Student errors in constructing concepts about the properties of integer operations are found from junior high school students to prospective elementary school teachers [1-2]. The error occurs because students only remember the properties of mixed operations, that there is a commutative, distributive and precedence multiplication rather than sum [1]. thus, they are wrong in answering and giving reasons to the question whether . Although prospective elementary school teachers can determine the true and false statements, they are still wrong in making arguments [2]. Because the properties of integer operations are first taught in primary school, this problem becomes an important issue to solve, considering that if the prospective elementary school teacher has the same error of thinking, then the error can be passed on to the student when he or she teaches later. Therefore, this article aimed to explain the results of an evaluation of the students' understanding of primary schools on the properties of integer operations. So that problems can be understood, and solved through the development of teaching materials design, the development of elementary school curriculum and the curriculum of prospective elementary school teachers.

 

The study was conducted in one of the primary schools in Bandung. Three students from grade 6 of elementary school were selected by teachers based on arithmetic ability to be the subject of research. S1 is a student who has high arithmetic ability. S2 is a student with average arithmetic ability. S3 is students who have low arithmetic skills. Interviews were conducted over 60 minutes to evaluate their understanding of the properties of integer operations. Interview results were analyzed based on Harel's knowledge classification [5].

 

The results are presented based on arithmetic ability. The results of the analysis related to commutative property that yield some findings. S1 initially did not use commutative property in the sentence of , but eventually the student can understand the commutative property of the sentence. S2 can immediately understand the commutative property of the given arithmetic sentence. Unlike S3 students, s/he does not understand commutative property in  sentences. S3 understand it as two different numbers placed in different positions. Since 2 is smaller than 3 or 3 is greater than 2, and the order of position 2 and 3 is different, then the sentence is considered wrong. In fact, S3 has understood that the sum of the two sides together. S3 also often shows errors in counting. Therefore, the questions given to students should consider students with low arithmetic ability, because if the student miscalculated, the commutative property cannot be conveyed easily.

 

The results of the analysis related to students' understanding of the associative property resulted in several findings. First, the three students initially still count from left to right on the sum of three numbers. However, after the interview continued, they finally understood and used the associative property. Second, the reasons S1 & S2 use the associative property of the sum of three numbers is to make it easier, but not easy for S3. This is because S3 has a low arithmetic ability. Third, they have not understood the use of associative and commutative property simultaneously, but they believe in the correctness of the use of both properties simultaneously.

 

The results of the analysis of the students' answers regarding distributive properties resulted in several findings. First, students did not initially use distributive properties to solve the problem of 34 × 5. They use the multiplication definition of repeated summation. They add up to 34 five times. Besides that, they also use ordered multiplication. This shows that in counting, students rarely use commutative property as a strategy to solve arithmetic problems. Secondly, when it was introduced how to calculate 34 × 5 = (30 × 5) + (4 × 5), students believe that the method can be used, because the results of both sides are the same. Third, students can also give some examples of multiplying the two numbers that the strategy to solve it like the example given. Another example that students can give is 29 × 5 = (30 × 5) - (1 × 5).

 

 

Figure 1. Students' method to do multiplication

 

From the results of this study, we could conclude that students who are proficient in calculating the sum and multiplication of integers can apply well the commutative, associative and distributive when completing a calculation. However, students who are not proficient in calculating sum and multiplication, find it difficult to apply commutative, associative and distributive. Therefore, teachers should pay attention to students with low arithmetic ability in designing instructional design and consider the number used in the calculation, so the students can understand the structure of commutative, associative, and distributive properties.