In this work, we obtained characterization curves of constant breadth in four dimensional Galilean space in terms of Frenet-Serret vector fields. First we investigate an explicit characterization of curves of constant breadth of four dimensional Galilean space.In addition,it has been observed that differential equation of 4th order of a constant breadth curve in four dimensional Galilean space.Finally,some differential and integral characterizations of the mentioned curves are expressed by the classical differential geometry methods.
Article History Received: 26 August 2015Mathematization competency is considered in the field as the focus of modelling process. Considering the various definitions, the components of the mathematization competency are determined as identifying assumptions, identifying variables based on the assumptions and constructing mathematical model/s based on the relations among identified variables. In this study, preservice elementary mathematics teachers' mathematization competencies are tried to be elicit by investigating their solution approaches while solving a modelling problem. It was seen the participants started to solve the problem by using only verbal explanations and then their expressions became more mathematical throughout the process. The participants, who made validations frequently in the process, displayed more comprehensive mathematization competencies by correcting their assumptions, mathematical models and solution. Accepted: 07 December 2015 KeywordsMathematical modelling Mathematization Competency Pre-service elementary mathematics teacher IntroductionMathematical modelling is defined as the process of expressing a real life problem situation mathematically and explaining this situation through the use of mathematical models (Blum & Niss, 1991). It is possible to define the real life situation mentioned in this definition as the whole world apart from mathematics in general (Pollak, 1979). In another definition which emphasizes the relationship between the real world and mathematical world, Heyman (2003) defines mathematical modelling as a simple way of applicability of mathematics, the relationship of mathematics with the real world, and revelation of this relationship, and it is stated that a mathematical model has to be constructed when mathematics is necessary to be benefitted from so as to explain real life problems, define and solve them (as cited in Peter Koop, 2004). In the modelling problems requiring mathematical models to be constructed, students mathematize real life situations and reach meaningful solutions depending on their experiences (Lesh & Doerr, 2003). In this context, in today's world in which only school success is not sufficient, students' performing mathematical modelling gains importance so that they will grow up as successful individuals in life and will be enabled to cope with the problems that they can encounter in real life.Because having knowledge on modelling process becomes important for students to engage in mathematical modelling, it seems to be important to discuss what modelling process means. Modelling process is defined as a cyclic process in which a real model is obtained through configuration of real life problems, in which mathematical model is constructed through mathematization of the real model, in which the mathematical model is solved and in which the obtained solution is interpreted and validated in the frame of real life (Borromeo Ferri, 2006; Maaβ, 2006). Many modelling processes and cycles are encountered in the literature depending on how...
This work deals with classical differential geometry of isotropic curves in the complex space C 4 . First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.
In this work, the concept of a slant helix is extended to Minkowski space-time.In an analogous way, we define type-3 slant helices whose trinormal lines make a constant angle with a fixed direction. Moreover, some characterizations of such curves are presented.
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