• Communications of Mathematical Education
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• v.36 no.1
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• pp.139-170
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• 2022

In relation to differentiated mathematics education, Russia has a longer experience in research and practice than Korea. The mathematics curriculum for 10-11 grades currently in use in Russia is a level-specific curriculum and consists of a basic level and an advanced level. And in Russia mathematics textbooks for 10-11 grades are also textbooks for each level. In this study, we analyzed basic level textbook and advanced level textbook written by the same author group among the textbooks 'Algebra and Introduction of Mathematical Analysis' of the 10th grade in Russia. To analyze the main learning contents and textbook descriptions that were added in advanced level the 'real numbers' and 'complex numbers' sections were studied. The main contents of basic and advanced level textbooks for 'functions', 'trigonometric functions', 'trigonometric equations', 'conversions of trigonometric expressions', and 'derivatives', which are included in both basic and advanced textbooks were compared and analyzed, and the descriptive characteristics of the definitions and theorems presented in the two levels of textbooks were also compared and analyzed. From the results of this study, it is expected that various information on the contents of various level textbooks of mathematics, the differences between textbooks for each level, and strategies for the composition of textbooks for various level can be accumulated.

• Advances in aircraft and spacecraft science
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• v.5 no.1
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• pp.51-71
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• 2018

This work proposes a theoretical and numerical study on the behavior of a tapered shaft rotor made of composite materials by the classical version h and the version p of the finite element method. Hierarchical form functions are used to define the model. The purpose of this paper is to determine the expressions of the kinetic and potential energies of the tree necessary for the results of the equations of motion. A comparison between the version h and the p version of the finite element method of the functions of polynomial and trigonometric hierarchical forms with six degrees of freedom per node, of a composite tapered and cylindrical shaft which rotates at a constant speed about its axis. It is found that when the number of functions of form (the version p) is increased, the solution converges. It is also observed that the conicity of the shaft increases the rigidity with respect to a uniform shaft having the same mechanical properties. The numerical simulation allowed us to determine the natural frequencies and the critical speeds of the composite shaft systems are compared with those available in the literature and the effectiveness of the methods used are discussed.

• Communications of Mathematical Education
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• v.21 no.2 s.30
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• pp.283-302
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• 2007

It is said that the university mathematics education in Korea faces critical situations due to the decreases of both qualities and quantities of students. In this paper we examine college students in order to know their basic ability for understanding about fundamental functions, such as polynomial, trigonometric, logarithm and exponential functions which have learned from highschool mathematics courses. The subject are 354 freshmen of 4 universities located in Daejeon and Chongchung area. The result of this study shows as follows. i ) More than half students are not able to draw graphs of given functions, except polynomial. ii ) More students do not fully understand about function properties such as domain, codomain, range, max and min value, cycle and parallel translation.

• Smart Structures and Systems
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• v.13 no.1
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• pp.1-24
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• 2014

The present study deals with two dimensional electro-elastic analysis of a functionally graded piezoelectric (FGP) cylinder under internal pressure. Energy method and first order shear deformation theory (FSDT) are employed for this purpose. All mechanical and electrical properties except Poisson ratio are considered as a power function along the radial direction. The cylinder is subjected to uniform internal pressure. By supposing two dimensional displacement and electric potential fields along the radial and axial direction, the governing differential equations can be derived in terms of unknown electrical and mechanical functions. Homogeneous solution can be obtained by imposing the appropriate mechanical and electrical boundary conditions. This proposed solution has capability to solve the cylinder structure with arbitrary boundary conditions. The previous solutions have been proposed for the problem with simple boundary conditions (simply supported cylinder) by using the routine functions such as trigonometric functions. The axial distribution of the axial displacement, radial displacement and electric potential of the cylinder can be presented as the important results of this paper for various non homogeneous indexes. This paper evaluates the effect of a local support on the distribution of mechanical and electrical components. This investigation indicates that a support has important influence on the distribution of mechanical and electrical components rather than a cylinder with ignoring the effect of the supports. Obtained results using present method at regions that are adequate far from two ends of the cylinder can be compared with previous results (plane elasticity and one dimensional first order shear deformation theories).

• Structural Engineering and Mechanics
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• v.27 no.2
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• pp.243-257
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• 2007

A numerically efficient superelement is proposed as a low degree of freedom model for dynamic analysis of rotating tapered beams. The element uses a combination of polynomials and trigonometric functions as shape functions in what is also called the Fourier-p approach. Only a single element is needed to obtain good modal frequency prediction with the analysis and assembly time being considerably less than for conventional elements. The superelement also allows an easy incorporation of polynomial variations of mass and stiffness properties typically used to model helicopter and wind turbine blades. Comparable results are obtained using one superelement with only 14 degrees of freedom compared to 50 conventional finite elements with cubic shape functions with a total of 100 degrees of freedom for a rotating cantilever beam. Excellent agreement is also shown with results from the published literature for uniform and tapered beams with cantilever and hinged boundary conditions. The element developed in this work can be used to model rotating beam substructures as a part of complete finite element model of helicopters and wind turbines.

• Journal of Educational Research in Mathematics
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• v.8 no.2
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• pp.453-474
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• 1998

This study is focused on the possibility if we can use graphic calculators in teaching and learning school mathematics. This study is consisted with four main chapters. In chapter II, the functions of the graphic calculator EL-9600 produced by Sharp Corporation was analyzed focused on the possibilities if the functions could be used in teaching and learning school mathematics. Calculating of real numbers and complex numbers, solving equations and system of linear equations, calculating of matrices, graphing of several functions including polynomial functions, trigonometric functions, exponential and logarithmic functions, calculation of differential and integrals, arranging of statical data, graphing of statistical data, testing of statistical hypotheses, and other more useful functions were founded. In Chapter III, a mathematics textbook developed by Core-Plus Mathematics Project was analyzed focused on how a graphic calculator was used in teaching and learning mathematics, In the textbook, graphic calculator was used as a tool in understanding mathematical concepts and solving problems. Graphic calculator is not just a tool to do complex computations but a tool used in the processes of doing mathematics, In chapter IV, the 7th mathematics curriculum for korean secondary schools was analyzed to find the contents could be taught by using graphic calculators. Most of the domains, except geometric figure, were found that they could be taught by using graphic calculators, In chapter V, a model of a unit using graphic calculator in teaching 7th mathematics curriculum was developed. In this model, graphic calculator was used as a tool in the processes of understanding mathematical concepts and solving problems. This study suggests the possibilities that we can use graphic calculators effectively in teaching and learning mathematical concepts and problem solving for most domains of secondary school mathematics.

• Journal of Educational Research in Mathematics
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• v.27 no.4
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• pp.833-855
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• 2017

This study aims to reinvestigate the reason for introducing radian as a new unit to express the size of angles, what is the meaning of radian measures to use arc lengths as angle measures, and why is the domain of trigonometric functions expanded to real numbers for expressing general angles. For this purpose, it was conducted historical, mathematical and applied mathematical analyzes in order to research at multidisciplinary analysis of the radian concept. As a result, the following were revealed. First, radian measure is intrinsic essence in angle measure. The radian is itself, and theoretical absolute unit. The radian makes trigonometric functions as real functions. Second, radians should be aware of invariance through covariance of ratios and proportions in concentric circles. The orthogonality between cosine and sine gives a crucial inevitability to the radian. It should be aware that radian is the simplest standards for measuring the length of arcs by the length of radius. It can find the connection with sexadecimal method using the division strategy. Third, I revealed the necessity by distinction between angle and angle measure. It needs justification for omission of radians and multiplication relationship strategy between arc and radius. The didactical suggestions derived by these can reveal the usefulness and value of the radian concept and can contribute to the substantive teaching of radian measure.

• Research in Mathematical Education
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• v.13 no.2
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• pp.151-170
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• 2009

This study reports a research which was conducted on how frequently and where the students use the unit circle method while dealing with trigonometric functions in solving the trigonometry questions. Moreover, the reasons behind the choice of the methods, which could be the unit circle method, the ratio method, or the use of trigonometric identities, are also investigated to get an insight about their understanding. In this study, the relationship between the students' choices of methods in solving questions is examined in terms of instrumental or relational understanding. This is a multi-method research which involves a range of research strategies. The research techniques used in this study are test, verbal protocol (think aloud), and interview. The test has been applied to ten tenth grade students of a public school to get students' solution processes on the paper. Later on, verbal protocol has been performed with three students of these ten who were of the upper, middle and lower sets in terms of their performance in the test. The aim was to get much deeper data on the students' thinking and reasoning. Finally, interview questions have been asked both these three students and other three from the initial ten students to question the reasons behind their answers to the trigonometry questions. Findings in general suggest that students voluntarily choose to learn instrumentally whose reasons include teachers' and students' preference for the easier option and the anxiety resulting from the external exam pressure.

• Steel and Composite Structures
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• v.34 no.4
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• pp.511-524
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• 2020

In this research, a simple four-variable trigonometric integral shear deformation model is proposed for the static behavior of advanced functionally graded (AFG) ceramic-metal plates supported by a two-parameter elastic foundation and subjected to a nonlinear hygro-thermo-mechanical load. The elastic properties, including both the thermal expansion and moisture coefficients of the plate, are also supposed to be varied within thickness direction by following a power law distribution in terms of volume fractions of the components of the material. The interest of the current theory is seen in its kinematics that use only four independent unknowns, while first-order plate theory and other higher-order plate theories require at least five unknowns. The "in-plane displacement field" of the proposed theory utilizes cosine functions in terms of thickness coordinates to calculate out-of-plane shear deformations. The vertical displacement includes flexural and shear components. The elastic foundation is introduced in mathematical modeling as a two-parameter Winkler-Pasternak foundation. The virtual displacement principle is applied to obtain the basic equations and a Navier solution technique is used to determine an analytical solution. The numerical results predicted by the proposed formulation are compared with results already published in the literature to demonstrate the accuracy and efficiency of the proposed theory. The influences of "moisture concentration", temperature, stiffness of foundation, shear deformation, geometric ratios and volume fraction variation on the mechanical behavior of AFG plates are examined and discussed in detail.

• Journal of Educational Research in Mathematics
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• v.18 no.2
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• pp.263-281
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• 2008

This study is to provide a base for discussions on teaching and teaming of radian through a theoretical analysis of it. Radian possesses two-fold comprehensive properties of measurement as a magnitude and a pure number. As a magnitude of an angle, it has some theoretical advantages in mathematics and in physics, in spite of its non-superiority to other angular measures in practical sense. As a pure number, it has some advantages in that it simplifies theoretical developments of trigonometric functions and justifies omitting the unit in the calculations and final expressions in physics. Radian should be taught and learnt with an appreciation of the advantages of the two-fold properties. Activities to measure angles from various viewpoints may be helpful for this. Students' awareness of the advantages of radian needs to be stimulated and deepened repeatedly as related content appears.