I. Background: Why focus on Key Stage 3?
When designing a mathematics scheme of work for Key Stage 3, the obvious move would be to try to adapt the official programme of study.1
However:
•the programme of study incorporates some startling omissions of essential content that simply cannot be skipped (to give just two examples: there is no reference to the subtleties of teaching the arithmetic of negative numbers, or of combining negatives and ‘minus signs’ in algebra; nor is there any explicit mention of isosceles triangles, or of deriving and using their properties in other settings);
•many of the officially listed themes require careful interpretation in other ways;
•in the official programme of study the connections between topics are rarely elaborated; and
•the grouping and sequencing of, and the progression through, topics is far from clear.
In short, the programme of study needs to be supplemented and ‘fleshed out’ (and sometimes corrected). Moreover, unlike the programmes for Key Stage 1 and Key Stage 2,
the programme of study for Key Stage 3 has no year-by-year structure and no accompanying Notes and guidance.
The fact that we need to think more carefully about mathematics teaching at Key Stage 3 has been a theme of the Ofsted triennial reports on mathematics:
Mathematics: Understanding the score (2008)2
and
Mathematics: Made to measure (2012).3
These reports have not been as widely read as they deserved. Their analysis is unusually forthright for official documents, and provides a sobering starting point for any school seeking to review its mathematics provision at Key Stage 3. The reports summarise observations from hundreds of inspections—but they do so in an unusually constructive spirit. For example, having classified half of secondary maths lessons, and more than half of the schemes of work, as being either ‘inadequate’ or ‘requiring improvement’, Ofsted went out of their way to provide down-to-earth advice.4
This down-to-earth Ofsted DIY guide begins with a four-page table contrasting
•the general features of “good mathematics teaching”
with
•those of “mathematics teaching deemed to require improvement”.
The Ofsted guide then presents a string of specific examples chosen to clarify the differences between ‘weak’ and ‘more effective’ mathematics teaching, and to challenge schools to reflect on, and to improve, their own teaching. Hence this collection of examples and advice should probably be taken seriously by any school seeking to revise its published scheme of work for Key Stage 3.
Key Stage 3 mathematics teaching is important because it marks a transition from the more informal approach in primary schools to the formal, more abstract mathematics of Key Stage 4 and beyond. Hence those teaching Key Stage 3 classes need a clear picture of how the constituent parts of secondary mathematics interlock, and how Key Stage 3 work can best support progression—first progression to Key Stage 4, and then to Key Stage 5 (at ages 16-18). In this regard the 2012 report Made to measure highlights the uncomfortable fact that (p. 4):
“More than 37,000 pupils who had attained Level 5 at primary school gained no better than grade C at GCSE in 2011. Our failure to stretch some of our most able pupils threatens the future supply of well-qualified mathematicians, scientists and engineers.”
This illustrates the extent to which current provision at Key Stage 2 and Key Stage 3 fails to lay the necessary foundations for subsequent stages, and raises the question of how to improve provision at Key Stage 3. The question is especially relevant given that so many schools feel unable to allocate their strongest mathematics teachers to Key Stage 3 classes. So there is clearly a need to provide more detailed guidance for those who teach at this level.
The quality of existing support and guidance at school level is summarised in the key findings of the 2008 report Understanding the score (p. 6):
“Schemes of work in secondary schools were frequently poor, and were inadequate to support recently qualified and non-specialist teachers.”
The ‘Executive Summary’ (p. 4) noted:
“Evidence suggests that strategies to improve test and examination performance, including ‘booster’ lessons, revision classes and extensive intervention, coupled with a heavy emphasis on ‘teaching to the test’, succeed in preparing pupils to gain the qualifications but are not equipping them well enough mathematically for their futures. It is of vital importance to shift from a narrow emphasis on disparate skills towards a focus on pupils’ mathematical understanding. Teachers need encouragement to invest in such approaches to teaching.” [emphasis added]
And the ‘Recommendations’ (p. 8) included:
“Schools should […]
•enhance schemes of work to include guidance on teaching approaches and activities that promote pupils’ understanding and build on their prior learning.”
Pages 19–25 of the 2008 report provide useful additional details: Figure 4 on p. 19, and Figure 5 on p. 24 summarise the observed weaknesses in secondary schools, and the surrounding paragraphs make clear suggestions as to what needs attention.
The 2012 report Made to measure echoes, and reinforces the concerns expressed in the 2008 report:
p. 9:
“Teaching was strongest in the Early Years Foundation Stage and upper Key Stage 2 and markedly weakest in Key Stage 3.” [emphasis added]
p. 18:
“Learning and progress […] were least effective in Key Stage 3, where only 38% of lessons were good or better and 12% were inadequate” [emphasis added]
p. 19:
“[…] Quick-fix approaches were particularly popular. Aggressive intervention programmes, regular practice of examination-style questions and extra provision, such as revision sessions and subscription to revision websites, allowed pupils to perform better in examinations than their progress in lessons alone might suggest.
These tactics account for the rise in attainment at GCSE; this is not matched by better teaching, learning and progress in lessons, or by pupils’ deeper understanding of mathematics. In almost every mathematics inspection, inspectors recommended improvements in teaching or curriculum planning, in most cases linked to improving pupils’ understanding of mathematics or their ability to use and apply mathematics.
[…] It remains a concern that secondary pupils seemed so readily to accept the view that learning mathematics is important but dull.” [emphasis added]
The analysis in this book may be seen as an attempt to help schools respond to one of the main ‘Recommendations’ in the 2012 report (p. 10):
“Schools should:
•tackle in-school inconsistency of teaching, making more of it good or outstanding, so that every pupil receives a good mathematics education
•increase the emphasis on problem solving across the mathematics curriculum
•develop the expertise of staff:
–in choosing teaching approaches and activities that foster pupils’ deeper understanding, including through the use of practical resources, visual images and information and communication technology
–in checking and probing pupils’ understanding during the lesson, and adapting teaching accordingly
–in understanding the progression in strands of mathematics over time, so that they know the key knowledge and skills that underpin each stage of learning
–ensuring policies and guidance are backed up by professional development for staff to aid consistency and effective implementation.”
The seriousness of the current situation summarised in these two reports, and the weaknesses in the published Key Stage 3 programme of study may explain why these notes and guidance grew into an ‘extended essay’, rather than being effectively distilled into a punchy DIY manual. Despite (or perhaps because of) this, we hope that all teachers (from those just beginning their careers, or those aiming to take responsibility as Head of Department, to the most experienced practitioners), and those who train teachers will find that what follows provides food for thought, and that schools will find what is presented here helpful in reviewing their current provision in lower secondary school.
1National curriculum in England: mathematics programmes of study, https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study; https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf
2http://webarchive.nationalarchives.gov.uk/20141124154759/http://www.ofsted.gov.uk/resources/mathematics-understanding-score
3https://www.gov.uk/government/publications/mathematics-made-to-measure
4http://webarchive.nationalarchives.gov.uk/20141124154759/http://www.ofsted.gov.uk/resources/mathematics-understanding-score-improving-practice-mathematics-secondary