Turtle’s navigation and manipulation of geometrical figures constructed by variable processes in ... more Turtle’s navigation and manipulation of geometrical figures constructed by variable processes in a 3d simulated space
This paper aims to contribute to the research into the way mathematical meaning-making can be int... more This paper aims to contribute to the research into the way mathematical meaning-making can be integrated with spatial navigation and orientation in 3d digital media. We report on the findings of a classroom research trying to shed light on 12 year-old pupils’ construction processes as they worked with a 3d Logo/Turtle Geometry environment called “MachineLab Turtleworlds.” It seems that the simulated 3d space was experienced by the pupils through two distinct perspectives: an intrinsic and an extrinsic one, which was depicted – among the others – on the way the available viewing angle manipulation tools were used. When the pupils were focusing on navigating and orientating the turtle, the 3d space was experienced through an intrinsic perspective, according to which the simulated space was viewed from inside, through the turtle’s viewpoint. In contrast, when the focus was shifted from the management of the turtle’s spatial movements to the construction of a graphic object, the 3d space was experienced through an extrinsic perspective, from the view point of an external observer who looked at the figural results of turtle’s movement.
Abstract. Issues related to 3d turtle’s navigation and geometrical figures ’ manipulation in the ... more Abstract. Issues related to 3d turtle’s navigation and geometrical figures ’ manipulation in the simulated 3d space of a newly developed computational environment, MaLT, are reported and discussed here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented 3d figures seem to offer new resources and to pose new challenges as far as geometrical activities and construction of meanings are concerned, which are strongly related to the representational infrastructure of MaLT. Abilities such as spatial orientation and spatial visualisation come into play and are interwoven with the software’s functionalities and semantics. Although the body-syntonic metaphor remains critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated with other – often conflicting one another – frames of reference. The strong link between spatial graphical and geometrical aspects, that was accentuated by the dragging functionalities...
Construction of meanings for fraction as number-measure is studied during the implementation of e... more Construction of meanings for fraction as number-measure is studied during the implementation of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. 12-year-old students were working collaboratively in groups of two with software that combines graphical and symbolic notation of fractions represented as points on the number line. Fractions as points and segments, ordering fractions as part of kinesthetic activities and abstracting the scaling of the numerical unit on the number line are some of the meanings developed. THEORETICAL BACKGROUND In this paper we report research [1] aiming to explore the mathematical meanings constructed collaboratively by 12 year-old students concerning the notion of fraction as a number-measure depicted on the number line within a specially designed computational environment. The students worked in collaborative groups of two using the ‘Fractions Microworld ’ (FM) [2], a piece of software which combines ...
The construction of meanings for the notion of angle in 3d geometrical space is studied during th... more The construction of meanings for the notion of angle in 3d geometrical space is studied during the implementation of specially designed geometrical tasks in the classroom. 13-year-old pupils were engaged in exploring the mathematical nature of angles while controlling and measuring the behaviours of geometrical objects in 3d simulations. Pupils worked in pairs using a specially designed computational environment that combines symbolic notation (by means of programming) and dynamic manipulation of graphically represented geometrical objects. The findings reveal a progressive coordination of different facets of angle in 3d space facilitated by the use of the available tools. THEORETICAL FRAMEWORK This contribution reports classroom research [1] aiming to e construction of meanings for the notion of angle in 3d geometrical space during activity involving the construction and manipulation of programmable geometrical figures. The students worked in collaborative groups of two using MaLT ...
Abstract. Issues related to 3d turtle's navigation and geometrical figures' manipulatio... more Abstract. Issues related to 3d turtle's navigation and geometrical figures' manipulation in the simu-lated 3d space of a newly developed computational environment, MaLT, are reported and discussed here. The joint use of meaningful formalism and the dynamic manipulation of ...
In this paper we report research [1] aiming to explore the mathematical meanings constructed coll... more In this paper we report research [1] aiming to explore the mathematical meanings constructed collaboratively by 12 year-old students concerning the notion of fraction as a number-measure depicted on the number line within a specially designed computational environment. The ...
In this paper we report findings from a design-based research aiming at shedding light on the way... more In this paper we report findings from a design-based research aiming at shedding light on the way eleven-year old students used gestures while constructing mathematical meanings in the framework of a 3d Turtle Geometry Environment. The results bring in the foreground the role of gestures, as signs that mediated the mathematical notions integrated in the computational environment, as well as the interconnection between gestures and the embodied turtle metaphor. Different kinds of gestures were used by students depending on their focus point during the construction processes. When focusing on turtle’s navigation and the graphical results of this navigation, students used dynamic representational gestures. On the contrary when viewing 3d space or 3d objects as external observers they used abstract deictic or static representational gestures.
The construction of meanings for the notion of angle in 3d geometrical space is studied during th... more The construction of meanings for the notion of angle in 3d geometrical space is studied during the implementation of specially designed geometrical tasks in the classroom. 13-year-old pupils were engaged in exploring the mathematical nature of angles while controlling and measuring the behaviours of geometrical objects in 3d simulations. Pupils worked in pairs using a specially designed computational environment that combines symbolic notation (by means of programming) and dynamic manipulation of graphically represented geometrical objects. The findings reveal a progressive coordination of different facets of angle in 3d space facilitated by the use of the available tools.
This paper aims to contribute to the research into the way mathematical meaning-making can be int... more This paper aims to contribute to the research into the way mathematical meaning-making can be integrated with spatial navigation and orientation in 3d digital media. We report on the findings of a classroom research trying to shed light on 12 year-old pupils’ construction processes as they worked with a 3d Logo / Turtle Geometry environment called ‘MachineLab Turtleworlds’ (MaLt). It seems that the simulated 3d space was experienced by the pupils through two distinct perspectives, an intrinsic and an extrinsic one, that was depicted –among the others- on the way the available viewing angle manipulation tools were used. When the pupils were focusing on navigating and orientating the turtle, the 3d space was experienced through an intrinsic perspective, according to which the simulated space was viewed from inside, through the turtle’s viewpoint. In contrast, when the focus was shifted from the management of the turtle’s spatial movements to the construction of a graphic object, the 3d space was experienced through an extrinsic perspective, from the view point of an external observer who looked at the figural results of turtle’s movement.
Issues related to 3d turtle’s navigation and geometrical figures’ manipulation in the simulated
... more Issues related to 3d turtle’s navigation and geometrical figures’ manipulation in the simulated
3d space of a newly developed computational environment, MaLT, are reported and discussed
here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented
3d figures seem to offer new resources and to pose new challenges as far as geometrical
activities and construction of meanings are concerned, which are strongly related to the representational
infrastructure of MaLT.
Abilities such as spatial orientation and spatial visualisation come into play and are interwoven
with the software’s functionalities and semantics. Although the body-syntonic metaphor remains
critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated
with other – often conflicting one another – frames of reference. The strong link between spatial
graphical and geometrical aspects, that was accentuated by the dragging functionalities of the software,
helped students go beyond an immediate perceptual approach, relating geometrical figures
with real 3d objects and the change of planes in 3d space with physical angle situations. In this
framework the concept of angle as turn and measure with emphasis on directionality but also as a
relationship between the planes defined by 2d figures has arisen as central.
… Journal for Technology in Mathematics Education, Jan 1, 2006
The new representations enabled by the educational computer game ‘the Juggler micro-world’ can pl... more The new representations enabled by the educational computer game ‘the Juggler micro-world’ can place vectors in a central role both for controlling and measuring the behaviors of objects, in a virtual environment simulating motion in three-dimensional spaces. The mathematical meanings constructed by 13 year-old students in relation to vectors as objects, as a set of properties and as representations of vectorial entities are reported and discussed in this document. It is argued that playing in the ‘Juggler’ micro-world could have a considerable contribution to the development of intuitions and to the generation of meanings related to vectors through the use of vectorial properties in a meaningful way.
This paper reports on a case-study design experiment in the domain of fraction as number-measure.... more This paper reports on a case-study design experiment in the domain of fraction as number-measure. We designed and implemented a set of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. Two groups of 12-year-old students worked collaboratively using paper and pencil as well as a specially designed microworld which combines graphical and symbolic notation of fractions represented as points on the number line. We used the students’ interactions with the available representations as a window into their conceptual understanding and struggles in making sense of fraction as number-measure. We report on the features of the available representations from an epistemological point of view, on the design of activities aiming at creating meaningful problem contexts for fractions as well as on the meanings generated by the students by some illustrative examples of their work indicating the potential of the activities and tools for expressing and reflecting on the mathematical nature of fraction as number-measure.
Turtle’s navigation and manipulation of geometrical figures constructed by variable processes in ... more Turtle’s navigation and manipulation of geometrical figures constructed by variable processes in a 3d simulated space
This paper aims to contribute to the research into the way mathematical meaning-making can be int... more This paper aims to contribute to the research into the way mathematical meaning-making can be integrated with spatial navigation and orientation in 3d digital media. We report on the findings of a classroom research trying to shed light on 12 year-old pupils’ construction processes as they worked with a 3d Logo/Turtle Geometry environment called “MachineLab Turtleworlds.” It seems that the simulated 3d space was experienced by the pupils through two distinct perspectives: an intrinsic and an extrinsic one, which was depicted – among the others – on the way the available viewing angle manipulation tools were used. When the pupils were focusing on navigating and orientating the turtle, the 3d space was experienced through an intrinsic perspective, according to which the simulated space was viewed from inside, through the turtle’s viewpoint. In contrast, when the focus was shifted from the management of the turtle’s spatial movements to the construction of a graphic object, the 3d space was experienced through an extrinsic perspective, from the view point of an external observer who looked at the figural results of turtle’s movement.
Abstract. Issues related to 3d turtle’s navigation and geometrical figures ’ manipulation in the ... more Abstract. Issues related to 3d turtle’s navigation and geometrical figures ’ manipulation in the simulated 3d space of a newly developed computational environment, MaLT, are reported and discussed here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented 3d figures seem to offer new resources and to pose new challenges as far as geometrical activities and construction of meanings are concerned, which are strongly related to the representational infrastructure of MaLT. Abilities such as spatial orientation and spatial visualisation come into play and are interwoven with the software’s functionalities and semantics. Although the body-syntonic metaphor remains critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated with other – often conflicting one another – frames of reference. The strong link between spatial graphical and geometrical aspects, that was accentuated by the dragging functionalities...
Construction of meanings for fraction as number-measure is studied during the implementation of e... more Construction of meanings for fraction as number-measure is studied during the implementation of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. 12-year-old students were working collaboratively in groups of two with software that combines graphical and symbolic notation of fractions represented as points on the number line. Fractions as points and segments, ordering fractions as part of kinesthetic activities and abstracting the scaling of the numerical unit on the number line are some of the meanings developed. THEORETICAL BACKGROUND In this paper we report research [1] aiming to explore the mathematical meanings constructed collaboratively by 12 year-old students concerning the notion of fraction as a number-measure depicted on the number line within a specially designed computational environment. The students worked in collaborative groups of two using the ‘Fractions Microworld ’ (FM) [2], a piece of software which combines ...
The construction of meanings for the notion of angle in 3d geometrical space is studied during th... more The construction of meanings for the notion of angle in 3d geometrical space is studied during the implementation of specially designed geometrical tasks in the classroom. 13-year-old pupils were engaged in exploring the mathematical nature of angles while controlling and measuring the behaviours of geometrical objects in 3d simulations. Pupils worked in pairs using a specially designed computational environment that combines symbolic notation (by means of programming) and dynamic manipulation of graphically represented geometrical objects. The findings reveal a progressive coordination of different facets of angle in 3d space facilitated by the use of the available tools. THEORETICAL FRAMEWORK This contribution reports classroom research [1] aiming to e construction of meanings for the notion of angle in 3d geometrical space during activity involving the construction and manipulation of programmable geometrical figures. The students worked in collaborative groups of two using MaLT ...
Abstract. Issues related to 3d turtle's navigation and geometrical figures' manipulatio... more Abstract. Issues related to 3d turtle's navigation and geometrical figures' manipulation in the simu-lated 3d space of a newly developed computational environment, MaLT, are reported and discussed here. The joint use of meaningful formalism and the dynamic manipulation of ...
In this paper we report research [1] aiming to explore the mathematical meanings constructed coll... more In this paper we report research [1] aiming to explore the mathematical meanings constructed collaboratively by 12 year-old students concerning the notion of fraction as a number-measure depicted on the number line within a specially designed computational environment. The ...
In this paper we report findings from a design-based research aiming at shedding light on the way... more In this paper we report findings from a design-based research aiming at shedding light on the way eleven-year old students used gestures while constructing mathematical meanings in the framework of a 3d Turtle Geometry Environment. The results bring in the foreground the role of gestures, as signs that mediated the mathematical notions integrated in the computational environment, as well as the interconnection between gestures and the embodied turtle metaphor. Different kinds of gestures were used by students depending on their focus point during the construction processes. When focusing on turtle’s navigation and the graphical results of this navigation, students used dynamic representational gestures. On the contrary when viewing 3d space or 3d objects as external observers they used abstract deictic or static representational gestures.
The construction of meanings for the notion of angle in 3d geometrical space is studied during th... more The construction of meanings for the notion of angle in 3d geometrical space is studied during the implementation of specially designed geometrical tasks in the classroom. 13-year-old pupils were engaged in exploring the mathematical nature of angles while controlling and measuring the behaviours of geometrical objects in 3d simulations. Pupils worked in pairs using a specially designed computational environment that combines symbolic notation (by means of programming) and dynamic manipulation of graphically represented geometrical objects. The findings reveal a progressive coordination of different facets of angle in 3d space facilitated by the use of the available tools.
This paper aims to contribute to the research into the way mathematical meaning-making can be int... more This paper aims to contribute to the research into the way mathematical meaning-making can be integrated with spatial navigation and orientation in 3d digital media. We report on the findings of a classroom research trying to shed light on 12 year-old pupils’ construction processes as they worked with a 3d Logo / Turtle Geometry environment called ‘MachineLab Turtleworlds’ (MaLt). It seems that the simulated 3d space was experienced by the pupils through two distinct perspectives, an intrinsic and an extrinsic one, that was depicted –among the others- on the way the available viewing angle manipulation tools were used. When the pupils were focusing on navigating and orientating the turtle, the 3d space was experienced through an intrinsic perspective, according to which the simulated space was viewed from inside, through the turtle’s viewpoint. In contrast, when the focus was shifted from the management of the turtle’s spatial movements to the construction of a graphic object, the 3d space was experienced through an extrinsic perspective, from the view point of an external observer who looked at the figural results of turtle’s movement.
Issues related to 3d turtle’s navigation and geometrical figures’ manipulation in the simulated
... more Issues related to 3d turtle’s navigation and geometrical figures’ manipulation in the simulated
3d space of a newly developed computational environment, MaLT, are reported and discussed
here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented
3d figures seem to offer new resources and to pose new challenges as far as geometrical
activities and construction of meanings are concerned, which are strongly related to the representational
infrastructure of MaLT.
Abilities such as spatial orientation and spatial visualisation come into play and are interwoven
with the software’s functionalities and semantics. Although the body-syntonic metaphor remains
critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated
with other – often conflicting one another – frames of reference. The strong link between spatial
graphical and geometrical aspects, that was accentuated by the dragging functionalities of the software,
helped students go beyond an immediate perceptual approach, relating geometrical figures
with real 3d objects and the change of planes in 3d space with physical angle situations. In this
framework the concept of angle as turn and measure with emphasis on directionality but also as a
relationship between the planes defined by 2d figures has arisen as central.
… Journal for Technology in Mathematics Education, Jan 1, 2006
The new representations enabled by the educational computer game ‘the Juggler micro-world’ can pl... more The new representations enabled by the educational computer game ‘the Juggler micro-world’ can place vectors in a central role both for controlling and measuring the behaviors of objects, in a virtual environment simulating motion in three-dimensional spaces. The mathematical meanings constructed by 13 year-old students in relation to vectors as objects, as a set of properties and as representations of vectorial entities are reported and discussed in this document. It is argued that playing in the ‘Juggler’ micro-world could have a considerable contribution to the development of intuitions and to the generation of meanings related to vectors through the use of vectorial properties in a meaningful way.
This paper reports on a case-study design experiment in the domain of fraction as number-measure.... more This paper reports on a case-study design experiment in the domain of fraction as number-measure. We designed and implemented a set of exploratory tasks concerning comparison and ordering of fractions as well as operations with fractions. Two groups of 12-year-old students worked collaboratively using paper and pencil as well as a specially designed microworld which combines graphical and symbolic notation of fractions represented as points on the number line. We used the students’ interactions with the available representations as a window into their conceptual understanding and struggles in making sense of fraction as number-measure. We report on the features of the available representations from an epistemological point of view, on the design of activities aiming at creating meaningful problem contexts for fractions as well as on the meanings generated by the students by some illustrative examples of their work indicating the potential of the activities and tools for expressing and reflecting on the mathematical nature of fraction as number-measure.
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Papers by Maria Latsi
3d space of a newly developed computational environment, MaLT, are reported and discussed
here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented
3d figures seem to offer new resources and to pose new challenges as far as geometrical
activities and construction of meanings are concerned, which are strongly related to the representational
infrastructure of MaLT.
Abilities such as spatial orientation and spatial visualisation come into play and are interwoven
with the software’s functionalities and semantics. Although the body-syntonic metaphor remains
critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated
with other – often conflicting one another – frames of reference. The strong link between spatial
graphical and geometrical aspects, that was accentuated by the dragging functionalities of the software,
helped students go beyond an immediate perceptual approach, relating geometrical figures
with real 3d objects and the change of planes in 3d space with physical angle situations. In this
framework the concept of angle as turn and measure with emphasis on directionality but also as a
relationship between the planes defined by 2d figures has arisen as central.
3d space of a newly developed computational environment, MaLT, are reported and discussed
here. The joint use of meaningful formalism and the dynamic manipulation of graphically represented
3d figures seem to offer new resources and to pose new challenges as far as geometrical
activities and construction of meanings are concerned, which are strongly related to the representational
infrastructure of MaLT.
Abilities such as spatial orientation and spatial visualisation come into play and are interwoven
with the software’s functionalities and semantics. Although the body-syntonic metaphor remains
critical while navigating the turtle in the 3d simulated space, it seems that it has to be co-ordinated
with other – often conflicting one another – frames of reference. The strong link between spatial
graphical and geometrical aspects, that was accentuated by the dragging functionalities of the software,
helped students go beyond an immediate perceptual approach, relating geometrical figures
with real 3d objects and the change of planes in 3d space with physical angle situations. In this
framework the concept of angle as turn and measure with emphasis on directionality but also as a
relationship between the planes defined by 2d figures has arisen as central.