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Visualisation in geometry education as a tool for teaching with better understanding
337-346Views:93In primary and secondary geometry education, some problems exist with pupils’ space thinking and understanding of geometric notions. Visualisation plays an important role in geometry education, and the development of pupils’ visualisation skills can support their spatial imagination. The authors present their own thoughts on the potential of including visualisation in geometry education, based on the analysis of the Hungarian National Core Curriculum and Slovak National Curriculum. Tasks for visualisation are also found in international studies, for example the Programme for International Student Assessment (PISA). Augmented reality (AR) and other information and communication technology (ICT) tools bring new possibilities to develop geometric thinking and space imagination, and they also support mathematics education with better understanding.
Subject Classification: 97U10, 97G10
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What does ICT help and does not help?
33-49Views:75Year by year, ICT tools and related teaching methods are evolving a lot. Since 2016, the author of the present lines has been looking for a connection between them that supports the development of mathematical competencies and could be integrated into Transcarpathian minority Hungarian language education too. As a doctoral student at the University of Debrecen, I experienced, for example, how the interactive whiteboard revolutionized illustration in Hungarian mathematics teaching, and how it facilitated students' involvement. During my research of teaching in this regard, in some cases, the digital solution had advantageous effects versus concrete-manipulative representation of
Bruner's too.
At the same time, ICT "canned" learning materials (videos, presentations, ...) allow for a shift towards repetitive learning instead of simultaneous active participation, which can be compensated for by the "retrieval-enhanced" learning method.
I have conducted and intend to conduct several research projects in a Transcarpathian Hungarian primary school. In the research so far, I examined whether, in addition to the financial and infrastructural features of the Transcarpathian Hungarian school, the increased "ICT-supported" and the "retrieval-enhanced" learning method could be integrated into institutional mathematics education. I examined the use of two types of ICT devices: one was the interactive whiteboard, and the other was providing one computer per student.
In this article, I describe my experiences, gained during one semester, in the class taught with the interactive whiteboard on the one hand, and in the class taught according to the "retrieval-enhanced" learning method on the other hand.
I compare the effectiveness of the classes to their previous achievements, to each other, and to a class in Hungary.Subject Classification: 97U70
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"How to be well-connected?" An example for instructional process planning with Problem Graphs
145-155Views:57Teachers’ design capacity at work is in the focus of didactical research worldwide, and fostering this capacity is unarguably a possible turning point in the conveyance of mathematical knowledge. In Hungary, the tradition hallmarked by Tamás Varga is particularly demanding towards teachers as they are supposed to be able to plan their long-term processes very carefully. In this contribution, an extensive teaching material designed in the spirit of this tradition will be presented from the field of Geometry. For exposing its inner structure, a representational tool, the Problem Graph is introduced. The paper aims to demonstrate that this tool has potential for analyzing existing resources, helping teachers to reflect on their own preparatory and classroom work, and supporting the creation of new designs.
Subject Classification: 97D40, 97D50, 97D80, 97G10, 97U30
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Balanced areas in quadrilaterals - Anne's Theorem and its unknown origin
93-103Views:45There are elegant and short ways to prove Anne's Theorem using analytical geometry. We found also geometrical proofs for one direction of the theorem. We do not know, how Anne came to his theorem and how he proved it (probably not analytically), it would be interesting to know. We give a geometric proof (both directions), mention some possibilities – in more details described in another paper – for using this topic in teaching situations, and mention some phenomena and theorems closely related to Anne's Theorem.
Subject Classification: G10, G30
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The tools for developing a spatial geometric approach
207-216Views:31Tamás Varga writes about the use of tools: "The rational use of tools - the colored bars, the Dienes set, the logical set, the geoboard, and some other tools - is an element of our experiment that is important for all students, but especially for disadvantaged learners." (Varga T. 1977) The range of tools that can be used well in teaching has grown significantly over the years. This paper compares spatial geometric modeling kits. Tamás Varga uses the possibilities of the Babylon building set available in Hungary in the 1970s, collects space and flat geometry problems for this (Varga T. 1973). Similarly, structured kits with significantly more options have been developed later, e.g. ZomeTool and 4D Frame. These tools are regularly used in the programs of the International Experience Workshop (http://www.elmenymuhely.-hu/?lang=en). Teachers, schools that have become familiar with the versatile possibilities of these sets, use them often in the optional and regular classes. We recorded a lesson on video where secondary students worked with the 4D Frame kit. We make some comments and offer some thoughts on this lesson.
Subject Classification: 97G40, 97D40
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Straight line or line segment? Students’ concepts and their thought processes
327-336Views:66The article focuses on students’ understanding of the concept of a straight line. Attention is paid to whether students of various ages work with only part of a straight line shown or if they are aware that it can be extended. The presented results were obtained by a qualitative analysis of tests given to nearly 1,500 Czech students. The paper introduces the statistics of students’ solutions, and discusses the students’ thought processes. The results show that most of the tested students, even after completing upper secondary school, are not aware that a straight line can be extended. Finally, we present some recommendations for fostering the appropriate concept of a straight line in mathematics teaching.
Subject Classification: 97C30, 97D70, 97G40
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Is it possible to develop some elements of metacognition in a Mathematics classroom environment?
123-132Views:58In an earlier exploratory survey, we investigated the metacognitive activities of 9th grade students, and found that they have only limited experience in the “looking back” phase of the problem solving process. This paper presents the results of a teaching experiment focusing on ninth-grade students’ metacognitive activities in the process of solving several open-ended geometry problems. We conclude that promoting students’ metacognitive abilities makes their problem solving process more effective.
Subject Classification: 97D50, 97G40
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Thoughts on Pólya’s legacy
157-160Views:83There is a saying, "the older I get, the smarter my parents become." What it means, of course, is that the more we learn, the more we appreciate the wisdom of our forebears. For me, that is certainly the case with regard to George Pólya.
There is no need to elaborate on Pólya's contributions to mathematics – he was one of the greats. See, for example, Gerald Alexanderson's (2000) edited volume The Random Walks of George Pólya, or Pólya's extended obituary (really, a
53-page homage) in the November 1987 Bulletin of the London Mathematical Society (Chung et al., 1987). Pólya was one of the most important classical analysts of the 20th century, with his influence extending into number theory, geometry, probability and combinatorics.