Haddas N., Hershkowitz R. (1998) Proof in geometry as an explanatory and convincing tool. PME XXII, Stellenbosh, South Africa. Vol 3, pp. 25 - 32.
	Abstract The goal of this paper is to show some mutual relationships between design of activities aimed to put students in geometrical situations where they feel the need for proof, and cognitive research on students' actions in such activities. We exhibit an example of the development process of such an activity in several cycles of design, experimentation and analysis, by describing the work of two pairs of students in two different cycles of the development. We also analyze the task as well as the work of the students. Finally, we discuss principles of activity design for leading students to feel the need for and to produce proofs by deductive reasoning.
© N. Haddas & Hershkowitz R.

Introduction For generations proofs were considered as tools for verifying mathematical statements and showing their universality. Hanna (1990) mentioned Leibniz who believed that "a mathematical proof is a universal symbolic script which allow one to distinguish clearly between fact and fiction, truth and falsity"(p.6). According to this approach the two classical roles of teaching proofs were to teach deductive reasoning as part of the human culture to be learned by human beings, and deductive reasoning as a vehicle for verifying and showing the universality of geometrical statements whereas experimenting, visualizing, measuring, inductive reasoning, and checking examples, were not counted for this purpose.    Recently there has been a change in this approach; it seems that there is a consensus that deductive reasoning -- proving -- still has a central role in geometry learning. However, the classical approach is now enriched by new facets and roles. The most important contribution is due to the development of dynamic geometry software such as the Geometry Inventor (Logal, 1994). A main pedagogical feature of many dynamic geometry based learning environments is that students are partners in the discovery of geometrical facts and in the reinvention of geometrical relationships, by means of exploration and inductive reasoning (Goldenberg & Cuoco, in press).    As a consequence of changes in mathematics and mathematics education, which where amplified by the existence of these tools, two roles of proof will be discussed here: Convincing - deductive reasoning as a vehicle for convincing oneself and others that a geometrical statement which was rediscovered or conjectured is true, and explaining why the rediscovered conjecture is true (Dreyfus & Hadas, 1996).    Recently, it has been argued that when Dynamic Geometry software is used in teaching geometry, conviction can be obtained quickly and relatively easily (by induction and experimentation), and therefore proof will become less important. For example, de Villiers (1997) argues that in learning with or without dynamic tools, convincing someone of the truth of a conjecture should take precedence over the process of proving, and the only important role of proof in a dynamic environment is as a tool for explaining.    We claim that, by careful design, based on experimentation and cognitive analysis of students' actions, situations can be constructed where students will had a need for proof, both for conviction and explanation: (i) Conviction by explanation in situations where the findings themselves, even while working in a dynamic geometry environment, are not convincing. (ii) The need for explanation in cases where the findings are convincing but surprising, and the surprise causes the need to understand why. In this paper, we will exhibit an example of the development process of such an activity which has several cycles of design, experimentation and analysis. We will describe the work of two pairs of students in two different cycles of the development, and analyze the task as well as the work of the students. Finally, we discuss principles of activity design for leading students to feel the need for and to produce proofs by deductive reasoning. The activity About 6 months into a geometry course with the use of the Inventor, two pairs of 9th graders, worked on the following activity:  Task 1. Which values can the base angle of an isosceles triangle have? explain! Task 2a. Follow the median and the angle-bisector from the same vertex, in a "dynamic triangle". What can you say about the triangle when both segments coincide? Task 2b. Draw the median and the angle-bisector from the other two vertices. Try to find a situation where two pairs coincide and the third pair does not. Explain! Task 3. Divide the side AC of a dynamic triangle into 3 equal segments. Connect the division points to the vertex B (Fig. 1). [Editor note: In this CabriJava figure, drag the vertices of the vertices of the triangle in order to explore the properties (to ensure the best animation, get MRJ 2.1.4). Click twice on the picture to get an enhanced interface.]   Figure 1 Investigate the relationships between the sizes of the 3 varying angles (<ABD, <DBE, and <EBC). Explain!   Interviews with two pairs of students working on this activity were videotaped and analyzed. Here we will discuss mostly the third task, for which the first two form part of the "pedagogical history". It is worth noting that in a previous lesson the students were engaged in another activity in which they concluded that if two triangles are equal in two sides and the angle opposite one of the two sides, then either the triangles are congruent, or the sum of the two angles opposite the second side equals 180o. The interviews First pair of students In task 1, Tammy and Shiri, two girls, concluded that the size of the basis angle in an isosceles triangle varies between 0o and 90o. In task 2a. they concluded that when the median and the angle bisector from a vertex coincide, then the triangle is isosceles.    After drawing Figure 1, they soon guessed that the three angles have to be equal. Using the measuring tools on the 3 angles, they were surprised for the first time. The following are some excerpts from the interview (I= interviewer): I: Try, if you can, to find situations where the 3 angles are equal and characterize these situations. The girls changed the triangle by dragging and watched the change of the angles visually as well as numerically. Shiri: There is a situation where all of them are equal. Tammy: When A and C (the vertices) are moving apart the middle angle takes all the angles (she means degrees) from others two. I: What will happen if you will drag only A? Tammy: Then the outside angle (<ABD) will become very small and will not be equal to the other two. May be when <B will be very small the angles will be equal (she dragged B until the triangle shrunk to a segment). Oh, but it is not a triangle anymore. Tammy who relates consistently to the process of the visual change she creates, accompanies all her claims by hand movements. Shiri, who was observing Tammy's actions quietly, interferes: The three triangles can't be congruent. I: Why? Shiri: Let's take an example where the two angles are equal (points on <ABD and <DBE) than AD is the angle bisector and median as well, and DABE is isosceles. If the other two angles are equal as well than DDBC is isosceles as well, and then the three triangles are congruent. In this situation we will have 6 equal angles (the marked angles in Fig. 2) ... but, we saw before (Task 1) that the base angles in isosceles triangle are always less than 90o, and here the two are adjacent angles and this is impossible situation. The two girls are discussing together, Shiri explains again her claims to Tammy and writes down her explanation.   Figure 2 We point out two main issues here: The students' need to be convinced by an explanation and the resources they make use of in their explanation. The students' need to be convinced arose from their surprise when obtaining unequal angles and, even more so, by their failure to find even one situation where the angles are equal. The fact that it is impossible to check all cases made the students feel the need for general considerations whether there is a situation with three equal angles, and if not, why not. Tammy, who was more active at the beginning, tried use the dynamic tool to visualize extreme cases. When both girls saw that they cannot get to a final conclusion by working with the software only, Shiri, the silent but very involved partner, took over and began to propose deductive arguments. The resources for her deductive explanation were the history of what they did in the previous two tasks (and in earlier activities). Another very crucial point in this situation is that the students were fully convinced only by the deductive explanation--proof -- even if it is not written formally, and if not all details are specified. Second pair of students. In task 1. Tommer and Lior, two boys, saw immediately (without using the software) that the size of the base angle in isosceles triangle is changing between 0o and 90o. In task 2a. they used the software to explain that one gets an isosceles triangle when the median and the angle bisector from the same vertex coincide: They did it by reflecting the triangle.    The interviewer presented task 3 by drawing a sketch of figure 1. Lior thought that the three angles were equal, and Tommer responded: "Why? It does not have to be like that because if triangle ABE is not isosceles, than BD the median does not have to be an angle bisector." The boys then used the software to find situations where the three angles are equal. They conjectured that this will happen when the triangle is equilateral and they checked their conjecture with the software . When they found that the angles are very close (19.1o, 21.8o, 19.1o), but not equal, they returned to the general triangle to look for cases in which the angles are equal. The interviewer suggested to use the graph option and asked: Suppose we will draw three graphs, describing the change of each angle when AC/3 (the opposite segment) is changing , in the same coordinate system. What on the graph will show situations where the three angles are equal? Note: In the graph option, the graph of the changing variables is drawn in real time, while dragging the geometrical shape. Tommer replied: This will happen when the three graphs will have a common point. Guided by the interviewer the boys built a triangle were AB=4 and <A=30o. While drawing the graphs they guessed and discussed the characteristics of the graphs. After drawing the first two graphs, (See Fig. 3), Tommer claimed: There is no need to draw the third graph ,because it will pass through the common point of the first two. So, we have the situation we look for.     Figure 3 The boys did not looked at the figure itself, nor at the measurements they had, to confirm or refute the above. However, Lior continued with drawing the third graph (Fig. 4), which "surprisingly" did not passed through the common point of the other two.   Figure 4 They concluded that they will never have three equal angles, and started to look for explanations, in spite of the fact that the Interviewer remarked that they had investigated a special case of triangles only. Lior dragged the triangle to match the intersection points of the graphs, and started to speak about congruence triangles, but was stopped by Tommer who exclaimed: I know, I know! and explained: If DB is both median and angle bisector in DABE, than the triangle is isosceles and BD is a height as well, and the same for BE in DDBC, and we cannot have two perpendiculars to AC from vertex B. It seems that Tommer realized that each intersection point expressed a situation of an isosceles triangle. Lior, like the two girls above, was expecting the 3 angles to be always equal, while Tommer realized from the beginning this is not necessarily the case. But Tommer was also convinced that there are situations where they are equal. The fact that the three angles in the equilateral triangle are very closed (19.1o, 21.8o, 19.1o), supported their intuition that they can find a case of equality, and therefore Tommer even suggested not to draw the third graph. The real surprise, in this interview, arose only when the boys realized that the three graphs do not have a common point. In spite of the fact that they were already convinced that the three angles will never be equal, they felt the need to understand why and to explain it deductively. Tommer's resources of explanations resulted from the action of matching between the different representations of the intersection points, and like Shiri's, from previous tasks, in particular task 2. Conclusions The goal of this paper is to show some relationships between the design of an activity with certain pedagogical purposes, and cognitive research on students' actions in this activity.    We started from a pilot version of the activity designed, according to our beliefs and experience, to fit the learning of proofs as a tool for explanation and conviction. This pilot version formed the basis for the first interview. The analysis of the interview lead us to a second version of the activity, according to which the second interview was planned and carried out, (as well as analyzed in view of a third version of the activity). We will now summarize the first two cycles, the final version (at least for now), and some global principles for the design of activities for meaningful learning of proofs in geometry with a dynamic tool.    As expected, the activity in its first design resulted in a surprise for the students. This surprise caused the need for an explanation in order to understand the surprising findings, and to be convinced of their truth. Clearly, the two girls in the first interview were not satisfied by visual considerations, while dragging the vertices and changing the triangle, and therefore moved quite smoothly to deductive explanations.    We decided to enlarge the investigated problem situation by inserting an additional tool -- the graphical representation of the varying angles. In this we had a twofold goal: (i) To create an additional potential source for explaining and/or convincing. (ii) To expand the students' conceptions of the variables, the way they are varying, and the relationships between the geometrical, the numerical, and the graphical representations of this variation. The graph option in the second version, was tried in the second interview. Although it had a great power of conviction for these students, the need for explanation arose even more explicitly. It is worth to note that the matching between the intersection points and the corresponding geometrical situations was an additional source for constructing students' explanation, which in the end was built deductively.    We decided to add another item in which we will try to persuade students to check the graphs for additional cases of the givens, for example, to attempt to make the three intersection points close by choosing a larger value for <A. After checking the 3 graphs for few similar situations, we expect students to realize that such checking can not solve their uncertainty, and that only the understanding why the three angles will never be equal has the power of convincing. Goldenberg, Cuoco, and Mark (in press) said that "a proof, especially for beginners, might need to be motivated by the uncertainties that remain without the proof, or by a need for an explanation of why a phenomenon occurs. Proof of the too obvious would likely feel ritualistic and empty". There is no doubts that a situation like in the above activity, where one can not find any example for her/his conjecture, has the potential to create uncertainty of this kind. In conclusion we would like to sum up global features of "activities that induce the need for explaining and convincing" which were demonstrated by the development of the above activity.    The need for explanation was raised by: a surprise caused by the contradiction between the conjectures and what students got (or could not get) while working with the dynamic tool. a situation where one can not find any example for a conjecture he or she made. the multiple representations of the situation (geometrical, numerical and graphical).    The resources for explanation and conviction. were constituted by: Conclusions based on previous tasks. The activity is planned as a sequence of interrelated from a contextual point of view, as well as by their potential to serve as resources for explanations . The possibility to have analogies in the three representations: in our activity the numerical and geometrical meanings of the intersection points of the graphs. References De Villiers M. (1997). The role of proof in investigative, computer- based geometry: some personal reflections. In J. King & D. Schattschneider (Eds.) Geometry Turned On! Dynamic Software in Learning, Teaching, and Research (pp. 15-24). Washington, DC: The Mathematical Association of America. Dreyfus T., Hadas N. (1996). Proof as an answer to the question why. Z.D.M. International Reviews on Mathematical Education, 96(1), 1-5. Goldenberg E. P., Cuoco A. A., Mark J. (in press). A role for geometry in general education? In R. Lehrer & D. Chazan (Eds), Designing Learning Environment for Developing Understanding of Geometry and Space. Hillsdale, NJ: Erlbaum. Goldenberg E. P., Cuoco A. A. (in press). What is dynamic geometry? In R. Lehrer & D. Chazan (Eds), Designing Learning Environment for Developing Understanding of Geometry and Space. Hillsdale, NJ: Erlbaum. Hanna G. (1990). Some pedagogical aspects of proof. Interchange, vol. 21, no 1 (pp. 6 - 13). Geometry Inventor (1994). Logal Educational Software and Systems Ltd.