Enrichment or acceleration?
Should we base our provision for high attaining pupils more on an accelerated curriculum or more on an enriched provision?
Is it about ‘enrichment or acceleration’ or ‘enrichment and acceleration’?
What does ‘enrichment’ really mean in mathematics?
Things to think about
“It is not unusual for those groups or individuals identified as able mathematicians to be allowed or encouraged to progress through the curriculum at a faster pace. Such acceleration in mathematics is often counterproductive. Acceleration encourages only a shallow mastery of the subject, and so promotes procedural learning at the expense of deep understanding. This shallow acquaintance can also lead to learners feeling insecure and fails to adequately promote a commitment to the subject in students. This approach therefore often leads to apparent success without students developing the depth and tenacity that is needed for long-term progression. In addition, the use of acceleration is in stark contrast to the successful practice in many of the world’s mathematically most highly performing jurisdictions”.
“Ofsted evidence shows that the most effective strategy for generating students’ interest in and commitment to mathematics is through planned enrichment and extension work with the minimum of acceleration.”
“Potential heavy users of mathematics should experience a deep, rich, rigorous and challenging mathematics education, rather than being accelerated through the school curriculum.”
Raising the bar: developing able young mathematicians, (ACME 2012, p. 2)
A school’s policy on how the needs of high attaining young mathematicians are to be met will need to address the issue of enrichment versus acceleration. Do such children work from mathematics content typical of older children (for example Year 6 children working from the Key Stage 3 programme of study) or is greater emphasis given to the development of children’s conceptual understanding, their mathematical reasoning and ability to apply their mathematics in more challenging problem solving contexts?
It’s probably not helpful to think in terms of either enrichment or acceleration, but to consider the balance between these two approaches. Approaches may vary depending on the age of children, or the mathematics topics, while there may be extra-curricular opportunities to meet the needs of high attaining children in other ways. In addition to considerations of which approach supports the best learning, there are practical issues to consider. For example, if high attaining Year 6 children make an early start on the Key Stage 3 programme of study, there are planning and organisational issues that need to be considered by the secondary school the children move on to. Close liaison between the phases would be essential if the children are to maintain their enthusiasm for the subject in year 7. Acceleration can also result in the attainment gap in year groups widening.
It is clear from the quotations above from ACME’s report that they consider enrichment to be significantly more important than acceleration. The 2013 draft of the mathematics programme of study identifies mathematical reasoning and problem solving (including the ability to persevere in seeking solutions) alongside fluency in the fundamentals of mathematics as key aims, and these areas give opportunities for enrichment.
What does research tell us?
ACME’s report Raising the bar (ACME, 2012) argues that it is essential that more young people continue to study mathematics at A level and beyond, but that this needs to be based on opportunities for children to develop a ‘deep mastery of the material’. This has echoes of the influential work of Richard Skemp, who discussed the distinction between instrumental and relational understanding in mathematics. When children learn mathematics in an instrumental way they tend to learn the subject as a set of discrete topics. Relational learning encourages children to see how these topics join together to create a richer conceptual structure. Skemp acknowledges that instrumental learning can at times be efficient (if children can remember the technique they have been taught), but this can lead to children failing to see the richness of the subject and can diminish children’s enthusiasm for the subject (Skemp, 1976).
Exactly what ‘enrichment’ is should not be taken for granted. Jennifer Piggott in her article Mathematics enrichment: what is it and who is it for? (Piggott, 2004) acknowledges that it is a complex concept and argues that it is as much about the development of positive attitudes to mathematical enquiry and problem solving as it is about the provision of, for example, rich tasks. She suggests that an enrichment curriculum consists of the use of engaging problems which develop and use problem solving skills and encourage mathematical thinking, whilst also requiring teachers to develop an ‘open and flexible’ environment in which the following are encouraged:
the valuing and utilisation of difference as a teaching tool,
the acknowledgment that mathematics is often hard.
She writes that the aim of an enrichment curriculum is to support:
a problem solving approach (either through, about or for problem solving) that encompasses the four elements of understanding the problem, devising a plan, carrying out the plan and looking back
improving pupil attitudes
a growing appreciation of mathematics as a discipline
the development of conceptual structures that support mathematical understanding and thinking.
Jennifer Piggott also refers to research which suggests that such an approach is of benefit to all learners, not just to high attainers. Her focus on improving pupil attitudes clearly relates to ACME’s report – if more young people choose to continue to study mathematics at a high level in the future, their own enjoyment of the subject will have been a significant factor in this decision.
Some of these issues are reflected in Ofsted’s 2012 report, Mathematics: made to measure. Referring to schools where mathematics teaching is consistently outstanding the report says:
“The schools focused on building pupils’ fluency with, and understanding of, mathematics. Pupils of all ages and abilities tackled varied questions and problems, showing a preparedness to grapple with challenges, and explaining their reasoning with confidence. This experience contrasts sharply with the satisfactory teaching that enabled pupils to pass tests and examinations but presented mathematics as sets of disconnected facts and methods that pupils needed to memorise and replicate.”
(Ofsted, 2012, p. 7)
You can read more about what ‘enrichment’ can mean in Wai Yi Feng’s article, Conceptions of Enrichment (Feng, 2005).
Implications for the use of the Year 6 Level 6 tests
Investigation of Key Stage 2 Level 6 Tests (Coldwell et al, 2013) discusses some of the issues that arise between primary and secondary schools. In their survey they found that some schools felt that secondary schools would question the validity of the L6 tests and ‘would re-test pupils on arrival in Year 7 and cover material previously learnt at primary school’ (p. 11). These reservations have some basis in truth. Evidence from secondary schools indicated that while the Level 6 test could be useful to help identify ‘high fliers’ but that they often felt that primary schools would ‘teach to the test’ and this would result in pupils having’ little depth and understanding of L6 working’ (p. 16).
These findings emphasise the importance of close liaison between primary and secondary schools as a part of making decisions about the use of Level 6 tests.
Working with colleagues
In order to develop consistency of approach in your school it is important that colleagues have a shared understanding of what acceleration and enrichment mean and of the strengths of each.
Ideas for staff meetings
Spend time together discussing whether enrichment or acceleration is more appropriate for different mathematics topics – can colleagues identify contexts where they feel that acceleration is the appropriate strategy? For example, what do they feel is the best way to meet the needs of:
a Year 1 child who is already good at times tables?
a Year 3 child who can already do long multiplication?
a Year 5 child who is already confident at solving algebraic equations?
Have they had experiences where they have found it difficult to plan because children have already experienced an accelerated curriculum?
Some colleagues may have a limited understanding of what it means to ‘think mathematically’. Spend staff meeting time doing some mathematics together. You could choose an activity from NRICH, and spend time discussing the mathematical reasoning within the task. Encourage colleagues to get children to submit their own answers to the live problems on the NRICH site – this could be an excellent voluntary homework for highly motivated children. Share with colleagues websites such as Brainbashers which they could use as homework activities for high attaining children
Ask colleagues to read Jennifer Piggott’s article Mathematics enrichment: what is it and who is it for? and use this as a prompt for a discussion about the nature of enrichment in mathematics and how the school can best embed this in its mathematics teaching.
Discuss with colleagues how best to work with parents are included in supporting the way the school teaches pupils identified as high attainers.
Other ways of working
Seek out the views of the children. What do the high attaining children think about their mathematics learning? What do they like and what do they dislike?
Contact the mathematics department at your local secondary school(s) and invite them to discuss issues of acceleration and enrichment with you in order to develop coherent cross-phase provision. Include discussion of the use of Level 6 tests with Year 6 children. Seek opportunities for you and your colleagues to visit the secondary school and for secondary colleagues to visit your school in order to develop teachers’ understanding of the curriculum in the different key stages.
Discuss with your senior leadership team the issues surrounding the use of Level 6 tests with high attaining Year 6 children. If children are going to be entered for these, how will the appropriate teaching be organised in upper key stage 2? How will your partner secondary schools acknowledge this in their Year 7 curriculum? What is your response to ACME’s view that ‘the current provision of level 6 tests at Key stage 2 has the potential to drive acceleration of pupils, and hinder a secure understanding of Key Stage 2 mathematics’?
Discuss with your senior leadership team extra-curricular opportunities for high attaining children. Consider taking part in the Mathematical Association’s Primary Mathematics Challenge. Discuss with colleagues in other local schools the possibility of setting up a local area mathematics challenge for high attaining children. Your local secondary school mathematics department may be able to help plan this.
Resources to use
Supporting gifted and talented learners in mathematics has further discussion material about enrichment and acceleration.
Working with highly able mathematicians is an article by Bernard Bagnall in which he reflects on his own experience of working with high attaining children. You could use this to develop discussion about the nature of enrichment.
Extension, enrichment and/or acceleration by Jennifer Piggott would also stimulate discussion with colleagues about how the needs of high attaining young mathematicians are best met.
Supporting highly able mathematicians is another NRICH resource that could be used to develop discussions with colleagues.
Use sections of Investigation of Key Stage 2 Level 6 Tests to develop discussion with staff about some of the issues about high attaining children starting to study Key Stage 3 mathematics in primary school, and to consider how secondary schools view this. Chapter 7 is about secondary schools’ views of the use of Level 6 tests in Year 6 and is particularly relevant.
ACME (2012) Raising the bar: developing able young mathematicians.
Coldwell, M., Willis, B. & McCaig, C. (2013) Investigation of Key Stage 2 Level 6 Tests. Centre for Education and Inclusion Research & Sheffield Hallam University.
Feng, W. Y. (2004) Conceptions of Enrichment. CamERA (Cambridge Educational Research Association) Conference, Cambridge, April 2005.
Ofsted (2012) Mathematics: made to measure.
Piggott, J. (2004) Mathematics enrichment: what is it and who is it for?. British Educational Research Association Annual Conference, University of Manchester, 16-18 September 2004.
Skemp, R. (1976) Relational Understanding and Instrumental Understanding, Mathematics Teaching, Volume 77, pp 20-26.