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The decision to focus on Key Stages 2 and 3 was made after an initial consultation period with teachers, academics, and other stakeholders. This consultation suggested that these were areas where guidance could make a big impact as not only were schools seeking advice on adjusting to a new curriculum, there is also concern about pupils making a transition between the stages.
This report is not intended to provide a comprehensive guide to mathematics teaching. We have made recommendations where there are research findings that schools can use to make a significant difference to pupils’ learning, and have focused on the questions that appear to be most salient to practitioners. There are aspects of mathematics teaching not covered by this guidance. In these situations, teachers must draw on their knowledge of mathematics, professional experience and judgement, and assessment of their pupils’ knowledge and understanding.
The focus is on improving the quality of teaching. Excellent maths teaching requires good content knowledge, but this is not sufficient. Excellent teachers also know the ways in which pupils learn mathematics and the difficulties they are likely to encounter, and how mathematics can be most effectively taught.
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Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas.
However, manipulatives and representations are just tools: how they are used is important. They need to be used purposefully and appropriately in order to have an impact. [1]
Teachers should ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently.
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| What are manipulatives and representations? |
A manipulative is a physical object that pupils or teachers can touch and move, which is used to support the teaching and learning of mathematics. Common manipulatives include Cuisenaire rods and Dienes blocks.
The term ‘representation’ refers to a particular form in which mathematics is presented. [2] Examples of different representations include:
-two fractions represented on a number line;
- a quadratic function expressed algebraically or presented visually as a graph; and
- a probability distribution presented in a table or represented as a histogram.
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| What does effective use of manipulatives look like? |
Manipulatives can be used across both Key Stages. The evidence suggests some key considerations:
- Ensure that there is a clear rationale for using a particular manipulative or representation to teach a specific mathematical concept. Manipulatives should be used to provide insights into increasingly sophisticated mathematics.
- Enable pupils to understand the links between the manipulatives and the mathematical ideas they represent. [3] This requires teachers to encourage pupils to link the materials (and the actions performed on or with them) to the mathematics of the situation, to appreciate the limitations of concrete materials, and to develop related mathematical images, representations, and symbols. [4]
- Try to avoid pupils becoming reliant on manipulatives to do a type of task or question. A manipulative should enable a pupil to understand mathematics by illuminating the underlying general relationships, not just ‘getting them to the right answer’ to a specific problem. [5]
- Manipulatives should act as a ’scaffold’, which can be removed once independence is achieved. Before using a manipulative, it is important to consider how it can enable pupils to eventually do the maths without it. When moving away from manipulatives, pupils may find it helpful to draw diagrams or imagine using the manipulatives.
- Manipulatives can be used to support pupils of all ages. The decision to remove a manipulative should be made in response to the pupils’ improved knowledge and understanding, not their age.
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| What about other types of representation? |
The evidence indicates that number lines are a particularly effective representation for teaching across both Key Stages 2 and 3, and that there is strong evidence to support the use of diagrams as a problem-solving strategy.
The specific evidence regarding the use of representations more generally is weaker; however, it is likely that the points above regarding effective use of manipulatives apply to all other representations.
While in general the use of multiple representations appears to have a positive impact on attainment, more research is needed to inform teachers’ choices about which, and how many, representations to use when. [6]
There is promising evidence that comparison and discussion of different representations can help pupils develop conceptual understanding. Teachers should purposefully select different representations of key mathematical ideas to discuss and compare with the aim of supporting pupils to develop more abstract, diagrammatic representations.
However, while using multiple representations can aid understanding, teachers should be aware that using too many representations at one time may cause confusion and hinder learning. [7]
Next email #3 - 'Recommendation 3: Teach strategies for solving problems'
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