PME (20). Study on the evolution of graduate students' concept images

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(xDf ( |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.
QI=.
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). Les principaux obstacles à l´apprentissage de la notion de limite, Bulletin, IREMA.P.M.E.P., Grenoble.
Delgado, C. (1995) Estudio de la evolución de los esquemas conceptuales de los alumnos
universitarios en su proceso de aprendizaje de los conceptos de límite y continuidad. Tesis de
maestría. Universidad Autónoma de Barcelona.
295
Habermans, J. (1984). The theory of communicative action. Vol.1. Reason and the rationalisation of
society. Boston: Beacon Press.
Piaget, J. & alt. (1981). Epistemología genética y equilibración. Fundamentos. Madrid.
Sierpinska, A. (1985). Obstacles épistémologiques relatifs à la notion de limite. Recherches en
Didactique des Mathématiques, Vol 6, Nº 1, pp. 5-67.
Sfard, A. (1991). On the dual nature of Mathematical conceptions: Reflections on processes and
objects as different sides of the same coin. Educational Studies in Mathematics. 22, Nº4, pp.1-32.
Tall, D. & Vinner, S. (1981). Concept images and concept definition in mathematics with particular
reference to limits and continuity. Educational Studies in Mathematics, Vol 12, 151-169.
Tall, D. (1994b). A versatile theory of visualisation and symbolisation in mathematics. Comission
Internationale pour l'Etude et l'Amélioration de Enseignement des Mathématiques, France.
Tall, D. (1995). Cognitive Growth in Elementary and Advanced Mathematical Thinking. Plenary
Lecture at the Annual Conference of the International Grup for the Psychology of Mathematics
Education, Recife, Brasil, July 1995.
Toulmin, S. (1972). Human understanding. Harvard: Oxford University Press.
Vygotsky; L. S. (1962). Thought and language. Cambridge: MIT Press.
Wertsch, J. V. (1985).Vygotsky and the social formation of mind. Harvard: Oxford University Press.
296