MISEIC 2018

Surabaya,  July 21, 2018

 

Students’s Strategies in Proof by Contradiction, Counter-example and Contrapositive in Terms of Mathematics Ability

 

N N K Sari¹, Y Fuad2 and R Ekawati3

 

^1, S2 Mathematics Program, Universitas Negeri Surabaya, Indonesia

(E-mail: nisari16070785005@mhs.unesa.ac.id)

^2,3 Department of Mathematics Education, Universitas Negeri Surabaya, Indonesia

(E-mail: yusuffuad@unesa.ac.id, rooselynaekawati@unesa.ac.id)

ABSTRACT

Proof in mathematics is very important for mathematical statement especially in contradiction, counter-example and contrapositive. Determining strategies in proving an argument is very important and useful for students. In proving an argument can using direct, indirect, contradiction, contrapositive and counter-example. Proving and denying is a very important skill in mathematical thinking because it helps students to show if the argument is true or false. Contradiction, counter-example and contrapositive argumentation can improve students’abilities to construct, critique, and validate the arguments. The approach is to first build a description of all possible counterexamples to a conditional claim and then use the description to discover a reasoning path for proving or disproving the claim. The reasoning path can include finding a counterexample or finding a reason why counter-examples can not exist. There are six levels of counter-example and contrapositive proving are direct/indirect proving, developing contrapositive arguments, proof concept and conception, pivotal intermediate conception, the mental models, and conceptualizing proof as eliminating counterexample.

The purpose of this study is to find out what strategies are used by students in solving mathematical problems in the form of proof with contradictions, counter-example, and contrapositif. The subjects of research comes from the 10^th grades students at SMA GIKI 1 in East Java, were selected from purpossive sampling. Six volunteer students with high, medium and low mathematics ability, were selected from mathematics ability test, and interview.

The data proof by contradiction, counter-example and contrapositive were categorized in 6 levels of  counter-example and contrapositive proving. In this study, students are still having difficulty in counter-examples in mathematical problems given. The results showed that one subject with low ability tends to proof the problem of proof by giving the answer in the form of example and do not include the answer that is not the example as strong evidence, one student with low ability give an answer without reason or examples.  While the high-ability students can give examples and counter-examples of refutation in solving the problem of proof by using method of solving Polya problem and 6 levels of counter-example and contrapositive proving and the medium ability student can proof by contradiction and contrapositive and give a wrong example in counter-examples.

The student’s contradiction, counter-example and contrapositive reasoning could be improved through a teaching experiment that encouraged the proving contradiction, counter-example and contrapositive approach. The research analysis also revealed several intermediate conceptions that appeared to be useful to students when developing more sophisticated contradiction, counter-example and contrapositive proving.

 

Keywords: Proving, Contrapositive, Counter-example, Contraposition, Mathematics Ability.

Acknowledgment: This research respond here in was supported by SMA GIKI 1 Surabay, especially for 10^th grades as a sample 6 students in this class who helped this research. The opinions expressed here in do not necessarily reflect teh position, policy, or endorsement of the supporting agency.