Proving and disproving play an important role for students in justifying the validity of mathematical propositions. Students may demonstrate how and why any proposition is true or false. In mathematics, proving and disproving can’t be separated and interrelated in a role to determine the trueness or falseness of a mathematical proposition. On the other hand, a counterexample had relation with disproving activity. Counterexample may illustrate how and why any proposition is false. If an counterexample was obtained relating to the given proposition, then the proposition is exactly false. Thus, proving and counterexample represent two kinds of matematics terms that are running together and demonstrate whether the given proposition is true or false.

Proving consists of constructing proof and validating proof. Students’ proof construction describes how students constructing a proof. Students’ proof construction is thinking schemes in arranging logical arguments to demonstrate truthness or falseness of statements. The students’ manner in constructing a proof refers to proof schemes.

This qualitative research concerned to assess students’ proof construction schemes in disproving maths propositions. The sample was the 11^th grade students of SMAN 1 Surabaya. All 38 students (13 boys and 25 girls) had to answer two tests, both maths’ ability and proof-by-counterexample tests. Four volunteer students (girls), with high maths’ ability and high score in proof-by-counterexample test, were volunteerly selected as research subjects. A semi-structured interview was individually conducted to every subject. The data of proof schemes was categorized in six levels of proof-by-counterexample criteria. The six levels and criteria are Level 0 (Irrelevant or minimal engagement in inferences), Level 1 (Novice use of examples for reasoning or logical reasoning), Level 2 (Strategic use of examples for reasoning), Level 3 (Deductive inferences with major flaws in logical coherence and validity), Level 4 (Construction of Proof-by-counterexample), and Level 5 (Construction of Proof-by-general-counterexample).

 

The result of two tests is depicted as follows.

Table 1. Frequency of Student’s for Two Tests

 

Gender

Mathematics Ability

Proof-by-counterexample score

High

Middle

Low

0 ≤ ss< 60

60 ≤ ss < 80

ss ≥ 80

Boys

4

6

3

5

6

2

Girls

14

7

4

8

13

4

 

All subjects were individually interviewed for clarifying how the students' proof schemes was constructing proof and validating proof, All subjects were able to prove the proposed maths’ proposition and there were significantly differences among their justification schemes. The high ability subject was able to demonstrate the falseness of maths’ proposition by giving a general counterexample with maths symbols and had achieved the 5^th level. Meanwhile, other subjects were in the 4^th level of proof schemes and could not state the falseness of maths’ propositions by providing specific counterexamples. They could not change the specific counterexample to  more general counterexample with maths’ symbols properly.

 

This study concluded that the students' proof schemes for disproving mathematical propositions were still low. Event most students were able to constructed such specific counterexample, but only one of 38 students was able to demonstrate general counterexample by using maths’ symbols correctly. Furthermore, there still need to increase proof-based learning so that students are able to improve logical reasoning and thinking in achieving higher order thinking skills. In addition, we suggest, to conduct a generalized research with considering gender and teachers’ competencies.