Introduction to the Space Strand of Mathematics

Introduction to Space Strand of Mathematics

Ch. 1, page 18

Introduction to the Space Strand of Mathematics

Making Sense of Words

Two Year 1 boys stood facing each other with the loop of braid around their necks. They were busy balancing cardboard squares on the space bridge between them. The teacher, Diane, was tickled by their creativity. She had introduced her lessons on area by giving each pair of students a braid to make a shape and then to cover the space inside the braid with squares. It was a valuable investigation that led to many ideas and a good discussion.

Like many people, the two boys thought of space as empty, and something you cannot see. People also think of space without boundaries, and out there in the universe. However, everyday words are often used in a specialised mathematical sense, and students need to appreciate the diverse meanings of words. The word area may have the meaning of the place to go for lunch or play but it also means a two-dimensional space within a boundary. In the playground, the area may be bounded by a fence and a wall or the boundary may be imagined. Areas can be big where the child is surrounded by the area or they can be small where the child looks down on the area. How confusing it can be for students. Space too has special mathematical meanings. It refers to one, two or three dimensions.

connecting ideas

The adjective from the term space also fails to bring to mind a mathematical meaning. It even has a spelling change. Spatial. Sounds like facial. Now dont jump to the conclusion that it may have to do with a make-over or the expression on the face of a person who is spaced out (with drugs or music or even maths)! Many people see the word spatial and do not link it to space at all, let alone mathematics. If they have studied spatial abilities in psychology, the phrase is left in the context of psychology.

For many people, the words numeracy and mathematics have to do with number although there is a syllabus strand, Space, to do with 3D, 2D, and position illustrated with pictures of blocks, shapes, and maps. A common definition of mathematics is the search for patterns in number and space. The symbol 3D refers to three dimensional space represented by blocks or objects or the three directions of sideways, up-and-down, and back-and-forward. 2D refers to two of these dimensions in space and is represented by flat surfaces and figures like triangles. Our English terminology comes from these perpendicular directions used down through the ages from ancient cultures like the Greeks and Persians through to the European cultures of the first two millennia after Christ (to 2000 AD).

The work in the Space strand can be appreciated as mathematics if it is seen as representative of ideas just as the symbol 3 is representative of the idea of threeness. These ideas are not just objects but involve relationships and actions just like the plus sign (+) can represent joining of groups. Space mathematics is linked with high-school work in geometry, for example, on triangles. Space mathematics can be viewed as including geometry although some people see Space mathematics as the visual abilities and knowledge needed to recognise geometrical equalities and similarities, proofs, and measurements. The question is how are the two ideas of mathematics and space related? How are spatial abilities related to mathematics?

NSW Mathematics Syllabus

Space and Geometry is the study of spatial forms. It involves representation of shape, size, pattern, position and

movement of objects in the three-dimensional world, or in the mind of the learner.

The Space and Geometry strand for Early Stage 1 to Stage 3 is organised into three substrands:

Three-dimensional space

Two-dimensional space

Position

The Space and Geometry strand enables the investigation of three-dimensional objects and two- dimensional shapes as well as the concepts of position, location and movement. Important and critical skills for students to acquire are those of recognising, visualising and drawing shapes and describing the features and properties of three-dimensional objects and two-dimensional shapes in static and dynamic situations. Features are generally observable whereas properties require mathematical knowledge eg a rectangle has four sides is a feature and a rectangle has opposite sides of equal length is a property. Manipulation of a variety of real objects and shapes is crucial to the development of appropriate levels of imagery, language and representation.

When classifying quadrilaterals, teachers need to be aware of the inclusivity of the classification system. That is, trapeziums are inclusive of the parallelograms, which are inclusive of the rectangles and rhombuses, which are inclusive of the squares. These relationships are presented in the following Venn diagram, which is included here as background information.

For example, a rectangle is a special type of parallelogram. It is a parallelogram that contains a right angle. A rectangle may also be considered to be a trapezium that has both pairs of opposite sides parallel and equal.

Three-dimensional Space

Students develop verbal, visual and mental representations of three-dimensional objects, their parts and properties, and different orientations

SGES1.1

Manipulates, sorts and represents three-dimensional objects and describes them using everyday language

SGS1.1 Sorts, describes and represents three dimensional objects including cones, cubes, cylinders, spheres and prisms, and recognises them in pictures and the environment

SGS2.1

Makes, compares, describes and names three dimensional objects including pyramids, and represents them in drawings

SGS3.1

Identifies three dimensional objects, including particular prisms and pyramids, on the basis of their properties, and visualises, sketches and constructs them given drawings of different views

Properties of Solids

SGS4.1

Describes and sketches three dimensional solids including polyhedra, and classifies them in terms of their properties

Two-dimensional Space

Students develop verbal, visual and mental representations of lines, angles and two-dimensional shapes, their parts and properties, and different orientations

SGES1.2

Manipulates, sorts and describes representations of two-dimensional shapes using everyday language

SGS1.2

Manipulates, sorts, represents, describes and explores various two-dimensional shapes

SGS2.2a

Manipulates, compares, sketches and names two dimensional shapes and describes their features

SGS2.2b

Identifies, compares and describes angles in practical situations

SGS3.2a

Manipulates, classifies and draws two-dimensional shapes and describes side and angle properties

SGS3.2b

Measures, constructs and classifies angles

Properties of Geometrical Figures

SGS4.3

Classifies, constructs, and determines the properties of triangles and quadrilaterals

SGS4.4

Identifies congruent and similar two dimensional figures stating the relevant conditions

Angles

SGS4.2

Identifies and names angles formed by the intersection of straight lines, including those related to transversals on sets of parallel lines, and makes use of the relationships between them

Position

Students develop their representation of position through precise language and the use of grids and compass directions

SGES1.3

Uses everyday language to describe position and give and follow simple directions

SGS1.3

Represents the position of objects using models and drawings and describes using everyday language

SGS2.3

Uses simple maps and grids to represent position and follow routes

SGS3.3

Uses a variety of mapping skills

Figure 1. Extracts from the NSW Mathematics K-6 Syllabus, p. 23 and p.117

Constructing Ideas

Students construct meaning as Piaget and many other psychologists have suggested through investigative tactics (often physical), visual imagery (initially based on things in front of them), and language. When students begin to put ideas together they will assimilate them, but there comes a time when there is cognitive conflict, and ideas have to be modified and rearranged so that new experiences can be accommodated. For example, a child who thinks that triangles are only equilateral triangles may hear adults referring to other three pointed shapes as triangles. The students will question and think about how to accommodate this new idea. The following task is a metaphor for this thinking process. The child may also hear that some pointy shapes that he thought were triangles because they were pointy had another name. He needs to link at least two ideas, that of three straight sides and at least one pointy corner to be called a triangle. Later, he refers to the corners as angles and vertices.

Learning Tasks for the Reader

Tangram Activities 1

experiencing

Look at the 6-piece Tangram in Lesson 7 at the end of this chapter.

Put pieces 1 and 2 together to make a simple four-sided shape. Keep making four-sided shapes by adding piece 3, then pieces 4 and 5, then piece 6.

connecting ideas

When adding the last piece did you need to rearrange your pieces? (You may have rearranged earlier depending on how you put the pieces together.) It is like accommodating a new piece of information in Piagets terms.

Try finding other solutions.

Look at the parallel lines.

What do you notice about the angles formed by the lines cutting across the parallel lines?

summarise

and record

Summarise your understanding of students constructing mathematical concepts. Use both educational psychology and mathematics education references. A summary of these can be found in Perry and Conroy (1996, pp. 61-71) or in papers by Cobb, Clements, Ernest, and others.

Early Learning about Space

Learning about three-dimensional space begins before a child is born. Movements help babies explore the objects in space around them. Their visual experiences combine with their physical movements and their touch to develop spatial knowledge. Young children enjoy putting objects inside containers and taking them out again. They enjoy putting things together, building up towers and watching them separate as they fall. The learning that arises from these activities is associated with language used by adults, other children, and later themselves. These experiences assist students to develop initial mathematical skills and knowledge about space.

By the time children enter school, they have already experienced spatial contexts and constructed some ideas about shape and space and images associated with these ideas. Structured play encourages cooperative actions with objects, and this play is effective in encouraging students to enjoy, investigate, visualise, and develop language in naturally occurring discussions with adults and children. For example, when students are making a person out of a collection of small boxes, they may talk about how they used long, thin boxes for the legs or found a round one for the head.

Mathematics actually has a delightful way of being pervasive in many aspects of our lives. It often helps us to describe, explain, and represent aspects of our lives and our environment. Mathematics is a tool for this. For example, if I were to give instructions for you to get to the shopping mall a few kilometres away, I may give you a series of right and left turns but it would be safe to draw a rough map, describe some landmarks, and draw the route to follow. Since we live in a three dimensional world, we will come across mathematical representations constantly but we need to develop the associated mathematics. Since we think constantly, we will think spatially, even if we vow and declare that we are left-hemisphere brain or word dominated and hopeless with diagrams (whatever that means). Teachers assist students to develop their spatial mathematical thinking. There are three aspects of learning that assist in the development of spatial mathematical thinking. They are language development, investigative tactics or movements, and visual imagery development.

Ways of Learning about Space

Language Development

Through family, television, and other pre-school experiences, students may learn the names of two-dimensional (2D) shapes like triangle, circle, and square. However, showing equilateral triangles and getting children to repeat the name triangle assists little in the development of the concept of a triangle. Students need to see and make a variety of examples, and pointy non-examples. They need to know why different pointy shapes are part of the triangle family or not.

Students are more likely to learn 2D shape words than the three-dimensional (3D) geometric names associated with the blocks used in play. Nevertheless, it is important to continue students visual development with three-dimensional shapes even though their language, classification, and analysis seemingly lags behind the 2D names.

Initially we know that students cannot always verbalise why a shape is, for example, a triangle(they seem to have a global understanding (van Hiele, 1986) much as they do that a chair is a chair in all its diverse manifestations. On the other hand, a young student may just focus on the pointiness without seeing the whole or noticing other important properties.

Both verbal and non-verbal, were shown by Owens to encourage students analysis of shapes and their monitoring during problem solving. Bell (1994) found vocabulary about 2D shapes among primary school students to be quite limited and Robertson (1992) showed that pre-service education students needed considerable development in understanding terminology.

Social Context, Play and Learning

Boulton-Lewis, Wilss, & Mutch (1994) noted the importance of prior experiences in reaching an intuitive understanding of measurement and Irwin (1995) had explained the effects of schooling and language or culture on making parts of shape designs. Past experiences and social contexts are key aspects of learning and problem solving. Zevenbergen (1992) particularly noted the effects of everyday experiences on spatial language and the implications of social equities for school learning of space mathematics. Like Owens and Clements (1998), and Boulton-Lewis et al. (1994), Zevenbergen emphasised the effects of expectation and social discourse on learning.

Thorpe (1995) conjectured, as a result of her observations, that childrens learning was strongly influenced by their language usage. Mainstream children used language mainly as a vehicle for communication among themselves whereas special-needs children used language mainly to summon adult help.

Play, including pretend play, involves students in mathematical thinking and discourse. Macmillan (1998) analysed the discourse of learning in a preschool play setting in which students were building with blocks. Children were using language about position and size of blocks to establish social position and cooperation. She coded events that occurred while the children were imagining the events surrounding the fire truck. Macmillan not only referred to possible motivational discourses but also to comments that indicated preschool students' use of mathematical language and understandings when involved in counting, measuring, locating, designing, explaining, and playing activities (Bishop, 1988). For example, students referred to "a really big" or "bigger than ever house". They referred to positions such as "go back in," "backwards" and they questioned asking for explanations "How do you get out?" Macmillan also developed a theoretical model showing that responsive and restrictive socio-regulative interactions could motivate interpersonal motivations in the mathematics learning situation. In Owens' (1996) study on angles, responsiveness in the learning situation was also seen as important.

Rogers (1999) carefully analysed video footage taken over a six week period in an early childhood XE "early childhood" centre in a large rural NSW town catering for 85 children aged three to five year. Actions, communication between children or with adults, and self-talk during block play were analysed. The analysis led to the following themes: (a) block play extending children's construction of knowledge, (b) communication and reasoning, and (c) social interaction of children. The themes were illustrated by tables of commonly used words, notes regarding non-verbal indicators, and transcripts. Much of the data was similar to the findings of Macmillan. The tables and transcripts indicated that early spatial and measurement XE "measurement" understandings included (a) position language of degree, e XE "measurement:volume" .g., halfway, near; (b) shape and line names and classification characteristics e XE "measurement:volume" .g. "like a window", (c) turns and corners, and (d) enjoyment at seeing and making spatial patterns. In measurement, children's actions illustrated informal measures and finding midpoints, and comments were on (a) strength and balance, size, comparisons decided by sight or direct matching (e XE "measurement:volume" .g., not equal, twice/half as big); (b) patterns of area; and (c) time and speed recognition.

Rogers developed a diagrammatic representation of the interactions between problem solving XE "problem solving" , reasoning and communication, and mathematical processes. These processes were (a) visualisation XE "visualisation" in estimation, preconceived planning, projection and refinement; (b) experimentation in concrete problem solving, and in creating balanced, symmetrical, and aesthetically pleasing structures, and (c) application to new structures, purpose, product-real and product-imagined. These interactions were illustrated by a vignette of the discussion between children on the building of a structure resembling a Greek temple. Not only was interaction between children important but adults' modelling, acceptance, positive responses, and questioning helped students to feel confident, to cooperate, and to express mathematical ideas and purposes. Rogers concluded by claiming that block play, building and cleaning up activities prepare children for school mathematics. Further she says block play encourages children to learn through sharing of knowledge and skills with peers, experimenting, practising new discoveries or techniques, and applying what they had learned to different situations. Rogers also noted that, once the girls began to participate in block play, there was no gender difference in the themes for play given above, nor was there a difference in whether blocks were used in imaginative play or other play. The girls among the six four-year-old children in Peters (1992) study showed a wide range of spatial skills. Those who scored better on the spatial tasks spent more time in play judged to be high in fostering spatial skills. Girls played with equipment likely to develop spatial skills even though it is traditionally regarded as belonging to the male domain.

Technological environments (e.g. with Geo-Logo) encourage young children to collaborate in spatial and measurement XE "measurement" tasks (Yelland & Masters, 1997). Results have indicated that encouraging young children to work collaboratively, scaffolding their learning via modelling, probing questions, and discussion facilitated problem solving and the more frequent use of higher order thinking skills. The studies have highlighted the importance of distinguishing between cognitive, affective XE "affective" , and technical scaffolding. When the young children were supported in this way the nature of their collaborations and interactions were characterised by more conflict resolution which in turn lead to more effective use of meta-strategic processes (Davidson & Sternberg, 1985).

Affective Factors

From two qualitative studies, one with primary school students (Owens, 1993) and one with teacher education students (Owens, Perry, Conroy, Geoghegan, & Howe, 1994), Owens concluded that affective processing was important in students responsiveness during problem solving including spatial problems. Affect was linked to self-monitoring and reassurance but also to willingness to tolerate open-ended problems and persistence in problem solving.

Two other studies looked at affect and geometry. Southwell and Khamis (1994) noted gender differences and Brodie (1993) noted poor teacher attitudes to the space curriculum.

Investigative Tactics

The best way for students to investigate triangles is through spatial problem solving (Owens, 1996). When they solve spatial problems that involve them making or using triangles, they attend to the features of a triangle, listen to others comments about the shapes, and try manipulating and checking ideas.

Students learn through their investigative tactics. Students will attempt certain actions that they think will assist in their solving of a problem or investigation of a concept. Young students solve shape problems more effectively when they not only turn representations of shapes but also flip them over (Mansfield & Scott, 1990). Movements with card shapes are precursors to tessellation work (like tiles) and to investigating further properties of shapes. Students touch and look at parts and later make comparisons. Young students, given the opportunity, will compare lengths of sides and different angles of shapes by overlaying card-cut out representations (Owens & Clements, 1998).

Visual Imagery

Students also develop visual mental imagery about shapes. Imagery associated with a concept, called concept imagery, forms part of the students summary of a particular conceptualisation. Static pictorial imagery constrained to only one or two examples of a concept (e.g. an equilateral triangle) may limit a students conceptualisation. By contrast, students might have a concept image of a triangle which is an equilateral triangle that changes in their minds so that the lengths of sides vary. This dynamic imagery can be assisted by physically changing a triangle made from string or elastic or on a computer using drawing packages or dynamic geometry software, such as Cabri Geometry, to represent a variety of examples of triangles. Each aid has a different limitation in making new shapes. Through active investigation, imagery is more likely to be dynamic or representative of a pattern, relationship or rule.

Spatial mathematics is more than just knowing that a particular shape is a rectangle or a triangle, and more than just knowing the names of three-dimensional (3D) shapes. It involves physical representations, relationships, position in space, mental representations, spatial thinking(including visualising(spatial purpose, and both artistic and intuitive creativity.

Learning Tasks for the Reader

Tangram Activities 2

experiencing

The folds on the square.

Figure 2. Seven piece tangram square.

Compare pieces and consider similarities and differences between the pieces. For example, think about sides, angles, similarity and areas.

Lay pieces on top of each other to help you compare.

How many small triangles are needed to make the largest triangle?

What fraction of the initial whole square is a small triangle?

Can you make a side from two other sides?

How many different ways can you make the large triangles? Sketch them.

Make many different-sized squares, rectangles, triangles, and other 2D shapes from the tangram pieces.

Sketch and label them.

Use geometric names like trapezium, parallelogram, rhombus, quadrilateral, scalene triangle, isosceles triangle for shapes that you have made?

Can you make the largest angle from the other angles? What is its size?

(You do not need a protractor to do this.)

Make a large and small parallelogram.

Are they similar? Why?

What other investigations did you try?

connecting ideas

When did you have trouble visualising and working with the instructions? Think about the experience or lack of experience that you may have had with this kind of activity.

Revise the names of different two-dimensional shapes. Then collaborate to make posters illustrating each with at least 4 different examples of each shape (changing dimensions and orientations). You may do this on the computer. You may divide the poster with a small section for shapes that are not the shape but small children may think are the shape (e.g. a pointy quadrilateral or an angle close to 180 is not a triangle).

What investigative tactics (actions with the pieces) did you use?

How did you use analysis to help work out sizes of angles?

How did we integrate number ideas to spatial mathematics? Did we make links to measurement of area or angles or length?

What parts of the shapes did you focus your attention on at different times?

Would you have thought about angles, sides, areas, fractions, or explored the pieces and shapes so thoroughly if you were not given both materials to work with and problems to investigate?

When do young children begin doing jigsaws? Why is the above activity more difficult? Are similar visualising skills being used?

Students often have an image of a concept that is very limited. For example, they might picture the triangle only as an equilateral triangle. This image might limit their concepts about triangles. Similarly, they might consider isosceles triangles as tall and thin without realising that they can have an obtuse angle.

summarise

and record

If a trapezium is defined as a quadrilateral with a pair of opposite sides parallel, why is a parallelogram a special kind of trapezium?

Look at NSW K-6 Mathematics Syllabus. Which outcomes did you achieve? Remember to consider outcomes for space and Working mathematically. What values and attitudes are you showing?

Spatial Skills (Abilities)

Visual imagery is a mental process so we can expect that psychologists will have something to say about it. Early factor-analytic work concentrated on theories of intelligence and mental abilities. There was discussion of visual ability along with a range of other abilities such as verbal ability, and of different kinds of visual or spatial abilities. There are many references to spatial abilities, while other areas of psychology also investigate visual imagery. The developmental psychologists such as Piaget developed theories about it, and more recent studies in cognitive psychology have studied it. In addition to this research, we will find educators discussing visual imagery.

The term visualisation is used in more than one way in the spatial-abilities literature and also as external visual imagery in other literatures (e.g., English, art or computers). Visualising or visual imagery is taken in this chapter as internal and as having a mediating mental role between the external input and the external output or representations of spatial concepts and thinking.

Examples of Spatial Skills

The literature on spatial abilities is very complex and the same name can be used for different skills. I am going to refer to two specific groups of skills.:

1. Orientation and movement.

2. Re-seeing

Orientation and movement. I follow the grouping used by Tartre (1990) to include recognising shapes in other orientations (a shape that is turned around) or from another perspective. Orientation is involved when the person considers the similarities of the relations among parts when viewed from another position. For example, the view of a building changes as you walk along the street or around the corner. It covers not only mental rotation, especially of three-dimensional objects, but all forms of transformation that is "mentally moving" or manipulating shapes (Eliot & Smith, 1983). Tartre also grouped mentally folding a shape as one does to make a net into a 3D shape.

Re-seeing or recognition (names from Tartre, 1990; Eliot & Smith, 1983 respectively) This included reorganisation of the whole, ambiguous figures (also called figure-ground perception like the urns or faces illusions), part of field (find part or fit part like you do in a jigsaw), and hidden figures (also called "disembedding"). The skill of being able to disembed shapes and parts of shapes (hidden figures, Figure 3a), is a different skill from those requiring mental manipulation of images. Closely associated with the disembedding skill, although producing the opposite result, is that associated with the task of completing figures (spatial relations in Figure 3b)

Hidden Figures

Look at the shape in the box

Can you find it in the drawings below?

Trace the figure in the drawing.

Figure Completion

Finish the figure on the right to look like the figure on the left.

Figure 3a. Some of Del Grande's items for spatial abilities

These puzzles are about squares. In each item, a piece has been cut out of the square at the left. The missing piece is one of the four on the right. Circle it."

From the JM Battery of Spatial Tests (Johnson & Meade, 1985) for lower primary school students, based on Thurstone and Thurstone 1941 tests for adults.

Figure 3b. Spatial Relations

Figure 3. Various examples of tasks that use different spatial abilities.

Both types of tasks although supposedly testing orientation or re-seeing may still be completed by analytic procedures (Egan, 1979; Carpenter & Just, 1986). Complex three-dimensional tasks, such as those requiring rotation of three-dimensional visual images, are likely to be done analytically (Burden & Coulson, 1982; Izard, 1987). This particular skill of rotating three-dimensional objects appears to be associated with gender differences (Linn & Hyde, 1989). Tasks requiring this skill are generally considered too difficult for young children and too many unexpected factors could impinge on performance.

Del Grande (1990) avoided making a distinction between recognition and transformation in the sense that recognition of shapes in other orientations can require both analytic and rotational skills. His classification of skills covered eye-hand coordination, visual memory, figure-ground perception (hidden figures), perceptual constancy (same shape in different contexts, sizes, or perspectives), perception of spatial relationships (requiring the recognition of the relationship between objects or between an object and oneself), position-in-space (requiring a distinction between reflections and rotations), and visual discrimination (recognising congruence under flips, slides, and turns).

If you are feeling somewhat overwhelmed by all this terminology, it is not surprising. The literature indicates that spatial abilities are quite complex and varied. One way of following the above discussion is to relate the various descriptors to your own personal imagery. Think about how you work out or use your imagery when you do visual tasks. Do you move images in your mind (turn, flip, slide, distort, enlarge)? Do you analyse the parts and their relationships? Do you focus on a part and pull it out from the whole?

Learning Tasks for the Reader

Visual Imagery Activities 1

experiencing

If you have not already done so, complete the visual tasks above.

Now look at the two pictures below. What do you see? Do you see a 3D shape or only 2D shapes.

Disembed shapes, turn shapes, flip shapes. Close your eyes and do the same in your head.

Figure 4. Three rhombus or a cube; A triangle or three lines.

The Role of Visual Imagery

Visualising is important in thinking and learning about many ideas such as the relationships between numbers but it is central to spatial thinking. Spatial thinking involves spatial skills (used in the above tasks) but it also involves the use of certain spatial concepts such as triangle or angle. Development of spatial concepts requires development of spatial skills (Owens, 1993). The term spatial thinking is used here to incorporate spatial conceptualising, spatial skills, and visual imagery. Support for holding this broad view of spatial thinking can be drawn from Piaget and Inhelder (1956, 1971) who discussed both the child's conception of space and the mental images which children employ. Part-whole analysis can be used in all the tasks above so visualisation (used in the broad sense of all visual imagery) is a skill which can involve analysis and checking and hence concepts (Clements, 1983; Krutetskii, 1976). This point was not recognised in the earlier factor-analytic literature on spatial abilities.

In trying to investigate the nature of visual imagery, some researchers (e.g. Lohman, Pellegrino, Alderton, & Regian, 1987; Poltrock & Agnoli, 1986) have concluded that there are differences between spatial ability, visual imagery, and visual memory. According to Lohman et al. (1987), "spatial ability may not consist so much in the ability to transform an image as in the ability to create the type of abstract, relation-preserving structure on which these sorts of transformations may be most easily and successfully performed" (p. 274). Some spatial abilities as well as visual memory appear to assist visual imagery; for example, in Poltrock and Agnoli's (1986) study, efficient image rotation, image integration, adding detail, and image scanning contributed to performance on spatial tests but image generation time did not. Spatial skills such as disembedding or re-seeing were noted as helpful in using imagery and in solving spatial problems.

There are many information processing studies that have explored how imagery might be stored in the mind. A common idea is that there are two ways, visual and verbal but not all research support this dual coding as espoused by Paivio (Kosslyn, 1983; Shepard & Metzler, 1971). Nevertheless, my synthesis of a position supported by both the dual coding and other theories is that:

1.Both verbal (analytic) and visual information can be processed.

2.Individuals vary in their preference for mode of mental representation whether by verbal, visual, or both mediums.

3.There is a means of mental storage which can be used either verbally or visually as needed in the working mind.

Information processing can involve the storage, retrieval and use of visual images. In brief, there are static images and there are images that require manipulation. Images can be analysed and synthesised. They can be prompted directly from external stimuli or created within the mind.

Different Types of Imagery

Presmeg (1986) has provided an analysis of the concept of visual imagery which is especially relevant to mathematics education research. She considered the reports of students who had completed some mathematical problems and classified responses into five groups: (a) concrete, pictorial imagery (pictures-in-the-mind), (b) pattern imagery (pure relationships depicted in a visual-spatial scheme), (c) memory images of formulae, (d) kinaesthetic imagery (imagery involving muscular activity), and (e) dynamic (moving) imagery. This classification by Presmeg has provided an expanded view of visual imagery which is both succinct and relevant to students' processing of information during classroom activities.

Interpreting figural information (diagrams) and visual processing were seen by Bishop (1983) as being two distinct processes. Students performance on visual tasks could result from either or both of these processes and it is not easy to provide tasks of visual processing which do not call on students abilities to interpret figural information (diagrams).

Effects of Training on Visual Imagery

Lean's (1984) comprehensive summary of training studies of three-dimensional spatial skills suggested general geometry courses are less likely to improve the skill of interpreting figural information than specific training courses. Furthermore, he noted that there is less conclusive evidence for being able to train visual processing. He warned that two major features could lead to misinterpretation of the value of training: (a) the training or testing may be indicative of skill in interpreting figural information or in some analytic skills rather than a visualisation skill (see also Deregowski, 1980), and (b) any improvement may merely be from practice rather than from a real improvement in visual skills as indicated by retention and transfer of skills to other tasks. (The latter argument was expounded by Piaget, Inhelder, & Szeminska, 1960.) Cultural factors will also influence development of spatial skills (Bishop, 1988).

Studies with Younger Children

Miller (1977, cited in Lean, 1984) carried out a series of studies in which young children were involved in training in visualising from alternative perspectives but the experience was primarily in interpreting figural information. Nevertheless, kindergarten children showed an improvement on a perspective task after eight training sessions. In another study by Cox (1977) young (five-year old) children were involved in training on either the taking of an alternative perspective or a two-way matrix classification task which was regarded, by the researcher, as requiring spatial skills. Cox reported significantly higher scores for the groups on the tests that used the same kind of items as in their training, and he found some transfer from the perspectives training to the matrices tasks but not the reverse. Cox found no transfer as tested by tasks requiring the prediction of a cross-section or the prediction of the water-level in a tilted jar, and he concluded that the basic requirement for learning and achieving on the spatial tasks was not just operational thinking but spatial skills specific to the task. Retention scores (after seven-months) on the tasks which were similar to those in their training were also significantly different from the other group. The training consisted of 20 sessions provided to children individually.

Training in problem solving appears to develop students spatial thinking. Moses (1977, cited in Lean & Clements, 1981) carried out a problem-solving training study in which grade 5 children, improved their scores on spatial-abilities tests as well as reasoning and problem-solving tasks as a result of the training. Lowrie (1992) found that students chose to use visual or analytic approaches to problems depending on the nature of the problem given that the student had sufficient ability to solve the level of problem. Other problem solving studies, for example, by Chinnapan & Lawson (1995) have shown that general training in metacognitive approaches assists in learning.

Over the years several programs have been developed to improve geometric and visual skills in younger children (Abe & Del Grande, 1983; Frostig & Horne, 1964; Kurina, 1992; Perham, 1978). The careful evaluation of a program by Del Grande (1992) showed that a course involving transformation of shapes did in fact improve the spatial visualisation (perception) of grade 2 students (see Figure 3a for items that he used). The activities involved concrete shapes, geoboards, other common classroom aids, and pencil-and-paper activities. He defined a range of skills, as mentioned earlier, which were tested individually on several occasions to show the effect of training by using a time-series research model.

Instruction in flips, slides, and turns (using activities involving tracing paper, geoboards, and free drawing as well as class and group discussion) assisted performance on tasks involving slides, flips, and reflections except those involving diagonal transformations, and some of those involving turns (Perham,1978; see also Genkins, 1975).

In Owens (1993) study, students in Years 2 to 4 of primary school were involved in early exploratory learning experiences spread over 12 sessions. They undertook a number of problem-solving activities using commonly available materials. The activities were chosen on the basis that they were likely to improve students' visual imagery and concepts about polygons, angles, areas, tessellations, similarity, and symmetry. Students used both (seven-piece) tangram pieces and pattern blocks to find similarities and differences between shapes, to make shapes from other shapes, to make outlines of these shapes with sticks, to sketch shapes and configurations, and to compare angles of these shapes. They used square breadclips to make pentomino shapes, folded paper with pentomino representations to find symmetries, and used cardboard replicas to explore tessellations of these pentomino shapes. The students were also asked to solve problems which required them to make different numbers of squares from a fixed number of matchsticks, to complete designs with matchsticks, and to find shapes within designs made of matchsticks.

The activities were intended to encourage the students to solve problems through discussion and through the invoking of visual imagery and pertinent spatial concepts. The activities provided a basis for challenging students to reflect on and, whenever necessary, to modify existing concepts, images, and skills. The open-ended and multifaceted activities catered for the needs of students with a range of prior experiences and existing concepts.

The study showed that these spatial learning experiences significantly affected students' scores on a delayed posttest used to measure spatial thinking processes (Owens, 1993). In addition, the study discussed how visual imagery and selective attention guided much of the problem-solving processes involved in solving the problems (Owens, 1996).

Computer programs can also be influential in childrens use of imagery. Clements and Battista (1992) note that drawing tools including Logo and Cabri have the potential of encouraging children to internalise actions, and construct new mental tools. Computer diagrams are more flexible than concrete materials and pencil diagrams, so computer drawing tools can also encourage more flexible concept images. However, Lowrie (1998) has shown how childrens use of spatial computer programs is greatly enhanced by interactions with a teacher. Appreciating perspective, angles, position, and the effect of a particular shape may not be easily interpreted or explored by children viewing two-dimensional representations of rooms. A teachers questions can greatly facilitate purposeful interactions with the computer and hence learning in the computer environment.

Studies with older students

Although Lean (1984) concluded that general geometry studies tended not to show improvements in spatial abilities, a study by Bishop (1973) provided evidence that active participation in a geometry course did positively affect spatial abilities. A significant feature of this course was the use of manipulatives. Bishop's (1973) result lends support to the van Hieles' (1986; Burger & Shaughnessy, 1986) theory that recognition should precede analysis in geometry and that manipulatives and everyday experiences have an important part to play in this. Saunderson (1973) is another to make use of concrete activities at the post-secondary level. His training program involved both three-dimensional and two-dimensional activities and his tests also covered both areas. He used informal activities including three two-dimensional activities(tangrams, pentominoes, and enlarging tile shapes. Both the nature of his tests and activities suggested that the improvement in spatial skills after training was linked to improvement in analytical skills. Wearne (cited in Lean, 1984) found that the greatest improvement in scores for secondary students was associated with an increased number of analytic solution strategies. Lean and Clements (1981) found that better problem solving was associated with verbal-analytic rather than visual strategies. Lowrie (1992) showed that students selected the method according to the problem, especially its difficulty.

Rowe (1982) carried out a training study which considered the effects of different types of spatial programs. The study involved grade 7 students, with one group undertaking training of spatial skills for transforming two-dimensional shapes, another group undertaking training on three-dimensional shapes, and a third group acting as a control. The group involved in the two-dimensional program improved statistically significantly more than those involved in the three-dimensional program but only on the test items involving two-dimensional shapes and easier spatial skills. The two-dimensional tasks may have been more suitable than the three-dimensional tasks for these students.

Developmental Studies - Piagetian Studies

Piaget and Inhelder (1956, 1971) provided a developmental theory describing different kinds of mental images held by children of different ages and explaining why these differences occur. They claimed that children who had not yet reached the concrete operational stage could not solve problems requiring mental rotation of images because this task required conservation skills.

Rosser, Lane and Mazzeo (1988) considered age as a predicting variable contributing to level of development. However, they showed that young children could solve rotation problems which were not difficult (such young children may not be conserving). The children reproduced the simple models of two rods, which formed a T or an L, and a circle placed at the end of a rod or in the right angle. The eight-year-olds were able to: (a) reproduce a model present in front of them, (b) reproduce from memory a model which was shown and then hidden (c) memorise and represent an anticipated rotation (which was indicated by hiding and rotating a model), and (d) represent another perspective by moving a model, but the last two of these spatial skills were significantly more difficult for them than the first two skills. Most children aged four and six were unable to demonstrate these last two skills, and they found that reproduction from memory was more difficult than mere reproduction.

Figure 5. Memorising the shapes in different arrangements.

Several studies (e.g. Kidder, 1978) have suggested that it is too simplistic to account for performance on tasks involving transformation (turns and flips) of shapes in terms of merely reaching a global level of thinking involving conservation. When conservation was considered as the determinant of level of development, it was found that students had more difficulty deciding whether parts of a shape had changed than they did with the Piagetian conservation of length task (the staggered lines test involving two equal horizontal sticks with non-vertical starting points). Kidder (1978) found that only a small percentage of conservers could choose the correct length of a side of a transformed triangle, while Thomas (1978) found that non-conservers (determined by the Piagetian task), irrespective of grade (1, 3, or 6), were less likely to be correct in assessing whether the length of the side of a triangle had changed or not under rotations, translations, and reflections than conservers in that grade.

In summary:

1.Descriptions of visualisation and its components in the literature have pointed to the complexity of visual imagery and to the questionable validity of the practice of using spatial-abilities tests to determine the extent of someone's visual imagery. Certain components of visual imagery may not be adequately measured in a spatial abilities test, particularly a test which uses language and configurations which are different from those commonly met in previous experiences.

2.Visual imagery is involved in problem solving, in information processing and storing, and in the generation of concepts at the perceptual and processing stages.

3.A series of general Space (geometry) learning experiences may improve spatial thinking processes. However, the complexity of visual imagery processes indicates that learning experiences need to be holistic and to involve the use of materials or computer images for manipulation.

4.Analytic-verbal procedures appear to be effective in improving problem solving, and this suggests that, while the skills of visualisation are to be encouraged, they need to be associated with analysis.

5.Recognition of the components of visualisation may help teachers to devise procedures which assist students to overcome blockages in problem solving and which develop visualisation skills.

6. Tasks can be made more difficult by incorporating shapes in different orientations and changing the reference lines (often vertical or horizontal).

Learning Tasks for the Reader

Tasks to Explore Young Childrens Spatial Thinking

experiencing

Give the worksheets to young students (Year 1 or above). It is necessary to make some equipment first to explain the items before giving the students the worksheets.

You will need to:

a) Cut-out L-shape pieces (about 10 and 13 cm long as shown in Figures 7 and four equal squares

b) Trace over the parallelogram in Figure 8 onto plastic like an overhead transparency sheet.

c) Four black squares (stickers) for the last section of the Worksheet. A pencil.

connecting ideas

What did you learn about how young students think spatially?

Which items on the worksheet were more difficult?

experiencing

Prepare and teach lessons on visual imagery and 2D spatial concepts at the end of the chapter.

connecting ideas

During the lessons, how did you use spatial skills to decide which shapes were subsets of other shapes?

Which cards led to discussion between members of your group? How did you use argumentation (Wood, 2003) or substantive communication (NSW DET, 2003)?

summarise

and record

Summarise your new knowledge on

spatial skills and visualization

2D shape concepts

Creating Space: Professional Knowledge & Spatial Activities for Teaching MathematicsKay Owens