Haddas N., Hershkowitz R. (1999) The role of uncertainty in constructing and proving in computarized environment. PME XXIII, Haifa Israel. Vol.3 pp. 57 - 64.
	Abstract The role of uncertainty in promoting the need to prove, in the sense of explaining and convincing in geometry, has been discussed by many researchers (e.f., Chazan, 1993; Dreyfus & Hadas, 1996; Goldenberg, Cuoco, & Mark, 1998). We demonstrated (Hadas & Hershkowitz, 1998) that uncertainty stems from a geometrical situation in which students cannot find any example confirming their intuitive conjecture, because such an example does not exist. In the present article we describe a case in which students are engaged in a construction task in a situation of uncertainty: the construction was possible, but was opposed to students' intuitive conjectures.
© Haddas N. & Hershkowitz R.

Background Yerushalmy, Gordon & Chazan (1993) distinguish two kinds of geometrical problems in computerized learning environments: construction problems and conjecture problems. In both, one has also to consider the different roles of proof; i.e., as a tool to show the universality of a statement and as a tool to explain (Hanna, 1990). In many conjecture problems students feel that the universality of the conjectured attribute of the geometrical object is confirmed by the computerized environment. The dragging operation on a geometrical object enables students to apprehend a whole class of objects in which the conjectured attribute is invariant, and hence they are convinced that their conjecture will be always true (De Villiers, 1998). Therefore the only motivation driving students to prove is to explain. We have already described (Hadas & Hershkowitz, 1998) an example of a conjecture problem, in which the conjectured attribute does not exist, hence the dynamic tool does not play a convincing role. Such a problem is a good didactic opportunity to make the student aware of the role of explaining by proving, as a convincing tool.    In construction problems the universality and the explanation are interwoven. Schoenfeld (1986) has described two construction problems to support his claim that "empiricism is an essential component of the machinery of deduction, conversely, however deduction makes possible a discovery that is inaccessible to insight or empiricism" (p. 249). In common construction problems in computerized environments, students are asked to construct a figure whose existence is obvious. In the following, we will show students working on a problem in which they are asked to investigate by construction, the existence of a figure satisfying given conditions. The problem - Congruent Triangles In this activity, we will investigate if and when, two triangles having several equal elements, are congruent. Task 1. Given a dynamic triangle ABC, build another triangle having two angles and the included side equal to two angles and the included side of ABC. Task 2. Is it possible to build a triangle with one side and two angles equal to those of a dynamic triangle ABC, but not congruent to ABC? If it is possible, build such a triangle; otherwise explain why . Task 3. Is it possible to build two non-congruent triangles, with five equal elements? Create a hypothesis. Task 4. Is it possible to build two non-congruent triangles with six equal elements? If yes construct two such triangles, otherwise explain why . Task 5. Is it possible to build two non-congruent triangles, with five equal elements ? If yes construct two such triangles, otherwise explain why . Problem analysis: a. Task 1 prepares students for Task 2: they realize that congruence is an invariant attribute under the dragging operation. This means that when one changes the original triangle ABC by dragging, the second congruent triangle changes accordingly. But, the second triangle cannot change when it is itself dragged by one of its vertices. b. When students complete Task 2, they are asked to identify the equal elements, and to their surprise, discover 4 equal elements in the two non-congruent triangles. c. The task of constructing a triangle non-congruent to ABC, with 5 elements equal to elements in ABC, is quite complex. We consequently decided to decompose it into three stages: a conjecture stage (Task 3), a discussion of the case of 6 equal elements (Task 4), and finally the construction of a triangle with 5 elements equal to elements of ABC, although non-congruent to it. Problem Characteristics: (i) As in all other construction problems in such an environment, the computerized tool elicits empirical actions followed by a prompt feedback from the tool. But, for complete success, deductive considerations are needed. (ii) The problem provokes surprise followed by uncertainty. (iii) Students' construction processes depend on their ability to analyze the problem deductively, leading to the elaboration of an existence proof. The last two characteristics are discussed in detail in our analysis of some students engaged in this activity. The Interviews Pairs of Grade 10 students with ability in the upper half of the population, were interviewed while working on the problem. The interviews were videotaped and analyzed. The students were towards the end of a two year geometry course, and were familiar with ways to prove. In the following, we describe two representative pairs of students. The first pair In Task 1 a girl and a boy, Gili and Nadav, constructed a triangle non-congruent to ABC with two angles and the included side equal to the same elements in ABC. They discovered that the new triangle can be modified only by dragging the vertices of ABC, and explained this modification by invoking the relevant congruence theorem. They then struggled with Task 2 for 15 minutes. First they constructed the second triangle (see Figure 1) where: DE=AC; <D=<A; and <F=<B.   The following excerpt expresses the dialectic process the pair underwent. Nadav: They were supposed to be non-congruent but they are [congruent]. [F is a random point on the line DF, They drag F in such a way that FE remains the triangle side. When they change DABC, they had to adjust F again by dragging.] Gili: It's the same again. Nadav: Ah! It is like that because if this one [pointing at <D and <A] and that one [pointing at <B and <F] are equal, the third must be equal as well...If two angles are equal even if they are not at the two ends of the side, the third is equal too. I: So what is your conclusion? Nadav: When we have two (equal) angles and side, and they [the triangles] are not congruent, it is impossible. [Gili starts looking for a way to show that the construction is possible. The interviewer suggests analyzing all possibilities for locating the "second angle".] Nadav: If we copy it [pointing at <B] here [pointing at a point on DF] we get stuck! Gili: If we copy <C here [she points at a point on DF, and starts to construct]. [Nadav who was hesitating so far, starts watching Gili, and after she copies the second angle, exclaims:] Nadav: Ahh! you did it the opposite [meaning not correspondingly]. [The pair complete the construction and check it by dragging. Nadav explains what they did explicitly. The discussion continues.] Gili: So, we succeeded? I: Did you discuss the idea of correspondence between two triangles in the context of congruence in the class? Nadav: No, we didn't. The teacher did and now I understand why. [To their surprise, the students find that the two non-congruent triangles have even 4 equal elements.] In Task 3, the pair hypothesize that a triangle with 5 elements equal to elements of ABC, but not congruent to ABC, can be constructed in a way similar to that in Task 2. The interviewer does not let them try to construct such a triangle and asks them to move to Task 4. They discuss the issue of 6 equal elements in non-congruent triangles. Gili claims that it is impossible to construct such a triangle, but did not justify her answer. In contrast Nadav argues that it is possible and applies the strategy used successfully in Task 2, on a piece of paper (see Figure 2).   The interviewer asks them to analyze their drawing using their geometrical knowledge. Nadav answers immediately: In fact there are 3 equal sides so the triangles are congruent, hence it is impossible with 6 equal elements. Their investigations on Task 5 are based on: (i) considerations they raised in the previous task, (ii) their hypothesis in Task 3 that it is possible to construct a triangle with 5 elements equal to elements of ABC, but not congruent to ABC. They try to apply similar methods to those they used in Task 2. First they plan the construction by erasing the "equal signs" of one pair of corresponding sides, on their drawing in Task 4 (Figure 2). They then try to construct DEF according to this drawing. They succeed in this endeavor, but with 3 equal elements only. They use measuring and dragging operations to try to obtain all the 5 equal elements, but without success.    The students then discuss what they have done with the interviewer and conclude that it is difficult or even impossible to obtain the 5 equal elements in the two triangles by dragging, even if such a triangle exists. They initiate deductive considerations as following: the triangles can have only two equal sides, so they must have 3 equal angles. This means that the triangles are similar, and therefore the ratio between corresponding sides in the two triangles must be constant. Following a suggestion by the interviewer, the students construct an example of two such triangles by choosing (with the help of the interviewer) particular lengths for two of the sides in each triangle (see Figure 3). They deduce that the invariant ratio between the corresponding sides in ABC and DEF must be 1.5, and hence that AB must be 4 and DE, 13.5.   The pair then constructed the two triangles with the software. The following discussion took place. Nadav: But the angles should be equal as well. [He thought for a while and added:] but it is O.K. I: Why are you sure that the angles should be equal? Nadav: Because they [the triangles] are similar. The second pair Two boys, Asaf and Asher answer Task 1 in the same way as the first pair. They also start Task 2 by constructing a second triangle (see Figure 1), in which DE=AC, <D=<A, and <F=<B. Once they realize that they obtain congruent triangles, they immediately conclude that they did it wrong, and replace the third equality by <F=<C. They then succeed in constructing the two non-congruent triangles. The following excerpt came at the end of their discussion concerning 4 equal elements: I: So, how many equal elements do you have in the two triangles? Asaf: Three I: No more? Asaf: The third [pair of] angles cannot be equal because then they [the two triangles] will be congruent. Asher: No because if the third [pair of] angles [are equal] they will be similar and not congruent. I: Can you construct two triangles with two pairs of equal angles in which the third pair is not equal? Asaf: Yes! Asher: No! Asaf: Wow! It's impossible!...Only the sides are not equal, because the sum of the angles in a triangle is 180š. Wow! four equal elements and the triangles are not congruent. In Task 3, they hypothesize that triangles with 5 equal elements are always congruent. This assumption is elaborated in Task 4, for 6 equal elements. The interviewer does not let them leave this issue too quickly, and brings the idea raised by the previous pair of students at this point by asking: Maybe one can construct equal angles not opposite to the equal sides? The students seem intrigued by this idea and draw a sketch similar to Figure 2, discuss it for a while, and conclude that because the triangles have 3 equal sides they must be congruent.    They come back to the issue of 5 equal elements in Task 5, and conclude that if such triangles do exist they must be similar, because the three angles are equal, and thus the ratio between corresponding sides must be constant. After a long dialogue between the two students, in which the interviewer suggests they focus on one special pair of triangles, they succeed in sketching one example, by calculating the ratio between the sides and applying it to calculate the third side in each triangle, - in the same way as the first pair (see Figure 3).    The interviewer checks their awareness of their own actions by asking: What do you think about the angles, must they be equal as well? This pair of students is unable at this point, to refer back to the similarity of the triangles in order to justify the equality of the angles, in spite of their intuitive feelings that this is true. So, they construct the two triangles with help of the software, measure the first pair of angles and find them equal. In the process, the similarity of the constructed triangles becomes visually obvious (see Figure 4).   This visual clue leads the pair, after a while, to deduce the triangles' similarity,as we can see from Asaf's concluding remark: If the ratio between the sides is fixed, then they [the triangles] are similar and then the angles are equal. But, they [the triangles] are not congruent. So, we did it! Discussion The dialectic feeling of uncertainty. As we mentioned above, one of the characteristics of this activity is the feeling of uncertainty concerning the existence of examples fulfilling the constraints. In Task 2, in contradiction to their conjecture, both pairs were surprised to find a way to construct the non-congruent triangle with the 2 angles and one side equal to those given, and even more, to discover that the two triangles have 4 equal elements. This way of copying an equal angle into DDEF in a non-corresponding position, is based on deductive geometrical reasoning. By using it to construct the required triangle and watching the result on the screen, students became confident that their construction was correct. This confidence led them to hypothesize that non-congruent triangles with 5 equal elements can be constructed. The students went even further to claim that when two triangles have 6 equal elements, they are not necessarily congruent, and they even tried to sketch such triangles. However, when they used a congruence theorem, they were confident that 6 equal elements always implies congruent triangles. This raised their suspicion in Task 5, and uncertainty characterized their first steps in this task. Only after they succeeded in designing their construction deductively did they become confident again that such triangles do exist, and insist on constructing them on the computer. The deductive explanations. Different kinds of deductive reasoning occurred in the various tasks. In Task 2, students constructed the DDEF by copying the two angles to two vertices which are not both on the given side, yet obtained a congruent triangle (see Figure 1). This action led them to a different construction, based on deductive considerations. This construction by itself is the deductive solution (an existence proof) to Task 2. The situation in Task 4 is similar to the one we have already described (Hadas & Hershkowitz, 1998), where students could not find any example confirming their intuitive conjecture, because such an example does not exist. Here a non-congruent triangle with 6 equal elements does not exist, so the only way to be convinced is to give deductive explanations. Task 5 has the same characteristics as Task 2; the construction is based on deductive considerations. And yet we saw that these deductive considerations are very fragile. Students needed to reflect on the whole process they underwent, before they became aware of the logical chain they themselves elaborated.    Deductive explanations were not the first weapon used by students in justifying their actions or their planned actions. This finding has been noticed by many researchers (e.f., Schoenfeld, 1986; Hoyles and Jones, 1998) Examples: (i) When asked, after accomplishing Task 2, how many equal elements the two non-congruent triangles have, the second pair spent a long time at arguing before being able to use deductive considerations, based on the sum of angles in a triangle. (ii) The two pairs had difficulties in seeing the logic chain on which the construction of the triangles in Task 5 was based. It was even more difficult to recreate the opposite chain from the constructed triangles to conclude that the triangles must be similar, and hence the angles must be equal.    As described above, students gave deductive explanations of their actions, a fact that made them confident in what they did and in the ways they justified why certain constructions are impossible. References Chazan D. (1993). Instructional implications of students' understanding of the difference between empirical verification and mathematical proof. In J. Schwartz, M. Yerushalmy & B. Wilson (Eds.) What Is It a Case of? (pp. 107-116). Hillsdale, NJ: Lawrence Erlbaum Associates. De Villiers M. (1998). An alternative approach to proof in dynamic geometry. In R. Lehrer & D. Chazan (Eds.), Designing Learning Environment for Developing Understanding of Geometry and Space. (pp. 369-394). Hillsdale, NJ: Lawrence Erlbaum Associates. Dreyfus T., Hadas N. (1996). Proof as an answer to the question why. Z.D.M. 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