you would rather answer in a different way, please do so on the back of the preceding page. YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 12, 13 OR 14 14. Assume you bought a 100 Watt A.C. power supply specified to supply cur-rents at a frequency of 50 cycles/sec. How could you test for the fre-quency using nothing else but some wire, and a calibrated stroboscope with adjustable frequency? (Please explain your answer with the aid of a drawing.) YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 15, 16 OR 17 15. (A Spy vs. Spy episode) Passing the Black Spy's spaceship with a rela-tive speed of 801! of the speed of light, the White Spy triggers a time bomb hidden in the Black Spy's ship. If the explosion is to occur 1,000 m distance from the White Spy's ship, at what time interval should the time bomb be set? YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 15, 16 OR 17 16. Spaco explorers discover a ring of charged particles orbiting around a mysterious cloud. The ring consists of positive hydrogen ions and nega-tive oxygen ions,.circulating in the same direction. The speed of the hydrogen ions is 1 km/sec, the.speed of the oxygen atoms is 2 km/sec, the radius of orbit is the same for both kinds of particles. The number of particles per cubic meter is too small to allow the ions to combine. For the same reason, no electric or magnetic forces between the ions could account for the motion. The explorers discuss the following explanations to account for the ci r -cular orbits of the ions. Try to rule out as many of these explanations as possible. a) The circular orbits are due to gravitational attrac-tion by a massive star within the cloud,/-/could be, J~~] cannot bo. Givo reasons for your choice. b) The circular orbits are due to a charged object hidden in the cloud. f~~j could be, j~~J cannot be. Give reasons for your choice. c) The circular orbits are due to a magnetic field at right angles to the plane of the orbits. f~~f could be, [~~Jcannot be. Give reasons for your choice. YOU HAVE THE CHOICE TO OMIT ONE OF THE QUESTIONS 15, 16 OR 17 17. Design a device to (indirectly) measure the wavelength of a given ultraviolet spectral line. No use may be made of interference (as e.g. by using gratings, s l i t s , standing wave patterns) or of refraction (e.g as by using a prism). Give a drawing of your design in sufficient detail. Explain what you observe with this device, and how you obtain the wavelength of the spec t r a l line from your observations. BASES OF SELECTION AND REJECTION OF EXAMINATION ITEMS Examination Item . . S e l e c t e d . . . Rejected Reasons f o r S e l e c t i o n or R e j e c t i o n 1 - T + Simple RECALL o f formulae . 8 + Student r e q u i r e d t o r e c a l l i n f o r m a t i o n on the p r o p e r t i e s of waves i n g e n e r a l . The student i s r e q u i r e d t o s e l e c t the appropr ia te r e c a l l e d i n f o r m a t i o n t o f i t the f a c t s - RECALL. 9 + Involves FORMAL OPERATIONS. Item con-cerns h y p o t h e t i c a l q u e s t i o n i n v o l v i n g n u c l e a r energy 10 + FORMAL OPERATIONS NOT INVOLVED 11 + FORMAL OPERATIONS NOT INVOLVED I n v 0 l v e s FORMAL OPERATIONS. Involves c o n v e r s i o n o f secondary emf i n t o pr imary c u r r e n t . I n s u f f i c i e n t a n a l y -s a b l e - m a t e r i a l i n student response . 13 + Involves FORMAL OPERATIONS. Involves s e l e c t i o n o f c o r r e c t response t o g i v e n c o n d i t i o n s . I n s u f f i c i e n t analysable m a t e r i a l i n student response . Ik + Involves FORMAL OPERATIONS. Involves design of method f o r t e s t i n g frequency of current under p a r t i c u l a r c o n d i t i o n s . Student L.W. d i d not respond. 15 + Involves FORMAL OPERATIONS. Item "based on understanding o f r e l a t i v i t y and a c a l c u l a t i o n i n s u f f i c i e n t analysable m a t e r i a l i n student response . 16 . + Involves FORMAL OPERATIONS. Involves s e l e c t i o n o f appropr ia te hypothes is t o f i t g iven s i t u a t i o n . 17 + FORMAL OPERATIONS NOT INVOLVED. 

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