**Overview and Qualitative results for the task**

Most of participants can draw the regular hexagon shape by using enclosing circle. In the diagnosis of color patterns some of them have difficulties. Horizontal or vertical axis of symmetry in the shape is not according expectations of them. The data in table 1 shows that there are some students who identify incorrect axis.

Most of the pre-service teachers, as it shows in table1, have a weak knowledge about symmetry axis. They use their own schooling geometry and are presented with opportunities to refine it through teacher preparation programs in university.

According to table2, most of in-service teachers don’t be able to do correctly. It shows that their teaching experiences just help them in the drawing the shapes, and it's not enough to solve the task.

Then the results were measured with a chi-square test to compare the results of both of them. A chi-square (*χ*2) statistic is a measure of the difference between the observed and expected frequencies of the outcomes of a set of events or variables.* χ*2 depends on the size of the difference between actual and observed values, the degrees of freedom, and the samples size. *χ*2 can be used to test whether two variables are related or independent from one another or to test the goodness-of-fit between an observed distribution and a theoretical distribution of frequencies.

The results of the Chi-square test for comparing the correct answers of pre-service and in-service teachers are presented in Table 3. For each of the 6 sections of the task presented to the subjects, in columns 3 and 5, the number of correct answers and columns 4 and 6, the number of incorrect answers in each group can be seen. There was no significant difference between the two groups in drawing the correct shape (χ2 (1) = 3.07, P> 0.05). The performance of the two groups of teachers in coloring was not different (χ2 (1) = 0.01, P> 0.05). There was no difference between the teachers in the four Chi-square tests related to the axis of symmetry (P> 0.05).

**Overview and qualitative results for the task**

The results were qualitatively classified into four types, which named codes. These categories were examined in the total answers and the type of response of the participants, regardless of their category (pre-service or in-service). Based on the detection of staining and the correct measurement of the axis of symmetry, these categories were defined. Following the patterns and finding the right answer and complete conclusion were the main materials of these categories.

**Code 1:** Losing the pattern and don’t understand what symmetry is in shape, most of them don’t know how can follow the painting patterns.

Figure a ,b

Participants don’t understand the template patterns for painting; therefore have misunderstanding on doing task. Most of in-service teachers did in this way. It shows that Experience alone is not enough for teaching the concepts.

**Code 2:** Participants who follow the painting pattern in correct way but can't develop finding right numbers symmetry axis:

Figure c ,d

They understand patterns and do it in correct way, yet finding the symmetry axis shows that they don’t have correct idea about it. In this code both of two groups of participants exist.

**Code 3:** Participants who follow the painting pattern in correct way and develop finding right symmetry axis just in horizontal and vertical:

Figure e, f

Most of pre-service teachers that take part in this study imagine that there is just horizontal and vertical symmetry axis.

**Code 4:** They do it in correct way. The correct painting followed and specified axis in right way.

Figure g ,h

They found a correct and deep understanding and perspective about the task and did their homework in the best way.