What should we be thinking about when planning mathematics for high attaining pupils?

How does this then link to how we teach and assess?

Things to think about

“The emphasis almost all of the schools placed on pupils using and applying their arithmetic skills to solving a wide range of problems was striking. Diverse opportunities were provided within mathematics, including measures and data handling, and through thematic and cross-curricular work. Pupils’ extensive experience of solving problems deepens their understanding and increases their fluency and sense of number.” Good practice in primary mathematics: evidence from 20 successful schools, (Ofsted, 2011, p.20)

“The Mathematical Association has also argued the need for the most able students to be routinely expected to master essentially the same material as their peers – but more robustly, fluently and deeply, and with a greater emphasis on making connections. They should also focus on communicating mathematically and on developing better problem solving skills both within and beyond mathematics.” Raising the bar: developing able young mathematicians, (ACME, 2013 p. 2)

“Many teachers continued to struggle to develop skills of using and applying mathematics systematically.” Mathematics: made to measure, (Ofsted, 2012, p.9)

The 2013 draft programme of study identifies three core aims for mathematics. These are to develop fluency in the fundamentals of mathematics, to ensure that all pupils reason mathematically and that pupils can solve problems by applying their mathematics to a variety of routine and non-routine problems.

High attaining children typically develop fluency in arithmetic relatively quickly. The main focus for ensuring they are challenged therefore needs to be within mathematical reasoning and problem solving. This means that teachers need to help children understand what mathematical reasoning consists of, and to develop understanding of the stages they need to go through to be able to solve more complex problems. It is through such approaches that high attaining children can be suitably challenged in their mathematics learning.

What does research tell us?

Previous versions of the National Curriculum have referred to the Using and Applying strand of the National Curriculum for mathematics. The 2006 Guidance Paper refers to five themes which remain a useful way to think about mathematical reasoning and problem solving:

This idea was developed by John Mason et al who suggested that it is helpful to think about seven phases:

getting started

getting involved

mulling

keeping going

insight

being sceptical

contemplating

(Mason et al, 1982)

Other researchers might come up with other lists, but the value here is that such lists help teachers think about the different elements that make up mathematical reasoning and problem solving. All such lists consist of verbs which indicate the sorts of things children should be doing when they are doing mathematics. Excellent teachers are clear about the types of mathematical thinking and problem solving strategies they are trying to develop, and this enables them to ensure that high attaining children are suitably challenged.

These ideas link to wider research on meta-cognition or ‘thinking about thinking’. Meta-cognitive strategies encourage children to think about their learning more explicitly, and one way to do this is to help children understand the different types of thinking they might use when reasoning mathematically and the different stages they need to consider in problem solving. This means that they need to be included explicitly in learning objectives and in peer and self assessment activities. The Sutton Trust’s report The teaching and learning toolkit (Higgins et al, 2013) identifies meta-cognitive strategies in general as having high potential impact on learning at low cost.

Guy Claxton has developed similar ideas. He argues that key learning dispositions are:

reciprocity – interdependence, collaboration, empathy and listening, imitation.

(Claxton, 2002)

and that teachers should build these explicitly into lessons so that children develop understanding of themselves as learners.

Implications for the use of the Year 6 Level 6 tests

Investigation of Key Stage 2 Level 6 Tests (Coldwell et al, 2013) found that one of the reasons that schools decided to enter children for the L6 test was to remove artificial ceilings from their assessment, and hence motivate children to aim higher. Some saw it as a way of demonstrating to stakeholders that they have high expectations of their pupils. But some secondary schools were concerned about the possibility of ‘teaching to the test’ and of the danger of the children then having little depth of understanding of what L6 mathematics is. The vast majority of secondary schools in the survey assessed the children again at the start of Year 7, using, for example, Cognitive Ability Tests.

This illustrates the importance of decisions about whether to enter children for the L6 test being incorporated into a wider policy on teaching and learning which maintains a focus on rich curriculum entitlement. It also highlights the importance of developing effective cross-phase liaison in order to develop understanding of teaching, learning and assessing in different key stages.

Working with colleagues

Some main areas to focus on with colleagues might be:

developing an understanding of meta-cognition and its significance;

helping colleagues think explicitly about objectives relating to mathematical reasoning and stages in problems solving when they plan;

developing the way colleagues make these objectives clear to the children and how they build them into peer and self assessment activities.

Policy decisions

discuss with the senior leadership team the possibility that your school’s mathematics policy states that mathematics lessons should consistently have objectives that encompass some aspect of mathematical reasoning and/or problem solving as well as content objectives.

Ideas for staff meetings

ask colleagues to read Robert Fisher’s article Thinking about thinking: developing meta-cognition in children. Use this as the basis for a discussion of what meta-cognition is, what they already do (possibly without realising it) and how they could develop this explicitly with their classes.

use the APP guidelines for mathematics to discuss with colleagues what progression in mathematical reasoning and problem solving might consist of. From this ask colleagues to choose one particular aspect (eg conjecturing and generalising) and try an activity (this NRICH page has several activities that can be used to develop conjecturing and generalising) that allows children to explore this and to focus in particular on the impact on the high attaining children. Ask them to feedback what happened in a subsequent meeting.

discuss with colleagues how they share objectives with children. Discuss the idea of not always sharing objectives at the start of a lesson but of asking children at an appropriate point to decide for themselves what the objectives are. Ask colleagues to try this out and report back on what they find out.

choose an activity from NRICH and work on this with colleagues in a staff meeting. Use this to get them to articulate what types of mathematical reasoning and problem solving strategies the tasks requires. Then use a contrasting activity in order to see how these can vary depending on the task.

ask colleagues to read the NRICH article Using low threshold high ceiling tasks in ordinary classrooms, and use this to share ideas about how NRICH type tasks can be used with whole classes to develop mathematical reasoning and problem solving. Get them to choose a specific task, consider exactly what aspect of mathematical reasoning or problem solving they will focus on with the children and discuss how they can express this as a success criterion. Ask them to get the children to use this for peer and self assessment, and then report back on what response they got.

These Mathemapedia entries describe some simple activities that can encourage high level mathematical reasoning. These are generic activity types that can be used again and again with all age groups. Try them out with colleagues and encourage them to use them in their classrooms. Ask them to reflect on how the high attaining children respond to them and get them to feedback at a later meeting.

There are several YouTube videos of Guy Claxton talking about his thinking about Building Learning Power. Try this one.

Dylan Wiliam has been a key proponent of formative assessment (Assessment for Learning). Listen to him explain aspects of his work on this video clip. With Paul Black he co-wrote Inside the black box (Black and Wiliam, 1988) which explores the rationale underpinning formative assessment.

Shirley Clarke has been highly influential through her work on formative assessment. Her book Active learning through formative assessment (Clarke, 2008) explores in very practical ways how to develop the use of learning objectives, success criteria and peer and self assessment. Her website has more information about her work.

Back, J., Piggott, J. & Liz Pumfrey, L. (2003) Using non-standard problems to challenge pre-conceptions: can they extend knowledge? International Symposium Elementary Maths Teaching, Charles University, Prague.