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KS2: Mathematics optional tasks for the more able


We fail to identify many of our most able children and we don’t challenge them enough. This can lead to disaffection and chronic underachievement. We owe it to these children to help them realise their potential. That means working with schools, parents and local authorities to establish and spread good practice. We must celebrate the abilities of our most able children and encourage them to achieve at the highest level. The attitude that gifted children can cope by themselves has let down too many young people.

Estelle Morris, School Standards Minister, September 1998
(used on the National Association of Gifted Children website)

There are a number of ways to measure high ability. In the past, IQ testing and good results in tests have been used to indicate mental achievement. However, more able pupils may not always do well in formal testing because their strengths lie in different areas, or because they process information in different ways.

Who is more able?

Within any school there are a number of pupils who are more able. Their ability may not be easily recognised because it is hidden behind their own low expectations, or a lack of focus or challenge.

More able pupils may produce work typical of older pupils. For instance you may find that a six-year-old works confidently with level 3 mathematics. Others may express their potential differently; for example pupils may use advanced reasoning when speaking, but do not express their ideas well in writing.

Pupils may do well in all subjects, or they could excel in particular subjects, such as English, mathematics or science. Teachers may find it useful to offer enrichment or extension activities to pupils in order to observe how they respond. More able pupils are likely to be able to recognise and solve problems.

All pupils benefit from developing and improving their problem-solving skills. The curriculum offers a variety of contexts in which to assess pupils’ problem-solving abilities. More able pupils may approach problems in unexpected ways. In these cases, formal testing alone is unlikely to be enough to identify them.

When considering which pupils are more able, it is also important to think about how they might manage their workload with extra challenges.

For more information on identifying more able learners, useful criteria are provided on the Guidance on teaching the gifted and talented website.

It is good practice for schools to have a written policy on how they look after their most able pupils. This should include how more able pupils are identified and what steps they take to extend their education.

Assessment

Assessment is used to inform teaching for all pupils, and with a clear focus it can help develop teaching and learning to help pupils demonstrate and expand their range of ability.

Formative assessment (assessment for learning) can be used as a dynamic process, informing teacher and pupil, and enabling them to reform their performance. Formative assessment involves using assessment in the classroom to raise pupils’ achievement. It is based on the idea that pupils will improve most if they understand the aim of their learning, where they are in relation to their aim and how they can achieve the aim (or 'close the gap in their knowledge'). Summative assessment involves judging pupils' performance against national standards (level descriptions). Teachers often make these judgements at the end of a unit of work, year or key stage. Test results also describe pupils’ performance in terms of levels. Summative assessment (assessment of learning) provides a summary of what has been learnt up to that point.

You can find out more about assessment for learning on this site. (Go to 3-14 and choose 'Assessment for Learning').

Formative assessment

Formative and summative assessment are best used in parallel. This allows teachers to assess as and when they need to, while monitoring and offering feedback to pupils to help them progress.

All pupils can benefit from this approach. In addition, teachers may find that using assessment in this way helps pupils reveal unexpected ability.

The key features and principles of formative assessment are outlined below:

Learning how to learn

Pupils need to develop the skills of how to learn, rather than simply being told what to learn.

To do this they need to:

  • locate and gather information
  • understand, organise and perform the task
  • record and communicate appropriately
  • evaluate the task
  • assess personal performance.
Using collaborative problem solving

Pupils will benefit from sharing and exchanging knowledge. Group problem solving helps pupils develop language skills, such as explaining and questioning, as well as social skills, such as sharing and supporting.

Developing awareness of process rather than product

Pupils will develop an awareness of process by understanding that the quality of thinking is as important as the quantity of material produced. They need to understand the way they process and use information. By considering learning in this way pupils may develop strategies and thinking skills to improve their performance.

Differentiating memory from thinking questions/tasks

Pupils need to learn skills and knowledge through practice and repetition, but they also need to understand the purpose and goal of doing so. It is useful to signal when a task involves remembering knowledge, or to signal when a pupil is required to think through tentative answers.

Allowing time for thinking

‘Stop a minute, let me think!’ is a useful phrase for pupils to learn and put into practice. Thinking time helps them develop reflection and avoid impulsive thought and action. It encourages pupils to search for a deeper answer and to mentally rehearse an idea.

Encouraging understanding links to prior learning

Learners need to make connections with prior knowledge in order to use existing skills and knowledge to help establish and extend their learning.

Providing an overview

Pupils benefit from an explanation of the purpose of learning a topic and how it is going to be tackled. By understanding the reason for incremental steps they can see the bigger picture and assess their own performance.

Involving pupils in establishing goals

A sense of ownership can increase pupil motivation. All learners need to feel that they are in real and active partnerships with their teachers. Setting personal goals and having a say in designing the tasks that they undertake boosts their self-esteem. This can help pupils persevere, accept mistakes, redraft and start again when necessary.

Discussing what good work is

By understanding what ‘better’ or ‘good’ work is, pupils can adjust their performance. Pupils benefit from being shown anonymous examples of good work and discussing how learners might improve their own performance. This can dispel any mystery surrounding successful achievement. Knowing what they are trying to do can help the learner navigate and develop from the process of self-evaluation.

Celebrating the process as much as the product

Schools often celebrate pupils’ achievements by displaying their work. Teachers and pupils also need to celebrate the thinking and effort that goes into drafting and working towards an end product. Work could be displayed using concertina booklets, which include every stage of development. This kind of display indicates the value of all the steps in learning, and gives other learners concrete examples of the process.

Giving positive hints and recommendations

School displays of work that include the process as well as the finished product can help learners revise and improve their work. They also remind pupils of the criteria for success that were set when they started a task, which they can use in redrafting. This revision process can also be stimulated by hints and recommendations from teachers and peers. Learners need to avoid any impulse to rush work, and must establish a work pattern of rethinking and improving their work. Achieving a piece of quality work in this way provides them with a good learning experience.

Developing tests with pupils

Tests should not be used to catch out learners, as this undermines their involvement in goal setting. Learners should know the purpose of tests before undertaking them, because they have agreed on the goals of learning within the topic. Even young learners can discuss the ‘why’ and ‘what’ of their learning experiences. They can understand the main skills and content they need to demonstrate.

Summative assessment

World Class Arena (see www.worldclassarena.org) offers support materials, including tasks, which can be used for summative assessment of mathematics and problem solving. It also offers World Class Tests aimed at the top 10 per cent ability range in mathematics, and problem solving in mathematics, science and design technology.

Assessment and the more able

More able pupils often demonstrate learning characteristics that could be indicators of their ability. These characteristics are evident regardless of the pupil’s age. The difference lies in the complexity of the contents and skills that they present, for instance the ideas and knowledge processed at reception age would be far less complex than the ideas and knowledge processed at the end of key stage 2. More able pupils process ideas and knowledge quickly.

Summary of the learning characteristics of more able learners, and specific learning experiences that can benefit these characteristics.


Learning characteristics

More able learners:

Specific learning experiences

More able learners need:

  • process ideas and knowledge quickly
  • retain, remember and recall rapidly
  • require minimum repetition – only enough to trigger what has already been learnt
  • function easily and naturally at a higher level
  • ask questions that search for deeper meanings
  • see connections and manipulate ideas
  • have unusual ideas and solutions
  • see the end almost before the beginning

  • the auditing of knowledge and skills already mastered to avoid repetition
  • an assessment and negotiation of how much basic practice is needed before the learner can accelerate to greater depth and breadth
  • targets set early at higher national curriculum levels
  • negotiation of the minimum explanation that can lead to the performance of the task
  • tasks set that require the use of advanced resource material
  • maximum thinking and minimum recording by using flowcharts, mindmaps, diagrams, keywords
  • challenging tasks that need higher order thinking skills, and that have assessment criteria linked to the demands of the tasks
  • to be challenged through high-level problem-solving activities evaluated against the degree of challenge
  • extended time for thinking, redrafting and completing work against assessment criteria negotiated at the beginning of the task.

Developing ways to assess the more able

You need to consider seven main areas when developing assessment procedures for the more able.

More able pupils need to use their capacity to organise and process high-level concepts and procedures within complex problem situations in order to hone and develop their skills.

Seven main areas to be assessed
  1. More able learners need appropriately challenging targets to feel a sense of progress and achievement.
Do a skills and knowledge audit using information from previous teachers and any record of achievements, to ensure pupils are not given work which repeats prior learning.

Discuss targets with the pupil. Set targets at the higher levels of achievement of the national curriculum framework early.

  1. More able learners need the opportunity to respond to questions using greater breadth and depth.
Use higher order questions to stimulate higher order thinking.
  1. More able pupils need work set within a problem-solving framework.
Set questions in complex problem situations so pupils can develop their capacity to organise and process high-level concepts and procedures.
  1. More able pupils need to develop and use high-level language.
Encourage pupils to develop a wide range of vocabulary so that they can express ideas clearly and creatively. Pupils need to acquire and use subject-specific vocabulary and technical terms to maximise their progress.
  1. More able pupils need to develop their abilities across the range of multiple intelligences.

    The full range of multiple intelligences is currently accepted as:
  • linguistic
  • logical
  • visual/spatial
  • musical
  • bodily kinaesthetic
  • naturalist
  • spiritual
  • intrapersonal
  • interpersonal.
Give pupils opportunities to develop a range of intelligences. Exceptional performance often results from using a range of interacting intelligences.
  1. More able pupils need determination and perseverance to achieve long-term goals.
Allow high performers opportunities to develop sensitivity and emotional strength as well as ability. Allow them to express their personal and ethical value system.
  1. More able pupils need to increase their range of abilities to encourage creative thinking and innovative outcomes.
Offer pupils the chance to innovate, take risks and test boundaries as part of the learning process.

Planning assessment of the more able

Everyone uses a range of thinking skills. Benjamin Bloom’s Taxonomy of educational objectives (1956) has been widely used to define a hierarchy of these skills. However it is more useful to consider these skills as an interactive collection of thinking processes. All learners can use the full range of higher-order thinking skills when the skills are embedded in relevant content at an appropriate level.

When planning for assessment or assessing responses to higher-order questions it is important to ensure that:

  • the learning tasks set do require higher-order thinking
  • pupils have access to extended resource material
  • pupils are given time to work in greater depth and breadth
  • the assessment is planned to enable demonstration of higher-order thinking.

More able pupils demonstrate greater flexibility across the range of higher-order thinking skills and use them in greater depth with more complex tasks. Even young learners can use the full range of higher-order thinking skills.

Stimulating higher-order thinking skills

Questions and open instructions are a useful way to encourage pupils to use higher-level thinking. The following table outlines the skill, context, purpose and questions or instructions that teachers can use to stimulate pupils’ higher-order thinking.

Skill Context Purpose Stimulus
Remembering

There is a core of relevant factual knowledge that pupils need in order to begin thinking. Pupils need to acquire and use a range of research skills and basic subject skills. For instance in mathematics and science this includes procedures and skills; in literacy, art and ICT it includes technical skills. They need examples, teaching and practice to develop these skills. They also need to present ideas using a variety of recording and communication skills. Know and recall information and skills as a basis for action
  • What happened when …?
  • Make a list.
  • Write an account. Make a summary.
Comprehension Able learners need to progress from lower-order skills, such as paraphrasing or selecting and explaining relevant information, to answer a question. They need to use higher-order comprehension skills. They need opportunities to interpret knowledge by presenting a new perspective, comparing and contrasting data and viewpoints, reordering information and examining consequences.

Demonstrate understanding
  • Write a summary of the main points.
  • Explain why that happened.
  • Discuss this from the point of view of …
  • What are the similarities between …?
  • Explain the differences between …
  • What would have happened if …?
  • How would this affect …?
  • Why did … react in this way?
  • What were the results of …?
Application Learners need to use their knowledge to: manipulate or construct something; reproduce it in a different format; apply it to a similar situation; build a model; illustrate or apply it to an example. Use knowledge to solve problems
  • How would you use this to …?
  • How does this rule apply?
  • How can you use what you have learnt to solve this?
  • Does the same principle apply?
  • Weigh up the evidence for …
  • Draw a conclusion giving reasons.
  • What else do you know that would apply?
  • Is this the same kind of …?
  • Construct a diagram to show …
  • Conduct an experiment to prove …
  • Paint a picture to show …
Analysis Learners need to fit pieces together and make a whole. They need to identify connections, patterns, sequences and themes. They should be aware that the elements they are learning contribute to a coherent plan. Understand overall relationships and patterns
  • In what ways are they the same, different, better, worse?
  • What was the overall plan?
  • How do the elements combine?
  • Discuss why the causes had inevitable consequences.
  • What is the general rule?
  • Explore the possible future consequences.
Synthesis Knowledge is best used for creative thinking and action. Learners need to use what they know to design, invent, change and improve. Create something new with the knowledge and skills learnt
  • Do you agree with …?
  • How would you change …?
  • What would happen if …?
  • Is there another way?
  • Is there another conclusion?
  • In how many ways can you …?
Evaluation Learners need to be taught how to balance decisions using reason and evidence, and avoid acting on impulse. Make decisions and judgements
  • How do you know?
  • On what grounds can you justify?
  • What is the evidence?
  • Why would you make that decision?
  • What are the arguments for and against …?
  • Why do you believe …?
  • Did … have a valid case?
  • Draw a conclusion giving reasons.


Sample task: key stage 1

This task uses the story of Goldilocks and the three bears to develop higher order thinking skills.

Tell the story of the three bears.

Remembering: know and recall information and skills as a basis for action.

Act out the story as I read it.

Comprehension: demonstrate understanding.

Was Goldilocks being sensible when she went into the three bears’ house?

Evaluation: make decisions and judgements.

Draw a picture of the three bears.

Application: use knowledge to solve a problem.

Make up a different ending for the story.

Synthesis: create something new with the knowledge and skills learnt.

Put the pictures in the same order as the story.

Analysis: understand overall relationships and patterns.


Sample task: key stage 2

This task uses the story The children of winter by Berlie Doherty to develop higher-order thinking skills.

Make a flowchart to show the storyline in Chapter 1.

Remembering: know and recall information and skills as a basis for action.

Pretend to be one of the main characters in Chapter 1 and make a short speech to explain your behaviour and actions.

Comprehension: demonstrate understanding.

Choose one of the characters in the story and say if you would have behaved in the same way as he or she did.

Evaluation: make decisions and judgements.

Outline the children's plan to gather food and water for the winter. What job did each of the children do? Which job would you have offered to do?

Application: use knowledge to solve a problem.

Write the outline of a story that would take you back to another period in history.

Synthesis: create something new with the knowledge and skills learnt.

Which are the most important events in the story that show how brave the children were?

Analysis: understand overall relationships and patterns.


Aids to assessment for sample task: key stage 1 and sample task: key stage 2

Children's performance needs to be assessed in response to the skills the task demands. The following questions might be used as an aid to assessment.

Remembering: know and recall information and skills as a basis for action.

Can the child:

  • recall information accurately?
  • list important events?
  • research information?
  • remember procedures?
  • use skills appropriately?

Comprehension: demonstrate understanding.

Can the child:

  • summarise main points?
  • provide clear explanations?
  • see something from another point of view?
  • discuss similarities and differences?
  • predict possible outcomes?
  • give appropriate reasons?
  • see possible consequences?
Evaluation: make decisions and judgements.

Can the child:

  • give sound reasons?
  • see more than one point of view?
  • argue for and against something?
  • draw conclusions?
  • justify their own point of view?

Application: use knowledge to solve a problem.

Can the child:

  • use the skills and knowledge in a different context?
  • apply the rule to a different problem?
  • give examples to show understanding?
  • use knowledge to construct or make something?

Synthesis: create something new with the knowledge and skills learnt.

Can the child:

  • create new patterns?
  • suggest unusual solutions?
  • find an alternate way?
  • change a routine for a better one?
  • predict consequences?

Analysis: understand overall relationships and patterns.

Can the child:

  • recognise and continue patterns?
  • see similarities and differences?
  • break down the whole into sections?
  • see the overall plan?
  • understand cause and effect?


Problem-solving and thinking skills

Problem solving is an accepted measure of ability. All pupils can use problem-solving strategies in the context of appropriate tasks.

More able learners are likely to have problem-solving skills and be able to work with advanced problems across the curriculum. It is useful to assess their level of skills in all curriculum contexts.

Assessing problem solving

The TASC problem-solving wheel is a well-researched generic framework outlining the universal processes of problem-solving and thinking skills.

Image of The TASC problem-solving wheel

Any method that claims to develop pupils’ problem-solving skills should include the following:

  • Learners need to gather and organise what they already know at the beginning of each new topic. In this way both teachers and pupils identify what is known and understood. This makes it possible to differentiate in lesson plans, and to avoid repetition of work for the more able. Repetition often leads to the more able becoming bored and frustrated.
  • Pupils need to establish the criteria for success at the beginning of the task. More able pupils also need examples of excellence to analyse so that they can identify what they are trying to achieve; this helps them to evaluate their own work and measure their own success against known criteria.
  • Since able pupils often easily satisfy the expectations of their teachers, they need to be challenged to achieve excellence beyond the norm.
  • Learning is crystallised when pupils reflect on the ‘what’ and the ‘how’ of their learning. All learners need to be able to see how they are developing and progressing. By analysing what new knowledge, skills and concepts they have learnt at the end of every topic, pupils can consider how they can be used elsewhere within and beyond school.

The TASC wheel can be used to help assess pupils’ problem-solving ability. This can be used in conjunction with the following.

Learn from experience

Can the pupil:

  • reflect on and learn from experience?
  • transfer skills and knowledge?
  • retain new knowledge?
  • articulate new skills?

Gather and organise

Does the pupil show:

  • a wide knowledge of the topic?
  • recall of extended information?
  • a sound understanding of advanced concepts?
  • the ability to organise data in complex groups?
Communicate

Can the pupil:

  • explain clearly to others?
  • share what is known?
  • select relevant information?
  • present in different modes?

Identify

Can the pupil:

  • demonstrate full understanding of the tasks set?
  • explain the task fully?
  • work to advanced/high-level objectives?
  • identify missing detail or knowledge?
Evaluate

Can the pupil:

  • evaluate work against criteria?
  • see ways to improve work?
  • carry out improvements?
  • reflect on the quality of the task?
Generate

Can the pupil:

  • suggest alternative methods?
  • collect and organise a variety of evidence?
  • see different perspectives?
  • find new ways of accomplishing tasks?

Implement

Can the pupil:

  • carry out a plan efficiently?
  • monitor and analyse progress?
  • change direction when necessary?
  • see the next steps clearly?

Decide

Can the pupil:

  • see the consequences of decisions made?
  • use evidence effectively?
  • select key ideas?
  • plan efficiently?



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