Students’ reasoning may implement to investigate how students learn to think mathematically in solving mathematical problems. Students motivated to express their reasoning without limitations to explain how they obtain such ways to answer an integer comparison. Some elementary students are often outwitted when comparing two integers with different sign, for instance a student’s answer is –3 > 3, and it is absolutely false because any positive integer is always bigger than any negative integer.

 

This study investigates the way of students’ reasoning in solving some integer comparisons. The sample were the 5^th grade of elementary students, consists of 4 girls and 14 boys. All students had to conduct mathematics test to know students’ mathematics ability and integer comparisons tests. Both tests and students’ portfolio were analyzed and were ranked into three groups, namely high, medium, and low mathematics abilities. Three students were voluntarily chosen from each group to assess their reasoning criteria in solving integers comparison problem. The data was obtained from the reasoning test about comparison temperature and a semi-structured interview based on reasoning’s criteria.

 

There were 4 high students’ ability, 7 medium students’ ability, and 6 low students’ ability . Three students were voluntarily choosen from each group to assess their reasoning behaviors in solving integers comparison problem. The selection of three volunteer based on consistency and uniqueness in solving integer comparison problem, fluently communication, and willingness. The description of students’ reasoning way based on different ability as follows:

Reasoning Way of High Mathematics Ability Student (HMAs)

Answering the integer comparison problem, HMAs graphed a number line to show the located integers that indicate the city’s temperature. In comparing integers, he shown the temperature of city by stating “less warmâ€, “warmerâ€, less coldâ€, “colder†for some problems. For example, when HMAs was asked to compare the temperature between Nagasaki and Hiroshima, HMAs stated that the temperature of Hiroshima was 6áµ’ C and it was warmer than Nagasaki. Beside that, HMAs provided explanation by using the given context for answering other problems. For example, there were some people of three cities that could not go anywhere because if the temperature was below 0áµ’ C, then the temperature would become colder than that of above 0áµ’ C.

Reasoning Way of Medium Mathematics Ability Student (MMAs)

Based on the answer sheet, MMAs interpreted the temperature by using number line. Eventhough MMAs were able to circle the number that indicated the given temperature, but he did not give more explanations in some problems. Moreover, he gave some arguments namely “Mikasa is warmer than Omura because it is near to 0áµ’ Câ€

Reasoning Way of Low Mathematics Ability Subject (LMAs)

In solving the given problem, LMAs compared the temperature by using given context. For example, LMAs was asked “Could the society of Nagasaki go anywhere?â€. Then LMAs responded that the society of Nagasaki could go anywhere because the temperature was not less than 0áµ’ C. If the temperature was more negative, then the society had to stay at home.

 

Figure 1 The way of students reasoning of HMAs

 

Figure 2 The way of students reasoning of MMAs

 

Figure 3 The way of students reasoning of LMAs.

This study had demonstrated the way of students reasoning in solving the integer problem. It is  underlined that students used a variety of strategy to draw their conclusions. Students presented appropriately ways and reasons in solving the problem. The student with high mathematics ability often used the number line to illustrate the problem, then answering it correctly. Meanwhile the student with medium ability utilized also the number line, and the student with low mathematics ability applied commonly the given context to solve the problem.