... Zachariades . 2005 . Conceptual change in advanced mathematical thinking . In Proceedings of ... more ... Zachariades . 2005 . Conceptual change in advanced mathematical thinking . In Proceedings of the Fourth Conference on European Research in Mathematics Education M. Bosch , 1727 1736 . Sant Feliu de Guixols, Spain. Available at. ...
In this paper we report the first phase of a study that aims to analyse curriculum content, pedag... more In this paper we report the first phase of a study that aims to analyse curriculum content, pedagogical practice and student perceptions of the complex, often counter-intuitive but significant mathematical concept of infinite sum (aka series in Calculus). Sources of student difficulty with the concept identified in previous, not very extensive, research include: certain student perceptions of infinity; limited exposure to visualisation, contextualisation and applications of infinite sums; and, teaching through reduction to an algorithmic approach. Here we report preliminary analyses regarding curriculum content and, in particular, the initial phases of a three-dimensional analysis (cognitive, epistemological, didactical) of mainstream texts used to introduce the concept to undergraduates in the UK.
Proceedings of the British Society for Research …, Jan 1, 2007
CalGeo is a three-year project supported by EU programme Comenius 2.1. Amongst the objectives of ... more CalGeo is a three-year project supported by EU programme Comenius 2.1. Amongst the objectives of this project is the design of an in-service teacher education programme which employs dynamic geometry tools for teaching Calculus in upper secondary education. In this paper we present the project, its main objectives and the produced material; an example of a learning environment/activity designed for the introduction to the notion of derivative at Year 12; and, some results of the application of this activity in a real classroom situation. In this activity we use the tangent line and the property of local straightness to introduce the formal definition of derivative. Several cases of differentiable and non-differentiable functions are discussed through their geometrical and symbolic representations.
This paper explores secondary teachers' views on the role of visualisation in the justif... more This paper explores secondary teachers' views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical ...
Proceedings of the Conference of the …, Jan 1, 2006
We examined the relationship between subject-matter knowledge and pedagogical content knowledge o... more We examined the relationship between subject-matter knowledge and pedagogical content knowledge of 53 in-service mathematics teachers in the context of their written responses to a question that involved: solving the equation |x|+|x-1|=0, examining a flawed student solution and providing feedback to the student. Here we focus on a group of scripts characterised by pedagogical sensitivity but constrained mathematically (substantively and meta-cognitively). Through examination of examples from the data we demonstrate, and discuss implications of, some of these constraints: insistence on standard procedural methods, inappropriate contextualisation of otherwise commendable pedagogical practices and inadequate reflection on student thinking.
This study concerns 182 first year mathematics undergraduates' perspectives on the tange... more This study concerns 182 first year mathematics undergraduates' perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils' perspectives. The aim was the investigation of tangency images that settle after undergraduates' distancing ...
The study we report here examines parts of what Chevallard calls the institutional dimension of t... more The study we report here examines parts of what Chevallard calls the institutional dimension of the students' learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is ...
This paper explores secondary teachers' views on the role of visualisation in the justif... more This paper explores secondary teachers' views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical ...
This study focuses on a teaching experiment applied in a Year 12 class for the introduction of th... more This study focuses on a teaching experiment applied in a Year 12 class for the introduction of the derivative and the tangent line of function graph. This experiment was based on research results concerning students' perspectives on tangents when they had met the notion of tangent in different mathematical contexts (Geometry and Analysis). For the experimental needs an electronic environment was developed, utilising Dynamic Geometry software. The analysis focuses on the evolution of classroom mathematical discourse with the mediation of the electronic environment and with specific examples. Here I focus on an incident of the experiment. This incident exemplifies: how an image of a curve magnified in order to look straight in the electronic environment did not act as visual mediator for a student towards a claim that a curve has a tangent; the conflict in mathematical discourse about tangents; and, its resolution through the discussion of a particular example.
In this paper, we propose an approach to analysing teacher arguments that takes into account fiel... more In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal
International Group for the Psychology of …, Jan 1, 2006
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N.(Eds.). Proceedings 30th Co... more 2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N.(Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 177-184. Prague: PME. 2-177 STUDENTS'THINKING ABOUT THE TANGENT LINE Irene Biza*, ...
In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek stu... more In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students' perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students' knowledge develops in specific social and cultural contexts, students' construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students' personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students' engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students' meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher.
... Zachariades . 2005 . Conceptual change in advanced mathematical thinking . In Proceedings of ... more ... Zachariades . 2005 . Conceptual change in advanced mathematical thinking . In Proceedings of the Fourth Conference on European Research in Mathematics Education M. Bosch , 1727 1736 . Sant Feliu de Guixols, Spain. Available at. ...
In this paper we report the first phase of a study that aims to analyse curriculum content, pedag... more In this paper we report the first phase of a study that aims to analyse curriculum content, pedagogical practice and student perceptions of the complex, often counter-intuitive but significant mathematical concept of infinite sum (aka series in Calculus). Sources of student difficulty with the concept identified in previous, not very extensive, research include: certain student perceptions of infinity; limited exposure to visualisation, contextualisation and applications of infinite sums; and, teaching through reduction to an algorithmic approach. Here we report preliminary analyses regarding curriculum content and, in particular, the initial phases of a three-dimensional analysis (cognitive, epistemological, didactical) of mainstream texts used to introduce the concept to undergraduates in the UK.
Proceedings of the British Society for Research …, Jan 1, 2007
CalGeo is a three-year project supported by EU programme Comenius 2.1. Amongst the objectives of ... more CalGeo is a three-year project supported by EU programme Comenius 2.1. Amongst the objectives of this project is the design of an in-service teacher education programme which employs dynamic geometry tools for teaching Calculus in upper secondary education. In this paper we present the project, its main objectives and the produced material; an example of a learning environment/activity designed for the introduction to the notion of derivative at Year 12; and, some results of the application of this activity in a real classroom situation. In this activity we use the tangent line and the property of local straightness to introduce the formal definition of derivative. Several cases of differentiable and non-differentiable functions are discussed through their geometrical and symbolic representations.
This paper explores secondary teachers' views on the role of visualisation in the justif... more This paper explores secondary teachers' views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical ...
Proceedings of the Conference of the …, Jan 1, 2006
We examined the relationship between subject-matter knowledge and pedagogical content knowledge o... more We examined the relationship between subject-matter knowledge and pedagogical content knowledge of 53 in-service mathematics teachers in the context of their written responses to a question that involved: solving the equation |x|+|x-1|=0, examining a flawed student solution and providing feedback to the student. Here we focus on a group of scripts characterised by pedagogical sensitivity but constrained mathematically (substantively and meta-cognitively). Through examination of examples from the data we demonstrate, and discuss implications of, some of these constraints: insistence on standard procedural methods, inappropriate contextualisation of otherwise commendable pedagogical practices and inadequate reflection on student thinking.
This study concerns 182 first year mathematics undergraduates' perspectives on the tange... more This study concerns 182 first year mathematics undergraduates' perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils' perspectives. The aim was the investigation of tangency images that settle after undergraduates' distancing ...
The study we report here examines parts of what Chevallard calls the institutional dimension of t... more The study we report here examines parts of what Chevallard calls the institutional dimension of the students' learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is ...
This paper explores secondary teachers' views on the role of visualisation in the justif... more This paper explores secondary teachers' views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical ...
This study focuses on a teaching experiment applied in a Year 12 class for the introduction of th... more This study focuses on a teaching experiment applied in a Year 12 class for the introduction of the derivative and the tangent line of function graph. This experiment was based on research results concerning students' perspectives on tangents when they had met the notion of tangent in different mathematical contexts (Geometry and Analysis). For the experimental needs an electronic environment was developed, utilising Dynamic Geometry software. The analysis focuses on the evolution of classroom mathematical discourse with the mediation of the electronic environment and with specific examples. Here I focus on an incident of the experiment. This incident exemplifies: how an image of a curve magnified in order to look straight in the electronic environment did not act as visual mediator for a student towards a claim that a curve has a tangent; the conflict in mathematical discourse about tangents; and, its resolution through the discussion of a particular example.
In this paper, we propose an approach to analysing teacher arguments that takes into account fiel... more In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal
International Group for the Psychology of …, Jan 1, 2006
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N.(Eds.). Proceedings 30th Co... more 2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N.(Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 177-184. Prague: PME. 2-177 STUDENTS'THINKING ABOUT THE TANGENT LINE Irene Biza*, ...
In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek stu... more In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students' perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students' knowledge develops in specific social and cultural contexts, students' construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students' personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students' engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students' meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher.
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