Major conclusions of the Working Group at the Palermo conference Presentations and discussions of... more Major conclusions of the Working Group at the Palermo conference Presentations and discussions of group 1 show that, in dealing with arithmetical word-problems, language understanding is not the only pre-request we need to develop in students, but as well mathematical concepts understanding. The understanding of mathematical concepts and propositions are the main pre-request to solve algebraic and geometric problems. The lack of geometric background of students is an obstacle to deal with other mathematical problems like integral calculus problems. In order to make connections in students' mathematical problem solving, the use of concrete, realistic, and authentic problem situations is recommended, especially for young children. However, it is more important that problems are meaningful for the learner. It has to be remarked that an abstract problem is realistic, once it is personally meaningful for the problem solvers. Moreover, we should keep in mind the dual nature of mathematics: on the one hand, mathematics offers tools for modeling and solving problems in everyday life, but on the other hand, mathematics is a culture of formal and abstract structure. From this perspective it is important to stimulate reciprocal interactions between the concrete and the abstract levels. As a result, one should also be aware of taking into account individual differences among learners. Disparate problem situations should be used. Using a contextual approach and relying on visualization we can develop young students understanding of algebraic concepts. Young students have to learn algebraic relations and solve algebraic problems through generalizing number properties. In addition besides problem solving, problem posing or problem generating by the learners themselves should be stimulated more than is hitherto the case. The teaching of problem solving should be more process-and strategy-oriented than productoriented. In view of stimulating in students' constructive and progressively more self-regulated learning, a change in the role of the teacher is essential. Instead of being the sole source of knowledge and solutions, the teacher should create a classroom climate and culture that encourages and facilitates students' own initiatives and stimulates interactive and collaborative problem solving. The teacher's input is thus becoming more essential than before as the architect of the learning process. Problem solving in an inquiry-based classroom activity helps students to create, modify, and promote their cognitive constructions. In solving problems, teacher should appreciate discussions and arguments of students. Well-planned dialogue led by teacher is an appropriate way to facilitate the reveal of different solutions for a particular mathematical problem. An important issue, however, is how such approaches to learning and teaching mathematics can be disseminated. Teachers' and students' beliefs can be an obstacle in improving the teaching of problem solving. In that perspective, substantial efforts are necessary to develop instructional materials that are in line with the so-called 'new conception of teaching and learning mathematics', and to introduce this conception in initial and in-service teacher education and training. To be successful it is also essential to change teachers' beliefs about mathematics and mathematics education. Finally, it is useful in view of disseminating the new conception of teaching and learning mathematics problem solving to identify and share what might be called "success stories", i.e. cases that illustrate in a convincing way how the new approach can be implemented appropriately and effectively. Initial discussions at the Palermo conference on "The importance of mathematics-related beliefs" Over the past decade many researchers have been studying students' beliefs aiming, on the one hand, at identifying the different kinds of students' beliefs that influence mathematical learning and problem solving, and on the other hand, at understanding the processes through which they develop and determine learning and problem solving. This work has revealed how different categories of
Mathematics has got roots in Finland in the last qu arter of the 19 th century and came to flouri... more Mathematics has got roots in Finland in the last qu arter of the 19 th century and came to flourish in the first quarter of the next century. In the first quarter of the 20 th century, mathematicians were involved in teaching mathematic s at schools and writing school textbooks. This involvement decreased and came to a n end by the launching of the ‘New Math’ project. Mathematics education for elite was of positive affect to higher education, and this has changed by the spread of education, the de crease of mathematics teaching hours at schools and the changes in school mathematical curr icula. The impact of curriculum changes is evident in Finnish students’ performance in the IEA comparative studies, PISA and IMO. 1. Brief and incomplete history of mathematics cultivation in Finland The University of Helsinki of today is a direct con tinuation of the first Finnish university established in Turku in 1640. This was the only uni versity Finland had till 1917. Relying on one university...
Including figures in textbooks had been a difficult task in the 19 th century. Therefore, in teac... more Including figures in textbooks had been a difficult task in the 19 th century. Therefore, in teaching mathematics, Geometric textbooks were the only ones having figures. The development of print technology in the 20 th century has encouraged publishers and authors to include figures in mathematics textbooks, up to using figures unrelated to the content discussed and even unrelated to mathematics. In Finland, our work with teachers through in-service and pre-service education, and as well our work in mathematical clubs, started from Joensuu, have made figures use of special meaning in visualizing mathematical ideas. Visualization developed, by us, has become of variety of roles. From one hand it assists children to construct and understand mathematical concepts and relations, and from the other hand it is a tool to demonstrate a mathematical problem to children and as well a tool for children to solve mathematical problems and pose mathematical problems. We do use a type of dynamic visualization, where the main property it has is being isomorphic to the visualized mathematical entity. For more accuracy, our approaches to achieve understanding and leading to discovery are not depending on only visualization, but among others functional materials and actions. Indeed, visualization, functional materials and actions could be alternatives, but in different cases two or the three are needed. In using more than one of these facilitators, one could have a major role, but also the used facilitator could be of equal value. These facilitators can be used effectively for not only Primary School, but as well, in some cases, in Secondary School and Higher Education. About the terms used by us, we do use functional materials instead of the commonly used term 'manipulatives'. This is related to our type of using these materials, where most of them have to be made by teachers and students for use in specific cases. About actions, we mean the physical participation of students in a performance designed by the teacher. Finally, we have to remember that there is no facilitator, which can bring alone understanding and leading to discovery of concepts and relations, the main facilitator is the teacher, who is the architect of the learning process.
George Malaty, University of Eastern Finland, [email protected] In 2010, I wrote a paper on th... more George Malaty, University of Eastern Finland, [email protected] In 2010, I wrote a paper on the developments of visualization, functional materials and actions in teaching mathematics. This paper was a historical survey, but in the recent paper we put emphases on practical issues, taking from the case of the commutativity of multiplication a model for the isomorphic type of visualization we have developed. From historical view to practical issues on the development of using visualization. In the previous paper of 2010, a discussion was given on the type of visualization used in Finnish mathematics textbooks in the 19th and 20th centuries (Malaty, G. 2010)*. In the recent work, 2011, we are moving to put emphasis on practical issues and present an example, through which we can explore some of the main roles visualization can have in teaching mathematics. Developing mathematics teaching approaches has been always one of my main interests, while working in teacher education and mathematical clubs, in Finland, has given me a chance to develop and test my teaching approaches. _______________ *Malaty, G. 2010, Developments in Using Visualization, Functional Materials and Actions in Teaching Mathematics: Understanding of Relations, Making Generalizations and Solving Problems (Part I). Matematika 4, Mathematical Education in a Context of Changes in Primary School, Conference Proceedings, Olomouc. (Plenary Address Paper)
Mathematics has got roots in Finland in the last quarter of the 19 th century and came to flouris... more Mathematics has got roots in Finland in the last quarter of the 19 th century and came to flourish in the first quarter of the next century. In the first quarter of the 20 th century, mathematicians were involved in teaching mathematics at schools and writing school textbooks. This involvement decreased and came to an end by the launching of the 'New Math' project. Mathematics education for elite was of positive affect to higher education, and this has changed by the spread of education, the decrease of mathematics teaching hours at schools and the changes in school mathematical curricula. The impact of curriculum changes is evident in Finnish students' performance in the IEA comparative studies, PISA and IMO.
... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the... more ... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the translation of the textbooks, the most crucial mistakes being in the translation of some ... The experience of teachers in relation to specific problems which occur more frequently in examinations ...
... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the... more ... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the translation of the textbooks, the most crucial mistakes being in the translation of some ... The experience of teachers in relation to specific problems which occur more frequently in examinations ...
Major conclusions of the Working Group at the Palermo conference Presentations and discussions of... more Major conclusions of the Working Group at the Palermo conference Presentations and discussions of group 1 show that, in dealing with arithmetical word-problems, language understanding is not the only pre-request we need to develop in students, but as well mathematical concepts understanding. The understanding of mathematical concepts and propositions are the main pre-request to solve algebraic and geometric problems. The lack of geometric background of students is an obstacle to deal with other mathematical problems like integral calculus problems. In order to make connections in students' mathematical problem solving, the use of concrete, realistic, and authentic problem situations is recommended, especially for young children. However, it is more important that problems are meaningful for the learner. It has to be remarked that an abstract problem is realistic, once it is personally meaningful for the problem solvers. Moreover, we should keep in mind the dual nature of mathematics: on the one hand, mathematics offers tools for modeling and solving problems in everyday life, but on the other hand, mathematics is a culture of formal and abstract structure. From this perspective it is important to stimulate reciprocal interactions between the concrete and the abstract levels. As a result, one should also be aware of taking into account individual differences among learners. Disparate problem situations should be used. Using a contextual approach and relying on visualization we can develop young students understanding of algebraic concepts. Young students have to learn algebraic relations and solve algebraic problems through generalizing number properties. In addition besides problem solving, problem posing or problem generating by the learners themselves should be stimulated more than is hitherto the case. The teaching of problem solving should be more process-and strategy-oriented than productoriented. In view of stimulating in students' constructive and progressively more self-regulated learning, a change in the role of the teacher is essential. Instead of being the sole source of knowledge and solutions, the teacher should create a classroom climate and culture that encourages and facilitates students' own initiatives and stimulates interactive and collaborative problem solving. The teacher's input is thus becoming more essential than before as the architect of the learning process. Problem solving in an inquiry-based classroom activity helps students to create, modify, and promote their cognitive constructions. In solving problems, teacher should appreciate discussions and arguments of students. Well-planned dialogue led by teacher is an appropriate way to facilitate the reveal of different solutions for a particular mathematical problem. An important issue, however, is how such approaches to learning and teaching mathematics can be disseminated. Teachers' and students' beliefs can be an obstacle in improving the teaching of problem solving. In that perspective, substantial efforts are necessary to develop instructional materials that are in line with the so-called 'new conception of teaching and learning mathematics', and to introduce this conception in initial and in-service teacher education and training. To be successful it is also essential to change teachers' beliefs about mathematics and mathematics education. Finally, it is useful in view of disseminating the new conception of teaching and learning mathematics problem solving to identify and share what might be called "success stories", i.e. cases that illustrate in a convincing way how the new approach can be implemented appropriately and effectively. Initial discussions at the Palermo conference on "The importance of mathematics-related beliefs" Over the past decade many researchers have been studying students' beliefs aiming, on the one hand, at identifying the different kinds of students' beliefs that influence mathematical learning and problem solving, and on the other hand, at understanding the processes through which they develop and determine learning and problem solving. This work has revealed how different categories of
Mathematics has got roots in Finland in the last qu arter of the 19 th century and came to flouri... more Mathematics has got roots in Finland in the last qu arter of the 19 th century and came to flourish in the first quarter of the next century. In the first quarter of the 20 th century, mathematicians were involved in teaching mathematic s at schools and writing school textbooks. This involvement decreased and came to a n end by the launching of the ‘New Math’ project. Mathematics education for elite was of positive affect to higher education, and this has changed by the spread of education, the de crease of mathematics teaching hours at schools and the changes in school mathematical curr icula. The impact of curriculum changes is evident in Finnish students’ performance in the IEA comparative studies, PISA and IMO. 1. Brief and incomplete history of mathematics cultivation in Finland The University of Helsinki of today is a direct con tinuation of the first Finnish university established in Turku in 1640. This was the only uni versity Finland had till 1917. Relying on one university...
Including figures in textbooks had been a difficult task in the 19 th century. Therefore, in teac... more Including figures in textbooks had been a difficult task in the 19 th century. Therefore, in teaching mathematics, Geometric textbooks were the only ones having figures. The development of print technology in the 20 th century has encouraged publishers and authors to include figures in mathematics textbooks, up to using figures unrelated to the content discussed and even unrelated to mathematics. In Finland, our work with teachers through in-service and pre-service education, and as well our work in mathematical clubs, started from Joensuu, have made figures use of special meaning in visualizing mathematical ideas. Visualization developed, by us, has become of variety of roles. From one hand it assists children to construct and understand mathematical concepts and relations, and from the other hand it is a tool to demonstrate a mathematical problem to children and as well a tool for children to solve mathematical problems and pose mathematical problems. We do use a type of dynamic visualization, where the main property it has is being isomorphic to the visualized mathematical entity. For more accuracy, our approaches to achieve understanding and leading to discovery are not depending on only visualization, but among others functional materials and actions. Indeed, visualization, functional materials and actions could be alternatives, but in different cases two or the three are needed. In using more than one of these facilitators, one could have a major role, but also the used facilitator could be of equal value. These facilitators can be used effectively for not only Primary School, but as well, in some cases, in Secondary School and Higher Education. About the terms used by us, we do use functional materials instead of the commonly used term 'manipulatives'. This is related to our type of using these materials, where most of them have to be made by teachers and students for use in specific cases. About actions, we mean the physical participation of students in a performance designed by the teacher. Finally, we have to remember that there is no facilitator, which can bring alone understanding and leading to discovery of concepts and relations, the main facilitator is the teacher, who is the architect of the learning process.
George Malaty, University of Eastern Finland, [email protected] In 2010, I wrote a paper on th... more George Malaty, University of Eastern Finland, [email protected] In 2010, I wrote a paper on the developments of visualization, functional materials and actions in teaching mathematics. This paper was a historical survey, but in the recent paper we put emphases on practical issues, taking from the case of the commutativity of multiplication a model for the isomorphic type of visualization we have developed. From historical view to practical issues on the development of using visualization. In the previous paper of 2010, a discussion was given on the type of visualization used in Finnish mathematics textbooks in the 19th and 20th centuries (Malaty, G. 2010)*. In the recent work, 2011, we are moving to put emphasis on practical issues and present an example, through which we can explore some of the main roles visualization can have in teaching mathematics. Developing mathematics teaching approaches has been always one of my main interests, while working in teacher education and mathematical clubs, in Finland, has given me a chance to develop and test my teaching approaches. _______________ *Malaty, G. 2010, Developments in Using Visualization, Functional Materials and Actions in Teaching Mathematics: Understanding of Relations, Making Generalizations and Solving Problems (Part I). Matematika 4, Mathematical Education in a Context of Changes in Primary School, Conference Proceedings, Olomouc. (Plenary Address Paper)
Mathematics has got roots in Finland in the last quarter of the 19 th century and came to flouris... more Mathematics has got roots in Finland in the last quarter of the 19 th century and came to flourish in the first quarter of the next century. In the first quarter of the 20 th century, mathematicians were involved in teaching mathematics at schools and writing school textbooks. This involvement decreased and came to an end by the launching of the 'New Math' project. Mathematics education for elite was of positive affect to higher education, and this has changed by the spread of education, the decrease of mathematics teaching hours at schools and the changes in school mathematical curricula. The impact of curriculum changes is evident in Finnish students' performance in the IEA comparative studies, PISA and IMO.
... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the... more ... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the translation of the textbooks, the most crucial mistakes being in the translation of some ... The experience of teachers in relation to specific problems which occur more frequently in examinations ...
... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the... more ... As a result of the involvement of 22 Arab and foreign experts on each textbook for ... in the translation of the textbooks, the most crucial mistakes being in the translation of some ... The experience of teachers in relation to specific problems which occur more frequently in examinations ...
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Papers by George Malaty
_______________
*Malaty, G. 2010, Developments in Using Visualization, Functional Materials and Actions in Teaching Mathematics:
Understanding of Relations, Making Generalizations and Solving Problems (Part I). Matematika 4, Mathematical Education in a Context of Changes in Primary School, Conference Proceedings, Olomouc. (Plenary Address Paper)
_______________
*Malaty, G. 2010, Developments in Using Visualization, Functional Materials and Actions in Teaching Mathematics:
Understanding of Relations, Making Generalizations and Solving Problems (Part I). Matematika 4, Mathematical Education in a Context of Changes in Primary School, Conference Proceedings, Olomouc. (Plenary Address Paper)