The development of concepts through thousands of years

There are several difficulties when teaching concepts; we have to consider its content, its abstract meaning, its visual representation, verbal and nonverbal aspects of it, etc. Also, the individual content of a concept changes with the knowledge of someone. However, the content of the concepts through the ages vary but education does not necessarily follow the changes. Not talking of the everyday meaning of a concept, this can get stuck at a certain level, while the scientific content changes. The concepts we teach in schools are not necessarily the actual ones. Thus, it is necessary to revise how we teach some of the concepts. One of the most important concepts of this kind is the prime number. Classification: D30, D39, C30, C39. Keywords: Prime number, concept development, goals of mathematics education, cognitive processes.

GeoGebra as a tool for visualisation in geometry education

There are problems in geometry education in lower and upper secondary school, which students have with the spatial imagination and with the understanding of some geometric concepts. In this article, we want to present tasks that show some advantages of the software GeoGebra. We use this software as a tool to visualize and to explain some geometric concepts, as well as to support students’ spatial imagination. Classification: D30, G10. Keywords: space imagination, GeoGebra, mathematics education at lower and upper secondary level.

Doppelte Diskontinuität, Fundamentale Ideen und Expository Style Teaching

Felix Klein coined the concept of double discontinuity. He rightly observed that students faced two major changes when they started to study mathematics at the university and when they returned as teachers to school feeling that university mathematics was not so useful. Recently a new book on this topic appeared (Ableitinger, Kramer und Prediger 2013). The didactical concepts of Fundamental Ideas (Schweiger 2010) and Expository Style Teaching (Schweiger 2016) could be a helpful addition to this problem. Classification: B50, D20, D40 . Keywords: Doppelte Diskontinuität, Fundamentale Ideen, Expository Style Teaching.

„Niemand kann erklären, was eine Funktion ist“

Starting from the quote from Hermann Weyl given in the title a ramble is undertaken through the development of the notion of function with special emphasis on the question whether the values are associated following a law. On the one hand, this shows a success story of the interplay of this notion and of infinitesimal calculus. On the other hand, one finds impressive examples of overgeneralizations. Classification: C30, D70, E40, I20, I30, M10. Keywords: notion of function, functional laws, overgeneralization.

Welche Aufgabenstellungen formulieren SchülerInnen und StudentInnen auf der Basis von Texten über reale Situationen?

Faced with real situations, pupils and students find questions of non-mathematical or mathematical character. What depends on the type and number of questions? What significance do previous experiences have? The article reports on studies in mathematics education and in teacher training. Classification: B50, C30, D50. Keywords: mathematics education, realistic mathematics, self-created questions, problem solving in context.

History of mathematics, mathematical histories, storytelling in Hungarian mathematics education

History of mathematics is rarely used in Hungarian mathematics education, and even more rarely goes beyond anecdotic mentions of history. In this paper I will argue that despite of this phenomenon, a historical perspective on mathematics, in a more general way, plays a crucial role in a specific Hungarian tradition of mathematics education, called felfedeztető matematikaoktatás (“teaching mathematics by guided discovery”). I will revisit the epistemological background of this approach, analyse the role of history in this view on the nature of mathematics and its teaching, and illustrate the analysis by some examples from written sources and nowadays teaching practice. Classification: A30, D20, D40. Keywords: History of mathematics, history in mathematics education, guided discovery in mathematics education.

Elementary problems with linear algebraic background

We discuss problems for students age between 14 and 18, solving in an elementary way, and by using higher mathematical tools. We also made a research asking university students to solve the problems. The aim of the research was, to see how students treat these elementary problems by using linear algebraic background. Classification: B40, H20, E50, F30. Keywords: proof methods, manipulation of expressions, linear algebra.

The long way a concept category develops

By the age of 6 every child has all the characteristics and capabilities what an individual dealing with Mathematics basically needs. The following are still missing when starting school: conscious way of gaining experience, knowledge acquired and processed the proper way. In this article I would like to show a possible sample of this proper, long way. Classification: C30, C70, D40, E60, U30. Keywords: Mathematics, mathematical thinking , long way, proper educational work, before the preschool age, Preschool age, Junior primary school, Secondary school, to gain experience and to build concepts based on actions and acts.

From the concrete activity to the exact mathematics

In this paper, we are dealing with problems that have been discussed with prospective mathematics teachers to make them discover methodological opportunities and pitfalls at different levels of problem solving. We have chosen topics which are in themselves interesting for children, because they can be introduced playfully with concrete activity, they require little mathematical knowledge at the start, and their conscious discussion is also important for mathematical activities and applications of mathematics. We will show in Chapter 2 how the games lead to the Fibonacci series and the Pascal triangle, and in Chapter 3, how to get from paper folding to dragon curves. Classification: A30, D50. Keywords: manipulative materials and their use in problem solving (visualizations, models, educational games, paper folding), Fibonacci sequence, Pascal triangle, dragon curve.

Zero – The contradiction between words and numbers

In this article I am dealing with the difference between the meaning and mental representation level of the number zero, when the word problem indicates the existence of a certain thing. This contradiction causes difficulty not only for the 10-year-old problem solvers, but also for university students. I give some statistics about the two different representation levels in the above mentioned two groups. There are huge differences between the two groups because university students were able to handle the problem but only two of them succeeded to do it on an ideal level. In my reasoning, I emphasize that the formulation of the task should not be necessarily short. Classification: C50, D50, F30. Keywords: number zero; word problem; problem solving; mathematics teaching; mental representation.