This study combines elements of the Japanese Lesson Study approach and teachers' professional dev... more This study combines elements of the Japanese Lesson Study approach and teachers' professional development. An explorative research design is conducted with three upper level high school teachers in the light of educational design research, whereby design activities will be cyclically evaluated. The Lesson Study team observed and evaluated two different research lesson cycles. The first one focused on the concept of the derivative. The second one deepened teachers' pilot experiences with regard to another mathematical concept. The Lesson Study revealed students' misconceptions with regard to the tangent line. Results of teachers' professional development are used to refine the Lesson Study observation instruments.
This study investigates the manner in which students build up meaning to draw the gradient of a g... more This study investigates the manner in which students build up meaning to draw the gradient of a given graph. Some researchers claim that students tend to use algebra to solve calculus questions. This research suggests this may happen when students are encouraged to ...
The editors of MERJ would like to express our sincere appreciation to the following people who re... more The editors of MERJ would like to express our sincere appreciation to the following people who reviewed manuscripts for Volume 20 of the Mathematics Education Research Journal. Also, we would like to thank the members of the Editorial Board (listed on the inside front cover), ...
we discuss the repercussions of the development of infinitesimal calculus into modern analysis, b... more we discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of mathematical thinking and imaginative visual interpretations of axiomatic proof.
The major idea in this paper is the formulation of a theory of three distinct but interrelated wo... more The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of newborn babies to the mathematics of research mathematicians. The title of this paper is a play on words, contrasting the act of 'thinking through' several existing theories of cognitive development, and 'thinking through' the newly formulated theory of three worlds to see how different individuals may develop substantially different paths on their own cognitive journey of personal mathematical growth.
Mathematics & Mathematics Education: Searching for Common Ground, 2014
This paper presents a global theoretical framework that complements cognitive and affective aspec... more This paper presents a global theoretical framework that complements cognitive and affective aspects of the increasing sophistication of mathematical thinking and proof, taking into account the nature of mathematics itself and the way in which learners mature by building on their previous experiences. It is based on our shared human facilities of perception, action and reason that mature in very different ways, to explain and predict how we may develop mathematical thinking in general and mathematical reasoning and proof in particular.
This chapter reflects on the evolution of the mathematics of change and variation as technology a... more This chapter reflects on the evolution of the mathematics of change and variation as technology affords the possibility of conceptualising and communicating ideas for a wider range of learners than the few who traditionally study the higher levels of the calculus. It considers the overall program of development conceived by Jim Kaput, instantiated in the software SimCalc as part of a full range of development using technology for 'expressing, communicating, reasoning, computing, abstracting, generalising, and formalising mathematical ideas.' The development begins with interactive representations of dynamic real world situations and extends the perceptual ideas of continuity and linearity through the operational symbolism of the calculus and on to the formalising power of mathematical analysis. It reveals that the Kaput program has the distinction that its overall framework contains the essence for continuing the complementary evolution of technology and the conceptions of mathematical change and variation. Furthermore, it envisages changes that we have, as yet, not implemented, such is the speed of technological change. In particular, new technology enables us not only to build more powerful ways of performing numerical and symbolic algorithms that may be represented visually and dynamically, it also provides new forms of input and gesture to offer an embodied, kinesthetic, and emotionally powerful experience of engaging with mathematics. This can be shared widely through fundamental human perception and action and can develop in the longer term through symbolism and human reason to the mathematical literacy required of today's citizens, the theoretical applications of mathematics essential for today's society and on to the boundaries of mathematical research that takes us into the future.
... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky &... more ... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky & Leron use the programming language ISETL (Interactive SET Language) to get the students to engage in programming mathematical constructs in group theory and ring theory. ...
The major focus of this study is to trace the cognitive development of students throughout a math... more The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these maps, schematic diagrams were constructed which strip the concept maps of detail and show only how they are successively built by keeping some old elements, reorganizing, and introducing new elements. The more successful student added new elements to old in a structure that gradually increased in complexity and richness. The less successful had little constructive growth, building new maps on each occasion. (Contains 19 references.) (Author/MM) Reproductions supplied by EDRS are the best that can be made from the original document. 1PERMISSION TO REPRODUCE AND
This presentation considers how the peculiar structure of the biological brain may be supported b... more This presentation considers how the peculiar structure of the biological brain may be supported by the computational power of the computer to enhance mathematical thinking. It considers how we think and learn mathematics with particular reference to the use of visualisation and symbol manipulation. Visualisation occupies a major portion of the brain’s cortex and enables Homo Sapiens to ‘see ’ how ideas can be formed and related. Mathematical symbols in arithmetic, algebra, calculus particularly suit the biological brain, acting as pivots between concepts for thinking about mathematics and processes to calculate and predict. We use the term ‘procept ’ to describe this particular combination of symbol as process and concept. Analysis of procepts reveals that the development of symbols does not follow an easy cognitive path for the growing individual because they operate in significantly different ways in arithmetic, algebra and the calculus. We therefore advocate a versatile approach ...
The conception of infinity as a process (potential infinity) or as an object (actual infinity) is... more The conception of infinity as a process (potential infinity) or as an object (actual infinity) is important for students to acquire understanding in many other related areas in mathematics. This study attempts to describe the infinite divisibility thinking of mathematics student teachers in an Institute of Teacher Education in Malaysia by making sense of mathematics through perception, operation and reason. Data were collected through a self-reporting questionnaire that was administered to 238 elementary school pre-service teachers from selected Teacher Education Institutes in Malaysia. Researchers categorised qualitatively different types of thinking and reported them by using descriptive statistics. The result revealed that the percentage of respondents who conceived infinity as an object was just slightly lower as compare to the percentage of respondents who conceived infinity as a process. Additionally, this study found that there were respondents with problematic conceptions as shown by their inconsistent answers. The open-ended explanations given by all the respondents revealed that most of the pre-service teachers used perception to make meaning on finite and infinite divisibility.
In their critique of " object as a central metaphor in advanced mathematical thinking "... more In their critique of " object as a central metaphor in advanced mathematical thinking " , Confrey and Costa (1996) describes members of the Advanced Mathematical Thinking Group, including myself, as " reification theorists ". By selective quation they attibute theores largely developed independently by Dubinsky and Sfard as being broadly shared. Whilst it is true that many share an interest in the relationship of process and object and the mediating role of symbols, the notion of " reification " is only part of the domain of discourse. In the book " Advanced Mathematical Thinking " to which Confrey and Costa refer, only two chapters out of thirteen can be considered as " reificationst " — a chapter by Harel and Kaput which focuses on the notion of " conceptual entity " as part of a wider theory and a chapter by Dubinsky on " Reflective Abstraction ". Quite different persepectives are also presented, for instance t...
This study combines elements of the Japanese Lesson Study approach and teachers' professional dev... more This study combines elements of the Japanese Lesson Study approach and teachers' professional development. An explorative research design is conducted with three upper level high school teachers in the light of educational design research, whereby design activities will be cyclically evaluated. The Lesson Study team observed and evaluated two different research lesson cycles. The first one focused on the concept of the derivative. The second one deepened teachers' pilot experiences with regard to another mathematical concept. The Lesson Study revealed students' misconceptions with regard to the tangent line. Results of teachers' professional development are used to refine the Lesson Study observation instruments.
This study investigates the manner in which students build up meaning to draw the gradient of a g... more This study investigates the manner in which students build up meaning to draw the gradient of a given graph. Some researchers claim that students tend to use algebra to solve calculus questions. This research suggests this may happen when students are encouraged to ...
The editors of MERJ would like to express our sincere appreciation to the following people who re... more The editors of MERJ would like to express our sincere appreciation to the following people who reviewed manuscripts for Volume 20 of the Mathematics Education Research Journal. Also, we would like to thank the members of the Editorial Board (listed on the inside front cover), ...
we discuss the repercussions of the development of infinitesimal calculus into modern analysis, b... more we discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of mathematical thinking and imaginative visual interpretations of axiomatic proof.
The major idea in this paper is the formulation of a theory of three distinct but interrelated wo... more The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of newborn babies to the mathematics of research mathematicians. The title of this paper is a play on words, contrasting the act of 'thinking through' several existing theories of cognitive development, and 'thinking through' the newly formulated theory of three worlds to see how different individuals may develop substantially different paths on their own cognitive journey of personal mathematical growth.
Mathematics & Mathematics Education: Searching for Common Ground, 2014
This paper presents a global theoretical framework that complements cognitive and affective aspec... more This paper presents a global theoretical framework that complements cognitive and affective aspects of the increasing sophistication of mathematical thinking and proof, taking into account the nature of mathematics itself and the way in which learners mature by building on their previous experiences. It is based on our shared human facilities of perception, action and reason that mature in very different ways, to explain and predict how we may develop mathematical thinking in general and mathematical reasoning and proof in particular.
This chapter reflects on the evolution of the mathematics of change and variation as technology a... more This chapter reflects on the evolution of the mathematics of change and variation as technology affords the possibility of conceptualising and communicating ideas for a wider range of learners than the few who traditionally study the higher levels of the calculus. It considers the overall program of development conceived by Jim Kaput, instantiated in the software SimCalc as part of a full range of development using technology for 'expressing, communicating, reasoning, computing, abstracting, generalising, and formalising mathematical ideas.' The development begins with interactive representations of dynamic real world situations and extends the perceptual ideas of continuity and linearity through the operational symbolism of the calculus and on to the formalising power of mathematical analysis. It reveals that the Kaput program has the distinction that its overall framework contains the essence for continuing the complementary evolution of technology and the conceptions of mathematical change and variation. Furthermore, it envisages changes that we have, as yet, not implemented, such is the speed of technological change. In particular, new technology enables us not only to build more powerful ways of performing numerical and symbolic algorithms that may be represented visually and dynamically, it also provides new forms of input and gesture to offer an embodied, kinesthetic, and emotionally powerful experience of engaging with mathematics. This can be shared widely through fundamental human perception and action and can develop in the longer term through symbolism and human reason to the mathematical literacy required of today's citizens, the theoretical applications of mathematics essential for today's society and on to the boundaries of mathematical research that takes us into the future.
... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky &... more ... experience. (Dubinsky & Leron, 1994, p. xiv) ... learning. Dubinsky & Leron use the programming language ISETL (Interactive SET Language) to get the students to engage in programming mathematical constructs in group theory and ring theory. ...
The major focus of this study is to trace the cognitive development of students throughout a math... more The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these maps, schematic diagrams were constructed which strip the concept maps of detail and show only how they are successively built by keeping some old elements, reorganizing, and introducing new elements. The more successful student added new elements to old in a structure that gradually increased in complexity and richness. The less successful had little constructive growth, building new maps on each occasion. (Contains 19 references.) (Author/MM) Reproductions supplied by EDRS are the best that can be made from the original document. 1PERMISSION TO REPRODUCE AND
This presentation considers how the peculiar structure of the biological brain may be supported b... more This presentation considers how the peculiar structure of the biological brain may be supported by the computational power of the computer to enhance mathematical thinking. It considers how we think and learn mathematics with particular reference to the use of visualisation and symbol manipulation. Visualisation occupies a major portion of the brain’s cortex and enables Homo Sapiens to ‘see ’ how ideas can be formed and related. Mathematical symbols in arithmetic, algebra, calculus particularly suit the biological brain, acting as pivots between concepts for thinking about mathematics and processes to calculate and predict. We use the term ‘procept ’ to describe this particular combination of symbol as process and concept. Analysis of procepts reveals that the development of symbols does not follow an easy cognitive path for the growing individual because they operate in significantly different ways in arithmetic, algebra and the calculus. We therefore advocate a versatile approach ...
The conception of infinity as a process (potential infinity) or as an object (actual infinity) is... more The conception of infinity as a process (potential infinity) or as an object (actual infinity) is important for students to acquire understanding in many other related areas in mathematics. This study attempts to describe the infinite divisibility thinking of mathematics student teachers in an Institute of Teacher Education in Malaysia by making sense of mathematics through perception, operation and reason. Data were collected through a self-reporting questionnaire that was administered to 238 elementary school pre-service teachers from selected Teacher Education Institutes in Malaysia. Researchers categorised qualitatively different types of thinking and reported them by using descriptive statistics. The result revealed that the percentage of respondents who conceived infinity as an object was just slightly lower as compare to the percentage of respondents who conceived infinity as a process. Additionally, this study found that there were respondents with problematic conceptions as shown by their inconsistent answers. The open-ended explanations given by all the respondents revealed that most of the pre-service teachers used perception to make meaning on finite and infinite divisibility.
In their critique of " object as a central metaphor in advanced mathematical thinking "... more In their critique of " object as a central metaphor in advanced mathematical thinking " , Confrey and Costa (1996) describes members of the Advanced Mathematical Thinking Group, including myself, as " reification theorists ". By selective quation they attibute theores largely developed independently by Dubinsky and Sfard as being broadly shared. Whilst it is true that many share an interest in the relationship of process and object and the mediating role of symbols, the notion of " reification " is only part of the domain of discourse. In the book " Advanced Mathematical Thinking " to which Confrey and Costa refer, only two chapters out of thirteen can be considered as " reificationst " — a chapter by Harel and Kaput which focuses on the notion of " conceptual entity " as part of a wider theory and a chapter by Dubinsky on " Reflective Abstraction ". Quite different persepectives are also presented, for instance t...
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