Framework for elementary school space mathematics

Kay Owens, University of Western Sydney,
Pat Leberne, and Ian Harrison, Department of Education and Training

Student’s early spatial learning

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, visualize, 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.

Language development. Interestingly, 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 verbalize why a shape is, for example, a triangle - they seem to have a global understanding (van Hiele, 1986) much as they understand 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.

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 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 turn them over (Mansfield & Scott, 1990). 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-cutout representations (Owens, 1992b, 1996; Owens & Clements, 1998).

Movements with card shapes are precursors to tessellation work and to investigating further properties of shapes. The use of grids or composite units seems to be important in developing area grids and hence an appreciation of the formula for the area of a rectangle (Owens & Outhred, 1997).

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 conceptualization. Static pictorial imagery constrained to only one or two examples can limit the development of a concept. 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 on a computer using dynamic geometry software, such as Geometer’s Sketchpad, to represent a variety of examples of triangles. Through active investigation, imagery is more likely to be dynamic or representative of a pattern, relationship or rule.

Spatial abilities have long been studied in psychology (Bishop, 1983; Eliot, 1988), and Piaget (Piaget & Inhelder, 1971) showed that visual imagery was an important aspect of development. Tasks for assessing spatial abilities and visual imagery skills (e.g. Del Grande, 1990, & Tartre, 1990) are akin to items found in the psychology literature. The work and writings of the van Hieles (van Hiele, 1986) and subsequent researchers (e.g. Burger & Shaughnessy, 1986) have assisted teachers to appreciate that students must recognize representations of concepts before they can work with them analytically or relate them to other concepts. However, a theoretical framework that can assist teachers to provide lessons for K—4 students is needed. Such a framework needs to be supported by tasks that reflect both the psychological literature and space mathematics.

The learning framework for space mathematics

There are many different ways of thinking about spatial skills and their impact on the learning of geometry. The organization of these skills can be in terms of the types of objects they relate to, such as 2D or 3D shapes or, alternatively, these skills can be arranged according to the types of actions involved, such as reflections or rotations.

The underlying theme for the learning framework for space mathematics presented in this paper is the increasing sophistication of the way we use imagery. The three aspects used to organize the framework are (a) orientation and motion, (b) part—whole recognition, and (c) classification and language. Each of these links to the imagery that students develop. The work of Presmeg (1986) and Owens (1996) influenced the names given to the visual strategies.

Orientation and motion

Students need to recognize shapes in different orientations and to develop the skill of appreciating what an object or group of objects might look like from another perspective. These changes in perspective and orientation are related to motion. Motions with manipulatives (e.g. card cutouts and tiles) that represent two-dimensional shapes include flips, slides, turns, and folds. These motions assist students to develop concepts such as (a) reflection symmetry (flips in horizontal, vertical, and diagonal lines or folding), (b) area (slide repetitions associated with covering of areas), and (c) rotational symmetries.

Movement is imaged by students as they make associations between shapes. For example, they can image one triangular shape moving to become another triangular shape as a point slides along a taut string. They might see how a triangular shape can become a quadrilateral by bending one side into two, or how a square can be pushed over to make a rhombus.

Movement necessarily involves position concepts. Actions are described in conjunction with directions such as left, right, or straight ahead. A particularly important change of direction or turn is associated with the concept of angle. Early learning is often stimulated by action and this turning of one arm of an angle away from the other does seem to be one of the ways that children first begin to learn about angles. However, they also begin to notice angles on shapes.

Part—whole recognition

All shapes are made up of parts. When students notice the parts, they develop their concepts about shapes. For example, a student might notice three corners on a triangle and decide that this is a defining feature of a triangle. The corners are at first just considered as the pointy parts. Gradually, the parts become specific in their features and students notice right angles or equal sides.

Students might only notice dominant features of shapes such as a pointy part or they might only see the overall shape. It is important that students develop both skills. Noticing the parts requires students to see the part within the context of a shape or configuration of lines. This is the skill of disembedding and it has a counter skill of embedding by which a student is able to complete shapes imagined in the mind.

The ability to see angles on shapes, to see differences in angles, to see shapes within other shapes, and to complete shapes using imagery, are necessary skills for students if they are to develop a repertoire of properties of shapes, or to apply geometry in real examples.

Classification and language

Students will realize that a variety of examples of a shape can be categorized as one particular shape. Students will begin to associate more and more properties of parts as necessary or not necessary for a shape to belong to a particular category. Verbal expressions are associated with visual imagery and help define it. As students group and regroup, they develop relationships between shape categories and properties of shapes and lines.

Students will associate particular words consistently with particular actions, shapes, and other spatial relationships. They can use words to represent their imagery. Students need to identify spatial features such as parallel lines, perpendicular lines, spatial patterns, slopes, shapes, and corners in their environment, and discuss what they see.

Through discussions, students begin to develop an abstraction for concepts such as shapes, to describe comparisons between parts, and to recognize why certain shapes make patterns and tessellate.

Imagery strategies

Each of the above aspects of spatial knowledge becomes evident in the way that students behave and respond to tasks. Their imagery strategies can be inferred from their actions and words. It is through student’s language and the selection and use of objects (including recognition of spatial features of card cutouts, pictures, and shapes in their environments) that decisions on children’s learning will be made.

By saying and pointing, students indicate that they notice parts and visualize their relationships. Students’ imagery develops with dynamic changes and patterns in their imagery. A young student may see the pattern of equilateral triangles covering a larger equilateral triangle as ‘one up one down’ or the student may mentally move one image of a triangle into another image (Owens & Clements, 1998).

Students also increase their visual memory, and sequential use of perceptions. For example, young students can mentally fold a net of an open cube (Owens, 1992a).

Nevertheless, students’ abilities to represent shapes by drawing or using materials like sticks can have different influences on their thinking. For example, some think that their drawing of a square is not a square because it no longer matches their mental imagery. They may then decide a problem cannot be solved because the square does not look right, but other students will accept that they just had trouble with the drawing and their mental imagery dominates. In a similar way, when students are making shapes, they make decisions about whether gaps and overlaps are important or not (Owens & Outhred, 1997). It is important, therefore, for teachers to wait and probe during assessment tasks in order to recognize how the student is thinking.

There are five groupings of strategies:

Emergent strategies. Students using emergent strategies are beginning to attend purposefully to aspects of spatial experiences, to manipulate and explore shapes and space, to select shapes like ones shown or named, and to associate words with shapes and positions.

Perceptual strategies. Students using perceptual strategies are attending to spatial features and beginning to make comparisons, relying on what they can see or do.

Pictorial imagery strategies. Students using strategies involving pictorial imagery are developing mental images associated with concepts, with increasing use of standard language.

Pattern and dynamic imagery strategies. Students using strategies involving pattern and dynamic imagery are using pattern and movement in their mental imagery and developing conceptual relationships.

Efficient strategies. Students using efficient strategies are beginning to solve spatial problems and constructions successfully by using imagery, classification, part—whole recognition, and orientation.

The order is more or less the order in which strategies are likely to emerge and be used by children. Intuitive and incidental learning can influence these strategies in unexpected ways. The casual use of a spatial term can be picked up by a young child in such a way that a well developed understanding of the concept is formed earlier than expected. This might occur if a child realizes that the term triangle is used over a number of different occasions to refer to shapes that are not all exactly the same.

Orientation and motion, part—whole recognition, and classification and language aspects of spatial thinking can be described for each strategy group to provide a framework of imagery for space mathematics (see
Table 1).

Assessment tasks

Tasks have been developed to assist in the assessment of students’ spatial thinking. They are used to illustrate how responses to the tasks indicate a student’s imagery strategies. They can also be modified and elaborated to provide interesting activities for students. For example, many puzzles are possible with different shapes - you can begin with the larger shape and cut it up in different ways. Task 5 has been influenced by Rosser, Lane, and Mazzeo (1988), Task 2 by Mansfield and Scott (1990), Task 2, 4 and 7 by Owens (1996), Task 6 by Lovitt and Clarke (1988). These tasks are not comprehensive but they do give the teacher the opportunity to observe the students’ creativity as well as mathematical knowledge.

The tasks are:

Task 1A: Recognizing shapes in the environment

Task 1B: Sorting shapes and identifying properties

Task 2A: Recognizing double tiling

Task 2B: Imagining triangle tiles

Task 2C: Imagining tiling of areas

Task 3: Imagining shape completion (partially hidden shape)

Task 4A: Seeing shapes within shapes (matchstick designs)

Task 4B: Seeing shapes within shapes

Task 5: Angle recognition, visual memory, and rotation skills (2 stick designs, in view & covered)

Task 6: Dynamic imagery (moving string to make different examples)

Task 7: Imagining, folding and turning nets to make three-dimensional shapes

Task 8: Visualizing turning three-dimensional shapes

General instructions and preparation for using the tasks

Try out the tasks before using them. When administering, sit the student next to you, set out the equipment and ask the questions. It may be necessary to ask the question another way but be careful not to teach or lead the student. If the task is too difficult initially, suggestions can be given to simplify the task in order to establish what the student knows.

It is important to allow the students the opportunity to use their imagery before they manipulate materials. For this reason, encourage students to explain or draw from their imagery before attempting to manipulate materials. This use of imagery may involve changes in orientation, movement in two or three dimensions, or visual analysis of the shapes involved, but the kinds of responses that are wanted must result from imagery in the mind. If students have difficulties, investigative tactics can be used, and manipulations with materials can be carried out. The more advanced students will, of course, use more advanced investigative tactics. An early development will be that of flipping pieces as well as turning pieces. Later, students show an interaction of imagery, concepts and perception. With more advanced investigative tactics, systematic trials and analyses are invoked.

It is recommended that at least three students in the class be videotaped so that you can later look at the students working, and analyse their more subtle responses such as facial expressions, slight finger movements, and where they look. These videotapes are also a good source for discussion with fellow teachers and consultants as you develop a more effective spatial mathematics program together.

Write the student’s name on all papers that the student draws on. Staple these papers to the student’s response sheet. The responses should be briefly but thoroughly recorded on the response sheet. Immediately after the student has completed the tasks, write down what imagery strategy their responses indicate in each task. In summary, record the strategies that the student is able to use, and how the student may be extended. This may be by tasks that are slightly more difficult, or tasks that will assist them to begin a new strategy.

Preparation of equipment for use with tasks

Tasks 1 and 2. Draw the shapes shown in Figure 1 on cards

Figure 1. Drawings on cards

Make card cutouts as shown in Figure 2. Start by cutting 3 large equilateral triangles (same as the drawing in Figure 1), and 3 large rectangles twice the square in Figure 1.

Cut one of the triangles in half. Cut another triangle into 4 smaller equal equilateral triangles. Leave the third triangle whole.

Cut one rectangle in half and use one square to make four smaller squares. Cut off another square and divide it into three equal rectangles. Leave the third rectangle whole.

In summary, you now have 2 large equal squares, 4 small equal squares which would equal one large square, one large rectangle equal to the 2 large squares, and 3 small rectangles.

The right-angled triangle is half the equilateral triangle. Four small equilateral triangles will make the large equilateral triangle. Similarly 4 small squares or 3 small rectangles make the larger square, and two large squares make the large rectangle. Most of these tiles will NOT be used except for probing when students cannot visualize, as indicated in the tasks.

Task 3. Use the small square under cardboard to reveal as shown in Figure 3.

Task 4. You need 7 equal length sticks. Draw an isosceles trapezium with 3 sides equal to the sticks and the long side equal to 2 sticks (Figure 4). The sticks should be narrow, preferably a little longer than matchsticks, and with flat faces so that they do not roll. After joining and drawing sticks to make 2 squares and then 2 triangles, and disembedding the rectangle and rhombus, the student is to see the trapezium in the design shown in Figure 4.

Task 5. Use three card circles with a position tab on them. Two pipe cleaners are each cut into a short and long length at matching points. These are used in different orientations, some shown, some hidden, some hidden and turned.

Task 6. Use a 30 cm string, joined to form a loop, and a firm stick. The teacher holds 2 points about 10 cm apart, and the student predicts changes in triangles.

Task 7. The nets of the open triangular prism and open cube as shown in Figure 6 should be prepared large enough for easy manipulation. Fold lines should have been folded and then opened. Practise folding up the open triangular box.

Task 8. Use a square pyramid.

Follow-up classroom activities

When the individualized assessment of the students has been undertaken, it is possible to plan suitable activities including whole-class teacher discussions and small-group learning experiences. Many of these small-group experiences will be open-ended so that students showing different strategies can attempt the activities. At the earlier stages, useful learning experiences can include shape making from smaller shapes, sorting a wide variety of objects (shells, pasta, etc.) and shapes, and structured play activities. It is also important to look for the shapes in the environment. Informal and formal discussions with the students will be essential. The teacher needs to ask about similarities and differences and then to ask students to see what is the same about all the shapes in the one group, allowing the student to abstract the concept.

The teacher needs to place shapes in different orientations, make different shaped cardboard cut-outs and different length geostrips, and encourage students to look for shapes in different environments. Students can make a variety of shapes of triangles or of other shapes on the geoboard or other form type of peg board or pins in polystyrene slabs. A teacher could ask students to begin with a dot on the page and then draw different straight-line intervals through it and then to complete triangles by joining the ends (this can be a delightful art activity too). In a similar way, students can make different parallel lines and then draw perpendiculars between the parallel lines to form different-sized rectangles.

At later stages, a range of tessellating and tangram-type puzzles can be used. Often the activities can be made more difficult by the use of more difficult shapes. The frequent opportunity to be creative with shapes which are turned, enlarged, proportionally changed, or tessellated will encourage the development of pattern and dynamic imagery, orientation and motion. Talking with peers and the teacher will turn the activities from just doing or playing with card cutouts into visual and conceptual learning experiences.

Joining points marked on a circle on paper, or students sitting in a circle with a ball of string to throw gently to one another can explain what shapes they are making as each new line is added. Students can also draw a range of shapes with a certain number of sides, e.g. all having six sides and discuss similarities and differences, concave shapes, and different sizes of angles. Students can discuss which are regular (i.e. all sides and angles equal) or irregular.

A length of rope, thin strip of cloth, or plasticene snake, computer drawing packages, and Cabri Geometry can all be used to modify shapes in a dynamic way. Cutting up and pulling pictures apart can also be useful, as well as making silhouettes using a strong spotlight such as an overhead projector or shadows in the sunlight with different body movements or over a period of time during a sunny day.

Opportunities to disembed shapes within shapes, complete shapes that are partially available, and to disembed and discuss parts of shapes will be important forerunners to discussions about properties and their relationships. Matchstick problems are just one kind of fun activity related to disembedding and embedding. These learning experiences assist imagery strategies and part—whole relationships to develop.

Students should have the opportunity to use materials such as polydrons with two-dimensional shapes that clip together to form three-dimensional shapes. They should pull boxes apart; and discuss, predict, wrap and make different boxes. Students should turn shapes over, draw them from different perspectives, and print and draw nets. Students can turn cubes marked with a different symbol on each face and then predict it. In addition, such activities as predicting an order of photographs taken when walking around a corner or in different parts of the school environment can encourage visualizing changes in position in relation to three-dimensional shapes.

References

Acknowledgements. Thanks are due to Peter Gould, Hillary Andrews, Jill Everett, Chris Francis, Maxelle Matthews, Mike Mitchelmore, Jan Stone, and many teachers and children for their input into the development of this framework.