# Double, double, toil without trouble;Mathematics

One of the best ways to work mathematically with very young or less able children involves using stories.One of my favourite and most successful ways of allowing children to explore number is through Wanda, the forgetful witch. Using a soft doll as a prop, a set of "cauldrons" made from black painted boxes, with varying numbers of sides, I tell the story of Wanda, who makes wonderful spells. For example, she can make nasty medicine taste nice, make your teacher smile all day or make holidays last longer.

Wanda is a special witch because she makes spells using numbers. For example, in her pot she will put the number of cubes that will make the spell work, and these cubes have come from the corners of her pot. Unfortunately, she forgets how many were at the corners in the beginning, and therefore her spells don't always turn out how she planned. For example, if the number of cubes in the pot is 7, and she has a square pot, the 7 cubes need to be shared among the 4 corners (2, 2, 2, 1 for example).

I ask the children to work with a friend and, using the given number of cubes in the pot, find a way of placing them at the corners so that all of the cubes have been used. Then we can find out how Wanda makes the spell work, or how the numbers can be taken apart and put together again. (It sometimes helps for each pair of children to have real four-sided pots and any given number of cubes so that they work with concrete apparatus.)

Once each pair of children has worked on the problem for a few minutes, I get the class to look at and discuss the results. They find that there are many ways of making the total 7. Each should be given the chance to contribute their own ideas. The problem for the children is, "How do we know which combination of numbers will make the spell work, and how do we know if we have found all of the combinations?"

In discussion, children can think and reason and offer proof for their arguments. What if zero is introduced into the equation? How many combinations will there be? Is 2+2+2+1 the same as 2+2+1+2 and 2+1+2+2? The teacher can allow children to make the rules, as long as they can give reasons for them. Once these ideas have been explored, further challenges can be set. What if we change the number of cubes in the pot? Will there be more ways of making that number or less? Does it depend whether the number is more than 7 or less? How do you know? What if we use only odd numbers at the corners, what totals are possible? What if we use only even numbers?

There is no need for every child to work through the same activity. Pairs can work together for a set time, then share their results with another pair. Each group of four can then share their results. Finally, everyone can come together as a class and discuss what has been discovered. How the children record their findings can be left open: some draw pictures, some write sentences, some use number. They should be encouraged to "use a variety of forms of mathematical presentation" (as the national curriculum says). Such work can be explored further. What if we change the shape of the pot? Suppose Wanda has a pot with three, five or six sides? How does that change the pattern of numbers? .

As the children become more confident, they may wish to move away from using concrete apparatus and use prepared sheets with the shapes already drawn to represent the pots. (As this is a number investigation, teachers should avoid wasting time having the children draw the shapes, when the drawing can take over as the main task.) Some children will want to design their own pots. Try to make sure that not all the pots are regular shapes. Introducing irregular two-dimensional shapes makes the children realise that a pentagon always has five sides, a hexagon always has six and so on.

Allow the children to make their own three-dimensional pots, and you're into a whole new ball game.

Analysing such an activity, it is interesting to see how many "sums" the children manage to get through. On top of this is discussion; selection and use of appropriate maths and materials; development of mathematical approaches and ways of overcoming difficulties.

Children also develop skills in organising and checking work, understanding and using the language of number and shape, relating numerals and symbols to a range of situations, discussing work, responding to and asking mathematical questions and using a variety of forms of mathematical presentation.

0Add to this, recognising relationships and making predictions, understanding general statements and investigating whether particular statements match them as well as asking questions such as "What would happen if?" and "Why?" then it almost does seem that Wanda works magic in the classroom.

Janet Rees is county advisory teacher for primary maths for Suffolk

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