Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education-first point that we make historical reference are theorems of Eudemus of Rhodes and Thales of Miletus and the second one is elaboration of these theorems in work of Serbian mathematicians Mihailo Petrović Alas. Fischbein's theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometric figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometric figures figural concepts because of their nature. We have analyzed the internal tensions of the concepts of angle and cube, which may appear in figural concepts because of their double nature, developmental aspects and didactical implications. The goal of the research was to examine the pre-service primary school teachers' geometric reasoning regarding the correlation between figural (picto-ral) and conceptual properties of geometric objects (angle and cube) in order to obtain a framework for creating didactic situations that reduce the gap between figure and concept. The results of this study show that the figural (pictoral) structure of an angle (and cube) dominates in the geometric reasoning of the pre-service primary school teachers over its formal conceptual constraints. There were some differences in situations when an image is helpful in solving tasks involving the concepts of the net of the cube and the cube itself.
Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education-first point that we make historical reference are theorems of Eudemus of Rhodes and Thales of Miletus and the second one is elaboration of these theorems in work of Serbian mathematicians Mihailo Petrović Alas. Fischbein's theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometric figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometric figures figural concepts because of their nature. We have analyzed the internal tensions of the concepts of angle and cube, which may appear in figural concepts because of their double nature, developmental aspects and didactical implications. The goal of the research was to examine the pre-service primary school teachers' geometric reasoning regarding the correlation between figural (picto-ral) and conceptual properties of geometric objects (angle and cube) in order to obtain a framework for creating didactic situations that reduce the gap between figure and concept. The results of this study show that the figural (pictoral) structure of an angle (and cube) dominates in the geometric reasoning of the pre-service primary school teachers over its formal conceptual constraints. There were some differences in situations when an image is helpful in solving tasks involving the concepts of the net of the cube and the cube itself.
Mental arithmetic plays an important role in teaching and learning mathematics. More precisely, ... more Mental arithmetic plays an important role in teaching and learning mathematics. More precisely, it develops problem-solving skills, helps students to develop a skill of making estimations in calculations, and contributes to a better understanding of the concept of numbers and decade system. The basic characteristics of mental arithmetic are as follows: 1) mental calculation uses numbers, not digits (without recording partial results); 2) strategic flexibility in terms of selecting a strategy relative to the characteristics of numbers in a mathematics task. The aim of this paper is to examine pupils’ ability to subtract two numbers without using paper and pencil (mentally) and to identify the strategies used by pupils while doing the calculations. The paper also focuses on the very important question of pupils’ flexibility in using calculation strategies, or more specifically, whether the choice of a strategy depends on the structure of a mathematical task. The research was conducted on a sample of 66 third-graders from two primary schools in Belgrade. A descriptive method and interview were used in the research. The obtained results indicate that pupils predominantly use the algorithm of digital calculation in trying to do a calculation without a paper and pencil, which is the cause of many errors. Equally important, the pupils lack flexibility in doing mental calculation, which may indicate the insufficient understanding of the structure of numbers and calculation procedures. In addition, the results point to the need to change the approaches to arithmetic content by shifting the focus from developing algorithm calculation skills to developing a more in-depth understanding and use of different calculation procedures.
Developing skills of calculating is one of the primary tasks of initial Arithmetic teaching. Fulf... more Developing skills of calculating is one of the primary tasks of initial Arithmetic teaching. Fulfill-ing calculating skills is one of the basic aims of initial Arithmetic teaching. Fulfilling this task means that students should be able to choose a suitable method in calculating. The most convenient strategy depends on the characteristic of numbers in an expression, age of students and situation in which it is being calculated. Students, who have a high degree of understanding the relation between addition and subtraction, characteristics of the decade number system and rules which describe it (arithmetic rules), show more flexibility in choosing adequate strategies of calculating. The aim of the research is studying aims and degrees of understanding sum and difference compensation strategy by the stu-dents and this kind of understanding is seen as an application of the stated rules in the procedures of calculating. The sample of the research is consisted of 39 fourth grade students of a primary school in Belgrade. Descriptive method was used, as well as techniques of testing and interviewing. Results of the research show that that knowledge of students about the examined rules is not operational, i.e. the students cannot apply it in suitable situations. Students apply strategies based on the stated rules more often as a mental strategy than in the situation paper-pen, but the analysis shows that students are not flexible in choosing the suitable strategy. Possible solution of this problem lies in concretiza-tion of methodological instructions in the Curriculum, and this would make clearer directions to both teachers and authors of the course books.
Proceedings of the Training Conference History of Mathematics in Mathematics Education, 2019
Drawing upon Fischbein’s theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein’s theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education. Fischbein’s theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometrical figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometrical figures figural concepts because of their nature. We shall analyze the internal tensions which, according to Fischbein, may appear in figural concepts because of their double nature, developmental aspects and didactical implications. In doing so, we have created a room for further research, the goal of which will be to examine the pre-service primary school teachers’ geometric reasoning regarding the correlation between figural and conceptual properties of geometric objects in order to obtain a framework for creating didactic situations in which a figure and a concept would merge into a unique mental object.
Keywords: History instruction, geometric object, geometric reasoning, notion of figural concept, pre-service primary school teachers.
Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education-first point that we make historical reference are theorems of Eudemus of Rhodes and Thales of Miletus and the second one is elaboration of these theorems in work of Serbian mathematicians Mihailo Petrović Alas. Fischbein's theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometric figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometric figures figural concepts because of their nature. We have analyzed the internal tensions of the concepts of angle and cube, which may appear in figural concepts because of their double nature, developmental aspects and didactical implications. The goal of the research was to examine the pre-service primary school teachers' geometric reasoning regarding the correlation between figural (picto-ral) and conceptual properties of geometric objects (angle and cube) in order to obtain a framework for creating didactic situations that reduce the gap between figure and concept. The results of this study show that the figural (pictoral) structure of an angle (and cube) dominates in the geometric reasoning of the pre-service primary school teachers over its formal conceptual constraints. There were some differences in situations when an image is helpful in solving tasks involving the concepts of the net of the cube and the cube itself.
Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education-first point that we make historical reference are theorems of Eudemus of Rhodes and Thales of Miletus and the second one is elaboration of these theorems in work of Serbian mathematicians Mihailo Petrović Alas. Fischbein's theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometric figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometric figures figural concepts because of their nature. We have analyzed the internal tensions of the concepts of angle and cube, which may appear in figural concepts because of their double nature, developmental aspects and didactical implications. The goal of the research was to examine the pre-service primary school teachers' geometric reasoning regarding the correlation between figural (picto-ral) and conceptual properties of geometric objects (angle and cube) in order to obtain a framework for creating didactic situations that reduce the gap between figure and concept. The results of this study show that the figural (pictoral) structure of an angle (and cube) dominates in the geometric reasoning of the pre-service primary school teachers over its formal conceptual constraints. There were some differences in situations when an image is helpful in solving tasks involving the concepts of the net of the cube and the cube itself.
Mental arithmetic plays an important role in teaching and learning mathematics. More precisely, ... more Mental arithmetic plays an important role in teaching and learning mathematics. More precisely, it develops problem-solving skills, helps students to develop a skill of making estimations in calculations, and contributes to a better understanding of the concept of numbers and decade system. The basic characteristics of mental arithmetic are as follows: 1) mental calculation uses numbers, not digits (without recording partial results); 2) strategic flexibility in terms of selecting a strategy relative to the characteristics of numbers in a mathematics task. The aim of this paper is to examine pupils’ ability to subtract two numbers without using paper and pencil (mentally) and to identify the strategies used by pupils while doing the calculations. The paper also focuses on the very important question of pupils’ flexibility in using calculation strategies, or more specifically, whether the choice of a strategy depends on the structure of a mathematical task. The research was conducted on a sample of 66 third-graders from two primary schools in Belgrade. A descriptive method and interview were used in the research. The obtained results indicate that pupils predominantly use the algorithm of digital calculation in trying to do a calculation without a paper and pencil, which is the cause of many errors. Equally important, the pupils lack flexibility in doing mental calculation, which may indicate the insufficient understanding of the structure of numbers and calculation procedures. In addition, the results point to the need to change the approaches to arithmetic content by shifting the focus from developing algorithm calculation skills to developing a more in-depth understanding and use of different calculation procedures.
Developing skills of calculating is one of the primary tasks of initial Arithmetic teaching. Fulf... more Developing skills of calculating is one of the primary tasks of initial Arithmetic teaching. Fulfill-ing calculating skills is one of the basic aims of initial Arithmetic teaching. Fulfilling this task means that students should be able to choose a suitable method in calculating. The most convenient strategy depends on the characteristic of numbers in an expression, age of students and situation in which it is being calculated. Students, who have a high degree of understanding the relation between addition and subtraction, characteristics of the decade number system and rules which describe it (arithmetic rules), show more flexibility in choosing adequate strategies of calculating. The aim of the research is studying aims and degrees of understanding sum and difference compensation strategy by the stu-dents and this kind of understanding is seen as an application of the stated rules in the procedures of calculating. The sample of the research is consisted of 39 fourth grade students of a primary school in Belgrade. Descriptive method was used, as well as techniques of testing and interviewing. Results of the research show that that knowledge of students about the examined rules is not operational, i.e. the students cannot apply it in suitable situations. Students apply strategies based on the stated rules more often as a mental strategy than in the situation paper-pen, but the analysis shows that students are not flexible in choosing the suitable strategy. Possible solution of this problem lies in concretiza-tion of methodological instructions in the Curriculum, and this would make clearer directions to both teachers and authors of the course books.
Proceedings of the Training Conference History of Mathematics in Mathematics Education, 2019
Drawing upon Fischbein’s theory of figural concepts, the starting point of the paper is the use a... more Drawing upon Fischbein’s theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education. Fischbein’s theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometrical figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometrical figures figural concepts because of their nature. We shall analyze the internal tensions which, according to Fischbein, may appear in figural concepts because of their double nature, developmental aspects and didactical implications. In doing so, we have created a room for further research, the goal of which will be to examine the pre-service primary school teachers’ geometric reasoning regarding the correlation between figural and conceptual properties of geometric objects in order to obtain a framework for creating didactic situations in which a figure and a concept would merge into a unique mental object.
Keywords: History instruction, geometric object, geometric reasoning, notion of figural concept, pre-service primary school teachers.
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Keywords: History instruction, geometric object, geometric reasoning, notion of figural concept, pre-service primary school teachers.
Keywords: History instruction, geometric object, geometric reasoning, notion of figural concept, pre-service primary school teachers.