Different Representations in Instruction of Vertical Angles
Tsamir, Pessia, Focus on Learning Problems in Mathematics
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". …