Triangle is one of fundamental concepts in geometry which has to be studied in secondary schools. Students have many difficulties to implement maths’ reasoning and to understand concepts of  triangle. In Chyprus, the 7^th grade students were reported  to have difficulties in utilizing the related formula of triangle’s areas, since they only memorized the formula without appropriated reasonings. If students’ difficulties has to be addressed and facilitated, then students might be conditioned to understand the concept of triangles.  One strategy is called a mathematics justification. It is an important skill for students to develop mathematical reasoning for learning and understanding mathematical concepts. In classroom, where students have opportunities to participate in mathematical argumentation and justification, the quality of students’ reasoning, and justification ability can be enhanced. Hence, students are motivated to utilize a justification strategy in explaining students' reasoning in solving triangle problems. A student who has an justification strategy is able to construct new knowledge independently and pushes students beyond a procedure to a deeper understanding of the math.

 

This study used a descriptive-qualitative approach that aimed to assess students’ justification strategies in understanding triangle concepts. This study was conducted in the MTs’ Hidayatul Hasan, Lumajang. On class, which was out of five classes of grade seven, were selected purposively as the sample of this study. All students’ were given both mathematical ability and justification tests. The justification test consisted of four problems which related to concepts of triangles. By involving students’ portofolio, the results were ranked into three categories namely low, medium and high maths’ abilities, respectively. Six volunteer students, each two students with low, medium and high maths’s abilities, respectively, were selected as respondents. All respondents were individually interviewed based on the answer of the justification test.

 

The results of the mathematics tests show that high maths’ abilities score 86, medium maths' ability 66  score 85, and low maths' sbility score  65. Average of final students’ result is 85. The result of justification tests, most students were still confused in distinguishing the types of triangles based on the triangles’ properties, i.e. length sides and angles

 

One of item tests, from which most students did mistake is given as follows.

Statement:   Given the triangle ABC which is congruent with the triangle PQR. There is at least one angle of ABC is . According to your knowledge,  the above statement is true or false. Please explain!

 

Students assumed that if there are two triangles of different sizes, then the angle were also different. Obviously, the item test provided a clearly hint that the two triangles were congruent. The result of the given test are presented in table 1 below.

Table 1. Subjects’ Responses of

Response Statement

Mathematical Ability

SH

SM

SL

The high maths’ ability demonstrated students’ justification more explicit. However, some justifications were mathematically incorrect  relating  to, for example, in utilizing the formula of triangle’s area.

Some parts of students’ justification were mathematically incorrect or contained insufficient details in explaining types of the given triangle based on its side lengths and angles.

Some parts of students’ justifications did not include a valid reasoning strategy or written work is missing.

Note that SH, SM and SL  denote the high, medium and low maths’ ability subjects, respectively.

 

Our results agreed to some findings that the geometry’s concepts are only focused on computational skills,  lack justifications in understanding of that concepts and without explaining why the used method is appropriate or correct.