Mathematics in the EYFS: Problem-solving - Within reason

In the Foundation Stage, mathematical problems are often just everyday events with a mathematical aspect: making sure that everyone has a biscuit; working out how to check that none of the teddies is missing; or deciding the best way to stack the large blocks. Children taking part in these activities are using the maths that they know to help them find solutions and are sometimes learning new facts and ideas on the way.

The important point about problem-solving is that it involves choices, so children have the opportunity to reason and make decisions.

Children's problem-solving strategies and skills develop through investigating and finding solutions for a wide range of different practical problems. Some will have quick solutions, while others will take much of the session time or may even need to be revisited during the course of a week.

It can be helpful to see problem-solving as two kinds of mathematical understanding. One is learning about some aspect such as counting, the other is developing the skills of mathematical reasoning and decision-making.

Problem-solving

How children learn

Children learn to solve practical problems by first recognising that there is a problem to be solved, and understanding what the problem is. For example, will there be enough fruit for everyone? They then learn to see the possibility of a solution: counting everyone.

Children progress in problem-solving by being systematic to some degree. In this case, they need to think of a way to make sure that everyone has been counted.

They then begin to see connections, realising that the fruit needs to be counted as well as the people, and realising that what they did yesterday, for example with the toys, is relevant to today's problem with the fruit.

Finally, children need to be persistent and try different approaches. They might, for example, estimate the number of apple pieces and the number of children, then check, and then share the fruit out.

Helping children learn

* Create an atmosphere where exploration and 'having a go' is seen as more important than getting the right answer.

* Provide a rich and interesting environment, with varied activities - don't make things too easy for the children.

* In your planning, include both set-up problems and everyday problems that occur as part of normal activities and involve some kind of maths, such as finding out who is taller, putting the right number of bulbs in each pot, or putting a cup back on each hook.

* Practise seeing the problem-solving and maths potential in everyday activities, and posing the problem in a way that appeals to children.

* Talk with children about what they are doing, with a wondering approach rather than too many direct questions:

* I wonder if there are enough for everybody to have one...

* I'm not sure if we've got three or four lumps of dough.

* How can you show us that you are shorter than Ryan?

* There should be five red skittles. Shall we see if we have them all?

* If the children get stuck, provide prompts:

* Show me what you have tried so far.

* I wonder if that one comes next.

* At the end of the task, you can recap the problem-solving process with the child or the whole group:

* Let's explain how you found which number shows how tall you are. First you... then you... And in the end you checked the numbers by counting them in order.

Progression

* Attempts to tackle problems without much awareness of the likely outcome, such as counting randomly to find out how many

* Begins to use acquired skills and knowledge, such as using tally marks to record a score in a game

* Begins to be more thorough and systematic, such as packing square blocks into rows rather than placing them together randomly

* Begins to use more sophisticated maths skills and knowledge, such as using a calendar to find out how many more days until the weekend

* Solves harder problems in unfamiliar contexts, such as working out that in order to have ten conkers they need another three

* Relates a problem to others they've solved, such as remembering to turn a shape round to see if it will fit in another way

* Uses what they already know in a new context, such as putting a cup on each hook to see if any are missing.

Child-initiated play

Stock check Children dress up in overalls and caps to do the weekly stock-take in an area of the setting. Provide clipboards and pencils.

Buy a brick Set up an outdoor builder's yard and sell large plastic bricks for £1 each. Discuss how many bricks children would get for £5.

Extend the play by constructing a builder's wall display: 'Build this wall for £10', 'Build this wall for £8'. Encourage the children to collaborate.

Decorating Make plans to decorate the home corner and provide wallpaper, flour paste, measuring tapes and rulers. Discuss measuring the walls and wallpaper lengths. Demonstrate how to paste the wall and press the paper on. Provide lengths of light material to cut as window curtains.

Adult-led activities

Gone missing Show the children a tray of assorted plastic animals - five of each type, with one missing. Ask for suggestions on how you can find out which one is missing. Extend the activity by removing three animals. How do you think you'll work it out?

Teddy toasts Arrange two teddies on a table with a plate each and ask the children to share out the six toast squares so that the teddies each have the same amount of toast to eat. Extend the activity by arranging for a third teddy to join the group so the toasts will need to be redistributed.

Do both teddies have the same amount of toast? How can you be sure?

Box makers Provide children with construction materials, such as Lego, and challenge them to make a lidded box that will hold ten marbles. These boxes look very different. Shall we see if they both hold just ten marbles?

Provision

Outdoor area Provide small articles to be priced, sticky labels, money, number carpet mats. Set up the number carpet mats as car boot sale areas.

Stick price labels on small items to be 'sold'. How much does that cost? I wonder if I've got enough money?

Water tray Provide transparent containers, pebbles, Lego blocks, elastic bands. Together, put ten pebbles in each container and slide an elastic band around them. Fill the containers to the top with water. Discuss making the level of the water rise and fall by adding or removing some pebbles.

Try to make the water in the containers level with the elastic bands. Do the same with Lego. I wonder if we need to add another or take one out? NW

Important words and phrases

What could we try next?

How do you think you'll work it out?

pattern, puzzle, listen, join in, say, think, imagine, remember, explain, describe, different way, another way

ASSESSING CHILDREN'S DEVELOPMENT

If a child

* comments that there isn't a paintbrush in each paint pot

* puts pegs in a pegboard, trying to fill all the holes; says they need more pegs

* tries to organise the wooden numerals when they are in a muddle

then they may be on this step

* uses number language in play

* shows an interest in shape and space by playing with shapes or making arrangements with objects

* shows willingness to tackle problems, choose challenges

If a child

* talks about patterns or other shape work on display

* tells you which of two dolls is taller

* comments that the numbers on the washing line aren't right

then they may be on this step

* shows interest by sustained activity or talking about shapes or arrangements

* initiates and manages developmentally appropriate tasks

If a child

* uses fingers to work out how many frogs would be left if two jumped off the log

* counts the spoonfuls of flour as they are put in the bowl and says 'There aren't enough, you need another'

one'

then they may be on this step

* sometimes shows confidence and offers solutions to problems

* uses own methods to solve a problem

If a child

* packs a suitcase with enough plates and mugs for a picnic with all the bears

* suggests a method for sharing the fruit pieces among the members of the group

then they may be on this step

* uses developing mathematical ideas and methods to solve practical problems

* works as part of a group, taking turns and sharing fairly

About this series

This series aims to:

* build on practitioners' knowledge of how children acquire numeracy skills

* offer ideas on how to help children develop these skills.

Each part will focus on an aspect of numeracy included within the 'Problem-solving, reasoning and numeracy' area of learning in the EYFS.

Authors Sheila Ebbutt and Carole Skinner are managing director and product development manager respectively of BEAM, which is dedicated to promoting excellence in mathematics education.