Volume XXV , issue 1 ( 2022 ) | back |

Geometry understanding assessment based on van Hiele theory using comparative judgment | 1$-$12 |

**Abstract**

The importance of improving and raising the understanding of core concepts in mathematics is well known. Geometry, along with sets, algebra, data and probability, is one of the basic mathematics fields taught in primary and high schools. We have noticed that lately, students' results in geometry are much worse than the results in other mathematical fields. One of the proven methods for improving teaching geometry is harmonising the learning process and content presented in school with the level of understanding on which students are. The mentioned method is based on Van Hiele theory. In order to monitor the effectiveness of the method, it is necessary to assess the level of understanding of students periodically. On the other hand, Comparative Judgment as an assessment method is efficient, fast, and has good outcomes. Our research aimed at investigating whether the Comparative Judgement method can be used to predict the level of understanding of geometric concepts according to the Van Hiele theory.

**Keywords:** Assessment; van Hiele theory; Comparative Judgement.

**MSC Subject Classification:** 97G40; 97D60

**MathEduc Subject Classification:** D44

Towards nonlinear equations -- a case teaching approach in mathematical analysis of functions of one variable | 13$-$20 |

**Abstract**

In this note we are concerned with the proposal of case study approach in teaching mathematical analysis. By describing a simple student's project about the solvability of (nonlinear) equations we aim at indicating that many possible further directions are also possible.

**Keywords:** Nonlinear equation; single variable function; solvability method; case-study.

**MSC Subject Classification:** 97I99

**MathEduc Subject Classification:** I20, C70

Two hidden properties of hex numbers | 21$-$29 |

**Abstract**

In this paper, we prove that the $n$-th hex number is exactly the sum of the number of pieces and the number of triple points associated with an `$n$-balanced' partition of a triangle obtained by $n-1$ cevians from each vertex. Moreover, we see via hex numbers an extension of a Feynman's result: the $(k+1)$-th hex number is the ratio of the area of a triangle $T$ and the area of central triangle associated with a regular partition of $T$ of order $2k+1$.

**Keywords:** Hex numbers; cevians; balanced partitions of triangles; regular partitions of triangles; Feynman's triangle.

**MSC Subject Classification:** 97G30

**MathEduc Subject Classification:** G34

Revisiting the first mean value theorem for integrals | 30$-$35 |

**Abstract**

We provide a proof of the first mean-value theorem for integrals using the Cauchy mean-value theorem, and give an interesting application of the mean-value theorem related to a Taylor remainder.

**Keywords:** Mean value theorem; Taylor remainder.

**MSC Subject Classification:** 97I50

**MathEduc Subject Classification:** I55

A short elementary proof of the infinitude of prime numbers | 36$-$37 |

**Abstract**

We present a new short proof of the infinitude of prime numbers. This is a proof by contradiction, and it is based on the prime factorization of a positive integer.

**Keywords:** Primes; prime factorization.

**MSC Subject Classification:** 97H40

**MathEduc Subject Classification:** H44

Root finding techniques that work | 38$-$52 |

**Abstract**

Several general techniques are described to incorporate the specific structure or properties of a nonlinear equation into a method for solving it. This can mean the construction of a method specifically tailored to the equation, or the transformation of the equation into an equivalent one for which an existing method is well-suited. The techniques are illustrated with the help of several case studies taken from the literature.

**Keywords:** Nonlinear equation; transformation; multiplier; approximation.

**MSC Subject Classification:** 97N40

**MathEduc Subject Classification:** N45

Teaching abstract algebra through programming | 53$-$59 |

**Abstract**

This note presents some programming exercise examples usable in a first course of abstract algebra. The topics include detecting properties of an algebraic operation in a finite set, implementing the Euclidean algorithm in different Euclidean rings, long division of polynomials over a finite field, finding all low-degree irreducible polynomials over a finite field.

**Keywords:** Teaching math through coding; algebraic operations; Euclidean rings; Polynomials over a field.

**MSC Subject Classification:** 97U50, 97H40

**MathEduc Subject Classification:** U55, H45