Programmable constructions in 3D geometrical space (familiar)

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • 3d geometrical figures
  • Spatial orientation / visualisation
  • Dynamic manipulation / transformation of 3d objects

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • turtle geometry
  • 3d space
  • 3d geometrical figures
  • dynamic manipulation / transformation
  • 3d simulations

Description show tooltip help

This pedagogic plan engages students in exploring the mathematical structure of 3d geometrical objects in a 3d computational environment. Students will have opportunities to construct 3d geometrical figures and dynamically manipulate, transform and animate 3d objects often encountered in everyday situations (e.g staircases, sliding doors) through Logo commands and variation tools. The tasks are designed to bring in the foreground issues concerning the mathematical nature of 3d geometrical objects and how these may be dynamically manipulated and tranformed in mathematically meaningful ways. This pedagogical plan involves two sub-scenarios aiming to engage pupils in exploring:

(a) moves and angles in 3d space,

(b) constructions and dynamic transformations of 3d geometrical figures.

Rationale show tooltip helpexplode

The teaching of 3d geometry in the primary education curriculum is primarily based on the presentation of static 3d objects -either in a concrete form or in printed form (e.g. from textbooks)- while students lack any chance of manipulating them. Therefore, it can be argued that the numbers and types of investigations for 3d mathematical concepts and processes that can be investigated in the traditional classroom (i.e. mathematical nature of shape, position and orientation) are rather limited. This can be an explanation for the difficulties that students encounter in higher grades when need to think of 3d geometrical notions in an abstract way.

This is in contrast to the number and types of 3d geometry investigations that can be carried out in digital media where multiple representations can place 3d geometry in a central role for both controlling and measuring the behaviours of objects in 3d spaces. However, relatively little research has been carried out on mathematical notions embodied in 3d geometrical objects based on functionalities, like the interactivity and the dynamic display, that these environments provide for the user. In designing MaLT we intend to explore some novel ideas for geometry by extending the integrated ‘Programming-Dynamic Manipulation’ idea to 3d spaces.

MaLT is a programmable environment for the creation and exploration of interactive 3d simulations providing rich semiotic representations and functionalities to facilitate spatial thinking, 3d visualisation and means for the dynamic manipulation of 3d geometrical objects. The design of MaLT suggests that 3d geometry is a field where mathematical formalism and graphical representation of objects and relations can be dynamically joined in interesting ways and that joint symbolic and visual control may have important potential for mathematical meaning-making processes.

In MaLT, students will be able to capitalise upon the power of Logo as a programming language to drive the turtle in body-syntonic and at the same time mathematical formal ways to construct geometrical figures and to get immediate visualization of it. MaLT as a Logo environment will thus allow learners to use their body movements to kinesthetically pace out a geometrical construction by speaking a mathematical language that includes vectors embedded in turtle’s moves (consisting of its position and heading).

Students will also be able to dynamically manipulate conventional representation 3d geometrical figures (e.g. angles, side lenghts, perimeters, area) by using specially designed variation tools rendering parametric procedures descriptors of evolving geometrical objects by means of dragging sliders and observing how an object changes in relation to the value of the variable. As a result students have the chance to observe the behaviour of the varying parts in relation to each other and to the invariant ones and to acquire a sense of generality and abstraction underlying some static instances of the mathematical structures.

Moreover, abilities such as spatial orientation and spatial visualisation come into play and are interwoven with the use of various frames of reference (e.g. an egocentric frame of reference related to the Logo commands ‘forward/backward, right/left’, a coordinate frame of reference, a display frame referring to the orientation and movement in the display screen etc.) and mathematical formalism.

These characteristics:

  1. make MaLT quite different from other 3d environments that allow pupils to handle movement using black-box algorithms that in many cases are hard for students to understand;
  2. foster a more holistic understanding about 3d geometrical objects by improving the chance to develop sound spatial abilities and transfer the meanings of a geometric construction among a variety of different representations.

This pedagogical plan consists of a set of proposed tasks that will take students from initial engagement and familiarisation with the computer environment through to construction challenges and opportunities for imaginative simulation and transformation of 3d geometrical objects.

Theoretical framework show tooltip help

MaLT was conceived as a constructionist microworld environment (Papert, 1980, Harel & Papert, 1991, Kafai & Resnick, 1996) for programmable constructions in 3d geometrical space. It is designed to provide opportunities for learners to engage in expression of mathematical ideas through meaningful formalism by means of programming and dynamic manipulation of graphically represented mathematical objects.

It constitutes a new digital medium within the context of more than a decade of ETL R&D work on designing constructionist exploratory media based on the principle of integrated and interdependent mathematical representations. A principal construct emerging from this work is that of half-baked microworlds (Kynigos, 1992 and in press). These microworlds are designed for instrumentalization (Guin &Trouche) through constructionist activity, i.e. they incorporate an interesting idea but at the same time invite changes to their functionalities and are mediated to the targeted users as unfinished artefacts which need their input.

ETL work is based on the idea of instrumentation though instrumentalization (see Verillon and Rabardel, 1995 for these terms) which gives rise to a dialectic by which learner and artefact are mutually shaped in action. We have focused particularly on the process by which implicit mathematical knowledge is constructed during shared student activity (theorems in action, Vergnaud, 1990 and socio-mathematical norms, Cobb & Yackel, 1996). We are strongly sensitive to the ways an instrument is internalised collaboratively by the students (Mariotti, 2002) while it is being changed often quite distinctly to what was designed by the researchers. Relatively, the implication of this perspective is that students' expressions can gain mathematical legitimacy, even if they differ from and/or they are shaped and structured by the artefact in ways that lead them to diverge from curriculum mathematics.

Constructionism is a theory of learning and designing that builds upon the connotation of learning as "building knowledge structures" in a context where learners are consciously engaged in constructing (or de/re constructing) something on the computer (such as animations simulations, geometrical figures etc.), implying that the notion of construction refers both to the ‘external’ product of this activity as well as to the theories constructed in pupil’s minds. As developed by Papert (1980) "better learning will not come from finding better ways for the teacher to instruct, but from giving the learner better opportunities to construct."

The constructionist environments designed at ETL so far provide dynamic visual means that support immediate visualization of multiple linked representations (i.e. any action carried on a specific representation provides immediate change and feedback in all representations, Kaput, 1992). A specific subset of these environments enables direct and sensual manipulations of mathematical objects with body syntonic metaphors (e.g. the turtle in the 2d space of E-slate ‘Turtleworlds’ and in the 3d space of MaLT). In such settings learners are engaged in constructing public entities (constructions) implying an explicit apreciation of the relationships between mathematical objects within any situation (i.e. a mathematical model of the situation).

The key point here is that students can build their models into the medium that can act as a support for developing new meanings by investigating their hypothesis and argumentating in social contexts. Manipulating objects and articulating relationships between them can thus provide an action/notation context which can be a new resource for activity and construction of meanings, not so dependent on the medium for its expression. Noss and Hoyles (1996) introduced the notion of situated abstraction to describe how learners construct mathematical ideas by drawing on the linguistic and conceptual resources available for expressing them in a particular computational setting which, in turn, shapes the ways the ideas are expressed. Yet, from a social constructivist perspective, psychological and social aspects of learning can never be considered separately and the term situated abstraction captures also the synergy between them: student’s activity within a community (Lave & Wenger, 1991) both shapes and is shaped by their interaction with the available tools and those around them.

A key feature of the approach of ETL is to design artifacts afforded with integrated graphical and symbolic representations and functionalities including dragging modalities which provide a kinesthetic sense of the dynamic manipulation of mathematical objects (e.g. geometrical figures). As an example, in the last decade ETL has been involved in the design of E-slate, an educational authoring system with which many different microworlds have been developed for mathematics and science. These microworlds can be characterized as hybrids between symbolic programming (such as Logo-based Turtle Geometry) and dynamic manipulation (such as Dynamic Geometry Environments) systems.

In designing MaLT, we intend to create learning environments extending the above design representations and functionalities to the 3d space. This decision is also related to the following facts (a) most of the powerful ICT tools for the learning of geometry (Logo, DGS) have been designed to operate within 2d environments, and (b) relatively little research has been carried out on spatio-visual mathematical concepts concerning geometry in 3d spaces.

New representations enabled by digital media can place spatial visualization concepts in a central role for both controlling and measuring the behaviours of objects and entities in virtual 3d environments. We have chosen the notion of vector as a means to represent the link between 2d and 3d representations, since vectors can be considered as basic components underpinning the study of geometry and motion in space facilitating the study of 3d spatial thinking. In MaLT the mathematical nature and identity of vectors is considered as embedded in turtle’s movement which takes place in a specially designed 3d Turtle Geometry Environment where students will be able to capitalise upon the power of Logo as a programming language. It is a dynamic visual environment in which learners will be able to create programmable 3d geometrical constructions and then using the provided tools as a means of manipulating them and investigating their properties. Our perspective is centered on the utilization of the different representations and the feedback that they can provide so as to facilitate multiple didactical decisions within open-ended exploratory tasks.

Our approach to learning promotes also investigation through the design of activities that offer a research framework to investigate purposeful ways that allow children to appreciate the utilityof mathematical ideas (Ainley & Pratt, 2002). In this context, our approach is to design tasks for either exclusively mathematical activities or multi-domain projects containing a mathematical element within the theme which can be considered as marginalized or obscure within the official mathematics curriculum (Kynigos & Psycharis, 2003, Keisoglou & Kynigos, 2006, Kynigos & Gavrilis, 2006). This educational plan is designed to bring in the foreground issues concerning the mathematical nature of 3d geometrical objects and how interactivity, control and experimentation in virtual reality microworlds can be a versatile vehicle for enhancing mathematically driven navigation, orientation and spatial visualization. One such idea is the dynamic manipulation of mathematical variation as it is realised in 3d geometrical objects.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This pedagogical plan is intended for students at lower secondary level who have already been taught the properties of basic geometrical figures (e.g. parallelograms, rectangles, triangles) at the primary level.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

  • exploring the notion of angle as turn and measure within the 3d space (e.g. the notion of angle as a change of direction and planes in 3d space, the notion of angle between two different planes, the notion of angle between two different 3d figures)
  • identifying the mathematical structure of 3d geometrical figures (e.g. distinguish the different 2d planes of the construction and relate it to the type/number of angles)

  • identifying the geometrical properties of 3d objects (logical arguments to justify conclusions, relationships among angle, side lengths, perimeter areas, and volume, develop intuitions and conjectures about the geometrical properties and relations of parallelepipeds)

  • understanding the relation between 2d and 3d representations when using the former to construct simulations of real 3d objects (e.g. identify the role of the ‘repeat’ command concerning 2d shapes in the construction of a 3d geometrical figure - between rectangle and revolving doors)

  • developing fluency with the mathematical expressions to describe a 3d geometrical construction with variables

  • identifying the role of variables in the construction/manipulation of 3d geometrical figures in different sizes

Cognitive goals show tooltip help

  • Analysing the structure of 3d geometrical figures.

  • Analysing and debbuging models of 3d geometrical figures.

Social-affective goals show tooltip help

  • Participating in collaborative group-work for the construction and dynamic transformation of 3d geometric simulations

  • Developing dialogic/argumentative practices

  • Since MaLT is an open environment different groups of students may follow different strategies for the same construction problem. Groups of students in a classroom may be asked to exchange their constructions so as to compare them to those of other pupils and to describe/analyse the differences/similarities concerning the mathematical content.

Instrumental goals show tooltip help

  • Writting, editing and executing Logo programs to construct 3d geometrical figures
  • Controlling and measuring the behaviours of geometrical objects in the 3d space through the use of variables and variation tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This is a sequence of tasks for students, taking them from an initial introduction to the software and its functionalities through to a number of problem solving challenges and opportunities for creative exploration.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

MaLT

MaLT is a programmable environment for the creation and exploration of interactive 3d simulations. The representational infrastructure provided by MaLT affords specially designed representations as tools to construct 3d geometrical objects -by navigating 3d agents (e.g 3d turtle)- and to dynamically manipulate them.

Three distinctive features of MALT that differentiates it from other Logo based 3d environments is that:

  • It combines graphical and symbolic representation with dynamic manipulation of variable procedure values.
  • Variation tool can be considered as providing a link between the mathematical variation and the geometrical representation of the variables representing geometrical objects.
  • In the development it was used a Java-based compiler.

Tool access show tooltip help

NIL

Turtle Scene (TS) [Component]

Turtle Scene is a 3d grid-like interface in which a 3d turtle is visualised. Whenever the turtle moves it leaves behind it a trace which is a selectable, 3d thin cylindrical line.It provides rich representations of geometric objects and behaviors allowing navigation within the 3d virtual space (e.g. by the use of different camera viewspoints).

Logo Editor [Component]

This component is the Logo-like programming interface and it is linked to the TS. The user is able to write, run and edit Logo procedures to ‘drive’ the construction of geometrical objects by the turtle.

Logo Commands in 3D Space [Component Feature]

There is an extension of Logo commands in 3D space including three kinds of turtle turns instead of one. These are:

  • LEFTROLL n (lr n)

Roll the turtle to its left side by n degrees.

  • RIGHTROLL n (rl)

Roll the turtle to its right side by n degrees.

  • UPPITCH n (up n)

Pitch the turtle upwards (nose up) by n degrees.

  • DOWNPITCH n (dp n)

Pitch the turtle downwards (nose down) by n degrees.

Variation Tools (VT) [Component]

This component is the dynamic manipulation feature of the computer environment providing means to represent and handle variation of variable values. It consists of three kinds of variation tools, the Uni-dimensional Variation Tool, the Two-dimensional Variation Tool and the Vector Variation Tool. The two latter tools are dependent on the first one and they are activated only through the Uni-dimensional Variation Tool.

Uni-dimensional Variation Tool (1dVT).
It consists of 'number-line'-like sliders, each corresponding to one of the variables used in a Logo procedure.
Two-dimensional Variation Tool (2dVT).
It allows the covariation of two variables at a time in the form of dragging on the xy-representation of the plane.
Vector Variation Tool (VVT).
It consists of three vector-like representations activated when a Logo procedure with at least three variables is defined. By using VVT the user is able to control the variation in 3D representations of geometrical objects in TS.
Dynamic manipulation using 1dVT [Component Feature]

After the variable procedure is defined and executed with a specific value, clicking the mouse on the turtle trace activates the 1d Variation Tool, which provides a slider for each variable (see the following Figure in which 1dVT is shown including six variables used in a procedure called SWIVELRECT). The user can change in each slider the initial value, the end value as well as the step of the variation (these numbers are shown in the small boxes beside the sliders). Dragging a slider has the effect of the figure dynamically changing as the values of the variable change sequentially.

Dynamic manipulation using 2dVT [Component Feature]

The 2dVT allows the co-variation of two variables at a time. It is activated through the 1dVT after selecting two variables and clicking on one of the two axes of an icon representing an orthogonal bi-axial system, which is placed next to each slider. Clicking on these axis changes its color and activates the 2dVT ( bottom right on the next figure).

In the example presented in the following figure variable :b is selected to represent the x-axis while variable :c is selected to represent the y-axis.

The numeric domain of the two variables is that defined in the 1dVT. Any change in the numeric domain and the step in the 1dVT is transferred automatically to the system of coordination on the 2dVT.

The tool is in the form of an orthogonal pad where the mouse can be freely dragged on the plane. Each position on the pad represents a value for each of the two selected variables, one representing the x-axis and one the y-axis, respectively. A trace is drawn when the mouse is dragged. The numeric changes in 2dVT can be seen simultaneously to the respective sliders on the 1dVT.

The effect of using the 2dVT is that of co-variation of two variables. It is critical for the user to understand that if the dragging is coordinated to approach –may be after experimentation/guess- a functional relationship underlying the co-variation of the two variables then the result would be the graph of this specific function.

Dynamic manipulation using VVT [Component Feature]

The VVT requires a procedure with at least three variables. Next to the little orthogonal bi-axial representation on the 1dVT tool we have another icon to activate the VVT. The semantic of this icon is (r, φ, θ) and the user can click on any of the three letters to change its color, in the same way that the color of the selected axis on the 2dVT representation changes. r stands for length, θ for the angle between the vector’s projection on the xy plane and the x-axis and φ for the angle between the vector and the xy plane.

The VVT vector tool consists of two vector-like representations, which appear as two square windows having a common side (see the next figure).

One representation stands for the projection of the vector on the horizontal plane xy. The user can dynamically manipulate the length of the vector’s projection on the xy-plane and rotate it with respect to its inclination with the x-axis, angle θ.

The second representation stands for the plane formed by the vector and the z-axis and the user can again manipulate the vector’s length and rotate it with respect to its inclination with the xy-plane, angle φ. These two representations sit side by side on the screen. This means that the VVT does not allow the manipulation of the resultant vector value, but only of the constituent projections on the two planes. The values seen on the VVT will thus be different to the ones observed on the 1dVT.

The VVT has a third component, where the resultant vector will appear in a cube-like box. This component will not be manipulable; its function will consist of merely representing what happens to the resultant vector as either of the two projections change. The two square window projections will contain both polar and Cartesian values. The polar values will be represented graphically by the vector itself and numerically by digits in corresponding text boxes. The Cartesian ones will be represented by the visible projections of the vector on the two axes and digitally again by text boxes.

In the following figure there is a schematic representation of the VVT. The second and third representations are manipulable while the first -which represents the vector in 3d space- changes only as a result of manipulating either of the latter two.

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

20 teaching sessions.

Introductory activity

Identity show tooltip helpexplode

Authors show tooltip help

ETL team

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Spatial orientation
  • Spatial visualisation

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • 3d space
  • 3d simulations
  • Turtle geometry

Description show tooltip help

This is an introductory sequence of tasks, aiming to familiarise students with the functionalities of MaLT and the basic Logo commands in the 3d space. Students 'drive' the turtle by using all the available commands for move and turning and then observe the graphical effects on the 3d Turtle Scene.

Rationale show tooltip helpexplode

This activity engages students in navigating the turtle scene of MaLT using Logo commands in order to be familiar with the turtle turns and moves in the 3d geometrical space. Initially students will be asked to explore turtle’s turns and moves by using different sets of 3d Logo commands and then to construct different kinds of rectangles or squares in at least two different successive planes of the 3d turtle scene of MaLT. In order this task to be more familiar to students, teacher could encourage them to imagine that the 3d turtle scene of MaLT is the internal part of a house (or a room). Then the teacher can ask them to create the simulation of the house's (or rooms') windows around its walls.

Theoretical framework show tooltip help

The constructionist theoretical perspective of the present pedagogical plan is based on the notion that children’s conceptions of space emerge from action rather than from passive ‘copying’ of sensory data. An implication is that programmable geometrical constructions designed to help children abstract the notion of turtle movement in the 3d space provide a useful environment for developing their conceptualizations of 3d geometrical objects.

Another fundamental principle of our approach is to encourage the construction of links between different forms of representation of a mathematical objects in 3d space. We have selected the version of Logo developed in MaLT as one context to explore students’ ideas around basic geometrical objects (e.g. angle, turn) in 3d space using interconnected graphical and symbolic representations. Because the intrinsic geometry of move in 3d space is closely related to real world experiences such as walking (or observe something flying), this kind of activities may be especially efficacious in developing students' conceptualizations of the basic geometrical objects acquired through bodily experience that allows the individual to ‘re-present’ relevant features of that experience.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already been taught the properties of basic geometrical figures (e.g. parallelograms, rectangles, triangles) at the primary level.

Student prerequisites show tooltip help

Familiarization with the basic 2d Logo commands.

Teacher prerequisites show tooltip help

Familiarization with MaLT and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Content-epistemological goals show tooltip help

  • Developing fluency with moves and turns in 3d space

  • Constructing 3d geometrical objects using 3d Logo commands

  • Exploring the notion of angle in 3d space (e.g. as change of planes and direction within the 3d space).

  • Exploring the role of the internal and external angles for the construction of a 3d geometrical figures.

Cognitive goals show tooltip help

  • Observing and discussing the ways by which a 3d geometrical construction can be visualised in the 3d space
  • Connecting the symbolic and the graphical representations with the visualisation of 3d geometrical constructions

  • Connecting spatial thinking with intuitions and everyday experiences

  • Developing and testing hypothesis through experimentation

Social-affective goals show tooltip help

Participating in collaborative group-work for the construction and dynamic transformation of 3d geometric simulations

Developing dialogic/argumentative practices

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Writing, editing and executing Logo programs to construct 2d geometrical figures in the 3d space

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This is a sequence of tasks for students, taking them from an initial introduction to the software and its functionalities through to a number of geometrical simulation challenges in the 3d space and opportunities for creative exploration through body syntonic activities. Initially students will be asked to explore turtle’s turns and moves by using different sets of 3d Logo commands and then to use them to demonstrate an aeroplane taking-off with the use of a relevant tangible concrete object (e.g. a model of a 3d aeroplane). Then students will be asked to construct different kinds of rectangles or squares in at least two different successive planes of the 3d turtle scene of MaLT. In order this task to be more familiar to students, teacher could encourage them to imagine that the 3d turtle scene of MaLT is the internal part of a house (or a room). Then the teacher can ask them to create the simulation of the house's (or rooms') windows around its walls.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Resources show tooltip helpexplode

General description show tooltip help

  In this introductory activity the resources available to students are the computational environment of MaLT and the two worksheets presented below. Students use the computational environment and try to carry out the tasks and answer the questions comprised in the two worksheets, which are given to students as hard copy.

1st worksheet [Resource for students]

In the 1st worksheet students are given a table with the basic Logo commands that can be used to ‘guide’ the turtle in the 3d simulated space of MaLT. Then are asked to move the turtle in such a way so as to simulate the take-off of an aircraft and to explain the way they worked in this activity.

 

 

2nd worksheet [Resource for students]

Having as a starting point their everyday experiences in this 2nd worksheet, students are asked to try to represent in the 3d computational environment of MaLT two windows in two adjacent walls of an imaginary room.

 

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

2 teaching sessions.

Actors' roles show tooltip help

Students

  • Work collaboratively in order to explore the tool’s functionalities

  • Experiment and develop their own constructions in MaLT

  • Express their intuitions and share ideas

Teacher

  • Facilitates/mediates learning

    • Supports students as far as the use of MaLT is concerned

    • Prompts students and provides hints

    • Poses questions and brings into the foreground points for consideration at group and classroom level

  • Moderates discussion at classroom level

  • Unifies learning experiences

What to do and how show tooltip help

Phase 1: Simulating an aeroplane taking off

The main aim of this task is to familiarise students with moves and turns in the 3d geometrical space of MaLT. For this reason it is better suggested that the teacher could invent a novel idea in order to introduce the turtle moves and turns and facilitating at the same time pupils’ engagement in body-syntonic activities. So, instead of giving to students the Logo commands for turning in printed form the teacher can ask them to demonstrate an aeroplane taking-off with the use of a relevant tangible toy (e.g. a model of a 3d aeroplane). Each possible movement or turn of the aeroplane during the take off can be considered as a corresponding one to the turtles’ moves and turns in the 3d geometrical space of MaLT. At the same time the teacher can also present to the students a photograph of an aeroplane during its take off and discuss with them its possible (kinaesthetic) movements in space. Students are expected to describe any possible aeroplane move and relate it to turtle’s moves and turns in MaLT.

Alternatively, the teacher can present to the students the commands concerning the turtle’s moves and turns in MaLT and then ask them to write procedures in order to simulate the aeroplane taking off demonstrated in the preceding activity. Moreover student’s introduction to writing procedures is an efficient way to execute specific lists of commands whenever they want.

As an extension of this activity the teacher can invite all the groups of pupils to play a game: Firstly, each team can present in another team a specific simulation of a ‘real’ plane taking off by using an aeroplane miniature. Then the other team is asked to try to drive the turtle so as to simulate that specific taking off in the Turtle Scene of MaLT by using appropriate commands.


Phase 2: Designing windows

Students are asked to create different kinds of rectangles or squares with the use of Logo commands in at least two successive planes. In order to make this task more interesting for the pupils, the teacher can ask them to create the simulation of the windows of a house (or a room) in two successive walls. For this construction the teacher can invite pupils to observe the windows of their classroom or to bring in mind windows of their own house (or room). The most important aim of this task is that students are expected to form initial conjectures about angular relationships.

Driving the turtle in successive planes students have the opportunity to use Logo commands, draw conclusions about the connection between an external and an internal angles in geometrical figures and about the necessary kinds of turns for the passage from one plane (wall) to another. Since the use of the paper and pencil is available for students during the whole task, in the end of this activity students may be asked to represent graphically and symbolically on paper the angular relationships that they have observed/found and then to discuss their observations with other working teams.

Angles in 3D space

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angles in 3d space
  • 3d simulations
  • Dynamic manipulation/transformation of 3d objects

Description show tooltip help

This pedagogic plan engages students in exploring the mathematical nature of angles in the 3d space which is an obscure part of the mathematics curriculum, not systematically studied up to now. Students will have opportunities to dynamically manipulate and transform 3d geometrical objects often encountered in everyday situations (e.g. staircases, sliding doors) through Logo commands and variation tools.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities the tasks are designed to integrate different angle domains (e.g. intersecting, turning, sloping) related to physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle.

In this context in MaLT the concept of angle will be explored:

- as a geometric shape, i.e. what different types of angle (e.g. 2d or dihedral angles) look like when visualised in 3d space (e.g. incorporated in 3d simulations);

- as a dynamic amount, indicating both a turn and the result of a turn which can also be represented by a variable;

- as a measure, in angle/turn measurement situations represented by a number.

The purpose of this pedagogic plan is

(a) to relate children’s angle concepts in the 3d space explicitly to their physical angle experiences

(b) to offer a framework in which to account specifically

- for meaning-making processes concerning angular relationships in the 3d space

- for student’s learning trajectories and potential difficulties in coordinating different aspects of angle/turn concepts in 3d space.

Theoretical framework show tooltip help

The concept of angle is central to the development of geometric knowledge. Angle is a multifaceted concept with several definitions that fit into different formal mathematical structures. Three particular definitions occur repeatedly: the region formed by the intersection of two half-planes (static definition); an amount of turning about a point between two lines (dynamic definition) and a pair of rays with a common end-point. Research demonstrates that students at school often learn static notions of angle measure while the facet of angle as a rotation or as the measure of a turning of a ray about a point from one position to another is often neglected. This situation is aggravated by the apparent difficulty of students to approach these dynamic aspects of angle with the use of static means, like protractors.

Student’s misconceptions and difficulties in learning relevant concepts and skills in these domains are well documented in the literature. For instance, children frequently relate the size of an angle to the lengths of the line segments that form its sides, the area enclosed by the triangular region defined by the drawn sides, the length between its sides (from points sometimes, but not always, equidistant from the vertex), the proximity of its sides or the turn at the vertex (Clements and Battista, 1989). It is also clear from the previous research that students have great difficulty coordinating the various facets of the angle embedded in various physical angle contexts involving slopes, turns, intersections, corners, bends, directions and openings (Mitchelmore & White, 1998). For example, students do not readily incorporate turning into their angle concepts and turn is seen to be similar to a corner by holding the initial position of the turning object in memory. Although such difficulties have been interpreted as parts of a developmental process, it has also been highlighted that a mature abstract angle concept depends essentiallyon learning to link the various physical angle contexts together through “the systematic attempt to investigate our spatial environment mathematically” (Mitchelmore & White, 2000, p. 233).

In this pedagogic plan we are motivated to relate parts from different physical angle situations reminding the ones that an individual experiences in everyday circumstances in 3d space where such situations need not be distinguished. We have selected the version of Logo developed in MaLT as one context to explore students’ ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure. Our purpose is to relate children’s angle concepts in the 3d space explicitly to their physical angle experiences and to offer a framework in which to account specifically for their difficulties in coordinating different aspects of the angle concept as well as to through light on the paths by which students might come to integrate their various angle concepts in 3d space. These are brought into relation by mathematical activities such as measuring and drawing in 3d space, which simultaneously leads to other abstract concepts such as point, line, direction, figure congruence. As noted by Skemp (1986) the meaning of ‘angular’ can be communicated only through specific characteristics such as sharpness, inclination, rotation, slope and direction which are embedded in the various angle situations.

The constructionist theoretical perspective of the present pedagogical plan is based on the notion that children’s conceptions of space emerge from action rather than from passive ‘copying’ of sensory data. An implication is that programmable geometrical constructions designed to help children abstract the notion of turtle movement in the 3d space provide a useful environment for developing their conceptualizations of 3d geometrical objects, like angles. Because angles and turns are critical to geometrical figures, and because the intrinsic geometry of move in 3d space is closely related to real world experiences such as walking (or observe something flying), this kind of activities may be especially efficacious in developing students' conceptualizations of angle and turn acquired through bodily experience that allows the individual to ‘re-present’ relevant features of that experience.

Research results strongly suggest that two major factor influencing students’ use of angles: the physical presence or absence of the lines which make up its arms and the conceptualisation of turning as a relationship between two headings of segments. The situation is aggravated in the 3d space by the apparent difficulty of students to recognise the basic elements of an angle. For example, in the simulation of an ‘opening and closing door’ the arms of the 3d angle are rectangles while in the ‘spiral staircase’ the arms are an equilateral triangle and a rectangle incorporating a more obscure similarity of the standard angle domain. Moreover, turning in 3d space can be considered as a sophisticated concept since the turning motion itself usually does not leave a trace (and the 'heading' must be reconstructed from memory). Also, turns are usually not clearly distinguished from other movements in most real-world contexts. For instance, when we turn a car or bicycle, we are also moving forward.

In the present pedagogic plan the main facets of angle are incorporated within and between geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000). More specifically, angles in MaLT are considered as

- directed turns between two similar geometrical figures, (e.g. in the construction of four successive doors of a ‘sliding door’);

- directed turns between two different geometrical figures (e.g. between an equilateral triangle and a rectangle in the simulation of a spiral staircase);

- directed turns between lines and planes (e.g. the vertical relationship between a line and the 2d ‘surface’);

- directed turns for the construction of 2d geometrical figures (e.g. rectangles);

- directed turns in functional angular relationships underlying specific geometrical constructions (e.g. the simulation of ‘opening and closing’ a book).

Trying to approach angle in a socio-constuctionist framework the educational goals formulated in this pedagogical plan are by design general and open-ended so as to leave space for students’ action, exploration and experimentation. Although there is a clear mapping between the educational goals of this pedagogical and greek national curriculum, much more attention is paid on meaning making processes rather than on certain a-priori specified performances. Our pedagogical intent is not to fulfil curricular goals but to investigate the potential of ICT to overturn many of the assumptions about what children can and cannot do and their readiness to understand specific mathematical concepts. However we don’t leave the students simply to interact with the software but we intervene and foster meaning construction through the careful development of activities and through teacher’s guidance.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space

    • angle as a change of direction within the 3d space
    • angles as results of specific turtle turns
    • angles between two different planes in 3d space
    • angle between two geometrical figures, each one of which defines a plane
    • angles as directed turns in functional angular relationships underlying specific geometrical constructions
  • Identifying the mathematical structure of real 3d objects
  • Identifying the mathematical structure of 2d and 3d geometrical figures

  • Identifying the relationships between 2d and 3d representations when using the former to construct simulations of real 3d objects

  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 3d objects and relate them to the abstract geometrical figures
  • Reconceptualising 3d objects in formal mathematical ways
  • Identifying the relation between visual and symbolic representations
  • Capitalising upon intuitions, bridging everyday experience and 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

18 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

Revolving door simulation

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angles in 3d space
  • 3d simulations
  • Dynamic manipulation/transformation of 3d objects

Description show tooltip help

After the familiarisation with the basic commands and functionalities of MaLT students are engaged in the construction of 3d programmable simulations with the use of variables and variation tools. We have chosen the sliding doors simulation since such a construction combines a multiplicity of situations in which the notion of angle is embedded (e.g. angles as results of specific turtle turns to construct the rectangles involved in sliding doors, angles between rectangles in the 3d space, angles as dynamic change of 'rolling' in planes perpendicular to the plane of the surface in simulating the motion of a sliding door).

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a sliding door simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

The implementation is designed to take place in phases. Initially students are experimenting with predefined variable procedures in the horizontal plane trying to investigate the properties of parallelograms. The they create a simulation of an opening-closing door which brings in the foreground the need to change planes and move from the horizontal (surface) to the other vertical ones making use of 3d Logo commands. Capitalising further upon 3d Logo commands and the use of 1d variation tool, they are expected to construct rectangles and then to place them in a plane perpendicular to the plane of the surface. Finally, the animation of a sliding door is expected to emerge by the need to turn the whole construction using a specific ‘rroll’ or ‘lroll’ turn represented with a variable.

Theoretical framework show tooltip help

Taking a constructionist approach the sliding doors simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In the sliding doors simulation the concept of angle will be explored:

- as a geometric shape, i.e. what different types of angle (e.g. 2d or dihedral angles) look like when visualised in 3d space (e.g. incorporated in 3d simulations);

- as a dynamic amount, indicating both a turn and the result of a turn which can also be represented by a variable;

- as a measure, in angle/turn measurement situations represented by a number.

In sliding doors the main facets of angle are incorporated within and between geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles)

- directed turns between two similar geometrical figures, (e.g. in the construction of four successive doors of a ‘sliding door’);

- directed turns between lines and planes (e.g. the vertical relationship between a line and the 2d ‘surface’).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space

    • angle as a change of direction within the 3d space
    • angles as results of specific turtle turns
    • angles between two different planes in 3d space
    • angle between two geometrical figures, each one of which defines a plane
    • angles as directed turns in functional angular relationships underlying specific geometrical constructions
  • Identifying the mathematical structure of real 3d objects
  • Identifying the mathematical structure of 2d and 3d geometrical figures

  • Identifying the relationships between 2d and 3d representations when using the former to construct simulations of real 3d objects

  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 3d objects and relate them to the abstract geometrical figures
  • Reconceptualising 3d objects in formal mathematical ways
  • Identifying the relation between visual and symbolic representations
  • Capitalising upon intuitions, bridging everyday experience and 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

12 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

Constructing parallelograms in 2d plane

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Construction of parallelograms
  • Properties of parallelograms

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Parallelograms
  • Rectangles
  • Angle and turn in 3d space

Description show tooltip help

The construction of parallelograms in 2d plane is the first task of the ‘Sliding doors simulation’ which introduces students to the construction of the main parts of a sliding door considered as geometrical figures.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks concerning the construction of parallelograms. Such a construction in MaLT has been chosen since it gives space for pupils’ engagement in expressing their ideas around the geometrical properties of parallelograms based on relations between specific turns and move in 2d plane. In this task students experiment with predefined variable procedures in the horizontal plane designed to trigger their attention to the underlying relations - either between the angles and/or the sides of a parallelogram. The implementation is designed to take place in phases. Initially students experiment with the measures of the sides of rectangles. Then they explore angle relations for the construction of parallelograms and finally they construct and dynamically manipulate different models of parallelograms.


Theoretical framework show tooltip help

Taking a constructionist approach the exploration of the parallelograms construction integrates different angle domains (e.g. intersecting, turning, sloping) related to these specific class of 2d geometrical figures familiar to pupils from the primary level. However, here the focus is on their properties and the use of variables for both angles and sides.

This stage is considered as an initial but necessary part of our research design which aims to capture students conceptualisation of angles in the transition stage between 2d plane and -as it will be apparent in the following tasks- and 3d space.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring the mathematical structure of 2d figures

    • Identifying the geometrical properties of parallelograms

      • The opposite angles and sides are equal

      • The sum of internal angles is 360 degrees

      • Rectangles can be considered as a subset of parallelograms

    • Exploring the relations of internal and the external angles on the construction of rectangles/parallelograms

    • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 2d objects (e.g. parallelograms) in the 3d space

  • Reconceptualising the construction of 2d geometrical figures in the 3d space in formal mathematical ways by relating visual and symbolic representations

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This is a sequence of tasks aiming to engage students in exploring the properties of parallelograms constructed in 2d planes in the 3d space of MaLT. Initially students are asked to experiment with predefined variable procedures so as to discover the equality of the opposite sides of a rectangle. Then students are engaged in exploring the relations between angles and turtle turns for the construction of parallelograms. Finally, students are asked to construct different kinds of parallelograms by using variables for their sides as well as for their angles.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Resources show tooltip helpexplode

General description show tooltip help

The resources available to students in this activity are the computational environment of MaLT, two ready made procedures (AMystery and BMystery) and the worksheet presented below. Students use the computational environment and try to carry out the tasks and answer the questions comprised in the two worksheets, which are given to students as hard copy.

 

3rd worksheet [Resource for students]

In the 3rd worksheet students are asked to open in MaLT the Amystery and the Bmystery logo procedures and to try to understand what kind of graphical representation are going to be constructed if the procedures are executed. Students are also asked to make changes and ‘amend’ the Logo code so as to develop initially a procedure that always constructs a parallelogram and then a procedure that constructs a parallelogram with the least possible variables

 

Amystery Logo procedure [Resource for students]

The following Logo variable procedure, when executed in MaLt, creates an orthogonal crooked line with four segments. The lenght of each segment is defined by a corresponding variable.

 

Resource contents show tooltip help

 

to Amystery :a :b :c :d

fd(:a)

rt(90)

fd(:b)

rt(90)

fd(:c)

rt(90)

fd(:d)

rt(90)

end

Bmystery Logo procedure [Resource for students]

The following Logo variable procedure, when executed in MaLt, creates a crooked line with four segments. While the lenght of each segment is fixed, the measure of each of the four angles is defined by a corresponding variable.

 

Resource contents show tooltip help

to Bmystery :a :b :c :d

fd(5)

rt(:a)

fd(10)

rt(:b)

fd(5)

rt(:c)

fd(10)

rt(d)

end

 

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

2 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Phase 1: Measuring the sides of rectangles

The teacher has load to the computers the following ‘Amystery’ procedure for the construction of a crooked line and invites students to draw a rectangle by dynamically manipulating with sliders the values of the four variables in the 1d variation tool corresponding to the lengths of it.

to Amystery :a :b :c :d

fd(:a)

rt(90)

fd(:b)

rt(90)

fd(:c)

rt(90)

fd(:d)

rt(90)

end

Initially students are asked to execute the given procedure with a random quadruplet of numbers. As a result an orthogonal crooked line is drawn in the Turtle Scene. Then students are asked to activate the 1d variation tool and to work in groups and experiment with the variable values so as to construct a rectangle.

The teacher can provoke students’ interest, focus their attention and facilitate discussion by asking questions like the following:

  • How does the turtle execute each line of the code?
  • Is there a quadruplet of variable values for which a rectangle is drawn? Are there any other? How many?

The main aim of the activity is to engage students in experimenting with the 1d variation tool so as to recognise the mapping between the values of the variables in the procedure and the lengths of the corresponding segments in the resulting figure. In order to achieve so students can go back and forth between the visual and the symbolic representations observing what changes on the screen whenever they kinaesthetically manipulate with the 1d variation tool the value of each variable representing particular lengths in the Logo code. The teacher should give them enough time to experiment and to find more than one quadruplet of variable values that when used a rectangle is constructed. Students can keep notes in their notebooks.

Within each team students discuss their observations and try to find out what relationships underly each quadruplet of numbers. The teacher can collect the quadruplets of numbers that each team has found and present them in a table in the blackboard or project them by using a data projector so as to discuss and form conjectures, at a classroom level, about the relationships underlying these numbers.

After unifying their observation and drawing certain conclusions about the relationship between the lengths of the segments of each rectangle students are asked to change the ready-made ‘Amystery’ procedure so as to have the least needed variables for the construction of rectangles. The procedures developed by students can be of the following form:

to rectangle :a :b

fd(:a)

rt(90)

fd(:b)

rt(90)

fd(:a)

rt(90)

fd(:b)

rt(90)

end

Phase 2: Exploring angle relations for the construction of parallelograms

In this second phase students are asked by the teacher to construct parallelograms using a second ‘BMystery’ procedure. In this procedure students have only to define the appropriate measures of the angles since the measures of the segments are given.

to Bmystery :a :b :c :d

fd(5)

rt(:a)

fd(10)

rt(:b)

fd(5)

rt(:c)

fd(10)

rt(d)

end

As in the previous phase, students are asked to execute the above given procedure with a random quadruplet of numbers. As a result a crooked line is drawn in the Turtle Scene. Then students are asked to activate the 1d variation tool, to work in groups and experiment with the variable values so as to construct a parallelogram.

The teacher can provoke students’ interest, focus their attention and facilitate discussion by asking questions like the following:

  • How does the turtle execute each line of the code?
  • Is there a quadruplet of variable values that when used the turtle draws a parallelogram? Are there any other? How many?

The main aim of the activity is to engage students in experimenting with the 1d variation tool so as to recognise the mapping between the values of the variables in the procedure and the measures of the corresponding angles in the resulting figure. In order to achieve so students can go back and forth between the visual and the symbolic representations observing what changes on the screen whenever they kinaesthetically manipulate with the 1d variation tool the value of each variable representing particular angles in the Logo code. The teacher should give them enough time to experiment and to find more than one quadruplet of variable values that when used a rectangle is constructed. Students can keep notes in their notebooks. Within each team students discuss their observations and try to find out what relationships underly each quadruplet of numbers. The teacher can collect the quadruplets of numbers that each team has found and present them in a table in the blackboard or project them by using a data projector so that the students discuss and form conjectures at a classroom level about the relationships underlying these numbers. The teacher should focus students’ attention in recognising relationships between the turtle turns and the (internal) angles of the figure or between the angles of the parallelogram (e.g. complementary angles, sum of all its angles) After unifying their observation and drawing certain conclusions about the relationship between the measures of the angles of a parallelogram students are asked to change the ready-made ‘Bmystery’ procedure so as to have the least needed variables. The procedures developed by students can be of the following form:

to parallelogram :a

fd(5)

rt(:a)

fd(10)

rt(180-:a)

fd(5)

rt(:a)

fd(10)

rt(180-:a)

end

Phase 3: Constructing parallelograms in 2d plane

In this phase, students are asked to synthesize their findings and their conclusions from the preceding phases and use it to edit a variable procedure with the least number of variables necessary to construct any kind of parallelogram. Such a procedure can be a synthesis of the procedures developed in the preceding phases taking the following form:

to parallelogramA :a :b :c

fd(:a)

rt(:c)

fd(:b)

rt(180-:c)

fd(:a)

rt(:c)

fd(:b)

rt(180-:c)

end


Working at a classroom level the teacher can help students elaborate their procedure by introducing the ‘repeat’ command. Using the blackboard or data projector to present some of the procedures developed by the various teams, teacher can focus students’ attention at the commands of their procedures that repeat. After the students have spot the sections that repeat, the teacher can introduce the use of the ‘repeat’ command in Logo programming. An indicative example of a procedure that may be developed is shown below:

to parallelogramB :a :b :c

repeat 2 [fd(:a) rt(:c) fd(:b) rt(180-:c)]

end

Simulating the opening and closing of a door

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angles in 3d space
  • 3d simulations
  • Dynamic manipulation/transformation of 3d objects

Description show tooltip help

After the construction of parallelograms in MaLT students are engaged in the construction of 3d programmable simulations with the use of variables and variation tools. For this part of the pedagogical plan we have chosen the opening and closing door simulation since such a construction combines a multiplicity of situations in which the notion of angle is embedded (e.g. angles as results of specific turtle turns to construct the rectangles involved in it, angles between rectangles in the 3d space, angles as dynamic change of 'rolling' in planes perpendicular to the plane of the surface in simulating the motion of a door).

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of an opening-closing door simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

The implementation is designed to take place in phases. Initially students are engaged in searching and describing the relations between a door with geometrical figures.Then they create the ‘frame’ of the door as a rectangle placed in a vertical position to the surface as well as the leaf of the door as another rectangle. Finally, they are expected to simulate the opening-closing of the door by recognising the need to use variables for the angle between the two rectangles.

Theoretical framework show tooltip help

Taking a constructionist approach the opening-closing door simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In opening-closing door the main facets of angle are incorporated within and between the geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles)

- directed turns between two similar geometrical figures, (e.g. in the construction of two rectangles representing the frame and the leaf of a door);

- directed turns between lines and planes (e.g. the vertical relationship between a line and the 2d ‘surface’).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space in simulating the opening and closing of a door

- angle as change of direction within the 3d space

- angles as results of specific turtle turns to construct a geometrical figure

- angles between two different planes in 3d space

- angle between two geometrical figures (rectangles), each one of which defines a plane

  • Identifying the mathematical structure of real 3d objects like a door

  • Identifying the relationships between 2d and 3d representations when using the former to construct simulations of real 3d objects (e.g. opening and closing a door)

  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 2d objects (e.g. parallelograms) in the 3d space

  • Reconceptualising the role of 2d geometrical figures (rectangles in the construction of a door) in constructing 3d simulations using formal mathematical ways

  • Relating interconnected visual and symbolic representations in the construction of 3d simulations

  • Capitalising upon intuitions, bridging everyday experience of opening and closing a door and 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This is a sequence of tasks for the construction of an opening - closing door simulation. Initially students observe the door of their classroom while opening and closing and discuss about the geometrical figures and the necessary turns/moves that are embedded in it (e.g. changing panes while turning). Then students experiment so as to ‘translate’ their intuitions in Logo codes using variables to construct the simulation of an opening and closing door.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Resources show tooltip helpexplode

General description show tooltip help

  In this activity the resources available to students are the computational environment of MaLT and the two worksheets presented below. Students use the computational environment and try to carry out the tasks and answer the questions comprised in the two worksheets, which are given to students as hard copy. However pupils’ constructions in previous activities can be also considered as a resource that students can capitalise upon.

4th worksheet [Resource for students]

  In the 4th worksheet students are firstly asked to try to construct a door in the 3d simulated space of MaLT while trying to connect its shape with the typical geometrical figures. As the changes of plane are a point of focus, students are asked to explain the sequence of commands used in the Logo editor component of MaLT and to write down the conclusions they have drawn. Finally they are asked to try to develop a variable procedure in order to construct the representation of a door.

5th worksheet [Resource for students]

In this final worksheet students are asked to construct the simulation of a door opening and closing and to write down the conclusions they have drawn after their engagement in this activity. Finally they are asked to try to develop a variable procedure in order to construct the simulation of a door opening and closing.

 

 

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

3 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Phase 1: Connecting real 3d objects with geometrical figures

Initially students are asked to observe the door of their classroom while opening and closing. The teacher informs the students that their final goal is to construct a simulation of a door that opens and closes in MaLT. The teacher provokes a classroom discussion by asking what geometrical figure the door looks like and what changes take place when the door opens and closes. Children can freely express their views using geometrical terms or not. From a mathematical point of view and taking into account that the standard angle concept first develops in situations where both arms of the angle are visible, the main difficulty while opening and closing a door is to identify door’s initial position as one arm of the angle. This concept may not arise in this phase but it may gradually be developed while searching for ways to reconceptualise 3d objects so as to describe its simulation with Logo commands in the next phases.


Phase 2: Changing planes in the 3d geometrical place of MaLT


Working in Malt students in this phase are trying to construct a door. Since they have been accustomed to construct variable procedures during the preceding activities, students are expected to realise that initially it is necessary to construct a rectangle in a vertical plane of the ‘surface’ in the 3d space. In other words, they have to command the turtle to change plane and move to the vertical one using the 3d Logo commands ‘uppith ndegrees’ or ‘downpitch ndegrees’. Students may experiment and draw conclusions about the measure of turtle’s uppith or downpitch by changing manually in the Logo editor the corresponding values and then run the procedures developed in the previous phases. Otherwise they can develop a simple variable procedure –like the following- where the degrees of ‘uppith’ or ‘downpitch’ are represented with variables. In such a procedure students can experiment while manipulating with sliders the degrees of uppith or downpitch.


to door :a :b :c :d

up(:d)

parallelogramB(:a :b :c)

end

Phase 3: Constructing an opening and closing door simulation


In order to animate their construction and create the simulation of an opening and closing rectangular door, students have to conjecture about what changes when a door opens and closes and to search for ways to command the turtle re-execute the procedure that constructs a rectangle in different planes around the x axis. In this phase they have to make use of another set of 3d Logo commands: ‘rightroll ndegrees’ and ‘leftroll ndegrees’. Students may experiment and draw conclusions about the measure of turtle’s rightroll or leftroll by changing manually in the Logo editor the corresponding values and then run the procedures developed in the previous phase. Otherwise they can develop a simple variable procedure –like the following- where the degrees of ‘rightroll’ or ‘leftroll’ are represented with variables. That way they can experiment while manipulating with sliders the degrees of rightroll or leftroll and animate their construction while sequentially changing planes around the x axis.

to movedoor :a :b :c :d

up(90)

lr(:d)

parallelogramB(:a :b :c)

end

lr(:d)

parallelogramB(:a :b :c)

end

Constructing a revolving door simulation

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angles in 3d space
  • 3d simulations
  • Dynamic manipulation/transformation of 3d objects

Description show tooltip help

Having achieved a familiarity with the construction of the use of rectangles in the simulation of an opening-closing door and the ways specific turtle commands describe the properties of it students begin to use their acquired experience to construct their own simulation of a sliding door.

As they develop a higher degree of control through the use of Logo formalism and dynamic manipulation of geometrical objects students begin to expand the types of simulations they can explore and construct.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a sliding door simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

The implementation is designed to take place in phases. Initially students are engaged in searching and describing the relations between a sliding door with geometrical figures. Then they create the main geometrical parts of a sliding door (e.g. four leafs placed in a vertical position to the surface). Finally, they are expected to animate the sliding door by recognising the need to turn the whole construction using a specific ‘rroll’ or ‘lroll’ turn represented with a variable.

Theoretical framework show tooltip help

Taking a constructionist approach the sliding-door simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In the sliding doors the main facets of angle are incorporated within and between the geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles)

- directed turns between two similar geometrical figures, (e.g. in the construction of two rectangles representing the frame and the leaf of a door);

- directed turns between lines and planes (e.g. the vertical relationship between a line and the 2d ‘surface’).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space

- angle as change of direction within the 3d space in simulating the motion of a sliding door

- angles as results of specific turtle turns to construct the parts (rectangles) of a sliding door

- angles between two different planes of 3d space in placing the successive doors of a sliding door

- angle between two similar geometrical figures (rectangles), each of which defines a plane

  • Identifying the mathematical structure of real 3d objects like a sliding door
  • Identifying the relationships between 2d and 3d representations when using the former to construct simulations of real 3d objects (e.g. sliding door)
  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the geometrical structure of 3d objects (e.g. a sliding door)

  • Reconceptualising the role of 2d geometrical figures (e.g. rectangles) in constructing a 3d simulation of a sliding door using formal mathematical ways

  • Relating interconnected visual and symbolic representations in the construction of 3d simulations

  • Capitalising upon intuitions, bridging the experience of moving a sliding door and the 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This is a sequence of tasks for the construction of a sliding door simulation. Initially students observe various types of sliding doors either by using concrete 3d models or relevant photos. Then students discuss about the geometrical figures and the necessary turns/moves that are embedded in it (e.g. changing planes while turning). Then students experiment so as to ‘translate’ their intuitions in Logo codes using variables to construct and dynamically manipulate the simulation of a sliding door.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

3 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Phase 1: Connecting real 3d objects with geometrical figures

Initially students are asked to observe various types of sliding doors either by using concrete 3d models or photos from various sources, e.g. from the internet. Ideally students can visit a nearby hotel or bank so as to have a more direct experience with this type of door. The teacher informs the students that their final goal is to construct a simulation of a sliding door and animate it through the use of the variation tools in MaLT. The teacher provokes discussion by asking what geometrical figure can be identified in a sliding door, how the parts of the door they are placed in relation to each other and in relation to the surface as well as how the door moves. Children are free to express their opinions. The aim of these questions is not to end up in definite answers but to trigger students’ attention on various aspects of sliding doors as geometrical constructions.


Phase 2: Constructing a sliding door

Students are asked to create sliding doors while working in groups. Initially students have to decide what geometrical figures a sliding door is composed of and then how they should be placed in relation to each other so as to construct a static 3d representation of a sliding door in the geometrical environment of MaLT. Drawing upon the Logo procedures that they have developed in the previous sub-scenarios and the conclusions they have drawn about moving around in the virtual 3d space, students have to place 4 rectangles in successive equal intervals around the x axis.

A critical point here is the realisation of the fact that a full turn around the x axis is 360 degrees. Students can reach this conclusion either by drawing upon their previous geometrical knowledge or by trial and error while experimenting with the measure of turtle’s RightRoll or LeftRoll turns in the Logo Editor. Alternatively, they can create a simple variable procedure and experiment with the dynamic use of 1d Variation Tool. According to the progress of the various teams, either at group or classroom level the teacher can give students the hind of constructing a variable procedure and use the 1d variation tool. For instance students may write a procedure like the following:

to sliding_doorA :a :b :c :d

up(90)

parallelogramB(:a :b :c)

lr(:d)

parallelogramB(:a :b :c)

lr(:d)

parallelogramB(:a :b :c)

lr(:d)

parallelogramB(:a :b :c)

lr(:d)

end


Then either at group or classroom level students are prompted to edit their procedures using the repeat command. In case that many students face difficulties the teacher can use the blackboard or the data projector so as to report the exact problems for a whole classroom discussion so as to help them realise what segments of the procedure repeat and how many times. The procedure developed may be of the following form:

to sliding_doorB :a :b :c

up(90)

repeat 4 [parallelogramB(:a :b :c) lr(90)]

end


Phase 3: Animating a sliding door

In this last phase of the Sliding doors activity the groups students are invited to find out ways to animate their constructions, i.e. to have the sliding doors moving around. Drawing upon their previous experience of the animation of a simple door they have to conclude initially that when a sliding door moves, the whole construction changes place around the x axis. In this phase they have to make use again the ‘rightroll ndegrees’ and ‘leftroll ndegrees’ Logo commands. Students may experiment and draw conclusions about the measure of turtle’s rightroll or leftroll by changing manually in the Logo editor the corresponding values and then run the procedures developed in the previous phase. Otherwise they can develop a simple variable procedure where the degrees of ‘rightroll’ or ‘leftroll’ are represented with variables. By adding in the beginning of their procedure a rightroll or leftroll command with a variable which changes the place of the turtle around the x axis , students can use the sliders of the 1d variation tool so as to sequentially change turtle’s position around the x axis. Then by reexecuting the procedure they can create the effect of having the doors moving around. Such a procedure that may the following:

to slide :a :b :c :d

up(90)

lr(:d)

repeat 4 [parallelogramB(:a :b :c) lr(90)]

end

In the end of each subscenario, the teams present their constructions in their classmates and explain the difficulties that they have faced and the conclusions they have drawn. Students’ comments and presentation can give teacher the opportunity for stirring up a fruitful discussion at classroom level which will help students express, share, unify and extend the meanings created in the construction process.

Extension: Simulating the opening-closing pages of a book

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angles in 3d space
  • 3d simulations
  • Dynamic manipulation/transformation of 3d objects

Description show tooltip help

Having achieved the 'Sliding doors' simulation using variables for both side lenghts and turns, students can be engaged in experimenting with the construction of other real world situations that involve turning. An example can be the simulation two opening-closing pages of a book.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a opening-closing pages of a book. The difference of this activity to the preceding one is that during the opening-closing turns both sides (rectangles) of the book move in opposite directions.

Such a construction in MaLT has been chosen since it gives space for pupils engagement in expressing angular functional relationships based on turning and directionality in 3d space using one variable. As it will be obvious in the Specifications part this activity can also be considered as an opportunity for using the functionalities of the 2d Variation Tool.

This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as movement in 3d space and turn as dynamic entity represented by variables and/or functional relationships.

Theoretical framework show tooltip help

Taking a constructionist approach the opening-closing pages simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In the opening-closing pages doors the main facets of angle are incorporated within and between the geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles)

- directed turns between two similar geometrical figures, (e.g. in the construction of two rectangles representing the frame and the leaf of a door);

- directed turns between lines and planes (e.g. the vertical relationship between a line and the 2d ‘surface’).

- angles as directed turns in functional angular relationships underlying the simulation of the opening-closing pages

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

As an extension of the previous 3d simulations, this activity requires student's familiarisation with writing/editing Logo procedures to construct 3d simulations with variables and dynamically manipulating them using 1d Variation Tool.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space

- angle as change of direction within the 3d space in simulating the motion of the ‘Opening-closing’ pages of a book

- angles as results of specific turtle turns to construct the parts (rectangles) of the 'Opening-closing' pages

- angles between two different planes of 3d space in designing the successive pages of a book

- angle between two similar geometrical figures (rectangles), each of which defines a plane

- angles as directed turns in functional angular relationships underlying the simulation of the opening-closing pages

  • Identifying the mathematical structure of real 3d objects like the ‘Opening-closing’ pages of a book
  • Identifying the relationships between 2d and 3d representations when using the former (e.g. rectangles) to construct simulations of real 3d objects (e.g. two pages of a book)
  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the geometrical structure of 3d objects (e.g. opening-closing pages of a book)

  • Reconceptualising the role of 2d geometrical figures (e.g. rectangles) in constructing a 3d simulation of the opening-closing pages using formal mathematical ways

  • Relating interconnected visual and symbolic representations in the construction of 3d simulations

  • Capitalising upon intuitions, bridging the experience of opening-closing a book and the 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This task aims to engage students in extending their experience to simulate in MaLT real world situations that involve turning. As an example students simulate the opening-closing two pages of a book while both pages are moving during the turning. Initially students show the different ways by which one can open/turn pages of a book. Then they discuss about the geometrical figures and the necessary turns/moves that are embedded in it (e.g. changing planes and directions while turning) and experiment so as to ‘translate’ their intuitions in Logo codes using variables to construct and dynamically manipulate this specific simulation.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

4 teaching sessions, since in the end of the activity the teacher can devote one more session to the student's experimentation to simulate situations of their own choice.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Phase 1: Connecting real 3d objects with geometrical figures


An indicative implementation of the activity should begin with students’ engagement in discussing at group or classroom level the relation between this simulation to the preceding one (‘Sliding doors’). Pupils can express their opinions after rehearsing the different ways by which one can open/close a book. Some pupils may insist that these two simulations are the same since one can imagine opening a book with moving only one page, which reminds opening a door. However, the teacher should stress that here student’s are to design a simulation in which both sides (pages) are moving while turning. Additionally, it can be said that this move reminds the moves that follows someone who keeps a book and opens it using both hands. After that pupils are asked to find ways to elaborate their procedures of sliding doors so as to simulate the book pages. They may observe that now they need two rectangles each one designed after a ‘leftroll’ and a ‘rightroll’ turtle turn respectively.


Phase 2: Constructing pages using variable turns


Students in this phase try to construct a simulation of the opening-closing pages in MaLT. Since they have been accustomed to construct simulations with variable procedures during the preceding activities, students are expected to realise that initially it is necessary to construct two rectangles in a vertical plane of the ‘surface’ in the 3d space using the command ‘uppith 90 degrees’ and to decide on the ways on which they have to turn each rectangle (page) in relation to each other using variables (leftroll and rightroll commands respectively). Students are expected to develop simple procedures –like the following- where the degrees of ‘leftroll’ and ‘rightroll’ are represented with variables.

to rectA
repeat 2 [fd(5) rt(90) fd(3) rt(90)]
end

to pagesA :b :c

up(90)

rr(:b)

rectA

lr(:c)

rectA

end

to rectB :x :y
repeat 2 [fd(:x) rt(90) fd(:y) rt(90)]
end

to pagesB :b :c :x :y

up(90)

rr(:b)

rectB(:x :y)

lr(:c)

rectB(:x :y)

end


The critical point for completing the simulation concerns the identification of the functional relationship (:b=:c/2) that underlying the ‘leftroll’ and ‘rightroll’ turns (corresponding to the variables :b and :c in the procedure pagesA and pagesB above). Students may have difficulties in even guessing that relationship since the use of the 1d variation tool does not offers adequate help for that. This is mainly due to the fact that the variation of variable :b with the 1d variation tool changes the ‘direction’ of the steady angle between the two page. (See the next two figures. The first one corresponds to the values :c=35 and :b=50 while the second to the values :c=35 and :b=133).

In a similar way, the variation of variable :c with the 1d Variation Tool changes only the angle between the two pages. See the next figures: The first one corresponds to the values :c=20 and :b=50 while the second to the values :c=60 and :b=50.

In a different teaching design the teacher can chose to give to the students the procedure pagesA and ask them (a) to experiment with the manipulation of variables and (b) to discuss on the role of each variable in the construction. Then they can proceed to the next phase.


Phase 3: Animating pages by discovering functional relations between turns

The functional relationship for which the simulation is completed is :b=:c/2. The teacher can exploit for that reason the 2d Variation Tool. It is activated through the 1d variation Tool after selecting two variables and clicking on one of the two axes of an icon representing an orthogonal bi-axial system, which is placed next to each slider.

In the example presented in the next figure we have selected variable b representing the x-axis and variable c representing the y-axis. The numeric domains of the two variables is (0, 90). The tool is in the form of an orthogonal pad where the mouse can be freely dragged on the plane. Each position on the pad represents a value for each of the two selected variables, one representing the x-axis and one the y-axis, respectively.

Pupils may be asked to find the relationship between variables :b and :c so as to animate the opening and closing of the pages. This can be achieved only if the dragging on the 2d pad is coordinated to design the graph of the linear functional relationship :b=:c/2 between the two variables (see in the next Figure). Here the student’s may initially experiment with freely dragging on the 2d pad. A critical point for the identification of the specific functional relationship concerns the observation of the changes in the graph according to the directions of dragging on the 2d VT. A dragging following a parallel direction the x axis, in particular, changes only the opening of the dihedral angle while a dragging following a parallel direction the y axis changes the direction of the dihedral angle around an axis vertical to the ‘surface’ without changing its measure.

After guessing this relationship pupils may be asked to try to write a procedure for the simulation of opening-closing pages with only one variable. Such a procedure may in the following forms:

to pages : a

up(90)

rr(:a/2)

rect

lr(:a)

rect

end


Or


to pages : a

up(90)

rr(:a)

rect

lr(2*:a)

rect

end


Phase 4: Student's choices to simulate situations that involve turning


In the end of the activity the teacher can devote one more teaching session with students experimenting to simulate situations of their own choice.

Spiral staircase simulation

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Angles in 3d space
  • Spatial visualisation/orientation
  • 3d geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

Description show tooltip help

Students are trying to construct a staircase having as ‘building components’ two geometrical figures: ‘a triangle and a rectangle’. When each stair of the spiral staircase is constructed , the concept of angle between two different figures, each one of which defines a plane, is explored, while when a succession of stairs is construced the focus is upon the notion of angle as change of planes and direction within the 3d space. The 1d variation tool gives students the chance to experiment with a sequence of values for the variables that define not only the size of the geometrical figures but also their place within the 3d space so as to construct a 3d object that represents a staircase. Moreover the ‘building components’, the triangle and the rectangle, can be used in logo programming within recursive procedures forming the base of a structured logo programme.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a spiral staircase simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

The implementation is designed to take place in phases. Initiating from real 3d objects, students are asked to construct a stair of a spiral staircase in MaLT. In order to achieve so students have to identify which are the ‘building components’ of a spiral staircase and to connect them to certain geometrical figures. Having identified that the rectangle and the equilateral triangle are the geometrical figures used in order to construct a stair, students have to search for ways to reconceptualise these geometrical figures in terms that can be explained to 3d turtle through Logo commands and variable procedures. Going on a step further they have to position their ‘building components’ in such a way in 3d space that a 3d stair is constructed.

Finally with the aim of developing a recurrent procedure that constructs a spiral staircase where the number of stairs is each time defined by the user, students explore and experiment with the kind of turn in 3d space needed and the appropriate measure of the relevant outlined angle. The 1d variation tool seems to be extremely helpful in experimenting with the appropriate measure of the turn in 3d space needed through the use of sliders and the sequential change of the respective values.

Theoretical framework show tooltip help

Taking a constructionist approach the spiral staircase simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In the spiral staircase simulation the concept of angle will be explored:

- as a geometric shape, i.e. what different types of angle (e.g. 2d or dihedral angles) look like when visualised in 3d space (e.g. incorporated in 3d simulations);

- as a dynamic amount, indicating both a turn and the result of a turn which can also be represented by a variable;

- as a measure, in angle/turn measurement situations represented by a number.

In spiral staircases the main facets of angle are incorporated within and between geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. triangles and rectangles)

- directed turns between two different geometrical figures, (e.g. in the construction of a stair of a spiral staircase);

- directed turns between lines and planes (e.g. the angular relationship between two successive stairs around the vertical axis).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring facets of the notion of angle in the 3d space

    • angle as a change of direction within the 3d space
    • angles as results of specific turtle turns
    • angles between two different planes in 3d space
    • angle between two geometrical figures, each one of which defines a plane
    • angles as directed turns in functional angular relationships underlying specific geometrical constructions
  • Identifying the mathematical structure of real 3d objects
  • Identifying the mathematical structure of 2d and 3d geometrical figures

  • Identifying the relationships between 2d and 3d representations when using the former to construct simulations of real 3d objects

  • Using variables to express the relations underlying the construction of 3d geometrical objects and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 3d objects and relate them to the abstract geometrical figures
  • Reconceptualising 3d objects in formal mathematical ways
  • Identifying the relation between visual and symbolic representations
  • Capitalising upon intuitions, bridging everyday experience and 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Constructing a stair

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • Construction of rectangles and equilateral triangles
  • Dihedral angle
  • 3d Geometrical figures

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Dihedral angle
  • Turn
  • 3d Space
  • Spatial Visualisation/Orientation

Description show tooltip help

Initiating from real 3d objects, students are asked to construct a stair of a spiral staircase in the 3d geometrical environment of MaLT. In order to achieve so students have to identify which are the ‘building components’ of a spiral staircase and to connect them to certain geometrical figures. The rectangle and the equilateral triangle, which are the ‘building components’ of a spiral staircase, have then to be placed in such a way in 3d space through the use of 3d logo commands and procedures so that the simulation of a 3d stair is contructed.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a stair of a spiral staircase simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

The implementation is designed to take place in phases. Initially students are engaged in searching to identify the geometrical figures used in real 3d spiral staircases. Having identified the building components of a spiral staircase, which are the rectangle and the equilateral triangle, they try to construct them in MaLT through variable procedures. Finally, they are expected to construct a 3d stair simulation by placing the triangle and the rectangle in the appropriate position in relation to each other in 3d space.

Theoretical framework show tooltip help

Taking a constructionist approach the construction of a stair of a spiral staircase is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In constructing a stair the main facets of angle are incorporated within and between the geometrical figures in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles and equilateral triangles)

- directed turns between two different geometrical figures, (e.g. in the construction of stair using a rectangle and an equilateral triangle)

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring the notion of angle within the 3d space

    • Exploring the intrinsic and the extrinsic to a geometrical figure angles, while constructing the building components of a stair, the triangle and the rectangle

    • Exploring the notion of angle between two different geometrical figures (a triangle and a rectangle), each one of which defines a plane

  • Identifying the mathematical structure of real 3d objects and in particular of a stair

  • Understanding the relationship between 2d and 3d representations when using the former to construct simulations of real 3d objects (of a stair)

  • Using variables to express the relations underlying the construction of 3d geometrical objects (of a stair) and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of a stair and connect it with abstract geometrical figures, with rectangles and triangles
  • Reconceptualising a 3d stair in formal mathematical way
  • Identifying the relation between visual and symbolic representations

  • Capitalising upon intuitions, bridging everyday experience and formal geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Having as a stepping off point real 3d spiral staircases students are asked to construct a stair of this kind of staircase in the computational environment of MaLT. Initially students have to investigate which are the ‘building components’ of a stair and connect them to certain geometrical figures. Having identified that the rectangle and the equilateral triangle are the geometrical figures used in order to construct a stair, students try to construct each figure in MaLT. This means that students will have to search for ways to reconceptualise geometrical figures in terms that can be explained to 3d turtle through LOGO commands and variable procedures. Going on a step further they have to place their building components in such a way that a 3d stair is constructed. It follows that students have to explore the kind of turn in 3d space needed and the appropriate angle measure between the two geometrical figures - each one of which defines a plane – so as to construct the simulation of a stair of a spiral staircase.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

3 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Phase 1: Connecting 3d objects with geometrical figures

Initially students are asked to observe various types of spiral staircases either by using concrete 3d models or photos from various sources, e.g. from the internet. Ideally students can visit a nearby building that has a spiral staircase so as to have a more direct experience of it. The teacher informs the students that their final goal is to construct a simulation of a spiral staircase in MaLT. The teacher provokes discussion by asking what geometrical figures can be identified in a spiral staircase and how they are placed in relation to each other. Children are free to express themselves. The aim of these questions is not to come out with definite answers but to arouse students’ interest and to focus their attention in the building components of a spiral staircase and their position in 3d space in relation to each other.

Phase 2: Constructing the building components of a stair

Students are asked to work in groups and construct in MaLT a stair of a spiral staircase. Initially students have to decide which are the basic geometrical figures needed so as to construct a stair. Careful observation of real spiral staircases in the previous phase is rather conducive to the identification of the main ‘building components’, which are the equilateral triangle and the rectangle. Then students will have to construct these ‘building components’ in MaLT. This can be done in various levels of abstraction: with simple commands executed line by line, with small procedures without variables, or with procedures with variables and repeat commands. The teacher goes around the various teams assisting students and giving relative hints. According to the progress of the various teams and at certain points in the course of the activity the teacher can bring discussion at a classroom level to help all the teams proceed with their task. It is important that the various teams develop variable procedure (even without the use of repeat command) because this will afford them flexibility in their experimentation especially in next phases. For instance, the variable procedures developed by students could be of the following type:

to rect :a :b :c

repeat 2 [fd(:a) rt(:c) fd(:b) rt(180-:c)]

end

to tri :b

rt(30)

repeat 3 [fd(:b) rt(120)]

lt(30)

end

Attention should be given in the development of the equilateral triangle procedure, as the measure of the angle defined is not the extrinsic to the figure but the extrinsic one which the turtle outlines as it turns

Phase 3: Constructing a stair

Finally students will have to explore the way the above developed ‘building components’ should be manipulated in order to construct a step. Again this can be done in various levels of abstraction: either manually - executing firstly the rectangle procedure, placing then by simple orders the turtle in the appropriate position and finally executing the triangle procedure - or through a variable procedure. It goes without saying that the teacher should help students with the appropriate hints - taking advantage of the interplay between the visual and the symbolic - to move to more abstract constructions. For instance students may develop the following procedure:

to step1 :a :b :c :d

rect(:a :b :c)

fd(:a)

dp(:d)

tri(:b)

end

Experimentation with the value of the variable: d initiates students into the experimentation with the notion of angle as turn between the two levels defined by the two geometrical figures in 3d space. The use of 1d variation tool could help students in their experimentation giving them the chance to sequentially change in a kinaesthetic way the respective variables and get visual feedback in Turtle Scene.

Constructing a spiral staircase simulation

Identity show tooltip helpexplode

Authors show tooltip help

Giorgos Psycharis, Maria Latsi, Chronis Kynigos

Subject domains show tooltip help

  • Geometry

Topics show tooltip help

  • 3d Geometrical Figures
  • Spatial orientation/visualisation
  • Angle and turn in 3d space

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Angle
  • Turn
  • 3d space
  • Spatial Orientation/Visualisation

Description show tooltip help

Based on the variable procedures they have developed in the previous activity, students try to construct a simulation of a spiral staircase in MaLT. In other words having already developed a variable procedure for each stair of the staircase, they are now experimenting with the aim of developing a recurrent procedure that constructs a spiral staircase where the number of stairs is each time defined by the user.

Rationale show tooltip helpexplode

Based on the use of the available representations and functionalities this part of the pedagogical plan consists of a sequence of tasks for the construction of a spiral staircase simulation. Such a construction in MaLT has been chosen since its construction gives space for pupils engagement in expressing their ideas around the concept of angle in 3d space based on turning and directionality. This context provides a situation that incorporates a dynamic visualization of turning integrating two schemes: turn as body movement in 3d space and turn as number– measure.

In the implementation phase students are initially asked to create a spiral staircase in the 3d geometrical space of MaLT following upon the variable procedures they have already developed. However a critical point of concern while developing a recurrent procedure of a spiral staircase simulation is the turn in space needed before constructing a new step using leftroll or rightroll commands.

Theoretical framework show tooltip help

Taking a constructionist approach the spiral staircase simulation is chosen since it integrates different angle domains (e.g. intersecting, turning, sloping) related to familiar physical angle experiences (e. g. corner, slope and turn) as well as to the main definitions of angle. In the spiral staircase the main facets of angle are incorporated within and between the geometrical objects in 3d space integrating also all the main clusters of physical angle situations that correspond to the everyday concepts of corner, slope and turning (Mitchelmore & White, 2000) and as such are considered as

- directed turns for the construction of 2d geometrical figures (e.g. rectangles and equilateral)

- directed turns between two different geometrical figures, (e.g. (e.g. in the construction of stair using a rectangle and an equilateral triangle);

- directed turns between lines and planes (e.g. the angular relationship between two successive stairs around the vertical axis).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The study of 3d geometrical objects (e.g. angles, turns, properties of 3d geometrical figures) are known to be an obstacle for many students beginning to study 3d geometry. This is an area of mathematics in which students' informal ways of experiencing the physical 3d space around them are excluded by the teaching approaches in the school which tend to focus on static instances of 3d geometrical figures. It is as if there is a lack of an appropriate mathematical language that will engage students in expressing their intuitive assumptions and arguments about the properties of 3d geometrical objects. This prepares them poorly for understanding the general principles of spatial visualisation and thinking.

This pedagogic plan provides an opportunity for students at this transition point in primary as well as lower secondary school to encounter and use 3d geometrical concepts in purposeful and meaningful ways while controlling and measuring the behaviours of geometrical objects in 3d spaces. Considering the teaching of geometry as a terrain in which students can be challenged to mathematize their everyday experiences (that includes move, turning etc.) using the formal power of a programming language, this educational plan is designed to link geometrical, graphical and algebraic aspects of the construction of 3d geometrical figures. The design of the MaLT microworld includes different kinds of representation of 3d spaces and objects enhanced by interactive and dynamic display affordances for the user (e.g. moving agents in 3d space, designing 3d objects, simulating 3d phenomena, dynamic manipulation of geometrical objects).

Theoretical framework show tooltip help

The objects and functionalities provided by the MaLT microworld are designed to behave in mathematically coherent ways. Building geometrical models in the MaLT microworld can provide students with the opportunity to construct and manipulate simulated dynamic representations of 3d geometrical objects and to test hypotheses about their properties.

Our theoretical perspective on learning suggests that interaction with multiple representations of geometrical objects can be a fruitful domain to challenge student’s intuitions and ideas concerning spatial thinking come into play. The joint use of visual and symbolic representational registers (Noss, Healy & Hoyles, 1997) can be considered as a means to connect body movement with geometrical problem solving which might go beyond the simple visual recognition of spatial relations to their expression and further elaboration. Thus, the proposed activities are designed to facilitate experimentation, expression of alternative ideas and personal forms of reasoning.

Population show tooltip helpexplode

School level show tooltip help

Lower secondary.

Age range show tooltip help

12-14 years old.

Population description show tooltip help

This task is intended for students at lower secondary level who have already taught angle at the primary level as an element for the construction of basic geometrical figures (e.g. parallelograms, rectangles, triangles). This means that these students have never studied angle as a specific geometrical concept on its own but just as a component of a geometrical figure.

Student prerequisites show tooltip help

No special programming skills needed, but a familiarisation with basic Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarisation with MaLT environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

Apat from the lesson of informatics -which is taught in the computer laboratory- the teaching of the various subjects in greek secondary schools usually takes place in a typical classroom where students sit in two at desks looking at the front of the classroom where the blackboard and teacher’s desk are placed. The implementation of the pedagogical plan will take place in the computer laboratory equipped with tablet PCs.

Institutional context show tooltip help

This scenario will be implemented at a secondary school in Athens. At the lower secondary level the teaching of mathematics takes place for four teaching sessions (45 minutes) a week. The school time schedule, the content and the curricular goals are determined by the National Curriculum, which suggests the use of computers concerning geometry. However, very few teachers follow these suggestions in their teaching practice for three main reasons:

- computer use for teaching mathematics is not officially part of the curriculum

- schools computer laboratories are usually occupied for the teaching of informatics

- the teacher training concerning the use of computers in the teaching of mathematics is rather limited

As a result the use of computers in mathematics is not concerned with the normal school practice and thus it can be conceived as an innovation.

Socio-cultural context show tooltip help

Students are accustomed to working individually in the traditional classroom where answers are validated by the teacher who is the only authority in classroom.

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of angle in 2d plane by the students:

  • Understanding of the concept of angle and its basic components
  • Understanding the definition of angle as a part of the plane between two rays meeting at their endpoint.

  • Being able to represent an angle graphically and symbolically

  • Identifying the role of angles in basic geometrical figures (e.g. triangles and rectangles)

The main focus of the present pedagogical plan concerns the student's conceptualisation of the concept of angle. We can thus consider the above goals as relevant to our content/epistemological goals concerning students' conceptualisation of the concept in 3d space.

Content-epistemological goals show tooltip help

  • Exploring the notion of angle within the 3d space

    • Exploring the notion of angle as change of planes and direction within the 3d space, while placing the stairs of a spiral staircase in relation to one another

    • Exploring the notion of angle between two geometrical figures, each one of which defines a figure

  • Identifying the mathematical structure of real 3d objects and in particular of spiral staircases

  • Understanding the relationship between 2d (triangles and rectangles) and 3d representations when using the former to construct simulations of real 3d objects and in particular of a spiral staircase

  • Using variables to express the relations underlying the construction of 3d geometrical objects (of a stair) and dynamically manipulate them

Cognitive goals show tooltip help

  • Analysing the structure of 3d objects (e.g. a spiral staircase) and connect it with abstract geometrical figures

  • Reconceptualising a spiral staircase in formal mathematical way

  • Identifying the relation between visual and symbolic representations

  • Capitalising upon intuitions, bridging everyday experience and 3d geometry

Social-affective goals show tooltip help

  • Collaborative group work in problem solving

  • Development of dialogic/argumentative practices

  • Active engagement in exploratory processes

    • Observe

    • Formulate guesses and conjectures

    • Test, verify or reject conjectures

    • Draw conclusions

    • Develop logical arguments to justify conclusions

Instrumental goals show tooltip help

  • Effective use of Logo commands to navigate turtle in the Turtle Scene

  • Understanding, interpreting, debugging and further developing ready-made Logo procedures or the procedures developed by other teams

  • Writing, editing and executing Logo programs with or without variables to construct 2d and 3d geometrical figures

  • Controlling and measuring the behaviour of geometrical objects in 3d space by the use of 1d and 2d Variation Tools

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

With the aim of developing a recurrent procedure that constructs a spiral staircase where the number of stairs is each time defined by the user, students explore and experiment with the kind of turn in 3d space needed and the appropriate measure of the relevant outlined angle. In other words the turtle should change place in relation to the vertical axis by using leftroll ndegrees or rightroll ndegrees commands before constructing each new step. The 1d variation tool seems to be extremely helpful in experimenting with the appropriate measure of the turn in 3d space needed through the use of sliders and the sequential change of the respective values.

Theoretical framework show tooltip help

MaLT is a programmable microworld for the creation and exploration of interactive 3d simulations. These simulations are based on Logo as a programming language to ‘drive’ the turtle in the 3d space. Students will be able to select objects and choose amongst representations allowing dynamic manipulation, programmable behaviors and properties.

The ontological principle underlying the design of MaLT is that all the properties and behaviours of 3d objects within the microworld are based on the notions of direction and position in the 3d space embedded in the turtle’s move which defines a vector. Moreover the turtle’s motion also represents a vector in the sense that the difference between two consecutive vector inclinations represents the (differential) curvature of the trajectory as the length of the vector tends towards zero. This way the Turtle Geometry is considered as Vector Geometry in the 3d space.

As students are engaged in navigating the turtle they gain a sense of the mathematical meanings related to the construction of 3d geometrical objects by a process of hypothesising, experimenting and reflecting on the empirical observation of the graphical feedback on the screen.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex geometric constructions simulating complicated 3d objects.

Work plan show tooltip helpexplode

Setting show tooltip help

The students will be working in pairs, sharing the same tablet PC. The implementation of the scenario will take place during normal school teaching sessions (45 minutes each).

Time show tooltip help

3 teaching sessions.

Actors' roles show tooltip help

Students

  • Experiment with their own constructions in MaLT

  • Express their intuitions and discuss their ideas

  • Formulate and test conjectures/hypothesis

  • Draw conclusions and form arguments

Teacher

  • Supports the use of the provided tools/functionalities of MaLT

  • Prompts students and gives hints

  • Poses questions and brings into the foreground points for consideration

  • Provokes discussion at group and classroom level

  • Moderates discussion at classroom level

  • Facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • Participant observation.

  • Data collection by taking field notes and using audio/video recording

  • Support to teachers and students in the use of MaLT

What to do and how show tooltip help

Students are asked to work in groups in order to create a spiral staircase in the 3d geometrical space of MaLT following upon the variable procedures they have already developed. However in this phase a critical point of concern in order to create a spiral staircase is the turn in space needed before constructing a new step. In other words the turtle should change place in relation to the vertical axis by using leftroll or rightroll commands before reexecuting the ‘step’ procedure. Again this can be done in various levels of abstraction: either manually - executing firstly the ‘step’ procedure and then placing by simple orders the turtle in the appropriate position before reexecuting it - or through a variable procedure. It goes without saying that teacher should help students with the appropriate hints - taking advantage of the interplay between the visual and the symbolic - to move to more abstract constructions.

For instance in the following procedure students can experiment with and observe how the position of the turtle changes in relation to the vertical axis dragging the sliders and changing sequentially the value of turn (:d):

to step2 :a :b :c :d

rect(:a :b :c)

fd(:a)

dp(90)

tri(:b)

up(90)

lr(:d)

end

After defining the position and orientation of the turtle in 3d space before drawing a new step, students are prompted to complete their construction and experimentation with a recurrent procedure that repeats as many times as defined the ‘step’ procedure.

For Instance:

to step3 :a :b :c

rect(:a :b :c)

fd(:a)

dp(90)

tri(:b)

up(90)

lr(60)

end

to ladder :a :b :c :e

up(90)

repeat :e [step(:a :b :c)]

end

Basic sterometrical objects

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Properties of geometrical objects

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