VISUALIZATION IN ITERATIVE PROCESSES
Flor M. Rodríguez, Gisela Montiel Espinosa, Ricardo Cantoral Uriza
CIMATE-UAGRO1, CICATA-IPN2, CINVESTAV-IPN3
The visualization topic in the classroom is important and interesting for the
community of didactics of mathematics because it appears in several
researches like an element that helps and allows to go and to come between
the graphical and algebraic representation frames. In this paper, we
present the principal ideas of a research in which visualization’s role was
to generate in the student a skill to predict the behaviour of an iteration
function based on the Fixed Point Theorem in R. It is necessary to say that
we used a computational tool, since it helps in the graphical representation
and even in the calculation of operations.
THE VISUALIZATION IN MATHEMATICS EDUCATION
The human being possesses the aptitude to give a different connotation to
the same fact, and in many cases it helps to enrich their perception of the
environment that surrounds him. Especially in mathematics, teachers and
students pupils have a diversity of forms for explaining one certain
concept, nevertheless, has been reported that there is not a strong link of the
above mentioned representations and consequently the above mentioned
concept is not “well learned”, therefore our interest is to observe the
visualization processes in mathematics education. Zimmermann and
Cunningham (1991) mentioned that visualization offers a method to see the
hidden, enriches the process of scientific discovery and foments deep and
unexpected penetrations.
Since Pythagorean times, whom consolidated maths as science, the study of
numbers and relations were studied across their diverse configurations
made with stones (operations), for them, visualization was something
totally connatural to the mathematics. However after several centuries,
rigor and formalization of the modern mathematics brought the
disappearance of explanations with pictorial representations or with figures.
Nevertheless, the interest for using visual representations re-arose for
diverse researchers of the mathematical occupation, undoubtedly among
them, who we deal with the educational mathematics, due to the fact that it
is considered that visualization is present in the development of
mathematical thought. (de Guzmán, 1996)
1
Centro de Investigación en Matemática Educativa-Universidad Autónoma de Guerrero.
2
Centro de Investigación en Ciencia Aplicada y Tecnología- Instituto Politécnico Nacional. México.
3
Centro de Investigación y de Estudios Avanzados-Instituto Politécnico Nacional. México.
The word visualization appears in the dictionary with several assertions, for
example the following: it is the action and effect of visualizing, that is, is
the action of imagine with visible features something that one does not see;
it is the formation in mind of the visual image of something abstract; it is
the representation with optical images of phenomena of another character.
Observe that a common question is that visualization allows an alternative
for representing a certain object.
In agreement with literature we have classified different approaches of
visualization in didactics of mathematics, for example we find that,

Visualization is the way that allows to link intuition and reasoning.
(Tall & Vinner, 1981; Dreyfus, 1994; Davis, 1993)

Visualization is the capacity of articulate inside a set of
representations of a same object to give it meaning. That is, the
formation of mental images is favoured. (Zimmerman &
Cunningham, 1991; Hodgson, 1996; Nelson, 1997; Presmeg,
1999; Arcavi, 1999; Hitt, 2003)

Visualization is the action of the individual to connect different
representations of a mathematical object. (Duval, 1999)

Visualization is a mental process that enables to represent,
transform, generate, report, document and reflect visual
information. (Cantoral & Montiel, 2001; Aparicio, RodríguezVásquez & Cantoral, 2003; Rodríguez-Vásquez, 2003)
Recapturing the latter approach, for this research we have characterized
visualization as a particular form that helps in social practice of prediction,
where we consider prediction as a social practice which is provided of
sense educational practice to deduce a mathematical result, specifically we
are focused in the process of observe visualization to predict when a
sequence converges or diverges.
Problem research
Elementary concepts in mathematics, in many cases are not systematically
treated in more abstract knowledge levels, which give us a problematic in
teaching–learning system. The problem we consider is to link to the
meaning that the derivative has on the fixed point theorem of a contraction
and consequently in prediction of convergence or divergence of a sequence
that comes from the function recursively. In particular we want that
visualization helps to generate relations, which give a new meaning to the
role of the derivative. Sierpinska (1994) reported that there is a difficulty
on the fixed point notion when students are asked to get the fixed point of
functions defined by parts, thus it is observed the absence of meaning about
the derivative role in such a theorem.
Theoretical and methodological referents
As members of a network of Research Centres in Educational Mathematics,
which have their bases on a relatively new theoretical approximation in our
discipline, we decided to continue under this sustenance, in order to
consolidate further its components theorists, namely socioepistemología.
This theoretical approximation4, assumes that the educational phenomenon
has a imminently social nature. (Cordero, 2001; Cantoral & Ferrari, 2004;
Bagni, 2004; Buendía, 2004, 2006; Camacho, 2006). About this theoretical
approximation, we mainly worked on four knowledge components: social,
didactical, cognitive and epistemology components.
Regarding to methodology, our main axis was the Ingénierie Didactique
(Artigue, 1995), so we proposed a design of activities appeared in 3 stages
(action-situation, formulation-situation and validation-situation), in general
the main objective was that, in a systemic way, the students recognize the
prediction property, that is, that they could establish relations between the
derivative magnitude and its graph form, in such a way that this going and
coming between the frames of graphical and algebraic representation was
guided by visual processes. The activities were applied to a sample of 15
students5 of the last semester of bachelor in mathematics.
The specific objectives of each stage were the following: in first stage we
wanted to know what notions students had about fixed point, convergence
and derivative concepts; in second stage, with the activities we wanted to
establish a cognitive unbalance in students for giving meaning to the slope
of a tangent in a point of a curve basing in the fixed point theorem and that
they get a consensuses of the first stage activities; finally in third stage the
objective was that students recognized the prediction property in a
systematic form, in other words, that students related the derivative
magnitude with the graphic form given in the activities of this stage.
The study object
We investigated about the following questions: how students can predict
when a sequence is convergent? How visualization helps in some
abstractions in conceptions of the students, in such a way that they build
their knowledge? In other words, the main question was, if visualization is
a useful form for the development of the mathematical thought, how
visualization can help students to predict convergence or divergence of a
sequence, determined by the iteration function?
Theoretical approximation in development by researchers of the Centro de Investigación y de Estudios
Avanzados del IPN, México, and by the Red de Centros de Investigación en Matemática Educativa,
México.
5
For details see Rodríguez-Vásquez (2003)
4
In particular we investigated these issues with regard to the following
theorem:
Let
be a closed finite interval and let
the following conditions:
i)
be a function that satisfies
is continuous in ;
ii)
for all
;
iii) satisfies Lipschitz condition, that is,
with a constant of Lipschitz
.
Then for any choice of
the sequence defined by
with
converges to the unique solution
equation
.
of the
We review how the previous theorem is presented in some of the
contemporary texts suggested for teaching in higher studies of
mathematics. We observed that though the derivative concept is present in
the demonstrations of this theorem for proving the uniqueness of the
solution, later it is forgotten completely, a fact that we observed of class
notes of the students group mentioned in the last section. Nevertheless, of
Lipschitz condition, it is possible to assure or not, the convergence of the
sequence, depending of the inclination of the tangent line to the function in
a point.
The most relevant activities in the design of the didactic situation were:
1. Do values exist of in the domain, that preserve their values when
they are iterated under the following representations? Indicates
which and indicates why.
a)
yx
5
b)
4
f ( x)  x 3
3
2
1
0
1
2
x
3
4
5
16
d) y  sin x  x
14
c)
12
6
10
4
8
2
6
-10
4
-2
0
-5
0
-2
-4
2
-4
8
-6
2
4
-8
5
x
10
e)
 5 x  8 if x  9 / 7
f ( x)  
 1 / 3 x  2 if x  9 / 7
f) In a visit to Macuiltepetl hill, a person begin his
tour at 10 a.m., and finish at 12 p.m., walking at a
constant speed by a 3 km of longitude road. The
hill’s vigilant was at the top when begin his walk
on the same road but with variable speed. Vigilant
started and finished at the same hours than visitor.
Can you state from this story the existence of some
meeting point for functions particularly chosen of
each tour? Show your example.
More than half of the students answered this activity relying on the notion
of intersection of the identity with the presented function, to argue they
resorted to the graphical representation, even in the subparagraph f) the
most, raised the situation of graphical form. The rest solved directly the
system
in the cases that they could. But obligatorily they resorted
to the graphical representation in the subparagraph f).
Also there were other activities that had as aim that students have a
cognitive unbalance, in such a way that they think in the search of new
arguments to give a solution to the raised exercises, for example, we asked
them to answer the following:
2. Make a table with the generated values by the following iteration
function


20
xn   2

 xn 1  2 xn 1  10 
Taking
x0  1
... (1)
3. Prove that it is equivalent to solve the equations A, B and C
Equation A.
x3  2 x 2  10 x  20  0
Equation B.


20
x
0
2
 x  2 x  10 
Equation C.
 20  2 x 2  x 3 
x
0
10


4. Determine the fixed point obtained by the iteration rule
xn 
(20  2 x n21  x n31 )
10
...(2)
5. The following tables show the value assigned to
in the first
column, the image for this
in the second column and the
derivative at the same point in the third column, of the functions:


20
Y1 =  2

 x  2 x  10 
and
Y2 =
 20  2 x 2  x 3 


10


Respectively.
Figure 5. Y1
Figure 6. Y2
With your teammates, take shifts and explain to your companions the
meaning of the elements of each table.
6. Observe that (1) and (2) have jointly a fixed point, guess why the
sequence generated by (1) does not far from that point whereas (2)
does it.
7. Make a 3-tuple with entries: the iterative graphs with its respective
iteration function and the type of the fixed point. Explain the
criterion that you adopted to do it.
1.
xn 
4.
xn 
7.
(20  2 xn21  xn31 )
10
x n  0 .5
xn  

xn
2
3. FIXED POINT
ATRACCTOR
5.
6. FIXED POINT
REPULSOR
8.


 n 1  2 xn 1  10 
10. xn  
x2
2.
20
11.
9. FIXED POINT
REPULSOR
12. FIXED POINT
ATRACCTOR
The activities were thought to generate in students actions that were
allowing us to observe processes of visualization in the sense of Cantoral &
Montiel (2001), i.e. the goal was that students represent, transform,
generate, report, document and reflect visual information. In fact, we could
look at the existing transition between different frames of representation as
the algebraic, numerical and graphical.
We show the following paragraphs6 in order to observe some transitions
between representation frames; they allowed to detect some of visualization
processes that undoubtedly helped to the understanding of the situation and
likewise of the mathematical study object.
For example, in the following four paragraphs, we allowed to observe the
change between the numerical and graphical frames, the students were
working in the third stage, trying to answer the question, how do you
explain to a high school student the convergence, into the context of
activities?
Alicia:
Evaluating each point in the function, we will be approaching
to a certain point?
Neri:
In this case you are approaching to a point that is being
intersected by a straight line... the identity
Jannet:
I see, take a point in the domain, evaluate it in the function,
then the value of this point in the ordinate is mapped to the
abscissa and evaluate it again in the function and so forth until
there will be a time when we will have to assesses very closed
the points evaluated, then they are approaching to a fixed point,
where the identity straight line and the evaluated point, are
almost the same one and it is that they converge
Alicia:
But also you have to show the graph, because if not then not,
textually you do not understand
With regard to the convergence, a team had the following discussion:
Interviewer: How do you explain the convergence to a student in higher
education?
Alonso:
See, I evaluated , begin to iterate under the given function
and, I am following the arrows... I realized that, it is
approaching to a point, an intuitive way to explain it to them
Interviewer: If you had only the rule of iteration
, how do you
explain the convergence to your team members?
Alonso:
So, with formal definitions and with the proof
Interviewer: How do you explain to your teacher?
Alonso:
In this case he would be for the minor values that 1, not, minor
and positive ... it is necessary to play with vicinities
We did not show all the transcriptions, only some parts for exemplify. For details see RodríguezVásquez (2003)
6
Raciel:
Yes, it converges to the fixed point, if given a fixed point, the
module of the derivative in the fixed point is less than K, with
K between zero and one
We saw in the discussion three ways in the form for answer, first they
answered into a graphic frame, after, they change their context.
With regard to the last activity, a team had the following discussion:
Nery:
Nery:
Alicia:
Nery:
Alicia:
Nery:
Jannet:
Nery:
Jannet:
It’s attractor, oh no, it’s repulse
So, would (1, 11, 9)
The other is root of
I’m not sure, this is this, oh no, this is the root plus a decrease
.5, it is a repulse
So it is (4, 8, 6)
Which is repulse?
This is attractor, I’m guided by the chart
So, the list is (10, 5) and it is attractor
For explaining the convergence at a fixed point, first we should
evaluated an initial point in the function, the value obtained,
it’s evaluated again in the function, and this is the iteration, it’s
approximating us to the intersection between the graphic and
the identity, oh I understood
We observed in these paragraphs, the systematic integration of the
graphical and algebraic frames, because the activity was raised dependence
of both frameworks, the students explications reflect this integrity.
In this sense, in our opinion, it was possible to state that visualization was
an element that was present, at all times in, the action, formulation and
validation situations, of one mathematician know.
Conclusions
Regarding the aim raised, how visualization can help the students to predict
convergence or divergence of the sequence determined by an iteration
function?, we can conclude that the above mentioned social practice of the
prediction, is present in a connatural way in mathematical thought, so that
the role of visualization was a way that allowed to students to go on and to
come from a context of representation to another. Besides the processes of
visualization generated, allowed to students to go out of a cognitive
unbalance (the activities 2-6 were raised, for the generation of the above
mentioned unbalance).
On the other hand, this didactic experience allowed us to think about one
process that somehow prevented to students to develop their capacity of
visualization, and we observed, that such impediment was due to the
context in which students were located, because the visualization it is not
emphasized in traditional teaching.
For example, the students who participated in this research, in one way or
another, they attempted to use formalization in their proofs, it causes
undoubtedly of their vocational training, however, although their
procedures were correct on this path, they made visualization processes for
surmise and anticipate about the future behaviour (convergence or
divergence) of a sequence, only by the form of the tangent line in the
intersection of the curve
given and the identity.
It is necessary to mention, that for didactic analysis, we reviewed the notes
given by their teacher in the course of numerical analysis and though they
worked at diverse contexts (graphical, algebraic and numerical), the notion
of inclination of a tangent line was debilitated at the moment of dealing in
the cases in which there is not a "nice" function, in the sense of derivate it
easily with the basic rules.
On the other hand the predominance of resorting to the form in how was
taught the theorem of the fixed point in class, was an invariant that the
students reflected in their argumentative speech, which gave place to
limitations in the development of visual processes, nevertheless, in our
opinion, we succeed that the student reflect about a new way of thinking to
solve the raised problems.
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