| A reconsideration of: ``Number systems characterized by their operative properties'' | 81$-$89 |
Abstract
The starting point of our approach to the number systems is the selection of basic operative properties of the system $N_0$ of natural numbers with~ $0$. This set of properties has proved itself to be sufficient for extension of this system to the systems of integers $Z$, positive rational numbers $Q_+\cup\{0\}$ and rational numbers $Q$ and for the formation of basic operative properties of these extended systems. \par In all these cases of number systems, the corresponding set of numbers with basic operative properties is an example of a concrete algebraic structure. These structures can be viewed abstractly as the structure $(S, +, \cdot, <)$, where $S$ is a non-empty set, ``$+$'' and ``$\cdot$'' are two binary operations on $S$ and ``$<$'' is the order relation on $S$, which satisfy the postulated conditions that are formed according to the basic operative properties of these systems. When matched up with $N_0$, $Z$, $Q_+\cup\{0\}$ and $Q$, the structure $(S,+,\cdot,<)$ is called ordered semifield, ordered semifield with additive inverse, ordered semifield with multiplicative inverse and ordered field, respectively. Then, these number systems are characterized as being the smallest semifield with which they fit together. Proofs of these facts require deduction of some properties of all mentioned types of this abstract structure upon which they will be clearly relied. Hence, the main aim of this paper is this deduction and some improvements of proofs contained in the paper whose reconsideration is this note.
Keywords: Number system; ordered semifield; operative property; characterization of number systems.
MSC Subject Classification: 97F40
MathEduc Subject Classification: F43
| Data science for novice students: A didactic approach to data mining using neural networks | 90$-$101 |
Abstract
This paper presents a way to introduce data science to college (business) students. To this end, data mining using neural networks may be practiced. After briefly clarifying the main differences between data science and data mining, the work with neural networks is examined in detail. The examination deals with the features and affordances of this work, as well as its expected challenges with possible reasons. The paper ends with a number of implications for practice, teacher education, and research.
Keywords: Neural networks; data mining; data science; college education.
MSC Subject Classification: 97K80, 97P80
MathEduc Subject Classification: K85, P85
| On some properties of triangle $OIG$ | 102$-$108 |
Abstract
Let $O$ be the circumcenter of a triangle $ABC$, $I$ the incenter and $G$ the centroid of $ABC$. In this paper, we study properties of the triangle $OIG$.
Keywords: Centroid; circumcenter; incenter.
MSC Subject Classification: 97G40
MathEduc Subject Classification: G44
| A case study of a student who created problems for a mathematics competition | 109$-$116 |
Abstract
In this article we present the conclusions drawn from our research on the subject of problem posing for mathematics competitions. We present the case study of a student who is familiar with problem solving, has understood what problem posing means and what is a mathematical problematic situation, has proven his mathematical skills, and is asked to create his own mathematical problem. The conclusions drawn from this research are, firstly, on the relation of mathematical creativity and mathematical problem posing, secondly, on the relation of mathematical education and self-education, and thirdly on the relationship between the creator of a problem and the potential solver of the problem.
Keywords: Problem solving; problem posing; mathematical creativity; problem situation.
MSC Subject Classification: 97D50
MathEduc Subject Classification: D54