• The Mathematical Education
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• v.50 no.3
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• pp.309-327
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• 2011

In this paper, we extended the division algorithm for the integers to the finite decimals. Though the remainder for the finite decimals is able to be defined as various ways, the remainder could be defined as 'the remained amount' which is the result of the division and as "the remainder" only if 'the remained amount' is decided uniquely by certain conditions. From the definition of "the remainder" for the finite decimal, it could be inferred that 'the division by equal part' and 'the division into equal parts' are proper for the division of the finite decimal concerned with the definition of "the remainder". The finite decimal, based on the unit of measure, seemed to make it possible for us to think "the remainder" both ways: 1" in the division by equal part when the quotient is the discrete amount, and 2" in the division into equal parts when the quotient is not only the discrete amount but also the continuous amount. In this division context, it could be said that the remainder for finite decimal must have the meaning of the justice and the completeness as well. The theorem of the division algorithm for the finite decimal could be accomplished, based on both the unit of measure of "the remainder", and those of the divisor and the dividend. In this paper, the meaning of the division algorithm for the finite decimal was investigated, it is concluded that this theory make it easy to find the remainder in the usual unit as well as in the unusual unit of measure.

• Journal of the Korean School Mathematics Society
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• v.10 no.2
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• pp.237-246
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• 2007

According to 7-th curriculum, irrational number should be introduced using non-repeating infinite decimals. A rational number is defined by a number determined by the ratio of some integer p to some non-zero integer q in 7-th grade. In 8-th grade, A number is rational number if and only if it can be expressed as finite decimal or repeating decimal. A irrational number is defined by non-repeating infinite decimal in 9-th grade. There are misconceptions about a non-repeating infinite decimal. Although 1.4532954$\cdots$ is neither a rational number nor a irrational number, many high school students determine 1.4532954$\cdots$ is a irrational number and 0.101001001$\cdots$ is a rational number. The cause of misconceptions is the definition of a irrational number defined by non-repeating infinite decimals. It is a cause of misconception about a irrational number that a irrational number is defined by a non-repeating infinite decimals and the method of using symbol dots in infinite decimal is not defined in text books.

• Journal of Educational Research in Mathematics
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• v.22 no.2
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• pp.163-178
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• 2012

The efforts to represent the numbers based on the decimal system give us fundamental understanding to construct and teach the concept network on the related knowledge of elementary and secondary school mathematics. In the process to represent natural numbers, integers, rational numbers, real numbers as decimal system, we will classify the extended decimal system. Moreover we will obtain the view to classify real numbers. In this paper, we will study the didactical significance of mathematical knowledge, which arise from process to represent real numbers as decimal system, starting from decimal system representation of natural numbers, and provide the theoretical base about the classification of real numbers. This study help math teachers to understand school mathematics in critical inside-measurement and provide the theore tical background of related knowledge. Furthermore, this study provide a clue to construct coherent curriculum and internal connections of related mathematical knowledge.

• Structural Engineering and Mechanics
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• v.39 no.1
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• pp.21-46
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• 2011

The natural frequencies of continuous systems depend on the governing partial differential equation and can be numerically estimated using the finite element method. The accuracy and convergence of the finite element method depends on the choice of basis functions. A basis function will generally perform better if it is closely linked to the problem physics. The stiffness matrix is the same for either static or dynamic loading, hence the basis function can be chosen such that it satisfies the static part of the governing differential equation. However, in the case of a rotating beam, an exact closed form solution for the static part of the governing differential equation is not known. In this paper, we try to find an approximate solution for the static part of the governing differential equation for an uniform rotating beam. The error resulting from the approximation is minimized to generate relations between the constants assumed in the solution. This new function is used as a basis function which gives rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam. The results of finite element analysis with the new basis functions are verified with published literature for uniform and tapered rotating beams under different boundary conditions. Numerical results clearly show the advantage of the current approach at high rotation speeds with a reduction of 10 to 33% in the degrees of freedom required for convergence of the first five modes to four decimal places for an uniform rotating cantilever beam.

• The Mathematical Education
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• v.56 no.2
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• pp.131-145
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• 2017

The representation of irrational numbers has a key role in the learning of irrational numbers. However, transparent and finite representation of irrational numbers does not exist in school mathematics context. Therefore, many students have difficulties in understanding irrational numbers as an 'Object'. For this reason, this research explored how mathematics textbooks affected to students' understanding of irrational numbers in the view of process and object. Specifically we analyzed eight textbooks based on current curriculum and used framework based on previous research. In order to supplement the result derived from textbook analysis, we conducted questionnaires on 42 middle school students. The questions in the questionnaires were related to the representation and calculation of irrational numbers. As a result of this study, we found that mathematics textbooks develop contents in order of process-object, and using 'non repeating decimal', 'numbers cannot be represented as a quotient', 'numbers with the radical sign', 'number line' representation for irrational numbers. Students usually used a representation of non-repeating decimal, although, they used a representation of numbers with the radical sign when they operate irrational numbers. Consequently, we found that mathematics textbooks affect students to understand irrational numbers as a non-repeating irrational numbers, but mathematics textbooks have a limitation to conduce understanding of irrational numbers as an object.