Building in 3D (alien)

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • 3D geometrical figures
  • spatial visualisation
  • 2D representation of 3D space
  • angles and turns in 3D

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

This pedagogical plan will engage students in designing and constructing a virtual building. The purpose of the building will be specified by the teacher with constraints or suggestions to encourage creative designs. In this case, the building specified is a new sports centre for the school, but other buildings relevant to the students' context and experience could be substituted with little effect on the overall plan.

Students will construct representations of 3D geometrical objects using both traditional forms of representation (building blocks, isometric drawings, plans and elevations) and a 3D computational environment.  

Rationale show tooltip helpexplode

3D geometry is an area of the curriculum that appears difficult both to learn and to teach.  2D representations of 3D objects are frequently used in order to support analysis of their geometrical features, yet the success of this approach relies on students' ability to make connections between the different representations. In practice, the focus of the curriculum is often more on developing skills in constructing particular type of 2D representations rather than on using them in order to develop understanding of the 3D objects themselves. The aim of this set of activities is to engage students in using a range of representations purposefully and making connections between them in order to gain a fuller understanding of  3D geometrical objects. A computational environment such as MachineLab provides new ways of making such connections.

The activities proposed in this plan make use of a range of modes of representation in order to encourage students to develop their visualisation skills and to enable them to choose the mode of representation best suited to the task in hand.

The set of activities as a whole has an overall purpose and a concrete outcome in the form of the design of a building. This project format is intended to motivate students through its opportunities for creativity and through competition with their peers. The need to communicate their final design to an audience also provides a realistic environment for students to reflect on the different forms of representation available to them and to make choices between them in order to communicate most effectively.

Theoretical framework show tooltip help

Multi-modal and multi-semiotic environments allow participants many opportunities for making meanings with the representations available to them and choices about the most apt representations to employ in order to communicate their desired meanings. Through the course of the set of activities, students will make use of a range of semiotic systems, both visual and symbolic, with different elements and grammars. Each of these semiotic systems has a different meaning potential (O'Halloran, 2005; Kress, 2001). Thus making use of one of the systems, for example, isometric drawings, offers a particular set of opportunities for developing understandings of the properties of 3D shapes, while another, such as nets, offers different opportunities. The symbolic logo-based programming of MachineLab provides a further semiotic system that makes more explicit use of angle and length relationships within figures than most paper and pencil based systems. 

The juxtaposition of multiple semiotic systems thus provides a rich environment for developing  understanding of 3D geometrical objects. Operating separately with the systems ensures that students encounter different aspects of the properties of such objects. Most importantly, however, operating with more than one system provides important opportunities for developing a fuller understanding of these properties and of the relationships between them. 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 computational environment provided by MachineLab not only juxtaposes different semiotic systems  for representation of 3D objects but also, by making one system depend on another, seems likely to facilitate 'conversion' and consequent abstraction of mathematical properties. 

The pedagogic plan takes the form of a project based on a 'real world' context that will have meaning for students within discourses from outside the classroom. They will thus be likely to draw on everyday discourses and forms of representation as well as on the formal mathematical discourses and representations encountered in the classroom. This provides opportunities for them to form links between these discourses, enabling sense to be made of the representations that are new to them. We understand such links between different domains of activity (sometimes referred to as 'transfer') to occur through the formation of chains of signification, where similar signifiers are encountered in different discourses (Carreira, et al, 2002).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Three dimensional geometry is commonly perceived to be a difficult area of the curriculum both to teach and to learn. The UK curriculum for lower secondary school focuses primarily on drawing and interpreting 2D representatios of 3D objects. However, much current teaching in this area makes very limited use of 3D resources, making the topic extremely abstract. This is thus an area of the curriculum recognised as in need of development.

The use of a 'real world' context is a currently common approach to motivating students within the UK curriculum for students in the lower secondary school. For example, a curriculum development project funded by the Bowland Trust and supported by the UK government Department for Education and Skills is producing a number of curriculum packages of pedagogic plans and materials for teacher training involving the use of mathematics within contexts of interest to students of this age (e.g. music and international aid). This congruence with current officially approved forms of pedagogy is likely to enable mathematics departments, teachers and students to perceive the pedagogical plan as an acceptable or even desirable innovation.

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

Familiarity with basic use of a computer - logging on, loading and saving files.

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 It is expected that different parts of the pedagogic plan will be carried out in three different physical contexts:

  • outside in the school grounds
  • a classroom with movable chairs and tables and an interactive whiteboard
  • a computer laboratory

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes; transform 2-D shapes by translation, rotation and reflection, recognising that these transformations preserve length and angle, so that any figure is congruent to its image under any of these transformations
  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • communicate mathematically, making use of geometrical diagrams and related explanatory text
  • use precise language and exact methods to analyse geometrical configurations
  • review and justify their choices of mathematical presentation

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis and use of properties of 2D and 3D figures
  • analysis of 3D figures into their component parts
  • understanding and use of angles in 2D and 3D

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

basic Logo programming including use of procedures with variables

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

MachineLab Turtleworld

Tool access show tooltip help

NIL

Work plan show tooltip helpexplode

Time show tooltip help

13 Hours in class plus homework

Process documentation show tooltip help

 Audio and video recording of 

  • one group of 5 students who worked together during the classroom based part of the scenario
  • one pair from this group who worked together during the computer based part of the scenario
  • some additional students during computer based work
  • the whole class (focused on the front of the room) during whole class instruction and discussion and student presentations

saved MaLT files
All paper and pencil work produced during the lessons
Posters produced in the final session

introduction: planning the overall design

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • scale drawing
  • plans
  • designing and conducting a survey

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

Students are introduced to the 'real world' aim of the project: to design a new sports centre for the school. They identify the space within which it is to be built and draw plans to scale of the space available. They discuss the features that such a building might have and design and conduct a survey of their peers to identify what facilities should be included in the building.

Rationale show tooltip helpexplode

The critical characteristic of the pedagogical plan as a whole and of this SNIPP in particular is the location of the students' task within a 'real life' context. This is intended both to motivate them by appealing to interests outside the mathematics classroom and to provide them with alternative means of making sense of the representations of 3D objects that they will encounter later in the project.

Theoretical framework show tooltip help

The pedagogic plan takes the form of a project based on a 'real world' context that will have meaning for students within discourses from outside the classroom. They will thus be likely to draw on everyday discourses and forms of representation as well as on the formal mathematical discourses and representations encountered in the classroom. This provides opportunities for them to form links between these discourses, enabling sense to be made of the representations that are new to them. We understand such links between different domains of activity (sometimes referred to as 'transfer') to occur through the formation of chains of signification, where similar signifiers are encountered in different discourses (Carreira, et al, 2002).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Three dimensional geometry is commonly perceived to be a difficult area of the curriculum both to teach and to learn. The UK curriculum for lower secondary school focuses primarily on drawing and interpreting 2D representatios of 3D objects. However, much current teaching in this area makes very limited use of 3D resources, making the topic extremely abstract. This is thus an area of the curriculum recognised as in need of development.

The use of a 'real world' context is a currently common approach to motivating students within the UK curriculum for students in the lower secondary school. For example, a curriculum development project funded by the Bowland Trust and supported by the UK government Department for Education and Skills is producing a number of curriculum packages of pedagogic plans and materials for teacher training involving the use of mathematics within contexts of interest to students of this age (e.g. music and international aid). This congruence with current officially approved forms of pedagogy is likely to enable mathematics departments, teachers and students to perceive the pedagogical plan as an acceptable or even desirable innovation.

Theoretical framework show tooltip help

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

none 

Teacher prerequisites show tooltip help

 none

Context show tooltip helpexplode

Physical context show tooltip help

Outside in the school grounds, on the site earmarked for a new sports centre.

On returning to the classroom, tables are arranged to allow 5 groups of 4-5 students to work together

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

 National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • communicate mathematically, making use of geometrical diagrams and related explanatory text

Content-epistemological goals show tooltip help

  • measurement of distance in metres and centimetres
  • visualisation of 3D shapes
  • basic understanding of the relationship between a 3D object and its plan

Cognitive goals show tooltip help

conceptualising and setting into motion a project to last over a period of time 

Social-affective goals show tooltip help

 discussing and coming to an agreed plan in a small group

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

measuring instruments

tape measures

trundle wheels

or other instruments suitable for measuring the dimensions of a plot of ground

Tool access show tooltip help

NIL

Work plan show tooltip helpexplode

Setting show tooltip help

The first part of this takes place outside in the school grounds.

The second part is based in the normal classroom. 

Time show tooltip help

two hours of class time plus homework

What to do and how show tooltip help

 Phase 1


In classroom:

Teacher explains overall purpose of the project and modes of working: to work in groups to create a design for a new sports centre for the school and to prepare a presentation of this design.

Teacher and class discuss: what needs to be done as first steps in the design process?

  • deciding the constraints on the design (space available, required elements)
  • finding out the needs the building has to meet

Outside:

Teacher identfies space designated for sports centre.

Students in groups of 4-5 choose measuring instruments, measure and record the dimensions of the ground space available.

Return to classroom:

Teacher leads discussion of how to produce a scale drawing of the space. Each group to agree on the necessary calculations and results and produce a scale drawing.

Phase 2

Whole class discussion of what might facilities might be put into the sports centre. Generate two possible lists: necessary (e.g. changing rooms) and optional.

In groups: Decide on two of the optional facilities that the group agrees should be included in the design. Draft a questionnaire gather opinions of students in other classes about desirable facilities.

Teacher led discussion: Each group shares the questions they have drafted; discussion of usefulness of the questions (issues that may arise: ambiguity, ease of answering, bias).

Groups revise their questionnaires. Each member of the group to ask five other students as homework before the next lesson.

Individuals make initial sketches of the outside of the building and discuss these within their group.

Homework

Each student collects answers to questionnaires from at least five students outside the class. The group should collate their results and make a decision about what facilities to include in their sports centre design.

2D representations of 3D space

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • 3D geometrical figures
  • spatial visualisation
  • 2D representation of 3D space
  • angles and turns in 3D

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

This sequence of lessons engages students with conventional and novel forms of representation of 3D objects in 2D. It focuses initially on developing visualisation and informal reasoning to familiarise students with the forms of representation and to develop their confidence. In later lessons, they are introduced to isometric drawings and to plans and elevations.

Students are thus introduced to a number of forms of representations that they may choose to use when presenting the designs of their buildings at the end of the project.

Rationale show tooltip helpexplode

3D geometry is an area of the curriculum that appears difficult both to learn and to teach.  2D representations of 3D objects are frequently used in order to support analysis of their geometrical features, yet the success of this approach relies on students' ability to make connections between the different representations. In practice, the focus of the curriculum is often more on developing skills in constructing particular type of 2D representations rather than on using them in order to develop understanding of the 3D objects themselves. The aim of this set of activities is to engage students in using a range of representations purposefully and making connections between them in order to gain a fuller understanding of  3D geometrical objects. A computational environment such as MachineLab provides new ways of making such connections.

The activities proposed in this plan make use of a range of modes of representation in order to encourage students to develop their visualisation skills and to enable them to choose the mode of representation best suited to the task in hand.

The set of activities as a whole has an overall purpose and a concrete outcome in the form of the design of a building. This project format is intended to motivate students through its opportunities for creativity and through competition with their peers. The need to communicate their final design to an audience also provides a realistic environment for students to reflect on the different forms of representation available to them and to make choices between them in order to communicate most effectively.

Theoretical framework show tooltip help

Multi-modal and multi-semiotic environments allow participants many opportunities for making meanings with the representations available to them and choices about the most apt representations to employ in order to communicate their desired meanings. Through the course of the set of activities, students will make use of a range of semiotic systems, both visual and symbolic, with different elements and grammars. Each of these semiotic systems has a different meaning potential (O'Halloran, 2005; Kress, 2001). Thus making use of one of the systems, for example, isometric drawings, offers a particular set of opportunities for developing understandings of the properties of 3D shapes, while another, such as nets, offers different opportunities. The symbolic logo-based programming of MachineLab provides a further semiotic system that makes more explicit use of angle and length relationships within figures than most paper and pencil based systems. 

The juxtaposition of multiple semiotic systems thus provides a rich environment for developing  understanding of 3D geometrical objects. Operating separately with the systems ensures that students encounter different aspects of the properties of such objects. Most importantly, however, operating with more than one system provides important opportunities for developing a fuller understanding of these properties and of the relationships between them. 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 computational environment provided by MachineLab not only juxtaposes different semiotic systems  for representation of 3D objects but also, by making one system depend on another, seems likely to facilitate 'conversion' and consequent abstraction of mathematical properties. 

The pedagogic plan takes the form of a project based on a 'real world' context that will have meaning for students within discourses from outside the classroom. They will thus be likely to draw on everyday discourses and forms of representation as well as on the formal mathematical discourses and representations encountered in the classroom. This provides opportunities for them to form links between these discourses, enabling sense to be made of the representations that are new to them. We understand such links between different domains of activity (sometimes referred to as 'transfer') to occur through the formation of chains of signification, where similar signifiers are encountered in different discourses (Carreira, et al, 2002).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

 none

Teacher prerequisites show tooltip help

none 

Context show tooltip helpexplode

Physical context show tooltip help

 a classroom with tables arranged in groups providing space for students to share physical models and drawings

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes;
  • select and combine known facts and problem-solving strategies to solve complex problems
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations

 

Content-epistemological goals show tooltip help

  • development of visualisation skills in 2D and 3D
  • recognising different representations of the same 3D object
  • recognition of functions of different forms of representation of 3D objects

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •   positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is intended to provide students with experience of interpreting a wide range of different forms of representation of 3D objects, skills in constructing conventional representations mandated by the curriculum and an appreciation of the functions of different forms. 

Theoretical framework show tooltip help

The juxtaposition of multiple semiotic systems provides a rich environment for developing  understanding of 3D geometrical objects. Operating separately with the systems ensures that students encounter different aspects of the properties of such objects. Most importantly, however, operating with more than one system provides important opportunities for developing a fuller understanding of these properties and of the relationships between them.

Work plan show tooltip helpexplode

Setting show tooltip help

Ordinary classroom with tables arranged so that a group of 5 or 6 students can work together with space to manipulate several resources.

Time show tooltip help

3 hours

Distinguishing forms of representation

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • 3D geometrical figures
  • spatial visualisation
  • 2D representation of 3D space
  • angles and turns in 3D

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

Students are introduced to the aim of the project as a whole: to construct a design for a building and to present this design to an audience of their peers, making use of a range of representational tools. They will examine, analyse and discuss examples of professionally produced designs in various forms and examples of virtual 3D environments.

Rationale show tooltip helpexplode

By introducing students to a range of professionally produced and published 2D representations of 'real world' 3D structures, they will have opportunities to interpret the representations and to reflect on the functions that these different kinds of representations may serve in real world contexts. This will provide them with a framework for thinking about their own production and use of such representations in the context of their design project.

Theoretical framework show tooltip help

In order to communicate mathematical ideas effectively, students need to make use of forms of representation and combinations of representations in ways that are accepted within a mathematical community. Explicit attention to the ways in which particular forms of communication are used for specific social purposes (genres) is argued to provide a means for students to develop powerful communication skills (Cope & Kalantzis, 1993; Marks & Mousley, 1990).

Cope, B., & Kalantzis, M. (Eds.). (1993). The Powers of Literacy: A Genre Approach to Teaching Writing. London: Falmer Press.
Marks, G., & Mousley, J. (1990). Mathematics, education and genre: Dare we make the process writing mistake again? Language and Education, 4(2), 117-135.


 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

 none

Teacher prerequisites show tooltip help

none 

Context show tooltip helpexplode

Physical context show tooltip help

A classroom with tables arranged in groups providing space for students to share physical models and drawings

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes;
  • select and combine known facts and problem-solving strategies to solve complex problems
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations

 

Content-epistemological goals show tooltip help

  • development of visualisation skills in 2D and 3D
  • recognising different representations of the same 3D object
  • recognition of functions of different forms of representation of 3D objects

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •   positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The introductory session will provide students with an overview of the project they will be working on. Its main aim is to provide a motivation for individuals and groups of students to engage with tasks involving 2D representations of 3D shapes both by providing a 'real life' context that they can relate to and by stimulating their aesthetic appreciation of 3D objects.

The stimulus materials will involve a number of different modes of representation in both 2D and 3D and discussion of the meaning of these representations will begin to develop the language of the domain

MachineLab Turtleworld

Tool access show tooltip help

NIL

1D variation tool [Feature]

The variation tool is activated by clicking on the turtle path formed by running a procedure. For every variable in the procedure, a slider appears. Manipulating this slider changes the value of the variable and thus changes the turtle path displayed in the turtle screen.

Work plan show tooltip helpexplode

Setting show tooltip help

Ordinary classroom with tables arranged so that a group of 5 or 6 students can work together with space to manipulate several resources.

Time show tooltip help

1 hour

What to do and how show tooltip help

1. Students seated in groups. Each group has a set of different representations of a selection of buildings including some or all of the following:

  • photographs, paintings and or drawings from different viewpoints
  • photographs or sketches of details (including interior features, doors, staircases
  • plans and elevations

Around the walls of the classroom are posters with large photographs and architects plans of each of the buildings.

Within the group, students discuss what these representations and sort them into sets related to the same building, identifying each building on the posters around the walls.

2. Teacher asks a member of each group to choose one of their photographs or drawings and to explain to the whole class:

  • which building does it show?
  • how do you know it is that building?
  • where was the photographer/artist viewing the building from?

3. Teacher leads a whole class discussion about the similarities and differences of these representations:

  • what can you see/ not see?
  • why might a particular representation be useful or interesting? to whom?

4. Teacher reminds the class of the project aim to design a sports centre. Outlines the kinds of drawings that they will be learning to make (plans and elevations, isometric drawings) and tells them that as part of this they will be using MachineLab to design the door for their building.

Demonstration to whole class of MachineLab:

(a) house - representation of a building that can be turned using the variation tools to see it from different angles.

Ask the class what they can see, what they cannot see as the house is moved.

(b) a simple door that opens/closes using the 1D variation tool

Ask the class to predict what angle will be shown on the variation tool when the door is half open, fully open.

Discuss what other kinds of doors they might choose to have (revolving, sliding, double swing doors).

4. Groups start the process of design, using the results of the questionnaires completed as homework to decide what components to include in their design and making initial rough drawings.
Their discussion should address (among other issues):

  • what rooms to include downstairs and upstairs
  • the overall shape of the building
  • what kind of front door to have (e.g. single or double, swing, sliding, rotating)

isometric drawings

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • 3D geometrical figures
  • spatial visualisation
  • 2D representation of 3D space
  • angles and turns in 3D

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

Students draw isometric representations of  3D objects viewed from various positions and reconstruct multilink buildings from isometric drawings.

Rationale show tooltip helpexplode

Isometric perspective is a standard form of representation, required by the curriculum. 


While students are generally able to interpret simple isometric drawings, they often lack awareness of possible ambiguities in more complex examples. They find it harder to construct their own drawings. Common errors include: drawing faces that should be hidden, mis-drawing angles between faces. 

Although the perspective provided in MachineLab Turtleworld is not isometric, we anticipate similar problems in interpretation and construction. This SNIPP thus provides an opportunity for students to develop familiarity with this form of representation and for teachers to identify and begin to address difficulties that may arise for individuals or groups of students.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

 none

Teacher prerequisites show tooltip help

none 

Context show tooltip helpexplode

Physical context show tooltip help

 a classroom with tables arranged in groups providing space for students to share physical models and drawings

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

 

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • use 2-D representations of 3-D shapes and analyse 3-D shapes (through 2-D projections and cross-sections, including plan and elevation)
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes
  • communicate mathematically, making use of geometrical diagrams and related explanatory text
  • use precise language and exact methods to analyse geometrical configurations

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis of 3D figures into their component parts
  • interpretation and construction of isometric representations

 

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •   positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

isometric paper [Resource for students]
Resource contents show tooltip help
illustration of views [Resource for students]

PowerPoint slides containing different views of the same three dimensional object (built from cubes):

  • two isometric drawings from different view points;
  • a plan, front elevation and side elevation (this is to be used in the next lesson).
multilink cubes [Resource for students]

2cm plastic cubes with projections that allow them to be connected together to build 3D models 

Resource contents show tooltip help

Work plan show tooltip helpexplode

Setting show tooltip help

 A classroom with tables arranged so that students can work in groups of 5-6.

A form of whole class display is required for 

  • pre-prepared isometric drawing (either using the PowerPoint provided or a large poster
  • teacher demonstration of isometric drawing technique

Time show tooltip help

1 hour

Actors' roles show tooltip help

At various points in the lesson the teacher leads discussions, demonstrates the technique of isometric drawing, supports students as they work independently.

Students participate in discussions. They set each other challenges and respond to challenges posed by others, thus engaging in practising the technique of isometric drawing.

What to do and how show tooltip help

 1. The teacher distributes to each group of students a 'building' consisting of 6-8 multilink cubes. This is positioned in the middle of the group's table. Using plain paper, each student draws what they can see of the 'building' and writes their name on the back of their paper.

2. All buildings and drawings are collected at the front of the classroom. The teacher chooses and holds up a drawing and asks the class to identify which building it is a picture of. Repeat this with several drawings. (It may be expected that different ways of representing the 3D aspect of the buildings will have been used and should be shared with the class.) The teacher leads a discussion about the differences and effectiveness of the various representations. Issues that may arise include:

  • whether/how hidden faces are shown
  • how the relationships between different faces are indicated
  • whether individual cubes are indicated or just the overall shape of the building

3. The teacher explains that a conventional way of drawing 3D objects may make it easier to communicate effectively. S/he shows the class the first slide of the PowerPoint presentation (or equivalent display), showing an isometric representation of a multilink model. Students use multilink to construct this model. It is possible that not all students will construct identical models as the representation is ambiguous. If this is the case, the teacher shows the whole class two different (but correct) versions of the model and invites discussion of the differences, establishing that both are correct given the available information. S/he then shows the second PowerPoint slide which shows an isometric representation of the same model from a different viewpoint. Students compare their models and decide which is the correct version.

4. The teacher selects one of the models made earlier and demonstrates on the board how to construct an isometirc drawing of it.

5. Each student now builds a model using 6-8 multilink cubes. Each model is placed in the centre of the group's table. Each student then uses isometric paper to draw an isometric representation of each of the models on their table. The teacher intervenes to support any students who are having difficulties with this.

6. Within each group, all the drawings are compared and sorted so that drawings of the same model from different view points are collected together. These sets of drawings are passed to another group. Each group of students now works togetherr to build models from the drawings they have been given.

7. Finally, the teacher selects some examples of sets of drawings and the models corresponding to them (the originals and those constructed from the drawings) and encourages the class to compare and validate the work that has been done.

plans and elevations

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

 Students make use of plans and elevations to construct buildings using multilink. They then design their own multilink buildings and draw plans and elevations for these.

Rationale show tooltip helpexplode

Plans and elevations are a standard form of representation, required by the curriculum. They are also widely used in the practical design process, providing a more convenient means to guide construction than is offered by other perspective-based representations. We thus anticipate that students will choose to use plans and/or elevations in developing and communicating their overall designs. This SNIPP provides the opportunity for students to recognise the relationships between measurements of corresponding parts of 3D objects, their plans and their elevations.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

 none

Teacher prerequisites show tooltip help

none 

Context show tooltip helpexplode

Physical context show tooltip help

 a classroom with tables arranged in groups providing space for students to share physical models and drawings

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • communicate mathematically, making use of geometrical diagrams and related explanatory text
  • use precise language and exact methods to analyse geometrical configurations

 

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis of 3D figures into their component parts
  • interpretation and construction of plans and elevations of 3D objects

 

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •   positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Resources show tooltip helpexplode

General description show tooltip help

 multilink cubes

drawing materials

plan and elevation cards [Resource for students]

A set of cards to be sorted. Each card shows a plan, side elevation or front elevation of an object consisting of combinations of cuboids.

The task is to find the three cards showing the same object.

Resource contents show tooltip help

Work plan show tooltip helpexplode

Setting show tooltip help

Ordinary classroom with tables arranged so that a group of 5 or 6 students can work together with space to manipulate several resources.

Time show tooltip help

1 hour

Actors' roles show tooltip help

At various points in the lesson the teacher leads discussions and supports students as they work independently.

Students participate in discussions. They set each other challenges and respond to challenges posed by others, thus engaging in practising the technique of converting between 3D models and plans and elevations.

What to do and how show tooltip help

1. The teacher reminds students of the different view of buildings they looked at during the lesson on 'Distinguishing Forms of Representation', drawing their attention to the architect's drawings (ideally the posters showing these buildings and their various representations should still be displayed around the classroom). S/he leads a discussion of what these are showing and what they might be useful for.

2.The teacher displays a pre-prepared set of plan, side elevation and front elevation (using the PowerPoint slide provided or an equivalent). Each student used multilink to construct a model corresponding to the plan and elevations. Each group of students compares the models built by their group members and agree on a single correct model to share with the rest of the class. The teacher leads a discussion validating the correct models, identifying and clarifying any difficulties or confusions.

3. The set of cards and multilink models is distributed to each group. Groups work together to sort the cards to match the models. The teacher supports any groups having difficulties.

4. Each student constructs a model using 6-8 multilink cubes and draws a plan, front and side elevation. The drawings from one group are exchanged with those of another group. Groups work together to construct models from the drawings they have been given.

5. Finally, the teacher selects some examples of sets of drawings and the models corresponding to them (the originals and those constructed from the drawings) and encourages the class to compare and validate the work that has been done.

constructing and manipulating in 3D

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

 Students use the MachineLab environment to construct and dynamically  manipulate 3D representations of components of their designs that are particularly hard to represent well using traditional forms of representation. They start with a sub-scenario that will introduce them to basic functioning of the microworld and then progress to three sub-scenarios focusing on specific components of their design.

  1. walls and windows
  2. roofs
  3. doors

Rationale show tooltip helpexplode

3D geometry is an area of the curriculum that appears difficult both to learn and to teach.  2D representations of 3D objects are frequently used in order to support analysis of their geometrical features, yet the success of this approach relies on students' ability to make connections between the different representations. In practice, the focus of the curriculum is often more on developing skills in constructing particular type of 2D representations rather than on using them in order to develop understanding of the 3D objects themselves. The aim of this set of activities is to engage students in using a range of representations purposefully and making connections between them in order to gain a fuller understanding of  3D geometrical objects. MachineLab provides new ways of making such connections.

The activities proposed here using MachineLab pose students challenges to construct representations of  3D objects, familiar to them within the built environment. By exploring the use of Logo commands in order to address these challenges, students will form and test hypotheses about the length and angle relationships in 3D structures, thus developing their understanding of 3D geometry.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

Familiarity with basic use of a computer - logging on, loading and saving files.

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 a computer laboratory with one computer for each group of two or three pupils

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes
  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations 

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis and use of properties of 2D and 3D figures
  • analysis of 3D figures into their component parts
  • understanding and use of angles in 2D and 3D

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

basic Logo programming including use of procedures with variables

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is integrated into the overall project of designing a sports centre, having as its overall aim the construction of an animated model of the doors of the main entrance. The earlier tasks are designed to introduce students to the Logo programming required to achieve this task, while developing their understanding and use of angle in 3D. 

Theoretical framework show tooltip help

 

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

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

By relating the construction challenges to real world objects and behaviours, students may form intertextual associations that will support their use of the representations offered by MaLT.

MachineLab Turtleworld

Tool access show tooltip help

NIL

Work plan show tooltip helpexplode

Setting show tooltip help

 A computer room with computers for pairs of students to work together and a display with computer for demonstrations

Time show tooltip help

8 hours

What to do and how show tooltip help

 

developing 3D sense of movement

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

Students are introduced to the Logo commands and syntax necessary to move in 3D space.

Rationale show tooltip helpexplode

Although students operate everyday within a 3D environment, this is generally done intuitively without a formal means of representation outside their own body. As they are introduced to MachineLab, they must begin to use formal representations in the form of Logo commands (turn, pitch, roll) which, though apparently similar to natural language, introduce new, distinctive ways of thinking about turning.

This introductory activity allows students to experiment with use of this formal language, to relate it to movements of their own bodies, other physical objects (real and imagined) and the visual feedback provided by MachineLab, in order to develop meanings for these terms.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Drawing in 3D in MaLT is related to students personal experience of flying in aeroplanes and simulation with a physical model of a toy aeroplane. 

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

Familiarity with basic use of a computer - logging on, loading and saving files.

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 a computer laboratory with one computer for each group of two or three pupils

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • communicate mathematically, making use of geometrical diagrams and related explanatory text
  • use precise language and exact methods to analyse geometrical configurations

 

Content-epistemological goals show tooltip help

  •   development of visualisation skills in 3D
  • understanding and use of angles in 3D

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

 basic MachineLab Logo instructions for movement and turn 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is integrated into the overall project of designing a sports centre, having as its overall aim the construction of an animated model of the doors of the main entrance. The earlier tasks are designed to introduce students to the Logo programming required to achieve this task, while developing their understanding and use of angle in 3D. 

Theoretical framework show tooltip help

 

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

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

By relating the construction challenges to real world objects and behaviours, students may form intertextual associations that will support their use of the representations offered by MaLT.

MachineLab Turtleworld

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

General description show tooltip help

  a toy aeroplane to be used for demonstration and discussion of 3D movement

MaLT help sheet [Resource for students]

 A worksheet providing descriptions of the Logo instructions required in MaLT to move and turn the turtle, lift and drop the pen and clear the screen.

Work plan show tooltip helpexplode

Setting show tooltip help

 A computer room with computers for pairs of students to work together and a display with computer for demonstrations

Time show tooltip help

1.5 hours

What to do and how show tooltip help

 

 1. The teacher explains the aim of the lesson - to learn how to use MaLT as a tool for drawing 3D shapes - and remnds students that they will be using this tool to construct animated doors for their sports centre design.

2. The teacher uses a toy aeroplane to demonstrate and discuss with the class how an aeroplane takes off and moves in three dimensions. In doing this, s/he introduces the vocabulary that will be used in MaLT to describe different kinds of turns in 3D space, turn, pitch and roll, relating these to different phases of the aeroplane's flight (e.g. take off, changing course, turning with a roll when coming in to land). It is anticipated that most students will have some experience of flying in aeroplanes and will be able to use this experience to make sense of the new vocabulary.

3. The teacher demonstrates the Logo commands corresponding to turn, pitch and roll, and takes instructions from the students to produce the path of an aeroplane taking off and changing course.

4. Pairs of students discuss how they want their own aeroplane to move and use the Logo instructions in direct drive mode to produce their paths. Support is provided by the teacher and by the hlep sheet as needed.

walls and windows

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

Students operate with basic Logo commands to construct representations of 3D structures involving configurations of rectangles. 

Rationale show tooltip helpexplode

The activities proposed here using MachineLab pose students challenges to construct representations of  3D objects, familiar to them within the built environment. By exploring the use of Logo commands in order to address these challenges, students will form and test hypotheses about the length and angle relationships in 3D structures, thus developing their understanding of 3D geometry.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

familiarity with the MachineLab Turtle world environment and basic Logo commands

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 a computer laboratory with one computer for each group of two or three pupils

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes
  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations 

Content-epistemological goals show tooltip help

 

  •  development of visualisation skills in 2D and 3D
  • analysis and use of properties of 2D and 3D figures
  • analysis of 3D figures into their component parts
  • understanding and use of angles in 2D and 3D

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

basic Logo programming including use of procedures with variables

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is integrated into the overall project of designing a sports centre, having as its overall aim the construction of an animated model of the doors of the main entrance. The earlier tasks are designed to introduce students to the Logo programming required to achieve this task, while developing their understanding and use of angle in 3D. 

Theoretical framework show tooltip help

 

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

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

By relating the construction challenges to real world objects and behaviours, students may form intertextual associations that will support their use of the representations offered by MaLT.

MachineLab Turtleworld

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

Walls and windows worksheet [Resource for students]

 Guidance for the task of constructing the walls of a room.

Work plan show tooltip helpexplode

Setting show tooltip help

 A computer room with computers for pairs of students to work together and a display with computer for demonstrations

Time show tooltip help

1.5 hours

Actors' roles show tooltip help

 Teacher introduces the task, then supports students as necessary.

Students build procedures, experiment with Logo commands in order to find the desired positions and angles.

What to do and how show tooltip help

The teacher introduces the task - to use MaLT to draw two adjacent walls of a room - and leads the class through the first step of imagining what their walls should look like:

Imagine one wall of a room.
What shape is it? How long, how high, what are the angles at the corners
Does it have a window in it? Where?
Go to the corner of your imaginary room. What happens when the first wall meets another wall?

Students then follow the guidance of the worksheet to make a procedure to draw one wall, then position the turtle appropriately to draw the second wall.

When students have succeeded in this task, they may draw a window or door in one of their walls.

Some students may chose to complete their rooms, adding further walls, windows and doors.

roofs

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

 Students are provided with procedures for making 'roof' shapes (triangular prisms). They manipulate the 1D variation tool to make the roofs join up correctly.

Rationale show tooltip helpexplode

 The activities proposed here using MachineLab pose students challenges to construct representations of  3D objects, familiar to them within the built environment. By exploring the use of Logo commands in order to address these challenges, students will form and test hypotheses about the length and angle relationships in 3D structures, thus developing their understanding of 3D geometry.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

familiarity with the MachineLab Turtle world environment and basic Logo commands 

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 a computer laboratory with one computer for each group of two or three pupils

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes
  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations 

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis and use of properties of 2D and 3D figures
  • analysis of 3D figures into their component parts
  • understanding of angle as a measure of turn

 

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

use and interpretation of MachineLab 1D variation tool 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is integrated into the overall project of designing a sports centre, having as its overall aim the construction of an animated model of the doors of the main entrance. The earlier tasks are designed to introduce students to the Logo programming required to achieve this task, while developing their understanding and use of angle in 3D. 

Theoretical framework show tooltip help

 

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

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

By relating the construction challenges to real world objects and behaviours, students may form intertextual associations that will support their use of the representations offered by MaLT.

MachineLab Turtleworld

Tool access show tooltip help

NIL

1D variation tool [Feature]

The variation tool is activated by clicking on the turtle path formed by running a procedure. For every variable in the procedure, a slider appears. Manipulating this slider changes the value of the variable and thus changes the turtle path displayed in the turtle screen. 

2D variation tool [Feature]

 If a procedure contains two or more variables, two of these may be chosen  by clicking on them within the 1D variation tool to be represented on the axes of a cartesian plane shown in the 2D variation tool. 'Drawing' with the mouse in this plane sets the values of the two chosen variables to the coordinates of the points drawn. Thus it is possible, for example to investigate what happens to the turtle path when the two variables are in a particular linear relationship (by drawing a straight line) or what the relationship between the variables is when the shape of the turtle path behaves in a particular way (by manipulating the tool carefully to produce the desired behaviour).

Resources show tooltip helpexplode

roofs worksheet [Resource for students]

 Instructions for loading and manipulating (with the 1D and 2D variation tools) pre-prepared constructions with variables. Questions to prompt reflection and explanation of the results.

prism1 [Resource for students]

 Logo procedure to produce a triangular prism with one variable angle.


Copy and paste into the MaLT Logo Editor.

prism2 [Resource for students]

 Logo procedure to produce a triangular prism with two variable angles.

Copy and paste into the MaLT Logo Editor.

prismX [Resource for students]

 Logo procedure to produce a triangular prism with two variable angles and two variable edge lengths.

Copy and paste into the MaLT Logo Editor.

Work plan show tooltip helpexplode

Setting show tooltip help

 A computer room with computers for pairs of students to work together and a display with computer for demonstrations

Time show tooltip help

1 hour

Actors' roles show tooltip help

 Students follow worksheet instructions, experiment, reflect on and explain their outcomes.

Teacher provides support as required, ensures students address the questions posed on the worksheet to prompt reflection and explanation, leads a final discussion to share explanatory insights.

What to do and how show tooltip help

1. Students load pre-prepared prism1 procedure, follow instructions to manipulate the 1D variation tool in order to make the prism join up, determine the angle required and explain it (using knowledge about equilateral triangles).

2. Students load pre-prepared prism2 procedure, follow instructions to manipulate the two variables using the 1D variation tool to make the prism join up, determine the angles required and explain it (using knowledge about equilateral triangles). They then follow instructions to initiate and manipulate the 2D variation tool in order to make the faces of the prism open like a double door. They discuss what is happening in their pairs and explain the required relationship between the angles.

3. Students load prep-prepared prismX procedure, follow instructions to manipulate the four variables using the 1D variation tool to make the shape into a prism. They discuss what is happening in their pairs and explain the required relationships between the angles and the side lengths.

4. The teacher leads a whole class discussion of the solutions to the final task. Points that may be drawn out of the discussion include:

  • what does each of the sliders control?
  • the sizes of complementary angles
  • alternative solutions that are and are not equilateral

doors

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

 Student build a representation of a door in 3D space that can be dynamically manipulated using the 1d variation tool to open and close.

They may adapt this design to construct a revolving door or a sliding door.

Rationale show tooltip helpexplode

The activities proposed here using MachineLab pose students challenges to construct representations of  3D objects, familiar to them within the built environment. By exploring the use of Logo commands in order to address these challenges, students will form and test hypotheses about the length and angle relationships in 3D structures, thus developing their understanding of 3D geometry. 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

familiarity with the MachineLab Turtle world environment and basic Logo commands 

Teacher prerequisites show tooltip help

 Familiarity with the MachineLab Turtleworld environment and Logo programming.

Context show tooltip helpexplode

Physical context show tooltip help

 a computer laboratory with one computer for each group of two or three pupils

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • explore the geometry of cuboids (including cubes), and shapes made from cuboids
  • recognise and visualise rotations, reflections and translations, including reflection symmetry of 2-D and 3-D shapes, and rotation symmetry of 2-D shapes
  • understand angle measure, using the associated language
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • select and combine known facts and problem-solving strategies to solve complex problems
    identify what further information is needed to solve a problem; break complex problems down into a series of tasks
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • use precise language and exact methods to analyse geometrical configurations 

Content-epistemological goals show tooltip help

  •  development of visualisation skills in 2D and 3D
  • analysis and use of properties of 2D and 3D figures
  • analysis of 3D figures into their component parts
  • understanding and use of angles in 2D and 3D

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Instrumental goals show tooltip help

  •  use of Logo procedures as building blocks for more complex models
  • animation using the 1D variation tool

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This sequence of tasks is integrated into the overall project of designing a sports centre, having as its overall aim the construction of an animated model of the doors of the main entrance. The earlier tasks are designed to introduce students to the Logo programming required to achieve this task, while developing their understanding and use of angle in 3D. 

Theoretical framework show tooltip help

 

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

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

By relating the construction challenges to real world objects and behaviours, students may form intertextual associations that will support their use of the representations offered by MaLT.

MachineLab Turtleworlds

Tool access show tooltip help

NIL

Resources show tooltip helpexplode

Door worksheet [Resource for students]

 Guidance for constructing a simple door that opens and closes.

Revolving Doors [Resource for students]

 Guidance for constructing and animating a multi-panelled revolving door.

Swing Doors [Resource for students]

Guidance for constructing and animating a pair of double swing doors. 

Sliding Doors [Resource for students]

 Guidance for constructing and animating a sliding door.

Work plan show tooltip helpexplode

Setting show tooltip help

 A computer room with computers for pairs of students to work together and a display with computer for demonstrations

Time show tooltip help

4 hours

Actors' roles show tooltip help

 Teacher introduces the task, supports students working independently, guides students in their choice of task and challenge.

Students debug procedures, decide on the designs for their doors and work to construct these using MaLT.

What to do and how show tooltip help

The teacher introduces the task in the overall context of the whol eproject to design a sports centre: groups are to decide what kind of main entrance they want for their sports centre - double swing door, revolving door or sliding door. The task for these sessions is to first construct a procedure for a single swinging door and then adapt this and/or use it as a sub-procedure to construct a more complex moving door.

1. Constructing a door

The teacher leads the class in imagining and describing the shape and movement of a swinging door. The discussion should include issues such as:

  • opposite edges of the door are equal
  • the corners of the door are all 90°
  • where is the angle that changes as the door opens and closes?
  • what is this angle when the door is fully closed? fully open?
  • relate the change in the angle to the previous lesson's use of the 1D variation tool to open and close prisms

Students follow the instructions in step 2  on the worksheet to make a procedure to draw a door. The instructions given are faulty, so students need to edit them to form a correct rectangle.
Once the procedure works, students follow the instructions in step 3 of the worksheet to incorporate their door procedure into a new procedure with a variable for the angle of turn of the door. They run (and if necessary debug) their procedure, using the 1D variation tool to animate the door.

2. An entrance for the sports centre

Each group of students discusses and makes a decision about the kind of main entrance they want for their sports centre design. The teacher may guide groups to help them to make achievable choices.

To help the teacher in providing such guidance: 

  • Revolving Doors involves combining several rectangular panels into a single unit, then constructing a super-procedure with a variable angle to make the unit rotate. (The introduction of the variable is very similar to that demanded by the simple door already constructed.)
  • Swing Doors involves constructing two independent opening doors and coordinating their positions, orientation and angles or turn in order to open them in opposite directions. An additional challenge is to control the animation of both doors by using a single variable, demanding use of complementary angles.
  • Sliding Door involves adapting the super-procedure used to open the simple door so that the variable represents the horizontal position of the base of the door rather than an angle of turn. An additional challenge is to construct a pair of doors that slide in opposite directions, requiring either two 'slide' variables to be manipulated separately or repeated use of a single ariable to animate two movements in opposite directions.

Groups work with the guidance provided by the worksheets to produce their doors.

When the animations are complete, each group saves their procedures to be demonstrated in the subsequent lesson and prints out copies of the screen display (Turtle screen and Logo editor) for possibel includion on their poster displays.

completing and presenting the design

Identity show tooltip helpexplode

Authors show tooltip help

Jehad Alshwaikh, Candia Morgan, Guinevere Dyker IOE/LKL

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

England

Keywords show tooltip help

Description show tooltip help

 Each group of students will collate the components of their design and prepare a poster presentation. This may include plans and elevations, isometric drawings, sketches, screen dumps and dynamic Machinelab demonstrations.

They present their completed designs to the rest of their class.

Rationale show tooltip helpexplode

3D geometry is an area of the curriculum that appears difficult both to learn and to teach.  2D representations of 3D objects are frequently used in order to support analysis of their geometrical features, yet the success of this approach relies on students' ability to make connections between the different representations. In practice, the focus of the curriculum is often more on developing skills in constructing particular type of 2D representations rather than on using them in order to develop understanding of the 3D objects themselves. The aim of this set of activities is to engage students in using a range of representations purposefully and making connections between them in order to gain a fuller understanding of  3D geometrical objects. A computational environment such as MachineLab provides new ways of making such connections.

The activities proposed in this plan make use of a range of modes of representation in order to encourage students to develop their visualisation skills and to enable them to choose the mode of representation best suited to the task in hand.

The set of activities as a whole has an overall purpose and a concrete outcome in the form of the design of a building. This project format is intended to motivate students through its opportunities for creativity and through competition with their peers. The need to communicate their final design to an audience also provides a realistic environment for students to reflect on the different forms of representation available to them and to make choices between them in order to communicate most effectively.

Theoretical framework show tooltip help

Multi-modal and multi-semiotic environments allow participants many opportunities for making meanings with the representations available to them and choices about the most apt representations to employ in order to communicate their desired meanings. Through the course of the set of activities, students will make use of a range of semiotic systems, both visual and symbolic, with different elements and grammars. Each of these semiotic systems has a different meaning potential (O'Halloran, 2005; Kress, 2001). Thus making use of one of the systems, for example, isometric drawings, offers a particular set of opportunities for developing understandings of the properties of 3D shapes, while another, such as nets, offers different opportunities. The symbolic logo-based programming of MachineLab provides a further semiotic system that makes more explicit use of angle and length relationships within figures than most paper and pencil based systems. 

The juxtaposition of multiple semiotic systems thus provides a rich environment for developing  understanding of 3D geometrical objects. Operating separately with the systems ensures that students encounter different aspects of the properties of such objects. Most importantly, however, operating with more than one system provides important opportunities for developing a fuller understanding of these properties and of the relationships between them. 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 computational environment provided by MachineLab not only juxtaposes different semiotic systems  for representation of 3D objects but also, by making one system depend on another, seems likely to facilitate 'conversion' and consequent abstraction of mathematical properties. 

The pedagogic plan takes the form of a project based on a 'real world' context that will have meaning for students within discourses from outside the classroom. They will thus be likely to draw on everyday discourses and forms of representation as well as on the formal mathematical discourses and representations encountered in the classroom. This provides opportunities for them to form links between these discourses, enabling sense to be made of the representations that are new to them. We understand such links between different domains of activity (sometimes referred to as 'transfer') to occur through the formation of chains of signification, where similar signifiers are encountered in different discourses (Carreira, et al, 2002).

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The selection and presentation of students' own designs completes the project and is essential to maintaining the students' sense of the purposefulness of their mathematical activity. Awareness of the functioning of different forms of representation has been a educational objective throughout the project and the necessity to make choices for inclusion in the final presentation highlights this aspect.

Population show tooltip helpexplode

School level show tooltip help

lower secondary

Age range show tooltip help

12-13

Population description show tooltip help

The class: 23 students of below average attainment (set 4 out of 5 sets in the year group). There is a preponderance of boys, as in the school as a whole. A high proportion of the students are of Afircan or Afro-Caribbean background - more than in the school as a whole. There are a number of students in the class who are hard to motivate.

As a newly established and relatively well resourced school, IT facilities are good and well supported by technical staff. All students are used to using computers in lessons, though not often in the context of mathematics lessons.

 

Student prerequisites show tooltip help

 none

Teacher prerequisites show tooltip help

 none

Context show tooltip helpexplode

Physical context show tooltip help

 a classroom with tables arranged to allow groups of students to work together

Institutional context show tooltip help

A comprehensive secondary school currently providing for students aged 11-16. It is a newly established school and will expand to take 16-19 students next year.

It is a voluntary aided (Church of England) school located in a South London borough. Its enrolment policy prioritises children from C of E church-going families but also admits a sizeable minority of students of other denominations. Students are admitted from a wider area than is usual for non-denominational state schools, including some from outside the borough.

The school is subject to the National Curriculum and to the same inspection and examination regimes as other state schools.

Socio-cultural context show tooltip help

The school is a located in a middle class area of South London. As a denominational school, it takes students from a wide catchment area, including more deprived working class areas, and has a substantial number of students from African and Afro-Caribbean backgrounds. The profile of attainment on national tests of students on entry to the school is similar to the national picture, though with few very high attainers. There is an unusually high proportion of boys in the school as there are several local girls' schools with strong reputations.

The culture of the school is highly regulated with formal relationships between teachers and students. The dominant pedagogy is predominantly 'traditional' with strong framing (Bernstein, 2000). Classrooms are mostly arranged to facilitate teacher-led whole class interaction followed by individual 'seat work'.

Goals show tooltip helpexplode

Curricular goals show tooltip help

National Curriculum Key Stage 3 MA3 Shape, Space and Measures

  • use 2-D representations of 3-D shapes and analyse 3-D shapes through 2-D projections and cross-sections, including plan and elevation
  • select problem-solving strategies and resources, including ICT, to use in geometrical work, and monitor their effectiveness
  • interpret, discuss and synthesise geometrical information presented in a variety of forms
  • communicate mathematically, making use of geometrical diagrams and related explanatory text
  • use precise language and exact methods to analyse geometrical configurations
  • review and justify their choices of mathematical presentation

 

Content-epistemological goals show tooltip help

  • purposeful selection and combination of different forms of representation of 3D shapes

Cognitive goals show tooltip help

 analysing a problem and breaking it into manageable parts

Social-affective goals show tooltip help

  •  positive attitudes towards mathematics and a vision of how mathematics may relate to problem solving in the real world
  • collaboration with peers
  • presentation and communication to peers

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

By discussing and evaluating alternative designs and presentations of designs, students will engage critically with the functions and effectiveness of different forms of representation of 3D shapes as well as with the social function of the sports centre design itself. 

MachineLab Turtleworld

Tool access show tooltip help

NIL

Work plan show tooltip helpexplode

Setting show tooltip help

 Classroom with:

  • tables arranged to allow groups to work together with space to manipulate paper-based materials
  • a display area for groups of students to present their completed posters
  • at least one computer to allow students to prepare and present the MaLT component of their designs.

Time show tooltip help

2 hours

Actors' roles show tooltip help

Teacher monitors and supports group work, coordinates presentation and discussion.

Stduents discuss, prepare and present their designs, observe and evaluate the designs of other groups.

What to do and how show tooltip help

1. Students review the components of their design of the sports centre so far and decide what further work they need to do, distribute the work among the group and complete the components. At this stage the teacher should intervene with each group to discuss their plans and to ensure that they are all clear about the overall task and about the role of each group member.

2. When all components of the design are ready, groups collate them, discuss and decide what to include in their poster presentation and how to incorporate the various components into the display and presentation. They plan how to present their design to the rest of the class. 

3. Each group in turn displays and justifies their design (including both a poster and a demonstration of their animated entrance in MaLT). The teacher and rest of the class may ask questions to clarify:

  • the meaning of particular representations
  • the function and rationale of particular parts of the sports centre design
  • the means of construction of the animated entrance

4. When all designs have been presented, the teacher leads a brief discussion reminding students about the criteria for a good design of a sports centre idenitfied at the beginning of the sequence of lessons.

5. Each group of students discusses the other groups' designs and agrees:

  • feedback for each group about the quality of their design
  • a vote for their choice of best design

6. The teacher calls on each group to share their feedback and the outcome of their vote  with the whole class.