Different representations in instruction of vertical angles.

It is widely accepted that teachers should be aware of and familiar with students' mathematical ideas and conceptions, and it is believed that such awareness and knowledge should play a significant role in planning and carrying out instruction (e.g., Kilpatrick, Swafford, & Findell, 2001; NCTM, 1989; 1991; 2000). One source of information regarding students' ways of thinking and their common errors in various mathematical topics is the vast amount of research findings reported in the literature. However, the shift from awareness of and familiarity with students' ways of thinking to designing research-based instruction, i.e., instruction that takes into consideration the findings reported in the literature, is not a trivial one. An important step in this direction might be the identification of general factors that play a role in students' mathematical reasoning. One such factor is the specific way in which the mathematical concepts included in the problems are represented. It has been reported that students tend to provide different and even conflicting solutions to different representations of the same mathematical problem (e.g., Janvier, Girardon, & Morand, 1993; Tirosh & Tsamir, 1996). Clearly, incompatible solutions to the same mathematical problem are not acceptable.

This article demonstrates a way of using knowledge about students' incompatible reactions to different representations of angles to raise students' own awareness of their intuitive ways of thinking, and to guide them towards proving the equality of vertical angles. More specifically, two main questions are addressed.

(1) How can research-based instruction promote students' awareness of their own intuitive thinking? and (2) How can research-based instruction promote students' appreciation of formal proof?

First, students' reactions to different representations of vertical angles are presented by means of a brief description of two related studies. Then, the Vertical Angles Conflict Activity and students' reactions to it are described. Finally, some concluding comments are made.

Students 'Reactions to Different Representations of Vertical Angles

Tsamir (1995) examined the responses of 204 students (Grades 4, 6 and 9) to tasks on comparing vertical angles. The vertical angles were presented in different representations, including two types of "equal arms" representations and one type of "different arms" representation (see Figure 1). In the "equal arms" representations the lengths of the arms of one vertical angle were equal to the lengths of the arms of the other. In the "different arms" representation the arms of one vertical angle were longer than the arms of the other.

It was found that both types of "equal arms" representations of vertical angles triggered "equal" responses (95% and 90% on average, to the "four equal arms" and the "two pairs of equal arms" representations, respectively). The "different arms" representation, however, triggered higher percentages of "unequal" responses (about 40%, 35% and 25% of the 4, 6, and 9th graders respectively).

On average, 45% of the participants correctly justified their correct responses to the two "equal anus" representations. Since only 9th-grade participants had studied the theorem regarding vertical angles, it was expected that only they would use the theorem in their justifications. The younger students' correct justifications were based on measurements and on the claim that the two angles had "the same opening" between their arms. Incorrect justifications were mostly based either on length or on area considerations. When relating to the 'equal sides' representations, typical claims were "the angles are equal because their arms are equal," or "the angles have an equal area enclosed by the angle's arms and the segment connecting the endpoints." When relating to the 'different sides' representation those students explained that "the angle with the longer arms is larger", "the angle with the longer segment connecting the end points of the drawn anus is larger", or "the angle with the larger area enclosed by the angle's arms and the segment connecting the end points is larger". While the application of any of the above mentioned justifications to each of the "equal sides" representations yielded "equal" responses, its application to the "different arms" representation yielded "unequal" responses.

The findings regarding students' reactions to the three representations of vertical angles were used in constructing a Vertical Angles Analogy Activity (Tasks 1-3 in the Appendix, see also Tsamir, 1999). The main aim of this study was to examine the possibility to trigger intuitive "equal" responses to the counter intuitive representation of vertical angles. Twenty-one 7th graders were individually interviewed. The first four interviewees were asked to justify their solutions to each of the tasks. They provided "equal" responses to the two "equal arms" representations and "unequal" responses to the "different arms" representation. During the fifth student's interview there were some unexpected interruptions, and counter to the original design, she was not asked to justify her responses. She was presented with Task 2 (the bridging task) immediately after being presented with Task I (the anchoring task), and immediately after responding to Task 2 she was given Task 3 (the target task). The entire process was ve ry fluent and tight, and the student intuitively gave "equal" responses to all three representations.

Since "the analogy" study aimed at examining the possibility to trigger intuitive "equal" responses to the counter intuitive representation, it was decided to investigate the impact of the "fluent version", which had accidentally evolved in the above described situation, on the other students. The Vertical Angles Analogy Activity was subsequently conducted by presenting the three tasks in close succession, with no distracting questions, asking merely for responses and no justifications. All 16 students confidently responded that the angles in Tasks 1 and 2 (first two "equal arms" representations) were equal. Twelve students also claimed that the angles in the last, "different-arms", representation were equal, but the other four hesitated and provided no response to the counter intuitive, "different-arms" representation in Task 3. A typical explanation was that "there are insufficient data for comparing the angles."

In conclusion, the findings of the two studies indicated that the "different arms" representation of vertical angles commonly triggered "unequal" responses, but participation in the Vertical Angles Analogy Activity led students to give "equal" responses. It was clear however, that the correct reactions in the latter case were not based on valid reasoning but on perceptual factors. Still, the fact that it was possible to trigger incompatible "equal" and "unequal" intuitive reactions to the same representation of vertical angles was regarded as valuable for instruction. It could, for instance, serve to promote students' awareness of their own inconsistent

responses, and of the need to be consistent in mathematics. Thus, the findings of these two studies were used in constructing a Vertical Angles Conflict Activity (see Figure 2) to promote students' awareness of inconsistencies in their own thinking about vertical angles, and to highlight the need in formal proof. The activity is fully described in the Appendix.

Students' Reactions to the Tasks in the Vertical Angles Conflict Activity

The Study

Twenty-six 7th-graders engaged in the Vertical Angles Conflict Activity. These students had studied different types of angles e.g., acute, straight and obtuse angles, and discussed the notion of adjacent angles. However, they had not been exposed in class to the notion of vertical angles or to the vertical angles theorem.

Each student was interviewed individually for about 30 minutes. The interviewer tried to say as little as possible, and usually addressed each student in a similar manner (see Appendix). During the interviews, the students had access to additional papers, pens, protractors, and a calculator for any necessary calculations. The interviews were audiotaped and transcribed.

Results

Seven of the 26 students provided consistent solutions to the "different arms" representation in Stages I and II (Tasks 3 and 4, see Figure 3). They either claimed that the vertical angles presented in the "different arms" representation were equal because "the length of the arms is irrelevant" (four students) or that "there is insufficient information to determine whether the angles are equal" (three students).

Obviously they were not aware of the issue of inconsistencies of one's own thinking about vertical angles.

The other 19 students gave contradictory responses to the "different arms" representation of the comparison of vertical angles tasks at Stages I and II. While at Stage I, following the Vertical Angles Analogy Activity, all claimed that the angles in the "different arms" representation were equal, at Stage II, after re-examining the problem, they changed their minds and explained that the angles were unequal. All these students accompanied this change by expressions of their awareness of the incompatibility of their solutions, and of the need to choose one solution. Typical reactions by the end of stage II were "Strange, for a moment I thought that the angles are equal, but they are probably not [pause]. They can obviously not be both [equal and unequal];" or "It is clear that 'equal' and 'unequal' [responses] are impossible for the same pair of angles. That would be a contradiction."

The interviewer challenged these 19 students by asking each of them to suggest a way to convince a hypothetical puzzled classmate about the correctness of their solution. Nine of them hesitated for several moments and then admitted they were still perplexed (see Table 1). Typical claims were, "For a moment my second solution seemed correct, but I am not sure. I feel that the answer is 'unequal', but I do not have a clue how to convince a friend"; or "Actually, I was quite sure of my solution at each stage, and I am afraid that you will provide a third suggestion that will look just as reasonable." Some suggested, "We can measure and see", but added upon reflection, "actually, measuring is no proof..." and one participant said, "Before convincing a friend I have to convince myself, which is more problematic".

Most of them expressed the need to study the topic, "I have not yet studied any related geometrical rules, so I do not know how to decide"; or "I probably have to study the relevant theorems."

Among the other ten students who gave contradictory responses and were asked to convince a friend, three ignored their "unequal" responses. They moved back to claiming that the angles were probably equal "because they look equal, and the length of the arms should not be considered in the comparison of angles." Seven students, however, remained impressed by their second stage, "unequal" responses, saying that their previous explanation (larger arms, area, etc.) should have sufficed to convince their friend.

At Stage III, all students reached "equal" solutions to the "different arms" representation of the vertical angles. They analyzed the process they went through and the contribution of the third stage saying, "I was actually proving by substituting. The different substitutions indicated how the general proof should run". In their explanations, students said that this stage "proved", "convinced" or "showed beyond any doubt" that the angles were equal. They commented that, "The adjacent angles idea provided was a reliable way to determine this equality;" or "This part suggested that this is definitely the correct solution."

Eleven students volunteered reflective remarks regarding the entire process they went through. They criticized, for instance, their own readiness to shift between responses: "Quite amazing how easily I shifted from 'equal' to 'unequal' responses;" or "It is shocking that I felt good about each of my different solutions;" or, "When I noticed that it took me no time to confidently come up with 'equal' and 'unequal' responses to exactly the same question I felt quite embarrassed [pause]. Even more troubling was the fact that I unwaveringly abandoned my correct response at Stage I for an incorrect response

Concluding Comments

In the introduction two main questions were posed: (1) How can research based instruction promote students' awareness of their own intuitive thinking? and (2) How can research-based instruction promote students' appreciation of formal proof. The discussion of the findings relates to these two questions.

The findings indicate that research-based instruction could encourage students to reflect on their own intuitive grasp of vertical angles. That is, participation in the Vertical Angles Conflict Activity indeed led a substantial number of students to notice their own tendency to give incompatible, representation-dependent responses, and to identify their readiness to neglect correct intuitive responses in favor of incorrect ones.

Traditionally, when the teaching by cognitive conflict approach is used, students are first presented with a task that is known to intuitively trigger an incorrect response, and then with another representation of the same task, known to intuitively trigger a correct response. As a result students will frequently come up with different solutions. During clarification the conflicting elements in the different solutions of the same task are identified and interpreted as problematic. Finally, the conflict is resolved according to the relevant mathematical theory (e.g., Swan, 1983; Tsamir & Tirosh, 1999). Teachers are challenged to convince students that the different representations relate to a task that is mathematically the same, to promote students' awareness of their own incompatible ideas, to clarify that such incompatibility is problematic in mathematics, and to help the students resolve the conflict in accordance with formal theory.

The activity presented here suggests another type of sequence for the teaching by cognitive conflict approach. We started by triggering intuitive, correct "equal" responses to the counter-intuitive representation. Then, Stage II triggered "unequal" responses, and Stage Ill led to formal proof of the "equal" responses. Consequently, here, two optional points of conflict are available. The one, for those who would give, for instance, "equal" responses at Stage I and "unequal" at Stage II. The other, for those who would give for instance "impossible to determine" responses at Stage I, "unequal" responses at Stage II, and "equal" responses at Stage III. This creates several possibilities for many students to experience conflict, or to take part in a vivid, relevant class discussion that highlights various intuitive pitfalls.

Moreover, as mentioned before, the first challenge that teachers face when using the traditional cognitive conflict approach is to convince the students that the different representations, to which they gave incompatible responses, are actually the same mathematical task. In the Vertical Angles Conflict Activity, by contrast, students were led to give clashing responses to the same, counter-intuitive representation of vertical angles. As a result, it was not hard for them to see that it was the same task to which they gave "equal" and "unequal" solutions. They also easily recognized the two solutions provided as incompatible, and commented on their own initiative that it is impossible to accept both solutions in the given situation.

Another marked advantage of the Vertical Angles Conflict Activity was the fact that students themselves requested formal proof. In Israel, students usually prove the equality of vertical angles by the end of Grade 8 or at the beginning of Grade 9. It is the first geometrical proof constructed in class.

The drawing of the vertical angles accompanying the proof is traditionally the 'identically represented' one. Consequently, students tend to intuitively regard the vertical angles as equal and do not feel a need for formal proof (see also. Fischbein, Tirosh & Melamed, 1981). Still, when such students are presented with "differently represented" vertical angles, a non-negligible number incorrectly claim that the angle with the longer arms or the one with the longer arc is larger (see, Tsamir, 1995).

In the case of the Vertical Angles Conflict Activity, students were exposed to their own intuitive thinking. Using Tall and Vinner's terminology (1981), the students were exposed to their own concept image of angles, which included reference to irrelevant characteristics such as the length of the arms and the enclosed area. They became familiar with their implicit tendencies to give incompatible responses even to the same representation of vertical angles, facing the fragility of intuitive, unsubstantiated, mathematical "knowledge" that relies merely on perceptual information. Participation in this activity gave rise to discussions about the important and often problematic role of intuitive thinking, and led to the realization of the need to scrutinize our basic assumptions and most self-evident responses. It was obvious that without such criticism we might give an incorrect response, or even abandon a correct response for an incorrect one, as many in this study did.

Students also realized that awareness of mathematical contradiction or of their confusion did not automatically lead to the correct solution. Consequently, many participants sensed the need for rules and theorems. They called for "something" to indicate the correct solution; occasionally, they explicitly asked for proof Thus, we can claim that they appreciated the formulated formal proof, the way it was presented and also the role it played in convincing them about the correctness of the solution. Discussions gave rise also to the need to go back to the definition of the notion of angle, and "going back to the definition" was accepted as an integral part of the validation process. The formal proof and the definitions were not part of a ritual to which students were occasionally exposed in geometry lessons, where teachers provide proofs to different types of claims, including apparently obvious claims. Here both definition and formal proof came in response to the students' own need for conclusive understanding .

The Vertical Angles Conflict Activity was designed for students about to embark upon the study of Euclidean geometry with reference to formal definitions and proofs in class. In terms of the van Hiele levels, these students were about to engage in activities requiring level 3 or level 4 of geometric thinking (e.g., Clements & Battista, 1992; Crowley, 1987). It should be noted that according to van Hiele, progress through the levels depends more on the instruction received than on age or maturity. The method and organization of instruction, as well as the content and materials used, are important areas of pedagogical concern (Clements & Battista, 1992, p. 427; Crowley, 1987, p. 5). Accordingly, one aim of the Vertical Angles Conflict Activity was to provide an instructional environment for promoting students' geometrical understanding.

The Vertical Angles Conflict Activity appears to be didactically valuable both for teaching the equality of vertical angles as well as for discussing various general, mathematical and psychological issues, such as contradictions, consistency and intuitive knowledge and its role in mathematical performance. Still, even in cases where this type of sequence is applicable, it is for the teacher to decide where, when and under what circumstances such activities should be integrated in teaching. As I said at the beginning, these decisions are not at all trivial. Planning and carrying out instruction is complex and demanding. This article shows how, in order to design instruction that is sensitive to students' ways of thinking and to their common mistakes, teachers need to be familiar with various representations of specific tasks, with common reactions to each representation, and with the possible instructional implications. Teachers should also judge whether a certain teaching approach is the most appropriate appr oach under given circumstances. For example, when teaching low achieving students, building their confidence in their own mathematical competence, one may assume the need to provide them a positive, learning experience with no conflict, so that an approach that includes only Stages I and III would be more suitable for them. Clearly, the impact of different representations on students' ways of thinking, and of taking into account students' common reactions to such different representations in instruction, should be further investigated.

APPENDIX: THE VERTICAL ANGLES CONFLICT ACTIVITY

STAGE I: Triggering "Equal" Solutions The Vertical Angles Analogy Activity

The tasks should be presented in close succession with no discussion.

Task 1: The anchoring task

Examine the following drawing and answer the questions.

Is [alpha] larger than, equal to, or smaller than [beta]?

Task 2: The bridging task

One of the arms of angle [alpha] and one of the arms of angle [beta] are extended by the same amount. Consequently, angles [alpha] and [beta] are represented in the following manner:

Is [alpha] larger than, equal to or smaller than [beta]?

Task 3: The target task

The drawing presented at Stage 2, before, is rotated by 90[degrees], in the following manner.

Is [gamma] larger than, equal to, or smaller than [delta]?

STAGE 11: Triggering "Unequal" Solutions and an Initial Need of Formal

Proof

Task 4:

* Here is the drawing given in Task 3. How would you explain to a classmate why your answer to Task 3 is correct?

* In case your friend believes that the angles are equal (the opposite of what the students suggested in his /her responses), but he / she is not sure why, how would you convince him / her about the correctness of your solution?

STAGE III: Triggering "Equal" Answers. and Formulating the Formal Proof

In Tasks 5 the students are given a transparency with a drawing of two adjacent angles [alpha] and [beta] and sheets of paper with a printed question. In Tasks 6 the students are given another transparency with another drawing of two adjacent angles [gamma] and [beta] (where [beta] is equal to [beta] from Task 5), and a relevant printed question. In Task 7 the students are asked to put the two transparencies one on top of the other to create a drawing of vertical angles, so that angle b from Task 5 covers angle [beta] from task 6. They create a drawing identical to the one presented in Task 3, and to answer the question.

The sequence of Tasks 5, 6, and 7 is performed, in this order, a number of times. Each time the student is asked to substitute another number for [beta] and relate, respectively, to the three tasks. In the first two rounds the substituted values are given by the interviewer. Then, the student is asked to suggest additional values for 8 and complete the solution of Tasks 5 to 7. Finally a possible generalization is discussed.

Task 5:

Here is a transparency with a drawing of two adjacent angles [alpha] and [beta].

Given that [beta]=55[degrees]

[alpha] = ? Why?

Task 6:

Here is a transparency with a drawing of another pair of adjacent angles [gamma] and [beta]. Given that [beta] is the same as in Task 5, that is to say that [beta] = 55[degrees]

y = ? Why?

Task 7:

Put the 2 transparencies in a manner that [beta] from Task 5 will cover [beta] from Task 6. (The following drawing of vertical angles is reached)

From Tasks 5 and 6 we know that y = ? [alpha] =?

Is a larger than, equal to or smaller than [gamma]?

Explain your answer.

STAGE IV: Reflection

Task 8:

Re-examine your responses to Tasks 4 and 7? What would you suggest?

[FIGURE 3 OMITTED]

STAGE I: Triggering "Equal" Solutions--The Vertical Angles Analogy Activity

Task 1: The "easy" (anchoring) task - the "four equal arms" representation

Task 2: The bridging task- the "two pairs of equal arms" representation

Task 3: The "difficult" (target) task- the "different arms" representation

STAGE II: Triggering "Unequal" Solutions and an Initial Need of Formal Proof

Task 4a: Re-examining the "difficult" (target) task

Task 4b: Convincing a friend who responded differently to Task 3

STAGE III: Triggering "Equal" Answers and Formulating the Formal Proof

Task 5: Relating to one pair of adjacent angles (belonging to the vertical angles).

Task 6: Relating to the other pair of adjacent angles belonging to these vertical angles.

Task 7: Combining the two pairs of adjacent angles - examining the vertical angles.

STAGE IV: Reflection

Task 8: Re-examining the students' own responses to Tasks 3, 4 and 7.

Figure 2. The Vertical Angles Conflict Activity

Table 1

Frequencies of students' inconsistent solutions to the
different-arms-representation (N=19)

Task 3                        Task 4a

[gamma] = [delta]  [gamma] [not equal to] [delta]

[gamma] = [delta]  [gamma] [not equal to] [delta]


[gamma] = [delta]  [gamma] [not equal to] [delta]


Task 3                         Task 4b                  No. of Students

[gamma] = [delta]  Do not know (need to study)                 9

[gamma] = [delta]  [gamma] [not equal to] [delta] (See
                   justification in Stage II)                  7

[gamma] = [delta]  [gamma] [not equal to] [delta]
                   (Ignore the lengths of the sides)           3

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