Volume XXI , issue 2 ( 2018 )back
Data modelling using interactive charts55$-$72
Djordje M. Kadijevich
Institute for Educational Research, Dobrinjska 11/III, 11000 Belgrade, P.O.B. 546, Serbia


This paper examines data modelling with interactive charts in upper secondary education. After defining data modelling per se, and explaining its relationship to mathematical modelling, data modelling with interactive charts is examined in detail, focusing on the main challenges in using them in the context of key data modelling activities. This examination is followed by a report of an empirical pilot study on the difficulty of these activities and the importance of this kind of modelling for upper secondary education. Implications for practice and research are included.

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Keywords: Data modelling; interactive charts; upper secondary education.

MSC Subject Classification: 97M10

MathEduc Subject Classification: M14

A greatest common divisor identity73$-$79
Yuanhong Zhi
School of Mathematics and Statistics, Yunnan University, Kunming, PR. China


In this paper, we present an identity involving the greatest common divisors of almost all possible subproducts of $n$ nonzero integers. Then we prove this identity, with the help of the fundamental theorem of arithmetic, and an identity concerning the minimum function $\min$. As a consequence, a new formula for the least common multiple is derived.

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Keywords: Greatest common divisor; fundamental theorem of arithmetic; least common multiple.

MSC Subject Classification: 97F60

MathEduc Subject Classification: F64

On ``one problem multiple change'' in Chinese ``bianshi'' mathematics teaching80$-$96
Xu Yanhui
Department of Mathematics, Wenzhou University, Zhejiang Wenzhou 325035, China


Bianshi teaching explains the paradox of Chinese learners very well. As an important form of bianshi teaching, ``one problem multiple change'' widely exists in classrooms of Chinese mathematics teachers. Based on three different examples of this method, the present paper shows how Chinese mathematics teachers do it in classroom, and further discusses the position, function and value of this method in mathematics classroom teaching in China. At the same time, this can further explain the phenomenon of ``Chinese learners paradox''.

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Keywords: One problem multiple change; bianshi teaching; mathematics problem; variation; invariant.

MSC Subject Classification: 97D40

MathEduc Subject Classification: D43

Bimatrix games have a quasi-strict equilibrium: an alternative proof through a heuristic approach97$-$108
Takao Fujimoto, N. G. A. Karunathilake and Ravindra R. Ranade
T.F.: Department of Economics, University of Kagawa, 760-8523, Japan,


We present an alternative proof for the existence of at least one quasi-strict equilibrium in every bimatrix game. While Norde [Bimatrix games have quasi-strict equilibria. Math Prog, {85}, 35--49] uses Brouwer's fixed point theorem, we employ Kakutani's fixed point theorem for multivalued maps, and make our proof shorter, thus teachable in a couple of lecture talks. Besides our approach admits of natural economic interpretations of some technicalities used in the proof. We also explain how we get to our method of proof. In addition, it is remarked that it is possible to adopt a field more general than that of real numbers.

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Keywords: bimatrix game; Kakutani fixed point theorem; multivalued map; quasi-strict equilibrium.

MSC Subject Classification: 97M40, 91A05, 91B62

MathEduc Subject Classification: M45