| Displaying parametric curves with virtual and physical tools | 61$-$73 |
Abstract
Parametric curves play an important role in mathematics and engineering education. The purpose of this paper is to introduce the hypotrochoid family of curves and to provide ideas and tools for teaching the topic. We give the general parametric equations of hypotrochoids and show that several special curves can be derived from these equations. To display the curves we use the Hypotrochoid Equation Tracer, which is a dynamic application made by Desmos graphic calculator. In addition we have created a LEGO robot called Spikograph, that models a Spirograph and draws different hypotrochoids providing a physical experience for students. Finally, we have compiled a student project to practice and deepen the knowledge about hypotrochoids.
Keywords: Hypotrochoid; Parametric equations for curves; Graphic calculator; Spirograph; LEGO robot.
MSC Subject Classification: 97I20, 14H50
MathEduc Subject Classification: I25, U75
| Students' success in solving mathematical problems depending on different representations | 74$-$92 |
Abstract
In order to solve mathematical problems, students often need to make the transition from one representation of mathematical concepts in those problem formulations to another representation. In this paper we explore the influence of the representations used in the problem formulation (problems with the same mathematical background with regards to solving easier or more complex equations and determining the unknown value of the proportion) on students' success in solving those problems. On a representative sample of 584 8th grade students, we tested whether there were differences in students' success in solving mathematical problems while using symbolic, graphic, or verbal representations in the formulations of problems belonging to different level of complexity. Results of this research indicate that there is significant impact of the representations of mathematical concepts used in problem formulation on students' success. Furthermore, the level of impact of using different representations in problem formulations depends on the level of the problem complexity when it comes to students' success in solving those problems.
Keywords: Mathematical problem posing; representations; transition between representations.
MSC Subject Classification: 97D50
MathEduc Subject Classification: D54
| Applications of logistic and generalized logistic difference equations in economics: AK model | 93$-$106 |
Abstract
In this paper, we demonstrate how logistic equation and generalized logistic equation can be applied in teaching economics with the aim that students easily grasp the AK model. To this end, the AK model is firstly modified into three versions, after which the first and the second model are reduced to the generalized logistic equations. The purpose of this paper is to present how through applying the mathematical modifications of mentioned versions of AK models, they become more cognizable to students, which should result by more deeply understanding of well-known economic models.
Keywords: Non-linear dynamics; logistic equation; generalized logistic equation; AK model; mathematics courses for economists.
MSC Subject Classification: 97M40
MathEduc Subject Classification: M45
| Gaussian integrals depending on a quantum parameter in finite dimension | 107$-$115 |
Abstract
A common theme in mathematics is the evaluation of Gauss integrals. This, coupled with the fact that they are used in different branches of science, makes the topic always actual and interesting. In these notes we shall analyze a particular class of Gaussian integrals that depend on the quantum parameter $\hbar$. Starting from classical results, we will present an overview on methods, examples and analogies regarding the practice of solving quantum Gaussian integrals.
Keywords: Gaussian integral; quantization; special functions; arithmetic-geometric mean.
MSC Subject Classification: 97I50, 97I80
MathEduc Subject Classification: I55, I85
| Relaxation of the sequential criteria for continuity and uniform continuity of a real function | 116$-$121 |
Abstract
We show how two fundamental sequential criteria for real functions, namely the sequential criterion for continuity and the sequential criterion for uniform continuity, are relaxed. Further, we derive a global continuity criterion, as an immediate consequence of the relaxed sequential criterion for continuity, and a fundamental property of uniformly continuous functions, as an application of the relaxed sequential criterion for uniform continuity.
Keywords: Continuity; uniform continuity; sequential criterion; Axiom of Choice; global continuity criterion.
MSC Subject Classification: 97I20, 97I30
MathEduc Subject Classification: I25, I35