This paper describes a professional development program for secondary school mathematics teachers... more This paper describes a professional development program for secondary school mathematics teachers. The issue of mathematical definition is the focus of the program. The program aimed to develop teachers ’ subject matter knowledge including knowledge of substance of mathematics and knowledge about the nature and discourse of mathematics. The rationale of the course is described as well as some outcomes of its implementation. The activities included in the program deal with some mathematical concepts and different didactical approaches to secondary school mathematics. The two activities presented in this paper, exemplify some of the ideas. Mathematics teachers make critical decisions about the mathematics they teach and the way they teach mathematics. This requires teachers to be aware of reform-oriented approaches for teaching mathematics as well as to be mathematically educated. Research in mathematics teacher education shows that teachers’ mathematical knowledge must be deep and ro...
This paper describes a professional development program for secondary school mathematics teachers... more This paper describes a professional development program for secondary school mathematics teachers. The issue of mathematical definition is the focus of the program. The program aimed to develop teachers ’ subject matter knowledge including knowledge of substance of mathematics and knowledge about the nature and discourse of mathematics. The rationale of the course is described as well as some outcomes of its implementation. The activities included in the program deal with some mathematical concepts and different didactical approaches to secondary school mathematics. The two activities presented in this paper, exemplify some of the ideas. Mathematics teachers make critical decisions about the mathematics they teach and the way they teach mathematics. This requires teachers to be aware of reform-oriented approaches for teaching mathematics as well as to be mathematically educated. Research in mathematics teacher education shows that teachers’ mathematical knowledge must be deep and ro...
The present study constitutes an attempt to check students' conceptions about the nature and ... more The present study constitutes an attempt to check students' conceptions about the nature and the significance of mathematical proofs. The setting of this study was a mathematical-historical discussion within the framework of a course dealing with the development of mathematics. The students-elementary school pre-service mathematics teachers-were exposed to some problems taken from the Egyptian mathematics. After the lesson – that included the presentation of a formal proof of the main statement discussed-the students were asked to answer individually and in writing questions concerning the Egyptian method to calculate the area of a quadrilateral. The analysis of their answers reinforces the conception that pre-service teachers may know how to perform the " ceremony " of proof but in general, they do not appropriately conceive its meaning or its role establishing truth in mathematics.
This article presents two classroom episodes in which students were exposed to the value of askin... more This article presents two classroom episodes in which students were exposed to the value of asking questions and to the different roles played by proof in mathematics. Having students ask questions is a way to address what de Villiers labels a ‘distorted perspective of mathematical creativity as being always purely deductive’ (de Villiers, 1997, p.15). He claims that ‘the false impression is sometimes created that mathematicians are only problems solvers who spend most of their time trying to solve already given problems’ (ibid). The conversation in the two episodes is outlined in the paper. The setting was a classroom of fifteen good high-school students, who were studying calculus. These episodes occurred spontaneously, after the discussion of certain theorems during the lessons. Each one of these theorems will be labelled trigger of the episode, since they inspired students to ask the initial question that led to the whole episode. All over her work, it appears that their teacher...
The International Commission on Mathematical Insttuction (ICMI) was first established at the Inte... more The International Commission on Mathematical Insttuction (ICMI) was first established at the International Congress of Mathematicians held in Rome in 1908 and its first President was Felix Klein .. ICMI was reconstituted in 1952, after an interruption of activity between the two World Wars, becoming an official commission of the International Mathematical Union (IMU) ICMI decided to invest effort in the identification and investigation of issues or topics of particular significance to contemporary mathematics education and to encomage the implementation of concrete studies on them. The emphasis of a given study may be on analytical or action-oriented aspects, but some analytical component should always be present. Built around an international seminar, each study culminates with the publication of a volume intended to promote discussion and action at an international, national, regional or institutional level. The tenth of the eleven ICMI studies which have been completed by 2001 (a...
This article describes three mathematical games suitable to be played by different groups of stud... more This article describes three mathematical games suitable to be played by different groups of students, from young elementary school students, learning the basics of arithmetic and geometry, to older students making their first steps in mathematical proving.
This article describes a mathematics lesson I had the pleasure to teach to prospective elementary... more This article describes a mathematics lesson I had the pleasure to teach to prospective elementary school teachers. I wanted to expose them to the need for algebra and to the different meanings the term "variable" can embrace.
“Mathematics at all levels is a box full of surprises,” according to Movshovitz-Hadar (1988, p. 3... more “Mathematics at all levels is a box full of surprises,” according to Movshovitz-Hadar (1988, p. 34). In general, however, students are not aware of this aspect of the subject.
This article describes an activity that is connected with mathematical definitions and that illus... more This article describes an activity that is connected with mathematical definitions and that illustrates the process of gradual refinement as a way to understand and construct knowledge. It presents a gradual construction of a specific geometry concept that was the result of the interaction among the participants in a mathematical discourse (Pimm 1987). This activity took place at the end of a professional development program for teachers of secondary school geometry. During this fourteen-week program, the participants were exposed to the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and to several activities adapted from two volumes in the Addenda Series: Geometry in the Middle Grades (Geddes 1992) and Geometry from Multiple Perspectives (Coxford 1991).
The activity described in this article illustrates that such adjectives as elegant, surprising, c... more The activity described in this article illustrates that such adjectives as elegant, surprising, concise, and challenging can be and in fact are attributed to proofs by a not very expert audience, even students of elementary algebra. You do not have to teach mathematics majors to hear someone in your class say something like “I don't like that proof…. Let's look for a more elegant way to justify the same result.”
This paper describes a professional development program for secondary school mathematics teachers... more This paper describes a professional development program for secondary school mathematics teachers. The issue of mathematical definition is the focus of the program. The program aimed to develop teachers ’ subject matter knowledge including knowledge of substance of mathematics and knowledge about the nature and discourse of mathematics. The rationale of the course is described as well as some outcomes of its implementation. The activities included in the program deal with some mathematical concepts and different didactical approaches to secondary school mathematics. The two activities presented in this paper, exemplify some of the ideas. Mathematics teachers make critical decisions about the mathematics they teach and the way they teach mathematics. This requires teachers to be aware of reform-oriented approaches for teaching mathematics as well as to be mathematically educated. Research in mathematics teacher education shows that teachers’ mathematical knowledge must be deep and ro...
This paper describes a professional development program for secondary school mathematics teachers... more This paper describes a professional development program for secondary school mathematics teachers. The issue of mathematical definition is the focus of the program. The program aimed to develop teachers ’ subject matter knowledge including knowledge of substance of mathematics and knowledge about the nature and discourse of mathematics. The rationale of the course is described as well as some outcomes of its implementation. The activities included in the program deal with some mathematical concepts and different didactical approaches to secondary school mathematics. The two activities presented in this paper, exemplify some of the ideas. Mathematics teachers make critical decisions about the mathematics they teach and the way they teach mathematics. This requires teachers to be aware of reform-oriented approaches for teaching mathematics as well as to be mathematically educated. Research in mathematics teacher education shows that teachers’ mathematical knowledge must be deep and ro...
The present study constitutes an attempt to check students' conceptions about the nature and ... more The present study constitutes an attempt to check students' conceptions about the nature and the significance of mathematical proofs. The setting of this study was a mathematical-historical discussion within the framework of a course dealing with the development of mathematics. The students-elementary school pre-service mathematics teachers-were exposed to some problems taken from the Egyptian mathematics. After the lesson – that included the presentation of a formal proof of the main statement discussed-the students were asked to answer individually and in writing questions concerning the Egyptian method to calculate the area of a quadrilateral. The analysis of their answers reinforces the conception that pre-service teachers may know how to perform the " ceremony " of proof but in general, they do not appropriately conceive its meaning or its role establishing truth in mathematics.
This article presents two classroom episodes in which students were exposed to the value of askin... more This article presents two classroom episodes in which students were exposed to the value of asking questions and to the different roles played by proof in mathematics. Having students ask questions is a way to address what de Villiers labels a ‘distorted perspective of mathematical creativity as being always purely deductive’ (de Villiers, 1997, p.15). He claims that ‘the false impression is sometimes created that mathematicians are only problems solvers who spend most of their time trying to solve already given problems’ (ibid). The conversation in the two episodes is outlined in the paper. The setting was a classroom of fifteen good high-school students, who were studying calculus. These episodes occurred spontaneously, after the discussion of certain theorems during the lessons. Each one of these theorems will be labelled trigger of the episode, since they inspired students to ask the initial question that led to the whole episode. All over her work, it appears that their teacher...
The International Commission on Mathematical Insttuction (ICMI) was first established at the Inte... more The International Commission on Mathematical Insttuction (ICMI) was first established at the International Congress of Mathematicians held in Rome in 1908 and its first President was Felix Klein .. ICMI was reconstituted in 1952, after an interruption of activity between the two World Wars, becoming an official commission of the International Mathematical Union (IMU) ICMI decided to invest effort in the identification and investigation of issues or topics of particular significance to contemporary mathematics education and to encomage the implementation of concrete studies on them. The emphasis of a given study may be on analytical or action-oriented aspects, but some analytical component should always be present. Built around an international seminar, each study culminates with the publication of a volume intended to promote discussion and action at an international, national, regional or institutional level. The tenth of the eleven ICMI studies which have been completed by 2001 (a...
This article describes three mathematical games suitable to be played by different groups of stud... more This article describes three mathematical games suitable to be played by different groups of students, from young elementary school students, learning the basics of arithmetic and geometry, to older students making their first steps in mathematical proving.
This article describes a mathematics lesson I had the pleasure to teach to prospective elementary... more This article describes a mathematics lesson I had the pleasure to teach to prospective elementary school teachers. I wanted to expose them to the need for algebra and to the different meanings the term "variable" can embrace.
“Mathematics at all levels is a box full of surprises,” according to Movshovitz-Hadar (1988, p. 3... more “Mathematics at all levels is a box full of surprises,” according to Movshovitz-Hadar (1988, p. 34). In general, however, students are not aware of this aspect of the subject.
This article describes an activity that is connected with mathematical definitions and that illus... more This article describes an activity that is connected with mathematical definitions and that illustrates the process of gradual refinement as a way to understand and construct knowledge. It presents a gradual construction of a specific geometry concept that was the result of the interaction among the participants in a mathematical discourse (Pimm 1987). This activity took place at the end of a professional development program for teachers of secondary school geometry. During this fourteen-week program, the participants were exposed to the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) and to several activities adapted from two volumes in the Addenda Series: Geometry in the Middle Grades (Geddes 1992) and Geometry from Multiple Perspectives (Coxford 1991).
The activity described in this article illustrates that such adjectives as elegant, surprising, c... more The activity described in this article illustrates that such adjectives as elegant, surprising, concise, and challenging can be and in fact are attributed to proofs by a not very expert audience, even students of elementary algebra. You do not have to teach mathematics majors to hear someone in your class say something like “I don't like that proof…. Let's look for a more elegant way to justify the same result.”
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