REPRESENTATIONS AND MATHEMATICS VISUALIZATION
WORKING GROUP PME-NA XXII, 2000, TUCSON, ARIZONA, USA
Departamento de Matemática Educativa del Cinvestav-IPN, México
The Group on Representations and Mathematics Visualization was constituted at the PME-NA XX meeting in the North Carolina State University (Hitt, 1998). In the last meeting in Mexico, the academic agenda of the WG included four presentations followed by corresponding discussions that addressed very important themes. 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. 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. 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. 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." Then, given a segment s as an independent variable, he constructed , and , and the Figures 1 and 2. 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 whole-class and small and small-group 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. At the end, all stated satisfaction that they all agreed on the meaning. 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". With respect to the responses from the other eleven students he said: "First, none of the responses is internally consistent. Five are relatively close. Six responses are conceptually incoherent, entailing internally conflicting meanings. 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 Sáenz-Ludlow (University of North Carolina at Charlotte):
Interpretation, Representation, and Signification: A Peircean Perspective
The work presented by Sáenz-Ludlow is related to representation and semiotics under a Peircean point of view. 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. She quotes: Charles Sanders Peirce (1839-1914) considers that "Semiotics is the doctrine of the essential nature and fundamental varieties of possible semiosis. That is, strictly speaking, semiosis and not the sign is, for him, the proper of semiotic studies. For Peirce " a sign, or representamen, is something that stands to somebody for something in some respect or capacity. It addresses somebody, that is, creates in the mind of that person an equivalent sign, or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. 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 Sáenz-Ludlow from the Peircean perspective. Figures 4 shows from the same point of view chains of signification.
Figure 5. Example given by Whitson (1997, p. 101-102): 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 rain 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 co-worker's decision not to go out for lunch.
Sáens-Ludlow 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. Semiosis, for him, involves the cooperation of the three elements in the triad object-representamen-interpretant. Such "tri-relative" 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."
Sáens-Ludlow 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 (Université du Lottoral Côte-d'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. His idea of register of representation is connected with the analysis of the productions of the students in a restricted system of representations. Duval (1993, p. 40) quotes: 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:
Š 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. It is the cognitive condition for transfer or for decontextualization.
Š Some conversions are congruent: The representation in the starting register is transparent (see-through) to the new representation in the target register. But in most cases, conversion is not congruent. That leads most of the students to an obstacle or to compartmentalized understanding.
Š They are factors which explain congruency or non-congruency 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: „every representation is partially cognitive with respect to what representsš and then: „The 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.š 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: „every representation is partially cognitive with respect to what it representsš and then: „The 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.š
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 technology-based 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. He illustrated, via an example, important components of mathematics activities.
Š Visualization. Arcavi and Hadas cited Hershkowitz (1998, p. 75) to characterize visualization: "Visualization generally refers to the ability to represent, transform, generate, communicate, document, and reflect on visual information." 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.
Š Experimentation. 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).
Š Surprise. 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.
Š Feedback. Surprises of the kind described above arise from a disparity between an explicit expectation from a certain action and the outcome of that action. The feedback is provided by the environment itself, which re-acted as it was requested to do.
Š Need for a proof and proving. 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. Two segments of length 5 with a common end point. Joining the two other end points, produces an isosceles triangle (see Figure 8). 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.
The problem situation -Second phase. 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.
Arcavi A. & Hadas N. (2000). Computer mediated learning: An example of an approach. International Journal of Computers for Mathematical Learning, 5,25-45. Kluwer Academic P.
Duval R. (1999). Figures' representational function in geometry and figure's multiple and parallel entries. Preliminary version for the Working Group on Representations and Mathematics Visualization. Twentieth-First 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. 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. North Carolina. USA. Columbus, OH: ERIC.
Sáenz-Ludlow A. (1999). Interpretation, Representation, and Signification: A Peircean Perspective. Preliminary version for the Working Group on Representations and Mathematics Visualization. Twentieth-First 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. Working Group on Representations and Mathematics Visualization, PME-NA XXII, Tucson, Arizona.
Whitson J. (1997). Cognition as a Semiosic Process: From Situated Mediation to Critical Reflective Transcendence. In D. Kirshner and J. Whitson (Eds.), Situated Cognition. Social, Semiotic, and Psychological Perspectives. LEA Publishers.