Early Geometrical Thinking in the Environment of Patterns, Mosaics and Isometries

Abstract

This survey confirms that one of the research trends in early geometrical reasoning has been a focus on creating a theoretical basis for research in this area. The main reason for the work on the theoretical foundations has been the fact that building geometrical concepts proceeds according to other regularities more than it takes place while building arithmetic knowledge. In this way, a broad research area was outlined. The first theory was created by Dina and Pierre Van Hiele (the Netherlands), but many on-going studies are related to this issue and many have focused on analysing and further expanding this theory.In parallel, new conceptual approaches that capture the question of the theoretical basis in a different way have been created. They should be considered complementarily, as they repeatedly point to the other aspects of the geometrical knowledge. The examples discussed in this paper come from the circles of scholars gathered around Alain Kuzniak (France) and Milan Hejný (Czech Republic). A networking between Van Hiele's theory and the description of geometrical paradigms created by Kuzniak examine ways that children work at the lowest levels. This allows analysis of how they gain experience, which is the basis for the transition in the area of further paradigms. This is therefore a departure from the theories that have a linear structure and focus only on a successive development.Milan Hejný indicates new aspects for descriptions of understanding of geometrical concepts, but first and foremost, he connects his proposals with the concept of schema-oriented education, where creating skills and procedures should be seen from a long perspective. In his theoretical background, a relatively new trend concerning the development of geometric reasoning is a pro-ceptual approach to the concepts. The formation of geometrical concepts is related to the empirical

Effect of GeoGebra-Aided REACT Strategy on Understanding of Geometry Concepts

“…Geometry is one of the best areas for a child to enter the world of mathematics. The geometrical world can be opened very early because geometrical knowledge correlates very well with children's natural cognition (Swoboda & Vighi, 2016).…”

Networked Analysis of a Teaching Unit for Primary School Symmetries in the Form of an E-Book

“…Although research published in the 1990s showed benefits of students programming Logo in their learning of transformations, in particular symmetries [1] and, more recently, it has been showed that students learning in dynamic manipulative environments (either real or electronic) are more successful than those learning in static environments [2], there is a lack of research on the application of the newly available technologies in the teaching and learning of geometry [3], either as purely technological classes or blended learning [4]. Furthermore, the educational constraints to in person classes imposed by the pandemic of covid-19 give an added value to technological teaching and learning environments that can be used in virtual classrooms, so research on new electronic learning environments and blended environments is still necessary.…”

“…Technological environments, like those based on DGS, have proved to be helpful to prevent or remove this kind of students' prototypical images when they are allowed to manipulate the axes of symmetry and get used to see them with any slope [16]. Then, although teaching and learning of isometries have not been frequently explored by mathematics education research [7], an interesting research direction is the design of technological environments where students can manipulate geometrical objects (in our case those related to symmetries) to gain experience and build on it a deep understanding of the related concepts and properties [2]. In this paper we present a teaching unit for symmetries implemented as an interactive e-book [17] that takes advantage of the possibilities of DGS played in computers and tablets.…”

“…Technological environments that can be designed to allow a diversity of learning objectives and learning speeds are adequate for primary and secondary school mathematics teachers to attend the diversity of pupils they usually have in their regular class groups [10,19]. Mathematical giftedness has not been a main focus of attention for researchers in mathematics education [20], although there is an increasing number of publications reporting research experiments with secondary school m-gifted students based on technological environments, but there is a need of such kind of research with primary school m-gifted students [21], since these students are ready to move from empirical tasks to abstract problems, and such transition poses troubles to them to gain deeper levels of understanding [2]. Most activities in the teaching unit we present are adequate for average students, but it also includes activities that are aimed to provide a deeper knowledge of symmetries to m-gifted students.…”

“…The teaching unit is adequate for upper primary or lower secondary school grades, depending on the grade where symmetries are studied for the first time. The learning objectives of the teaching unit include (1) the basic properties of plane symmetry, (2) procedures to draw the symmetric of a point or shape respect to a given axis of symmetry and draw the axis of two given symmetrical points or shapes, (3) the definition of plane symmetry, and (4) products of two symmetries and decompositions of translations and rotations into products of two symmetries. We present the teaching unit in the form of an Apple's iBook which takes advantage of the interactivity of e-books and some specific features of iBooks, like the possibilities of modifying the sequence of activities to be solved by a student and the availability of evaluation questionnaires.…”