### Report of Mathematical Transgressions conference

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The study presented in this paper is a part of a substantive analysis of different didactical phenomena in relation to several mathematical concepts, conducted at the preparatory stage of a project aimed at investigating substantive and didactical competencies of pre-service teachers of mathematics. What I report here, relates to the exploration of didactical phenomena in relation to the tangent concept. Since school textbooks are the most common resource referred to by students and teachers, they...

This article is devoted to the different representations of real numbers. In particular, the following types are distinguished and discussed: (1)representations based on theorems referring to the axiomatic characterization of the field of real numbers, (2) genetic representations – related to the construction of real numbers, (3) visual representations – mainly related to the geometrical way of presenting numbers. The paper addresses different representations of real numbers from a higher standpoint...

In this paper we explore different ways of solving quadratic equations. Our main goal is to review traditional textbooks methods and offer an alternative, often side-stepped method based on the area model. We conclude that whereas traditional methods offer effective algorithms that quickly lead to the desired results, alternative methods may enhance meaningful and joyful learning.

This article draws on the work of Wittmann and his followers who conceived and developed the notion of substantial learning environment (SLE). The paper contains a proposal of a teaching unit based on the definition of Factorial Number System (FNS). First, we illustrate the process of conversion from FNS to the Decimal Number System (DNS) and back. Secondly, we provide theorems on the divisibility rules for several numbers in FNS. The main aim of this paper is to present FNS as an example of a~mathematically...

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