Working with high attainers

REMOVE THE CEILING

High expectations, problem solving and collaborative learning are all key to increasing the attainment of your most able pupils, says Peter

In May, Ofsted published a report into the teaching of mathematics in maintained primary and secondary schools in England – Mathematics: made to measure. From evidence gathered through inspections and visits to schools identified as having good practice in mathematics teaching, the report concluded that “the aim of all schools should be to secure high calibre, ‘made-to-measure’ mathematics provision to optimise every pupil’s chance of the best mathematics education.”

One of the key findings of the report was that more able pupils were not effectively and consistently challenged. The report recommended the need to raise ambition for more able pupils, and to increase the emphasis on developing problem solving skills across the mathematics curriculum.

This is, of course, a subtle balancing act for the teacher. Just as too low a level of expectation results in boredom and is demotivating, so too can unrealistically high levels of challenge produce stress and frustration which, in the long run, can be counter-productive.

What is most important is finding a level of expectation that makes the learning experience motivating and challenging. It is in a learning environment that encourages ambitious but realistic expectations, and where there are no fixed ceilings on learning, that more able pupils are most likely to thrive.

Developing skills, not just knowledge

The most important aspect of meeting the needs of more able pupils is the day-to-day experience in the classroom. This can be best achieved through a combination of:

• incorporating breadth across the whole curriculum;
• increasing depth within the subject;
• accelerating the pace of learning.

While it is important to extend and enrich more able pupils’ mathematical knowledge and understanding, it is equally, if not more, important to develop and enhance their skills base. In particular, more able pupils need to be encouraged to produce independent and original thinking. Learning in mathematics is all about making connections between previously unrelated concepts, and it is thinking that helps make the connections. These connections lead to a wider and deeper understanding of the concepts involved. As a teacher, it is important to find opportunities to increase the use of the higher order thinking skills as defined by Harold Bloom (1956) which encourage more able pupils to ‘think about their thinking’ (metacognition) and to question their own learning. By explicitly introducing more able pupils to the concept of higher order thinking skills, 126 using child friendly versions of Bloom’s Taxonomy, teachers and pupils will see that ‘thinking’ can be taught and learnt, rather than being something that might just happen.

Not afraid to fail

As well as developing their higher order thinking skills, more able pupils also need to develop their ability to persevere. Failure, especially if it is public, is something that more able pupils find difficult to embrace, given they are generally seen to be the ones who always get the right answer, and quickly. However, they need to experience failure so they become aware that, along with success, it is a consequence of exploring different avenues of thinking and there is no disgrace in it. They also need to come to appreciate that there is often no ‘right’ or ‘wrong’ answer. It is only through opportunities for trial and improvement that they can begin to truly learn from their mistakes.

Mathematical problem solving and investigations

If a child is truly mathematically able, then not only will they have a good grasp of pure mathematical knowledge, but they should also be able to use and apply this knowledge to solve problems.

Problem solving is what you do when you don’t know what to do. If you know how to get the answer then it’s not problem solving. If we are genuinely to motivate and challenge our more able pupils, and equip them with the skills necessary to be autonomous learners, then we need to provide opportunities to problem solve and reason in mathematics. This means that the activities we give our more-able pupils need to:

• puzzle;
• challenge;
• be problematic;
• require thought;
• lead somewhere mathematically.

However, applied mathematics is more than just solving problems – it also involves making and investigating general statements; searching for patterns and relationships; identifying and applying rules; making and testing predictions; conjectures or hypotheses; reasoning and investigating mathematics itself; and explaining and justifying results, solutions, conjectures, conclusions and generalisations. This is why collaboration is so important.

Promoting collaboration and communication

One of the three aims of the proposed National Curriculum for Mathematics (2014) is to ensure that all pupils can “reason mathematically by following a line of enquiry and develop and present a justification, argument or proof using mathematical language”.

True reasoning is most effective when predictions, conjectures, hypotheses, results, solutions and conclusions are shared. It is through meaningful collaboration, when a pupil’s opinion is reinforced and/or counter-arguments are offered, that pupils learn to justify their reasoning with a greater degree of certainty and authority.

Collaboration over challenging activities promotes skills of communication between learners. The process of mutual support in tackling such activities helps develop positive interdependence between pupils and also with teachers. Providing opportunities for more able pupils to work in pairs or as a group also:

• increases pupils’ motivation and engagement in learning;
• promotes speaking and listening skills;
• prevents intellectual isolation;
• encourages the sharing and acceptance of the ideas and reasoning of others;
• stimulates critical and analytical thinking skills;
• improves behaviour and relations with peers.

One of the major benefits of collaboration is that it inherently gives opportunities for pupils to practice their questioning skills. All pupils, but particularly more able pupils, need to be explicitly shown how to develop effective questioning skills in order to build on responses and explore ideas. They need to be taught the value of questioning, and how to question effectively, rather than being left to acquire such skills for themselves.

So, as we embark on a new era in primary mathematics education and aim to ensure that pupils develop their conceptual understanding alongside their fluent recall of knowledge, let us not forget that at the heart of how we use and apply this understanding and knowledge, as adults, is our confidence in problem solving.

Let us ensure, therefore, that we hold high expectations of all our pupils, not just the most able, and that we integrate into the heart of our teaching the problem solving and thinking skills necessary to equip our pupils to be autonomous learners of the future.