Prospective teachers constructing dynamic geometry activities for gifted pupils: Connections between the frameworks of Krutetskii and van Hiele

The Swedish educational system has, so far, accorded little attention to the development of gifted pupils. Moreover, up to date, no Swedish studies have investigated teacher education from the perspective of mathematically gifted pupils. Our study is based on an instructional intervention, aimed to introduce the notion of giftedness in mathematics and to prepare prospective teachers (PTs) for the needs of the gifted. The data consists of 10 dynamic geometry software activities, constructed by 24 PTs. We investigated the constructed activities for their qualitative aspects, according to two frameworks: Krutetskii’s framework for mathematical giftedness and van Hiele’s model of geometrical thinking. The results indicate that nine of the 10 activities have the potential to address pivotal abilities of mathematically gifted pupils. In another aspect, the analysis suggests that Krutetskii’s holistic description of mathematical giftedness does not strictly correspond with the discrete levels of geometrical thinking proposed by van Hiele.

MENTAL ACTS OF MATHEMATICALLY GIFTED STUDENTS WHEN SOLVING FRACTIONS PROBLEMS

In the classroom, we will find various types of students with their special learning needs. One group of learners who have different learning needs are gifted students. The paper will focus on the study of mathematically gifted students. This research aims to obtain a description of the mathematically gifted students’ mental acts when solving fractions problems. The respondents were two students of the 7th graders in junior high school, in the West Java Province, Indonesia. The research approach was qualitative. The data were collected through paper and pencil measure, observation, and interview. The data were analyzed by grounded theory with coding and constant comparison. The results show four types of mental acts, those are interpreting, explaining, problem-solving, and inferring. The results of this study can be made as one of didactic anticipation when teachers teach the concept of fractions to the mathematically gifted student. These findings are significant to be considered by the teacher when teaching the mathematically gifted student. Teachers should anticipate how students think when they teach gifted students. So that teachers and students can achieve optimal learning outcomes.

PECULIARITIES OF MODELLING OF SPECIALIZED METHODICAL CASES IN THE CONTEXT OF PROFESSIONAL DEVELOPMENT OF MATHEMATICS TEACHERS

The practice of mathematics teaching and its scientific, methodical and didactic support in the system of in-service education of modern teachers generates a topical problem of modernization of operational and technological and reflexive functionality of competence-oriented learning of solving high complexity problems, among which the tasks of mathematical olympiads as an indicator of the quality of the established professional competence stand out.In the competency-based and methodical context of working with mathematically gifted students and preparing them for mathematical competitions, the transformation and genesis of the problem material, which is discussed with teachers on in-service training courses, are consistently considered from the perspective of forming productive convolved didactic structures with regard to the features of flexibility, differentiation of levels, algorithmic and structural recognizability, essential for creating convoluted associations.Implementation of the convergence for theoretical approaches to these methodical problems is hampered, for example, by the internal contradictions caused by the subject-object status of teachers undergoing professional development.Our researches and scientific and practical findings, including those aimed at overcoming such contradictions, consolidate the comprehensive use of balanced dynamic synergetic mechanisms based on the emergent effect (as opposed to more traditional mechanisms of dynamic transitions such as “educational activity ⟶ quasi-professional activity ⟶ educational-professional activity ⟶ professional activity”) in the practice of teacher professional development. Such interpretation fundamentally changes the significance and functions of the case method (a form of situational learning), depriving it of the features of an intermediate organizational form in the interpretation of other studies.In the course of the research the methods of systematic scientific and methodological analysis, synthesis, generalization of theoretical positions, modelling and practical conclusions are used.The article highlights and clarifies the structure and interaction of the components of professional competencies of the teacher, the specifics of the approach to designing and developing effective specialized competency-based cases, aimed at stimulating work with mathematically gifted students. The article pays attention to some differences between developing the methodical competencies of a future teacher of mathematics and improving the competencies of a practising teacher. The article presents a model example of a tested situational training devoted to an important class of olympiad-type geometry problems, accumulating a significant layer of mathematical skills of both teachers and students. Keywords: professional teacher development, scientific and methodical support, case method, basic competencies of teachers, productive didactic structures, mathematics teaching methodology, mathematically gifted students, olympiad-type problems in geometry.

COMPONENTS OF THE TRAINING PROGRAM FOR EDUCATORS WORKING ON THE DEVELOPMENT OF THE MATHEMATICALLY GIFTED STUDENTS POTENTIAL

The article is devoted to the formation and improvement of competencies of teachers and psychologists of secondary schools to identify and develop mathematically gifted students. It has been identified the components of the training program of basic competencies that psychologists and subject teachers must have to recognize and develop mathematical talent. The results of an empirical study of an educational project are online training for educators to deepen their theoretical knowledge of mathematical talent and the development of practical skills of organizing the educational process for students with a high level of ability in the field of exact sciences. It was found that training in the development of competencies is an effective way to improve the skills of teachers to understand the essence of talent, the peculiarities of its detection in students, prevention of loss of potential, development of individual educational trajectories, use of new learning technologies and ways to develop personal skills.

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

In mathematics education, technology offers many opportunities to enrich curricular contents. Plane symmetries is a topic often skipped by primary teachers. However, it is important and may be worked in attractive ways in dynamic geometry software environments. In any regular classroom there are students with different levels of mathematical attainment, some needing easy tasks while others, particularly mathematically-gifted students, need challenging problems. We present a teaching unit for plane symmetries, adequate for upper primary school grades, implemented in a fully interactive electronic book, with most activities solved in GeoGebra apps. The book allows student to choose which itinerary to follow and attention is paid to different levels of students’ mathematical attainment. The research objective of the paper is to make a networked analysis of the structure and contents of the teaching unit based on the Van Hiele levels of mathematical reasoning and the levels of cognitive demand in mathematical problem solving. The analysis shows the interest of networking both theories, the suitability of the teaching unit, as the Van Hiele levels and the cognitive demand of the activities increases, and its usefulness to fit the needs of each student, from low attainers to mathematically-gifted students.

A Gifted High School Student’s Generalization Strategies of Linear and Nonlinear Patterns via Gauss’s Approach

Mathematically gifted students have a high potential for understanding and thinking through mathematical relations and connections between mathematical concepts. Currently, it is thought that generalizing patterns algebraically can serve to provide challenges and opportunities that match their potential. This article focuses on a mathematically gifted student’s use of generalization strategies to identify linear and nonlinear patterns in the context of a matchstick problem. Data were collected from a 10th-grade gifted student’s problem-solving process in a qualitative research design. It was observed that the gifted student’s ways of generalizing the linear and nonlinear patterns were different. In a generalization process, the student used figural reasoning in the linear pattern and numerical reasoning in the nonlinear patterns. It was noted that the student explored using Gauss’s approach in structuring the general rules of nonlinear patterns. Accordingly, aside from assisting their more gifted students, mathematics teachers may want to consider ways to introduce Gaussian thinking to the benefit of all their students.

Olympiad potential for identifying mathematical giftedness in elementary schoolers

The purpose of this article is to demonstrate the possibilities of identifying the mathematical giftedness in elementary schoolers with the help of Olympiad problems. For this, the authors clarify the concept of “mathematical giftedness”, show the relationship between the concepts of “mathematical giftedness” and “mathematical abilities”, and indicate the most significant abilities of elementary schoolers from the set of mathematical giftedness. The role of mathematical Olympiads in identifying mathematically gifted elementary schoolers is substantiated. This role consists in creating situations where the participants of the Olympiad are forced to make mental efforts to perform the following actions: analysis of a problem situation to identify essential relationships, modeling a new way of action to solve the proposed problem, combining available methods of action to apply in a new situation in limited time. The criteria for compiling Olympiad tasks for identifying mathematically gifted students are formulated, the most important of which is the clear focus of each task on demonstrating a mathematical ability of a certain type, as well as the selection of the mathematical content of the Olympiad problems strictly from the elementary course of mathematics. The problems of one Olympiad should be based on the content of different sections of the elementary mathematics course. The examples of the Olympiad problems based on the content of the elementary mathematics course are provided and the substantiation of their role in demonstrating the mathematical abilities of the Olympiad participant in solving them is given. The results of observing the educational achievements of students (during their entire stay at school) who showed mathematical abilities at the Olympiads are provided as well as the prospects and certain difficulties of further research on ways to solve the problem.