Bolite Frant J., Rabello de Castro M. (2000) Proofs in Geometry: Different concepts build upon very different cognitive mechanisms. Contribution to: Paolo Boero, G. Harel, C. Maher, M. Miyazaki (organisers) Proof and Proving in Mathematics Education. ICME9 TSG 12. Tokyo/Makuhari, Japan.
	Abstract The introduction of computer technology in school mathematics, particularly in Geometry, brought back a discussion about proofs. Many topics are related to this issue: student's conceptions of proof; intuitive proof; axiomatic proof; usage of technology to enhance proof skills in Geometry, among others. In this paper, I focus on the idea of using CABRI as a media for promoting student's proof skill and, I also discuss the difference between teachers and students needs of proofing. Based on cognitive science findings, semantic field model and argumentative strategy model, I argue against the idea of using geometry software to promote a smooth passage from intuitive, pictorical or informal proof to axiomatic proof because, in Geometry, there are two radically different fields of meaning production for proof: one deals with movement and the other is static. Since they are not always consistent there is, if exists any, rather a complex path than a simple path relating both approaches.
© J. Bollit Frant & M. Rabello de Castro

Mathematics History and Beliefs: Pedagogical Implications. It is important to go back in time to remember that mathematics was regarded as an utilitarian tool for a long period in man kind history, and it was only with the Greeks that math came into being as an area of study. Geometry was related for a long time to utilitarian needs like, for example, measuring the boundaries of Nile, sharing lands. From 500 BC to 300 AD Geometry stop being only utilitarian and start being an intellectual pursuit. Thales introduce the idea of proof that was based on formal argument. This approach culminated in the publcation of Euclides'Elements.   About pedagogical implications we could say that dependent on which view are teachers rooted in there is a different impact on classroom math lessons. An intuitive or pictorical proof would be closer to the utilitarian view than to a science view. But history plays only a role in the scholl mathematics proof, another important role plays cognitive science. I argue that there are two different approaches involved: one is intuitive, related to motion, and another is static, related to symbols. One can argue that a symbol is not disconnected from its image, for us it is important to rise other questions : In school mathematics who makes the connection? How those connections are made?   Moreover, lessons are based in teachers' conception about knowledge, about how children learn, even though it is not always explicited. Conception of Knowledge and Learning There are two strongs beliefs that come into view when reading articles about using dynamic geometry to enhance students axiomatic proof skills: Knowledge comes in stages so if we start with manipulatives or similar there will be a passage or transfer from the experience with the object (first stage) to a more abstract level (second stage). I am strongly against that "container metaphor view" (Lakoff 1980, 1987), lugages and containers can be carried all over, from here to there, information can be stored in video, audio, computers but knowledge is different. Knowledge is not a "thing" that can be transferred so there is no need to find a way to delivery knowledge. How is it different?   According to Lins (1997) knowledge is defined by a pair (belief, justification) and when two persons share the same belief but justify it in different ways we can say that they have different knowledge. What is most relevant for our discussion, in this presentation, is that knowledge is produced by the learner (individual) while talking, and is directly related to enunciation. Meaning production is "a set of things that can be said about an object. It is not the set of what could be said but what actualy is said within an activity". In this way, arguments used by the subjects to express their beliefs and justifications compose the corpus of our analysis. Argumentative Strategy Perelman (1993) and Rabello (1997) provide us a ground to study students arguments. The Argumentative Strategy Model was developed to describe how students engender their arguments within an episode while working in an activity(Frant, Rabello and Araujo in press).   Douek (1999) found analogies between argumentation and ordinary mathematical proof and argued that formal proof is very distant from the activity of conjecturing and proving. Case Study Two case studies were carried out in Rio de Janeiro, Brasil: both involved 4 people. One was a teacher development (TDC) course on Geometry and CABRI, and the other was a Geometry course for future teachers (GFT). TDC TDC Proposed Task: What can be said about the area of ABCD knowing that it was constructed by the intersection of two squares such that one has a vertice at the center of the other. The 4 teachers worked in pairs in the computer lab, the teachers started to drag some points, compared the figures before and after moving them, tried to establish some relations and to proof them. Interesting to note that their using of movement helped them to rise conjectures, verify whether their hypothesis were right but there was still a need to proof, formally. Bia: OK. The area [ABCD] is _ of the square. I can see it Ana: OK, but we need to proof it. Ana was talking about formal proof. Each pair worked in different ways: a geometric argument- if two angles have perpendicular sides they are equal, and a more algebric argument - solving a four linear equation system. Both pairs finished almost at the same time and screamed with a smile in their aces: Proved! GFT Proposed Task: It is given two disjoint circles with the same radius, and n1 and n2 are tangent lines to both circles at the same time. Is it true that the intersection point of n1 and n2 is on the line that connect the centers of the two circles? If we change the radius of one circle, what will happen to the intersection of n1 and n2? Is it true that the intersection point will be on the line that connect the centers of each circle?   The students made some lines, drag the circles in order to modify their radius and were satisfied, convinced, that thhe three points will always be on the same line. G- I drew the line connecting the centers of each circle, then I asked for the intersection of this new line with the point of intersection of lines n1 and n2. From then on we modify the radius and we prooved that the three points were on the same line We found that proof in this case was understood as a way of convincing the self and others that an affirmation is true. It is important to note that, as David Reid (1998) observed in relation to proof by contradiction, for the students there is no need to proof (again) something that they already know that is right. Summary Analysing TDC case, we can say that to rise and test conjectures about the area be _ of the first square, it was important to use CABRI possibility of moving figures, but as soon as they decided to proof it (formal) they looked at the static figure and used axioms to do it. There is a strong difference in proof conception between teachers (TDC) and students (GFT). For mathematics teachers proof is synonym of formal proof while for students a proof means having the capacity to convince the self and other persons that something works or is true. Moreover, if a student knows that an affirmation is true (by drawing, or being told by peers) why bother with another prof? Consistent with our view of meaning production, the learner is responsible to make connections between symbols and images.. However, we found researchers and teachers advocating that there is a smooth path going from the maniulative and intuitive proof to a more advanced formal proof. This attitude has strong implications for math education, if students are exposed to manipulative geometry but is not able to write a formal proof they should fail. We propose that the role of teacher in classroom is similar to the filmaker in a montage process. The film-maker takes two different fragments and by assembling them together he/she produce a third piece that is no longer any of the others. It is a new piece. Helping students understand that informal or intuitive mathematical proof is different from formal mathematical proof can help them build connections, because if things are the same (similar) there is no need for connecting them. References Douek N. (1999) Argumentative aspects of proving. Proceedings of the 23rd PME. Technion. Israel Eisenstein S. (1968) Structure, Montage, Passage. Change 1, Paris. 17-41 Frant J., Rabello M., Araujo J. (in press). Cabri: a Formação e o Desenvolvimento Profissional de Professores de Matemática. Proceedings of the Cabriworld Congress. PUC-SP Lakoff G., Mark Johnson (1980) Metaphors we live by. The University of Chicago Press. Lakoff G. (1987) Women, fire, and dangerous things: What categories reveal about the mind. The University of Chicago Press. Lins R, Gimenez J. (1997) Perspectivas em aritmética e álgebra para o século XXI. Ed. Papirus. Perelman Ch. (1993) O império retórico - Coimbra, Edições ASA. Rabello M. (1997) Retóricas da rua: criança, educador e diálogos - Rio de Janeiro, Ed Amais/EDUSU