Take a look under the tree
Last October, a group of primary teachersmet at the Wakefield in-service training centre to consider how they could use the second half of the autumn term to workon practical maths activities involving shape and space. Such work, they hoped, would contributeto the end-of-term Christmas traditions of decorating the school and classrooms and the exchange of greetings cards, calendars and presents with parentsand friends.
Christmas is a special time for children. Teachers recognise this and, understandably, are reluctant to abandon established traditions because of pressure from the national curriculum. However, everyone recognises the need to use time productively. Our main aim, therefore, was to work out a progression of activities so that different year groups throughout the school could purposefully be involved. We started by looking at the national curriculum programmes of study for art and design technology as well as mathematics, and we found considerable scope for cross-curricular work.
The strands in the maths document on Shape, Space and Measures we thought particularly relevant are: Pupils should be given opportunities to"gain a wide range of practical experience using a variety of materials" and "use purposeful contexts for measuring" (at key stage 1) and "extend their practical experience ..." and "apply their measuring skills in a range of purposeful contexts"(at key stage 2) Pupils should be taught to describe and discuss shapes and patterns...make common 3-D and 2-D shapes and models, working with increasing care and accuracy: recognise and use geometrical features of shapes...rectangles (including squares) circles, triangles, cubes, cuboids, progressing to hexagons, pentagons, cylinders and spheres; recognise reflective symmetry in simple cases (key stage 1) Visualise and describe shapes and movements, developing precision in using related geometrical language; make 2-D and 3-D shapes and patterns with increasing accuracy, recognise their geometric features and properties . . . understand the congruence of simple shapes: recognise reflective symmetry of 2-D and 3-D shapes (key stage 2) The sort of activities we were thinking about offered pupils opportunities for practical experiences using a wide variety of 2-D and 3-D shapes, as well as measuring in meaningful situations to different degrees of accuracy.
Teachers could introduce materials - card, papers of different kinds, fabrics and threads - according their budget. Colour, design, textures could be discussed with the children as part of their work in art, and extension work planned to involve technology.
One thing we all agreed on was the valuable opportunity such work gave to children for developing their language skills by sharing, developing and explaining their ideas to each other, as well as the teacher.
Recent national initiatives in literacy and numeracy have raised public awareness of the importance of such aspects of learning, and two recently-published books will be very useful for future work of this kind: the School Curriculum and Assessment Council booklet on developing language across the curriculum, and the more detailed booklet on mathematical vocabulary from the National Numeracy Project identifies words and phrases that pupils need to understand and use in each year group if they are to make good progress in mathematics.
We started by deciding to concentrate on those particular aspects of Christmas which we thought would provide most opportunities for links with mathematics. These would include: * making decorations * making greetings cards and calendars * making and wrapping presents * planning and organising parties.
We made a matrix using these headings in order to organise our ideas and plan for a progression of activities, from the younger infants through to Years 5 and 6.
We discussed the importance of the appropriate use of pupils' skills. For example, we should not give young children tasks which demand accurate cutting; slightly irregular edges should be acceptable in the finished design.
However, we should expect pupils in Year 6 not only to be able cut shapes, but also to measure and construct them to a high degree of accuracy.
With these thoughts in mind we then considered the motifs most often used in decorations and decided to focus on three in particular: simple geometric shapes, stars and Christmas trees.
SIMPLE GEOMETRIC SHAPES
Displays of patterns made from 2-D geometric shapes which the children cut out from coloured paper can be very effective, particularly if the number of colours is restricted. Shapes can be arranged in a variety of patterns, some showing "positive" and "negative" views (see Diagram 1).
Children can explore geometric shapes with Mathematical Activity Tiles and with supervision, they can glue shapes together to make 3-D models. For example, they can make pyramids with squares and triangles and a dodecahedron from pentagons. They love the sound of "dodecahedron" and can see it is made up of 10 pentagons, and they remember the name "pentagon".
Three-dimensional star shapes can be made by joining square-based pyramids to a cube (Diagram 2).
Groups of 3-D shapes such as cylinders and spheres can be decorated and hung as mobiles around the school.
We thought a "Find the Shape Trail" would be useful to get children to use "position" words to describe where certain shapes could be found. For example, a Year 3 child might say "I saw some spheres hanging above the doorway, opposite the cylinders which were between the squares. By Year 6 you could expect not only a wider knowledge of geometric shapes but also a wider vocabulary of position and direction words including parallel, vertical and horizontal.
The star is a particularly significant motif, and star shapes have a fascination for children not only at Christmas. Talking about different star shapes, the number of points, how they are constructed, the symmetries which can be found encourages the use of mathematical language.
The simplest star for the youngest children can be made from overlapping two paper squares for an eight pointed star or two equilateral triangles for a six pointed star (Diagrams 3 and 4). An equilateral triangle can be made from a piece of A4 paper as explained in Diagram 5.
The special "magic" of A4 paper depends on the ratio that exists between the sides of the rectangle. The same ratio of 1:1.4142 is found in all sizes of metric paper from A6, the smallest, to A0 the largest, which has an area of one square metre - hence the name metric paper.
Shapes folded in the same way from different sizes of paper will be similar and can be "stacked" or "nested" according to size, and patterns can be created by using contrasting colours of paper.
Older pupils can make stars which demand more practical skill as well as mathematical knowledge. They can create five- pointed stars by extending the sides of a pentagon (Diagram 7). They can make eight-pointed stars by folding paper on the diagonal and cutting out a rhombus (Diagram 8).
A large, six-pointed star can be made by using the equilateral triangles made from A4 paper to make a hexagon and then extend it.
There are many more stars which can be made, described and discussed. The Tarquin book Making Star Shapes is full of exciting ideas.
One group on the course looked at Christmas trees. Triangles, both equilateral and isosceles, can be used to represent trees very effectively. If trees are cut from folded paper then there can be an interesting discussion about symmetry. Children need to meet this concept in many different situations before they are fully confident (see diagrams of Christmas trees).
A 3-D tree can be created from 2-D by using a cone made from a semi-circle of card; and pupils in Years 56 will enjoy making an Advent Calendar "tree" out of hexagons for use in assembly. Younger pupils would need considerable help.
Similar decorations could be used to make cards, calendars and wrapping paper. The group working on ideas on the theme of "presents" recognised the opportunities for talking about 3-D shapes and their relationships with 2-D shapes. Whereas the youngest children would talk about sizes and shapes of boxes and describe attributes such as curves and corners, Year 5 and 6 pupils would be able to make their own boxes and discuss such ideas as "cross-section" and recognise "concave" and "convex" surfaces.
Involving pupils in the planning and organising of parties provides many opportunities for discussing 2-D and 3-D shapes. Table decorations need to be appropriate and party hats need to fit properly so children have a purpose for measuring accurately.
Everyone went back to school more aware of the possibilities for linking Christmas activities with worthwhile work in maths. Although in this particular instance we had been concerned with Christmas we could see that links could also be made with other festivals by shifting the focus slightly. For example, the eight-pointed octagonal star is found in many Islamic patterns, and the patterns used for decoration at Diwali have many symmetries which could be explored.
* Language Across the Curriculum SCAA
* Mathematical Vocabulary Literacy and Numeracy Project available from BEAM, Barnsbury Complex, Offord Road, London N1 1QH * Mathematics through Art and Design by Ann Woodman from Unwin Hyman
* Making Star Shapes by Hine Limbrick from Tarquin Publications
* Winter Mathematics Shropshire Maths Centre
* Mathematical Activity Tiles from the Association of Teachers of Mathematics, 7 Shaftesbury Street, Derby DE3 8YB