Navigation in Geographical Space (familiar)

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • Functions
  • 3d representations
  • Geographical coordinates
  • Sperical coordinates

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Function
  • Geographical coordinates
  • Spherical coordinates
  • 3d representations

Description show tooltip help

Cruislet environment is a microworld designed to provide learners with the ability to be involved in exploratory activities focusing on the use of vectors navigating in 3d large scale spaces. In the number of tasks that are included in this hierarchical pedagogical plan the experimentation with the Cruislet environment focuses on the study of the development of student’s conceptions concerning the mathematically driven navigations in virtual 3-d geographical spaces. The general mathematical issues are those underlying the use of analytic and/or vector-differential geometry, including functions, co-variation and rate of change. These mathematics are integrated with geo-spatial representations and information, providing opportunities for processes of mathematisation of geographical space

Rationale show tooltip helpexplode

The notion of navigational mathematics is used to describe the mathematical concepts that are embedded and the mathematical abilities the development of which is supported within the Cruislet microworld. Particularly, we could distinguish two main components concerning the notion of navigation, the process of moving around and the actual space within which this action take place.

Specifically, the process of navigation is possible through gradual movements, displacements. Each of these displacements constitutes an action which is defined by its previous condition (position and direction) and is represented by a vector. Consequently, a displacement is defined by employing either

–        a geographical latitude-longintude-height coordinate system by setting the end point of the vector of the displacement or

–        a spherical coordinate system by setting the direction and the length of the vector of displacement (θ,φ, r) .

In the case of using the geographical system, the end position of the vector of the displacement is defined  by the geographical coordinates lat-long-height. A geographical coordinate system expresses every location on Earth by two of the three coordinates of a spherical coordinate system which is aligned with the spin axis of Earth. Therefore, the geographical coordinates are invented for spherical geometry but are projected on orthogonal geographical planes because of maps. Specifically, Latitude (Lat.) is the angle of the intersection of the equator and the perpedicular to the tangent at an arbitrary point on Earth. Longitude(Long.) is the angle east or west between an arbitrary point on Earth and the meridian of Greenwich (UK) which is the international zero-longitude point (longitude=0 degrees). By combining these two angles, the position of any location on Earth can be specified. For example, the city of Sparti has a latitude of 37.2° North, and a longitude of 22.3° East (37.2° N 22.3° E). Traditionally, degrees have been divided into minutes ( ′ ) and seconds ( ″ ). There are several formats for degrees, all of them appearing in Lat.-Long :

DMS Degree:Minute:Second (e.g. 49:30:00-123:30:00)

DD Decimal Degree (e.g. 49.5000-123.5000), generally with 4 decimal numbers.

In the Cruislet micoworld the movement of an airplane is defined in DD format. To completely specify a location above Earth, one has to also specify the height i.e. the distance from the earth's surface. Thus, Height is the vertical distance to Earth below, (the height above mean sea level).

On the other hand, the displacement could be defined using a spherical coordinate system. In spherical coordinate system, spherical coordinates define a vector in terms of its length (R), an angle between the vector and the z-axis (φ) and the angle between the vectors projection onto the xy plane and the x-axis (θ). Correspondingly, the displacement/movement is defined by the definition of 3 variables, the measure and the two angles which determine the length of the displacement and its direction.

The terrain scene of Cruislet within which this process of navigation is actually take place is actually a Map Projection of Greece. The term "map projection" actually means any mathematical transformation of the globe onto a flat surface. Some distortions of conformality, distance, direction, scale, and area always result from this process. Some types of projection minimize distortions in some of these properties at the expense of maximizing errors in others. In Cruislet the Transverse Mercator Projection is used. This projection is actually a cylindrical projection, meaning that the globe is encircled by an imaginary cylinder touching at the equator, and the earth is projected onto the cylinder. This is a conformal projection, meaning that angles and small shapes on the globe project as the same angles or shapes on the map. This type of projection has been selected because it is  very accurate in narrow zone with larger north-south than east-west extent such as the form of the map of Greece. Moreover, this type of projection is often used for marine navigation.

Having defined the context of the geometrical space within which the navigation process take place, the variation of the vector of the displacement could become the subject of the study. In particular, the variation of vector variables which define the vector of the displacement, either the lat, long and height in geographical system or the angles (φ, θ) and its length (R) in spherical coordinate system, imply the process of navigation upon the 3d map projection of Greece.  Mathematical concepts such as the notion of function as covariation or the interrelation between the geographical and the spherical coordinate system could be study within the mathematical context of Cruislet microworld.

Theoretical framework show tooltip help

Cruislet was conceived as a digital medium for mathematically driven navigations in virtual 3-d geographical spaces. Users can view avatar positions and define their displacements by employing either a Cartesian lat-long-height system or a vector-differential (φ,θ,ρ) system where ρ is the length of the vector of displacement. The users also have at their disposal:

(a)   a geographical (non-mathematical) option of defining displacements by naming locations.

(b)   a programming language (Logo) which can be used for programming multiple or relative displacements.

Cruislet is primarily a navigational medium but it is also constructionist (Papert, 1980, Harel & Papert, 1991, Kafai &  Resnick, 1996) since avatar trips can be constructed by using as well as displacement rules in 3d geometrical space (Papert, 1980, Harel & Papert, 1991, Kafai &  Resnick, 1996). It is designed to provide opportunities for learners to engage in expression of mathematical ideas through meaningful formalism by means of programming and interdependent representations of Cartesian and Vector-differential geometrical systems.

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, 1999) 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. With respect to mathematical content, the approach is to identify conceptual fields (Vergaud, 1990) which with the use of this kind of media becomes rich in the potential to generate mathematical meanings, irrespective of the ways in which they might be structured (or fragmented) in the mathematics curricula.

ETL work is 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 artifact in ways that lead them to diverge from curriculum mathematics. Work and play with cruislet is based on the idea of instrumentation and instrumentalization since displacement rules can be questioned and re-defined by learners resulting in a variety of artefacts (see Verillon and Rabardel, 1995 for these terms) which gives rise to a dialectic by which learner and artifact 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).

The constructionist environments designed at ETL 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). In such settings learners are engaged in constructing public entities (constructions) implying an explicit appreciation of the relationships between mathematical objects within any situation (i.e. a mathematical model of the situation). In the case of Cruislet, learner constructions are avatar trips as well as the rules of displacement. The mathematics are those underlying the use of analytic and/or vector-differential geometry, including functions, co-variation and rate of change. However, these mathematics are integrated with geo-spatial representations and information, providing opportunities for processes of mathematisation of geographical space.

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 argumentation in social contexts. Displacing avatars and articulating rules of and relationships between the displacements 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 representations. 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), dynamic manipulation (such as Dynamic Geometry Environments), simulations, information handling and geographical  systems.

In designing Cruislet we wanted to integrate programming, mathematical and geographical concepts, relations and representations.

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 Cruislet, a vector-differential geometrical system co-exists with a Cartesian-geographical one in an inter-dependent way. 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.

Moreover, navigations in virtual 3d geographical spaces within Cruislet could be conceived as game play simulations. There is a growing interest about the ways in which game–based learning environments facilitate new ways of learning (Gee, 2003). The key feature of this approach is that games can provide a context for the development of valuable skills (Kirriemuir and McFarlane et al., 2004) in the transitional stage between intuitions (informal) and formal mathematics. Using games with an appropriate set of tasks and pedagogy, students can be engaged in exploration, problem solving, rule-based thinking and other forms of mathematical thinking (Goldstein et al, 2001; Mor et al, 2004). From this point of view, the process of building game play activities involving navigation within the 3d representational space can be seen as the design of the terrain within which instrumentation/instrumentalisation processes may take place by student’s interactions with the microworld and the mathematical concepts and rules embedded in it.

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 & Yiannoutsou, 2002, Yiannoutsou & Kynigos, 2004). This educational plan is designed to bring in the foreground issues concerning the mathematical nature of 3d navigation and how interactivity, control and experimentation in virtual reality microworlds can be a versatile vehicle for enhancing mathematically driven navigation, orientation and spatial visualization.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

Mathematics: Familiarity with the concept of function. 

Programming: No special programming skills needed, but a basic understanding of (basic) Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • The exploration of the concept of function as covariation using the geographical coordinates as a system of reference.
  • The development of the notions of dependency between  the positions of two airplanes.
  • The study of the existence of a rate of change of relative displacements on the 3d space.
  • The development of the notions of dependency, covariation and the rate of change through visual, numeric and symbolic representations. 
  • The development of the concept of vector as displacement using a geographical system of reference.
  • The exploration and comparison of geographical and spherical coordinates.
  • The correlation of geographical and mathematical concepts.

 

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with expressing symbolically using Logo.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This pedagogical plan involves the students' engagment to a a number of activities, focusing on the development of student’s conceptions concerning the mathematically driven navigations in virtual 3-d geographical spaces. The general mathematical issues are those underlying the use of analytic and/or vector-differential geometry, including functions, co-variation and rate of change. These mathematics are integrated with geo-spatial representations and information, providing opportunities for processes of mathematisation of geographical space. 

Theoretical framework show tooltip help

Meaningful formalism, constructionism, half-baked microworlds as well as the idea of instrumentation and instrumentalization are the theoretical constructs concerning the Cruislet microworld. In particular, Cruislet is designed to provide students for instrumentalization through constructionist activity in the context of half-baked microworlds (Kynigos, 1992 and in press). In particular we use the idea of half – baked games. These are games that incorporate an interesting game idea, but they are incomplete by design in order to poke students to finish or change their rules. Thus students play them, and change them and thus adopt both roles of player and designer of the game. From this point of view the work and play with Cruislet is based on the idea of instrumentation and instrumentalization (Guin &Trouche, 1999) since displacement rules can be questioned and re-defined by learners resulting in a variety of artefacts.

 

Cruislet DDA

The Cruislet environment is a digital medium based on GIS (Geographic Information Systems) technology that incorporates a Logo programming language. It is designed for mathematically driven navigations in virtual 3d geographical spaces and is comprised of two interdependent representational systems for defining a displacement in 3d space, a spherical coordinate and a geographical coordinate system.

Terain scene [Component]

The terrain scene is consisted of a geographic map of Greece which could be seen to an analysis ranged from a satellite view up to 1:50.000. The scene can be seen from two different windows, which are two map viewers. The first window contain is the 3d representation of the map whilst the second contains its 2d representation. These windows could be simultaneously displayed by splitting the screen (the first above the second).

Avatar Tab [Component]

Avatar tab consists of:

  1. Avatar properties: The user is able to create, delete or define the properties of an avatar.
  2. Systems of reference: The user is able to choose among different representations of geographical and spherical coordinates and displace the airplane in 3D space. In particular, the geographical coordinates are defined either by editing the latitude, longitude and height values of a place, or by selecting among a set of predefined cities and displace the airplane there. Spherical coordinates can be defined either by editing the angles (θ, φ) and the magnitude or the displacement (R), or by using the 3D representation of shperical coordinates (3D controller). 
  3. Camera properties: The user  is able to change the camera view, resulting in displaying the map from a different viewpoint.
Contents Tab [Component]

The Contents tab contain a thematic browser with the configure tree of the layers’ visibility of the geo – coded information. The real geocoded information contained in the terrain scene is descriptive information of the various land features and geographic entities.
Using content tab, the user is able to display specific geo-coded information at the terrain scene (map) such as rivers, mountains, cities, roads, airports etc.

 

Logo Tab [Component]

The Logo editor enables user to edit and run Logo programs and thus define the behavior of objects.

 

Work plan show tooltip helpexplode

Function as covariation in Geographical Space

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • Functions
  • 3d representations
  • Geographical coordinates
  • Sperical coordinates

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Function
  • Geographical coordinates
  • Spherical coordinates
  • 3d representations

Description show tooltip help

In the number of tasks that are included in this hierarchical pedagogical plan students' engagement focuses on the exploration of the concept of function as covariation using the geographical coordinates as a system of reference. In particular, students are actually asked to study the existence of a rate of change of relative displacements of points on the plane, i.e. when aeroplane A moves for a length of x and direction a then aeroplane B moves for a length of a function of x and the same direction. Another simpler version  is  linear functional relationships between  changes in  lat -long coordinate values. 

 

Rationale show tooltip helpexplode

Functions are a central feature of mathematics curricula, both past and present. Many research studies indicate students’ difficulty in understanding the concept of functions. This difficulty comes from a) the static media used to represent the concept, b) the introduction of function mainly as a mapping between sets in conventional curricula, c) the use of formalisation and function graphs as the only representations. With digital media, students can dynamically manipulate informal representations of function defined as co-variation and rate of change, which is an interesting and powerful mathematical concept. Tall(1996) points out a fundamental fault-line in “calculus” courses which attempt to build on formal definitions and theorems from the beginning. Moreover, he suggests that enactive sensations of moving objects may give a sense that “continuous” change implies the existence of a “rate of change”, in the sense of relating the theoretically different formal definitions of continuity and differentiability. The enactive experiences provide an intuitive basis for elementary calculus built with numeric, symbolic and visual representations.

Cruislet is a navigational medium which provide learners with the ability to be involved in exploratory activities navigating in 3d large scale spaces. These activities involve the exploration of the concept of function as covariation using the geographical coordinates as a system of reference. The focus is on the study of the existence of a rate of change of relative displacements of airplanes on the space.

 

Theoretical framework show tooltip help

A number of research studies suggest that students of all grades, even undergraduate students, have difficulties modeling functional relationships of situations involving the rate of change of one variable as it continuously varies in a dependent relationship with another variable (Carlson et all, 2002; Carlson, 1998, Monk & Nemirovsky, 1994). This ability is essential for interpreting models of dynamic events and foundational for understanding major concepts of calculus and differential equations.  On the other hand, the VisualMath curriculum (Yerushalmy & Shternberg, 2001) is an a example of a function based curriculum that involves the moving across multiple views of symbols, graphs, and functions. VisualMath uses specially designed software environments such as simulations' software, or other modelling tools that include dynamic forms of representations of computational processes. Yerushalmy (2004) suggest that such emphasize on modeling offers students means and tools to reason about differences and variations (rate of change). Moreover, Kaput (1993) and Nemirovski (1996) using computers simulations study aspects of calculus at an earlier stage. These simulations (MBL tools), allow the study of change and the ways it relates to the qualities of the situation. In their study Nemirovsky, Kaput and Roschelle (1998) show that young children can use the rate of change as a way to explore functional understanding. In studying the process of understanding of dynamic functional relationships, Thompson (1994) has suggested that the concept of rate is foundational.

Confrey and Smith (1994) choose the concept of rate of change as an entry to thinking about functions. They introduce introduce two general approaches to creating and conceptualizing functional relationships, a correspondence and a covariation approach. They suggest that “a covariational approach to functions makes the rate of change concept more visible and at the same time, more critical (p. 138). They explicate a notion of covariation that entails moving between successive values of one variable and coordinating this with moving between corresponding successive values of another variable.

Moreover, Carlson, Larsen and Jacobs (2001) stress on  the importance of covariational resoning as an important ability for interpreting, describing and representing the behavior of dynamic function event.. They consider covariational reasoning to be the cognitive ability involved in coordinating images of two varying quantities and attending to the ways in which they change in relation to each other. On the same line, Saldanha and Thompson (1998) introduced a theory of developmental images of covariation. In particular, they considered possible imagistic foundations for someone’s ability to see covariation. Carlson et all (2001) in their study exploring the role of covariational reasoning in the development of the concepts of limit and accumulation, suggest a framework including five categories of mental actions of covariational reasoning:

  1. An image of two variables changing simultaneously
  2. A loosely coordinated image of how the variables are changing with respect to each other
  3. An image of an amount of change of one variable while considering changes in discrete amounts of the other variable
  4. An image of the average rate-of-change of the function with uniform increments of change in the input variable
  5. An image of the instantaneous rate of change of the function with continuous changes in the independent variable for the entire domain of the function

 The proposed covariation framework contains five distinct developmental levels of mental actions.  Using this particular framework we will try to classify student’s covariational reasoning while studying navigation within the context of Cruislet microworld. In the number of tasks included in the pedagogical plan we consider navigation as a dynamic function event. The function’s independent variable is the geographical coordinates of the position of the first airplane, which student are asked to navigate, while the dependent variable is the geographical coordinates of the position of the second airplane.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

Mathematics: Familiarity with the concept of function. 

Programming: No special programming skills needed, but a basic understanding of (basic) Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • The exploration of the concept of function as covariation using the geographical coordinates as a system of reference.
  • The development of the notions of dependency between  the positions of two airplanes.
  • The study of the existence of a rate of change of relative displacements on the 3d space.
  • The development of the notions of dependency, covariation and the rate of change through visual, numeric and symbolic representations. 
  • The development of the concept of vector as displacement using a geographical system of reference.
  • The exploration and comparison of geographical and spherical coordinates.
  • The correlation of geographical and mathematical concepts.

 

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with expressing symbolically using Logo.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This pedagogical plan involves the students' engagment to a a number of activities, focusing on the development of student’s conceptions concerning the mathematically driven navigations in virtual 3-d geographical spaces. The general mathematical issues are those underlying the use of analytic and/or vector-differential geometry, including functions, co-variation and rate of change. These mathematics are integrated with geo-spatial representations and information, providing opportunities for processes of mathematisation of geographical space. 

Theoretical framework show tooltip help

Meaningful formalism, constructionism, half-baked microworlds as well as the idea of instrumentation and instrumentalization are the theoretical constructs concerning the Cruislet microworld. In particular, Cruislet is designed to provide students for instrumentalization through constructionist activity in the context of half-baked microworlds (Kynigos, 1992 and in press). In particular we use the idea of half – baked games. These are games that incorporate an interesting game idea, but they are incomplete by design in order to poke students to finish or change their rules. Thus students play them, and change them and thus adopt both roles of player and designer of the game. From this point of view the work and play with Cruislet is based on the idea of instrumentation and instrumentalization (Guin &Trouche, 1999) since displacement rules can be questioned and re-defined by learners resulting in a variety of artefacts.

 

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Learning to fly (Familiarization)

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Geographical coordinates
  • Spherical coordinates
  • Logo programming language

Description show tooltip help

The task is consider as introductory to the Cruislet environment and the provided representations. The task is divided in three faces each one aims to capture different ways of specifying a position in space according to the representations.

  • Geographical coordinates
  • Spherical coordinates
  • Logo programming language

Within this task, students become familiar with geographical coordinates (lattitude, longitude) and the third coordinate (height) and thus come in contact with 3d coordinates as a way to specify a position in space. They also become familiar with editing and running basic Logo commands in Cruslet environment.

 

Rationale show tooltip helpexplode

In this task students get familiarized with the functionalities of the software. In paticular, they will asked to experiment with the creation and the displacement of an avatar either by using the Avatar Tab functionalities or by editing Logo commands. 

Theoretical framework show tooltip help

Following the constructionist perpspective students are encouraged to experiment with the software and explore its functionalities. Moreover, reflecting on their actions they encouraged to make links between the provided multiple representations

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

Mathematics: Familiarity with the concept of function. 

Programming: No special programming skills needed, but a basic understanding of (basic) Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • Come in contact with geographical and spherical coordinates as a way to specify a position in 3D space.
  • Associate the two systems of reference.

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with programming using Logo.
  • Associate airplanes displacement with variation in geographical coordinates or/and spherical coordinates.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Cruislet

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

General description show tooltip help

In each teaching hour the teacher give students the corresponding worksheets.

 

Worksheet 1: Familiarization [Resource for students]

The activities engage students with geographical and spherical coordinates as a way to specify a position in space. There is also an activity where students come in contact with basic Logo commands for displacing the airplane, apart from using the representations at Avatar tab. 

 

Worksheet 2: Mapping Greece [Resource for students]

There are several activities where students get accostumed in using both the map and the systems of reference for displacing the airplane in specific places. 

 

Worksheet 3: The instruments are broken [Resource for students]

These activities engage students in comparing the two systems of reference, as they are asked to use the geographical coordinate system to make a particular flight and spherical coordinates to make another flight and then compare these two flights. The last activity engage students in using Logo to make these flights.

 

Worksheet 4: Take - off and Landing [Resource for students]

Students engage in taking-off and landing the airplane using either the avatar tab (define geographical or spherical coordinates to displace the airplane) or Logo tab (define multiple displacements through Logo commands).

 

Students fill in the worksheets, write down notes, create Logo files. [Resource produced by students]

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Time show tooltip help

8 teaching hours (Can be restricted to 2 hours if there is no time available. In this case, only the first worksheet should be done.)

Actors' roles show tooltip help

Teacher


Introduce the environment to students by demonstrating some basic functionalities:

  • create an airplane and move it to another place by changing the geographical and spherical coordinates,
  • demonstrate functionalities of the environment (e.g. using the camera, using content tab, etc.)
  • give some examples of Logo commands of Cruislet and shows how commands are edited and executed,
  • support students with their experimentation,
  • ask questions in order to understand students' strategies while working on the tasks.


Researcher

  • collect data using audio / video recording
  • collect background data, such as students’ notes on papers
  • write down observational notes
  • support teacher and students at the technical level

What to do and how show tooltip help

Demonstration of Cruislet (1 hour)

At the beginning, the teacher use a projector to demonstrate the main functionalities of the Cruislet environment. To be more specific:

  • create and displace the airplane using geographical and spherical coordinates.
  • use the camera properties to change the viewpoint of the terrain scene (map).
  • show the way Logo commands are created (using the 'export' button in avatar tab), edited and executed.

The demonstration of the environment is proposed to be followed by another teaching hour where the worksheet 1 will be given to students. In this way students can experiment with what is demostrated.

In the following hours the corresponding worksheets are given to each team and the teacher prompt students to cooperate in order to fill in them. Each worksheet is meant to get through during a teaching hour or more (if necessary). 

Worksheet 1: Familiarization (1 hour)

Students create their own avatar (airplane) using the Avatar tab. By default, the avatar is placed upon sea near the island of Crete. At the beggining, students have to move their airplane to a destination of their choice using either the geographical or/and spherical system of reference.

During their experimentation students come in contact with:

  • position and direction properties that define the position and direction of the airplane. In this way, students come in contact with geographical coordinates as a way to specify a position in space, and shperical coordinates as a way to define the direction and the the magnitude of the displacement.
  • geo-coded information provided by the environment in the Content Tab (e.g. in order to define specific coordinates such as the airport of Athens). In this way they associate geographical coordinates with the absolute system of reference that we use to specify coordinates of earth.
  • the systems of reference and their interdepenence.

Worksheet 2: Mapping Greece (1 hour)

Students experiment with the environment while trying to accomplish specific goals in each activity. In this way  students get accostumed in using both the map and the systems of reference for displacing the airplane in specific places. Each activity aims to capture different aspects of the interdependence between the use of the systems of reference and the navigation in geographical space.

Worksheet 3: The instruments are broken (1 hour)

The teacher explain the activity at the students. These activities engage students in comparing the two systems of reference, as they are asked to use the geographical coordinate system to make a particular flight and spherical coordinates to make another flight and then compare these two flights. The teacher may propose to students to accomplish these flights either cooperatively (both students of the team work together), or each student of the team choose a flight and work on its own. If the team prefer to work according to the second way, the teacher prompts each student of the team to rationalize his/her desicion in choosing this system of reference.
The last activity engage students in using Logo to make these flights. In this way students come in contact in using Logo procedures for accomplishing multiple displacements. The teacher help them if necessary.

Worksheet 4: Taking - off and Landing (1 hour)

A conversation is taking place in the classroom for about 5 minutes about the take-off and landing of the airplanes. After that the teacher prompt students to choose a system of reference and try to take-off and land the airplane.At the Logo activity, the teacher provide help with Logo syntax if necessary.

Conversation at the classroom (2 hours)

After completing the worksheets (at the last 2 hours), a conversation is taking place between the teacher and the whole class. This aims to capture students conceptions about geographical and spherical coordinates and the ways these are internalised by students or the possible difficulties students confronted when utilized them to navigate the airplanes. They may also talk about Logo programming, possible difficulties that may emerged, etc.

Process documentation show tooltip help

  • Audio and screen capture recordings of each team.
  • Observation sheets and researcher's notes.
  • Worksheets by students.
  • Logo files created by students.

 

Airplanes' chase

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • functions
  • geographical coordinates
  • navigation in 3d space

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Function
  • Geographical coordinates
  • Spherical coordinates
  • 3d representations

Description show tooltip help

In the tasks included here, students are actually engage with the  study of the existence of a rate of change of the displacements of the airplanes which are defined in the geographical coordinate system. In particular the dispacements of two airplanes are relative according to a linear function. This function will be hidden and the students will have to guess it at the first phase of the activity based on repeated moves of aeroplane A and observations of the relative positions and moves of planes A and B. At the second phase, the students will be able to change the function of relative motion and play games with objectives they may define for themselves such as move plane A from Athens to Thessaloniki and plane B from Athens to Rhodes and then to Thessaloniki in the same time period.

Rationale show tooltip helpexplode

We base our scenario on the idea of relative displacements of points in space. Students interacting with these tasks explore the impact of the variation of the geographical coordinates to the displacement of an airplane and study the relevance between the positions of two airplanes, generating meanings of the notions of dependency, covariation and the rate of change. 

Theoretical framework show tooltip help

This scenario is based on the idea of half – baked games, an idea taken from microworld design (Kynigos, 2001). These are games that incorporate an interesting game idea, but they are incomplete by design in order to encourage students to change their rules. Students play and change them and thus adopt both roles of player and designer of the game (Kafai, 2006). 

Initialy, students are asked to study the relation between the two airplanes, the rate of change of their displacements and consequently find the linear function (decode the rule of the game). In order to decode "the rule of the game", they should give various values to coordinates (Lat, Long, Height) that define the position of the first plane.They will be encouraged to communicate their observations about the position of the second plane to each other and form conjectures about the relationship between the positions of the two airplanes. 

At the second phase students are encouraged to build their own rule of the game by changing the function of the relative displacements of the two airplanes.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

Mathematics: Familiarity with the concept of function. 

Programming: No special programming skills needed, but a basic understanding of (basic) Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • The exploration of the concept of function as covariation using the geographical coordinates as a system of reference.
  • The development of the notions of dependency between  the positions of two airplanes.
  • The study of the existence of a rate of change of relative displacements on the 3d space.
  • The development of the notions of dependency, covariation and the rate of change through visual, numeric and symbolic representations. 
  • The development of the concept of vector as displacement using a geographical system of reference.
  • The exploration and comparison of geographical and spherical coordinates.
  • The correlation of geographical and mathematical concepts.

 

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with expressing symbolically using Logo.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This pedagogical plan involves the students' engagment to a a number of activities, focusing on the development of student’s conceptions concerning the mathematically driven navigations in virtual 3-d geographical spaces. The general mathematical issues are those underlying the use of analytic and/or vector-differential geometry, including functions, co-variation and rate of change. These mathematics are integrated with geo-spatial representations and information, providing opportunities for processes of mathematisation of geographical space. 

Theoretical framework show tooltip help

Meaningful formalism, constructionism, half-baked microworlds as well as the idea of instrumentation and instrumentalization are the theoretical constructs concerning the Cruislet microworld. In particular, Cruislet is designed to provide students for instrumentalization through constructionist activity in the context of half-baked microworlds (Kynigos, 1992 and in press). In particular we use the idea of half – baked games. These are games that incorporate an interesting game idea, but they are incomplete by design in order to poke students to finish or change their rules. Thus students play them, and change them and thus adopt both roles of player and designer of the game. From this point of view the work and play with Cruislet is based on the idea of instrumentation and instrumentalization (Guin &Trouche, 1999) since displacement rules can be questioned and re-defined by learners resulting in a variety of artefacts.

 

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Time show tooltip help

21 teaching hours (Rather flexible for the teacher, as some of the activities can be restricted to less hours if there is no time available.)

Actors' roles show tooltip help

Avoid the spy

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • functions
  • geographical coordinates
  • navigation in 3d space

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Function
  • Geographical coordinates
  • Spherical coordinates
  • 3d representations

Description show tooltip help

The task is based on the idea of the "Guess my function" game, in order to provoke children to discuss, compare and experiment with dependence relations such as linear functions. In particular, students are actually asked to study the existence of a rate of change of the displacements of the airplanes which are defined in the geographical coordinate system.

Rationale show tooltip helpexplode

 In this task students are encouraged to experiment with the displacements of the two airplanes by varying the geographical coordinates of their new positions. Reflecting on their actions they encouraged to explore the rate of change of these positions and formulate the function that defines this dependent relationship.

Theoretical framework show tooltip help

Based on the principal construct of half-baked microworlds, students in this tasks incorporate they idea of rate of change between the positions of the two airplanes and the concept of function that relates their displacements on the space.

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

Mathematics: Familiarity with the concept of function. 

Programming: No special programming skills needed, but a basic understanding of (basic) Logo commands and use of variables in Logo procedures.

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • The exploration of the concept of function as covariation using the geographical coordinates as a system of reference.
  • The development of the notions of dependency between  the positions of two airplanes.
  • The study of the existence of a rate of change of relative displacements on the 3d space.
  • The development of the notions of dependency, covariation and the rate of change through visual, numeric and symbolic representations. 

 

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with expressing symbolically using Logo.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Logo program

A predefined Logo program is used. The procedures included are a black box to students and students experiment with the result of the execution of these procedures. 

The 'Radar' program

to begin
createavatar("|white| 37.94 23.94 5000 "|Plane 1|)
createavatar("|red| 37.89 23.92 5000 "|Plane 2|)
activateavatar("|white|)
setupcamera(15000 0 -87 -41 0)
end

to radar :a :b :c
activateavatar("|white|)
setpos(:a :b :c)
wait(1)
activateavatar("|red|)
setpos(:a-0.1 :b-0.05 :c-2500)
print("Συντεταγμένες oppos())
if and(and(and(:a>40.73 :a<40.74) and(:b>22.99 :b<23.1)) :c<3001)
[print("Ξέφυγες! ) removeavatar("|red|)]
setupcamera(25000 0 -87 -41 0)
end

 

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

Worksheet 5 [Resource for students]

 The worksheet contains the rules of the 'Guess my function' game.

Completed worksheet [Resource produced by students]

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Time show tooltip help

2 teaching hours

Actors' roles show tooltip help

Teacher

  • explains the task
  • provides technical support if necessary
  • prompts students to experiment with the environment
  • ask questions in order to understand students' strategies for achieving the goal of the game

Researcher

  • collect data using audio / video recording
  • collect background data, such as students’ notes on papers
  • write down observational notes
  • support teacher and strudents at the technical level

What to do and how show tooltip help

In this task students' engagement focuses on the exploration of the concept of function as covariation using the geographical coordinates as a system of reference. In particular, students have to explore linear functional relationships between  changes in  lat -long - height coordinate values, between the positions of two airplanes. This function will be hidden and the students will have to guess it based on repeated moves of first airplane and observations of the relative displacements and positions of the two airplanes.

The activity is based on the "Guess my function" game, in order to provoke children to discuss, compare and experiment with dependence relations such as functions. The game has been used as an introduction to functions and to formal algebraic notation as it provokes a discussion of algebraic nature through using it.

The object of the game

The activity could be conceived as a game as students move the airplanes having a particular goal. The object of the game is to land the second airplane at Thessaloniki, where the second airplane (the spy) disappear and the first is getting free from the espial. In order to do this they must find the dependence relationship and move the first airplane in a position where the second airplane would be at Thessaloniki.
 

The hidden function 

The relationship is that the latitude of the second plane’s position is equal to the latitude of first plane’s position minus 0.05, the longitude of the second plane’s position is equal to the longitude of first plane’s position plus 0.05 and the second plane flies 1000 meters under the first plane. For instance,

if the first airplane is at (37   28   10000) →  (x y z)

the second airplane will be at (x-0.05 y+0.05 z-1000) → (36.95 28.95 9000).

Students must find the dependence relation by moving the first airplane and see what happens to the other one.

There is a Logo procedure (named flight) that determines this relation but will be a black box to students. They will move the first airplane using this procedure and the environment will give feedback about second airplane’s position. For example, if students write:

flight(37 28 10000) → define first plane’s position (Lat Long Height)

they will get the answer:

Second airplane's position (36.95 28.95 9000)

Decoding the rule (2 hours)

In order to decode the rule, students are experimenting by giving various values to the coordinates (Lat, Long, Height) that define the position of the first plane, they communicate their observations about the position of the second plane and they form conjectures about the relationship between planes’ positions (figure). The correlation of airplanes' position could be possibly written by students in a piece of paper that will be available. For instanse they might make a table with the values in order to compare them and derive the relationship underline the covariation of the coordinates' values.  

With this activity students get involved with sevaral concepts including, navigation in 3d space, geospatial concepts and the concept of function in two ways:

i) function as a local dependency relationship as the coordinates of the second airplane depends on the coordinates of the first airplane and

ii) function as a dynamic process of co-variation as moving the first airplane between successive positions cause the second plane to move between corresponding successive values.

Regarding navigation in 3d space and geospatial concepts,  students experiment with:

  • position properties (Lat, Long, Height) that define the position of the airplane and are related to geographical coordinates. In this way, students come in contact with geographical coordinates as a way to specify a position in space, and experiment giving several values to them.
  • geo-coded information provided by the environment. For instanse at the beggining of their experimentation they have to move the first airplane and experiment by giving several values to the coordinates. As they are possibly not familiar with coordinates of Greece they might choose to use coordinates that are not included in the border lines of Greece. The environment will give them feedback that the movement is not allowded and they will have to make conjectures about this restriction. In this way they associate geographical coordinates with the absolute system of reference that we use to specify coordinates of earth and in particular the coordinates of Greece.

Process documentation show tooltip help

  • Audio and screen capture recordings of each team.
  • Observation sheets and researcher's notes.
  • Worksheets by students.

 

Create rules for the chase

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • functions
  • geographical coordinates
  • navigation in 3d space

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Function
  • Geographical coordinates
  • Spherical coordinates
  • 3d representations

Description show tooltip help

In this task students actually have to formulate another rule and change function’s relations of planes coordinates by intervening in the Logo code and by creating through this process another game to challenge another team. The process of finding another rule, alternate the game and challenge others could engage students with mathematical and geographical concepts as well as with Logo while experimenting with the environment. While students reflect on rules, engage in the process of instrumentalisation (Guin &Trouche, 1999), since displacement rules can be questioned and re-defined by them resulting in a variety of artefacts, which are actually the different games they create.

 

Rationale show tooltip helpexplode

In this task students are encouraged to construct their own dependent relationship between the displacements of the two airplanes. They are motivated to build a complex relationship in order to challenge their classmates. Moreover, they have to explore the provided possibilities taking into account the actual size of the map and the range of the geographical coordinates within which the airplanes should be positioned. Consequently, the students are encouraged to build their own function reflecting on these limitations of the software.

Theoretical framework show tooltip help

Extending the idea of half-baked microworlds, students are encouraged to reach instrumentalization through constructionist activity and actually build their own microworld.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

There were 12 students in the classroom (6 pairs of two).

Computer use: The students were not accustomed in using computers for doing mathematics, but they were familiar with computers and liked using them, as almost the whole class particpated in the computer class (available as a course to choose at this school level). 

Geographical and spherical coordinates: None of the students had previous knowledge or experience with spherical coordinates and only 4 of them told us that they were familiar with geographical coordinates from the geography lesson.

Logo programming language: Some of the students were familiar with the basic Logo commands (movement of the turtle, such as front, right, etc.) but all of the students were not experienced in using programming languages.

Map software: Only few were familiar with such a computational environment and specifically Google Earth.

3D environments / Games: Almost all of the students were playing computer games and most of them were familiar with 3D game environments.

Student prerequisites show tooltip help

 

Mathematics: Familiarity with the concept of function. 

Cruislet environment: Students must be well familiarized with the environment and the Logo programming from previous phases of the pedagogical, so as to create rules for the game using the Logo language. 

Teacher prerequisites show tooltip help

Familiarity with Cruislet 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 lab- the teaching of the various subjects in greek high 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 lab equipped with tablet PCs.

 

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

Computer use for teaching mathematics.

  • Although computer use for doing mathematics is suggested in high school curriculum, teachers usually don't use computational environments as it isn't  officially part of the curriculum.
  • Students are not familiar in using environments to explore mathematical concepts.
  • Schools computer laboratories are usually occupied for the teaching of informatics.

Teaching of mathematical concepts (included in this particulat pedagogical plan).

  • Students study the concept of cartesian, geographical and spherical coordinate systems within abstract mathematical contexts in a rather static way.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.
  • Students are introduced and study the concept of vectors mainly in physical sciences for the description of a numbr of physical properties such as velocity, force, acceleration.

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

Our aim when designing this pedagogical plan was to create activities based upon the available computational environment, that allow students to get engage with concepts like geographical and spherical coordinates. Thus, we didn't use the curriculum to specify the concepts, as we think that technology may provide us with means to reconceptualize the concepts that can be taught in school using computers. In this way, our main focus was at the mathematics we are able to do with Cruislet environment as a vehicle, rather than focus on mathematics that are fragmented by the curriculum.

 

Content-epistemological goals show tooltip help

  • The exploration of the concept of function as covariation using the geographical coordinates as a system of reference.
  • The development of the notions of dependency between  the positions of two airplanes.
  • The study of the existence of a rate of change of relative displacements on the 3d space.
  • The development of the notions of dependency, covariation and the rate of change through visual, numeric and symbolic representations. 
  • The development of the concept of vector as displacement using a geographical system of reference.
  • The exploration and comparison of geographical and spherical coordinates.
  • The correlation of geographical and mathematical concepts.

 

Cognitive goals show tooltip help

  • Express in a symbolic way using Logo language.
  • Identify the relation between visual and symbolic representations.
  • Use intuitions to bridge mathematical and geographical concepts.

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with expressing symbolically using Logo.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Logo Tab

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

Logo program and basic Logo commands [Resource for students]

The Logo program that was hidden at the previous phase is given to students. Additionally some basic Logo commands are given.

 

Worksheet 6 [Resource for students]

In this worksheet students may write down their ideas in order to create the new rules of the game.

 

New game [Resource produced by students]

Students interfere into the Logo code and change it according to the rules they decided to create. As a result they come up with new Logo procedures that define the rules of the game they created. At the last phase, these Logo programs act as a resource for another team's students. 

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Time show tooltip help

19 teaching hours

Actors' roles show tooltip help

Teacher

  • provides technical support if necessary
  • prompts students to experiment with Logo procedures
  • gives feedback about Logo language (as students are not already familiar with editing Logo commands)
  • prompts students to cooperate

Researcher

  • collect data using audio / video recording
  • collect background data, such as students’ notes on papers
  • write down observational notes
  • support teacher and strudents at the technical level

 

What to do and how show tooltip help

In this activity students actually have to formulate another rule and change function’s relations of planes coordinates by intervening into the Logo code and through this process create a new game to challenge another team. The process of finding another rule, alternate the game and challenge others will engage students with mathematical and geographical-geospatial concepts while experimenting with the environment as well as with programming with Logo.

Students’ modifications of the game could be either at the imaginative level where students reflect on the game and use their imagination to create new rules, or at the creative level, where students edit their new rules using Logo and through this process create a new game. In the first case we refer to students’ ideas while in the second to rules that students actually created. These two games may be different as the expression and application of rules are mediated by the tools available and as students are not familiar with Logo, the implementation of their ideas will be  limited.

Reflect on the game (2 hours)

While students reflect on rules, engage in the process of instrumentalisation, since displacement rules can be questioned and re-defined by them resulting in a variety of artefacts, which are actually the different games they will create. In this phase, students are engaged with the process of finding alternative rules in order to change the game. During this process they come in contact with:

  • concepts embeeded in the game such as geospatial concepts and the concept of function,
  • the Logo language as a way to express the rules.

Create a new game (12 hours)

After reflecting on rules, students will probably focus on specific rules and try to edit them. In order to specify and implement the rules, students have to extend the predefined Logo procedure that was a black box to them when they were playing the game.

//procedure for creating the airplanes

to creation
createavatar("|first| 37.9 23.7 10000 "|plane 1|)
createavatar("|second| 37.9 23.7 10000 "|plane 2|)
activateavatar("|first|)
setupcamera(20000 0 -87 -41 0)
end

//procedure that defines the rule

to flight :a :b :c
activateavatar("|first|)
setposition(:a :b :c)
activateavatar("|second|)
setposition(:a-0.05 :b+0.05 :c-1000)
print("second "airplane :a-0.05 :b+0.05 :c-1000)
end

During the process of making the rule, students will get involved with the process of mathematisation of the geographical space and with several issues regarding the concept of function. In particular:

      Function 

  • Alternative function: The process of finding another function will engage students with the notion of function as they  experiment with alternative functions and try them on the environment. During this procedure students discuss about functions and get involved in the process of argumentation as they have to support their decisions. 
  • The domain of numbers that can be values of the function is restricted as it is related to geographical coordinates of Greece. Using this as a base, students will get involved with the domain as a notion, in order to find a function that can be applied on a limited by specific boundaries domain. 

      Mathematical and geospatial concepts

The activity will provide opportunities to students to associate issues regarding both navigation in geographical space and mathematical concepts underlying 3d space. 

Exchanging games (5 hours)

  • Explain rules (1 hour): After the completion of the games created, each team come up with a Logo program. The teams exchange the game with another team and explain the rules, as the Logo code will be a black box to the other team. 
  • Playing the new game (2 hours): Relatively to a previous phase of the pedagogical plan (named 'Avoid the spy'), each team play the game of the other team and try to reach the goal.
  • Reflection on the games / Conversation in the class (2 hours): A conversation is taking place in the classroom where students reflect on the new games and share their thoughts about them. The teacher prompts students to rationalize their opinion and communicate their thoughts.

Process documentation show tooltip help

  • Audio and screen capture recordings of each team.
  • Observation sheets and researcher's notes.
  • Worksheets by students.
  • Logo files created by students.

 

The instruments are broken

Identity show tooltip helpexplode

Authors show tooltip help

ETL Team

Subject domains show tooltip help

  • Mathematics
  • Geography

Topics show tooltip help

  • functions, inverse function,identity function, geographical coordinateς

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

Description show tooltip help

 

In the tasks included here there are three phases

1) Students  engage with the  study of the function that relates the coordinates of an out of order airplane.

2) Students drive the out of order plane in a given place i.e the city of Rhodos (inverse function)

3) They  discuss what happens if the instruments of the plane will repaired (identity function)

    They  write their observations in a paper and express them   verbally sympolically and graphically

  

Rationale show tooltip helpexplode

 Students interacting with these tasks explore the impact of the variation of the geographical coordinates to the displacement of an airplane and study the relation between the “wrong” positions that  an out of order airplane actually goes(dependent variable) and the “right” coordinates that we gave it (independent variable). In the second phase they conceiving of inverses as reversing a process so that the old outputs become new inputs and vice-versa.Finally they express the idental function since the instruments of the plane are repaired. 

 

Theoretical framework show tooltip help

Several studies report student difficulties in representing and modeling a system that involves two quantities that change in tandem (Carlson, 2002; Monk, 1992; Monk & Nemirovsky, 1994; Thompson, 1994a; Kaput, 1994). Such difficulties are related to the concept of function generally applied: the correspondence concept or Dirichlet-Bourbaki definition of function, which usually leads to a strong dependence on algebraic representations (y = f(x)) and to a static idea of the concept (Kaput, 1994). A more intuitive approach which rests on dynamic aspects of functions is the covariational approach (Confrey & Smith, 1994, 1995; Thompson, 1994 b). Central to this concept is the coordination between two varying quantities and the development of images of covariation (Saldanha & Thompson, 1998; Thompson, 1994b). Carlson et al. (2002) developed the notion of covariational reasoning and proposed a framework for describing and analyzing the cognitive activities involved in coordinating two varying quantities while attending to the ways in which they change in relation to each other

Mental Action 1(MA1)  Coordinating the dependence of one variable on another variable

Mental Action 2 (MA2) Coordinating the direction of change of one variable with changes in the other

Mental Action 3 (MA3) Coordinating the amount of change of one variable with changes in the other

Mental Action 4 (MA4) Coordinating the average rate of change of the function with uniform increments of  change in the input

Mental Action 5 (MA5) Coordinating the instantaneous rate of change of the function with continuous changes in the independent variable for the entire domain of the function

     Using this framework  of covariation we will try to classify student’s covariational reasoning while studying navigation within the context of Cruislet microworld. In the number of tasks included in the pedagogical plan we consider navigation as a dynamic function event. The function’s independent variable is the geographical coordinates  that  students give to the  airplane while they navigate it, and the dependent variable is the geographical coordinates of the position that  the airplane actually goes. They also observe the variation of height(dependent variable) as  the avatar goes upwards  every time they push the buttom "go" of direction.

Students also engage with the concept of function  as a process that may be reversed (to obtain the inverse of a function). Without a generalized view of inputs and outputs, students  are limited to understanding only the related procedural tasks such as switching x and y and solving for y or reflecting the graph of f across the line y = x. A recent study of over 2000 precalculus students at the end of the semester (Carlson et al., submitted) showed that only 17% of these students correctly determined the inverse of a function for a specific value, given a table of function values.Students with a process view are   able to understand aspects of functions such as composition and inverses. They are consistently able to correctly answer conceptual and computational questions about composition in a variety of representations by coordinating output of one process as the input for a second process. Similarly, students conceiving of inverses as reversing a process so that the old outputs become the new inputs and vice-versa or by asking “What does one have to do to get back to the original values?” were able to correctly answer a wide variety of questions about inverse functions (Carlson et al., submitted)


 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Upper high school

Age range show tooltip help

15-16 years old

Population description show tooltip help

14 students in pairs of 2

 

Student prerequisites show tooltip help

Students should be familiar with:

  • the concept of function,
  • drawing graphs.

Teacher prerequisites show tooltip help

 Familiarity with Cruislet enviroment

Context show tooltip helpexplode

Physical context show tooltip help

Children  work in pairs with a computer. The teacher will provide them with technical support.

Institutional context show tooltip help

This scenario is introduced to Greek educational system that encompasses the following characteristics:

  • Computer use for doing mathematics is not officially part of the curriculum of primary school and students are not familiar in using environments to explore mathematical concepts.
  • Students are introduced to the concept of function through  static representations in their textbooks without the chance of manipulating or change them.

 

Socio-cultural context show tooltip help

Students are already familiar, from their out-school activities, with 3D game environments. Cruislet has game-like features such as navigation in 3D space by avatars, thus students are already familiar with the representations provided by the environment. 

 

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

The secondary curriculum indicates the following aims concerning the understanding of the concept of function by the students:

  • Express a value relative to another one.
  • Fill in a table.
  • Identify the domain of the function.
  • Identify the inverse function.
  • Be able to represent a function in a symbolic and graphical way.

Τhe main focus of the present task concerns the student's conceptualisation of the concept of function. We can 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

  • The exploration of the concept of function as a general process that includes inputs and outputs
  • The exploration of the concept of function as covariation
  • Understanding of the function as a process that may be reserved

Cognitive goals show tooltip help

  •  Observing and analyzing the dynamic visual representations in a verbal symbolic and graphical way

Social-affective goals show tooltip help

Students are engouraged to get accustomed to collaboration for achieving a goal and to argumentation procedure so as to confront different views and construct shared understandings in small groups.

 

Instrumental goals show tooltip help

  • Explore and manipulate dynamically 3D representations.
  • Familiarise with the enviroment of Cruislet
  • Associate airplanes displacement with the variation of the geographical coordinates.

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Logo program

A predefined Logo program is used. The procedures included are a black box to students and students experiment with the result of the execution of these procedures.

The fly1 procedure
to fly1 :a :b :c
setposition(:a-0.5 :b-0.7 :c*:c)
print("airplane's "position :a-0.5 :b-0.7  :c*:c)
end

The fly2 procedure
to fly2 :a :b :c
setposition(:a-0.1 :b-0.2 :c*:c-1000)
print("first "airplane "position :a-0,1 :b-0.2 :c*:c-1000)
end

Tool access show tooltip help

NIL

Logo program [Feature]

A Logo program is used as a black box to students at the first phase. 

 

Resources show tooltip helpexplode

Introductory activity [Resource for students]

This activity aims to introduce students to the concept of function.

Activity 1 [Resource for students]

The activity carried out at the first phase.

 

Activity 2 [Resource for students]

The activity carried out at the second phase.

 

Activity 3 [Resource for students]

The activity carried out at the third phase.

 

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs of two, sharing the same tablet PC.

 

Time show tooltip help

6 hours

Actors' roles show tooltip help

Teacher

  • explains the task
  • prompts students to experiment with the environment
  • ask questions in order to understand students' strategies for achieving the goals
  • provokes discussion at group and classroom level
  • facilitates further elaboration of student's ideas in a mathematical framework

Researcher

  • collect data using audio / video recording
  • collect  students worksheets
  • write down observational notes
  • support teacher and students at the technical level

Students
  • experiment with their own constructions in Cruislet
  • express their intuitions and discuss their ideas.
  • express their findings in their worksheets in a graphical and symbolically way


What to do and how show tooltip help

Students work in pair, first with cruislet and then with paper and pencil.

Initially, as an indroductory activity,they try to find the variation of the height of an airplane every time they push the bottom "go"of the direction In particular,students define the vector of a vertical upward displaysment and observe that the height is the only element that changes in the position of the avatar.Through a number of displacements every time they push the buttom "go", students identify and express verbally sympolically and graphically the relationship between the height of the airplane(h)  and the direction functionality (go)

PHASE 1
Students  explore linear and quadratic functional relationships between  changes in  lat -long - height coordinate values, between the positions of an out of order airplane. These functions will be hidden and the students will have to guess  based on repeated moves of the airplane and observations of the relative displacements and positions that it actually goes.There are  Logo procedures (named fly1 or fly2 ) that determine this relation but will be a black box to students .
 For example, if students write in logo tab:

fly1(37 28 100) → define the plane’s position (Lat Long Height)

they will see (in avatar or in logo tab) that the plane goes :

 airplane's position (36.5 27.3 10000)

The hidden relationship is that the latitude of the   position that the plane goes is equal to the latitude of the position that we have sent the plane minus 0.5, the longitude is minus  0.7 and the plane flies x times the hight x that we have sent it. Students must find the dependence relation by giving several values to the coordinates (Lat, Long, Height) that define the position of the  plane and see where the plane actually goes.They  fill  the values in a table in their work sheets in order to compare them and find the correct relationship between coordinates . They also express their findings symbolically and graphically in their parers.

PHASE 2

Students know the hidden function .They know what is the plane's fault and now they are asked to drive the plane to the city of Rhodos They interacting  with the software consider the inverse function as a reverse process in a way that the old outputs could become new inputs. What values they have to put in fly1 procedure  in order to go to a specific place;For example they have to plus or minus 0,5 in lat coordinate;They make assumptions concerning the values they have to input. Their wrong guesses cause the misleading of the plane.The immediate feedback provided by the DDA support the students to think the concept of the inverse function as a process that may be reserved

.

PHASE 3
The out of order plane is repaired. What it will happened? Students discuss and discover  the new function.(identity function). They write their findings  in their work sheets and express them in graphical and symbolic way

Variables as displacements in Geographical Space

Identity show tooltip helpexplode

Subject domains show tooltip help

Topics show tooltip help

Keywords show tooltip help

Description show tooltip help

Rationale show tooltip helpexplode

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode