• The Mathematical Education
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• v.55 no.1
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• pp.91-106
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• 2016

The aim of this study was to analyze characteristics of teachers' knowledge about reductio ad absurdum. In order to achieve the aim, this study conducted didactical analysis about reductio ad absurdum through examining previous researches and developed a questionnaire with reference to the results of the analysis. The questionnaire was given to 34 high school teachers and qualitative methods were used to analyze the data obtained from the written responses by the participants. This study also elaborated the framework descriptors for interpreting the teachers' responses in the light of the didactical analysis and the data was elucidated in terms of this framework. The specific features of teachers' knowledge about reductio ad absurdum were categorized into five types as a result. This study raised several implications for teachers' professional development for effective mathematics instruction related to reductio ad absurdum.

• Journal for History of Mathematics
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• v.14 no.1
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• pp.47-60
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• 2001

The present study develops the given theme “Mathematics and Reality” along two lines. First, we explore the answers, in its various facets, to the following question: How is it possible that mathematics shows such wondrous efficiency when explaining nature\ulcorner In addition to a comparative analysis between empiricism and rationalism, constructivism as a function of idealism is compared with realism within the frame provided by rationalism. The second step involves limiting our discussion to realism. We attempt to explain the various stages of mathematical realism and their points of difficulty. Postulate of parallels, Godel's theorem, continuum hypothesis and choice axiom are typical examples used in demonstrating undecidable propositions. They clearly show that it is necessary to mitigate the mathematical realism which depends on bivalent logic based on an objective exterior world. Lowenheim-Skolem theorem, which states that reality is composed not of one block but rather of diverse domains, also reinforces this line of thought. As we can see the existence of undecidable propositions requires limiting the use of reductio ad absurdum proof which depends on the concept of excluded middle. Consequently, it becomes obvious that bivalent logic must inevitably cede to a trivalent logic since there are three values involved: true, false, and undecidable.

• The Mathematical Education
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• v.49 no.2
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• pp.161-174
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• 2010

The purpose of the study is to analyze various types of errors appeared in true-false proof problems of matrix and to find out correction method. In order to achieve this purpose, error test was conducted to the subject of 87 second grade students who were chosen from D high schoool. It was shown from this test that the most frequent error type was caused by the lack of understanding about concepts and essential facts of matrix(35.3%), and then caused by the invalid logically reasoning (27.4%), and then caused by the misusing conditions(18.7%). Through three hours of correction lessons with 5 students, the following correction teaching method was proposed. First, it is stressed that the operation rules and properties satisfied in real number system can not be applied in matrix. Second, it is taught that the analytical proof method and the reductio ad absurdum method are useful in the proof problem of matrix. Third, it is explained that the counter example of E=$\begin{pmatrix}1\;0\\0\;1 \end{pmatrix}$, -E should be found in proof of the false statement. Fourth, it is taught that the determinant condition should be checked for the existence of the inverse matrix.