| A Classroom Note: Entropy, Information, and Markov Property | 1$-$12 |
Abstract
How to introduce the concept of the Markov Property in an elementary Probability Theory course? From this author's teaching experience, it appears that the best way that gives a natural intuitive flavor and preserves the mathematical rigor, is to use concepts of entropy and information from the classical Shannon Information Theory, as suggested in the brilliant monograph of A. Rényi [5]. Following this path, the connection between Entropy and Markov Property is presented.
Keywords: Random variable, Independence, Entropy, Information, Conditional Probability, Sufficient Function, Markov Chain.
| Solving inequalities in primary school | 13$-$30 |
Abstract
When sums and differences with a variable component are compared with a fixed number, simple inequalities suitable for solving in the set N of natural numbers are obtained. Solving of these inequalities bases upon the properties of such expressions to increase or decrease, depending on the change of values of their components. To establish the meaning of an inequality, children have to be stimulated to see it as a representation of a whole set of numerical relations, some of which are true and some false. Then, the search for those values of the variable component for which these relations are true is the procedure of solving an inequality. Thinking of prerequisite knowledge and skills for this procedure, some exercises have to be planed that will help children assimilate the meaning of some operative terms: expression, value of an expression, to take value, etc., as well as to instruct them in using first set-theoretical notations properly. Not counting general observations, the entire text of this paper resembles a concrete elaboration, appropriate for school practice.
Keywords: Expression, Value of an expression, Variable summand (minuend) (subtrahend), Inequality, Set of solutions.
| Challenging Mathematics by ``Archimedes'' | 31$-$39 |
Abstract
This paper presents the activities of the Serbian mathematical club ``Archimedes'' aiming at challenging mathematics in and beyond the classroom. It also summarizes main achievements of the club throughout three decades of its existence, focusing on the main findings from its rich experience in challenging mathematics. Examples of challenging tasks with some didactical remarks are given in the appendices.
Keywords: Mathematical club, out-of-school education, teacher education.
| Solutions of some classes of congruences | 41$-$44 |
Abstract
We present a method for solving nonlinear congruences. It is different from methods often met in the literature, and it is based on reduction of nonlinear congruence to either linear or quadratic congruence. This paper is written for teachers and talented gymnasium students, interested in solving nonlinear congruences.
Keywords: Congruence, congruences of higher degree.
| Students' understanding of the variable as general number and unknown: a case study | 45$-$51 |
Abstract
This study aims at determining the misconceptions and learning difficulties that students experience regarding general number and unknown which are different aspects of the use of the concept of variable. Two sub-problems were examined for this purpose. The first one is the comments of students on general number and the second one is their comments on unknown. A qualitative method (case study design) was used in this study. The data presented in this paper were obtained through the analysis of the students' answers to the tests designed in accordance with the examined issues, their work on the variable-related questions in the exam documents, and the interviews with some of them. The sample of this research consisted of 158 eight grade students. The results show that the subjects of this study experience a set of difficulties and have certain misconceptions in understanding the concept of variable. This paper presents these learning problems and considers means to improve the matters.
Keywords: Different uses of variable, General number, Unknown.