Volume XXV , issue 1 ( 2022 )back
Geometry understanding assessment based on van Hiele theory using comparative judgment1$-$12
Aleksandar Balašev-Samarski and Manuela Muzika Dizdarević and Selma Zeković
Faculty of Natural Sciences and Mathematics, University of Sarajevo, Bosnia and Herzegovina


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.

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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 variable13$-$20
Marek Galewski and Jakub \L ompie\'{s}
Institute of Mathematics, Lodz University of Technology, Al. Politechniki 8, 90-924 Lodz, Poland


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.

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

MSC Subject Classification: 97I99

MathEduc Subject Classification: I20, C70

Two hidden properties of hex numbers21$-$29
Silvano Rossetto and Giovanni Vincenzi
Centro Ricerche Didattiche ``U. Morin'', Paderno del Grappa, Treviso, Italy
nd Dipartimento di Matematica, Universit\`a di Salerno, Via Giovanni Paolo II, Fisciano, Salerno, Italy


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 $2k+1$.

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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 integrals30$-$35
Humberto Rafeiro and Sehjeong Kim
United Arab Emirates University, College of Sciences, Department of Mathematical Sciences, P.O. Box 15551, Al Ain, Abu Dhabi, United Arab Emirates


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.

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Keywords: Mean value theorem; Taylor remainder.

MSC Subject Classification: 97I50

MathEduc Subject Classification: I55

A short elementary proof of the infinitude of prime numbers36$-$37
Romeo Meštrović
Maritime Faculty Kotor, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro


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.

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Keywords: Primes; prime factorization.

MSC Subject Classification: 97H40

MathEduc Subject Classification: H44

Root finding techniques that work38$-$52
Aaron Melman
Department of Applied Mathematics, Santa Clara University, Santa Clara, CA 95053


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.

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Keywords: Nonlinear equation; transformation; multiplier; approximation.

MSC Subject Classification: 97N40

MathEduc Subject Classification: N45

Teaching abstract algebra through programming53$-$59
Natalia Saealle and Lauri Tart and Indrek Zolk
Institute of Mathematics and Statistics, University of Tartu, Narva mnt 18, 51009 Tartu, Estonia


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.

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

MSC Subject Classification: 97U50, 97H40

MathEduc Subject Classification: U55, H45