One of the key abilities that should be learned by every student is mastering math skills. It is particularly difficult for blind students who have problems with access to information that is usually presented graphically for common pupils. In mathematics one such information is the presentation of graphs. However there are several drawing techniques for the blind, but they are expensive, and hard for common using. One way to solve this problem may be using a special sound representation of math graphics for the blind students. In this article, an approach allowing to provide making such presentation possible is presented. Briefly it is grounded on the conversion of graphic data of the math drawing to sound wave with proper assets.
In this article I intend to present implementation of the CLIL method at the International Faculty of Engineering of Lodz University of Technology and share my experience I have gained while working as an academic teacher there. I will point out challenges facing lecturers who teach in a foreign language as well as satisfaction this work brings.
It is evident for many people that the area of a rectangle can be calculated according to the very well known formula: P = a ∙ b. We, mathematicians do believe in no statement without the proof. Then we can ask whether it is possible to prove that this formula is correct. This article answers to that question.
A graphic display calculator (GDC) is becoming more and more popular in teaching mathematics as it is used to examine some mathematical activities of students of almost all ages. Various modes of GDC are considered to be a useful tool in understanding of particular parts of mathematics. In most cases the properties of functions are examined by observation of their graphs. However, there are some properties of the functions which one cannot see during the graphs analysis (for example properties concerning complex roots of polynomials). The aim of this paper is to analyse how 17-and-18-year-old students for whom GDC is an obligatory device can generalize some relations between polynomials and so called “shadows” of these functions. The whole paper is concerned with in investigation of properties of quadratic, cubic and quartic functions with both real and complex roots.
The author introduces one possible option for the analysis of the textbooks. It introduces an analysis of those teaching methods used in four textbooks. The presentation is focused on the evaluation of two parallel mathematical textbooks for secondary schools of four different publishers by two publishers Didaktis and Prometheus. It evaluates and compares the textbooks. The aim of this evaluation is to help teachers with the selection of the most suitable textbook for teaching purposes.
The CLIL method is a relatively new trend in education. It is a combination of content and language learning (from the English: Content and Language Integrated Learning). The article is focused on using the CLIL method in the Czech Republic. The article explores the influence of this method on students’ motivation, too.
The current core curriculum in mathematics for lower secondary school (3-rd educational level in Poland) omits formal definitions of concepts related to geometric transformations in the plane and is based on their intuitive sense. Practice shows that the current approach makes teaching very difficult and the students solve the typical tasks, not understanding the meaning of geometrical concepts. The article contains basic concepts connected with geometric transformations and examples of geometric tasks that are solved in the third and also in the fourth educational level in an intuitive way, sometimes deviating or even incompatible with the mathematical definition. We show how they could be solved in easier way with introducing definitions of geometric transformations in a simple and understandable for students way sometimes using vector calculus. We take into account isometries: reflection and point symmetry, rotation and translation and similarities with particular consideration on homothetic transformation.
Mathematical modelling is a concept that covers a wide range of activities. Mathematical modelling can be understood both as formulation of an equation, a function, etc., describing a given situation and as a whole process of creating a model, starting from the real-world situation to the creation of a ready-to-use optimized tool. The work presents different approaches to mathematical modelling from the point of view of teaching mathematics. It presents the results of the research conducted on students (future teachers) regarding their theoretical knowledge and skills related to mathematical modelling.
In the paper we present selected outcomes of a research into possible acceptance of inquiry based mathematics education as a method used by primary school teachers. We suggest some objective as well as subjective reasons for explaining its so far low level of usage and aim at identifying obstacles, which prevent the enquiry based techniques in the primary school educational environment.
The article is concerned with the influence of modern technologies on learning mathematics and other subjects. It presents Marc Prensky’s theory of digital natives and digital immigrants in the roles of the present-day pupils and teachers. The article includes opinions with regard to the necessary change in the educational process, especially in the approach of teachers to the present-day pupils and students.
In the paper we present the main idea of the concept which we have called confrontational concept of mathematical epistemology. We refer it to philosophy of mathematics (in the context of epistemology of research) as well as to didactic problems (in the context of teacher preparation). Although we tried not to involve our discussion directly with any existing concepts of the philosophy of mathematics, however, in the paper one can notice some elements of modern formalism as well as Lakatos quasi-empiricism or a modern approach to structuralism.
Elements of theory of divisibility are present in many various interesting mathematical tasks, especially in tasks that are addressed to talented pupils taking part in mathematical competitions. Good understanding of it lets solve very interesting and difficult (at first glance) issues. On the other hand, there are a lot of problems with understanding such terms as: multiple, divisor, divisibility, prime number, LCM, GCD etc. The purpose of the article is presenting the base terms of theory in the language understood for pupil of the gymnasium (3-rd educational level, 13-16 years old). In addition, we present some algorithms that are used to solve problems from the theory of divisibility and we discuss the influence the choice of the algorithm on its effectiveness (so we analyse its complexity). Presented algorithms let us create computer programs that solve the problems mechanically. We also enlarge a bit some topics for those ones which can be taught in the class of pupils which are interested in Mathematics.
The article shows a specific example of extracurricular activities conducted in secondary school, how the graphing calculator helped the first class student in learning mathematics to solve a very difficult task (math problem): How many elements has the equation: a x = log a x. The article describes the reasoning and attitude of the student who voluntarily of his own accord, inspired by other students to experiment, putting, generalizing and verifying hypotheses coped with the solution of this task. It describes the impact of this teaching mean that triggered activity and aroused student’s interest with the task on the degree of knowledge and skills in mathematics, the student’s skills in the use of mathematical language, self-reliance in solving a mathematical problem.
Development in technology, changing labour market and the need for lifelong learning necessitate the evolution in the education of children and youth. Blended learning is about combining the online learning and traditional methods in order to personalize the learning process. The intensive application of this teaching method in the United States seems to make the American students more engaged in solving the tasks assigned to them. It is believed that it has a positive impact on their final exam results. The article describes the basic principles of blended learning from the perspective of the theory of hybrid. It also includes some personal experience of working with this method.
Many times our students make some errors in definitions, especially when we must apply some quantifiers. The definition of a limit is one of the definition with many quantifiers, so one can observe many mistakes in it. We want to present one of possible mistakes and show how to improve the understanding of this difficult but one of most important notions.
Difficult subjects in the course of study may have beneficial influence on future life of graduates. Authors of the article are pointing out attention on connections that exist between teaching mathematics and choice of career path for graduates of Management Faculty AT University of Technology in Częstochowa. In first part of the article is briefly presenting an influence of mathematical thinking on growth of various civilizations. Than focus in turned on benefits of mathematical thinking, personal development and positive personality traits of people working with mathematics. Succeeding part of the article is describing the topics covered by the scope of studies on a bachelor's degree in Management Faculty. Educational goals were presented as well as their usage in economic realities. Final chapter consist of different paths of life choices for graduates of university of technology with emphasis of those traits, which was developed during intercourse with mathematic.
Trudne przedmioty w toku studiów mogą mieć wręcz zbawienny wpływ na przyszłe życie absolwentów. Autorzy artykułu zwracają uwagę na związki, jakie występują między nauczaniem matematyki, a wyborem zawodowej drogi absolwentów Wydziału Zarządzania Politechniki Częstochowskiej. W pierwszej części artykułu krótko przedstawiony jest wpływ matematyki na rozwój różnych cywilizacji. Następnie zwrócono uwagę na korzyści matematycznego myślenia, rozwój osobowości i pozytywnych cech charakteru ludzi „uprawiających” matematykę. W dalszej części artykułu opisany został zakres materiału realizowany w ramach studiów pierwszego stopnia na Wydziale Zarządzania. Zaprezentowane zostały cele kształcenia oraz zastosowania w realiach gospodarczych. Na zakończenie przedstawione zostały różne możliwości wyboru drogi życiowej absolwenta wyższej uczelni ze szczególnym uwzględnieniem tych cech, które nabył w wyniku „obcowania” z matematyką.
The paper signals the foundations of the new method of teaching mathematics that is currently emerging from the concept of human cognition and the constructivist paradigm. The presented examples of the hermeneutic research conducted for 17 years are concerned with an analysis of the formulated mathematical problems in the language of photographic metaphors. Thoughts expressed through the photographic image and text consisting of the caption and the description (dual coding) reveal the structure of cognitive networks of authors of photographs, which has a special significance in creation of the new didactics that will fulfil the needs of the contemporary photosociety. Mathematical photoeducation free transition between art and mathematics lies on the student’s artistic sensitivity and on enlivening the student’s cognitive expression in a space distant from the classroom (at a lake, in the playground, on the skating ring or during a field excursion to a mineral museum). It utilizes the student’s natural interest in observable natural phenomena and in man-made objects. This kind of creativity, which relies on independent uncovering or constructing of knowledge with the help of a photographic camera, opens the gates to an entirely new space of mathematical didactics, as it brings to students’ awareness specific ways of association leading to accomplishment cognitive processes in relation to abstract mathematical objects.
W artykule zasygnalizowano podstawy nowej metody nauczania matematyki powstającej na bazie koncepcji poznawczej człowieka i w oparciu o paradygmat konstruktywistyczny. Przedstawione przykłady prowadzonych przez 17 lat badań hermeneutycznych dotyczą analizy formułowanych problemów matematycznych w języku metafor fotograficznych. Myśli wyrażone za pomocą obrazu fotograficznego oraz tekstu składającego się z podpisu i opisu (dual coding) odsłaniają strukturę sieci poznawczych autorów, co ma szczególne znaczenie w tworzeniu nowoczesnej dydaktyki, która ma odpowiadać na potrzeby dzisiejszego fotospołeczeństwa. Fotoedukacja matematyczna pozwala na swobodne przemieszczanie się pomiędzy sztuką a matematyką, zakłada artystyczną wrażliwość ucznia i ożywienie jego ekspresji poznawczej w przestrzeni odległej od sali szkolnej (nad jeziorem, na podwórku, lodowisku czy podczas wycieczki do muzeum minerałów). Wykorzystuje naturalne zaciekawienie ucznia obserwowanymi zjawiskami świata przyrody i obiektami utworzonymi przez człowieka. Twórczość tego typu, opierająca się na samodzielnym odkrywaniu lub konstruowaniu wiedzy przy pomocy aparatu fotograficznego, otwiera zupełnie nową przestrzeń dydaktyki matematyki, gdyż uświadamia uczącym konkretne drogi asocjacji, prowadzące do zrealizowania procesów poznawczych w odniesieniu do abstrakcyjnych obiektów matematyki.
In contemporary concepts of school education one suggests a far-reaching integration of teaching contents. The integration is aimed to help school children to gain a comprehensive world picture, stimulate their activeness, develop some creative attitudes of the school children toward mathematics and elaborate various forms of organizing classes. According to the experience with integrating mathematics with other teaching subjects, there are difficulties in realizing the aims mentioned above. In this article some reasons for these difficulties will be discussed at large.
We can create many activities when we place geometrical problems to solve in class. For example: analyzing questions, making observation, discovering property of geometrical objects, searching the dependences among them, making hypotheses, making argumentation, proving, investigation of special cases and many others. Using traditional didactic resources causes that the abovementioned activities may be difficult for pupils at every level of education. It is also hard to explore the considered problems. That is why it is legitimate to search how to use ICT to support pupils in this process. Using GeoGebra based materials is an interesting idea for this purpose. Well made interactive materials can be used during normal lessons as well as during additional activities. Such a solution may increase understanding of the problem, and also allows a teacher to discuss with pupils extra curricular mathematics contents.
The problem of students' understanding and mastering mathematics is currently one of the most emphasized topics of expert public discussion. New ways of teaching mathematics more effectively and attractively are being searched for. The information society brings new alternatives nowadays; it enables us to change established and rigid forms and methods in teaching. While searching an answer to the question why the student attitudes toward mathematics are so often negative and what could influence this situation in a positive way, we conducted a survey realized under the program ITMS -Flexible and attractive study at the University of Žilina for the needs of the market and knowledge-based society. The paper presents some of the out comes which have ensued from the data obtained in the project realization. 200 students from three faculties of the University of Žilina were addressed in the survey (Faculty of Civil Engineering, Faculty of Operation and Economics of Transport and Communications, Faculty of Special Engineering), in which their attitudes toward mathematics were detected as well as the factors determining these attitudes.