Some ideas for combining Mathematics, Science and English by problem-solving from picture books.

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Setting out to make learning meaningful for children through problem-solving is not new; but during the last ten years it has been much discussed particularly in connection with Mathematics, Science and Technology in the primary curriculum. And, rumour has it, those commissioned to come up with ideas for the Standard Attainment Tests (SATs) in the National Curriculum are considering a problem-based approach.

What exactly is meant by problem-solving? It has become something of a buzz-word in education but one which in practice is often applied too narrowly only to 'design-make' technological activities or, in Mathematics, to 'computation in words'. This is to deny its real value as a thinking and working process which can be applied right across the primary curriculum. The potential for integration through problem-solving is huge (something to hang on to as we start rationing out time for curriculum coverage). To make the most of it though, it is necessary to recognise and understand which of the processes children go through in solving problems are common to all areas and which are distinctive to learning Mathematics. Science or other curriculum subjects - of this, more later.

Across the curriculum, the main difference between this and more 'traditional' learning experiences offered to children lies in the challenge to think and to resolve: the notion of 'right' answers is exchanged for the search for a satisfactory solution. There are, in many instances, no right answers and, in others, the possibility of many right answers (and wrong ones). This frees children to explore for themselves, without the constraint of 'one right answer' and without construing the task as trying to match what the teacher thinks. Through an essentially practical task children are put in a position which requires them* to apply knowledge and use skills*; in the process they may well also *acquire new knowledge* and *develop *both *skills and understanding*. Central, to all this, of course, is talking and listening (50% of the English curriculum).

For children to engage with a problem, they must first be stimulated by the context in which it is introduced and feel a need or inclination to get involved. Secondly, it is important that the 'solver' acknowledges, recognises or identifies the 'problem. Without these the child has no way into the problem and the processes of analysing, categorising, hypothesising, etc. which (at any level) are involved in its 'solution'. The most successful problems therefore are those which the children identify for themselves as they arise from what they are doing. Problems of this sort are 'real'. They should he seized when they occur. In practice, however, teachers (who cannot always he sure that the 'real' problem will arise) will choose to create situations or present problems for children to tackle. Stories in picture books offer a rich source of possibilities for this kind of stimulus-based problem -solving. The activities involved can lead to 'real' problems when the children take new directions of their own.

Working on problems arbitrarily invented from the events of the story, some might claim, is unrelated to the essential literary encounters with the text, which should he paramount. This view compartmentalises curriculum (not a helpful move) and underestimates the central importance of 'story' in all life and learning, particularly for children. Using published stories to explore Mathematics and Science offers children a way into learning which is natural and meaningful; the story context encourages children to think imaginatively and laterally, something they can be more reluctant to do in 'realistic' situations. In addition in taking on 'the problem' children become engaged imaginatively with the story and its characters, a process which for many enhances their experience of the book. Book, used for problem-solving frequently become favourites for reading and talking about.

**The Books**

The picture books which were the starting point for the problems suggested here are by Ronda and David Armitage. Their stories are ideal for the purpose in that each one is essentially about a problem which the characters have to solve. In **The Lighthouse Keeper's Lunch** the problem is how to stop the greedy seagulls stealing Mr Grinling's lunch when Mrs Grinling sends it out to him from the cottage on the cliff. Even more dramatic, **The Lighthouse Keeper's Catastrophe** has Mr and Mrs Grinling locked out of the ligthhouse (one key inside, the spare one lost). Night is coming, a storm is brewing: how will they get the lighthouse lamp switched on? **One Moonlit Night** poses a problem of communication between the two boys camping out in the garden and feeling a hit insecure, and dad asleep upstairs in the house. In **Ice Creams for Rosie** the problem is how to get fresh supplies of ice cream to the island when Rosie runs out of stock before the usual delivery date. Appealing illustrations, a strong storyline and (most of all) the humour that emerges from a skilful blend of text and pictures, combine to make these books 'work' across a wide age range.

**Introducing the Problems**

The problems can he introduced in different ways. The most straightforward is to read the story to the point where the problem occurs, or where it is about to he resolved; in **Catastrophe**, for instance, to the point where the only spare key sinks to the sea bed. 'There it lay amongst the rocks and the seaweed where only the octopus, the crab and other sea creatures would ever find it again.' How will Mr Grinling get into the lighthouse? Or read on through the unsuccessful attempts to the point where Mrs Grinling says 'I've got an idea -a perfectly brilliant idea.' What might it he'?

The problems suggested here are also *extensions* of the story. involving the characters in new problems and could he posed after reading the whole story.

**The Problems**

The problems included here have been tried and tested in a number of classrooms and, although we have grouped them separately, there are many overlaps between Maths and Science in the activities. We have categorised the problems as:

** open** - where the children are in control of the variables and the materials (the nearest thing to a 'real' problem)

** semi-open** - where some of the variables and the materials are controlled by the teacher

** closed** - where all the variables and the materials are given and the specific learning is deliberately defined by the teacher. Even a 'closed' problem can he presented as a challenge. In this way thinking and action work together to strengthen the learning experience.

We have also suggested the kind of mathematical or scientific processes involved, the ideas contained within the problems and the National Curriculum Attainment Targets (ATs) and Levels (as they stood in February) covered in these activities. Materials and equipment which you will need to provide are listed.

Problems are labelled for 'earlier' or 'later' stages of development but most are capable of adaptation as starting points for any children.

**Into Practice**

However the story and the problems are introduced, the ground rules for the activity are the same.

**I.** The children work in groups as a team.

**2.** It is made clear that no-one is going to help them or give them any answers. They are going to work 'like scientists and mathematicians in real life'.

**3.** Teachers' role is to encourage, help and remotivate by asking*questions *to trigger a thought process.

**4.** Children are encouraged to articulate their problem and report (and evaluate) possible solutions to others in the class (who may he working on the same or different problems).

**Materials**

*How* materials/equipment are made available needs to be carefully considered.For a genuinely 'open' investigation resources should be in a central general location - available for children to select from as they see fit. If the resources are set out as part of the activity the children are more likely to see it as a puzzle and try to find a use for whatever is provided, making it not their own investigation. In 'closed' problems the teacher chooses to narrow the focus by deliberately providing specific resources with which children will face the challenge of the problem.

Centrally located resources should include - as well as the materials suggested - scissors, glues, measuring devices, timers, junk materials, etc.

**MATHEMATICAL PROBLEMS**

*'The ability to solve problems is at the heart of mathematics.' (Mathematics Counts, The Cockcroft Report)*

The problems invite application of spatial and numerical concepts and skills; they involve looking for pattern, making generalisations, and logical thinking.

**THE LIGHTHOUSE KEEPER'S CATASTROPHE**

**1. Packing the lunch**

Mrs Grinling 'prepared cold chicken sandwiches, a fruit salad with lots of strawberries and a chocolate milkshake for his lunch'. She put the fruit salad and the milkshake into square cartons and started to wrap everything up to put it in the basket .. .

** Problem**: Mrs Grinling can wrap up the food (represented by 4 multilink cubes) in five different ways (or if time, let the children discover how many ways).

She's going to make four packets the same shape (four sets of 4 cubes). Here is the 2D shape of the basket she is going to put them in (a 16-square grid in a simple 4 x 4 pattern - or use a more complex shape). The four packets can only go in one layer or they will poke out of the basket and the seagulls will get the food!

Choose one of the ways Mrs Grinling could wrap up the food. Will all four packets fit into the basket in one layer'? If they do. can you colour the basket grid to show where each one will go? Choose another shape. Will this one fit'? How many of the five shapes will fit into the basket four times in one layer'?

** Type of problem**: Semi-open

** Mathematical ideas**: Spatial (involving prediction)

** Nat Curric**: AT9 Level 1-2, AT 10 Level I

** Range**: Earlier stage but can be complicated. Possibility of extending the idea with 3D shapes.

(Adapted from *Mathematical Investigation**. *Somerset CC)

**2. Crossing the gap**

'Every morning Mr Grinling rowed Out to the lighthouse to clean and polish the light.' On 'high days and holidays when the sun shone' Mrs Grinling, Mr Grinling and Hamish the cat all go to the lighthouse taking a huge basket of delicious lunch.

- Mr Grinling has broken his arm so Mrs Grinling has to row and there is only room for Mr Grinling or the basket or Hamish in the boat with her.
- Hamish will cat the lunch if he is left alone with it.
- Mr Grinling won't be left alone with Hamish - it was Hamish who tripped him up when he broke his arm.

** Problem**: How can they all get across? How many trips are needed?

** Type of problem**: Semi-open

** Mathematical ideas**: Spatial and logical thinking

*Nat Curric*:

*AT1*Level 3

** Range**: Later stage

** Resources**: Dolls, models, a toy boat, a real or paper sea: things to make concrete this problem in logic.

(Adapted from Brian Boult, ** Mathematical Activities**, Cambridge)

**ONE MOONLIT NIGHT**

**1. Dressing in the dark: how many ways?**

On another night when they were in the tent Tony took his red bobble hat, scarf and gloves; Sam took his green bobble hat, scarf and gloves. In the tent the clothes got mixed up. In the middle of the night the boys woke up, decided to go back into the house and got dressed.

** Problem**: How many different outfits could each one put on? Can you record all the combinations of hat, scarf and gloves'?

** Type of problem**: Semi-open

** Mathematical ideas**: Logical thinking, looking for pattern

** Nat Curric**: ATI Level 1-2

** Range**: Earlier

** Resources**: Real sets of clothes for experiment. Red and green crayons for recording. Paper cut-outs or duplicated sheets are alternatives.

(Adapted from ** Number** magazine, RLDU, Avon CC)

**2. Making the tent**

Children are invited to draw the tent in which the boys camped in the garden, on a VDU screen using a Logo chip or the program DART

OR They might make a skeleton of the tent using straws and plastic joints from the Orbit Kit

OR What about trying to design a different type of tent'? Will it stand up?

** Type of problem**: Open

** Mathematical ideas**: Spatial involving length, distance and angle

** Nat Curric**: ATE) Level 4, ATI1 Level 4

** Range**: Later

**ICE CREAMS FOR ROSIE**

**1. Rosie Posie's ice cream cornets**

'Rosie sold the only ice cream on the island. Creamy delectable, luscious lip-smacking ice cream ... The islanders and visitors came in droves to eat Rosie Posies ice cream. Rosie was delighted. She liked to see people munching and licking contentedly.'

** Problem**: Today Rosie Posie has three different flavours of ice cream to sell:

Frangipani Frost (yellow)

Mountain Mint (green)

Whopple Dopple (purple)

Each cornet holds a double scoop.

How many different combinations of flavours can you make?

Suppose Rosie also has Praline Peach (orange), making four flavours. How many combinations can you make now?

Rosie sometimes has Texas Twirl (pink) and Choc-o-Chip (brown). Can you investigate how many different cornets you can make up with five, six or even more flavours?

Could you investigate triple top cones?

** Type of problem**: Semi-open, investigational

** Mathematical ideas**: Looking for pattern and generalisation

** Nat Curric**: AT1 Level 1-4

** Range**: Capable of adaptation for all stages

** Resources**: Younger children work with 2D cones and coloured scoops: older or more mathematically capable children can work in the abstract with coloured pens.

(Adapted from *Investigator**No. *11, SMILE Centre, London)

**SCIENCE PROBLEMS**

*`The ability to communicate, to relate science to everyday life and to explore are essential elements of an initial experience of science.' **(Draft Statutory **Orders for Science, NCC 1989)*

The problems involve processes of investigation. experimentation. fair testing, design-make, and include a variety of scientific ideas.

**THE LIGHTHOUSE KEEPER'S CATASTROPHE**

**1. What was Mrs Grinling's brilliant idea?**

**2. Tidying up**

Searching for another key Mr and Mrs Grinling 'opened old tins and jars. they emptied out drawers and they peered into cupboards but to no avail

Lots of things came out of the tins, jars, drawers and cupboards.

** Problem**: How many different ways can you find of sorting all these things' Can you find a way of recording what you decide without using words? Can you make anything useful - something for Hamish to play with, perhaps-from these things?

** Type of problem**: Open (though materials provided by teacher)

** Science ideas and processes**: Investigation, sorting and classifying. Communicating ideas. Design-make. Other ideas dependent on materials, perhaps mechanisms, energy transfer.

** Nat Curric**: Depends on resources selected

** Range**: A starting point for all ages

** Resources**: Collection of objects could include mechanical toys, household articles (egg whisk, bicycle bell, hand drill), cotton reels, elastic hands. etc.

For a variation on this problem, take as a starting point the 'glorious muddle in Rosie Posie's shop in *Ice Creams for Rosie.*

**3. Packing the lunch**

Some days Mrs Grinling puts the chocolate milk shake in a tall beaker with a lid and the fruit salad into a jar with a screw top. She puts the sandwiches in polythene boxes. She chooses something in which to carry it all.

** Problem**: Which is the best thing to carry the lunch.' A basket? A plastic carrier'? A cardboard box?

Why is it the best'? Easiest to carry'? Strong enough'? Holds most? Which would he the best for keeping the seagulls out'?

** Type of problem**: Semi-open

** Science ideas and processes**: Investigation, sorting, experimenting, shape/spatial, nature of materials

** Nat Curric**: AT6 (Also mathematical)

** Range**: Earlier, but can easily be complicated for more advanced problem-solving, e.g. how could we send a cardboard box down the wire?

** Resources**: Different shaped containers; baskets, plastic carrier hag, cardboard box.

**4. Across the gap**

'At lunchtime Mrs Grinling packed the lunch in a basket and sent it down the wire to the lighthouse.'

** Problem**: ('an you devise a system for getting Mr Grinling's lunch to the lighthouse? Sometimes in bad weather Mr Grinling gets stranded on the lighthouse for several days. How can he get the basket back to Mrs Grinling so she can send more food?.

** Type of problem**: Open

** Science ideas and processes**: Design-make, structures and mechanisms

** Nat Curric**: AT6, ATI3, AT 10

** Range**: Later

** Resources**: Junk materials, construction kit, etc.

**5. Turning on the light**

'Think of all the ships that aright he lost because your light isn't shining.'

** Problem**: Could you help Mr Grinling by fixing a switch on the mainland that would turn on the light in an emergency'.'

What materials would you need'? Perhaps some of the things you have turned out of the drawers, tins and cupboards might help'.) Can you find a way to test your ideas" Can you make a switch to turn the light on and off'?

** Type of problem**: Semi-open

** Science ideas and processes**: Investigate, experiment, electricity (conductivity)

*Nat Curric**:AT 11, AT *13, *AT6*

** Range**: Later

** Resources**: Bulb and battery (the lighthouse), assortment of wire, metal clips, wood, plastic. paper, etc. - available in a central resource for selection.

**NOTE**

*Before* giving children this problem it is *essential* that they have experienced a series of *closed *problems as challenges.

**I.** Can you make the bulb light? (Bulb and battery)

**2.** Can you make the bulb light using all these pieces'? (Bulb, battery, two pieces of wire)

**3.** Can you connect the wires to the bulb holder and make the bulb light? (Bulb. battery. two pieces of wire, one bulb holder)

**4.** Which of this collection of things do you think electricity will travel through? Find a way to test them to see if you are right. (Add to the above metal clips, pins, toy cars, plastic, wood, card, etc.)

**ONE MOONLIT NIGHT**

**1. Telling the difference**

'It took most of the afternoon for Tony and Sam to move in (to the tent)… Mum brought a delicious camping tea. "Just in case you're starving after moving house all afternoon." she laughed.'

On the tray were burgers, crisps, apples and milk shakes (see picture). Tony and Sam like some brands of crisps more than others. Do you think they could *really *tell the difference? Could *you* tell the difference?

** Problem**: Can you design a fair test to see it your friend can really tell the difference?

Ask first which one your friend would choose - now try your test.

** Type of problem**: Semi-open

** Science ideas and processes**: Fair testing - identification of and dealing with range of possible variables

** Nat Curric**: AT3 Processes of Life, AT13 Energy

** Range**: Early - capable of complication

** Resources**:

*Two*sorts of crisps.

**2. Toys in the dark**

The boys move into the tent. "'I brought my gumboots," explained Tony. "just in case it rains." "I've brought something to drink," said Sam. "just in case we're thirsty, and I thought we might wake up early so I brought some toys."'

** Problem**: If theboys do wake up early and it is still dark, which toys will they be able to see?

** Type of problem**: Open

** Science ideas and processes**: Light, shape, colour, prediction. Investigation can lead to the designing of a test, exploring qualities of objects, identifying variables.

** Nat Curric**: AT6 Nature of Materials, AT15 Using Light

** Range**: Adaptable to any stage. Children may identity many variables. Accept those the children understand and alloy them to explore. Too many variables can confuse the issue and make the activity 'teacher-directed' rather than 'teacher led'.

** Resources**: Centrally available for selection: large cardboard boxes (dark room?). pieces of fabric (to simulate tent), collection of toys (see picture).

**ICE CREAMS FOR ROSIE**

**1. Ice cream drop**

Rosie had a bright idea for getting the ice cream to the island. 'Early next morning at the airport on the mainland an aeroplane took on a rather precious cargo. When the aeroplane was over Kotuku island. the precious cargo was parachuted down.

** Problem**: Can you design and make a parachute to carry the precious packages?

** Type of problem**: Semi-open

** Science ideas and processes**: Design-make, investigating strength of materials, weight in relation to capacity to catch air

** Nat Curric**: AT6 Materials, AT1O Forces, A T13 Energy

** Range**: Later

** Resources**: Packages to represent ice cream. Centrally available for selection - fabrics, papers, polythene, string, cotton, etc.

**2. Saving the ice cream**

'Poor Rosie Posie Hubble. Something bad to be done or there would be nothing left of her idea but a river of melted ice cream.'

** Problem**:

*How*quickly do ice creams melt? Is one kind of wrapping better than another to help stop the ice cream melting? Can you find out?

** Tvpe of problem**: Semi-open

** Science ideas and processes**: Investigation leading to a test of design - make, nature of materials. insulation

** Nat Curric**: AT6, AT13 Energy

** Range**: Variable

** Resources**: Ice cubes (to represent ice cream!). Centrally available for selection: newspaper, tissue paper. waxed paper, brown paper. different sorts of card - dull, shiny, etc.

Ideas and problems for this feature contributed by

**Christine Thomas**, Advisory Teacher for Mathematics, Maths, Science and Technology Centre, Avon CC.

**Chris Ollerenshaw**, Lecturer, Bristol Polytechnic; former Advisory Teacher for Science. Maths, Science and Technology Centre, Aeon CC.

Ronda and David Armitage's books are published in hardback by Andre Deutsch and in paperback as Picture Puffins

**Grandma Goes Shopping**, 0 1 233 97627 2, £5.95; 0 1 14 050.4601 5, £1.95 pbk

**Ice Creams for Rosie**, 0 233 97361 3, £5.50

**The Lighthouse Keeper's Catastrophe**, 11 233 97891 7, 0.25; 0 140150.663 2, £ l.95 pbk

**The Lighthouse Keeper's Lunch**, 0 233 96868 7, £5.25; 0 1 14 050.327 7, £2.25 pbk

**One Moonlit Night**, 0 233 97540 3, £5.50; 0 1 14 050.461 3, £1.75 pbk