## 4 Findings
### Defining Congruence in General and the Congruence of Triangles in Particular
#### 4.1.2 Defining the Congruence of Triangles  
Through concrete and visual work with ornamentation, students in all the groups were able to define congruent triangles. The students cut out and covered triangles in the ornamentation, and drew new ornamentation using triangles and their reflections. They concluded, as presented in Episode 4, that there are many different congruent triangles.  
Episode 4: The definition of congruent triangles in Group A  
(54)  
[PERSON]:  
Triangles 1 and 2 (see Figure 1) are congruent, because they cover each other completely.  
(After cutting out the numbered triangles in the ornamentation.)  
(68)  
[PERSON]:  
There are other congruent triangles in the ornamentation.  
(74)  
[PERSON]:  
The vertex of the triangle must be D. (Drawing a triangle that is congruent to triangle ABC; see Figure 2)  
(82)  
[PERSON]:  
I drew a triangle in the opposite position to the ABC triangle.  
(83)  
Teacher:  
You can check if they are congruent.  
(84)  
[PERSON]:  
I can do reflection.  
(91)  
[PERSON]:  
All these triangles are congruent.  
(94)  
[PERSON]:  
All these triangles are equal in their sides and angles.  
Episode 4 shows that [PERSON] (54) understood the meaning of congruent triangles through her concrete experience, enabling her to define congruent triangles correctly. [PERSON] (68) showed her understanding, too, by indicating that there are congruent triangles different to the ABC triangle. [PERSON] (74) inferred what the position of the vertex should be, in order for two triangles to be congruent. [PERSON] (82) showed her understanding, by indicating that she succeeded in drawing congruent triangles that were in different positions. This indicates that she knows the conditions for congruent triangles. [PERSON] (84), using her imagination, argued that reflection could help her in checking the congruence of triangles. [PERSON] (91,94) formulated the properties of congruent triangles.  
Our analysis of all the students' answers in questionnaires 1 and 2 indicated changes in how they defined congruent triangles. Before the learning process, students' definitions were organized into the following six categories: equal sides, equal angles, similarities, complete covering, covering of the sides, and covering of the angles. However, after the learning process, we identified only four categories of definition. Table 1 presents the distribution of students' definition categories before and after the learning process.  
Table 1 shows that, before the learning process, students either did not provide definitions of congruent triangles, or their definitions related to similarity only; whereas, after the learning process, students' definitions related more to the coverage of triangles.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Category** & **Before Learning** & **After Learning** \\ \hline Equal sides: When the sides of the triangles are equal, & 6.6\% & 0 \\ \hline Equal angles: When the angles of the triangles are & 13.3\% & 6.6\% \\ equal, they are congruent. & 20\% & 10\% \\ \hline Coverage of sides and angles: When the sides and & & \\ angles of two triangles cover each other, the two & 10\% & 33.3\% \\ triangles are congruent. & & \\ \hline Complete cover: Triangles are congruent when one & 6.6\% & 50\% \\ covers the other precisely. & 43.3\% & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Students’ definitions of the congruent triangles, before and after the learning process.  
Figure 1: First ornamentation.  
Figure 2: Second ornamentation.