1. Visualisation in the Curriculum: Australian Curriculum- understand the concepts of size, shape, relative position, and movement of 2D and 3D shapes- investigate properties of shapes, and use these properties to define compare and contrast shapes and objects- choose appropriate units to measure objects, and recognise the connectedness between measures. : VELS- Shape- 2D and 3D shapes, and their transformations. Properties of shapes- continuity, edge, surface, region, boundary, connectedness, symmetry invariance, congruence and similarity Expected that students will be able to identify and represent shapes, but also construct and transform them. - Measurement- estimate measures using comparison methods, prior knowledge and experience, or spatial manipulations.
1.1. Geometry"involves shape, size, position, direction, andmovement and describes and classifies the physical world we live in" (Copley, 2000, p.105)
1.1.1. NCTM (2000), considers Geometry as one of the five basic mathematics concepts that students should have the opportunity to learn. This concept is made up of four instructional areas:
1.1.1.1. SHAPEAnalyse characteristics and properties of shapes: The van Hiele model for geometric thought highlights the stages of cognitive development that are related to shape
1.1.1.1.1. Level 0; Visualisation. Identify shapes on their appearances, but little attention is paid to attributes
1.1.1.1.1.1. Reys, Lindquist, Lambdin & Smith (2007) suggests that at this stage it is important to show students both examples and non-examples of shapes or else students will have a fixed idea of shape (p. 368). This could then hinder their further shape understandings.
1.1.1.1.1.1.1. Good Activities for this level:
1.1.1.1.1.1.1.1. Shape sorting activities. according to properties they notice. Eg. for 3D shapes: which shapes stack, which roll ect.
1.1.1.1.1.1.1.2. Making shapes from other shapes: - Using pattern blocks/ tangrams- With tessellations- Shape Maker (Reys, Lindquist, Lambdin & Smith, 2007, p. 380)
1.1.1.1.1.2. Way (n.d) suggests that this level is characterised by "nonverbal thinking", as shapesare recognised as "a whole, rather than by distinguishing parts" (para. 3)
1.1.1.1.2. Level 1: Analysis.Begins to recognise shape properties andidentifies shapes by these properties
1.1.1.1.2.1. Properties of 2D shapes:- Number of sides and corners- Symmetry: there are two types of symmetry- reflectional (when two sides are the same) & rotational- Length of sides- Angle size- Parallel lines (when two lines will never meet)- Perpendicular lines (when two lines intersect at 90 degree angles)`- Concave and convex features- Altitude (height) is important whenfinding the area of shapes.
1.1.1.1.2.1.1. Good activities:- Sorting shapes based on properties- Grouping shapes with similar properties- Mystery bag- students have to guess the hidden shape based on described properties- Mystery definition (Van de Walle, 2010, p. 413)- Discovering Pi (Van de Walle, 2010, p. 415)- Folding a piece of paper in half and then drawing only on one side, then folding the paper to create a symmetrical picture.- How many lines of Symmetry (Reys, Lindquist, Lambdin & Smith, 2007, p. 371)
1.1.1.1.2.2. Reys, Lindquist, Lambdin & Smith (2007) suggest that students first need to develop their ideas of 2D shapes in order to "describe [3D] shapes" (p. 362).
1.1.1.1.2.2.1. - Edges- Faces- Vertex- Corners
1.1.1.1.2.2.2. Good Activities to develop ideas of properties of 3D shapes:From Reys, Lindquist, Lambdin & Smith, 2007, p. 367-368)- Gumdrops & Toothpicks- Build your own Pyramid
1.1.1.1.3. Level 2: Informal Deduction.Begin to see the relationships among shapes.
1.1.1.1.3.1. Relationships between 2D and 3D shapes
1.1.1.1.3.2. Relationships among quadrilaterals
1.1.1.1.3.3. Way (n.d) suggests that at this levelstudents can "explain the relationships between shapes" and "formulate definitions" (para. 5)
1.1.1.1.4. Level 3: Formal Deduction. Use rules to prove statements
1.1.1.1.5. Level 4: Rigorunderstand geometry in an abstract way
1.1.1.2. LOCATIONSSpecify Locations
1.1.1.2.1. Vocabulary:- Position words (eg. under, on top of, beside ect.) - Movement words (eg. up, down, sideways, left ect.)- Distance words (eg. near, far, close ect.)
1.1.1.2.2. Understanding the function and use of a compass
1.1.1.2.3. Reading maps and diagrams
1.1.1.2.3.1. Lowrie & Logan (2006) suggest that information in maps today is represented in a Dynamic way. Information can be presented for example through colour or scale
1.1.1.2.3.1.1. Good Actvties:- Treasure hunts- Battleship- Interpreting weather maps (Lowrie & Logan, 2006)
1.1.1.2.4. Networks:"Represent relationships involving connectedness" (DEECD, 2009, para 5)
1.1.1.2.4.1. Examples: tree diagrams, flowcharts, road maps, family trees
1.1.1.2.4.1.1. Good Activity:- Turtle Geometry / Ladybug leaf (Utah State University,2010)- K?nigsberg: from nrich (2007)
1.1.1.3. TRANSFORMATIONSApply transformations
1.1.1.3.1. Transformations
1.1.1.3.1.1. -Turns (rotation)-Flip (reflection)-Slide (translation)
1.1.1.3.1.1.1. Good Activities:- Patch Tool (NCTM 2012)- Visualizing Transformations [6.4.1] (NCTM)- ITP Symmetry (Mathsframe.co.uk, 2012)
1.1.1.3.1.1.2. Reys, Lindquist, Lambdin & Smith (2007) suggest that students should first be asked to predict the outcome of transformations, as this will help "develop a deeper understanding" (p. 378)
1.1.1.3.1.1.2.1. NCTM (2012) sugges that when teaching transformations it is important for students to consider the difference between the original and transformed image (para. 5).
1.1.1.3.1.2. Congruence:Same size and shape & area
1.1.1.3.1.2.1. Reys, Lindquist, Lambdin & Smith (2007) indicate that 2D shapes are congruent when they have the same area
1.1.1.3.1.3. Ratios: Enlarging or Reducing images
1.1.1.3.1.3.1. Good Activity:Using isometric paper students have to proportionally enlarged/ reduce an image (Board of Studies NSW, 2003,p.148)
1.1.1.3.1.4. Dynamic Imagery is the mental "[manipulation] and changing [of] shapes" (Office of Shool Education,Department of Education, Employment and Training, 2001, p.52). Eg. stretching shapes.
1.1.1.4. VISUALISATIONUse Visualisation to solve problems
1.1.1.4.1. To develop mental imagery of 2D shapes: - Peeking over/ shape reveal- What's under my blanket (Anne Downtown- Genertic Tasks)- Look make & Fix (Copleu, 2000, p. 105)- Quick Draw (Copley, 2000, p.119)
1.1.1.4.2. Perspectives of 3D objects:- Constructing structures & representing them through drawings- Photo Sort (Lovitt & Clarke (1988))- Baarrier Game (Board of Studies NSW, 2003,p.144)
1.1.1.4.3. Visualise 2D nets, making 3D objects
1.1.1.4.3.1. - A Puzzling cube (nrich 2007). - Investigating the different types of nets that create a cube, and developing a rule. (Lesson by Rose Knight).
1.1.2. Spatial SenseDEECD (2009) states that for many years it was argued that spatial sense was something innate. However recent research highlights that spatial sense can be improved and taught, especially in geometry, through activities that include concrete materials, require students to predict and conduct transformations and imagine and experience different perspectives. Such activities can help to develop an understanding of the surrounding world. Copley (2000) states that spacial sense is childrens "awareness of themselves in relation to the people and objects around them" (p.105)Van de Walle (2010) suggests that spatial sense also includes a comfort with geometric descriptions and objects.
1.1.2.1. Spatial Orientationthe ability to see and "operate on the relationships between objects in space" (DEECD, 2009, para 7)
1.1.2.2. Spatial Visualisation/ Spatial ReasoningThis is the process of forming mental images. It's being able to "carry out mental movements of two and three-dimensional objects in space" (DEECD, 2009, para. 7)
1.1.2.2.1. Lowrie & Logan (2006) suggest that Spatial Visualisation & Reasoning skills include "building & manipulating mental representation of objects, perceiving an object from different perspectives & interpreting & describing physical environments"
When teaching shape, it is important to provide students with concrete materials to explore and manipulate shapes. Students will not learn about shapes from formal descriptions or properties, they need to be given experiences to "handle, manipulate, draw & represent shapes in a variety of ways" (Copley, 2000, p.111-2).
Visual Imagery:Being able to create images of what is seen mentally or the "ability to create a picture in the mind" (Mulligan, Prescott & Mitchelmore, 2003, p.24)
Representations
See how shapes can be made up of other shapesTo extend understanding Reys, Lindquist, Lambdin & Smith (2007) students to make predictions before carrying out ideas and describing, feeling and manipulating are all important experiences to help foster this idea (p.380).
Gould (2003) states that "recognising the parts within a shape...is an essential component of spatial sense" (p.6)
Copley (2000) suggests that spacial understandings that must be taught with shape includedirection, distance, location and representation.
The aim of geometry education should be to "enhance a child's ability to solve problems, both in mathematics and other disciplines and to apply geometry after leaving school" (Davey & Holiday, 1992, p. 26)
Skills to develop in GeometryDavey & Holliday (1992)
Visual Skills: eg. recognising things from pictures or in embedded situations, seeing similarities and differences, reading maps, interpreting diagrams, recognising properties, visualising objects from oral descriptions
Verbal Skills: eg. developing vocabulary to describe (orally or written) an object or spatial situation or relationship, understanding and developing definitions
Drawing Skills: drawing different perspectives, diagrams and shapesmaking models
Logical Skills: understanding properties of shapes and using these to determine similarities and differences, and classifying and sorting them, and testing and making conjectures
Application skills: being able to apply geometry understandings to everyday life.
Visualisation in Number: Subitising: "the ability to see a number of objects instantly without counting them one by one" (Mulligan, Prescott & Mitchelmore, 2003, p. 24)
Measurement:Van de Walle (2010) defines a measurement as:"a number that indicates a comparison between the attribute of the object (or situation or event) being measured and the same attribute of a given unit of measure" (p.370).
Process of Measuring:Van de Walle (2010, p. 370)1. Decide on an attribute to measure2. Selecting a unit of measure appropriate to the attribute 3. Compare the units to the attribute
Estimation:Van de Walle (2010) defines estimation as "the process if using visual information to measure or make comparisons without the use of measuring instruments.
Strategies. Van de Walle (2010) suggests that there are four strategies that can help improve estimating skills (p. 390)
Developing benchmarks for units
Less Than, More Than, About the Same (Board of Studies, 2003,p.138)
Chunking by breaking down an attribute to be measured into smaller parts
Visually or physically iterate units
Van de Walle (2010) suggests that estimation is important to help students "focus on the attributes being measured", help to develop benchmarks and unit familiarity, and can help to motivate students (p. 373)
Students need to be able to recognise the characteristics of the attribute being measured. This therefore links in with recognising shapes in the environment. Attributes to be measured:
LengthMeasures one-dimensional space
Comparing and ordering object based on length
Being able to use a ruler to accurately align and measure objects
Concepts of Perimeter of shapes
Good Activities:- Getting students to prove conjectures about the relationship between Perimeter and Area. Activities are accessed through: National Stem Centre (n.d).
AreaDefinition: The space inside a region. Measures two-dimensional space
Students need to visualise the boundaries when tiling to cover an area
Volume & Capacity
Definitions: McDonough, 2004:Volume: "the amount of space occupied by an object"Capacity: "the amount a container can hold" (p. 283)Measures three-dimensional space
In Volume and Capaity, students have to attend to the properties of the shape of the object such as the height, width and depth
Good Activities for students to understand how the attribute of an object will effect the Volume & Capacity:- Teddy Containers (McDonough, 2004, draws on McDonough, Cheeseman & Clarke (2003), p. 286)- Making different shpaed containers from the same piece of paper- Estimating, checking and ordering the volume of containers
Weight & Mass
Linsay & Scott (2005) suggest that unlike in measurement it is very difficult "distinguish between two masses" by looking, unlike distinguishing between the length of two objects (p. 5)
However estimation in Mass & weight is important, so students need experiences with hefting, and developing benchmarks.
Good Activity:- comparing and ordering objects in a lunchbox from heaviest to lightest (Ann Downton, 2012).
Angles
Visualising a right angle to determine whether a angle is obtuse or acute
Jacobb (2008) suggests that the can Hiele model of geometric thinking can also be related to the cognitive development of students understanding the concept of a measurement. For example he suggests that "the concept of units would be considered Level 1" as students would have to be able to visualise attributes, and then select appropriate units (p. 24)
Early Numeracy Research Project (DEECD, 2006) related growth points:F: Length measurementG: Mass measurement frameworkH: Properties of shapeI: Visualisation and orientation
Spatial Relations:How an object is located in relation to other objects
Yackel & Wheatley (1990) draw on NCTM to support the idea that to foster spatial sense in students "they must have many experiences that focus on geometric relationships; the direction, orientation, and perspectives of objects in space; the relative shapes & sizes of figuers and objects" (p. 52)
Yackel & Wheatly (1990) suggest that it is important to get students to discuss what they observe, other students are able to learn from these explanations & begin "to think about the visually presented images in more than one way & to elaborate on and extend their own ideas" (p.54). In the classroom this can be done by asking students "What do you notice?"
Visual Reasoning:"the way in which an image, in either a physical or mental representation, is used to complete a problem-solving task"(Lowrie & Smith , 2003, p. 2)
Lowrie & Smith (2003) also suggest that since society is becoming more "reliant on visual and spatial reasoning skills" with the use of computers (p.2).
Visualisation is important for
Constructing. Eg: building cupborads
Creating and Analsing routes
Designing plans/ layouts/ blueprints
Following instructions/ Plans
Explaining the location of a person or object
Piggott & Woodham (2008) suggest that Visualisation is important when solving problems. It can be used to:1. understand "what the problem is about" 2. "model a situation" 3. Visulaising to plan ahead". This involves considering the consequences of an action (p. 27-8)
Visualisation skills used when problems solving
Internally representing important information
Identifying an useful images to represent & help solve the problem
Being able to communicate to others the created representation
Spatial sense also related to measurement as it helps us to compare, using measures, ourselves in relation to other objects.
Students need to have an idea of the relative size of units compared to attributes, so that they can "select appropriate units for measuring. Jacobbe (2008) suggests that this is a "major part of understanding measurement". (p.24) This size recognition is reliant on visualisation and estimation.
Good Activities:Sorting shapes using Venn Diagram/ Carrol Diagram ect. based on properties. Good IWB activity can be accessed on Mathsframe.co.uk (2012)
Good Resources for developing visualisation in the classroom
PHYSICAL RESOURCES:- Tangrams- Attribute blocks- Grid paper, isometric paper, cm paper- MAB blocks- Geoboards- Interlocking cubes- construction kits- Soma cubes- 3D solid shapes- Containers-unifix cubes
Way (2006) suggests that where possible students should be first ask to "complete a similar hands-on activity" before using virtual manipulative as it will help "students form mental images that can support more abstract on-screen tasks" (p. 15)
VIRTUAL MANIPULATIVES:- IWB- Interactive websites such as nrich, illiuminations, NCTM, mathsframe ect
Moyer, Salkind & Bolyard (2008) suggests that when choosing a virtual manipulative it is important to consider and assess a programs "mathematical fidelity, cognitive fidelity, pedagogical fidelity and externalized representations" (para. 5)
Moyer, Salkind & Bolyard (2008) advocates that the use of virtual manipulatives can help to extend students skills and knowledge as "virtual manipulatives have unique characteristics that go beyond the capabilities of physical manipulatives" (para. 3)
Good Activity:- Counting Triangles from nrich (2007)
Tesselations: Furner, Goodman & Meeks (2004) state that tessellations can help to develop "concepts like to-dimensional shapes, area, symmetry, rotations, reflections, translations and repetition" (p.26)
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