Juggling with equations (alien)

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Algebraic equations
  • Models
  • Animation
  • Behaviors and properties

Description show tooltip help

MoPiX constitutes a highly visual and highly interactive constructionist learning environment that provides learners and teachers the opportunity to use it as an authoring tool and develop microworlds embedding a variety of knowledge domains - often in combination - such as mathematics, physics and mechanics. It is designed to foster the construction of virtual models consisting of objects whose properties and behaviours are defined and controlled by equations assigned to them. The models can be animated to provide a visual/graphical representation, while the equations ascribed to the objects are fully accessible and available for inspection and modifications.

The "Juggling with Equations" Pedagogical Plan wishes to engage students in the deconstruction, editing and the (re)construction of MoPiX models representing phenomena such as collisions and motions. The manipulation and the construction of MoPiX-compatible equations, so as to control and define the models' behaviour, will engage students in the interpretation and use of the equations' mathematical formalism, while the animation of the models will offer them the chance to connect the mathematical formalism to its visual/graphical representation.

Rationale show tooltip helpexplode

For more than 15 years the Educational Technology Lab (ETL) has been engaged in the development of exploratory software educational tools in a variety of subjects, such as mathematics, physics, geography, history, language and ICT and the design of corresponding innovative pedagogical plans (“scenarios”). Designed to be used both as research instruments for on-site research in the school environment and as open-ended material for teachers and students, the Pedagogical Plans' main characteristic is that they incorporate the constructionist-exploratory and the social-constructivist perspective for learning with technology.

Thus, the activities proposed by the Pedagogical Plans are mainly based on an interdisciplinary thematic perspective (project-based learning) encouraging and requiring a close collaboration between small groups of students working together and are specifically designed so as to support students in a personal construction of meanings. They encompass:

  • exploration and experimentation activities (e.g hypotheses formulation and testing)

  • data/information handling activities (e.g gathering, evaluating information)

  • communication and co-operation activities (e.g peer to peer discussion)

  • creating collaborative and personally meaningful artifacts activities (e.g creating collaborative documents)

  • sharing artifacts activities.

The Pedagogical Plans developed by ETL are not exclusively curriculum oriented. Therefore, some of them may explicitly address goals directly associated to the National Curriculum, whereas others could be considered as game-oriented or open-ended and address solely content - epistemological, cognitive, social-affective and instrumental goals.

The "Juggling with Equations" Pedagogical Plan embraces the aforementioned perspectives, wishing to support the students’ explorations and experimentations with the MoPiX environment. The activities we have designed for this Pedagogical Plan intend to engage students in the manipulation and use of the environment's mathematical formalism in order to define and control the behaviour of animated models. The use and editing of equations that already exist in the environment or the microworlds provided as well as the construction of new ones will engage students in the interpretation and use of the available algebraic formalism, while the models’ generated animations will offer them opportunities to connect the equations’ formalism to visual/graphical representations. In this process, we wish to provide a meaningful context for the construction of mathematical meanings concerning the role of the algebraic formalism in the creation and control of animated models.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

Basic familiarity with equations

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of using the technology to by-pass conventional practices –such as transmitting the Curriculum’s predefined content- and address open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic formalism in order to control and create animated models. Drawing on the constuctionist and socio-constructivist paradigm, we particularly focused on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researchers so as to a priori correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and social activity.

Content-epistemological goals show tooltip help

  • Developing fluency with the algebraic formalism used in the MoPiX environment

  • Interpreting the existing equations' algebraic formalism

  • Connecting the equations' algebraic formalism to the visualised behaviours animated on the Stage

  • Manipulating existing equations to control the animated models (adding-withdrawing them)

  • Manipulating the equations' algebraic formalism to control the animated models
    (editing the symbols, modifying the structure)
  • Constructing new algebraic equations employing the available formalism to define new behaviours

Cognitive goals show tooltip help

  • Observing, interpreting and analysing the behaviour of an already developed models in terms of the algebraic equations used to develop them and the graphical representation generated by their animation.

  • Connecting the mathematical formalism (symbolic representation) used to describe a model's behaviour to the graphical representation produced when the model is animated.

  • Using the environment's feedback to their actions to further decide on their actions or debug a flawed model
  • Editing equations in order to define properties and behaviours

  • Constructing new equations in order to define properties and behaviours

  • Modifying the behaviour and the properties of the object by applying or removing newly constructed or edited one-object algebraic equations

  • Modifying the behaviour and the properties of the objects by applying removing existing one-object algebraic equations to the object

Social-affective goals show tooltip help

Collaborating in pairs discussing, forming and testing hypotheses, arguing, negotiating and reaching in joint conclusions/decisions concerning:

  • the models’ behaviour as they observe the animations generated

  • the models’ behaviour and properties as they interpret the equations’ mathematical formalism and link it to the animations generated

  • the desired new behaviours and the properties to be assigned to newly inserted or existing objects

  • the strategies to be implemented so as produce the new behaviours (e.g add/remove equations, construct equations)

  • the models’ new behaviour in relation to their equations and the animations generated

Instrumental goals show tooltip help

  • Using the MoPiX environment functionalities and features to create, control and animate models

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The "Juggling with Equations" Pedagogical Plan is designed to engage students in activities that give will them the opportunity to interpret, manipulate, use the algebraic formalism available in MoPiX in order to create and control animated models. The students will deconstruct already saved models, edit saved models to embed their own ideas and use and expand the available mathematical formalism, constructing their own equations so as to attribute new behaviours to their models. The animation generated on the screen when the model is executed, will allow students to connect the equations' algebraic formalism to its visual/graphical representation. In this process, the students are expected to construct mathematical meanings concerning the role of the algebraic equations in creating and controlling animated models.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist learning environment (Harel and Papert 1991), designed to allow learners to explore, manipulate and construct animated models consisting of objects whose properties and behaviours are defined and controlled by algebraic equations assigned to them. The students’ interactions with the equations’ mathematical formalism and the visual/graphical representation generated on the screen, as well as their interactions with their peers, will give them the opportunity to express ideas and construct mathematical meanings, while engaging in collaborative activities personally meaningful to them (Kafai and Resnick 1996).

The equations attributed to the objects and underpin the models’ behaviour in MoPiX do not constitute “black boxes”, unavailable for inspection or modifications by the user (for a discussion on black and white box approaches see Kynigos 2004). Therefore, the learner in MoPiX is an empowered user who has deep structural access (diSessa 2000) to the models animated. The user may explore a model’s behaviour in terms of its symbolic representation (i.e the equations attributed to the object) and the visual representation displayed when the model is executed, edit it to embed his/hers own ideas or create new models from scratch. Any action performed to a model’s symbolic facet (e.g. inserting/removing equations or editing/modifying equations) produces a direct change to the visual result generated on the Stage. Although the visual representation can not be manipulated to induce changes to the model’s symbolic facet (i.e the representations are nor interdependent), both representations may provide students meaningful feedback, supporting them in their explorations and experimentations with the MoPiX environment, especially in case they attempt to link the representation to each other .

Taking into account those aspects of the environment, ETL considers MoPiX as a computational tool with which one may construct half-baked microworlds (Kynigos 2007) i.e. microworlds that incorporate an interesting idea but are incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. Under this perspective, the half –baked microworld are considered to be improvable objects (Bereiter & Scardamalia 2003), in the sense that they do not constitute end artefacts that can not be further modified, revised or refined, that foster student’s engagement in constructionist, meaning making activities.

Half - baked microworlds are by nature designed for instrumentalisation (Guin and Trouche 2002) since they serve as intriguing starting points and idea generators, stimulating students to transform them from a plain artefact (i.e. a piece of software constructed by humans) to an instrument (Rabardel 2001), engaging in the way in meaning-making processes. Developing the “Juggler” half-baked microworld in the MoPiX environment, we focused on the idea of designing microworlds for instrumentalisation in a collaborative context. The “Juggler” is a game–like microworld consisting of three interrelated objects: a ball and two rackets. What is easily noticeable about the ball’s behaviour is fact that it seems to move rather slowly. The equations attributed to the ball and define its behaviour are not exclusively ready made equations found in the environment’s Library, but also equations created by us using the MoPiX formalism. The rackets do not actively participate in the animation but as the model is executed they can be freely moved around using the mouse. When a racket hits the ball, it forces it to bounce off and move away in a manner that doesn’t allow the user to play with the ball and the rackets like a real juggler. Consequently, the Juggler half-baked microworld constitutes a challenging venue for the students who, in the process of exploring the initial microworld’s behaviour and modifying its functionalities to integrate their own personal ideas, express mathematical meanings as they use algebraic equations to define and control their model’s behaviours.

The half-baked microworlds are also perceived as boundary objects (Cobb et al. 2003), in the sense that can be used to convey meaning among of the members of the same community and thus enhance the meaning generation process as it occurs in the social context of the classroom (Cobb and Yackel 1996). The students in the Juggling with Equations PP are expected to work in groups of two or three discussing, forming and testing hypotheses, engaging in joint decision-making processes, developing strategies to verify the validity of their conclusions, arguing and negotiating. During plenary sessions, all the workgroups come together to present their work, justify their choices and get meaningful feedback on the decisions they have made from their classmates. In both processes, the half baked microworld operates as a tool of communication among the members of the students’ community and constitutes the focal point around which the students organise their activities (Kynigos 2007).

The design of the Juggling with Equations Pedagogical Plan is based on the idea of engaging students in constructionist activities such as observing, exploring, deconstructing, manipulating, changing and (re)constructing animated models, as they are invited to use and change a half-baked microworld. The “Juggler” microworld serves as a communication tool among the students of the same or different workgroups, enhancing in this way the meaning making process. The generation of meanings is perceived as a process  through the feedback received not merely by the computational tool but also by their peers as they present their ideas, argue and negotiate on their choices, reflect on their actions to support their decisions.

MoPiX v1

MoPiX v1 constitutes a programmable environment that provides the user the opportunity to build, manipulate and share models. These models are created as the user associates algebraic equations - constructed by himself or already available in the environment - with the objects inserted on the Stage. By generating the animations of the models constructed, the user has at his disposal not only the symbolic representation of the behaviour of his model (the equations he used) but also the visual one.

MoPiX v1 can be run in the Internet Explorer or the Mozilla Firefox browser provided that there is a Flash Player installed (version 8 or latest).

MoPiX features and functionalities [Feature]

The MoPiX features will be described separately in one each of the Pedagogical Plan's phases according to when and if the designed activities call for their use.

Resources show tooltip helpexplode

General description show tooltip help

User's Manual, Microworlds, Models, Work Sheets, Notes

User's Manual - Translations in Greek [Resource for students]

A User's Manual for MoPiX v1 and translations in Greek.

A printed version of the manual will be available to each workgroup while an electronic version will be available at the ETL website.

Microworlds [Resource for students]

The microworlds introduced in each phase of this Pedagogical Plan will be described in detail at the corresponding sections. 

Resource contents show tooltip help

The Microworlds used in this Pedagogical Plan are saved in the form of MoPiX models in the Remath Portal's DDA Library.

http://www.remath.cti.gr

Work sheets [Resource for students]

In this phase of the Pedagogical Plan we provided two worksheets in order to support students in their experimentations with MoPiX. The two worksheets aimed at helping students to develop some fluency with the MoPiX equations' formalism and link this formalism to the behaviours generated on the screen. A detailed description of these worksheets will be presented in sub-phases where they were given to the students

Resource contents show tooltip help
Models [Resource produced by students]

During the experimentation process, we expect students to produce their own models, employing different strategies such as editing already existing equations or -at least at first- just attributing Library equations to their objects. The students' models will be saved in the ReMath portal's DDA Library so as to be shareable and accessible by all the workgroups.

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs, sharing the same PC. Under certain circumstances the workgroups could possibly consist of three members.

The furniture layout will allow face-to-face communication between the formed workgroups during the plenary sessions. A portable PC connected to a video projector and having access to the Internet will be available on the table around which students are gathered for the plenary sessions. A white board and markers will also be at the students' disposal.

The implementation of the Pedagogical Plan will take place during school hours since it is extremely difficult for Vocational Education students who also have a professional life to stay at school for extra hours. The teacher involved in the experimentation will provide the number of school hours required for the implementation of the Plan. If necessary other teachers will be asked to provide school hours.

The duration of each school hour is 40 minutes. For each session we will need 2 or 3 school hours according to the designed activities.

Time show tooltip help

10 sessions - 25 school hours.

Actors' roles show tooltip help

During their interaction with the computational and the social environment students will engage in activities permitting them to express their ideas and original intuitions, discuss and share these ideas with other workgroup members or with the whole class during plenary sessions, formulate, test, reject or accept hypotheses, draw conclusions/assumptions, negotiate and argue about validity of their conclusions, deconstruct and reconstruct artefacts.

The teacher is expected not to intervene in this process by providing the “correct answer” to the students when asked or giving out instructions about what to do next and how. On the contrary, the teacher is expected to guide students discretely by posing meaningful- often intriguing- questions, urging them to continue their experimentation, collaborate and discuss their ideas with their peers (the members of the workgroup or other workgroups). At the same time the teacher has to support the students both intellectually and emotionally so as not to get discouraged when they encounter problematic situations requiring task solving skills.

The researcher will provide technical support in the use of MoPiX both for the teacher and the students. During the experimentation process he will be observing the classroom activity and collecting data while handling the video and audio recorders. Under certain circumstances the researcher could focus on particular workgroups taking detailed data (notes or recordings) of their interactions.

What to do and how show tooltip help

For each phase of the Pedagogical Plan there will be a detail description of "What to do and how".

Process documentation show tooltip help

A screen capture software like HyperCam would be suitable for an experiment with MoPiX. It allows the researcher to record the students' voices and at the same time capture their interactions with the environment. Apart from the HyperCam files, video recordings from the students' plenary sessions will also be available. The students' notes and MoPiX models as well as the researchers' field notes and logs will also constitute the research data.

One-object equations

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Equations (algebraic formalism)
  • Saved model
  • Animation
  • Behaviours and properties
  • Interpreting equations' mathematical formalism
  • Manipulating/using equations
  • Editing equations

Description show tooltip help

The “One-Object Equations” sequence of activities introduces students to the MoPiX environment and aims at giving them the opportunity to achieve a certain level of familiarity with the environment and its functionalities as they interpret, manipulate and edit equations in order to control and create animated models. The activities proposed in this phase of the Pedagogical Plan are of two kinds:

(a) Activities aiming to engage students in the deconstruction of the one-object equations that underpin the behaviour of an already saved model and

(b) Activities aiming to engage students in the editing of one - object equations that already exist in the "Equations Library" and the construction of new one-object equations.

Rationale show tooltip helpexplode

The familiarisation process is not perceived by ETL as a process during which the teacher gives out a fixed set of instructions for the students to follow or as a process during which the teacher demonstrates the main functionalities of the environment and the students repeat the teacher's actions step-by-step, following the exact same procedure.

On the contrary, we believe that such a process should take place in a context that allows students to explore the computational environment by themselves and at the same time permits them to engage in activities personally meaningful to them (Kafai & Resnick 1996). This means that we regard the students as being empowered users of the environment instead of passive users having no access to any other functionalities apart from the ones presented by the teacher.

Drawing on the notion of layered learning design (Kahn et al. 2000), we consider dividing the sequence of activities into two phases and developing two microworlds -one for each phase- to be purposeful for the students especially with regard to the frustration such an environment could cause to inexperienced users during their initial explorations (Kahn 2004). Taking into account the natural distinction in MoPiX between one–object and multi–object equations, the microworld we developed for the first phase of the experimentation consists exclusively of one-object equations while the one developed for the second phase contains both one and multi-object equations.

Taking under consideration these perspectives we select to introduce students to an already saved model consisting of a single bouncing ball whose behaviour is defined solely by one-object equations. The designed activities in this phase will allow them to explore the functionalities of the environment while experimenting with the entities and the representations available in the environment, expressing their initial intuitions and ideas, discussing and sharing these ideas with other members of the workgroup or with the whole class, deconstructing and reconstructing artefacts using the environment's available formalism.

The activities proposed in this phase of the pedagogical plan will give students the opportunity to construct mathematical meanings about the role of the equations' algebraic formalism as they interpret, manipulate, use and edit the already existing equations in the process of controlling and creating animated models.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

Basic familiarity with equations

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of using the technology to by-pass conventional practices –such as transmitting the Curriculum’s predefined content- and address open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic formalism in order to control and create animated models. Drawing on the constuctionist and socio-constructivist paradigm, we particularly focused on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researchers so as to a priori correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and social activity.

Content-epistemological goals show tooltip help

  • Developing fluency with the algebraic formalism available in the MoPiX environment

  • Interpreting the existing one - object equations' algebraic formalism and structure

  • Connecting the existing one - object equations' algebraic formalism to the behaviours animated on the Stage

  • Manipulating existing one - object equations to assign properties and behaviours to the objects or to withdraw properties and behaviours from the objects

  • Constructing new algebraic equations to define properties and behaviours using the available formalism

Cognitive goals show tooltip help

  • Observing, interpreting and analysing the behaviour of an already developed one-object model in terms of the algebraic equations used to develop it and the graphical representation generated by its animation.

  • Interconnecting the mathematical formalism (algebraic equations) used to describe an one-object model's behaviour to the graphical representation produced by the animation of the model.

Social-affective goals show tooltip help

  • Discussing and analysing in pairs the behaviour and the properties of the models

  • Collaborating in pairs in problem solving situations (form and test hypotheses, negotiate, develop arguments)

Instrumental goals show tooltip help

  • Using the MoPiX environment functionalities and the MathDiLS library features

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

For the first phase of the activities we developed, using exclusively one-object “Library” equations, the “One Red Ball” microworld which consisted of a single red ball performing a combined motion in the vertical and the horizontal axis. This means that at this phase the students come across equations that are not extremely complicated and are classified in specific categories in the "Library"  (e.g. Horizontal Motion Equations) which makes it easier to interpret their symbols and predict/pinpoint the behaviours they describe.

In this phase our main aim is not to introduce the students to the MoPiX environment by just demonstrating and explaining them its main functionalities and features available. On the contrary, we believe that such a process should take place in a context that allows students to explore the computational environment and at same time give them the opportunity to get involved in activities that are personally meaningful to them.

Thus, we select to ask the students to load the “One Red Ball”, execute it and observe the animation generated. They will be invited to discuss with their teammates and with other workgroups the behaviours animated and write down their remarks and observations on a worksheet that we will have prepared for them. Since, at this time, the students will not be acquaint with the MoPiX features and functionalities, during one of the plenary meetings, we will carefully try to draw the students’ attention to the “Equations Library”, aiming to provoke discussions regarding the equations’ specific role in the animation generated. In order to stimulate students to start using the equations themselves, we will asked them to insert a new object on the Stage and try to reproduce the ball’s motion. In this process, we will encourage the students to interpret and use equations from the “Library”, add and remove equations from their objects so as to explore the model’s changes of behaviour and link the equation’s formalism to the behaviours they would have previously noted down on their worksheet or/and to the graphical representation produced on the screen. As we have deliberately made the original red ball move rather slowly, near the end of this phase, we expect students to start expressing their personal ideas about their own object’s motion (e.g. make it move faster) and thus start editing the model’s equations, using the “Equations Editor”, so as to describe the new behaviours they may have in mind.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist learning environment (Harel and Papert 1991), designed to allow learners to explore, manipulate and construct animated models consisting of objects whose properties and behaviours are defined and controlled by algebraic equations assigned to them. The students’ interactions with the equations’ mathematical formalism and the visual/graphical representation generated on the screen, as well as their interactions with their peers, will give them the opportunity to express ideas and construct mathematical meanings, while engaging in collaborative activities personally meaningful to them (Kafai and Resnick 1996).

The equations attributed to the objects and underpin the models’ behaviour in MoPiX do not constitute “black boxes”, unavailable for inspection or modifications by the user (for a discussion on black and white box approaches see Kynigos 2004). Therefore, the learner in MoPiX is an empowered user who has deep structural access (diSessa 2000) to the models animated. The user may explore a model’s behaviour in terms of its symbolic representation (i.e the equations attributed to the object) and the visual representation displayed when the model is executed, edit it to embed his/hers own ideas or create new models from scratch. Any action performed to a model’s symbolic facet (e.g. inserting/removing equations or editing/modifying equations) produces a direct change to the visual result generated on the Stage. Although the visual representation can not be manipulated to induce changes to the model’s symbolic facet (i.e the representations are nor interdependent), both representations may provide students meaningful feedback, supporting them in their explorations and experimentations with the MoPiX environment, especially in case they attempt to link the representation to each other .

Taking into account those aspects of the environment, ETL considers MoPiX as a computational tool with which one may construct half-baked microworlds (Kynigos 2007) i.e. microworlds that incorporate an interesting idea but are incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. Under this perspective, the half –baked microworld are considered to be improvable objects (Bereiter & Scardamalia 2003), in the sense that they do not constitute end artefacts that can not be further modified, revised or refined, that foster student’s engagement in constructionist, meaning making activities.

Half - baked microworlds are by nature designed for instrumentalisation (Guin and Trouche 2002) since they serve as intriguing starting points and idea generators, stimulating students to transform them from a plain artefact (i.e. a piece of software constructed by humans) to an instrument (Rabardel 2001), engaging in the way in meaning-making processes. Developing the “Juggler” half-baked microworld in the MoPiX environment, we focused on the idea of designing microworlds for instrumentalisation in a collaborative context. The “Juggler” is a game–like microworld consisting of three interrelated objects: a ball and two rackets. What is easily noticeable about the ball’s behaviour is fact that it seems to move rather slowly. The equations attributed to the ball and define its behaviour are not exclusively ready made equations found in the environment’s Library, but also equations created by us using the MoPiX formalism. The rackets do not actively participate in the animation but as the model is executed they can be freely moved around using the mouse. When a racket hits the ball, it forces it to bounce off and move away in a manner that doesn’t allow the user to play with the ball and the rackets like a real juggler. Consequently, the Juggler half-baked microworld constitutes a challenging venue for the students who, in the process of exploring the initial microworld’s behaviour and modifying its functionalities to integrate their own personal ideas, express mathematical meanings as they use algebraic equations to define and control their model’s behaviours.

The half-baked microworlds are also perceived as boundary objects (Cobb et al. 2003), in the sense that can be used to convey meaning among of the members of the same community and thus enhance the meaning generation process as it occurs in the social context of the classroom (Cobb and Yackel 1996). The students in the Juggling with Equations PP are expected to work in groups of two or three discussing, forming and testing hypotheses, engaging in joint decision-making processes, developing strategies to verify the validity of their conclusions, arguing and negotiating. During plenary sessions, all the workgroups come together to present their work, justify their choices and get meaningful feedback on the decisions they have made from their classmates. In both processes, the half baked microworld operates as a tool of communication among the members of the students’ community and constitutes the focal point around which the students organise their activities (Kynigos 2007).

The design of the Juggling with Equations Pedagogical Plan is based on the idea of engaging students in constructionist activities such as observing, exploring, deconstructing, manipulating, changing and (re)constructing animated models, as they are invited to use and change a half-baked microworld. The “Juggler” microworld serves as a communication tool among the students of the same or different workgroups, enhancing in this way the meaning making process. The generation of meanings is perceived as a process  through the feedback received not merely by the computational tool but also by their peers as they present their ideas, argue and negotiate on their choices, reflect on their actions to support their decisions.

MoPiX v1

MoPiX v1 constitutes a programmable environment that provides the user the opportunity to build, manipulate and share models. These models are created as the user associates algebraic equations - constructed by himself or already available in the environment - with the objects inserted on the Stage. By generating the animations of the models constructed, the user has at his disposal not only the symbolic representation of the behaviour of his model (the equations he used) but also the visual one.

MoPiX v1 can be run in the Internet Explorer or the Mozilla Firefox browser provided that there is a Flash Player installed (version 8 or latest).

Equations Library [Feature]

The “Equations Library” provides the user with a set of pre-defined equations. These equations can be directly dropped onto the objects present on the “Stage” so as to assign them behaviours and properties or be transferred to the “Equations Editor” so as to be modified.

The “Equations Library” may also contain equations constructed by the user himself. After constructing a new equation or editing an existing one, the user may drop the equation into the library so as to store it temporarily into the “Your equations” section.

Picture 1: Equations Library

Drag and Drop [Feature]

With the assistance of the mouse, the user can drag the equations situated in the “Equation Library” and drop them onto the objects present on the “Stage”. The dragging and dropping procedure could be also activated when wishing to remove an equation from an object and discard it or place it in the “Equations Library”.

Picture 1: Drag and Drop

Flipping Objects [Feature]

Each object presented on the “Stage” may be “flipped” by right clicking on it and selecting the “Flip Object” option. Any animation in progress pauses and a new differently coloured window appears. The window contains the equations assigned to the object by the user and the equations assigned to the object by the MoPiX environment (e.g equations defining its x and y position in a Cartesian plane assigned to the object by default once inserted on the “Stage”).

Picture 1: Flip Object

Equations Editor [Feature]

The “Equations Editor” is situated at the upper part of the “Stage” and consists of several buttons as well as an editing area. It may be used to edit already existing equations or construct new equations from scratch.

An equation can be edited when dragged from the library or any other point of the Stage and dropped onto the editing area. As the equation to be edited opens up, its structural components become accessible to the user.

By pressing the “New Equation” button, a template appears at the editing area. The fourteen “Operator buttons” and the “Function button” are to be used so as to complete the fields on the emerging template. In order to generate an acceptable MoPiX equation a binary tree structure has to be followed.

The new equations created may be dragged and dropped onto the objects or dragged and dropped into the “Equations Library” to be stored temporarily. Alternatively, they could be stored for future use in the MathDiLS library.

Picture 1: Equations Editor

Resources show tooltip helpexplode

General description show tooltip help

User's Manual, Microworlds, Models, Notes, Work sheets

"One Red Ball" microworld (saved model) [Resource for students]

The “One red ball” is a saved in the MathDiLS library model. It consists of a single object -a ball- performing a combined motion (horizontal and vertical). Each time the ball hits the side boarder lines or the line separating the “Stage” from the “Equations Library”, it bounces and returns in the main area of the “Stage”. The equations associated to the object's behaviour and the object's properties are one-object equations and derive exclusively from the “Equations Library”.

Picture 1: The "One Red Ball" microworld

This model derived from the "Bouncing ball" model that the LKL team had created with MoPiX.

The "One Red Ball" model is saved in a txt file. To run the model open the file (see the Resource Contents section), copy the content and paste it in the window that opens after clicking on the XML button situated at the upper right corner of the MoPiX environment.

User's Manual - Translations in Greek [Resource for students]

The User's Manual for MoPiX v1 in Greek.

An appendix will be added to serve as a quick guide for the MoPiX interface providing the translation in Greek of:

  • the names of the equations available in the "Equations Library" (e.g the "amIHittingASide" equation)

  • the names of the categories of the equations available in the "Equations Library"

  • the names of the buttons available on the "Equations Editor" area

A printed version of the manual will be available to each workgroup while an electronic version will be available at the ETL website.

Work sheets [Resource for students]

In this phase of the Pedagogical Plan we provided three worksheets in order to support students in their experimentations with the MoPiX formalism and the environment's features and functionalities. The three worksheets aimed at helping students to develop some fluency with the MoPiX equations' formalism and link this formalism to the behaviours generated on the screen. A detailed description of these worksheets will be presented at the sub-phases where they were given to the students.

Resource contents show tooltip help
Notes [Resource produced by students]

The notes the students will keep while collaborating in pairs or collaborating with other workgroups during plenary sessions.

Models [Resource produced by students]

Each workgroup could develop one or several models consisting of equations deriving from the "Equations Library" and/or equations that they have constructed themselves. The models are saved at the MathDiLS library so as to render them shareable and accessible to other workgroups.

The models the students produce in each one of the Pedagogical Plan' s phases could serve as a basis on which new models will be developed.

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs, sharing the same PC. Under certain circumstances the workgroups could possibly consist of three members.

The furniture layout will allow face-to-face communication between the formed workgroups during the plenary sessions. A portable PC connected to a video projector and having access to the Internet will be available on the table around which students are gathered for the plenary sessions. A white board and markers will also be at the students' disposal.

The implementation of the Pedagogical Plan will take place during school hours since it is extremely difficult for Vocational Education students who also have a professional life to stay at school for extra hours. The teacher involved in the experimentation will provide the number of school hours required for the implementation of the Plan. If necessary other teachers will be asked to provide school hours.

The duration of each school hour is 40 minutes. For each session we will need 2 or 3 school hours according to the designed activities.

Time show tooltip help

10 hours

Actors' roles show tooltip help

In this phase of the pedagogical plan students are expected to explore the MoPiX environment and its basic functionalities while working in pairs. They will be guided to load an already saved model from the MathDiLS library and during their interaction with the computational and the social environment they will engage in activities permitting them to express their ideas and original intuitions, discuss and share these ideas with other workgroup members or with the whole class during plenary sessions, formulate, test, reject or accept hypotheses, draw conclusions/assumptions, negotiate and argue about validity of their conclusions, deconstruct and reconstruct artefacts.

The teacher is expected not to intervene in this process by providing the “correct answer” to the students when asked or by giving out instructions about what to do next and how. On the contrary, the teacher is expected to guide students discretely by posing meaningful - often intriguing - questions, urging them to continue their experimentation, collaborate and discuss their ideas with their peers (the members of the workgroup or other workgroups). At the same time the teacher has to support the students both intellectually and emotionally so as not to get discouraged when they encounter problematic situations requiring task solving skills.

The researcher will provide technical support in the use of MoPiX both for the teacher and the students. During the experimentation process he will be observing the classroom activity and collecting data while handling the video and audio recorders. Under certain circumstances the researcher could focus on particular workgroups taking detailed data (notes or recordings) of their interactions.

What to do and how show tooltip help

The "What to do an how" is described in details in the "Adding - removing equations" and "Constructing equations" section.

Process documentation show tooltip help

A screen capture software like HyperCam would be suitable for an experiment with MoPiX. It allows the researcher to record the students' voices and at the same time capture their interactions with the environment. Apart from the HyperCam files, video recordings from the students' plenary sessions will also be available. he students' notes and MoPiX models as well as the researchers' field notes and logs will also constitute the research data.

Manipulating one-object equations

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Equations (algebraic formalism)
  • Saved model
  • Animation
  • Behaviours and Properties
  • Interpreting the equations' formalism
  • Manipulating/using equations
  • Reproducing animations

Description show tooltip help

After loading from the MathDiLS library an already saved one-object model (the “One Red Ball”) students get engaged in activities that permit them to become familiar with some of the basic functionalities of MoPiX environment. The activities proposed for this phase could be:

(a) Executing the saved model and observing the animation generated

(b) Discussing with peers about the object's behaviour and properties

(c) Adding equations from the equations library - assigning new behaviours to the object

(d) Removing equations - withdrawing behaviours previously assigned to the object

(e) Reproducing the original object's behaviour

The order of the proposed activities is not obligatory as it is not unlikely for the students who work in pairs to decide to explore the functionalities of the environment in a different sequence. For example, students could first start deleting equations and discussing about how this action changes the object's behaviour or/and properties and then experiment with adding equations to the object.

Rationale show tooltip helpexplode

The familiarisation process is not perceived by ETL as a process during which the teacher gives out a fixed set of instructions for the students to follow or as a process during which the teacher demonstrates the main functionalities of the environment and the students repeat the teacher's actions step-by-step, following the exact same procedure.

On the contrary, we believe that such a process should take place in a context that allows students to explore the computational environment by themselves and at the same time permits them to engage in activities personally meaningful to them (Kafai & Resnick 1996). This means that we regard the students as being empowered users of the environment instead of passive users having no access to any other functionalities apart from the ones presented by the teacher.

Drawing on the notion of layered learning design (Kahn et al. 2000), we consider dividing the sequence of activities into two phases and developing two microworlds -one for each phase- to be purposeful for the students especially with regard to the frustration such an environment could cause to inexperienced users during their initial explorations (Kahn 2004). Taking into account the natural distinction in MoPiX between one–object and multi–object equations, the microworld we developed for the first phase of the experimentation consists exclusively of one-object equations while the one developed for the second phase contains both one and multi-object equations.

Taking under consideration these perspectives we select to introduce students to an already saved model consisting of a single bouncing ball whose behaviour is defined solely by one-object equations. The designed activities in this phase will allow them to explore the functionalities of the environment while experimenting with the entities and the representations available in the environment, expressing their initial intuitions and ideas, discussing and sharing these ideas with other members of the workgroup or with the whole class, deconstructing and reconstructing artefacts using the environment's available formalism.

The activities proposed in this phase of the pedagogical plan will give students the opportunity to construct mathematical meanings about the role of the equations' algebraic formalism as they interpret, manipulate, use and edit the already existing equations in the process of controlling and creating animated models.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

Basic familiarity with equations

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of using the technology to by-pass conventional practices –such as transmitting the Curriculum’s predefined content- and address open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic formalism in order to control and create animated models. Drawing on the constuctionist and socio-constructivist paradigm, we particularly focused on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researchers so as to a priori correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and social activity.

Content-epistemological goals show tooltip help

  • Interpreting the Library's equations' algebraic formalism and mathematical structure
  • Identifying the role of the symbols in the equations used to describe the behaviour of the objects
  • Identifying the role of the variable of time in the equations

  • Identifying the role of the variable ME in the equations

  • Connecting the existing one - object equations' algebraic formalism to the behaviours animated on the Stage

  • Manipulating existing one - object equations (adding/removing) to assign properties and behaviours to the objects or to withdraw properties and behaviours from the objects so as to verify the equations' role in the animation

Cognitive goals show tooltip help

  • Observing, interpreting and analysing the behaviour of an existing one-object model in terms of the algebraic equations used to develop it and the graphical representation generated by its animation.

  • Modifying the behaviour and the properties of the objects by attributing "Library" one-object equations to the object

  • Modifying the behaviour and the properties of the objects by removing "Library" one-object equations from the object

  • Interconnecting the mathematical formalism (algebraic equations) used to describe an one-object model's behaviour to the graphical representation produced by the animation of the model.

Social-affective goals show tooltip help

  • Discussing and analysing in pairs the behaviour of an one-object saved model

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the object' s behaviour after adding equations to the object

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the object' s behaviour after removing equations from the object

  • Collaborating in pairs in order to decide on the strategies implemented so as to assign the object specific behaviours or deprive it from specific behaviours

Instrumental goals show tooltip help

  • Using the MoPiX environment functionalities and the MathDiLS library to load an already saved model

  • Navigating through the Library categories and equations
  • Adding one-object equations to an object

  • Removing one-object equations from an object

  • Discarding existing equations and objects

  • Introducing new objects on the Stage

  • Using the Flip Object feature to inspect and manipulate equations

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The activities designed in this part of the pedagogical plan wish to introduce the students to the basic features of the MoPiX environment. Instead of giving out a set of instructions for the students to follow, we decided provide them with a meaningful interactive context. Thus, students will be asked to load to the MoPiX environment the “Bouncing Ball” saved model.

This saved model will act as a microworld permitting the students to develop their basic knowledge and skills concerning the use of MoPiX while manipulating the entities present in the microworld (equations and objects), exploring their properties, behaviours and structure and attempting to define the governing rules. Moreover, during their exploration activities, students will have the opportunity to make hypotheses and test them by themselves as they receive feedback from the environment and not from the teacher confirming or rejecting their answers.

The exploration of the microworld will take place in social context, as students are expected to collaborate in pairs. Socially mediated activities in combination with the computational tool offer students the opportunity to share with peers the artefacts constructed and their current understanding so as to make them objects of discussion and reflection.

The sequence of the activities proposed is not strictly predetermined as students are likely to decide on exploring different aspects and functionalities of the microworld. For example, students could choose to explore the “Flipping Object” feature first and thus start removing equations from the object and discuss about the impact of their actions to the object's behaviour and properties.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist learning environment (Harel and Papert 1991), designed to allow learners to explore, manipulate and construct animated models consisting of objects whose properties and behaviours are defined and controlled by algebraic equations assigned to them. The students’ interactions with the equations’ mathematical formalism and the visual/graphical representation generated on the screen, as well as their interactions with their peers, will give them the opportunity to express ideas and construct mathematical meanings, while engaging in collaborative activities personally meaningful to them (Kafai and Resnick 1996).

The equations attributed to the objects and underpin the models’ behaviour in MoPiX do not constitute “black boxes”, unavailable for inspection or modifications by the user (for a discussion on black and white box approaches see Kynigos 2004). Therefore, the learner in MoPiX is an empowered user who has deep structural access (diSessa 2000) to the models animated. The user may explore a model’s behaviour in terms of its symbolic representation (i.e the equations attributed to the object) and the visual representation displayed when the model is executed, edit it to embed his/hers own ideas or create new models from scratch. Any action performed to a model’s symbolic facet (e.g. inserting/removing equations or editing/modifying equations) produces a direct change to the visual result generated on the Stage. Although the visual representation can not be manipulated to induce changes to the model’s symbolic facet (i.e the representations are nor interdependent), both representations may provide students meaningful feedback, supporting them in their explorations and experimentations with the MoPiX environment, especially in case they attempt to link the representation to each other .

Taking into account those aspects of the environment, ETL considers MoPiX as a computational tool with which one may construct half-baked microworlds (Kynigos 2007) i.e. microworlds that incorporate an interesting idea but are incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. Under this perspective, the half –baked microworld are considered to be improvable objects (Bereiter & Scardamalia 2003), in the sense that they do not constitute end artefacts that can not be further modified, revised or refined, that foster student’s engagement in constructionist, meaning making activities.

Half - baked microworlds are by nature designed for instrumentalisation (Guin and Trouche 2002) since they serve as intriguing starting points and idea generators, stimulating students to transform them from a plain artefact (i.e. a piece of software constructed by humans) to an instrument (Rabardel 2001), engaging in the way in meaning-making processes. Developing the “Juggler” half-baked microworld in the MoPiX environment, we focused on the idea of designing microworlds for instrumentalisation in a collaborative context. The “Juggler” is a game–like microworld consisting of three interrelated objects: a ball and two rackets. What is easily noticeable about the ball’s behaviour is fact that it seems to move rather slowly. The equations attributed to the ball and define its behaviour are not exclusively ready made equations found in the environment’s Library, but also equations created by us using the MoPiX formalism. The rackets do not actively participate in the animation but as the model is executed they can be freely moved around using the mouse. When a racket hits the ball, it forces it to bounce off and move away in a manner that doesn’t allow the user to play with the ball and the rackets like a real juggler. Consequently, the Juggler half-baked microworld constitutes a challenging venue for the students who, in the process of exploring the initial microworld’s behaviour and modifying its functionalities to integrate their own personal ideas, express mathematical meanings as they use algebraic equations to define and control their model’s behaviours.

The half-baked microworlds are also perceived as boundary objects (Cobb et al. 2003), in the sense that can be used to convey meaning among of the members of the same community and thus enhance the meaning generation process as it occurs in the social context of the classroom (Cobb and Yackel 1996). The students in the Juggling with Equations PP are expected to work in groups of two or three discussing, forming and testing hypotheses, engaging in joint decision-making processes, developing strategies to verify the validity of their conclusions, arguing and negotiating. During plenary sessions, all the workgroups come together to present their work, justify their choices and get meaningful feedback on the decisions they have made from their classmates. In both processes, the half baked microworld operates as a tool of communication among the members of the students’ community and constitutes the focal point around which the students organise their activities (Kynigos 2007).

The design of the Juggling with Equations Pedagogical Plan is based on the idea of engaging students in constructionist activities such as observing, exploring, deconstructing, manipulating, changing and (re)constructing animated models, as they are invited to use and change a half-baked microworld. The “Juggler” microworld serves as a communication tool among the students of the same or different workgroups, enhancing in this way the meaning making process. The generation of meanings is perceived as a process  through the feedback received not merely by the computational tool but also by their peers as they present their ideas, argue and negotiate on their choices, reflect on their actions to support their decisions.

MoPiX v1

MoPiX v1 constitutes a programmable environment that provides the user the opportunity to build, manipulate and share models. These models are created as the user associates algebraic equations - constructed by himself or already available in the environment - with the objects inserted on the Stage. By generating the animations of the models constructed, the user has at his disposal not only the symbolic representation of the behaviour of his model (the equations he used) but also the visual one.

MoPiX v1 can be run in the Internet Explorer or the Mozilla Firefox browser provided that there is a Flash Player installed (version 8 or latest).

Equations Library [Feature]

The “Equations Library” provides the user with a set of pre-defined equations. These equations can be directly dropped onto the objects present on the “Stage” so as to assign them behaviours and properties or be transferred to the “Equations Editor” so as to be modified.

The “Equation Library” may also contain equations constructed by the user himself. After constructing a new equation or editing an existing one, the user may drop the equation into the library so as to store it temporarily into the “Your equations” section.

Picture 1: Equations Library

Drag and Drop [Feature]

With the assistance of the mouse, the user can drag the equations situated in the “Equation Library” and drop them onto the objects present on the “Stage”. The dragging and dropping procedure could be also activated when wishing to remove an equation from an object and discard it or place it in the “Equations Library”.

Picture 1: Drag and Drop

Flipping Objects [Feature]

Each object presented on the “Stage” may be “flipped” by right clicking on it and selecting the “Flip Object” option. Any animation in progress pauses and a new differently coloured window appears. The window contains the equations assigned to the object by the user and the equations assigned to the object by the MoPiX environment (e.g equations defining its x and y position in a Cartesian plane assigned to the object by default once inserted on the “Stage”).

Picture 1: Flip Object

Resources show tooltip helpexplode

General description show tooltip help

User's Manual, Microworlds, Notes, Worksheets

User's manual - Translations in Greek [Resource for students]

The User's Manual for MoPiX v1 in Greek.

An appendix will be added to serve as a quick guide for the MoPiX interface providing the translation in Greek of:

  • the names of the equations available in the "Equations Library" (e.g the "amIHittingASide" equation)

  • the names of the categories of the equations available in the "Equations Library"

  • the names of the buttons available on the "Equations Editor" area

A printed version of the manual will be available to each workgroup while an electronic version will be available at the ETL website.

"One Red Ball" microworld (saved model) [Resource for students]

The “One red ball” is a saved in the MathDiLS library model. It consists of a single object -a ball- performing a combined motion (horizontal and vertical). Each time the ball hits the side boarder lines or the line separating the “Stage” from the “Equations Library”, it bounces and returns in the main area of the “Stage”. The equations associated to the object's behaviour and the object's properties are one-object equations and derive exclusively from the “Equations Library”.

Picture 1: The "One Red Ball" microworld

This model derived from the "Bouncing ball" model that the LKL team had created with MoPiX.

The "One Red Ball" model is saved in a txt file. To run the model open the file (see the Resource Contents section), copy the content and paste it in the window that opens after clicking on the XML button situated at the upper right corner of the MoPiX environment.

Worksheet A [Resource for students]

The worksheet A is designed to support students in describing the red ball's behaviour as they observe the animation generated on the screen. It is given to the students just after loading the "One red ball" model, asking them to describe as accurately as possible the ball's behaviour in their own words. At this phase the students are not yet acquaint with the MoPiX formalism and the environment's features.

Worksheet B [Resource for students]

The worksheet B is designed to support students in connecting the red ball's -already described in the worksheet A- behaviour animated on the screen to the equations' formalism. It is given to the students after asking them to reproduce the ball's motion and after having discussed in plenary sessions the "Flip Object" feature.

Notes [Resource produced by students]

The notes the students keep while collaborating in pairs or collaborating with other workgroups during plenary sessions

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs, sharing the same PC. Under certain circumstances the workgroups could possibly consist of three members.

The furniture layout will allow face-to-face communication between the formed workgroups during the plenary sessions. A portable PC connected to a video projector and having access to the Internet will be available on the table around which students are gathered for the plenary sessions. A white board and markers will also be at the students' disposal.

The implementation of the Pedagogical Plan will take place during school hours since it is extremely difficult for Vocational Education students who also have a professional life to stay at school for extra hours. The teacher involved in the experimentation will provide the number of school hours required for the implementation of the Plan. If necessary other teachers will be asked to provide school hours.

The duration of each school hour is 40 minutes. For each session we will need 2 or 3 school hours according to the designed activities.

Time show tooltip help

6 hours

Actors' roles show tooltip help

The teacher provides guidance to the students so as to load the necessary saved model from the MathDiLS library and start the animation by clicking on the “Play” button.

Students are asked to observe the animation generated and discuss within their workgroups about the ball's behaviour and about the actions performed by the person who created this model in the MoPiX environment in order to make the ball behave the way it does. The students are expected to interact with both the computational environment and their peers, share their ideas, form hypotheses, develop strategies to test these hypotheses and engage in joint decision-making processes, draw conclusion, negotiate and argue about the validity of their conclusions. As we have already distributed the Worksheet A, we ask the students to note down their ideas regarding the ball's behaviour.In this phase of intra workgroup communication (i.e. between the members of each workgroup), the teacher doesn't intervene by giving out information concerning the “correct answer” or by indicating a certain procedure to follow. She/he encourages the students to continue their explorations and participates in workgroup discussionns, posing intriguing questions.

During the second phase an inter workgroup communication takes place. The workgroups gather together, forming a kind of learning community. After having built, tested and evaluated their explanations, students are asked to make public, discuss, explain and justify these explanations. In this phase, the teacher act as a facilitator and co-ordinator, encouraging the members of each workgroup to question both their own ideas and the ideas presented by others and to reflect on their current understating. At this phase, it is important for the students to express their ideas regarding the way in which this model was created in the MoPiX environment. Since they have no familiarity with the MoPiX environment, the teacher at this phase carefully draws their attention to the "Equations Library", aiming to provoke a plenary discussion on the equations' specific role in the animation generated. 

To stimulate students to further use the equations themselves, the teacher asks the students to reproduce the ball's motion by inserting a new object on the Stage. For this phase, the students have at their disposal the Translations of certain equations and categories of equations in Greek. The teacher doesn't intervene, giving out informations about which of the equations should be used, but encourages students to use the translations and select the categories of equations they would use to describe their object's behaviour.

Moreover, the teacher supports students in the exploration of the MoPiX environment and answers questions regarding its functionalities and features. If the members of a workgroup ask questions about the MoPiX environment (e.g. how they could change the Library equations), the teacher provides the necessary answers leaving them room to experiment with MoPiX.

What to do and how show tooltip help

Initially, the teacher will guide the students into accessing the MoPiX environment by visiting the ReMath website (http://remath.cti.gr). The necessary access passwords will be provided to the students so as to access the MoPiX ReMath forum and load from the MathDiLS library the saved model “One red ball”.

In order to load the aforementioned model students will have to:

  • press the “Load” button which is situated on the upper part of the MoPiX environment

  • provide the necessary “username” and “password” on the corresponding fields and click “OK”

  • type the model' s name (not necessary the full one) on the “Enter the tag” field

  • scroll down using the arrows situated next to the “Equation Library” so as to locate the desired model if more than one saved model contain the phrase/word typed on the “Enter the tag” field

  • click “OK” to load the selected model

The students will be asked to press the “Play” button that is situated in the lower left part of the MoPiX environment so as to run the model and generate an animation of a bouncing ball on the “Stage”.

Students will be asked to observe the animation generated and discuss within their workgroups about the ball's behaviour and about the actions performed by the person who created this model in the MoPiX environment in order to make the ball behave the way it does. The Worksheet A will be distributed, and the students will be asked to note down their ideas regarding this issue. The students will be are expected to interact with both the computational environment and their peers, share their ideas, form hypotheses, develop strategies to test these hypotheses and engage in joint decision-making processes, draw conclusions, negotiate and argue about the validity of those conclusions.

Each workgroup's conclusions will be presented to the whole class during a face-to-face discussion. The members of each workgroup will analyse the reasons why they reached such a decision/explanation, while other workgroups will intervene and ask questions concerning the issue discussed. A portable PC, in which the "One red ball" model will be running, will be available for the students to use on the table around which they are gathered.

After arguing, negotiating and deciding on what makes the object behave the way it does and on what kind of actions have been performed on the part of the person who has saved the model, the teacher will ask the students about the role of the equations provided in the “Equations Library” and about the way they could be integrated in the animation of the moving object. To urge students to start using the equations from the "Equations Library", the students will  be asked to create a new object that will move in the exact same way as the original one.

Students - working within their team again- will be encouraged to insert a new object on the Stage and use the MoPiX User's Manual and the Translations so as reproduce the red ball's behaviour. The students are expected to attempt to interpret the Library's equations' formalism and start adding equations to their object so as to evaluate their interpretations through the animation generated.

Picture 1: Adding equations - Dragging equations from the equation library and dropping them onto the object

Some of the equations dragged from the library may not provide an immediate visual effect on the animation. The students will be encouraged to discuss in pairs so as to determine the reasons why the animation isn't affected by the equation assigned to the object and try to locate equations that could have a direct visual effect following a trial-and-error procedure.

The students are also expected during their experimentation with the MoPiX environment to locate the equations that have already been assigned to the object by the person who has produced the saved model. The “Flipping Objects” feature, available by right clicking on the object whose animation is generated, offers this functionality.

Picture 2: The "Flipping Objects" feature

To motivate students to further interpretthe equations' formalism, even after having located the "Flip Object" feature and made the equations underpinning the model's behaviour visible, we will provide the WorkSheet B. The Worksheet B provides all the equations attributed to the ball in a written form and asks the students to describe the behaviour each of these equations gives to the ball. The students have also at their disposal the initial answers they have given in the Worksheet A. In this process we expect the students to use and manipulate the MoPiX equations (add and/or remove them from their object) so as to refine their understanding on the kind of behaviour they attribute to their object and establish links between the animated behaviour generated on the screen and the equations' formalism.

A plenary session will follow so as for the students to discuss and share their understanding on the meaning of the equations' symbols and the behaviours these symbols attribute to the objects when formed into equations. A portable PC, in which the "One red ball" model will be running, will be available for the students to use on the table around which they are gathered.

Process documentation show tooltip help

A screen capture software like HyperCam would be suitable for an experiment with MoPiX. It allows the researcher to record the students' voices and at the same time capture their interactions with the environment. Apart from the HyperCam files, video recordings from the students' plenary sessions will also be available. The students' notes and MoPiX models as well as the researchers' field notes and logs will also constitute the research data.

Editing one-object equations

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Equation (algebraic formalism)
  • Saved model
  • Animations
  • Behaviors and properties
  • Manipulating/using equations
  • Editing equations

Description show tooltip help

After having familiarized with a fist set of the basic functionalities and features of the MoPiX environment, such as adding and removing one-object equations, students get engaged in activities that allow them to create one-object equations for themselves, by editing the Library equations. The activities proposed for this phase could be:

(a) Editing equations that already exist in the "Equations Library"

(b) Adding equations to the "Equations Library"

(c) Creating new objects

(d) Deleting existing objects and equations

After having constructed a new model, each workgroup is asked to provide a name for the new model and save it at the MathDiLS library.

The order of the proposed activities is not obligatory as it is not unlikely for the students, who work in groups of 2 or 3, to decide to explore the environment's functionalities in a different sequence. For example, students could start from creating new objects and assigning them behaviours and properties using equations from the equations library and then edit equations.

Rationale show tooltip helpexplode

The familiarisation process is not perceived by ETL as a process during which the teacher gives out a fixed set of instructions for the students to follow or as a process during which the teacher demonstrates the main functionalities of the environment and the students repeat the teacher's actions step-by-step, following the exact same procedure.

On the contrary, we believe that such a process should take place in a context that allows students to explore the computational environment by themselves and at the same time permits them to engage in activities personally meaningful to them (Kafai & Resnick 1996). This means that we regard the students as being empowered users of the environment instead of passive users having no access to any other functionalities apart from the ones presented by the teacher.

Drawing on the notion of layered learning design (Kahn et al. 2000), we consider dividing the sequence of activities into two phases and developing two microworlds -one for each phase- to be purposeful for the students especially with regard to the frustration such an environment could cause to inexperienced users during their initial explorations (Kahn 2004). Taking into account the natural distinction in MoPiX between one–object and multi–object equations, the microworld we developed for the first phase of the experimentation consists exclusively of one-object equations while the one developed for the second phase contains both one and multi-object equations.

Taking under consideration these perspectives we select to introduce students to an already saved model consisting of a single bouncing ball whose behaviour is defined solely by one-object equations. The designed activities in this phase will allow them to explore the functionalities of the environment while experimenting with the entities and the representations available in the environment, expressing their initial intuitions and ideas, discussing and sharing these ideas with other members of the workgroup or with the whole class, deconstructing and reconstructing artefacts using the environment's available formalism.

The activities proposed in this phase of the pedagogical plan will give students the opportunity to construct mathematical meanings about the role of the equations' algebraic formalism as they interpret, manipulate, use and edit the already existing equations in the process of controlling and creating animated models.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

Basic familiarity with equations

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of using the technology to by-pass conventional practices –such as transmitting the Curriculum’s predefined content- and address open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic formalism in order to control and create animated models. Drawing on the constuctionist and socio-constructivist paradigm, we particularly focused on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researchers so as to a priori correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and social activity.

Content-epistemological goals show tooltip help

  • Using the MoPiX equations' algebraic formalism to create new equations
  • Identifying the MoPiX equations' mathematical structure used to create new one-object algebraic equations
  • Identifying the role of the variable of time in editing one-object equations

  • Identifying the role of the variable ME in editing one-object equations

  • Developing new algebraic equations to define properties and behaviours using the available formalism
  • Connecting the new one - object equations' algebraic formalism to the behaviours animated on the Stage

Cognitive goals show tooltip help

  • Observing, interpreting and analysing the behaviour of an existing one-object model in terms of the algebraic equations used to develop it and the graphical representation generated by its animation.

  • Editing one-object existing equations in order to define properties and behaviours

  • Interconnecting the mathematical formalism (algebraic equations) used to describe the new model's behaviour to the graphical representation produced by the animation of the model.
  • Testing the effect of a newly developed equation by adding/removing it from an object and editing it further

Social-affective goals show tooltip help

  • Discussing in pairs and deciding on the desired behaviours and the properties to be assigned to newly inserted or existing objects

  • Collaborating in pairs in order to decide on the strategies implemented (create an one - object equation from scratch or edit an existing one) so as to assign the objects with specific behaviours and properties or deprive them from specific behaviours and properties

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the modification in the object' s behaviour after editing equations previously assigned to the object

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the modification in the object' s behaviour after constructing new equations and assign them to the object

Instrumental goals show tooltip help

  • Using the “Equations Editor” so as to edit one-object Library equations
  • Discarding newly formed equations and objects present on the “Stage”

  • Introducing new objects on the “Stage”

  • Saving equations in the Equations Library
  • Using the MoPiX environment functionalities and the MathDiLS library to save a new model

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

After having achieved a level of familiarity with some of the basic functionalities of MoPiX, students could engage in activities that will permit them to construct their own one-object model. Once again, instead of giving out a set of instructions for the students to follow, we choose to let them decide on which equations they would be interesting for them to edit and which behaviours and properties they would like their object to hold. Thereby, we offer the students the opportunity to work with the equation's algebraic formalism in a meaningful for them context.

To stimulate students to start editing equations, we have made the “One red ball” move rather slowly. We expect students to be intrigued by this fact and attempt to edit -at least initially- velocity equations so as to make the ball move faster. As students become more and more familiar with using the Equations Editor, we expect them to start expressing their own ideas about their object's motion -not just the velocity- and experiment, modifying and adjusting equations, so as to describe the new behaviours they have in mind.

In the previous phase, by deconstructing the equations assigned to the original object, the students have already develop their understanding on the structure of the equations used in the MoPiX environment and the meaning of the symbols comprising those equations. However, the editing of the equations will give them the opportunity not just to interpret the MoPiX algebraic formalism, but also to use this formalism in order to develop their own equations.

In the process of creating new behaviours for their objects, students are expected to develop hypotheses, to negotiate and argue about the strategies they would follow in order to achieve those new behaviours (e.g. which equation to edit in order to make it move faster: the Vx or the Ax) as well as about the kind of behaviour a new equation may attribute to one of their object when ascribed to it. The feedback received from the computational environment and the social context will help the students to confirm their hypotheses, evaluate the strategies they followed, draw conclusions concerning the effect a newly developed equation may have on their object and decide on further changes to be made so as to accurately describe a new behaviour using the MoPiX formalism. Thus, in this phase, the students are expected to further develop their understanding on the equations' algebraic formalism and structure as well as to use this understanding in creating new artefacts in a meaningful for them context.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist learning environment (Harel and Papert 1991), designed to allow learners to explore, manipulate and construct animated models consisting of objects whose properties and behaviours are defined and controlled by algebraic equations assigned to them. The students’ interactions with the equations’ mathematical formalism and the visual/graphical representation generated on the screen, as well as their interactions with their peers, will give them the opportunity to express ideas and construct mathematical meanings, while engaging in collaborative activities personally meaningful to them (Kafai and Resnick 1996).

The equations attributed to the objects and underpin the models’ behaviour in MoPiX do not constitute “black boxes”, unavailable for inspection or modifications by the user (for a discussion on black and white box approaches see Kynigos 2004). Therefore, the learner in MoPiX is an empowered user who has deep structural access (diSessa 2000) to the models animated. The user may explore a model’s behaviour in terms of its symbolic representation (i.e the equations attributed to the object) and the visual representation displayed when the model is executed, edit it to embed his/hers own ideas or create new models from scratch. Any action performed to a model’s symbolic facet (e.g. inserting/removing equations or editing/modifying equations) produces a direct change to the visual result generated on the Stage. Although the visual representation can not be manipulated to induce changes to the model’s symbolic facet (i.e the representations are nor interdependent), both representations may provide students meaningful feedback, supporting them in their explorations and experimentations with the MoPiX environment, especially in case they attempt to link the representation to each other .

Taking into account those aspects of the environment, ETL considers MoPiX as a computational tool with which one may construct half-baked microworlds (Kynigos 2007) i.e. microworlds that incorporate an interesting idea but are incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. Under this perspective, the half –baked microworld are considered to be improvable objects (Bereiter & Scardamalia 2003), in the sense that they do not constitute end artefacts that can not be further modified, revised or refined, that foster student’s engagement in constructionist, meaning making activities.

Half - baked microworlds are by nature designed for instrumentalisation (Guin and Trouche 2002) since they serve as intriguing starting points and idea generators, stimulating students to transform them from a plain artefact (i.e. a piece of software constructed by humans) to an instrument (Rabardel 2001), engaging in the way in meaning-making processes. Developing the “Juggler” half-baked microworld in the MoPiX environment, we focused on the idea of designing microworlds for instrumentalisation in a collaborative context. The “Juggler” is a game–like microworld consisting of three interrelated objects: a ball and two rackets. What is easily noticeable about the ball’s behaviour is fact that it seems to move rather slowly. The equations attributed to the ball and define its behaviour are not exclusively ready made equations found in the environment’s Library, but also equations created by us using the MoPiX formalism. The rackets do not actively participate in the animation but as the model is executed they can be freely moved around using the mouse. When a racket hits the ball, it forces it to bounce off and move away in a manner that doesn’t allow the user to play with the ball and the rackets like a real juggler. Consequently, the Juggler half-baked microworld constitutes a challenging venue for the students who, in the process of exploring the initial microworld’s behaviour and modifying its functionalities to integrate their own personal ideas, express mathematical meanings as they use algebraic equations to define and control their model’s behaviours.

The half-baked microworlds are also perceived as boundary objects (Cobb et al. 2003), in the sense that can be used to convey meaning among of the members of the same community and thus enhance the meaning generation process as it occurs in the social context of the classroom (Cobb and Yackel 1996). The students in the Juggling with Equations PP are expected to work in groups of two or three discussing, forming and testing hypotheses, engaging in joint decision-making processes, developing strategies to verify the validity of their conclusions, arguing and negotiating. During plenary sessions, all the workgroups come together to present their work, justify their choices and get meaningful feedback on the decisions they have made from their classmates. In both processes, the half baked microworld operates as a tool of communication among the members of the students’ community and constitutes the focal point around which the students organise their activities (Kynigos 2007).

The design of the Juggling with Equations Pedagogical Plan is based on the idea of engaging students in constructionist activities such as observing, exploring, deconstructing, manipulating, changing and (re)constructing animated models, as they are invited to use and change a half-baked microworld. The “Juggler” microworld serves as a communication tool among the students of the same or different workgroups, enhancing in this way the meaning making process. The generation of meanings is perceived as a process  through the feedback received not merely by the computational tool but also by their peers as they present their ideas, argue and negotiate on their choices, reflect on their actions to support their decisions.

MoPiX v1

MoPiX v1 constitutes a programmable environment that provides the user the opportunity to build, manipulate and share models. These models are created as the user associates algebraic equations - constructed by himself or already available in the environment - with the objects inserted on the Stage. By generating the animations of the models constructed, the user has at his disposal not only the symbolic representation of the behaviour of his model (the equations he used) but also the visual one.

MoPiX v1 can be run in the Internet Explorer or the Mozilla Firefox browser provided that there is a Flash Player installed (version 8 or latest).

Equations Library [Feature]

The “Equations Library” provides the user with a set of pre-defined equations. These equations can be directly dropped onto the objects present on the “Stage” so as to assign them behaviours and properties or be transferred to the “Equations Editor” so as to be modified.

The “Equation Library” may also contain equations constructed by the user himself. After constructing a new equation or editing an existing one, the user may drop the equation into the library so as to store it temporarily into the “Your equations” section.

Picture 1: Equations Library

Drag and Drop [Feature]

With the assistance of the mouse, the user can drag the equations situated in the “Equation Library” and drop them onto the objects present on the “Stage”. The dragging and dropping procedure could be also activated when wishing to remove an equation from an object and discard it or place it in the “Equations Library”.

Picture 1: Drag and Drop

Flipping Objects [Feature]

Each object presented on the “Stage” may be “flipped” by right clicking on it and selecting the “Flip Object” option. Any animation in progress pauses and a new differently coloured window appears. The window contains the equations assigned to the object by the user and the equations assigned to the object by the MoPiX environment (e.g equations defining its x and y position in a Cartesian plane assigned to the object by default once inserted on the “Stage”).

Picture 1: Flip Object

Equations Editor [Feature]

The “Equations Editor” is situated at the upper part of the “Stage” and consists of several buttons as well as an editing area. It may be used to edit already existing equations or construct new equations from scratch.

An equation can be edited when dragged from the library or any other point of the Stage and dropped onto the editing area. As the equation to be edited opens up, its structural components become accessible to the user.

By pressing the “New Equation” button, a template appears at the editing area. The fourteen “Operator buttons” and the “Function button” are to be used so as to complete the fields on the emerging template. In order to generate an acceptable MoPiX equation a binary tree structure has to be followed.

The new equations created may be dragged and dropped onto the objects or dragged and dropped into the “Equations Library” to be stored temporarily. Alternatively, they could be stored for future use in the MathDiLS library.

Picture 1: Equations Editor

Resources show tooltip helpexplode

General description show tooltip help

User's Manual, Microworlds, Models, Notes, Work sheets

User's manual -Translations in Greek [Resource for students]

The User's Manual for MoPiX v1 in Greek.

An appendix will be added to serve as a quick guide for the MoPiX interface providing the translation in Greek of:

  • the names of the equations available in the "Equations Library" (e.g the "amIHittingASide" equation)

  • the names of the categories of the equations available in the "Equations Library"

  • the names of the buttons available on the "Equations Editor" area

A printed version of the manual will be available to each workgroup while an electronic version will be available at the ETL website.

"One Red Ball" microworld (saved model) [Resource for students]

The “One red ball” is a saved in the MathDiLS library model. It consists of a single object -a ball- performing a combined motion (horizontal and vertical). Each time the ball hits the side boarder lines or the line separating the “Stage” from the “Equations Library”, it bounces and returns in the main area of the “Stage”. The equations associated to the object's behaviour and the object's properties are one-object equations and derive exclusively from the “Equations Library”.

Picture 1: The "One Red Ball" microworld

This model derived from the "Bouncing ball" model that the LKL team had created with MoPiX.

The "One Red Ball" model is saved in a txt file. To run the model open the file (see the Resource Contents section), copy the content and paste it in the window that opens after clicking on the XML button situated at the upper right corner of the MoPiX environment.

Worksheet C [Resource for students]

The worksheet C is designed to support students in defining new behaviours and editing Library equations to describe those behaviours. It is given to the students after having a plenary discussion on the "Equations Editor" feature.

Notes [Resource produced by students]

Notes kept by the students while collaborating in pairs

Models [Resource produced by students]

Each workgroup develops a one-object equations model and saves it at the MathDiLS library so as to render it shareable.

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs, sharing the same PC. Under certain circumstances the workgroups could possibly consist of three members.

The furniture layout will allow face-to-face communication between the formed workgroups during the plenary sessions. A portable PC connected to a video projector and having access to the Internet will be available on the table around which students are gathered for the plenary sessions. A white board and markers will also be at the students' disposal.

The implementation of the Pedagogical Plan will take place during school hours since it is extremely difficult for Vocational Education students who also have a professional life to stay at school for extra hours. The teacher involved in the experimentation will provide the number of school hours required for the implementation of the Plan. If necessary other teachers will be asked to provide school hours.

The duration of each school hour is 40 minutes. For each session we will need 2 or 3 school hours according to the designed activities.

Time show tooltip help

4 hours

Actors' roles show tooltip help

In this phase of their experimentation, the teacher's main role is to support the students in their experimentations with the Equations Editor feature. Since the students have at their disposal the User's Manual, the teacher urges the students to use it -instead of asking the teacher to provide instructions at each step of their explorations- and intervenes just to make sure that the students' have taken into consideration the computational environment's limitations.

The students discuss within their workgroups the new behaviours they would like to attribute to their objects, form their hypotheses and argue about the strategies they would follow in order to achieve those new behaviours. A plenary session will give students the opportunity to present their ideas on their model's behaviour, justify their choices on how to implement their ideas and get feedback regarding the feasibility of those ideas from other workgroups. Both during the discussions within the same work group and during the discussions among the workgroups, the teacher doesn't interfere with the students' decisions, allowing them to create artefacts that would be meaningful to them. She/he encourages the students to continue their explorations implementing trial-and-error procedures and participates in the discussions, posing intriguing questions that could help students refine their artefacts.

In case the animation generated after attributing an equation to an object does not correspond to the students' ideas when forming the equation, the teacher urges the students to reconsider their choices, to deconstruct the equation to examine its content and structure and reflect on the links they have made between the symbols of their equation and the interptretations they have given to those symbols.

Moreover, the teacher continues to support students in the exploration of the MoPiX environment and answers questions regarding its functionalities and features. If the members of a workgroup ask questions about the how to construct a new equation from scratch, the teacher provides the necessary answers leaving them room to experiment with MoPiX and their models.

What to do and how show tooltip help

Initially, the teacher offers guidance to the workgroups so as to access the MoPiX environment by visiting the ReMath website (http://remath.cti.gr) and load from the MathDiLS library the saved model “One red ball”.

During the previous phase the students have already explored some of the basic functionalities of the microworld and have already been introduced to some of the main features of the MoPiX environment. Thus, they are expected to start the animation without requiring any specific instructions from the teacher.

After giving the students some time so as to interact with the microworld and observe the animation generated, the teacher will ask the students what they would change in the behaviour or the properties of the object on the “Stage” if they had the chance to do so. Students are expected to discuss such an option while working in pairs and come up with some ideas about making the object behave in a different way and therefore produce a different animation of its motion. Since there is a high possibility that some of the students' suggestions could be infeasible or extremely time - consuming the teacher asks the students to discuss not only about making the object behave differently but also about how this will be achieved after taking into consideration certain limitations. The WorkSheet C will be distributed.

While discussing in pairs about the new desired behaviour assigned to the object and how it will be attained, students could use some of the features of the MoPiX to which they have already been introduced during the previous phase, such as the “Flipping Objects” feature. This feature's main purpose is to render visible the equations attributed to the object. Making visible the equations defining the objects initial behaviour and trying to deconstruct them has a twofold effect on students understanding.

Firstly, they develop their understanding about what makes the object behave the way it does e.g the x (ME,t) defines the objects position on X axis and secondly, they develop their understanding on the structure of the equation e.g in the x (ME,t) equation ME represents a variable and it has to be included in the function on the left part of the equation.

In order to attribute a new behaviour to the object, it is likely for the students to think at first place to modify the existing equations assigned to the object and particularly those equations that include numerical values. For example they could think of editing the equation Vx (ME,0) = 3 and changed it into Vx (ME,0) = 4. However, after acquiring some familiarity with the Editor and having developed an understanding regarding the structure of the one-object equations and MoPiX algebraic formalism, students are expected to start editing other elements of the originals equations, such as the variables.

Picture 1:Using the "Flipping Objects" feature and editing an equation using the "Equations Editor"

The students are expected to assign the newly constructed equations to the object and observe its behaviour. The animation generated by the tool will help them test their original hypotheses concerning the object's behaviour after assigning it the equation they constructed and evaluate the strategies they implemented so as to construct the equation.

In case the behaviour of the object is not the desired one, students are urged by the teacher to deconstruct their equations, examine its content and its structure. A trial-and-error procedure, the animation offered by the tool and the intra workgroup discussions will help students to refine their models.

The students are expected to assign the newly constructed equations to the object and observe the new behaviours. The animation generated by the tool will help them test their original hypotheses concerning the object's behaviour after assigning it the equation they created, and evaluate the strategies they implemented so as to produce the equation. In case the behaviour of the object is not the desired one, students are urged by the teacher to deconstruct their equations, examine its content and its structure. A trial-and-error procedure, the animation offered by the tool and the intra workgroup discussions will help students to refine their models.

Near the end of this phase, we expect the students to use the “New Equation” button so as to create from scratch new one-object equations. The User's Manual and Translations on the Editor's Buttons' names could be proved to be helpful for the students' new explorations.

Picture 2: Using the "New Equation" and the "Function" button to construct an equation

Finally, in order to save the one-object equations model the students created themselves and render it shareable through the MathDiLS library, with the assistance of the teacher, they will follow the procedure mentioned below:

  • press the “Save” button which is situated on the upper part of the MoPiX environment

  • provide the necessary “username” and “password” on the corresponding fields an click “OK”

  • type the model' s desired name on the “Enter the tag” field

  • click “OK” to save the new model

Process documentation show tooltip help

A screen capture software like HyperCam would be suitable for an experiment with MoPiX. It allows the researcher to record the students' voices and at the same time capture their interactions with the environment. Apart from the HyperCam files, video recordings from the students' plenary sessions will also be available. The students' notes and MoPiX models as well as the researchers' field notes and logs will also constitute the research data.

Multi - object equations

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Equations (algebraic formalism)
  • Saved model
  • Animation
  • Behavior and properties
  • Multi - object equations
  • Deconstructing models
  • Reconstructing models
  • Sharing and integrating models

Description show tooltip help

The “Multi-Object Equations” sequence of tasks introduces students to the “Juggler” microworld and aims at giving them the opportunity to play with the “Juggler” game according to the rules governing the microworld, deconstruct it, reconstruct it according to their personal ideas and finally share parts or the complete customized microworld. The activities with which students get engaged are of three kinds:

(a) Activities aiming at deconstructing the already existing multi-object equations model

(b) Activities aiming at modifying the already existing multi-object equations model and using it as a starting point to construct a new ones

(c) Activities aiming at sharing the newly developed models

Rationale show tooltip helpexplode

The activities designed in this part of the pedagogical plan wish to introduce the students into the “Juggler” microworld and its functionalities. The microworld consists of three objects whose behaviours are linked to each other's. This connection between the objects is perceived both by means of symbolic representation and by means of the visual representation.

Working in this meaningful context (Kafai and Resnick, 1996) students are expected to explore the functionalities of the microworld and deconstruct the saved model so as to define the equations underpinning the behaviour and properties of each object both as a standalone object and with regard to others. The deconstruction procedure will give the students the opportunity to gain understanding of the equation's formalism used to express behaviours and properties and of the mathematical structure used to express conditionals. The equations that underpin the microworld's behaviour are both "Library" equations as well as equations that we have constructed ourselves, using the MoPiX formalism.

Students will have the chance to act as game designers and modify the initial model. After reaching a joint decision concerning the behaviour and properties to be assigned to each object, the connection between the objects and the conditions under which a certain behaviour will be demonstrated, the students will save their model and make it public using the MathDiLs library.

Finally, the members of each workgroup decide on what objects they will give to other workgroups and ask them to integrate those objects into their model. The interaction between the workgroups will afford the opportunity for the students to reflect on their models and identify their pitfalls and inadequacies. This kind of reflection will lead the students into refining their models and acquiring a deeper understanding of the MoPiX formalism and  the ways in which the equations underpin the models' behaviour.

Viewing the construction of mathematical meanings as a social phenomenon (Cobb and Yackel 1996), the exploration of the microworld will take place in social context, as students collaborate in pairs. Socially mediated activities in combination with the computational tool offer students the opportunity to share with peers their current understanding and make it an object of discussion and reflection as feedback is received both from the microworld and the social context.

Moreover, while working with this microworld students elaborate their skills in using the MoPiX environment as they are introduced to one more feature i.e. the “Multi - Object Equations” feature.

The sequence of the activities proposed is not strictly predetermined as students are likely to decide on exploring different aspects and functionalities of the microworld. For example, students could detect the existence of multi - link equations as they explore the “Flip Object” feature first and then discuss how this equations influence the visual representation generated when executing the model.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

  • Basic familiarity with equations

  • Basic familiarity with motion equations

  • Familiarity with basic MoPiX functionalities

  • Famililarity with the MoPiX algebraic formalism

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of using the technology to by-pass conventional practices –such as transmitting the Curriculum’s predefined content- and address open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic formalism in order to control and create animated models. Drawing on the constuctionist and socio-constructivist paradigm, we particularly focused on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researchers so as to a priori correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and social activity.

Content-epistemological goals show tooltip help

  • Interpreting the multi - object equations' algebraic formalism and structure
  • Interconnecting the multi - object equations' algebraic formalism to the behaviours animated on the Stage
  • Identifying the mathematical structure used in the MoPiX algebraic equations to express conditions
  • Manipulating existing one-object and multi-object equations to assign properties and behaviours to the objects or to withdraw properties and behaviours from the objects

  • Editing existing one-object and multi-object equations to describe new behaviours
  • Constructing new algebraic equations to define properties and behaviours using the available formalism

Cognitive goals show tooltip help

  • Observing, interpreting and analysing the behaviour of the objects in an multi-object equations model -perceiving them both as standalone objects and with regard to the other objects present in the same model- in terms of the algebraic equations used to develop it and the graphical representation generated by its animation

  • Interconnecting the mathematical formalism (algebraic equations) used to describe the behaviour and the properties of each object to the graphical representation produced by the animation.
  • Interconnecting the mathematical formalism (algebraic equations) used to describe the relations between the objects to the graphical representation produced by the animation.

Social-affective goals show tooltip help


  • Discussing and analysing in pairs the behaviour and the properties of the objects constituting the multi-object saved model

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the objects' behaviour as a standalone object

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the objects' behaviour with regard to the behaviour of other objects

  • Collaborating in pairs discussing, forming and testing hypotheses, negotiating and reaching in joint conclusions concerning the conditions under which the objects behave in certain ways

Instrumental goals show tooltip help

  • Using the MoPiX environment functionalities at their full extend and the MathDiLS library features

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

For the second phase of the activities we designed a half–baked microworld (Kynigos 2007), i.e. a microworld that incorporates an interesting idea but it is incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. In this case we built a game–like microworld –called “Juggler” (Kynigos 2007) – consisting of three interrelated objects: a red ball and two rackets with which the ball interacted (using multi-object equations). The ball’s behaviour was partially the same as the “One Red Ball’s”. However, certain equations underpinning the ball’s behaviour with regard to the rackets, don't derive from the environment’s “Library” but were created by us, using the MoPiX formalism. The rackets do not actively participate in the animation (i.e their attributes do not vary over time), but it is possible to move them around using the mouse and make the ball bounce on them, forcing it to move away in specific ways.

We will ask the students to execute the Juggler’s model, observe the animation generated and identify the conditions under which each object interacts with the others as well as the objects’ possible changes of behaviour because of these interactions. The students will be encouraged to discuss with their teammates on how they would change the “Juggler” microworld and embed in it their own ideas regarding its behaviour. In the process of changing the half–baked microworld, students are expected to deconstruct the existing model so as to link the behaviours generated on the screen to its equations’ formalism and reconstruct the microworld, employing strategies that depict their ideas about the new model’s animated behaviours.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist learning environment (Harel and Papert 1991), designed to allow learners to explore, manipulate and construct animated models consisting of objects whose properties and behaviours are defined and controlled by algebraic equations assigned to them. The students’ interactions with the equations’ mathematical formalism and the visual/graphical representation generated on the screen, as well as their interactions with their peers, will give them the opportunity to express ideas and construct mathematical meanings, while engaging in collaborative activities personally meaningful to them (Kafai and Resnick 1996).

The equations attributed to the objects and underpin the models’ behaviour in MoPiX do not constitute “black boxes”, unavailable for inspection or modifications by the user (for a discussion on black and white box approaches see Kynigos 2004). Therefore, the learner in MoPiX is an empowered user who has deep structural access (diSessa 2000) to the models animated. The user may explore a model’s behaviour in terms of its symbolic representation (i.e the equations attributed to the object) and the visual representation displayed when the model is executed, edit it to embed his/hers own ideas or create new models from scratch. Any action performed to a model’s symbolic facet (e.g. inserting/removing equations or editing/modifying equations) produces a direct change to the visual result generated on the Stage. Although the visual representation can not be manipulated to induce changes to the model’s symbolic facet (i.e the representations are nor interdependent), both representations may provide students meaningful feedback, supporting them in their explorations and experimentations with the MoPiX environment, especially in case they attempt to link the representation to each other .

Taking into account those aspects of the environment, ETL considers MoPiX as a computational tool with which one may construct half-baked microworlds (Kynigos 2007) i.e. microworlds that incorporate an interesting idea but are incomplete by design so as to invite students to deconstruct it, build on its parts, customize and change it, eventually constructing a new artefact that could be distinctly different than the original one. Under this perspective, the half –baked microworld are considered to be improvable objects (Bereiter & Scardamalia 2003), in the sense that they do not constitute end artefacts that can not be further modified, revised or refined, that foster student’s engagement in constructionist, meaning making activities.

Half - baked microworlds are by nature designed for instrumentalisation (Guin and Trouche 2002) since they serve as intriguing starting points and idea generators, stimulating students to transform them from a plain artefact (i.e. a piece of software constructed by humans) to an instrument (Rabardel 2001), engaging in the way in meaning-making processes. Developing the “Juggler” half-baked microworld in the MoPiX environment, we focused on the idea of designing microworlds for instrumentalisation in a collaborative context. The “Juggler” is a game–like microworld consisting of three interrelated objects: a ball and two rackets. What is easily noticeable about the ball’s behaviour is fact that it seems to move rather slowly. The equations attributed to the ball and define its behaviour are not exclusively ready made equations found in the environment’s Library, but also equations created by us using the MoPiX formalism. The rackets do not actively participate in the animation but as the model is executed they can be freely moved around using the mouse. When a racket hits the ball, it forces it to bounce off and move away in a manner that doesn’t allow the user to play with the ball and the rackets like a real juggler. Consequently, the Juggler half-baked microworld constitutes a challenging venue for the students who, in the process of exploring the initial microworld’s behaviour and modifying its functionalities to integrate their own personal ideas, express mathematical meanings as they use algebraic equations to define and control their model’s behaviours.

The half-baked microworlds are also perceived as boundary objects (Cobb et al. 2003), in the sense that can be used to convey meaning among of the members of the same community and thus enhance the meaning generation process as it occurs in the social context of the classroom (Cobb and Yackel 1996). The students in the Juggling with Equations PP are expected to work in groups of two or three discussing, forming and testing hypotheses, engaging in joint decision-making processes, developing strategies to verify the validity of their conclusions, arguing and negotiating. During plenary sessions, all the workgroups come together to present their work, justify their choices and get meaningful feedback on the decisions they have made from their classmates. In both processes, the half baked microworld operates as a tool of communication among the members of the students’ community and constitutes the focal point around which the students organise their activities (Kynigos 2007).

The design of the Juggling with Equations Pedagogical Plan is based on the idea of engaging students in constructionist activities such as observing, exploring, deconstructing, manipulating, changing and (re)constructing animated models, as they are invited to use and change a half-baked microworld. The “Juggler” microworld serves as a communication tool among the students of the same or different workgroups, enhancing in this way the meaning making process. The generation of meanings is perceived as a process  through the feedback received not merely by the computational tool but also by their peers as they present their ideas, argue and negotiate on their choices, reflect on their actions to support their decisions.

MoPiX v1

MoPiX v1 constitutes a programmable environment that provides the user the opportunity to build, manipulate and share models. These models are created as the user associates algebraic equations - constructed by himself or already available in the environment - with the objects inserted on the Stage. By generating the animations of the models constructed, the user has at his disposal not only the symbolic representation of the behaviour of his model (the equations he used) but also the visual one.

MoPiX v1 can be run in the Internet Explorer or the Mozilla Firefox browser provided that there is a Flash Player installed (version 8 or latest).

Equations Library [Feature]

The “Equations Library” provides the user with a set of pre-defined equations. These equations can be directly dropped onto the objects present on the “Stage” so as to assign them behaviours and properties or be transferred to the “Equations Editor” so as to be modified.

The “Equation Library” may also contain equations constructed by the user himself. After constructing a new equation or editing an existing one, the user may drop the equation into the library so as to store it temporarily into the “Your equations” section.

Picture 1: Equations Library

Drag and Drop [Feature]

With the assistance of the mouse, the user can drag the equations situated in the “Equation Library” and drop them onto the objects present on the “Stage”. The dragging and dropping procedure could be also activated when wishing to remove an equation from an object and discard it or place it in the “Equations Library”.

Picture 1: Drag and Drop

Flipping Objects [Feature]

Each object presented on the “Stage” may be “flipped” by right clicking on it and selecting the “Flip Object” option. Any animation in progress pauses and a new differently coloured window appears. The window contains the equations assigned to the object by the user and the equations assigned to the object by the MoPiX environment (e.g equations defining its x and y position in a Cartesian plane assigned to the object by default once inserted on the “Stage”).

Picture 1: Flip Object

Equations Editor [Feature]

The “Equations Editor” is situated at the upper part of the “Stage” and consists of several buttons as well as an editing area. It may be used to edit already existing equations or construct new equations from scratch.

An equation can be edited when dragged from the library or any other point of the Stage and dropped onto the editing area. As the equation to be edited opens up, its structural components become accessible to the user.

By pressing the “New Equation” button, a template appears at the editing area. The fourteen “Operator buttons” and the “Function button” are to be used so as to complete the fields on the emerging template. In order to generate an acceptable MoPiX equation a binary tree structure has to be followed.

The new equations created may be dragged and dropped onto the objects or dragged and dropped into the “Equations Library” to be stored temporarily. Alternatively, they could be stored for future use in the MathDiLS library.

Picture 1: Equations Editor

Linking objects [Feature]

The behaviour and the properties of each object present on the “Stage” may be linked to the behaviour and the properties of other objects, using multi-object equations. These equations contain both the ME and OTHER variable (possibly OTHER1, OTHER2,…) are dragged and dropped first to the object whose behaviour is to be defined (ME), then to the other object(s) (OTHER1, OTHER2,…) and finally back to the first object (ME).

Resources show tooltip helpexplode

General description show tooltip help

User's Manual, Microworlds, Notes, Saved object(s)

“Juggler” microworld (saved model) [Resource for students]

The “Juggler” is a saved in the MathDiLS library model. It consists of three objects, one ball and two rackets. The ball performs a two-dimensional combined motion (both in the horizontal and vertical axis). Each time the ball hits the side boarder lines, the line separating the “Stage” from the “Equations Library” or one of the two rackets, it bounces and returns in the main area of the “Stage”. Some the equations associated to the ball's behaviours and properties derive from the “Equations Library” while others were built from scratch by us.

Picture 1: The Juggler microworld

The "Juggler" model is saved in a txt file. To run the model open the file (see the Resource Contents section), copy the content and paste it in the window that opens after clicking on the XML button situated at the upper right corner of the MoPiX environment.

User's manual -Translations in Greek [Resource for students]

The User's Manual for MoPiX v1 in Greek.

An appendix will be added to serve as a quick guide for the MoPiX interface providing the translation in Greek of:

  • the names of the equations available in the "Equations Library" (e.g the "amIHittingASide" equation)

  • the names of the categories of the equations available in the "Equations Library"

  • the names of the buttons available on the "Equations Editor" area

A printed version of the manual will be available to each workgroup while an electronic version will be available at the ETL website.

Work Sheet [Resource for students]

In this phase of the Pedagogical Plan we provided a worksheet in order to support students in the deconstruction of the "Juggler" microworld. A detailed description of this worksheet will be presented at the "Deconstructing multi-object equations".

Resource contents show tooltip help
Notes [Resource produced by students]

The notes the students keep while collaborating in pairs or collaborating with other workgroups during plenary sessions

Work plan show tooltip helpexplode

Setting show tooltip help

Students will be working in pairs, sharing the same PC. Under certain circumstances the workgroups could possibly consist of three members.

The furniture layout will allow face-to-face communication between the formed workgroups during the plenary sessions. A portable PC connected to a video projector and having access to the Internet will be available on the table around which students are gathered for the plenary sessions. A white board and markers will also be at the students' disposal.

The implementation of the Pedagogical Plan will take place during school hours since it is extremely difficult for Vocational Education students who also have a professional life to stay at school for extra hours. The teacher involved in the experimentation will provide the number of school hours required for the implementation of the Plan. If necessary other teachers will be asked to provide school hours.

The duration of each school hour is 40 minutes. For each session we will need 2 or 3 school hours according to the designed activities.

Time show tooltip help

15 hours

Actors' roles show tooltip help

During their interaction with the computational and the social environment students will engage in activities permitting them to express their ideas and original intuitions, discuss and share these ideas with other workgroup members or with the whole class during plenary sessions, formulate, test, reject or accept hypotheses, draw conclusions/assumptions, negotiate and argue about validity of their conclusions, deconstruct and reconstruct artefacts.

The teacher is expected not to intervene in this process by providing the “correct answer” to the students when asked or giving out instructions about what to do next and how. On the contrary, the teacher is expected to guide students discretely by posing meaningful- often intriguing- questions, urging them to continue their experimentation, collaborate and discuss their ideas with their peers (the members of the workgroup or other workgroups). At the same time the teacher has to support the students both intellectually and emotionally so as not to get discouraged when they encounter problematic situations requiring task solving skills.

The researcher will provide technical support in the use of MoPiX both for the teacher and the students. During the experimentation process he will be observing the classroom activity and collecting data while handling the video and audio recorders. Under certain circumstances the researcher could focus on particular workgroups taking detailed data (notes or recordings) of their interactions.

What to do and how show tooltip help

The "What to do an how" is described in details in the "Deconstructing multi-object equations", "(Re)Constructing multi-object equations" and the "Exchanging hands and ballls" section.

Process documentation show tooltip help

A screen capture software like HyperCam would be suitable for an experiment with MoPiX. It allows the researcher to record the students' voices and at the same time capture their interactions with the environment. Apart from the HyperCam files, video recordings from the students' plenary sessions will also be available. The students' notes and MoPiX models as well as the researchers' field notes and logs will also constitute the research data.

Deconstructing multi - object equations

Identity show tooltip helpexplode

Authors show tooltip help

Foteini Moustaki, Chronis Kynigos

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • Equations
  • Algebraic formalism
  • Modeling
  • Equations of motion

Language show tooltip help

English

Country show tooltip help

Greece

Keywords show tooltip help

  • Equations (algebraic formalism)
  • Saved model
  • Animation
  • Behavior and properties
  • Multi - object equations
  • Deconstructing models

Description show tooltip help

After loading from the MathDiLS library the “Juggler” microworld students engage in activities that permit them to deconstruct a multi-object model consisting of several objects linked to each other with algebraic equations. The majority of these equations are constructed by the author of the microworld and they are not included in the “Equations Library”. The activities proposed for this phase could be:

(a) Executing the saved model and observing the animation generated

(b) Discussing with peers about the objects' behaviour and properties as standalone objects

(c) Discussing with peers about the objects' behaviour and properties with regard to others objects' behaviour and properties

(d) Deconstructing the model's equations and interpreting the formalism use to develop those equations

(e) Identifying the conditions under which the behaviour or the properties of each object are influenced by the behaviour and the properties of other objects and linking them to the corresponding equations

The order of the proposed activities is not obligatory as it is not unlikely for the students who work in groups of two or three to decide to explore the functionalities of the microworld in a different sequence. For example, students could come across the equations that link the objects and identify them before discussing how these equations affect the behaviour and the properties of the objects at hand.

Rationale show tooltip helpexplode

The activities designed in this part of the pedagogical plan wish to introduce the students into the “Juggler” microworld and its functionalities. The microworld consists of three objects whose behaviours are linked to each other's. This connection between the objects is perceived both by means of symbolic representation and by means of the visual representation.

Working in this meaningful context (Kafai and Resnick, 1996) students are expected to explore the functionalities of the microworld and deconstruct the saved model so as to define the equations underpinning the behaviour and properties of each object both as a standalone object and with regard to others. The deconstruction procedure will give the students the opportunity to gain understanding of the equation's formalism used to express behaviours and properties and of the mathematical structure used to express conditionals. The equations that underpin the microworld's behaviour are both "Library" equations as well as equations that we have constructed ourselves, using the MoPiX formalism.

Students will have the chance to act as game designers and modify the initial model. After reaching a joint decision concerning the behaviour and properties to be assigned to each object, the connection between the objects and the conditions under which a certain behaviour will be demonstrated, the students will save their model and make it public using the MathDiLs library.

Finally, the members of each workgroup decide on what objects they will give to other workgroups and ask them to integrate those objects into their model. The interaction between the workgroups will afford the opportunity for the students to reflect on their models and identify their pitfalls and inadequacies. This kind of reflection will lead the students into refining their models and acquiring a deeper understanding of the MoPiX formalism and  the ways in which the equations underpin the models' behaviour.

Viewing the construction of mathematical meanings as a social phenomenon (Cobb and Yackel 1996), the exploration of the microworld will take place in social context, as students collaborate in pairs. Socially mediated activities in combination with the computational tool offer students the opportunity to share with peers their current understanding and make it an object of discussion and reflection as feedback is received both from the microworld and the social context.

Moreover, while working with this microworld students elaborate their skills in using the MoPiX environment as they are introduced to one more feature i.e. the “Multi - Object Equations” feature.

The sequence of the activities proposed is not strictly predetermined as students are likely to decide on exploring different aspects and functionalities of the microworld. For example, students could detect the existence of multi - link equations as they explore the “Flip Object” feature first and then discuss how this equations influence the visual representation generated when executing the model.

Theoretical framework show tooltip help

ETL perceives MoPiX as a constructionist artefact integrating the use of mathematical formalism with the use of graphical representations and their dynamic manipulation. What is particularly interesting for the team is the potential to create half-baked game-like microworlds (Kynigos, 2007) designed for instrumentalization, i.e. so that users will want to change the rules of the half-baked games and thus engage in constructionist activity. We perceive that activity to involve integration of verbal and written communication, negotiation and argumentation between users working in small groups. What is also central to our approach is that the driving force behind constructionist activity with the artefact is the use of mathematical formalism in the form of equations in the role of programs for the behaviour and properties of virtual objects. We have been engaged for years in the design and development of such artefacts integrating different representations but always with mathematical formalism in a driving role (Turtleworlds and many other E-slate microworlds). In MoPiX, we see this characteristic with emphasis on the ways in which formalisms are handled as entities which apply to an object when they are drag-dropped on it and on the semantic choice of the status of formalisms, i.e. that they are standalone programs in themselves which the user just aggregates on an object -- this means that there is limited emphasis on a language to connect or structure formalisms. We would like to see a functionality which would help structure the equations and ‘hide' them from users so that black-white box design decisions can be made. However, in our scenario design, we will focus on the idea of design for instrumentalization where engineer students from a vocational school change a half baked microworld and informally use equations in the process.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

One of ETL's main fields of interest has been the development of technology-enhanced innovative activities. Both during the phase of designing such activities and the implementation process, the Greek educational system's dominant characteristics have always been considered as an integral part of the institutional/cultural context of the teaching experiment.

The Greek educational system is considered to be highly centralised (Kontogiannopoulou - Polydorides, G. & Kynigos, C. 1993). The National Curriculum constitutes a pre-prescribed set of instructions for the teacher to follow, imposing uniformity in the educational practice and leaving no room for initiatives on behalf of the teacher or the school. Thus, the teacher is perceived as the technical implementer of the curriculum (Kynigos, 2004) having officially no right to implement his personal educational agenda or put into practice any teaching methods alternative to the traditional ones.

Although the use of technology is partially integrated in the National Curriculum, the educational use of technology is limited and depends exclusively on the teacher's will to integrate innovative practices in his teaching agenda. Even so, most of times, the use of technology is perceived -both by the teachers and the school administration- as a new, fancy way to deliver the defined by the curriculum content and not as means for expression and construction. The “revelatory” educational paradigm (reveal the students the information considered to be right or true) -deeply rooted in the educational system (Kontogiannopoulou-Polidorides, 1996) and explicitly reflected on the classroom norms- remains is salient even when teachers integrate in their agenda the use of educational software.

Inevitably, in this context, both students and teachers consider learning to be a merely individualistic procedure during which the teacher attempts to transmit knowledge, usually through problem solving activities, leaving no room for experimentation, meaning generation, personal expression, meaningful constructions and collaboration among peers. The implementation of innovative activities is extremely rare and bound to generate perturbation (Laborde, 2001) both in the school-level and the classroom-level.

ETL perceives the implementation of a Pedagogical Plan of this type as an opportunity to generate turbulence in a context in which teachers strictly follow of the National Curriculum’s goals, leaving no room to their students for explorations and experimentations with purposefully designed computational tools. The design of our activities was based on the idea of addressing open didactical goals that would not directly correspond to specific National Curriculum educational goals. Our aim was to create a meaningful context in which students could interpret, manipulate and use the algebraic mathematical formalism in order to control and create animated models. We particularly focus on the meaning generation process taking into account the students’ interactions with the available computational tool and the social environment. The meanings generated are not predefined by the researcher so as to correlate them with specific curricular goals, but are considered to be emergent and shaped by the students’ mathematical and technological activity.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 21 years old

Population description show tooltip help

Vocational education students studying mechanical engineering.

The pedagogical plans after the essential modifications may also be appropriate for lower secondary education students or upper secondary General education students.

Student prerequisites show tooltip help

  • Basic familiarity with equations

  • Basic familiarity with motion equations

  • Familiarity with basic MoPiX functionalities

  • Famililarity with the MoPiX algebraic formalism

Teacher prerequisites show tooltip help

The teacher should be familiar with the MoPiX environment its features and functionalities. In order to achieve a degree of familiarity with MoPiX environment, a username and a password to the ReMath portal will be provided to the teacher.

Moreover, we wish the teacher to be familiar with the educational objectives addressed by he Pedagogical Plan. For this reason, the teacher could participate in the final configuration of the goals addressed by the Plan's activities.

Context show tooltip helpexplode

Physical context show tooltip help

The implementation of the Pedagogical Plan will take place in a networked computer laboratory equipped with PCs. All PCs will be connected to the Internet and all the students will hold connection usernames and passwords.

Institutional context show tooltip help

The Pedagogical Plan will be implemented in a Secondary Vocational Education school in Athens. The secondary vocational education schools comprise three grades and accept students who have graduated from the Lower Secondary Education schools. At 3rd grade, mathematics courses are held for three school hours every week. The school time-schedule, the educational goals addressed and the materials used in classroom are exclusively determined by the National Curriculum, leaving no room for action to the institution or the teacher itself.

Socio-cultural context show tooltip help

The Vocational Educational students use mathematics both in their everyday school and professional life. They perceive mathematics mainly as means to carry out their professional routine activities such as calculating, measuring, sorting out, designing and programming. In order to do so as accurately, effectively and quickly as possible, students often develop modelling techniques. Most of the times they appear to be unaware of the fact that they are actually constructing mathematical models and seem to make no connections between the mathematics the use at work and school mathematics (for an extended discussion on workplace mathematics see Noss 2000).

Goals show tooltip helpexplode

Curricular goals show tooltip help

ETL perceives the implementation of the "Juggling With Equations" Pedagogical Plan as an opportunity to generate turbulence in a context in which teachers strictly follow the National Curriculum’s goals, leaving no room to their students for collaborative explorations and experimentations