|Volume XXVI , issue 2 ( 2023 )
|Teaching perspectives of the Frobenius coin problem of two denominators
Let $a,b$ be positive, relatively prime, integers. Our goal is to characterize, in an elementary way, all positive integers $c$ that can be expressed as a linear combination of $a,b$ with non-negative integer coefficients and discuss the teaching perspectives of our methods.
Keywords: Coin problem; Frobenius number; Diophantine equations; Number Theory.
MSC Subject Classification: 97F60, 97D50
MathEduc Subject Classification: D54, F64
|Some optimization problems with calculus
Starting from the well-known and elementary problem of inscribing the rectangle of the greatest area in an ellipse, we look at gradually more and more complicated variants of this problem. Our goal is to demonstrate to an average but motivated student of Calculus how to start from an inconspicuous textbook problem and to arrive at considerably more interesting and complicated problems, some of which can be subjects of independent research.
Keywords: Optimization problems; Calculus.
MSC Subject Classification: 97I40, 97I60
MathEduc Subject Classification: I45, I65
|Correlation analysis of students' success in solving analytic geometry and multiple integral problems
In this paper, we aim to contribute to the planning and implementation of education in higher mathematics education for students from non-mathematics study programs, specifically focusing on multivariable calculus, i.e., multiple integrals. Indeed, the outcomes of various empirical studies indicate that students from non-mathematical faculties struggle to grasp and comprehend multiple integrals and multivariable functions in general. The research presented in this paper aims to ascertain whether there is a significant correlation between students' achievements in multiple integrals and their achievements in applying knowledge and skills from analytical geometry (to define sets of points in the plane and space, determined by lines, curves, planes and surfaces). Additionally, the study investigates whether this correlation potentially varies based on the various instructional teaching approaches. The presented empirical research was conducted at the Faculty of Engineering, University of Kragujevac, with 72 second-year students, divided into two groups. The results indicate that the given linear correlation is statistically significant and positive. Moreover, the differences in correlation coefficients calculated for two groups of students who acquired knowledge in multiple integrals through different instructional approaches are not statistically significant. These findings underscore the need to devote substantial attention to the teaching of multiple integrals, especially in devising methods that enable students to visualize specific mathematical concepts in both plane and space. Additionally, a precise definition of integration domains, and an accurate specification of variable bounds, should be emphasized in the multiple integrals teaching and learning process.
Keywords: Multiple integrals; visualization; calculus; multiple representations; analytic geometry.
MSC Subject Classification: 97D60
MathEduc Subject Classification: D65
|Maple as a tool for the teaching of cyclotomic polynomials elementarily
Cyclotomic polynomials are an interesting topic and play an important role for other topics in Number Theory. For special values of $n$, computing a cyclotomic polynomial is not difficult; this can be done by using properties of the polynomial. For a prime value of $n$, the polynomial can be written quickly. However, for values which are multiples of odd prime numbers, say $85=17\cdot5$, the task can be quite difficult if it must be done manually. The polynomial has 41 terms and the degree of 64. Software for cyclotomic polynomials is available; Maple for example, can solve the problem of cyclotomic polynomials very easily. How\-ever, understanding of how to compute polynomials is very important for a student in applying various properties of the cyclotomic polynomial. Here, we will use Maple to help students understand cyclotomic polynomials from the basic. Answer is not obtained directly but step-by-step using properties of the polynomial. So, Maple is used to simulate the process of obtaining the polynomial. Once the students grasp the skill, they will be able to use the software for advanced applications.
Keywords: Cyclotomic polynomial; irreducible; polynomial multiplication; polynomial division; Maple.
MSC Subject Classification: 97C70, 97F40
MathEduc Subject Classification: C75, F45