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STUDY ON THE EVOLUTION OF GRADUATE STUDENTS' CONCEPT IMAGES WHILE LEARNING THE NOTIONS OF LIMIT AND CONTINUITY1 César Delgado García. Departamento. de Matemáticas. Universidad del Valle, Cali, Colombia Carmen Azcárate Giménez. Departamento de Didáctica de la Matemática. Universidad Autónoma de Barcelona, Spain. Abstract This is a report about a case dealing with the evolution of students' concept images as related to the definitions of Limit and Continuity. The theoretical ontogenetic-sociocultural and pedagogical framework defines analytical units in order to scrutinize the student's development and concept image as related to the initial and potential cognitive stage. Teaching situations are used to originate cognitive conflicts, and the evolution of concept schemes during the process of teaching-learning is analysed. Conclusion is reached which tell us of the feasibility of setting out pedagogical situations which allow for controlling the conceptual change and for an efficient relationship between learning to cognitive development. 1. Introduction Research hereby presented is linked to the pursuit started in 1986, at the Universidad del Valle (Cali, Colombia). A teaching-learning strategy was devised and applied for Calculus I course, aiming at achieving a comprehensive learning of concepts. The aim was to have the target subject establish relationships between the new knowledge object and elements already established inside his cognitive structure, all done in an essential and coherent manner. This pursuit led to Delgado's (1995) research, subject-matter of this report. The intended problem to be investigated is the following: In a learning situation, is it possible to control the cognitive structure evolution of the student, as related to conceptual difficulties and to cognitive hindrances arising in connection with the notions of Limit and Continuity? Is it possible for a pedagogical sequence where conceptual difficulties and cognitive hindrances are taken into account, to be efficient enough as to boost the evolution of cognitive structure related to those notions? We cannot find any study facing this problem within the available literature. Research is focused on the study of written outcomes from a student considered as a member of a group. These outcomes are structured according to a guide acting as a booster for situations capable to provoke cognitive conflicts. This conflicts are generated, regulated and strengthened by means of the intercommunicating action of debate in the classroom (interpersonal processes) and are internalized by means of the individual cognitive processes: assimilation, accommodation and balancing (inner-personal processes). Activities, as much as th Proceedings of 20 Conference of the International Group for the Psychology of Mathematics Education, , Vol 2, pp. 289-296, (1995). Valencia. España. 1 289 objects and methods, are common for the group, and the group states its requirements and regulates actions, taking into account theoretically established basic principles. As a consequences of consecutive coordinations between the different concept images, the student is expected to manage encapsulating the definition of continuity inside the following terminology: (>0)(0(xDf ( |x-p|<|f(x) - f(p)|< 2. Theoretical Framework Works by Piaget (1981) provide with elements of analysis for the study of conceptual change: structure, outline and its relationship with assimilation processes, accommodation and balancing. This last process gives reasonable explanations about how and why the subject's cognitive enhancement is produced. Balancing is the mechanism which maintains cognitive organization and it appears because of disturbances caused by the subject's activity, and because of compensations the organism opposes to disturbances achievement of various new possible conceptions and coordination among what is possible, necessary and real. As a process, balancing is the factor unifying conceptual change and development adaptation . As a state (in equilibrium) it is the continuously changing balance of active compensations selection. This theory eases the analysis of the subject's outcomes from the stand point of inner processes (inner-personal), but ignores the external processes (inter-personal) affecting the cognitive development. We complement the ontogenetic approach by adding Vygotsky's (1962) socio-cultural approach, while always keeping in mind Habermas' (1984) and Wertsch's (1985) updates, paying special attention to concepts like mediated action, measure instruments and close development zone: "Vygotsky argued that, starting at a specific moment of development, biological forces cannot be seen as the only, or even the main, changing force; a fundamental arrangement in development forces is brought about as well as a need for its corresponding rearrangement of principles explaining the system [...]. The center of the explanation switches from biological factors to social factors," (Wertsch 1985). These theories provide with analytical units in order to study conceptual change from an ontogenetic and socio-cultural standpoint. We will use the definition of "concept image" (C.I.) (Tall and Vinner, 1981; Azcárate, 1995) when talking about those elements of the cognitive structure put into action in connection to mathematical notions. We define conceptual change as the consequence of solving a conceptual problem derived from a mistaken or insufficient C.I. in front of the "ideal explanations" required to face the situation. Conceptual change comes from awareness of conflict in recognizing C.I.'s limitations to confront the need to find the "rational explanations" required by the group taking part in the teaching pursuit. This definition takes into account Toulmin's (1977) views and links socio-cultural and ontogenetic domains. 290 We use the above definitions, units and theories in order to study the following variables: conceptual change (selection) and balancing (selection and adaptation). As far as epistemological hindrances linked to the notions of limit and continuity are concerned, we consider Brousseau's (1983), Sierpinska's (1985) and Cornu's (1986) researches. As for the development of advanced mathematical thought, we contrast thesis by Sfard (1991) against Tall's thesis (1994, 1995), on the role of algorithms and concept formation in mathematical thought, adapting Tall's notion of "procept" in order to explain the strategy in constructing the (-) definition of Limit. 3. Research Method This research consists of a case study based on a student's written outcome in a learning situation, and corresponds to the introduction of the notions of continuity and limit. Tools used in the classroom are: Text (T. Apostol, Calculus) and a guide. Data have been analyzed from: Systemic Networks (figure 1 presents whatever concerns continuity) Protocols: for the guide and for student's answers A chart of codes, where the student's reactions are recorded and coded in terms of: C.I., applications of C.I., conflicts, difficulties, epistemological hindrances and interpretations of instructions. A diagram of C.I. evolution, developed from the chart of codes. C1 :if his graph can be draw by a single line stroke C3: if when x V(p) f(x) V(f(p)) Geométrical Image < C : if for all V (f(p)) there exists V(p) so that 6 Continuity (of f at x=p) f(x) V(f(p)) x V(p) < C : if when x se approaches p, f(x) approaches f(p) 2 Analytical image< C4 : if | x-p | < | f(x)-f(p) | < Imagen >0 existe >0 so that C5 : if for all | x-p | < | f(x)-f(p) | < Figure N º 1: S ys temic netw ork The guide includes 25 situations, through which questions are submitted, and comments, hints 2 and instructions are offered. Eg. G1 shows a drawing of a ball rolling down over an inclined plane, then going on over a horizontal plane and lastly climbing up a final plane; then it asks to draw the corresponding graphs for the speed and acceleration of the mobile object and to 2 Guide´s situation number i (i=1,2,3,....,25) will be noted as G i. 291 characterize those graphs at the precise points when movement changes. Attention is drawn towards the aspect we are interested the student to define. G 1 (d) What are the s peed v(t) and the acceleration a(t) graphs like at the ins tant t1 w hen the ball cros s es B? At the ins tant t2 w hen the ball cros s es C? A mathematician w ould s ay: • "S peed function is continuous at all points , w hile the acceleration function is not continuous at t 1 y t 2 . " G 3 . Write down your idea for the definition of a function’s continuity at point x=p. G5. cons ider the following functions: f(x)=x ; g(x)= x x Q Using your own 2 x I definition, determine whther: a) f is continuous at x=1 b) g is continuous at x=2 Hint: cons ider two s ituations . One where f will be continuous at x=p and other where f will not be continuous . Note what value f(x) tends to when x approaches p. Situations such as those stimulate the appearances within the group of distinct Cis (Fig.1). Simbolic Simbolitation Analitic g( x) = ‹ x x Q 2 x I 2 Visual Logical Defined concept: C1 2 Geometric Verbalisation Cognitive processes take part in the generation of conflict. Assimilation: in drawing the graph, pencil is uplifted. Conflict Reality: f is continuous at x=2. Accommodation: if pencil is uplifted, it is not continuous. Conflict amplifies the world for possible: is erroneus. –Response –It is broken but not –It is broken but f(x) values should be close to briskly. f(p) Previous facts lead to construct what is need: –Graph should not brake. –Braking should not briskly. –f(x) should take values close to f(p) wen x takes values close to p. Possible, necessary and real are coordinated to counterbalance disturbance and to install equilibrium: Beta equilibrium: Beta equilibrium: Alfa f is continuous at x=p if, when x –Ifequilibrium: it brakes is not continuous. –when braking is not brisk, f takes values close to p, then f(x) is continuous –Response is closed. takes values close to f(p). A gama equilibrium corresponding to C5 or to C6 is not to be expected. Figure N º 2: S ituations and cognitive proces s es is not to be expeted 292 Debate in the classroom calls for the subject's definition to account "rationally" for every situation. Thus, in case C1 (Fig 1) is stated in response to G3, its transposition to situation G5 will not withstand debate inside the group. C1 is expected to induce a conflict in considering that function g graph cannot be drawn "in a single line stroke" even though it is continuous at x=2. Conflict comes as a result of cognitive processes originating awareness of C1 global geometric C.I. limitation to account for the situation. This should lead to construct C2, local analytical, concerning "nearby" numerical values, only suggested in G5 hint. We illustrate (fig. 2) the constructive process, following Tall-Piaget's theory. Each equilibrium state is controlled by the "rational" requirements of the group at each one of the 25 situations, thus generating a dynamics for conceptual change which can be perceived through the student's written outcomes. 4. Instance from the analysis As a result of interaction with G1 and G2, the student writes about a G3 the following statement "F is continuous when we do not observe brisk breaks in its values. Its graph can be drawn without uplifting the pencil Protocol of response to G3 This definition corresponds to C1 (geometrical, global); C.I. is activated by the graphical context of the situation. In G4, question is about continuity of the function "minor integer" at x=1/2 and x=1; the student applies properly his own definition C1 for the affirmative and negative cases. This is a control question. In relation to G5, the student wrote: a) b) f(x)=x 1 g(x)= ‹ x 2 x Q x I 2 1 Function f(x) is continuous at point x=1 since it does not show any brake. Protocol of res pons e to G 5 2 ¿is g continuous?. (a) Draw the graph and apply C1 properly. (b) He does not respond. He only asks himself: is g continuous? Application of negative C1 is empty. His graph does not correspond to the one defined by the algebraic expression of g. The required conflict (P1) between "continuity without brake" and "continuity with brake" to evolve from C1 to C2 does not emerge. 293 C2 definition has not emerged. He has not followed No. 5 instructions. 5. Results From analysis of the first five situations we gather: 1. The student has understood and assumed the task of constructing the definition 2. Initial C.I. corresponds to C1 and the student does hold another C.I. 3. There exists a conceptual difficulty concerning the interpretation of the algebraic expression of a function defined by "portions" 4. The student became aware of C1's limitation but was incapable to forward an alternative 5. Hints given in G5 came out being inoperative; this suggests the expected conflict does no sit inside "area of immediate development". The student does not possess the required theoretical elements to with P1. Similar analysis to those above allowed to extract from the rest of the situations results such as: 6. The interpretation error about the algebraic expression of a function defined by portions is due to the C.I. stating that “the expression of a function in portions is a simplified notation way to actually note two functions”. (CI1) The student is not aware of inconsistency between CI1 and the function definition [f(p)=y1≠y2=f(p)]! Here CI1 appears as an epistemological hindrance linked to the notion of function! (O1F). This has prevented the emergence of the required conflict (P1) to evolve from C1 to C2. 7. The C.I leading to believe that "Qs 'fill' the straight line" or that "Is 'fill' the straight line" is an epistemological hindrance O2N which prevents "seeing" the following characteristics of real numbers R: Rational numbers, Q, are dense/compact within Irrational numbers, I. Irrational numbers are dense/compact within rational numbers. QI=. Q I = R. The student needed first to compensate the disturbances occurred as a consequence of conceptual deficiencies such as: definition of a function in portions; definition of the notion of function, algebraic expression for f and its graph, and numerical structure of the straight line. These deficiencies were linked to the epistemological hindrances O1F and O2N. Both deficiencies and hindrances prevented conflict (P1) between "braking" and "proximity" to appear in considering question No. 5, as would have been required to evolve conceptual scheme, hereby known as C1, towards C2 scheme; the student only sees the ideas to be conflicting starting from question No. 7, in as much as he has developed and balances his conceptual schemes related to the notions of function and of real number. At the end, the student managed to achieve beta equilibrium for C2. Evolution towards C5 and C6 could no be reached because of the presence of other hindrances, already highlighted by Sierpinska (1985) and Cornu (1986), and confirmed through Delgado's (1995) research, although not analyzed in this report. 294 6. Conclusions This research infers the possibility of setting up pedagogical situations which allow to keep control over the student's conceptual change, thus performing an efficient teaching through the linkage of learning to cognitive progress. The student manages to improve his mathematical foundations by applying the guide, and gaining awareness of conflict within the situations generated by such a tool. The student's behavior unveiled: that he identifies the value of his own outcomes and errors, of academic discussion, of mathematical rigor, of reasoning and of mathematical prove. This identification generated a set of requirements which developed his motivation for an active involvement in mathematical debates and construction. This motivation was to be seen due to maintaining and improving an activity focused on setting out some practical relationships between the following mathematical structures, theories and approaches: structure of real numbers ordered field and numerical structure of real straight line; theory of sets and theory of functions; mathematical logics and proving methods. The systemic network used for establishing the defined notions the students would be likely to formulate, proved to be a very useful tool for tracing conceptual changes and in the setting out of situations for conceptual conflict. Evolution diagrams were designed to allow identifying conflicting situations and centering attention on the elements involved in them, or else, in case no conflict would arise, the reasons preventing it to do so were identified. Within those diagrams, level of balancing of conceptual schemes at the end of the process was also observed. The chart of codes eased the ranging of students' outcomes in view for a further analysis. Bibliography Azcárate, C. (1995). Sistemas de representación. Uno, No 4, pp. 53-61. Brousseau, G. (1983). Les obstacles epistémologiques et les problèmes en mathématiques. Recherches en Didactique des mathématiques, Vol.4, Nº 2, pp. 165-198. Cornu, B. (1986). 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