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Numeracy: Step to it

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By John DabellMaths Coordinator at Lawn Primary School in Derby

Use practical challenges and a simple four-step procedure to develop effective problem-solving skills

young girl writing

Provide children with a problem-solving method to help them investigate maths problems

Maths is knee-deep in problems, yet how many learners know a logical and systematic approach to solving them? It is essential that we make explicit a problem-solving method for children to follow in order that they approach problems and investigations with a structured plan of action in mind. There are four steps to learn and apply:

Step 1 – Understand the problem

Step 2 – Plan a strategy

Step 3 – Solve the problem

Step 4 – Reflect

Troubled waters

To illustrate the problem-solving process step-by-step, take a look at the following problem:

Farmer Fog went to the market in Sandford and bought a fox, a duck and a bag of grain for £65. To get home, he has to cross Loch Angle. He can take himself and one other thing in the boat at a time. However, unless he is supervising everything, the duck will gobble the grain and the fox will guzzle the duck. How can he get all three across the Loch successfully and how many crossings will it take him?

Step 1

Understanding the problem

In this first stage, children need to ask themselves two questions:

  1. What do we know?
    • The farmer has a fox, a duck and a bag of grain.
    • He has to cross Loch Angle, but can ferry only one thing across at a time.
    • The duck cannot be left alone with the grain.
    • The duck cannot be left alone with the fox.

This is an opportunity to identify the important and redundant information. For example, where the market was held isn’t important and neither is the cost of the fox, duck and grain. Both pieces of information are superfluous. Encourage children to cross out any information that isn’t important to solving the problem and ensure that they write down what is important.

  1. What do we need to find out?
    • How the farmer can get the fox, duck and grain across the Loch without losing any of them.
    • How many crossings he will have to make.

Considering these points will encourage children to share their understanding about the task in hand and focuses their mind on the ‘nitty gritty’ of the problem itself.

Step 2

Plan a strategy

At this stage, children will need to think of and discuss different strategies for solving the problem. They could try to act the problem out and dramatise the challenge. Alternatively, they could draw a sketch or organise their thoughts into a list to document each crossing. Other strategies to apply to different problems include guessing and checking, using logical reasoning, working backwards, identifying a pattern and breaking the problem down into smaller parts. Children should decide which strategy is the most appropriate and be aware that there might be more than one solution.

Once children start to see a problem as a process, they will begin to tackle problems in a prepared, planned and meaningful way

Step 3

Solve the problem

This problem can be solved mentally, but is probably best solved by acting out the situation physically and orally and then recording it as follows:

  • Bring the duck over
  • Return
  • Bring the fox or grain over
  • Bring the duck back
  • Bring the grain or fox over
  • Return
  • Bring the duck back.

There are seven crossings – four forward and three back.

Step 4

Reflect

In the reflection stage, children are encouraged to RIP (Refine, Improve and Polish) their strategy, consider how successful they have been and whether the problem could have been solved more efficiently using another method. You might want to encourage children to think about using the popular method of evaluation ‘two stars and a wish’ (two stars equal two pieces of positive feedback and a wish equals one idea for potential development) in this final step to focus their minds on how well they have resolved the problem.

A problem shared is a problem solved

Have a go at solving a similar problem to ‘Troubled waters’ online at www.mathcats.com/explore/river/crossing.html A problem-solving challenge like this works best when children work together in small groups of four. Their collaboration is vital because they can discuss and debate ideas and share alternative viewpoints that might not have been thought of independently. A problem shared is often a problem solved when tackled in a group and children soon realise the importance of working as a team. Point out that: T*ogether *Everyone A*chieves *More and use this as your problem-solving mantra.

Jumping frogs

After you have tried the online problem with your class, consolidate the four-step process, challenging children with other problems using an ‘acting out’ strategy. The jumping frogs puzzle is an excellent problem to try and can be enjoyed with learners of all ages:

Three red frogs each sit on a lily pad. To the right of the red frogs, three yellow frogs each sit on a lily pad. There is an empty lily pad between the groups. One day, the red and yellow frogs decide to swap sides with each other. Can you work out the least number of moves it will take for the red frogs to end up sitting on the right and the yellow frogs to end up sitting on the left? There are also some additional rules:

  • The frogs can only move one at a time.
  • There can only be one frog on a lily pad at any time.
  • The frogs can move onto an adjacent lily pad.
  • The frogs can only jump over one other frog at a time onto an empty lily pad.
  • The frogs cannot move backwards.

Differentiate the problem by increasing or decreasing the number of frogs. Try online versions at www.hellam.net/maths2000/frogs.html and www.nzmaths.co.nz/BrightSparks/frogs.asp?applet

No problem!

Once children start to see a problem as a process, they will begin to tackle problems in a prepared, planned and meaningful way. Only by practising this process regularly and building up a repertoire of strategies, can we expect children to approach problems with confidence and authority.

Reviews

  1. Hannah Cartwright
    on 13 March 2011

    Thank you!

    A brilliant, informed article on problem solving. There is so little guidance on how to teach problems and what stategies to use and I found this article to be of great use for my lessons. I am going to try the frog lesson next week. Thank you!

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