Volume XXIII , issue 1 ( 2020 ) | back |

Paying attention to students' ideas in the digital era | 1$-$16 |

**Abstract**

This paper demonstrates how recognition of a hidden potential of rather involved mathematical explorations in a student's unintentionally far-reaching response to an open-ended question about constructing a visual pattern allows for the development of the so-called TITE problem-solving activities that require concurrent use of computing technology and mathematical reasoning. The paper begins with the presentation of such a response by an elementary teacher candidate and it continues towards revealing the potential of the response as a springboard into the development of various TITE generalization activities with ever increasing conceptual and symbolic complexity. It is argued that whereas one of the goals of moving from particular to general is to assist in understanding special cases, the construction of workable computational algorithms for spreadsheet-supported problem solving and posing is not possible without experience in generalization. The mathematical content of the paper deals with polygonal numbers and their partial sums. Computer programs used are Wolfram Alpha (free interface) and Microsoft Excel spreadsheet.

**Keywords:** TITE curriculum; polygonal numbers; teacher education; generalization; problem solving; technology.

**MSC Subject Classification:** 97D40, 97I30

**MathEduc Subject Classification:** D45, I35

A note on Brauer's theorem | 17$-$19 |

**Abstract**

We present an elementary proof of Brauer's theorem, which shows how knowledge of an eigenpair can be used to change a single eigenvalue of a matrix.

**Keywords:** Eigenvalue; eigenvector; rank-one modification; Brauer.

**MSC Subject Classification:** 97H60, 15A18

**MathEduc Subject Classification:** H65

Combining math ideas via probability problems | 20$-$29 |

**Abstract**

The paper presents a collection of problems from the Chernorizec Hrabar math tournament that takes place in Bulgaria during the last 28 years. The formulation of a question as a probability problem is used for different math ideas to be linked in a complex test unit. The complexity of the problem requires searching for the correct answer by applying not only advanced math knowledge and skills, but also a synthetic-type thinking, including some sense of mathematics. The multiple-choice format of the tournament problems allows students to skip a lot of routine work and to focus on the purified mathematical ideas. On one hand, the expected reasoning for the answer choice appears as an extract of the comprehensive problem solution. On the other hand, the reasoning forms the carcass of a sophisticated math task that challenges student’s synthetic competence.

**Keywords:** Probability test items; advanced students in mathematics; sense of mathematics; synthetic competence.

**MSC Subject Classification:** 97U40, 97K50

**MathEduc Subject Classification:** D44, D54

Proofs for old and new triangle inequalities | 30$-$34 |

**Abstract**

In this paper, we give new elementary proofs for some old triangle inequalities and we also present proofs for new inequalities in triangle.

**Keywords:** Problem solving; BergstrÃ¶m's inequality; Radon's inequality; Euler's inequality; MitrinoviÄ‡'s; Padoa's inequality; B\u atine\c tu's inequality.

**MSC Subject Classification:** 97H30

**MathEduc Subject Classification:** H34

Recursive formulas for root calculation inspired by geometrical constructions | 35$-$50 |

**Abstract**

This article describes a method for calculating arithmetic, geometric and harmonic means of two numbers and how they can be represented geometrically. We extend these mean values to arithmetic, geometric and harmonic thirds, fourths, etc. For this we will only use the tools of the affine planar geometry. Also, we will make allusion to the more general interpretation in the projective plane. \par From the relations between these means we can deduce a multitude of recursive formulas for $n$-th root calculation and represent them by geometric constructions. {These formulas give a solution for reducing the power of the root}. Surprisingly, one of these algorithms turns out to be the same as the one using Newton's tangent method for calculating zero values of functions of the form $f(x)=x^n-c$, but obtained without use of analysis. Moreover, regarding speed of convergence these algorithms are faster than Newton's tangent method. \par This geometric interpretation of mean values and root calculation fits into the larger context of affine geometry, where we use multi-projections as generating transformations for building up all the affine transformations. Our focus will primarily be on mean values and roots.

**Keywords:** Arithmetic mean; geometric mean; harmonic mean.

**MSC Subject Classification:** 97G40

**MathEduc Subject Classification:** G44

Some approximations of the Euler number | 51$-$56 |

**Abstract**

In this paper, we find new approximations of the Euler number $e$ and using Matlab we compare the existing approximations and the new approximations by testing their convergence rate to the Euler number for some terms.

**Keywords:** Number $e$; approximation; Carleman's inequality.

**MSC Subject Classification:** 97I30

**MathEduc Subject Classification:** I35

Some remarks on Ptolemy's theorem and its applications | 57$-$70 |

**Abstract**

The subject matter of this article is a beautiful theorem developed by Ptolemy (about 100--178) in Chapters 10 and 11 of the first book of Almagest, the Great Collection of Astronomy in 13 books. In order to solve astronomical problems, Ptolemy used mathematical tools to calculate the length of the chord in the circle of radius $60$ as a function of the central angle. The application of Ptolemy's theorem is presented by giving some interesting examples. The paper analyzes the results in solving problems from the competitions in which Ptolemy's theorem is applied.

**Keywords:** Ptolemy's theorem; cyclic quadrilateral.

**MSC Subject Classification:** 97G40

**MathEduc Subject Classification:** G44

Modern statistical literacy, data science, dashboards, and automated analytics and its applications | 71$-$80 |

**Abstract**

With regard to the internationalization of statistics education, this papers considers first a global context concerning modern statistical literacy, data science, and dashboards. Then, it examines data discovery using automated analytics, whereby data insights may be indicated by suitable signals generated by the computer environment used. This theoretical paper, directed towards statistics educators, as well as other educators in relevant high school subjects, should make them (more) aware of this context and such analytics, supporting them to identify issues that need be considered in their teaching (and research) in order to have their students better prepared for the jobs of tomorrow.

**Keywords:** Automated analytics; data science; statistics education; upper secondary.

**MSC Subject Classification:** 97K80, 97R40

**MathEduc Subject Classification:** K44, R44