Balanced simplicial complexes are important objects in combinatorics and commutative algebra. A d -dimensional simplicial complex is balanced if its vertices can be coloured into d +1 colors, so there is no monochromatic edge. In this article, we establish two results concerning balanced simplicial complexes assigned to tilings of m × n board in a plane and a torus by I p -omino tile

The main aim of this review is to systematize and present evidence from recent scientific studies regarding the role of early mathematical competencies in later mathematics achievement, as well as to analyze characteristics of educational and family environments that facilitate the development of these competencies in children. Presented longitudinal studies provide clear evidence that early math competencies significantly predict later mathematical outcomes. Reviewed studies also suggest that how adults and children interact in terms of mathematics activities is related to children’s mathematics development. In both formal education and family environment, adults who use everyday situations and activities for reasoning, communicating, and mathematical thinking seem to be the most important factor for fostering early math competencies. These findings are also discussed in the context of BiH where most children do not attend formal early education.

Lindström – Gessel – Viennot theorem connects linear algebra and combinatorics with graph theory. We will present proof of LGV theorem and its application on Cauchy – Binnet and generalized Cauchy – Binnet formula as well as calculation of binomial determinants and some other specific determinants.

A wide range of mathematical ideas could be used to develop and justify a formula for calculating the area of trapezoid. Those ideas lead to different strategies for finding out area of trapezoid that we classify in three groups: decomposing, enclosing and transforming strategies. Those strategies should be part of mathematics content knowledge for teaching. In this study we trace a change in structure of mathematics content knowledge of nine pre-service mathematics teachers as a result of using GeoGebra applets that visualize different approaches in finding out the area of trapezoid. We argue that engaging pre-service mathematics teachers to develop and justify formula for calculating the area of trapezoid using GeoGebra applets is a worth task that enhances pre-service mathematics teachers’ content knowledge for teaching. Our experiment confirmed that the use of Geogebra encourage pre-service mathematics teachers to uncover new ideas that lead them towards clearer justifications and easier way of proving formula for area of trapezoid. Keywords: Area of trapezoid, GeoGebra, content knowledge for teaching