Zdalne nauczanie i uczenie się matematyki w szkole średniej podczas pierwszej fali pandemii Covid-19 z perspektywy wybranych nauczycieli i uczniów

The article presents the beginnings of remote learning in Poland (March-June 2020) in the context of teaching mathematics at the secondary level, as recounted by selected students and teachers. The empirical research upon which this article is based was conducted for the purposes of a master's thesis. The research was carried out in two stages. The first stage presents remote mathematics teaching as perceived by selected teachers and their students, based on two summative questionnaires. The second stage consists of a qualitative analysis of teaching polynomials from the perspective of three pairs of participants: teacher and student. The article shows various issues concerning remote teaching, analyses the effectiveness of the forms and methods of this approach, signals selected shortcomings and defects both in the work of teachers and students as well as selected causes for the difficulties in the implementation of remote teaching, while also highlighting its positive aspects.

Reformy programów i podręczników szkolnych sprzed pół wieku inspirowane prądami tzw. nowej matematyki

The purpose of this paper is to outline the reforms of mathematics education in the spirit of “New Math” in USA, France and to document their features in Poland. Activities and achievements of Zofia Krygowska are listed. Particular attention is paid to two radical reforms: the 1967 textbook on geometry based on set-theory for grade IX and far-reaching changes in primary math education in the 1970s. Excerpts from articles, curricula and textbooks are included.

How to solve third degree equations without moving to complex numbers

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

Wspomnienie o profesorze Bogdanie Janie Noweckim

Informations

Geometria w pierwszych latach szkolnej edukacji w świetle wybranych programów nauczania i poradników metodycznych dla nauczycieli z lat 1773 – 2020

Currently there is little geometry content in the early years of school. In my doctoral thesis Geometry in teaching children since the times of the Commission of National Education until today. Analysis of the successionof education concepts, supervised by Prof. Edyta Gruszczyk-Kolczynska, I examined how the teaching of geometry to younger children has changed since the time of KEN.In the article, I discuss curricula and methodological guide books for teachers in terms of the geometric content they cover in the first years of school education. I focus on three periods: the second half of the 19th century, the 1920s and the 1970s. These periods stand out from the others I studied n that there was a lot of geometric content in the first years of school. However, in the first one of these, the content was mainly included in the subject “drawings”, while in the others the main aim of teaching was to develop pupils’ computational skills. To a large extent, geometry has also served this purpose.

Obraz pojęcia ciągłości – wyniki badania przeprowadzonego wśród studentów

This article addresses the issue of university students' images of the concept of continuity. The aim of the reported study was learn how the students understand continuity of a function and what kind of difficulties they experience when dealing with this concept. The author hopes that this report will help academic teachers to better understand students and their mistakes, and hence will facilitate academic teaching.

Students’ approaches while solving a non-typical geometrical problem

In the paper we present the results of two teaching episodes, which took place in two middle school classes with 13- and 14-year-old students. The students in both classes were asked to solve the same geometrical problem;then a discussion followed, in which they had to justify their solutions. In both cases the students had no prior experience in solving non-typical mathematical problems. Additionally, the students were asked to justify theiranswers, which is not a common characteristic of a ‘typical’ mathematics classroom at that level. The problem was chosen from a wider study, in which twenty classes from twenty different schools were analysed. One of theaims of the present study was to analyse the skills that require a deeper understanding of mathematical concepts and properties. Particularly, we aimed to investigate students’ different solution methods and justifications duringproblem solving. The results show considerable differences among the two classes, not only concerning the depth of investigating (which was expected due to the different age groups), but also concerning the relationship betweenachievement (as assessed by the mathematics teacher) and success in solving the problem. These results demonstrate the need for re-directing mathematics education from a pure algorithmic to a deeper thinking approach.

PISA, TIMSS and Swedish students’ knowledge of linear equations: A ‘telling’ case of a system fixing something that may not be broken

In this paper, we construct a ‘telling’ case to highlight a problematic inconsistency between the results of international large-scale assessments (ILSAs) and other studies of Swedish students’ knowledge of linear equations.In this context, a ‘telling’ case, based on the scrutiny of appropriately chosen cases, is presented as a social science counter-example to the prevailing view that ILSAs’ assessments are not only valid but should underpin systemicreform. Our ‘telling’ case comparison of the different forms of study shows that Swedish students, in contrast with the summative assertions of the different ILSAs, have a secure and relational understanding of linear equationsthat persists into adulthood. We conclude with a cautionary message for the curriculum authorities.

Decoding Book II of the Elements

The paper is a commentary to the Polish translation of the Elements Book II, included in this volume.We focus on relations between figures represented and not represented on diagrams and identify rules which enableEuclid to bridge these two kinds of objects. Also, we argue that the main mathematical problem addressed in Book II is constructing a leg of a rightangled triangle, given its hypotenuse and the other leg. In proposition II.14,Euclid solves it through the construction called the geometric mean. We trace the problem in Book III and beyond the Elements: in Heron’s Metrica, Descartes’ La Géométrie, and modern foundations of mathematics. We showthat Descartes, by novel interpretation of the Pythagorean theorem, provides a modern solution to this problem.

Application of Banach contraction principle to approximate the golden number

This work is devoted to the application of selected fixed point theorems in the problems of convergence of certain sequences to the golden number. It contains the theorem about the fixed point of so-called ψ-contraction specified on the closed interval <a, b> and the local version of Banach Contraction Principle as a conclusion. It will also be used to approximate the golden number.