This study explores young children’s strategies while transforming polygons, through the use of geometrical models. Data were collected from 291 children ranging from 4 to 8 years of age in Cyprus. Children were asked to draw a stairway of specific polygons, with each shape being bigger or smaller than its preceding one. Relationships between children’s responses in the transformation tasks, their ability to recognize geometric shapes and their IQ level were investigated. Results showed that children used three alternative strategies in the transformation tasks. Children’s IQ score was directly associated with their transformation strategies, while only a low recognition ability was associated with the use of a defective strategy.

The present study examines secondary school students’ geometrical figure apprehension based on Duval’s theoretical framework regarding perceptual, operative, and discursive apprehension. The aim is to explore the cognitive structure of the geometrical figure apprehension dimensions (operative, discursive, and perceptual) in three grades of secondary school students. The tasks in the present study were completed by a sample of 881 students attending public secondary education in Cyprus. Confirmatory factor analysis indicated the stability of the structure of the model concerning secondary school students’ geometrical figure apprehension. However, differences were found in the interrelations among the three main aspects of the model in the examined grades (9, 10, and 11). Moreover, it was observed that students find it easier to solve tasks involving perceptual apprehension compared to discursive apprehension tasks, indicating a possible hierarchical structure of figure apprehension. The present study acts as a pilot study of the constructed instrument. Finally, the results are interpreted in relation to the type of geometrical paradigm in which students work at each hierarchical level.

Este trabajo analiza las diferentes maneras en que los estudiantes miran las figuras geométricas al resolver tareas geométricas y los diferentes tipos de razonamiento que tienen lugar en relación con los diferentes tipos de aprehensión figural, en el sentido de Duval, que se movilizan. El espacio de trabajo geométrico personal de los estudiantes de secundaria y bachillerato en Chipre se define con respecto a su forma de mirar las figuras y el tipo de razonamiento que producen.

The present study investigates in-service secondary teachers’ geometrical figure apprehension in relation to their ability to construct geometrical proofs and to predict didactically their students’ difficulties and mistakes. The theoretical framework of analysis is based on Duval’s geometrical figure apprehension which was taught to the teachers as part of an in-service training course in didactics of mathematics, offered by one of the researchers. As part of the course final assessment, a written test consisting of various geometrical tasks was constructed and administered to the sample. Participants’ answers in both solving and interpreting difficulties related to the tasks were analyzed. The results of the study indicate that theories and concepts of didactics of geometry can shed light to various facets of teaching and learning of geometry. A teacher who presents a correct solution at a task does not necessarily identify or understand the possible difficulties faced by students. Discussion concentrates on teaching implications about geometry and geometrical proof.

This study aims to investigate high school students’ geometry learning by focusing on mathematical creativity and its relationship with visualisation and geometrical figure apprehension. The presentation of a geometrical task and its influence on students’ mathematical creativity is the main topic investigated. The authors combine theory and research in mathematical creativity, considering Roza Leikin’s research work on Multiple-Solution Tasks with theory and research in visualisation and geometrical figure apprehension, mainly considering Raymond Duval’s work. The relations between creativity, visualization and geometrical figure apprehension are examined through four Geometry Multiple-Solution Tasks given to high school students in Greece. The geometrical tasks are divided into two categories depending on whether their wording is accompanied by the relevant figure or not. The results of the study indicate a multidimensional character of relations among creativity, visualization and geometrical figure apprehension. Didactical implications and future research opportunities are discussed.

The notion of negative numbers is one of the most fundamental in mathematics. For many years, they have caused confusion and they have been an object of controversy among great researchers, until the development of symbolic algebra led to their acceptance in the form they have today. When students are first introduced to the concept of negative numbers and the operations between them, various difficulties occur and misconceptions arise, as expected. Their comprehension is one of the most challenging tasks of teaching mathematics due to their complexity and abstract nature. For this purpose, various models have been devised and employed. In the present study we expose some of the epistemological obstacles commonly observed in understanding negative numbers and then exhibit and compare two different models frequently used for introducing the four basic operations between them.

There are numerous studies about the teaching and learning of mathematics at different educational levels. In the case of higher education most studies were conducted at pedagogical departments for prospective teachers and mathematical departments. The present study concentrates on university students who attend a course on mathematics as part of a program at the Faculty of Economics and Management. It examines aspects of students’ affective and cognitive behavior in solving representation tasks concerning their understanding of exponential and logarithmic functions. Results confirmed the existence of a comprehensive model with significant interrelations among general beliefs, self-efficacy beliefs and cognitive behaviour about the use of representations in general and, in the case of the specific concept. Regression analysis indicated the predominant role the self-efficacy beliefs play in the use of representations in defining the concept of function and solving recognition and translation tasks. Implications about the teaching of mathematics in higher education are discussed.

This study aims to examine the relation between spatial ability and creativity in Geometry. Data was collected from 94 ninth graders. Three spatial abilities were investigated: spatial visualization, spatial relations and closure flexibility. As for students' creativity, it was examined through a multiple solution problem in Geometry focusing on three components of creativity: fluency, flexibility, and originality. The results revealed that spatial visualization predicted flexibility and originality while closure flexibility predicted all creativity components. Additionally, it was deduced that auxiliary constructions played an essential role in the problem-solution process. Finally, further study opportunities for the teaching and learning of Geometry are discussed.

This paper analyzes the content of Arithmetic (1632), a manuscript of commercial arithmetic written by the Greek-Cypriot Petros Argyros. Arithmetic was probably influenced by the first Greek printed arithmetic Logariastiki (Glyzonios 1569), a work with a great deal of similarity to the Italian Abaci. Consequently, the paper devotes attention on the one hand to the Abaci tradition and on the other hand to the early formative stages of Greek and Cypriot mathematics education. The paper's main aim is to draw attention to the analysis of patterns presented in the book. Recreational mathematics was typical for the Abaci tradition and analysis of these problems raises questions about their history and the approaches that were used centuries ago. In the authors' opinion, the very existence of such problems, which presuppose analytic thought and reasoning, compels us to verify the widespread conception of the nature of Abaci mathematics and to define it with greater precision.

In this chapter, we address children's geometry learning in the early years with a focus on visualization. We start the chapter with some background information about visualization in mathematics and geometry and its relationship with language and gestures paying special attention to the early years. The next parts of the chapter aim to give insight into how young children solve geometrical activities with emphasis on the uses of visualization in developing understanding of space and shape concepts. In particular, we discuss three different research approaches which investigated young children's development of geometrical thinking when dealing with shapes' transformations, imaginary perspective taking, and space and shape aspects with the use of gestural and verbal acts. Finally, in light of the above, a number of conclusions are drawn about the multiple qualities and uses of visualization in the development of the understanding of shapes and space and the diverse factors that may intervene in early geometry learning which involves the use of visualization.

This article focuses on exploring students' understanding of the concept of function concerning three main aspects: secondary students' ability to (1) define the concept of function and present examples of functions, (2) solve tasks which asked them to recognize and interpret the concept of function presented in different forms of representation, and (3) solve function problems. Confirmatory factor analysis verified 4 dimensions comprising the conceptual understanding of functions: definition, recognition, interpretation, and problem solving. Furthermore, the important role of the ability to define the concept on the rest abilities was revealed, leading to important didactic implications.

Abstract The study focuses on the cognitive level of Mathematical Working Space (MWS) and the component of the epistemological level related to semiotic representations in two mathematical domains of rational numbers: fraction and decimal number addition. Within this scope, it aims to explore how representational flexibility develops over time. A similar developmental pattern of four distinct hierarchical levels of student representational flexibility in both domains is identified. The findings indicate that the genesis of the semiotic axis in fraction and decimal addition is not automatic, but a long process of developmental steps that could be referred to as MWS1, MWS2, MWS3, MWS4 (final). There is not a clear and stable correspondence between developmental levels of representational flexibility and school grades. Didactical implications in order to foster representational flexibility in the MWS of fraction and decimal addition are discussed.

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.

This study investigates students’ conceptions of absolute value (AV), their performance in various items on AV, their errors in these items and the relationships between students’ conceptions and their performance and errors. The Mathematical Working Space (MWS) is used as a framework for studying students’ mathematical work on AV and the obstacles that hinder their work in Turkey and Cyprus. A comparative study between the two countries is undertaken, by which a deeper understanding on students’ personal MWS on AV is gained. Specifically, a survey was carried out in Turkey, following a similar survey in Cyprus, in which secondary school students’ performance was assessed using a test. Findings showed a discrepancy in the conception of AV that was most prevalent in each country, indicating the differences in the reference and suitable MWS between the two countries. For Turkey, the conception of AV as distance from 0, which was the most widely used definition, gave a positive support to the solution of items involving discursive reasoning. This was not the case for Cyprus, in which the most prevalent conception of AV was ‘number without sign’. An analysis of the Turkish students’ errors revealed a distinction between errors in students’ discursive genesis and semiotic genesis, which were a consequence of either didactic or epistemological obstacles that intervened in students’ personal MWS.

Abstract The study focuses on the cognitive level of Mathematical Working Space (MWS) and the component of the epistemological level related to semiotic representations in two mathematical domains of rational numbers: fraction and decimal number addition. Within this scope, it aims to explore how representational flexibility develops over time. A similar developmental pattern of four distinct hierarchical levels of student representational flexibility in both domains is identified. The findings indicate that the genesis of the semiotic axis in fraction and decimal addition is not automatic, but a long process of developmental steps that could be referred to as MWS1, MWS2, MWS3, MWS4 (final). There is not a clear and stable correspondence between developmental levels of representational flexibility and school grades. Didactical implications in order to foster representational flexibility in the MWS of fraction and decimal addition are discussed. Key words : Representational Flexibility. Mathematical Working Space. Fractions. Decimals.