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#### The Need for Vision in Effective Implementation

Why we need to know the end from the beginning

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by Bradford Research School

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With problem solving being one of the main aims of the mathematics national curriculum, it has certainly been high on the agenda in schools for the last few years. But exactly what problem solving is, and how best to teach it, is not always clear. Sometimes the ability to problem solve might be perceived as the preserve of only the most talented mathematicians – the ones who ‘just get it’ – but the curriculum makes it clear that this shouldn’t be the case:

*“The national curriculum for mathematics aims to ensure that all pupils can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions”*

The latest EEF guidance report ‘Improving Mathematics In Key Stages Two And Three’ has some recommendations that will help school leaders and teachers to clarify what their expectations might be when it comes to the teaching and learning of problem solving.

The bulk of the advice comes under the heading ‘Teach Pupils Strategies For Solving Problems’, which is the third strategy suggested in the guidance report, however there are useful ideas in other parts of the guidance too.

__What is problem solving?__

Children will be solving problems when they don’t know a specific way to solve a problem. When children are answering word problems they are not necessarily solving problems. The guidance report says:

*“Problem solving generally refers to situations in which pupils do not have a readily-available method that they can use”* (page 14)

*“Sometimes problem-solving is taken to mean routine questions set in context, or ‘word problems’, designed to illustrate the use of a specific method. But if students are only required to carry out a given procedure or algorithm to arrive at the solution, it is not really problem solving; rather, it is just practising the procedure.”* (page 14)

__How should we teach problem solving?__

To answer this question the guidance report suggests several best bets based on research evidence. In summary we should consider the following when developing problem solving skills:

- Select genuine problem-solving tasks that pupils do not have well-rehearsed, ready-made methods to solve;
- Consider organising teaching so that problems with similar structures and different contexts are presented together, and, likewise, that problems with the same context but different structures are presented together;
- Teach pupils to use and compare different approaches;
- Teach pupils to interrogate and use their existing mathematical knowledge to solve problems;
- Encourage pupils to use visual representations;
- Use worked examples to enable pupils to analyse the use of different strategies; and
- Require pupils to monitor, reflect on, and communicate their reasoning and choice of strategy.

__What might this look like in practice?__

A teacher may begin by presenting a worked example to the pupils. This worked example might be complete, incomplete, correct or incorrect (modelling errors and/or misconceptions). To model this we are going to use a **complete and correct worked example** (see bullet point 6 above) which has a **similar structure** (see bullet point 2 above) to the problem that the pupils themselves will later work on.

The teacher would first show the pupils the above sequence and the algebraic formula that can be used to find out the total number of cubes in any tree in the sequence:

n^{2} + 1

The teacher would demonstrate that for the first tree this formula would be used to show:

1^{2} + 1

(1 x 1) + 1 = 1 + 1 = 2

And that for the second tree:

2^{2} + 1

(2 x 2) + 1 = 4 + 1 = 5

The teacher should demonstrate the process of looking for relationships and patterns between the numbers. They could represent the numbers in a table:

Tree number | Green cubes | Brown cubes | Total cubes |

1 | 1 | 1 | 2 |

2 | 4 | 1 | 5 |

3 | 9 | 1 | 10 |

To model identifying patterns and relationships they might point out that:

- there is always one brown cube in each tree;
- the number of green cubes and the total number of cubes grows by adding consecutive odd numbers; and
- the number by which the number of green cubes grows is always a square number.

The teacher should take this opportunity to show that the +1 is there for the brown cube, or the tree stump, so to find the number of green cubes you multiply the tree’s number in the sequence by itself. The benefit of representing this well-known number pattern (2, 5, 10, 17, 26) as trees made out of cubes is that the +1 can easily be seen to be the brown cube, or the stump, allowing children to begin to explore the n^{2}aspect of the pattern (more on that later). The teacher then might model what the next tree in the sequence would look like by building it with cubes and proving that the matching formula works by carrying out the calculations.

The children should then be introduced to the problem that they will have to solve:

The pupils now have a starting point for their own problem solving as they have seen a problem with a similar structure. In their problem the first term has been removed and the first tree/pattern becomes the tree/pattern that was the second tree in the worked example. The children will not have a ready-made method to apply to this problem, but they will have had an insight into how they might go about solving it because they have seen a worked example – they should be able to *“search their knowledge of similar problems they have encountered for strategies that were successful, and for facts and concepts that might be relevant.” *(see bullet point 4)

A set of simple challenges can be issued for the children to work through:

**Describe the pattern (in words)****Find the number of green, brown and total cubes for the next trees in the pattern****Write a formula to find the number of cubes in any tree in the sequence**

At this point teachers play the role of facilitator, encouraging children to do the following:

- represent the next terms in the sequence using cubes and to record results in a table (see bullet point 5);
- ask pupils questions and
*“…encourage pupils to ask questions like, ‘What am I trying to work out?’, ‘How am I going about it?’, ‘Is the approach that I’m taking working?’, and ‘What other approaches could I try?’”*(see bullet point 6); and - look for patterns and relationships between the numbers in their table.

Pupils should follow the structure that the teacher demonstrated earlier and, if they are on the right track, will work out the relationship between n and the number of green cubes:

n | Green cubes | ||

1 | 4 | 4 is 2^{2} | To get to 2 from n: n+1 |

2 | 9 | 9 is 3^{2} | To get to 3 from n: n+1 |

3 | 16 | 16 is 4^{2} | To get to 4 from n: n+1 |

So they should arrive at the fact that all the green cube values are square numbers and that to work out the number of green cubes in each tree they need to add 1 to n and then square the product of that:

(n+1)^{2}

Then, because each pattern only has one brown cube as a tree trunk, they will need to add one:

(n+1)^{2} + 1

The guidance report then recommends that*“when the problem is completed…” *pupils should be encouraged “*to ask questions like, ‘What worked well when solving this problem?’, ‘What didn’t work well?’, ‘What other problems could be solved by a similar approach?’, ‘What similar problems to this one have I solved in the past?’ Pupils should communicate their thinking verbally and in writing — using representations, expressions, and equations — to both teachers and other pupils.” *By modelling this style of self-questioning and reflection the guidance report outlines how this can develop metacognition which in turn develops pupils’ independence and motivation (pages 20 and 21).

Pupils can then be given another similar problem for which they will use the processes and formulae they have already developed in order to find a solution. For this further example pupils should be able to work more independently.

The only difference in this problem is that instead of adding just one cube for the stump, the number of brown cubes = n, so the formula becomes:

(n+1)^{2} + n

The whole of the above activity is an example of the guidance report’s sixth recommendation: ‘Use Tasks And Resources To Challenge And Support Pupils’ Mathematics’ (pages 24 – 27). The use of manipulatives supports understanding throughout the problem solving process outlined above; this addresses the guidance report’s second recommendation: ‘Use Manipulatives And Representations’ (pages 10 – 13). This approach to teaching also encompasses the gradual release of responsibility model which is outlined in the EEF’s KS2 Literacy guidance:

- an explicit description of the strategy and when and how it should be used;
- modelling of the strategy in action by teachers and/or pupils;
- collaborative use of the strategy in action;
- guided practice using the strategy with gradual release of responsibility; and
- independent use of the strategy

A further question to really challenge the thinking of the pupils would be to ask:

*Why is the number of green cubes always a square number?*

This question gives children the opportunity to further reflect on the process they have undertaken and really capitalises on the visual nature of the way the teacher has presented the problem. Pupils will be able to take the green cubes that made up the tree and then rearrange them into squares, thus demonstrating with physical objects that the number of green cubes can always be used to make a complete square (with the same area) and is always (n + 1)^{2}.

Teachers can then guide pupils into further exploration, leading the children to discover another pattern:

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

The above pattern is actually what they have been representing all along with the green parts of their trees. Pupils could be encouraged to explain the above pattern:

When you add consecutive odd numbers (beginning at 1) the answer is a square number

They can then be shown how these patterns can be represented using the cubes:

Pupils should be able to then explain that:

- The first square = 1
- To make the next square you need to add 3 more cubes to get to 4
- To make the next square you need to add 5 more cubes to get 9

By carrying out this follow-up process children will be adding to their network of mathematical knowledge, the development of which is another suggested strategy of the guidance report: recommendation 5, ‘Enable Pupils To Develop A Rich Network Of Mathematical Knowledge’ (pages 16 – 19).

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