Modelling with Equations (familiar)

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • equations
  • kinematics
  • dynamics
  • graphs

Language show tooltip help

English

Country show tooltip help

United Kingdom

Keywords show tooltip help

Description show tooltip help

This pedagogic plan will engage students in using an innovative interactive learning environment, MoPiX, in which it is possible to construct animated models defined and controlled by equations. Through this activity students will have opportunities to develop their understanding of equations and of how motion may be described and defined. Collaborative problem solving is encouraged both by face-to-face group work and through electronic communication.  Students may make use of ready-made models from a web-based repository/library and may also contribute their own models to this library to be used by others.

The plan is structured in three parts:

(i) introductory familiarisation with the use of MoPiX

(ii) straight line motion

(iii) motion with acceleration

Rationale show tooltip helpexplode

The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

MoPiX is a mobile game  environment for mathematics learning. It  allows learners to build, inspect, edit, execute, monitor, and share dynamic visual  models. These models can be executed to produce animations, interactive simulations, or  small physics-based games. Characters/objects are constructed from a toolset provided. Properties and behaviours, defined by sets of equations, can  be associated with objects, modified and investigated. Thus algebraic equation notation is linked with dynamic visual representations. It is also possible to use equations to construct objects which leave a trace as they move, thus creating graphic representations.

This scenario consists of a set of proposed tasks that will take students from initial engagement and familiarisation with the tool through to problem solving challenges and opportunities for imaginative creation.

Some of the tasks may be worked on by students individually, but a particular feature of MoPix will be its capacity to facilitate student-student interaction and collaborative problem  solving among small groups of students. Student constructions will be shareable between machines  and inter-group communication will be facilitated by the open nature of knowledge sharing. 

Theoretical framework show tooltip help

MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

  • The task model, the constituents of activity derived from the inquiry based learning  literature.
  • Obstacles encountered by learners.
  • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
  • Scaffolding strategies, more specific implementation approaches
  • Examples    

While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

 The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

Theoretical framework show tooltip help

The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 19 years old

Population description show tooltip help

 Students in tertiary college studying Advanced level mathematics.

The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

Student prerequisites show tooltip help

Familiarity with equations.

A basic understanding of the concepts of velocity and acceleration.

Teacher prerequisites show tooltip help

  • no specialised mathematical knowledge beyond that normally required for teaching at this level
  • basic familiarity with computer technology
  • familiarity with MoPiX and the educational objectives of the tasks.

Context show tooltip helpexplode

Physical context show tooltip help

A rectangular classroom with:

  • PCs around three edges of the room
  • a digital projector and screen on the fourth wall
  • movable chairs and tables in the centre of the room

Institutional context show tooltip help

Tertiary college:

The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

Socio-cultural context show tooltip help

 In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

Goals show tooltip helpexplode

Curricular goals show tooltip help

 Key Stage 4 (higher) National Curriculum (England) MA2:

1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

1g    present and interpret solutinos in the context of the original problem

1h    use notation and symbols correctly and consistently within a given problem

3e    set up simple equations

4d    discuss and interpret graphs modelling real situations

Advanced level:

parametric equations (C4)

the basic ideas of mathematical modelling as applied in mechanics (M1)

kinematics of a particle moving in a straight line or plane (M1, M2)

the concept of a force; Newton's laws of motion (M1)

Content-epistemological goals show tooltip help

developing fluency with equation notation

constructing equations to define properties and behaviours

understanding and operating with horizontal and vertical components of position, velocity and acceleration

understanding and using velocity and acceleration to define motion

Cognitive goals show tooltip help

 analysing physical systems

analysing and debugging models

Social-affective goals show tooltip help

 collaboration in problem solving by sharing components

Instrumental goals show tooltip help

 learning to use the MoPiX tool and its web-based library

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

 This is a sequence of tasks for students, taking them from an initial introduction to the software and its functionalities through to a number of problem solving challenges and opportunities for creative exploration. Students are provided with the means to construct animated models. In order to control the motion of these animations, students need to make use of formal definitions of velocity and acceleration.

Theoretical framework show tooltip help

 The ontological principle underlying the design of MoPiX is that all the properties and behaviours of objects within the microworld are determined by equations and sets of equations. As students engage with the microworld they gain a sense of the meanings of the equations by a process of hypothesising, experimenting and reflecting on the empirical functioning of the microworld.

This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex problem solving and from structured guided exploration to independent creative authoring.

MoPiX

MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

Equations may be selected from a library or may be user defined.

Objects and equations may be shared between users to facilitate collaborative problem solving.

Resources show tooltip helpexplode

pre-questionnaire [Resource for students]

A paper-and-pencil questionnaire designed to gather initial data about students' understanding of motion

post-questionnaire [Resource for students]

 A paper-and-pencil questionnaire completed after completion of the course, with free access to MoPiX if desired.

Some questions are identical to those in the pre-questionnaire, in order to identify any differences. Others address specific aspects of motion addressed during use of MoPiX.

Work plan show tooltip helpexplode

Setting show tooltip help

 A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

Time show tooltip help

15 hours in total

Process documentation show tooltip help

  • audio records of all pairs throughout
  • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
  • all paper and pencil work produced by students throughout
  • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

Familiarisation

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • equations
  • kinematics
  • cartesian graphs

Language show tooltip help

English

Country show tooltip help

United Kingdom

Keywords show tooltip help

Description show tooltip help

The activities proposed in this section are intended to introduce students to the basic functionalities of the tool and to the fundamental idea that equations are used to describe and control the behaviour of objects.

Following an initial introduction to basic MoPiX functions, this task guides students:

  • to explore and make sense of the relationship between equations and the characteristics and behaviour of objects
  • to develop fluency with the syntax of MoPiX equations
  • to become aware of the different kinds of equations available and their functions.

It consists of two sub-tasks:

  1. Introductory Tasks - familiarising student with use of  the equations library
  2. Editing and creating equations - providing students with greater freedom to design their own animations

As they become more familiar with the meanings and effects of the various kinds of equations, students' activity may become more creative, setting and working towards their own goals.

Rationale show tooltip helpexplode

The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

MoPiX is a mobile game  environment for mathematics learning. It  allows learners to build, inspect, edit, execute, monitor, and share dynamic visual  models. These models can be executed to produce animations, interactive simulations, or  small physics-based games. Characters/objects are constructed from a toolset provided. Properties and behaviours, defined by sets of equations, can  be associated with objects, modified and investigated. Thus algebraic equation notation is linked with dynamic visual representations. It is also possible to use equations to construct objects which leave a trace as they move, thus creating graphic representations.

This scenario consists of a set of proposed tasks that will take students from initial engagement and familiarisation with the tool through to problem solving challenges and opportunities for imaginative creation.

Some of the tasks may be worked on by students individually, but a particular feature of MoPix will be its capacity to facilitate student-student interaction and collaborative problem  solving among small groups of students. Student constructions will be shareable between machines  and inter-group communication will be facilitated by the open nature of knowledge sharing. 

Theoretical framework show tooltip help

MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

  • The task model, the constituents of activity derived from the inquiry based learning  literature.
  • Obstacles encountered by learners.
  • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
  • Scaffolding strategies, more specific implementation approaches
  • Examples    

While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

 The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

Theoretical framework show tooltip help

The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

Population show tooltip helpexplode

School level show tooltip help

Upper Secondary

Age range show tooltip help

17 to 19 years old

Population description show tooltip help

 Students in tertiary college studying Advanced level mathematics.

The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

Student prerequisites show tooltip help

Familiarity with equations.

A basic understanding of the concepts of velocity and acceleration.

Teacher prerequisites show tooltip help

  • no specialised mathematical knowledge beyond that normally required for teaching at this level
  • basic familiarity with computer technology
  • familiarity with MoPiX and the educational objectives of the tasks.

Context show tooltip helpexplode

Physical context show tooltip help

A rectangular classroom with:

  • PCs around three edges of the room
  • a digital projector and screen on the fourth wall
  • movable chairs and tables in the centre of the room

Institutional context show tooltip help

Tertiary college:

The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

Socio-cultural context show tooltip help

 In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

Goals show tooltip helpexplode

Curricular goals show tooltip help

 

Key Stage 4 (higher) National Curriculum (England) MA2:

1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

1g    present and interpret solutions in the context of the original problem

1h    use notation and symbols correctly and consistently within a given problem

3e    set up simple equations

Advanced level:

parametric equations (C4)

Content-epistemological goals show tooltip help

developing fluency with equation notation

constructing equations to define properties and behaviours

understanding the notation f(object, t) as a value belonging to a given object, varying with time

understanding the relationship between parametric equations in t and changes in the values of x and y over time

 

Cognitive goals show tooltip help

observing and analysing behaviour of dynamic visual representations

making connections between different representational systems and recognising how changes in one are paralleled by changes in another

Social-affective goals show tooltip help

developing confidence and independence in using mathematical representations

collaboration in joint problem solving

Instrumental goals show tooltip help

 learning to use MoPiX and its web-based library

applying equations and sets of equations from the library to objects

editing equations

authoring new equations

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

 A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

Time show tooltip help

2.5 hours

Introductory Tasks

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

Subject domains show tooltip help

  • Mathematics
  • Mechanics
  • Physics

Topics show tooltip help

  • equations
  • forms of representation

Language show tooltip help

English

Country show tooltip help

United Kingdom

Keywords show tooltip help

    Description show tooltip help

    This scenario forms a first introduction to MoPiX. students are introduced in general terms to the capabilities of MoPiX as a tool for creating animations. They then begin to create their own models using 'ready made' equations. The first equations to which they are introduced produce act independently to produce an immediately visible effect e.g. change in colour, size, shape, orientation.

    Rationale show tooltip helpexplode

     This introductory activity has the dual purpose of introducing the students to the MoPiX tool and its modes of operation and beginning the process of developing their understanding of formal notation and of equations as descriptors of properties behaviours.

    The combination of teacher demonstration and student exploration is intended to support students' use of the unfamiliar tool and representational features while allowing them opportunities to interact purposefully with the representations, forming and testing hypotheses about the relationships between equations and object behaviours.

    Theoretical framework show tooltip help

     A fundamental principle of our approach is to encourage the construction of links between different forms of representation of mathematical object. Different semiotic systems have different meaning potentials (Halliday, O'Halloran) and combining and making links between these systems has the potential to expand and enrich the meanings that students make in interaction with such a multi-semiotic system.

    In this case, moving between the symbolic medium of equations and the dynamic visual medium should enable students to deepen their understanding of equations and symbolic representation of behaviours.

    The constructionist approach leads us to design the task in such a way that students are able to act upon the MoPiX entities (equations and objects) and construct understanding of their properties and behaviours by forming and testing hypotheses about the likely effects of applying a given equation or set of equations to an object.

    At the same time, learning is seen as a social process and the role of the teacher/researcher as a relative 'expert' in the use of MoPiX at this stage in the scenario is important in supporting student access to and understanding of use of the tool.

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

     

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutions in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    Advanced level:

    parametric equations (C4)

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding the notation f(object, t) as a value belonging to a given object, varying with time

    understanding the relationship between parametric equations in t and changes in the values of x and y over time

     

    Cognitive goals show tooltip help

    observing and analysing behaviour of dynamic visual representations

    making connections between different representational systems and recognising how changes in one are paralleled by changes in another

    Social-affective goals show tooltip help

    developing confidence and independence in using mathematical representations

    collaboration in joint problem solving

    Instrumental goals show tooltip help

    applying equations and sets of equations from the library to objects 

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

     This set of tasks is intended to introduce students to the basic components of the microworld and the functions they need to be familiar with in order to get started.

    Initially students will be provided with specific instructions and clearly structured tasks in order to allow them to develop basic knowledge and skills for using MoPix. For this activity, the microworld is limited in so far as students are only able to use the equations provided in the library.

    As they become more familiar and confident with the technical procedures, they will be able to explore the equations available in the library more independently.

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

    Equation library [Feature]

    A minimal set of equations, always present at the bottom of the MoPiX window, from which users may select one or more to allocate to an object. 

    Dropping Equations [Feature]

     Users may use a mouse (or finger or other pointing tool enabled by the device they are using) to drag an equation from the Equation Library and drop it onto an object. This action allocates the equation to the object which, once the play button is pressed, defines a property or behaviout of the object.

    save and load [Feature]

     models, objects and equations may be saved to MathDils and then loaded by any user

    Equation editor [Feature]

     Equations may be dragged from the library to the editor to be changed or new equations can be created directly in the editor.

    Resources show tooltip helpexplode

    Introduction to MoPiX [Resource for students]

     A worksheet to guide students' initial familiarisation with the use of equations in MoPiX

    Work plan show tooltip helpexplode

    Setting show tooltip help

    Each student has their own tablet PC (though desktop or laptop machines may also be used for this introductory task). 

    Time show tooltip help

    1.5 hours

    Actors' roles show tooltip help

     

    Initial and introduction to editing phases:

    Teacher/researcher demonstrates choosing an equation from the library, dropping it on an object and pressing play to see its effect. Supports students to follow the same procedure.  Similarly, they demonstrate editing at an appropriate point during the lesson.

    Students reproduce teacher actions for themselves, then try other equations, following suggestions in materials.

    Challenge and exploration phase:

    Teacher/researcher supports student exploration by: answering questions; questioning student understanding of the functioning of equations; posing challenges.

    Student set a challenge to their partner then try to solve their pa.

    What to do and how show tooltip help

     

    Initial phase:

    Teacher demonstrates the kind of animation that can be produced in MoPiX by loading and running a complex model (e.g. ...) from the web-based library. 

    procedure:

    •Go to website.
    •Click load. Enter username and password.
    •Enter the tag for the desired model (e.g. ...) (scroll through models if more than one is listed) and click select.

    Explains that they will learn how to construct their own animations using this tool. ‘Flip’ an object within the model to show the set of equations defining its properties and behaviour. Teacher and students discuss what kinds of animations they might be able to construct; clarify what may or may not be possible using this tool.

    Teacher demonstrates how to drag an object onto the stage and assign it properties by dragging and dropping equations and pressing play. Start with equations for shape, size, colour, rotation which have an immediately visible effect with the application of a single equation. Students follow the same procedures to reproduce the same effects.

    Students follow the instructions in the first part of the worksheet.

    Exploration:

    Once students have tried the equations specified on the worksheet, the teacher encourages them to  try out other equations, to articulate what the effect of each equation and to inspect equations to try to understand how they produce their effect. This should include combining several equations and considering their combined effect.

    Anticipated problems: 

    1) Students will expect single equations to have an immediate visible effect. For example, they are likely to expect that adding the equation Vx(object, 0)=3 to an object will make it move with a velocity of 3. Motion equations will be addressed comprehensively in future sessions so the teacher has a choice of how to deal with this problem. It may be deferred to the next session or individual students or groups may be prompted to examine the equations for horizontal motion and consider how they  (individually and together) affect the horizontal position  (x coordinate) of the object. There are two issues that need to be clarified here: 

        (i) motion is defined as change in position, implying that an equation of the form x(object, t)=x(object, t-1)+velocity is required, and 

        (ii) Vx(object, 0)=3 only defines the velocity when t=0 so a further equation defining the change in velocity over time is also required. These issue could be clarified by inspection of a complete model or by discussion of the meaning of the equations.

    2) Difficulty in understanding and using multi-object equations.At this stage, it is probably best to avoid introducing multi-object equations. However, this issue may arise spontaneously as students explore the equation library. A simple example, such as joining two objects by making their x- and y-coordinates equal, may be demonstrated by the teacher - or the teacher could show how to make the x-coordinates equal and ask the students to copy this procedure and to make the y-coordinates equal. 

    Introduction of editing procedure:

    At some point the teacher will judge that students are confident with the basic process of dropping equations on objects in order to achieve desired effects and that they are ready to start editing and possibly authoring their own equations.

    Either individually/ to pairs or to the whole class the teacher will suggest that different colours, sizes and rotation speeds are possible. In interaction with the students, decide a specific desired effect (e.g. make the colour of an object dark purple). Discuss the values that might achieve this effect (e.g. the ratio of the quantities of red and blue colour). Demonstrate or give instructions for a student to follow: 

    • drag a copy of the relevant equation to the editor; 
    • change the value(s) as required; 
    • put away the editor by dragging to the bin; 
    • drag the new equation onto the object.

    Challenge and exploration phase:

    Students design their own object with a set of characteristics defined by equations of their choice. They follow the instructions on the worksheet to save the model. They then load their partner's model and attempt to reproduce it by choosing and applying relevant equations.
     

    Process documentation show tooltip help

     

    • audio records of all pairs throughout
    • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
    • all paper and pencil work produced by students throughout
    • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

    Creating/Editing equations

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    • Mathematics
    • Mechanics
    • Physics

    Topics show tooltip help

    • equations
    • kinematics
    • cartesian graphs

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

     Having achieved a basic familiarity with the format of equations in MoPiX and the ways equations describe object properties and behaviours, students begin to construct their own equations, initially by editing equations chosen from the library and then constructing new equations.

    As they develop a higher degree of control and expand the library of equations available to them, students begin to be more independent  in exploring and creating behaviours.

    Rationale show tooltip helpexplode

    The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    MoPiX is a mobile game  environment for mathematics learning. It  allows learners to build, inspect, edit, execute, monitor, and share dynamic visual  models. These models can be executed to produce animations, interactive simulations, or  small physics-based games. Characters/objects are constructed from a toolset provided. Properties and behaviours, defined by sets of equations, can  be associated with objects, modified and investigated. Thus algebraic equation notation is linked with dynamic visual representations. It is also possible to use equations to construct objects which leave a trace as they move, thus creating graphic representations.

    This scenario consists of a set of proposed tasks that will take students from initial engagement and familiarisation with the tool through to problem solving challenges and opportunities for imaginative creation.

    Some of the tasks may be worked on by students individually, but a particular feature of MoPix will be its capacity to facilitate student-student interaction and collaborative problem  solving among small groups of students. Student constructions will be shareable between machines  and inter-group communication will be facilitated by the open nature of knowledge sharing. 

    Theoretical framework show tooltip help

    MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

    MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

    Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

    The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

    Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

    • The task model, the constituents of activity derived from the inquiry based learning  literature.
    • Obstacles encountered by learners.
    • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
    • Scaffolding strategies, more specific implementation approaches
    • Examples    

    While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

     

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     Although students may intially use MoPiX equations as a 'black box' in order to explore their effects, this mode of use is limited both in the complexity and interest of the models produced and in the learning that may be achieved. This SNIPP therefore seeks to develop students' understanding of the relationship between the equations and the properties and behaviours they produce.

    The development of fluency in reading and writing within the algebraic semiotic system is a general aim of the curriculum at this level as a prerequisite for more advanced mathematical work. In the context of use of MoPiX it is an important foundation for engaging in construction of any but the most basic animations.

    Theoretical framework show tooltip help

    As students begin to use a new expressive medium such as MoPiX - or indeed more conventional mathematical forms of representation, there is a dialectical relationship between the use of the medium and the development of meaning for it: through using it and experiencing and analysing its effects, meaning emerges; as meaning develops, operation with the medium becomes more purposeful. 

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

     

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutions in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    Advanced level:

    parametric equations (C4)

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding the notation f(object, t) as a value belonging to a given object, varying with time

    understanding the relationship between parametric equations in t and changes in the values of x and y over time

     

    Cognitive goals show tooltip help

    Analysing the parts of the equation representation and editing one or more part in order to make specific changes to the behaviour defined by the equation.

    Social-affective goals show tooltip help

     persistence in the face of error

    confidence to critique the productions of others and skill in doing this constructively

    Instrumental goals show tooltip help

     learning to use MoPiX and its web-based library

    applying equations and sets of equations from the library to objects

    editing equations

    authoring new equations

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Students learn the formal rules for structuring equations in MoPiX by editing existing equations and creating new ones in order to create different effects. This allows them to continue to develop their understanding and control of both the syntax and the semantics of the equation representational system and its relationship to the properties and behaviours of objects.

    Theoretical framework show tooltip help

     Growing control over the tool enables students to begin to use it more creatively and independently. This allows the process of hypothesising, testing and theorising within the microworld and hence the ongoing construction of domain specific knowledge as students interact with objects and their mathematically defined behaviours.

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

    Equation Editor [Component]

    The Equation Editor is situated above the stage. It may be used either to create new equations (by clicking on New Equation) or to edit existing ones (by dragging an equation from the Equation Library to the Equation Editor area).

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    1 hour

    Actors' roles show tooltip help

     Teacher models the processes of editing an equation and creating a new equation. The decisions about which equations to edit and how should be negotiated with the students.

    Teacher suggests types of models that students might build, involving using a combination of existing and new equations.
    Students work independently to construct their models and share the results.
    Teacher provides support while students work.

    What to do and how show tooltip help

    Basic editing may have already been introduced as part fo the 'Introductory Task' scenario. If this is the case, this scenario may start with creating new equations. 


    Teacher initiates this scenario by inviting students to suggest limitations of the equations offered in the equation library and to identify aspects that they would like to be able to change.

    Possible variables that might be suggested: colour, size of objects (note shape cannot be changed as only circle/ellipse and square/rectangle are currently defined); point at which objects are joined; velocity; acceleration.
    Teacher chooses one of these, e.g. colour.

    Look at the colour equations that are available in the library. Ask how these might be changed? (The value 100 indicates full saturation with the named colour, lowering this value changes the saturation.)

    procedure:
    •    drag and drop the equation redColour(ME,t)=100 in the Equation Editor area at the top of the screen - the Editor appears, parsing the equation
    •    change the value 100 to a smaller number  - the changed equation appears on the stage
    •    drag and drop the changed equation on an object
    •    ‘put away’ the Editor by dragging the equals sign to the bin - the changed equation remains on the stage and may either be discarded in the bin or may be dragged to the Equation Library to be saved for the rest of the current session.

    Suggest to students that they experiment with other changes of colour equations and with combinations of colours.

    Once students have grasped the method of editing, teacher proposes a construction task, e.g.:

    Make a ‘person’ dressed in different coloured clothes. This could involve changing equations of size, colour, connecting two objects together, rotation.

    Alternatively, students can set their own challenges.

    As students work on their construction tasks, at some point it is likely to become clear that the need to locate a relevant equation to edit is slowing them down. This is the point at which the teacher may intervene with individuals or with the whole group to suggest that creating a new equation from scratch might be quicker.

    Using an equation  relevant to a problem one or more of the students is working on, demonstrate the creation of a new equation, e.g. Vx(ME,t)=Vx(ME,t-1)+0.5

    procedure:
    •    Click on New Equation - an equation template appears
    •    The structure of the equation has to be defined before entering any terms so: click in the left hand side of the equation and choose Function; click on the right hand side and choose +, followed by Function in the box to the left of the + sign. The template should now have the correct structure for the desired equation.
    •    The terms of the equation may now be entered into the relevant positions in the template - the equation itself appears on the stage, turning from red to black when the equation is syntactically complete.
    •    Drag and drop the equation on the object and/or into the Equation Library.
    •    Put away the Editor as before by dropping the equals sign into the bin.

    When students complete their models, or at the end of the time available for this scenario, these can be shared among the group (using Save and Load) and discussed.

    Process documentation show tooltip help

     

    • audio records of all pairs throughout
    • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
    • all paper and pencil work produced by students throughout
    • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

    Straight line motion

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

    This scenario is in four parts.

    1. Students are introduced to the idea that the direction and speed of motion are determined by the composition of horizontal and vertical velocities. 
    2. Once they are able to construct models involving straight line motion, they then investigate relationships between the velocities of objects moving on perpendicular paths. 
    3. They use changes in velocity to create changing paths of motion consisting of straight line segments.
    4. Finally, they are introduced to equations that detect interaction between objects and use these to construct models with bouncing motions.

    The definition of discrete changes in velocity is introduced in two forms:

    • changing the value of velocity at a given point in time
    • interaction between two objects causing a change in direction

    Rationale show tooltip helpexplode

     

    Mechanics is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also often fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    The notation of MoPiX provides a structure that can support the development of critical ideas about velocity:

    1. horizontal and vertical components of motion are defined separately, supporting establishment of the idea that velocity is a vector quantity;
    2. the equation x(ME,t)=x(ME,t-1)+Vx(ME,t) symbolically embodies the definition of velocity as change in position.
    The requirement to use these notations in order to address purposeful problems and the direct association between the formal notation and the observed behaviour of dynamic models are seen to provide a rich multi-semiotic environment that enables students to make connections between the various forms of representation and to develop ideas about motion that are compatible with these two key ideas.

    Theoretical framework show tooltip help

    MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

    MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

    Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

    The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

    Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

    • The task model, the constituents of activity derived from the inquiry based learning  literature.
    • Obstacles encountered by learners.
    • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
    • Scaffolding strategies, more specific implementation approaches
    • Examples    

    While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

     

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Basic familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

     

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position and velocity 

    understanding and using velocity to define motion

     

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

    developing fluency with the use of horizontal and vertical motion equations from the MoPiX library

    using the MoPiX editor

     

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited.  

    motion equations [Feature]

    The equation library contains sets of equations for horizontal motion and for vertical motion that form the basis for the activities in the rest of the pedagogical plan. 

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    6.5 hours

    Introduction to straight line motion

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

     Students are introduced to the MoPiX equations used to describe straight line motion. They work with the idea that the direction and speed of a moving object are determined by a combination of horizontal and vertical velocities and develop their control over the motion of MoPiX objects through a sequence of directed tasks and through more open challenges

    Rationale show tooltip helpexplode

     

    Mechanics is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also often fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    The notation of MoPiX provides a structure that can support the development of critical ideas about velocity:

    1. horizontal and vertical components of motion are defined separately, supporting establishment of the idea that velocity is a vector quantity;
    2. the equation x(ME,t)=x(ME,t-1)+Vx(ME,t) symbolically embodies the definition of velocity as change in position.
    The requirement to use these notations in order to address purposeful problems and the direct association between the formal notation and the observed behaviour of dynamic models are seen to provide a rich multi-semiotic environment that enables students to make connections between the various forms of representation and to develop ideas about motion that are compatible with these two key ideas.

    Theoretical framework show tooltip help

    MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

    MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

    Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

    The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

    Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

    • The task model, the constituents of activity derived from the inquiry based learning  literature.
    • Obstacles encountered by learners.
    • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
    • Scaffolding strategies, more specific implementation approaches
    • Examples    

    While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

     

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Basic familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

     

    Content-epistemological goals show tooltip help

    understanding the notation f(object, t) as a value belonging to a given object, varying with time

    using x, y notation to define position in a Cartesian plane

    understanding that the movement of an object has horizontal (x) and vertical (y) components

    understanding the relationship between parametric equations in t and changes in the values of x and y over time

     

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

    development and consolidation of skill in using the MoPiX equation library, editing and creating equations

    familiarity with the 'basic' set of equations for defining straight line motion

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    This sequence of tasks introduces students to the equations necessary for straight line motion, starting separately with horizontal and vertical components, then combining these to form motion in other directions. Students are then challenged to construct animated models involving several objects moving with different velocities.

    MoPiX

     MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

    Resources show tooltip helpexplode

    General description show tooltip help

    Straight Line Motion 1 [Resource for students]

    Work plan show tooltip helpexplode

    Setting show tooltip help

    Students sit in pairs, each with their own tablet PC.

    Time show tooltip help

    2 hours

    Actors' roles show tooltip help

    Introductory phase: Teacher demonstrates the addition of equations to an object in order to define horizontal motion and, in interaction with the students, develops meaning for the components of the basic set of equations.

    Main phase: Students work individually or in pairs on directed tasks from the worksheet. Teacher monitors this work, intervening to provide support where necessary and to ask students to articulate their strategies.

    Challenge phase: Students work in pairs to decide on their design and to allocate component tasks between them. They then work individually to produce their components, eventually sharing and combining these electronically. The teacher monitors and provides support where necessary.

    What to do and how show tooltip help

    Introductory phase:

     The teacher demonstrates dragging the basic set of equations for horizontal motion to an object (this is task I on the worksheet):

    A minimum set of equations chosen from the library for horizontal motion is:

        Vx(ME,0)=3
        x(ME,t)=x(ME,t-1)+Vx(ME,t)
        Vx(ME,t)=Vx(ME,t-1)+Ax(ME,t)  

    After setting the object moving and observing its motion, flip the object to display the equations and ask the students to explain what each of the equations is doing. Critical points to emerge from this discussion should include:

    • the role of t and the difference between Vx(ME, 0) and Vx(ME,t)
    • the need to specify an initial velocity
    • the operationalisation of the definitions of velocity as change in position and of acceleration as change in velocity
    • Ax(ME,t) has the default value of zero

    It is not expected that all these points will be firmly established for all students but that the vocabulary and the forms of reasoning should be established as part of the classroom discourse to scaffold students' activity during the session and as a reference point for further teacher interventions.

    Main Phase

    Students follow the directed tasks on the worksheet.

    Task I asks them to reproduce what the teacher has done in defining horizontal motion but then poses some additional questions to challenge and deepen their thinking. Students may discuss these questions in their pairs or the teacher may intervene with some pairs to ensure that they engage with the questions.

    Task II asks students to produce an object with vertical motion. Most students will find this straight forward as the equations are provided in the library and are essentially identical to those used for horizontal motion in Task I. They are then asked to produce an object that moves 'diagonally'. Some students may need help to get started with this as it is likely to be the first time they have considered the idea of  motion as a combination of horizontal and vertical components.

    Task III initially introduces students to the equations needed to make an object draw a trace. Having tried this out, students then move on to the challenge phase of the session.

    Challenge phase

    Students discuss in pairs what they would like to draw, design it, define its separate components and divide the task of creating these components between them.

    The teacher may need to intervene to ensure that:

    • the design itself is manageable i.e. is not too large to be completed in a reasonable amount of time
    • the components of the design can be defined using MoPiX equations that the students are already familiar with (Depending on what additional equations might be required, the teacher may decide to introduce a pair of students to the new equations. However, this should not distract from the main focus of the task.)
    • the pair divide the task in an equitable and practical way

    Each student defines the objects for their part of the overall design, then saves their partial model.

    The pair of students loads both partial models onto one machine and play it to create their complete design.

    The teacher coordinates the whole class so that each pair demonstrates their completed design to the rest of the class, describing how they divided the work between them and, following any questioning from other students, explain how they constructed  parts of their design. The teacher may pose a question for clarificaton of methods used if s/he judges any pair to have used ideas or equations that should eb shared with the rest of the class.

     

    Perpendicular motion

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

    Rationale show tooltip helpexplode

    One of the key aspects of the notion of velocity that many students have difficulty grasping is its vector nature, having both magnitude (speed) and direction. This scenario intends to focus students' attention on this aspect by considering a special case, constructing and comparing the motion of two objects moving in perpendicular directions.

    Students may already be familiar with the relationship between gradients of perpendicular lines and the notation of MoPiX supports use of this knowledge to construct objects moving in perpendicular directions. Thus if one object has horizontal and vertical components of velocity Vx=2, Vy=5, the other may have components Vx=-5, Vy=2. However, this is not the only possible solution; any pair of components in the same ratio will produce motion in the same direction but with different magnitudes.

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Familiarity with the basic equations needed to create straight line motion in MoPiX.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

     

    Content-epistemological goals show tooltip help

    eveloping fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position and velocity 

    understanding and using velocity to define motion

    understanding and using the relationship between the directions of perpendicular lines

     

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

    developing fluency with the use of horizontal and vertical motion equations from the MoPiX library

    using the MoPiX editor

     

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Students are posed the challenge of creating an object that moves in a direction perpendicular to another object. They are then asked to formally describe the relationship between the velocities of perpendicular motions.

    Theoretical framework show tooltip help

    Students already know how to use horizontal and vertical velocity to define the direction of movement of an object. In the new context of this challenge, they are able to explore possible combinations of horizontal and vertical components with immediate visual feedback, to form and test hypotheses in order to construct new knowledge about perpendicularity.

    The notation of MoPiX makes the numerical values of horizontal and vertical components highly visible. It is these that must be manipulated in order to change the direction of motion. This context thus provides strong potential for a generalisation about perpendicularity to be expressed quantitatively.

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

     

    Resources show tooltip helpexplode

    Straight Line Motion Review [Resource for students]

    student worksheet 

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    1 hour

    Actors' roles show tooltip help

    Each student works independently to construct an object moving perpendicularly to that given to them by their partner. 

    The teacher's role in the initial stages of the scenario is to facilitate students' entry into the task, to maintain their focus on the challenge and to respond to any difficulties in using MoPiX. Once a student has constructed a correct object, the teacher will prompt them to generalise the relationship and to express it in words.

     

    What to do and how show tooltip help

     

     The initial problem is posed in the worksheet resource given to the students.

    Each student constructs an object that moves in a straight line (not horizontal or vertical). If necessary, the teacher will provide support to remind students how to do this. They then save these objects and load the object constructed by their partner.

    The challenge for each student is to construct a second object that moves in a direction perpendicular to that of their partner's object. Students may proceed analytically or by trial and improvement.

    Possible interventions that the teacher may make if necessary to enable students to progress with the challenge:

    • suggest flipping the partner's object to look at its velocity
    • ask student to consider how to make an object move in the opposite direction
    • suggest holding one component of velocity constant and seeing the effect of changing the other component

    Once students have succeeded in constructing a model with two objects with perpendicular motion, they are asked to write a generalised statement explaining the relationship between the velocities of the two objects.

    Students are likely to intially express their solutions in an operational form, e.g. 'swap the values of the x and y velocities and change the sign of one of them'. In order to develop this into a relational form the teacher will need to raise students awareness of alternative solutions. This can be done in discussion with individual students, with pairs and/or with the whole class, sharing the conclusions of this task. Issues that may arise in this discussion or that the teacher may wish to raise include:

    • there are more than one possible solution to the task - what variations are possible? (movement in  opposite senses; movement with different speeds)
    • how can the different possible solutions be incorporated into a single generalised explanation? (e.g. refer to the ratios of the horizontal and vertical components.)

    Process documentation show tooltip help

    • audio records of all pairs throughout
    • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
    • all paper and pencil work produced by students throughout
    • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

     

    Changing direction

    Identity show tooltip helpexplode

    Subject domains show tooltip help

    Topics show tooltip help

    Keywords show tooltip help

    Description show tooltip help

    Rationale show tooltip helpexplode

    This scenario continues the development of students' understanding of the quantitative vector definition of velocity and its relationship to the dynamic visual representations of motion provided by MoPiX. By working purposefully to construct paths of different shapes, students have the opportunity to strengthen their operational understanding of the ways in which the relationships between the horizontal and vertical components of velocity determine the motion of an object.

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Familiarity with the basic equations needed to create straight line motion in MoPiX.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

     

    Content-epistemological goals show tooltip help

     developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position and velocity 

    understanding and using velocity to define motion

    quantifying and using relationships between the directions of straight lines

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

    developing fluency with the use of horizontal and vertical motion equations from the MoPiX library

    using the MoPiX editor

     

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Students are introduced to the idea that the direction of motion of an object can be changed by specifying a new 'initial' velocity at a specific time. They use this approach to construct moving objects that leave traces to draw shapes composed of straight segments.

    Theoretical framework show tooltip help

    The problems that students are posed and those they set themselves in this scenario are intended to provide them with opportunities to explore the behaviours of objects and the relationships between them, constructing new knowledge about these behaviours by forming and testing hypotheses. 

    MoPiX

     MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    1.5 hour

    Actors' roles show tooltip help

     In the initiation phase: 

    • students follow directions, then share, discuss and reflect on the outcomes
    • the teacher coordinates class discussion of students' answers, ensuring that key ideas emerge that will enable students to engage with the exploration phase of the lesson. 

    in the exploration phase:

    • students are challenged to produce specific outcomes; they address these challenges independently and then may pose their own challenges
    • the teacher supports students' problem solving, solving technical difficulties with MoPiX and prompting students to reflect on the behaviour of their models and the equations determining this behaviour.

    What to do and how show tooltip help

    Initiation

    Students should first create an object moving in a straight line.

    They are then posed the following question (by the teacher or on a worksheet).

        What happens if you add the equations:

            Vx(ME,20)=0
            Vx(ME,30)=10

        to you original object?

    The teacher coordinates discussion of answers to this question. The answers are likely to differ, especially if some students have started with objects moving either horizontally or vertically:

    • original movement horizontal - the effect is likely to be seen as stopping then starting
    • original movement vertical - the first equation has no effect, so students are likely to notice only one change in direction

    From this discussion, two important points should emerge:

    • 20 and 30 are specific values of t, so the equations will take effect at specific instants
    • change in the velocity in the horizontal direction can have the effect of changing the direction of motion

    Exploration

    Students are now challenged to construct objects that will draw shapes consisting of straight line segments.

    Can you draw a triangle?
        What about other shapes (square, hexagon, circle, …)?

    They may work individually or in pairs on this task.
    Initially they may attempt to draw a triangle by choosing the values of Vx and Vy by trial and error. They may need to be directed to consider the relationships between the values that determine the direction of each of the sides.

    When they have succesfully completed a triangle, they may choose what other shapes to attempt.

    Process documentation show tooltip help

    • audio records of all pairs throughout
    • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
    • all paper and pencil work produced by students throughout
    • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

     

    Bouncing Ball

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    • Mathematics
    • Physics
    • Mechanics

    Topics show tooltip help

    • equations
    • velocity and acceleration
    • gravity

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

     Students are challenged to construct a bouncing ball. This may be extended to create a simulation of a ball thrown against a wall or to explore the effects of different gravitational forces.

    Rationale show tooltip helpexplode

     Students make use of their knowledge of the MoPiX tool, developed through the earlier familiarisation activities, in order to produce a dynamic model of a familiar physical phenomenon. Through this activity, they will:

    1. gain insight into motion and gravity
    2.  solve problems through exploration of the action of equations

    The problem solving process is supported by the social context which involves both teacher support and possibilities for collaboration between students.

    Theoretical framework show tooltip help

     

    MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis.


    The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPix thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct.

    This task is designed to allow students to engage in such constructive activity, while providing them with support structures to assist in this process.

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

     Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Familiarity with the basic equations needed to create straight line motion in MoPiX.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

     

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position and velocity 

    understanding and using velocity to define motion

     

    Cognitive goals show tooltip help

     using feedback from a dynamic model to evaluate and debug the model

    Social-affective goals show tooltip help

     providing effective feedback and support for peers within the group

    sharing and evaluating each other's ideas and models

    developing perseverance in problem solving

    Instrumental goals show tooltip help

     developing fluency with use of MoPiX

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Having had an initial introduction to the software and its functionalities, students are challenged to solve a modelling problem. Building on their solution to this, they will then have  opportunities for creative exploration of similar situations.

    Theoretical framework show tooltip help

     The ontological principle underlying the design of MoPiX is that all the properties and behaviours of objects within the microworld are determined by equations and sets of equations. As students engage with the microworld they gain a sense of the meanings of the equations by a process of hypothesising, experimenting and reflecting on the empirical functioning of the microworld.

    This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex problem solving and from structured guided exploration to independent creative authoring.

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited.  

    drag and drop equations [Feature]

    Resources show tooltip helpexplode

    tablet pc / mobile device [Resource for students]

    student worksheet

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    2 hours

    Actors' roles show tooltip help

    Students initially follow the procedures offered on the worksheet but then need to work to understand the way the equations function in order to adapt them to achieve the desired effects.


    The teacher's role  is to support students' problem solving processes, in particular:

    • help students to reflect on feedback from the models they are constructing and on the structure and meaning of the equations they are using
    • respond to students' technical questions or problems with using the software

     

    What to do and how show tooltip help

    Up to this point, objects in motion have 'wrapped around' the screen. This is likely to have frustrated many students both because it does not connect with their experience in the real world and because, when a trace is made, it causes a messy and confusing display.

    Students are asked to make an object move horizontally but bounce off the 'walls' of the screen (the vertical sides). The worksheet draws students' attention to two equations provided in the library:

    Having applied these equations, students are then challenged to make an object bounce of vertical walls positioned elsewhere on the screen. In order to do this, they will have to analyse the two new equations in order to determine which parts of these would need to be changed.

    The teacher may discuss the meaning of these equations with pairs or with the whole class:

    • what function does each of the equations have?
    • what does 799 represent?
    • what values does the function amIHittingASide take? (It is likely that, for many students, this will be the first time that they have encountered a truth function of this type with value 0 or 1.)
    • what happens to the value of Vx (as defined by the second equation) when amIHittingASide has the value 1?

    Students then adapt their models to construct a new object that bounces vertically.

    Challenge

    Students are challenged to create an object that bounces off all four edges of the screen.

    Note that the vertical bouncing equations only make the object bounce off the bottom edge of the screen. In order to make it bounce off both top and bottom edges, the equations will need to be edited or new equations created. If students are having difficulties seeing how to start this process, the teacher may suggest that they refer to the structure of the horizontal bouncing equations.

    Extension

    Students work on new models of their own choice, adapting the initial bouncing ball model. Pairs may work on individual challenges or may work collaboratively or in competition on the same challenge.

    Possible extensions:

    • add furtehr balls to the model
    • consider what happens when two objects collide
    • graph some aspects of the model. For example, students may choose to graph velocity against time.

    Acceleration

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

     This scenario is in three parts. 

    1. The first introduces students to models of motion involving constant acceleration, including acceleration due to gravity. 
    2. The second part presents students with a 'buggy' model to correct. This involves comparing the motion of an incorrect model of a projectile with expectations from the 'real world', analysing and editing the set of equations in order to achieve a 'correct' effect. 
    3. The final part introduces the idea that a force applied at a specific moment effects an instantaneous change in velocity.

    Rationale show tooltip helpexplode

     

    Mechanics is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also often fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Just as was the case with velocity, the notation of MoPiX provides a structure that can support the development of critical ideas about acceleration:

    1. horizontal and vertical components of motion are defined separately, supporting establishment of the idea that acceleration is a vector quantity;
    2. the equation Vx(ME,t)=Vx(ME,t-1)+Ax(ME,t) symbolically embodies the definition of acceleration as change in velocity.
    With the introduction of acceleration, students begin to be able to model situations in the real world that involve forces. The visual feedback provided by running dynamic models allows immediate testing of their hypotheses by comparing the behaviour of their models with that anticipated. Specific issues that we anticipate may be addressed through this scenario include:
    • consideration of the forces acting upon projectiles
    • the notion that motion continues in a straight line unless acted upon by a force

    Theoretical framework show tooltip help

    MoPiX is conceived as a constructionist toolkit. The constructionist approach to learning (Papert, 1980; Harel & Papert, 1991; Kafai &  Resnick, 1996) promotes investigation through the design of microworld environments, i.e.  technology-enhanced educational tools and activities, and the observation of learners’ actions,  developments and communication within these environments. As developed by Strohecker and Slaughter (2000) constructionist toolkits are very much  based on these principles. They are dynamic visual environments that support building  activities in social contexts. Learners build constructs with fundamental elements and then  activate these constructions as a means of investigating their hypothesis. The fundamental elements of MoPiX are equations and objects whose properties and behaviours are defined by the equations assigned to them. Working with MoPiX thus provides students with opportunities to explore and develop their understanding of equations and relationships within a Cartesian plane as well as to investigate the behaviours of the objects they construct. 

    MoPiX is a multi-semiotic environment involving formal notation of equations and visual animated models. Specialised animations may also be constructed to leave a trace in the form of a cartesian graph. These three semiotic systems, with their various elements and grammars, have different meaning potentials (O'Halloran, 2005; Kress, 2001). In addition, students may use pencil and paper-based representations involving conventional or informal notations or diagrams. They can also  communicate with each other using 'natural' language in face-to-face speech and by sharing MoPiX objects, equations and models electronically. Again, these modes of communication have distinctive elements, grammars and meaning potentials. The multi-semiotic nature of the environment thus provides rich possibilities for students to interpret mathematical ideas and for them to express their own mathematics through constructing new animations. 

    Duval (2006) argues that conversion between semiotic systems (which he names representational systems or 'registers') is of fundamental importance to mathematical learning. Conversion demands that the student distinguishes what is mathematically relevant in each system and separates the mathematical object from its representation. The MoPiX environment not only demands that students engage in conversion (using different forms of representation for the 'same' mathematical object) but also that they actively use the representations available in the system of equations to effect changes in the visual forms of representation. In the opposite direction, the process of 'debugging' faulty animations again demands conversion: identifying those equations which are responsible for the 'buggy' behaviour. We hypothesise that activities in this environment will enable students to develop their understandings of  algebraic notation and of definition of motion.

    The tasks proposed in this pedagogical plan are designed to allow students to engage in constructive activity, while providing them with support structures to assist in this process. We draw on the idea of learner-centred design (Soloway, Guzdial and Hay, 1994), based on socio-cultural and  constructivist theories of learning and the user-centred approach to interaction design. Learner-centred design is based on the premises that a user of technology constantly changes through learning, and that their needs from the technology change in the process. In  particular, the user learns through using the technology, and the design of the technology  needs to account for that learning. This leads to the question of how can environments  support learners and learning? LCD suggests that students learn  through an active, social process of meaning construction (Vygostky, 1962). Critically,  understanding is built up through the acts of conversing with others, constructing  artefacts, and reflecting on those conversations and artefacts. Soloway, Guzdial and Hay  (1994) see scaffolding as the main role of teachers in constructivist learning, and propose that  this should be the role of the interface in technology-rich environments. We see a more critical role of the teacher in mediating the communication among a group of learners (see below). 

    Quintana et al (2005) propose a framework for designing scaffolding structures. Position this  framework in the context of inquiry-based learning. Consequently, organize the framework  around three processes: ‘sense making’, which involves the basic operations of testing  hypotheses and interpreting data; 'process management', which involves the strategic  decisions involved in controlling the inquiry process; and articulation and 'reflection', which is  the process of constructing, evaluating, and articulating what has been learned. From these  principles, they derive a framework that includes several elements:    

    • The task model, the constituents of activity derived from the inquiry based learning  literature.
    • Obstacles encountered by learners.
    • Scaffolding guidelines provide principles for designing scaffolds to help learners  overcome the obstacles.
    • Scaffolding strategies, more specific implementation approaches
    • Examples    

    While each student will have their personal tablet pc with which to  build and investigate scenes, communication between the devices will allow them not only to  share what they have built but also to investigate, modify and play each other’s constructions  in a game-like manner. The multi-semiotic environment, including equations, animation, graphical representation, oral  communication within the group and pencil and paper representations, allows opportunities  for participants to contribute in different ways to the construction of a problem solution. For  example, while one student might focus their attention and arguments on the behaviour of an  animation proposed as a solution to a problem, another could corroborate or criticise the  solution by reference to the properties of a graph showing relationships between chosen  variables or by constructing a conventional paper and pencil forces diagram. Alternative  modifications to the construction could be carried out by different students and the results  compared immediately either by sending them to each other’s personal devices or by laying  several devices together to be viewed simultaneously by the group. This environment aims to encourage collaboration and  ‘exploratory talk’, enabling students to exchange ideas and to engage critically with each  other’s contributions - a form of talk that, it is argued, supports learning through group  interaction (Barnes, 1976; Barnes & Todd, 1995; Edwards, 2005; Mercer, 1995). 

     

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Familiarity with the basic equations needed to create straight line motion in MoPiX.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

    the concept of a force; Newton's laws of motion (M1)

     

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position, velocity and acceleration

    understanding and using velocity and acceleration to define motion

     

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

     learning to use the MoPiX tool and its web-based library

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    Students are first introduced to the effects of adding a constant acceleration to an object. This allows them in particular to model objects moving under the influence of gravity.

    The second part of this scenario addresses the application of a force at an instant. This is effected in MoPiX by applying an acceleration at a specified value of t.

    A theme throughout the scenario is the development of the concept of acceleration as change in velocity.

    Theoretical framework show tooltip help

     The ontological principle underlying the design of MoPiX is that all the properties and behaviours of objects within the microworld are determined by equations and sets of equations. As students engage with the microworld they gain a sense of the meanings of the equations by a process of hypothesising, experimenting and reflecting on the empirical functioning of the microworld.

    This process of meaning construction is built into all the tasks in this pedagogic plan, from initial introduction and familiarisation to complex problem solving and from structured guided exploration to independent creative authoring.

    MoPiX

     MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    6 hours

    Using constant acceleration

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

    Topics show tooltip help

    Language show tooltip help

    English

    Country show tooltip help

    United Kingdom

    Keywords show tooltip help

    Description show tooltip help

    This scenario introduces students to models of motion involving constant acceleration. 

    Initially they construct models that move horizontally or vertically. They then coordinate horizontal an vertical components to further develop a vector-based conception of velocity and acceleration. Finally a group challenge is posed to construct a complex model of an exploding firework.

    Rationale show tooltip helpexplode

     

    Mechanics is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also often fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Just as was the case with velocity, the notation of MoPiX provides a structure that can support the development of critical ideas about acceleration:

    1. horizontal and vertical components of motion are defined separately, supporting establishment of the idea that acceleration is a vector quantity;
    2. the equation Vx(ME,t)=Vx(ME,t-1)+Ax(ME,t) symbolically embodies the definition of acceleration as change in velocity.
    With the introduction of acceleration, students begin to be able to model situations in the real world that involve forces. The visual feedback provided by running dynamic models allows immediate testing of their hypotheses by comparing the behaviour of their models with that anticipated. Specific issues that we anticipate may be addressed through this scenario include:
    • consideration of the forces acting upon projectiles
    • the notion that motion continues in a straight line unless acted upon by a force

    Target show tooltip helpexplode

    Rationale show tooltip helpexplode

     The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

    It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

    Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

    Theoretical framework show tooltip help

    The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

    Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

    Population show tooltip helpexplode

    School level show tooltip help

    Upper Secondary

    Age range show tooltip help

    17 to 19 years old

    Population description show tooltip help

     Students in tertiary college studying Advanced level mathematics.

    The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

    Student prerequisites show tooltip help

    Familiarity with equations.

    A basic understanding of the concepts of velocity and acceleration.

    Familiarity with MoPiX: applying equations to objects, editing and creating equations.

    Familiarity with the basic equations needed to create straight line motion in MoPiX.

    Teacher prerequisites show tooltip help

    • no specialised mathematical knowledge beyond that normally required for teaching at this level
    • basic familiarity with computer technology
    • familiarity with MoPiX and the educational objectives of the tasks.

    Context show tooltip helpexplode

    Physical context show tooltip help

    A rectangular classroom with:

    • PCs around three edges of the room
    • a digital projector and screen on the fourth wall
    • movable chairs and tables in the centre of the room

    Institutional context show tooltip help

    Tertiary college:

    The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

    Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

    The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

    Socio-cultural context show tooltip help

     In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

    The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

    Goals show tooltip helpexplode

    Curricular goals show tooltip help

    Key Stage 4 (higher) National Curriculum (England) MA2:

    1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

    1g    present and interpret solutinos in the context of the original problem

    1h    use notation and symbols correctly and consistently within a given problem

    3e    set up simple equations

    4d    discuss and interpret graphs modelling real situations

    Advanced level:

    parametric equations (C4)

    the basic ideas of mathematical modelling as applied in mechanics (M1)

    kinematics of a particle moving in a straight line or plane (M1, M2)

    the concept of a force; Newton's laws of motion (M1)

     

    Content-epistemological goals show tooltip help

    developing fluency with equation notation

    constructing equations to define properties and behaviours

    understanding and operating with horizontal and vertical components of position, velocity and acceleration

    understanding and using velocity and acceleration to define motion

     

    Cognitive goals show tooltip help

     analysing physical systems

    analysing and debugging models

    Social-affective goals show tooltip help

     collaboration in problem solving by sharing components

    Instrumental goals show tooltip help

     learning to use the MoPiX tool and its web-based library

    Specifications show tooltip helpexplode

    Rationale show tooltip helpexplode

    This scenario introduces students to the use of acceleration in MoPiX.

    MoPiX notation operationalises the definition of acceleration as change in velocity and it is this notion that is at the heart of the scenario.

    The vector challenge relates the MoPiX programming environment directly to a problem taken from the conventional textbook. This is intended both to enable students to understand how their work with MoPiX may support their learning within the examined curriculum and to draw their attention explicitly to the vector nature of velocity and acceleration by 'translating' between the two vector based notations.

    The final group challenge allows students an opportunity for creativity in making use of the motion equations as well as prompting them to think about the forces acting on a projectile such as a firework.

    MoPiX

    MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited.  

    Resources show tooltip helpexplode

    Acceleration 1 (constant acceleration) [Resource for students]

    Work plan show tooltip helpexplode

    Setting show tooltip help

     A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

    Time show tooltip help

    2.5 hours

    Actors' roles show tooltip help

    In the introductory phase, students follow instructions on the worksheet  provided. The teacher intervenes with pairs of students to invite them to reflect, analyse and explain their models.

    In the vector challenge phase, students work in pairs on a set problem. The teacher supports their work as necessary.

    In the group challenge phase, students work as a pair or group to design a desired model and to plan how it will be implemented. They then work independently to complete their own components of the design. The teacher supports their work as necessary.

    What to do and how show tooltip help

    Introductory phase

    Students start by constructing an object with horizontal motion, adding the acceleration equation provided in the equation library Ax(ME,t)=2. They play this model and observe the resulting motion.

    Following the instructions on the worksheet, they edit the equation and try other values of Ax, including negative numbers.

    As students do this, the teacher should encourage them to describe what is happening to their models and to explain their observations in relation to the value of Ax and its roel within the set of equations.

    Issues that may arise:

    • values of Ax that are too large or too small may make the motion hard to analyse - teacher may guide students to choose more convenient values
    • when using negative values of Ax, students initial expectation is likely to be that the moving object will slow down. If too large a negative number is chosen, it may be difficult analyse what is happening. It is useful to try a value that allows one to see the initial slowing before Vx becomes zero. For example, if the initial velocity is 3, try a constant acceleration of -0.1.

    The teacher should discuss with students, either individually or as a group, what effect the acceleration equation is having, drawing out the structure and operation of the equation Vx(ME,t)=Vx(ME,t-1)+Ax(ME,t) and relating this to the formal definition of acceleration as change in velocity.

    Students then work with an object moving vertically. In this case, rather than being led through a sequence of tasks, they are challenged to construct an object that first moves upwards and then downwards. This requires that they make connections with the action of the negative acceleration encountered when working with horizontal motion.

    Issues that may arise:

    • as in the horizontal case, students may need guidance to choose suitable values of Vy and Ay that will make the desired up and down motion obvious
    • some students may attempt to address this problem by adding separate equations for the upwards and the downwards motion. This will, of course, be effective if, for example a positive initial velocity is added and then an equation such as Vy(ME,10)=-3 to change the velocity at time t=10. Students who do this should then be encouraged to construct a second object whose motion is defined by just initial velocity and an acceleration - and to compare the motion of the two objects.

    Vector challenge phase

    Students are posed a problem presented using conventional i, j vector notation.

    A particle P has velocity (3i + 2j)m/s when t = 0 and the velocity (7i + 4j) m/s at time t=20. The acceleration of P is constant. Find the acceleration.

    The teacher should explain to them that this is a question taken from a standard Advanced level text book and that using MoPiX can help them to understand how to solve it.

    The worksheet guides students to load a pre-prepared model which includes numerical objects that display the current values of t, Vx and Vy as the model runs. They then add the given initial velocity and attempt to find values for Ax and Ay that will produce the desired velocity at time t=20.

    Issues that may arise:

    • Students may initially work by trial and improvement. This can be an important stage in making sense of the situation. Eventually, however, they should be encouraged to think more analytically about the problem, thinking about the effect of the value of Ax on the value of Vx as t increases.
    • This is likely to be the first time that some students have met the i, j notation. They will need to be told that the i and j components represent the horizontal and vertical components respectively.
    • A further difference between the standard notation used in this question and MoPiX is the use of units in th question - not specified in MoPiX. If it seems appropriate, the teacher may wish to discuss this with students and consider what unit should be given in the answer to the question and how these might be 'translated' into the MoPiX context.

    Group firework challenge phase

    Pairs or groups of three students are challenged to construct a model of an exploding firework consisting of several small objects moving outwards from a single starting point.

    Pairs/groups start by negotiating the overall design of their firework and divide the construction work between them. One way of dividing the work is for each student to take responsibility for a sub-set of the objects to be combined into the whole. Once these are completed, each student saves their contribution and then loads the contributions of the other(s) into a single model. If students are unclear about how to collaborate, the teacher shoudl suggest this approach. There are, however, other ways of collaborating and students may be free to choose their own approach to sharing the work on this task.

    The 'think' points on the worksheet provide hints to help students to get started.

    • What should you make the initial velocity in the vertical direction?
    • How can you make objects start off in different directions?
    • What forces should be acting on the objects?

    The teacher may need to direct students' attention to these questions if they appear to be stuck. 

    The final question relates to one of the key learning objectives of this task: to recognise that, after the initial explosion, the only force acting on the objects is gravity. It may be fruitful to encourage students to experiment with other accelerations (including a horizontal acceleration and/or different values of vertical acceleration) and to reflect on the nature of the motion they observe when running the model.

    Process documentation show tooltip help

    • audio records of all pairs throughout
    • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
    • all paper and pencil work produced by students throughout
    • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

     

    Debugging models

    Identity show tooltip helpexplode

    Authors show tooltip help

    Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

    Subject domains show tooltip help

      Topics show tooltip help

        Language show tooltip help

        English

        Country show tooltip help

        United Kingdom

        Keywords show tooltip help

        Description show tooltip help

        This task aims to develop students' ability to examine models critically and identify the  meanings and functions of the equations attached to objects. 

        Students are given a model that is not functioning as intended and are challenged to find the source of the malfunction and to correct it.

        The model used is intended to be a projectile. In order to produce a 'realistic' path of motion, the solution will thus include deciding initial velocity to provide an appropriate starting direction and speed of motion and deciding on the appropriate acceleration.

        Rationale show tooltip helpexplode

        The process of debugging a model is a key way of encouraging analysis of the set of equations and of the behaviour of the model.

        In this scenario, the 'buggy' model provided as a starting point is designed to provoke students to address the question of what forces act upon a projectile.

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

         The process of debugging is essential to allow students to progress to building more complex models of their own and to persevere when (inevitable) errors occur.

        Theoretical framework show tooltip help

         The notion of debugging is fundamental to constructionist approaches to learning. By making and testing hypotheses about the environment they are working with, students develop their understandings of the nature of the objects in the microworld.

        Population show tooltip helpexplode

        School level show tooltip help

        Upper Secondary

        Age range show tooltip help

        17 to 19 years old

        Population description show tooltip help

         Students in tertiary college studying Advanced level mathematics.

        The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

        Student prerequisites show tooltip help

        Familiarity with equations.

        A basic understanding of the concepts of velocity and acceleration.

        Familiarity with MoPiX: applying equations to objects, editing and creating equations.

        Familiarity with the basic equations needed to create straight line motion in MoPiX.

        Teacher prerequisites show tooltip help

        • no specialised mathematical knowledge beyond that normally required for teaching at this level
        • basic familiarity with computer technology
        • familiarity with MoPiX and the educational objectives of the tasks.

        Context show tooltip helpexplode

        Physical context show tooltip help

        A rectangular classroom with:

        • PCs around three edges of the room
        • a digital projector and screen on the fourth wall
        • movable chairs and tables in the centre of the room

        Institutional context show tooltip help

        Tertiary college:

        The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

        Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

        The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

        Socio-cultural context show tooltip help

         In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

        The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

        Goals show tooltip helpexplode

        Curricular goals show tooltip help

        Key Stage 4 (higher) National Curriculum (England) MA2:

        1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

        1g    present and interpret solutinos in the context of the original problem

        1h    use notation and symbols correctly and consistently within a given problem

        3e    set up simple equations

        4d    discuss and interpret graphs modelling real situations

        Advanced level:

        parametric equations (C4)

        the basic ideas of mathematical modelling as applied in mechanics (M1)

        kinematics of a particle moving in a straight line or plane (M1, M2)

        the concept of a force; Newton's laws of motion (M1)

         

        Content-epistemological goals show tooltip help

        • understand that velocity can be considered as a vector with horizontal and vertical components and that change in the ratio of these components corresponds to a change in the direction of travel
        • reinforce the notion of gravity as a negative vertical acceleration
        • understanding of the forces acting on a projectile

        Cognitive goals show tooltip help

        analyse the components of a solution provided by another and identify errors 

        Social-affective goals show tooltip help

        develop skills in critiquing one's own and other's work constructively 

        Instrumental goals show tooltip help

        • 'flip' MoPiX objects to reveal their defining equations
        • edit objects 

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        By working with a relatively simple model in a familiar context, built by someone else, students will have the opportunity to develop their analytical skills in a relatively unthreatening environment where they are likely to experience success.

        MoPiX

        MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

         

        "flipping" objects [Feature]

        Any object may be flipped over (using right click) to reveal the set of equations defining its properties and behaviours. 

        Equations may be removed or replaced by new or edited versions.

        Flipping back (right click on the set of equations) returns to the object.

        Resources show tooltip helpexplode

        buggy projectile [Resource for students]

         A MoPiX model representing a projectile fired from a cannon. 

        The model contains errors which mean that the projectile's trajectory is not as anticipated.

        The model can be loaded from within MoPiX running in MathDils

        Resource contents show tooltip help

        Work plan show tooltip helpexplode

        Setting show tooltip help

         A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

        Time show tooltip help

        30 minutes

        Actors' roles show tooltip help

         The teacher explains the task and leads a whole class discussion of the initial approach to the problem.

        Students participate in this intial discussion, then work individually or in small groups to complete the task. During this work, the teacher supports students as necessary and proposes extension tasks as appropriate.

        What to do and how show tooltip help

         Instruct students to load and run pre-prepared model buggy projectile

        Explain that this is supposed to be a simulation of a projectile fired from a cannon.

        Run the simulation. Discuss how it is behaving and how this is different from what should be expected.

        Points to arise from the discussion:

        • initial angle of the trajectory of the projectile - how might this be defined?
        • shape of the trajectory - what is wrong with this and why?

        Ask which of the objects should be examined first. (Expect students to choose the projectile as this is what is malfunctioning. It will also be necessary at some point to flip the cannon to see its angle.)
        'Flip' the projectile (right click) to see the equations.

        Students examine the equations defining the properties and behaviours of the projectile and discuss what the function of each equation might be.

        Students edit the equations, remove and/or add equations and test the results.

        Procedure:

        • click on the equation to be removed and/or edited
        • right click to flip back over
        • drop the chosen equation on the Editor
        • make desired changes and drop back on the object
        • put Editor away by dropping equals sign on the bin

        The (minimum) changes needed are:

        • initial velocity in x and y directions (approximately 9 and 5 respectively or defined using sin and cos of the angle of elevation of the cannon)
        • value of acceleration due to gravity (should be negative)

        Once a correctly working model has been achieved, suggest to students that they try changing the angle of rotation of the cannon and make the corresponding necessary changes to the projectile.

        In general, this will require changes in the initial velocity in x and y directions:

        • Vx(ME,0)=cos(rotation(OTHER1,0))
        • Vy(ME,0)=sin(rotation(OTHER1,0))

        Process documentation show tooltip help

        • audio records of all pairs throughout
        • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
        • all paper and pencil work produced by students throughout
        • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)

         

        Applying a force to change direction

        Identity show tooltip helpexplode

        Authors show tooltip help

        Jehad Alshwaikh, Ken Kahn, Candia Morgan, Dusanka Nicolic, Niall Winters. Institute of Education - London Knowledge Lab

        Subject domains show tooltip help

        Topics show tooltip help

        Language show tooltip help

        English

        Country show tooltip help

        United Kingdom

        Keywords show tooltip help

        Description show tooltip help

         This scenario introduces the idea that a force applied at a specific moment effects an instantaneous change in velocity.

        Students initially apply accelerations for specific values of t in order to change the motion of objects. They use this method to construct paths of shapes such as squares and triangles. Finally a group challenge is posed to construct a complex model of 'dancing' objects.

        Rationale show tooltip helpexplode

         

        Mechanics is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

        Traditional approaches to teaching mechanics as a part of advanced mathematics also often fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

        Just as was the case with velocity, the notation of MoPiX provides a structure that can support the development of critical ideas about acceleration:

        1. horizontal and vertical components of motion are defined separately, supporting establishment of the idea that acceleration is a vector quantity;
        2. the equation Vx(ME,t)=Vx(ME,t-1)+Ax(ME,t) symbolically embodies the definition of acceleration as change in velocity.
        With the introduction of acceleration, students begin to be able to model situations in the real world that involve forces. The visual feedback provided by running dynamic models allows immediate testing of their hypotheses by comparing the behaviour of their models with that anticipated. Specific issues that we anticipate may be addressed through this scenario include:
        • consideration of the forces acting upon projectiles
        • the notion that motion continues in a straight line unless acted upon by a force

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

         The use of algebraic notation is known to be an obstacle for many students beginning to study more advanced mathematics in the UK. This pedagogic plan provides an opportunity for students at this transition point in upper secondary school to encounter and use equations in purposeful and meaningful ways.

        It is also at this point that students begin formal study of mechanics. This is an area of applied mathematics/science in which students' informal ways of experiencing the world have often prepared them poorly for understanding scientific principles. Indeed, students' intuitive assumptions and arguments about motion, forces, acceleration, etc. are often discontinuous with the principles and styles of reasoning underpinning the Newtonian model. Everyday physical experience does not sufficiently challenge these assumptions and arguments.

        Traditional approaches to teaching mechanics as a part of advanced mathematics also fail to challenge student intuitions. These approaches tend to focus on a formal approach based on the construction of conventional static diagrams of physical systems which are then translated into algebraic representations. The components and relationships in such systems and the principles underpinning their construction are often poorly understood, leading to errors and poor student engagement with the subject. Building models in the MoPiX microworld, however, can provide students with the opportunity to construct and manipulate simulated dynamic representations of physical objects and to test hypotheses about the behaviour of physical systems.

        Theoretical framework show tooltip help

        The objects and equations provided by the MoPiX microworld are designed to behave in mathematically coherent ways. This provides an environment that, by exploring and building models within the microworld, allows students to construct mathematical meanings. By setting students tasks to build dynamic models that behave in ways consistent with their experience of the physical world, students' intuitions about motion, forces, etc. may be challenged.

        Our theoretical perspective on learning suggests that interaction with physical representations is not by itself sufficient for effective learning, but that students need to make sense of their experiences of manipulating representations in the context of social interaction with peers and with teachers in order to be able to challenge and test alternative conceptualisations and forms of reasoning. Thus the plan proposes that a group of students should work together (in some cases on collaborative tasks). Collaboration is facilitated by the physical context and by the tool.

        Population show tooltip helpexplode

        School level show tooltip help

        Upper Secondary

        Age range show tooltip help

        17 to 19 years old

        Population description show tooltip help

         Students in tertiary college studying Advanced level mathematics.

        The pedagogic plan may also be suitable for younger students, though with less pre-existing knowledge some of the tasks may be less accessible for them and they may interact with the tasks in different ways.

        Student prerequisites show tooltip help

        Familiarity with equations.

        A basic understanding of the concepts of velocity and acceleration.

        Familiarity with MoPiX: applying equations to objects, editing and creating equations.

        Familiarity with the basic equations needed to create straight line motion in MoPiX.

        Teacher prerequisites show tooltip help

        • no specialised mathematical knowledge beyond that normally required for teaching at this level
        • basic familiarity with computer technology
        • familiarity with MoPiX and the educational objectives of the tasks.

        Context show tooltip helpexplode

        Physical context show tooltip help

        A rectangular classroom with:

        • PCs around three edges of the room
        • a digital projector and screen on the fourth wall
        • movable chairs and tables in the centre of the room

        Institutional context show tooltip help

        Tertiary college:

        The college takes students aged 16+ from the local area: a borough in West London. Some of the students have come from secondary schools that do not take students beyond the age of 16. Others have chosen to come to the college rather than to stay in their secondary school, either because the college is able to offer a wider choice of subjects to study or because they prefer the more 'adult' social environment.

        Almost all of the students on this site are studying 2-year 'Advanced Level' courses and most of these intend to go to university on completing their courses. Other sites offer vocational courses.

        The catchment area is very mixed socially and racially with a high number of recent immigrants (especially from Eastern Europe and Somalia) as well as well established populations originating from Pakistan, India and the Caribbean. Some richer middle class families choose to send their children to another college in a different area. The college has a substantial programme of language support for students with English as an additional language.

        Socio-cultural context show tooltip help

         In the UK context, the mathematics curriculum for the target group is very pressurised with frequent high-stakes examination. While one the one hand, the pedagogic plan has been designed to have an explicit relationship to the official curriculum, it has had to be implemented in an extra curricular setting.

        The game playing, animation construction aspect of MoPiX use makes links with youth culture where game playing and game construction are high value activities.

        Goals show tooltip helpexplode

        Curricular goals show tooltip help

        Key Stage 4 (higher) National Curriculum (England) MA2:

        1f     use a variety of strategies and diagrams for establishing algebraic or graphical representation of a problem and its solution: move from one representation to another to get different perspectives on the problem

        1g    present and interpret solutinos in the context of the original problem

        1h    use notation and symbols correctly and consistently within a given problem

        3e    set up simple equations

        4d    discuss and interpret graphs modelling real situations

        Advanced level:

        parametric equations (C4)

        the basic ideas of mathematical modelling as applied in mechanics (M1)

        kinematics of a particle moving in a straight line or plane (M1, M2)

        the concept of a force; Newton's laws of motion (M1)

         

        Content-epistemological goals show tooltip help

        • understand force as acceleration 
        • appreciate that motion continues in a straight line unless acted upon by a force
        • operationalise the notion of acceleration as change in velocity

        Cognitive goals show tooltip help

         analysing physical systems

        analysing and debugging models

        Social-affective goals show tooltip help

         collaboration in problem solving by sharing components

        Instrumental goals show tooltip help

         learning to use the MoPiX tool and its web-based library

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        The central concept of acceleration as change in velocity is extended by considering acceleration applied at an instant rather than constantly. This may metaphorically be considered as a 'kick' given to an object that changes the speed and/or direction of its motion.

        Students initially observe the effects of such a 'kick'. They use similar 'kicks' to cause changes in the direction of motion of an object in order to create paths of specified shapes.

        Theoretical framework show tooltip help

        The progression from observation to exploration and problem solving is consistent with the constructionist framework, enabling students to develop their understandings of the effects of applying acceleration by posing and testing hypotheses and engaging in analysis of the motion of their objects during the debugging process.  

        MoPiX

         MoPiX is a program that run stand-alone or within most any browser (that has Flash 8 Player installed). A MoPiX user can associate equations with objects and run the resulting animation, simulation, or game. Equations can also be created and edited. 

        Resources show tooltip helpexplode

        Acceleration 2 - applying a force [Resource for students]

        Work plan show tooltip helpexplode

        Setting show tooltip help

         A pair or group of three students, each with their own tablet PC will be positioned (probably around a single table) in such a way that they can communicate face-to-face. They will also be able to share their MoPiX constructions both electronically and by moving their displays to make them directly visible to each other.

        Time show tooltip help

        3 hours

        Actors' roles show tooltip help

         During the introductory phase, students folow the instructions on the worksheet to construct a model and explain its behaviour. The teacher monitors this work and intervenes to make student explanations explicit. If appropriate, the teacher leads a whole class discussion in which students share their ideas.

        During the challenge phase, students work in pairs or small groups to complete the challeng tasks. The teacher supports this work as necessary.

        In the group challenge phase, students work as a pair or group to design a desired model and to plan how it will be implemented. They then work independently to complete their own components of the design. The teacher supports their work as necessary. When models are complete, students demonstrate their models to the whole class.

        What to do and how show tooltip help

        Introductory phase

        Students construct an object moving with constant velocity in a diagonal direction (i.e. with non-zero horizontal and vertical components of velocity). They then add acceleration to the object at time t=20:

        Ax(ME,20)=2

        Ay(ME,20)=-2

        The worksheet proposes some questions to prompt students to analyse and explain the observed effect of these equations. The teacher should monitor these discussions or lead a whole class discussion. Points that should arise in the discussion include:

        • the values of acceleration change the values of velocity
        • the values of velocity change only once - at the designated time t=20
        • motion continues in a straight line and at a constant speed unless it is acted upon by a force, i.e. the application of an acceleration

        It may be useful for the teacher to introduce the metaphor of giving a 'kick' to an object.

        Challenge phase: changing direction

        Students are now challenged to use the application of acceleration at an instant in order to create paths of motion with instantaneous changes of direction. The suggested paths are:

        • a right angle
        • a square
        • a triangle
        • and finally a design of the student's own choice

        Students may initially proceed by trial and error and this exploratory phase can play an important role in helping them to make sense of the effects of applying a force.

        Issues that may arise:

        • students change velocity directly by adding equations of the type Vx(ME,20)=3 rather than by adding acceleration equations. While this may be helpful initially to enable them to identify the desired values of Vx, the teacher should eventually direct them to use acceleration instead.
        • making the motion of the object stop when it has completed the desired path. Again, the teacher should direct students' attention to the idea of using a force to stop motion rather than simply setting the velocity components directly to zero.

        Group challenge phase

        Students are challenged to form a complex model with several objects 'dancing' on different paths.

        Each pair or group plans a design for their 'dance', deciding how each object in the model will change direction and/or speed and at what time such changes will happen. Pairs/groups start by negotiating the overall design of their dance and divide the construction work between them. One way of dividing the work is for each student to take responsibility for a sub-set of the objects to be combined into the whole. Once these are completed, each student saves their contribution and then loads the contributions of the other(s) into a single model. If students are unclear about how to collaborate, the teacher shoudl suggest this approach. There are, however, other ways of collaborating and students may be free to choose their own approach to sharing the work on this task.

        When models are complete, they may be shared with the rest of the class, with discussion of  issues arising during the construction process.

        Process documentation show tooltip help

         

        • audio records of all pairs throughout
        • audio and video records of one pair throughout, focusing the video to capture the screen and student interaction with the screen
        • all paper and pencil work produced by students throughout
        • pre-questionnaire and post-questionnaire (see resources at top level of pedgaogical plan)