REPRESENTATIONS
AND MATHEMATICS VISUALIZATION
WORKING
GROUP PMENA XXII, 2000, TUCSON,
ARIZONA, USA
Fernando Hitt
Departamento de Matem‡tica Educativa del
CinvestavIPN, MŽxico
fhitta@data.net.mx
The Group on Representations and Mathematics
Visualization was constituted at the PMENA XX meeting in the North Carolina
State University (Hitt, 1998).Ê In the
last meeting in Mexico, the academic agenda of the WG included four
presentations followed by corresponding discussions that addressed very
important themes.Ê Here, we intend to
summarize ideas discussed in each presentation (a full version will be
available at each meeting of the WG and at WWW.
Cinvestav.mx/mat_edu/PMENAXXI.html).
Patrick W. Thompson's (Vanderbilt University):
Some Remarks on Conventions and Representations
The work presented by Thompson raised something
important dealing with group production versus the memories about the group
discussion from the individual perspective.Ê
Thompson said: "When we claim that agreement has been reached on a relatively
complex idea because disagreement hasn't been expressed, we must consider the
possibility that students haven't analyzed their own or others expressions
sufficiently to detect severe inconsistencies."
Thompson's study did not stop where usually other
research studies stop.Ê That is, usually
the research undertaken concerned with groups discussions of a mathematical
problem stops when the teacher or the researcher thinks that a consensus among
students has been reached.Ê In
Thompson's case, he addressed the importance of being cautious about concluding
what individuals understand by only considering evidence based on group
agreement.
The mathematical problem he studied was to have
students "interpret a formula that defined an 'average rate of change'
function in regard to the variation in a square's area expressed as a function
of the square's side length."Ê
Then, given a segment s as an
independent variable, he constructed _{}, and _{}, and the Figures 1 and 2.Ê
The question to students was: "What is represented by the point on
r's graph having coordinates (0.7667,1.635)?" Nineteen students
participated in wholeclass and small and smallgroup discussions of the
question for 90 minutes in order to arrive at a consensus, both among the whole
class and within small groups, as to for what (0.767,1.635) stood.Ê At the end, all stated satisfaction that
they all agreed on the meaning.Ê Thompson
then asked them to each, individually, write the interpretation with which they
all agreed.


Figure 1.
Side length and area varies. 
Figure 2. Graphs of A(s) and r(s). 
Thompson reported: "eight of 19 couldn't remember what their group
had said before, that they couldn't reconstruct it, or they couldn't come up
with an interpretation".Ê With
respect to the responses from the other eleven students he said: "First,
none of the responses is internally consistent.Ê Five are relatively close.Ê
Six responses are conceptually incoherent, entailing internally
conflicting meanings.Ê Second, no
interpretation even remotely resembles those that they spent 50 minutes
developing and to which they each expressed satisfaction that they had said
what they intended".
He finally stated that the two aspects together, lack of internal
coherence in students' interpretation and lack of agreement between private and
public stated interpretations, points to a matter worth considering.
Adalira S‡enzLudlow (University of North Carolina at Charlotte):
Interpretation, Representation,
and Signification: A Peircean Perspective
The work presented by
S‡enzLudlow is related to representation and semiotics under a Peircean point
of view.Ê Her primary questions were:
1.
Is
a representation a thing or a process?
2.
Is
a representation a dynamic process?
3.
Is
a representation a sign?
Her approach to discuss
these questions was from a Peircean perspective.Ê She quotes: Charles Sanders Peirce (18391914) considers that
"Semiotics is the doctrine of
the essential nature and fundamental varieties of possible semiosis.Ê That is, strictly speaking, semiosis and not
the sign is, for him, the proper of semiotic studies.Ê For Peirce " a sign, or representamen,
is something that stands to somebody for something in some respect or
capacity.Ê It addresses somebody, that
is, creates in the mind of that person an equivalent sign, or perhaps a more
developed sign.Ê That sign which it
creates I call the interpretant of
the first sign.Ê The sign stands for
something, its object.Ê It stands for that object, not in all
respects, but in reference to a sort of idea" and "nothing is a sign
unless it is interpreted as a sign. "
Figures 3 shows the Triadic
Sign Relations presented by S‡enzLudlow from the Peircean perspective.Ê Figures 4 shows from the same point of view
chains of signification.


Figure 3 
Figure 4 


Figure 5. Example given by Whitson
(1997, p. 101102): Suppose I look
abarometer, say "Let's go," pick up my umbrella, and start for the
door. You pick up your umbrella and follow. The barometer reading is being
interpreted as a sign of rain (the object represented). It is functioning as a
sign when it produces as its interpretant the event (me picking up my umbrella)
in which the reading is interpreted as a sign of rain. That interpretant can,
in turn, function as a sign of rainÊ
producing a subsequent interpretant (for example, you taking you
umbrella). The two of us both leaving with umbrella can function as a sign
producing (as an interpretant) a coworker's decision not to go out for lunch.
S‡ensLudlow continued with
the Peircean perspective and quotes: Peirce defines semiosis as "the
triadic action" of the sign in which a sign has a cognitive effect on its
interpreter.Ê Semiosis, for him,
involves the cooperation of the three elements in the triad
objectrepresentameninterpretant.Ê Such
"trirelative" influence is not resolved by the isolated action
between any of the pairs in the triad, but only by the synergistic action of
all elements in the three way relationship."
S‡ensLudlow finished her presentation
giving a reflection about these questions:
1.
Is
it useful for us in mathematics education to think of a representation as a
process that presents something other than itself?
2.
Is
it useful for us in mathematics education to think of a representation as a
dynamic triadic process or as a dynamic dyadic process?
3.
Is
it useful for us in mathematics education to think of a representation as a
sign?
Raymund Duval
(UniversitŽ du Lottoral C™ted'Opale):
Figures'
representational funtion in geometry and Figure's multiple and parallel entries
The two sessions led by
Duval (plenary and working group sessions) were based on theoretical
perspective related to semiotic representations and mathematics visualization
and specifically in the second one about figures and the learning of geometry.
Five questions were at the
core of his presentations:
1.
Why
should semiotic representations be taken into account in order to analyze the
learning of mathematics?
2.
How
do semiotic representations work in mathematics?

Are
some semiotic representations characteristic of mathematics?

In
the case where some representations are similar to those within other areas,
are they used in the same way?
3.
Does
the students' use of semiotic representations involve a difficulty that they
need to overcome?

How
can a student learn to recognize a mathematical object through its various
representations?

How
can a student learn to distinguish a mathematical object from any particular
semiotic representation?
4
Does
visualization in mathematics work like ordinary iconic representations?
5
What
kind of variables, connected with semiotic representations, must the study of
learning in mathematics take into account?
The ideas discussed by Duval
seem to be framed through a Sausserian approach rather than a Peircean perspective.Ê His idea of register of representation is
connected with the analysis of the productions of the students in a restricted
system of representations.Ê Duval (1993,
p. 40) quotes:Ê A semiotic system could
be a representation register, when it permit three cognitive activities related
to the semiosis:
1)
The
presence of an identifiable representation...
2)
The treatment of a representation which
is the transformation of the representation within
the same register where it has been formed ...
3)
The conversion of a representation which is
the transformation of the
representation in other representation of another register in which it
conserves the totality or part of the meaning of the initial representation...
From a cognitive point of
view, conversion (and not treatment) is the central process of mathematical
thinking for three reasons:
á
Related
to Question 3 above, the distinction between the representation and the
represented object is possible only when students become able to convert a
representation of a mathematical object in another representation.Ê It is the cognitive condition for transfer
or for decontextualization.
á
Some
conversions are congruent: The representation in the starting register is
transparent (seethrough) to the new representation in the target
register.Ê But in most cases, conversion
is not congruent. That leads most of the students to an obstacle or to
compartmentalized understanding.
á
They
are factors which explain congruency or noncongruency of conversion. They
depend on the opposite starting and target registers (Visual or not, Language
or Symbolic) thus we can study experimentally these complex phenomena and we
can define cognitive variables for the different areas in mathematics
education.
On the construction of
concepts Duval (Ibidem, p. 46) states that: ãevery representation is partially
cognitive with respect to what representsä and then: ãThe understanding
(integral) of a conceptual contents based on the coordination of at least two
registers of representation, and this coordination is revealed by the rapid use
and spontaneity of the cognitive conversion.äÊ
The last paragraph as Duval quotes, needs another description of the
structure of the semiotic representation and his performance (see Figure 6
(Duval, 1993, 1995) and 7 (Duval, 1999)).
On the construction of
concepts Duval (Ibidem, p. 46) states that: ãevery representation is partially
cognitive with respect to what it representsä and then: ãThe understanding
(integral) of a conceptual content is based on the coordination of at least two
registers of representation, and this coordination is revealed by the rapid use
and spontaneity of the cognitive conversion.ä
The last
paragraph, as Duval quotes, needs another description of the structure of the
semiotic representation and his performance (see Figure 6 and 7).
Figure 6. Duval
1993, 1995
Figure 7. Duval 1999
Abraham Arcavi  Nurit Hadas
(Weizmann Institute of Science):
Computer Mediated Learning: An example of an approach
Arcavi's presentation addressed
directly a theme of our working group: The influence of technologybased
multiple linked representation in the students' construction of mathematical
concepts and also in regard to the role of visual thinking (see Hitt, 1998).
In his presentation Arcavi
showed clear advantages in the use of dynamic computerized environments.Ê He illustrated, via an example, important
components of mathematics activities.
á
Visualization.Ê Arcavi and Hadas cited Hershkowitz (1998, p. 75) to characterize
visualization: "Visualization generally refers to the ability toÊ represent, transform, generate, communicate,
document, and reflect on visual information."Ê And they added, visualization not only organizes data at hand in
meaningful structures, but it is also an important factor guiding the
analytical development of a solution.
á
Experimentation.Ê Playing with dynamic environments allows students to learn to
experiment, and to appreciate the ease of getting many examples·, to look for
extreme cases, negative examples and non stereotypic evidence· (Yerushalmy,
1993, p. 82).
á
Surprise.Ê The challenge is to find situations in which the outcome of the
activity is unexpected or counter intuitive, such that the surprise (or
puzzlement) generated creates a clear disparity with explicitly stated
predictions.
á
Feedback.Ê Surprises of the kind described above arise from a disparity
between an explicit expectation from a certain action and the outcome of that
action.Ê The feedback is provided by the
environment itself, which reacted as it was requested to do.
á
Need for a proof and proving.Ê Following a surprise, many students may require a proof, maybe
not explicitly, but by demanding from others or from themselves an answer to
their 'why' (or 'why not').
The Problem Situation Arcavi and Hadas Analyzed was in Two Phases
First phase.Ê Two segments of length 5 with
a common end point.Ê Joining the two
other end points, produces an isosceles triangle (see Figure 8).Ê The dragging of, for example, the vertex C,
yields many possible isosceles triangles whose equals sides are 5.
Question: Predict the shape
of the graph of the area of that triangle as the function of its base.
In order to check the
prediction, they asked students to make use of the software to draw the graph.
Figure 8
The problem situation Second phase.Ê
Following
again the "what if?" approach, the question became: "So far, we
explored isosceles triangles where the equal sides have a fixed given value.
What would happen if we make a 'small' change, so that the triangle is not
isosceles, but "close to being one?"
On the basis of the above
tasks and the experience with them, Arcavi and Hadas made a reflection about
(see large version of the Working Group on Representations):
1.
The
role of computerized tools
2.
Mathematics
and mathematical activity
3.
A
new way of thinking.
References
Arcavi A. & Hadas N.
(2000).Ê Computer mediated learning: An
example of an approach.Ê International Journal of Computers for
Mathematical Learning, 5,2545.
Kluwer Academic P.
Duval R. (1999). Figures'
representational function in geometry and figure's multiple and parallel
entries.Ê Preliminary version for the
Working Group on Representations and Mathematics Visualization. ÊTwentiethFirst
Annual Meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education, Cuernavaca, Mexico.
Hitt F. (1998). Working
Group on Representations and Mathematics Visualization.Ê In S. Berenson, K. Dawkins, M. blanton, W.
Columbe (Eds.), Proceedings of the
Twentieth Annual Meeting of the North American Chapter of the International
Group for the Psychology of Mathematics Education.Ê North Carolina. USA.Ê Columbus, OH: ERIC.
S‡enzLudlow A. (1999).Ê Interpretation, Representation, and
Signification: A Peircean Perspective.Ê
Preliminary version for the Working Group on Representations and
Mathematics Visualization.Ê
TwentiethFirst Annual Meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education, Cuernavaca,
Mexico.
Thompson P. (2000). Some
remarks on conventions and representations.Ê Working Group on Representations and
Mathematics Visualization, PMENA XXII, Tucson, Arizona.
Whitson J. (1997).Ê Cognition as a Semiosic Process: From
Situated Mediation to Critical Reflective Transcendence.Ê In D. Kirshner and J.
Whitson (Eds.), Situated Cognition.
Social, Semiotic, and Psychological Perspectives.Ê LEA Publishers.