EXPERIENCING THE NECESSITY OF A MATHEMATICAL STATEMENT Catherine SACKUR, Jean-Philippe DROUHARD, Maryse MAUREL GECO - IREM, Université de Nice, France Teresa ASSUDE DIDIREM, IUFM de Versailles, France A teaching device tends to permit students to experience the necessity of mathematical statements (here in spatial geometry). We emphasize the role of the confrontation between students to have them come up against what could be called “the mathematical reality“. We describe four steps in such a teaching, going from a personal work of students to a sequence when the whole class collects through work in small groups of four students. In front of the whole class, the teacher plays an important role to bring into light the learned knowledge, its necessity in mathematics and the way some students experienced this necessity.

Introduction : In this presentation, we will be interested with the teaching of the character of necessity of mathematical knowledge Let us observe the statement: “ In a parallelogram, if the sides are equal, the diagonals are perpendicular ”. All mathematicians know that the content of this statement cannot be different. The truth of this statement comes from the mathematics themselves, from the axioms and from the rules of demonstration in mathematics. It comes, neither from observation of nature, nor from any arbitrary choice of the mathematicians. This is the case with the mathematical knowledge which is taught at school or in the first years of university. From this point of view, mathematical statements differ from many other statements such as: “ Mount Fuji is 3776 meters high ”. We observe that many students don’t know this characteristic of mathematical knowledge. They quite often think that things could be different if we decide so, especially in Algebra where the rules of computation seem arbitrary to them; thus their knowledge is not coherent. We are then interested in teaching this characteristic of “ necessity ” in mathematics and we believe that it cannot be taught directly through a discourse of the teacher. We shall describe here a school-room situation in which the necessity of a mathematical statement is brought into light and is then institutionnalized by the teacher as the same time as the knowledge itself. In the first part of our presentation we shall present the theoretical frame of our work, then describe the teaching device and in the third part we shall analyse the activity of the students and interpret it. The knowledge on which we have been working is the following: in an Euclidian space , the equation ax + by = c must be the equation of a plane.

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I. Theoretical Background Mathematicians are well aware that mathematical truth has a character of necessity in the meaning we explained in the introduction (we speak here of properties, theorems etc., obviously not of definitions or axioms). Few students however, show such awareness. We even interviewed a 16 years old student, average level, who complained having been bored with algebra because “it was made of a stack of unrelated rules”! Most commonly, teachers think that demonstrations do convince students that mathematical properties are necessarily true. It is clear indeed that a given demonstration may establish that a given property is necessary. However, it is the case just with the few students who are already aware that - in general - demonstrations have something to do with necessity. We tend to say that to demonstrate may reinforce the conviction that in mathematics, truth is necessary, but cannot initiate this conviction. One can say that something is necessary, only if one may think that things could be different, but are not. In the realm of physics for instance, two isolated magnets will be said necessarily attracting one another, insofar as one can imagine nonattracting magnets, meanwhile it is not possible that they do not attract one another. Now, what could be a mathematical equivalent of the physical world, where the subject can imagine that things could differ, and experiment how things behave? This point is related to the question of the “milieu” in G. Brousseau’s Theory of Situations (1997) or to “real” world in constructivist theories (Piaget, in his studies on necessity (1981, 1983)) even though we don’t know this “real” world, (Von Glasersfeld 1996). We claim that the interaction with ‘Others’ (which may be not convinced, disagree, have another point of view etc., Drouhard, 1997) permits both to make the different possibilities thinkable and to experience a conflict (with the Others’ disagreement). This experience is constitutive of the experience of the necessity (Drouhard, Sackur, Maurel, Paquelier & Assude, 1999). We meet here authors like Ernest (1997) in the importance given to the social interaction in the subject’s construction of mathematics. The “mathematical discussions“ as described by Bartolini-Bussi (1991) are a good example of teaching devices taking into account this importance. We are then interested in teaching situations centered on necessity, in which peer interactions play an important role. Wittgenstein helped us to figure out that social and language aspects are literally essential to understand the very nature of the necessity of mathematical knowledge. We found in Wittgenstein (1978) a subtle characterisation of necessity, related to resistance, although apparently paradoxical: the idea (expressed here in a very sketchy way) that mathematical objects resist us to the very extent that we want them to resist, and not because of their physical nature as walls do. But, what is the

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origin of this willpower? It seems that, for Wittgenstein, mathematicians need objects that resist them because doing mathematics is precisely working on such resisting objects. Surely they could work with ‘weaker’ mental objects but then, what they would do, could no longer be called “mathematics”(Drouhard & al 1999). In the same way, the phenomenological approach (Husserl) permitted us to better understand how the subject can experience something in mathematics and then to devise some experiments as the one we present in this paper. The emphasis given to peer interaction must not lower the teacher’s role. Not only is he supposed to set up a convenient ‘milieu’ for the learning of the (necessary) knowledge, but he also plays an essential role in helping students to be conscious of this character of necessity (during the phase of “institutionalization” in the terms of the theory of didactical situations). II. The Work in the Class 1. purpose of the teaching

Three groups of forty students from the first year of university (age 19 and20) worked following this device. Their knowledge about equation of planes and lines was the following: They had knowned for a long time that in the plane, ax + by = c is the equation of a straight line. During the last year of high school they learn that in the space the equation of a plane is ax + by + cz = d , and that a straight line is defined by a set a two equations as it is the intersection of two planes. Of course when one variable is missing in the equation, although there are taught that they are working in a space, the old knowledge comes back and they say that ax + by = c is the equation of a straight line. The purpose of this work is then to make them work on their old knowledge, correct it, and, at the same time, teach them that the correct knowledge is necessary. 2. schedule

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There were five phases in the device: first and second phases (the same session): 20 minutes of personal work, followed by one hour of work in small groups of four students, third phase (second session, one or two days later): synthesis in each large group; report of the small groups, agreement on the result, institutionnalisation by the teacher, fourth phase (one week later): work on the link between planes in R 3 and linear systems. Work on change of settings. fifth phase (one week later): the three large groups together in a “ regular ” class, course on equations of lines and planes in R 3. R.R PME 2000 C.S.

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3. a-priori analysis

We will explain here the role of the three first phases of the work. During the first phase, the students are working alone; this is the time when their own knowledge is activated, when they make up their mind about the answer to the problem, using what they know and when some of them produce the expected error. During the second phase, when they are working together in small groups of four students, they have time to: • confront their opinion with the opinions of others and choose an answer; here one finds the role of the conflict between the students. • become sure of the validity of this answer through the discussion with others. An important point of the device is that the teacher has asked that at the end of the discussion any of them should to be able to give the report of their work. So they have to agree and be convinced. • find an agreement based on mathematics. In a way, one could say that the mathematics “ decide ” what is the correct answer to the problem. At that moment the students experience the “ necessity ” of the statement. In this case they experienced the necessity that the given equations were equations of planes and surfaces and not of lines and curves. In the third phase, the synthesis takes place. The large group gathers, and each small group tells about its work; it is the moment when the teacher can institutionnalize this experience of necessity as it has been experienced in some of the groups. It is clear that all the groups do not reach the same state of knowledge, and do not experience the necessity in the same way. The teacher can take advantage of the experience of some of them to make clear the way which led to this experience. Together with the mathematical knowledge, this is what is brought to light by the teacher. Following our theoretical elaboration, this characteristic of the knowledge cannot be separated from it. A mathematical knowledge is “ necessary ” otherwise it is not mathematical. One can sum up the film of what takes place in the following diagram: other students ↓ 1 activate the actual K → error → confrontation and conflict → reorganisation of K ↑ ↑ →→→→ necessity 4. the question

Here is the question given to the different groups of students:

1

K. stands for knowledge

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Let’s consider the sets of points of space whose coordinates (x, y, z) are linked by the following relations: E1 2x - y = -1 E2 x = 3 E3 x + y + z = 1 E4 x2 + y2 = 1 E5 xy = 1 Describe and represent these sets of points as precisely as possible. You can use any method you wish to solve this problem. Please write the different steps of your research. Give the result which seems really correct to you and explain how you know that it is correct. Find the way in which you could convince someone that you result is the correct one.

III. Analysis 1. the work in small groups

The teacher observed the students while they were working alone. At first they all wrote that E1 was the equation of a line. Some students changed their mind later; it seems that the study of set E2 was the reason for this change. Of course some other students did not change their mind, so the confrontation was possible2. Here are some exemples of what happened in the small groups, as it has been observed by the teacher: Lorinne started to prove with gestures. She came to the corner of the room and described the position of the three axes: Ox was at the bottom, on the right hand-side, Oy was vertical and Oz at the bottom on the left. She showed, using her right arm, the position of the line 2x - y = -1 in the plane xOy and said, moving her arm: “ as we are in a space, all points have three coordinates, so my line can move ahead ”. She went on, doing the same gestures and answering the questions of the members of her small group, as long as necessary for them to be convinced. It took quite a long time, but eventually they were convinced. Laurent used an analogy: “ when we are on a line, x=3 is a point, when we are in a plane, x=3 is a line, here we are in a space, x=3 is a plane. (The hyperplan is behind this finding, one linear equation for a subspace which dimention is n-1). He went on: “ in x=3, there is no y, in a plan it is still a line and not a point and y can have whatever value we want. Necesserally in a space, even though there is neither y nor z it is a plane. So in a space, 2x - y = -1, where there is no z, z can be whatever we want, it is also a plane ”. This explanation, along with drawings is oriented to his group mates. Edouard explained why he was convinced with gestures and speach, telling he could see the points: “ piled up one on the top of the others, it can go up, because z has no value ”. Nobody in the group asked him what it meant that z had no value. 2

In one of the large groups no student at all changed is mind. They all believed that the two first sets where lines. Nevertheless when they gathered in small groups and started to explain “ how they knew that it was a line ” the changes occured. R.R PME 2000 C.S.

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2.the evolution in the small groups

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When the small groups reported about their work in front of the large group, the following steps appeared quite clearly: 2x - y = -1 is the equation of a straight line x = 3 is the equation of a plane come back to 2x - y = -1, it is the equation of a plane x + y + z = 1, I don’t know what it is, or it is the equation of a plane, by I don’t know how to represent it. The first sets being studied and correctly identified, the students could turn to the other sets. They started again with the gestures, this time with no difficulty. For x2 + y2 = 1, they said that it was a tube, a pipe, that you could pile up circles, they could draw it and some found the word cylinder. For xy = 1, they could not say much about it, but they used gestures and sheets of paper put upright on an hyperbola drawn on the table. It was clear for all of them that it was not a line but a surface. 3. the meaning of the missing variable

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We can trace z is not in the equation, so z equals zero, z is not in the equation, so z has no value, z is not in the equation, so z has any value, or, z is not in the equation, so z has whatever value we want, it varies from -∞ to+∞, there is no z, so z is free, I would not make mistakes if the equation was written 2x - y +0z = -1. This sentence showed clearly that the students had understood that the missing z was not equal to zero, but that it was its coefficient which was equal to zero. And in fact all students agreed to this explanation. In the questionnaire, we found some statements identical to these. 4. four steps in the resolution

This teaching device was ment to bring some changes in the knowledge of the students, through four steps corresponding to the different phases of the work. We shall examine these steps and describe the story of the knowledge: 1. first step: “ E1 is a straight line ”. This knowledge comes immediately to the mind of the student, almost in an instinctive way. 2. second step, at the end of the personal work: “ E1 is a straight line ” or “ E1 is a plane ”. This knowledge can be different for different students; from this difference comes the possibility of a conflict. This knowledge is already the result of some work of the student; it generally comes from the work on the set E2. As we saw, the students who changed their knowledge can talk about this change.

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3. third step, during the work in small groups: “ E1 is a plane ”. The knowledge is no longer the knowledge of one isolated person. It is shared by the members of the small group, at least in some groups if not in all of them.The knowledge has an history inside the group as it can be seen by the person who observes the work of the group. The shared knowledge has been constructed through verbalisation or through gestures. This is the time when the students experience the necessity of the mathematical statement: in an Euclidian space, the equation 2x - y = -1 has to be the equation of a plane. 4. fourth step, when the teacher operates in the large group: “ E1 is a plane and it cannot be anything else”. At this moment the knowledge is a shared knowledge, even for those who had not constructed it in their small group. There is something more: the teacher, taking argument of what had been said by the students in their reports, makes visible for every one the fact that some groups experienced the necessity of this statement. 5. the role of the fourth step

This fourth step has a most important role in this device, and we will examine it now. We would like to emphasize the fact that this step is made possible by the three previous ones and that without them all it would be impossible for us to reach our aim. We don’t content ourselves with some statement like: “ we came to the result that, in a space, ax + by = c is the equation of a plane, as well as ax + by + cz = d ”, possibly adding a demonstration of it. We also tell the story of this knowledge, in the way that the students experienced it. If we don’t talk about this story, we take the risk that the shared knowledge of the fourth step will very quickly vanish as it has already vanished several times. Remember that our students have learned about equations of planes several times already. We hope that the link between the knowledge and the story of the way in which it had been built, and especially the experience of its necessity, will give this knowledge a special quality and make it more stable. Conclusion • It sems impossible to organize such a teaching for each concept that the students have to learn. In fact this would not be useful. We want the students to know that mathematical statements are necessary and that this character of necessity is a quality of mathematical knowledge which cannot be taken apart, otherwise mathematics would not be mathematics anymore. It seems quite obvious to us that, if we take the opportunity to make the students experience this necessity for some statements, they will know that this characteristic is common to all mathematics.Then we can choose some knowledge for which a conflict of the type that we described here can occur. In R.R PME 2000 C.S.

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fact, depending on the level of the students, we work in the same way on different exercices. For instance, with 15 years old, we used the same device to have the students work on inequalities. At this moment, we are working on functions and derivatives with students in the last year of high-school. • One month after the end of this teaching we gave a questionnaire to the students asking what they remembered of their work, and what had been the most important moments for them during this teaching. Their answers to the questionnaire gave us three types of information: First of all, concerning the knowledge itself, they remember that they changed their mind about the missing variable: “ when a variable is not present in the equation, this doesn’t mean that it is zero, but, on the contrary, it means that this variable can have any value ”. Concerning the necessity, some of them wrote: “ stepping back, and discussing with others, it became obvious to me that 2x - y = -1 had to be a plane ” . Finally, concerning the role of others, we found this type of remark: “ I found out that, working together, we managed to find the correct answer, although at the beginning we were all wrong ”. This is important to us, as it illustrates the fact that the “mathematical reality“ is met through the discussion with others. • In the introduction, we explained, that the character of necessity of a mathemetical statement has to be experienced, and cannot be just “told“ by the teacher. This doesn’t mean that the teacher has no role to play in such a teaching device. Quite the contrary. During phase four the teacher does two things: he says what is the correct knowledge and he says that this knowledge could not be different. The first thing could, in some situations, be said by some students who could give a demonstration. When saying this second thing about necessity, the teacher can take argument of the experience that some groups have had of this fact. This makes a difference from a situation where the teacher would say it and no student really knowing what it ment. In this device quite a lot of small groups already know that such a set “has to be a plane“. When the teacher says it, he can refer to this experience which is common to many students. And from that moment on, when the mistake appears in the classroom, it is generally a student who says: “ remember , we all agreed that it could not be a line, it had to be a plane ”. The teacher is no longer the only person who “knows“; the knowledge, together with its necessity, is a common knowledge. In that way, one can say it is a mathematical knowledge. References Bartolini-Bussi M. G. (1991): “ Social Interaction and Mathematical Knowledge ”, Proceedings of PME XV, (Assisi), Furinghetti F., Dir., Università di Genova, Italy.

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Brousseau, G. (1997). Didactique des mathématiques, 1970-1990. Translated and edited by Balacheff, M. Cooper, R. Sutherland & V. Warfield. Kluwer (Dordrecht). Drouhard J-Ph. (1997): “ Communication in the classroom on necessary mathematical statements: The double didactic pyramid model. ”, Proceedings of the 1st European Research Conference on Mathematics Education, Podébrady, J. Novotna & M. Hejny´ (Eds.), CharlesUniversity, Prag. Drouhard J-Ph. & Sackur C. (1999) : “ The CESAME Project: Mathematical Discussions and Aspects of Knowledge ”, in Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education, O. Zaslavsky (Ed.), Technion, Haifa, ISSN 0771-100X, 1-272. Drouhard J-Ph., Sackur C., Maurel M., Paquelier Y. & Assude T. (1999): “ Necessary Mathematical Statements and Aspects of Knowledge in the Classroom ”, CERME-1, Osnabrück, The Philosophy of Mathematics Education Journal, 11, Paul Ernest (Ed.), http://www.ex.ac.uk/~PErnest/, ISBN No. 0 85068 195 2. Ernest P. (1996) “ Varieties of Constructivism: A Framework for Comparisons ” In Theories of Mathematical Learning, Steffe L., Nesher P., Cobb P., Goldin G. & Greer B. (Eds). Mahwah, NJ: Lawrence Erlbaum Associates. Ernest P. (1997): Social Constructivism as a Philosophy of Mathematics, Albany, New York: SUNY Press. Piaget, J. (1981 & 1983) Le possible et le nécessaire I & II. Paris: PUF. Von Glasersfeld E. (1996) “ Aspects of Radical Constructivism and its Educational Recommendations ”. In Theories of Mathematical Learning, Steffe L., Nesher P., Cobb P., Goldin G. & Greer B. (Eds). Mahwah, NJ: Lawrence Erlbaum Associates. Wittgenstein L. (1978) Remarks on the Foundations of Mathematics. MIT Press (Cambridge, Mass.).

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