Introduction to Space Strand of Mathematics
Ch. 1, page 1
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.
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 don’t jump to the conclusion that it may
connecting ideas
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?
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Ch. 1, page 2
Introduction to Space Strand of 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 twodimensional 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
Two-dimensional Space
Position
Students develop verbal, visual and
mental representations of threedimensional objects, their parts and
properties, and different orientations
Students develop verbal, visual and
mental representations of lines,
angles and two-dimensional shapes,
their parts and properties, and
different orientations
Students develop their
representation of position through
precise language and the use of
grids and compass directions
SGES1.1
Manipulates, sorts and represents
three-dimensional objects and
describes them using everyday
language
SGES1.2
Manipulates, sorts and describes
representations of two-dimensional
shapes using everyday language
SGES1.3
Uses everyday language to
describe position and give and
follow simple directions
SGS1.3
SGS1.1 Sorts, describes and
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Kay Owens
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Introduction to Space Strand of Mathematics
represents three dimensional objects
including cones, cubes, cylinders,
spheres and prisms, and recognises
them in pictures and the environment
SGS2.1
SGS1.2
Manipulates, sorts, represents,
describes and explores various twodimensional shapes
SGS2.2a
Makes, compares, describes and
names three dimensional objects
including pyramids, and represents
them in drawings
Manipulates, compares, sketches
and names two dimensional shapes
and describes their features
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
SGS2.2b
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
Identifies, compares and describes
angles in practical situations
SGS3.3
SGS3.2a
Uses a variety of mapping skills
Manipulates, classifies and draws
two-dimensional shapes and
describes side and angle properties
SGS3.2b
Properties of Solids
Measures, constructs and classifies
angles
SGS4.1
Properties of Geometrical Figures
Describes and sketches three
dimensional solids including
polyhedra, and classifies them in
terms of their properties
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
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
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 4
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

connecting ideas

summarise
and record
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.
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 Piaget’s 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 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
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
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Introduction to Space Strand of Mathematics
Ch. 1, page 5
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 student’s 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 children’s 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
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Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 6
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 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 understandings included (a) position language of degree, e.g.,
halfway, near; (b) shape and line names and classification characteristics e.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.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, reasoning and communication, and mathematical processes. These processes
were (a) 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-yearold children in Peter’s (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.
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Introduction to Space Strand of Mathematics
Ch. 1, page 7
Technological environments (e.g. with Geo-Logo) encourage young children to
collaborate in spatial and 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, 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 student’s 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 student’s 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
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Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 8
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
How many small triangles are needed to
and differences between the pieces. For
make the largest triangle?
example, think about sides, angles,
What fraction of the initial whole square
similarity and areas.
is a small triangle?
Can you make a side from two other
Lay pieces on top of each other to help
sides?
you compare.
How many different ways can you make
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Introduction to Space Strand of Mathematics
Make many different-sized squares,
rectangles, triangles, and other 2D shapes
from the tangram pieces.
Sketch and label them.
Can you make the largest angle from the
other angles? What is its size?
(You do not need a protractor to do this.)
What other investigations did you try?
the large triangles? Sketch them.
Use geometric names like trapezium,
parallelogram, rhombus, quadrilateral,
scalene triangle, isosceles triangle for
shapes that you have made?
Make a large and small parallelogram.
Are they similar? Why?

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?
connecting ideas
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.
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
summarise
and record
Ch. 1, page 10

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)
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Introduction to Space Strand of Mathematics
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
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Ch. 1, page 12
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, relationpreserving 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
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 13
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).
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 14
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
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 15
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 children’s 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 children’s 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 teacher’s 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
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 16
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 twodimensional 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,
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 17
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 Children’s 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.

What did you learn about how young students think spatially?

Which items on the worksheet were more difficult?

Prepare and teach lessons on visual imagery and 2D spatial
concepts at the end of the chapter.

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
connecting ideas
experiencing
connecting ideas
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens
Introduction to Space Strand of Mathematics
Ch. 1, page 18
substantive communication (NSW DET, 2003)?
Summarise your new knowledge on

spatial skills and visualization

2D shape concepts
summarise
and record
Creating Space: Professional Knowledge & Spatial Activities for Teaching Mathematics
Kay Owens