Knowledge of the common errors and misconceptions in mathematics can be invaluable when designing and responding to assessment, as well as for predicting the difficulties learners are likely to encounter in advance. A non-example is something that is not an example of the concept. Manipulatives and representations can be powerful tools for supporting pupils to engage with mathematical ideas.
However, manipulatives and representations are just tools: They need to be used purposefully and appropriately in order to have an impact. The aim is to use manipulatives and representations to reveal mathematical structures and enable pupils to understand and use mathematics independently. 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. Manipulatives can be used across both Key Stages. The evidence suggests some key considerations:. The pupils tried this with other two-digit numbers ending in 0 and discovered that the result was always a multiple of 9.
What does it look like if we remove 4 cubes? Another pupil removed 4 cubes in a different way. If we take away one from each 10 then we are left with four 9s. 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.
The teacher noticed that some pupils were incorrectly adding fractions by adding the numerators and the denominators. She gave the class this task:. The pupils then invented their own examples of incorrect and correct fraction additions using number lines to make sense of it. 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.
Problem solving generally refers to situations in which pupils do not have a readily-available method that they can use. Instead, they have to approach the problem flexibly and work out a solution for themselves. To succeed in this, pupils need to draw on a variety of problem-solving strategies which enable them to make sense of unfamiliar situations and tackle them intelligently. A problem-solving strategy is a general approach to solving a problem.
The same general strategy can be applied to solving a variety of different problems. For example, a useful problem-solving strategy is to identify a simpler but related problem. Discussing the solution to the simpler problem can give insight into how the original, harder problem may be tackled and the underlying mathematical structure.
A strategy is different from an algorithm, which is a well-established sequence of predetermined steps that are executed in a particular order to carry out a commonly-required procedure. The evidence suggests that teachers should consider the following when developing these skills. So their ages are: This recommendation presents the evidence regarding teaching specific topics in mathematics. Although this recommendation concerns particular topics, teaching should emphasise the many connections between different mathematical facts, procedures, and concepts to create a rich network.
Currently, the evidence about effective teaching approaches is stronger regarding number including fractions, ratio and proportion and algebra than for other areas such as geometry. Teachers should adopt such approaches while drawing on their knowledge of maths, their own professional experience, and the other recommendations in this guidance.
Quick retrieval of number facts is important for success in mathematics. Pupils are able to apply procedures most effectively when they understand how the procedures work and in what circumstances they are useful. One reason for encouraging understanding is to enable pupils to reconstruct steps in a procedure that they may have forgotten. Teachers should help pupils to compare and choose between different methods and strategies for solving problems in algebra, number, and elsewhere.
Pupils should be taught a range of mental, calculator, and pencil-and-paper methods, and encouraged to consider when different methods are appropriate and efficient. In fact, studies have shown using a calculator can have positive impacts, not only on mental calculation skills, but also on problem-solving and attitudes towards maths.
The aim is to enable pupils to self-regulate their use of calculators, consequently making less but better use of them. Multiplicative reasoning is the ability to understand and think about multiplication and division. Fractions are often introduced to pupils with the idea that they represent parts of a whole—for example, one half is one part of a whole that has two equal parts.
This is an important concept, but does not extend easily to mixed fractions that are greater than 1. Another important concept is often overlooked: They have magnitudes or values, and they can be used to refer to numbers in-between whole numbers.
Understanding that fractions are numbers, and being able to estimate where they would occur on a number line, can help pupils to estimate the result of adding two fractions and so recognise, and address, misconceptions such as the common error of adding fractions by adding the numerators and then adding the denominators.
Paying attention to underlying mathematical structure helps pupils make connections between problems, solution strategies, and representations that may, on the surface, appear different, but are actually mathematically equivalent. Some examples of teachers supporting pupils to recognise mathematical structure are: Teachers should encourage pupils to take responsibility for, and play an active role in, their own learning.
Ultimately the aim is for pupils to be able to do this automatically and independently, without needing support from the teacher or their peers, however, these are complex skills which will initially require explicit teaching and support. Teachers should model metacognition see example below by simultaneously describing their own thinking or asking questions of their pupils as they complete a task.
Teachers can provide regular opportunities for pupils to develop independent metacognition through: While demonstrating the solving of a problem, a teacher could model how to plan, monitor, and evaluate their thinking by reflecting aloud on a series of questions. Developing metacognition is not straightforward and there are some important challenges to consider.
The development of positive attitudes and motivation is, of course, itself an important goal for teaching. It can also support the development of self-regulation and metacognition, as these capabilities require deliberate and sustained effort, which can require motivation over a long period of time.
Motivation is complex and may be influenced by a like or dislike of maths, beliefs about whether one is good or bad at maths, and beliefs about whether mathematics is useful or not. Although positive attitudes are important, there is a lack of evidence regarding effective approaches to developing them. It is likely to be important to model positive attitudes towards mathematics throughout the whole school.
School leaders should ensure that all staff, including non-teaching staff, encourage and model motivation, confidence, and enjoyment in maths for all children.
Teachers should engage parents to encourage their children to value, and develop confidence in, mathematics. Maths anxiety is a type of anxiety that specifically interferes with mathematics, and is not the same as general anxiety.
Mathematics anxiety tends to increase with age, but there are signs of it appearing even in children in Key Stage 1. Tasks are critical to the learning of mathematics because the tasks used in the classroom largely define what happens there.
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