Ofsted subject reports 2003/04 Mathematics in secondary schools HMI 2326 February 2005 Page 1
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Standards in mathematics In the 2004 national test results there was a two percentage point rise, to 73%, in the proportion of pupils reaching the expected level (level 5) or above. This builds on the increase in the previous year and represents an increase of six percentage points since 2002. Girls outperformed boys at this level. There was also a noticeable rise of three percentage points, to 52%, in the proportion of pupils achieving at least Level 6, with no difference between boys and girls performance at this level. In GCSE mathematics, 50% of pupils gained grades A* C, with girls performing slightly better than boys; 96.3% of pupils gained A G grades. Although the A* C figure represents a two percentage points increase on the previous year, it remains the case that too many of the pupils who achieve five or more higher grades in GCSE courses do not attain at this level in mathematics. At GCE A level, 54.4% of candidates achieved grades A or B and 95.2% gained a pass grade, in each case a small increase on the previous year. Although the number of entries has increased slightly, this has done little to arrest the longer term decline in mathematics at this level. As at GCSE level, girls generally outperform boys, but with only 9,000 girls taking the examination compared with almost 15,000 boys. The results of AS examinations show a similarly improved picture in mathematics with 38.8% of candidates gaining higher grades and an overall pass rate of 77.3%. When compared with 2003, these represent an increase of approximately three percentage points, although the success rate in AS mathematics remains the lowest of all subjects. Overview The teaching and learning of mathematics continues to improve in Key Stage 3, with the Key Stage 3 National Strategy having a positive impact. For 11 to 14 year olds, the quality of teaching and learning in mathematics is good or better in seven schools in ten, slightly above the average for all subjects, as is the achievement of pupils. However, provision for 14 to 19 year olds remains beset with problems. In Key Stage 4 and on post-16 courses the quality of teaching and learning in mathematics is below the average for all other subjects. The dip is particularly pronounced for 14 to 16 year olds where it is good or better in only half of schools. This is likely to be a major contributory factor to the low take-up of post-16 mathematics courses, which continues to cause significant concern for employers and higher education (HE) providers. Page 3
Strengths in secondary mathematics Schools that have developed effective practice in mathematics demonstrate some or all of the following characteristics. The department, despite the current national recruitment and retention difficulties in mathematics, is fully staffed and well managed. Senior managers allocate time for mathematics staff to plan collaboratively and share good practice effectively; hence teachers are provided with a good range of resources and teaching ideas. The head of department monitors the quality of teaching and learning and has a positive influence on practice across all mathematics classes. Teachers have good subject knowledge and plan collaboratively with subject colleagues to ensure that mathematics expertise is shared for the benefit of all pupils. The department s schemes of work are of high quality, incorporating the best practice across all topics, including using and applying mathematics, and making this available to all teachers in the department. Teachers use nationally produced mathematics support materials and teaching resources critically, ensuring that they match both the prior attainment of their group and the current learning intentions. Classroom tasks challenge pupils to think and reason mathematically. Teachers expect pupils to develop a secure understanding of key mathematical concepts; pupils progress in these areas is monitored thoroughly and is used to help them understand what they are aiming towards and how well they are achieving. Pupils attitudes to mathematics are good and pupils take their share of responsibility for achieving to their full potential in the subject. Homework is used to good effect to extend and enrich mathematical understanding. Page 4
Areas for development in secondary mathematics Improving provision in mathematics for 14 to 19 year olds The report of the Inquiry into Post-14 Mathematics Education, Making mathematics count, and the government s subsequent response have put the spotlight on several key problems. 1 The report highlighted many young people s perception of mathematics as being boring and irrelevant and too difficult, compared with other subjects. It made it clear that some of the responsibility for this lies with a mathematics curriculum and qualifications framework that fail to meet the needs of many learners or the expectations of employers and HE providers. While inspection evidence supports this view to some degree, it also highlights significant failings associated with classroom practice. In particular, many lessons employ an approach to teaching that is based on imitation: the teacher demonstrates a technique and then the pupils practise it by completing exercises from a textbook or worksheet, usually without contexts that are meaningful, or without any opportunity to appreciate the use of these techniques to solve problems. Last year s Ofsted report on secondary mathematics raised this and other significant concerns regarding post- 14 mathematics. 2 Analysis of this year s evidence underlines the factors at work. Teaching and learning styles in post-14 mathematics lessons are often too limited to motivate and engage pupils compared with approaches in other areas of learning. While many high attaining pupils manage to maintain some interest in order to secure high grades, many others lack any real motivation to learn and have little confidence in their ability to secure a grade at GCSE or A level that they judge to be worthwhile. Inflexible setting arrangements in Key Stage 4 lead many pupils to believe that their GCSE goals in mathematics are limited in nature and ambition. Many become disaffected and choose to channel their energy and enthusiasm into other subject areas in which they believe they will achieve better results. For pupils who achieve lower than average levels at the end of Key Stage 3, subsequent provision or support is not sufficiently tailored to their needs; this then reinforces their fears and fails to rebuild their self belief or self esteem. Too many teachers taking middle and lower attaining groups have significant gaps in their subject knowledge, so they tend not to probe pupils levels of understanding or invite questions from the class in areas where they are not secure. Instead, they confine pupils to practising routine exercises, so that pupils have limited engagement with their work. Pupils who fail to achieve a GCSE grade C or above at the end of Key Stage 4 rarely have the opportunity to follow an alternative, appropriate course beyond this point; many therefore follow a re-sit GCSE course with the majority still failing to achieve the result they desire with subsequent attempts. Page 5
Changes to modular arrangements for AS and A2 courses since the introduction of Curriculum 2000 have meant that many of the strongest post-16 mathematicians have not taken further mathematics. To raise achievement in Key Stage 4 and improve take-up of mathematics at post- 16, schools need to address these issues actively. Ensuring secure understanding for all pupils in Key Stage 3 Since the introduction of the Key Stage 3 National Strategy much whole class teaching is of good quality, with shared objectives, good pace and use of stimulating resources. It has now reached a point where the quality of teaching and learning is good or better in seven lessons in ten. However, this quality is not equally distributed across teaching groups and some pupils are not well served. In particular, teaching is stronger in higher ability classes, where very few unsatisfactory lessons were observed. In contrast, for middle and lower ability classes, teaching is good in fewer lessons and the quality of learning is more often unsatisfactory. This is often related to limitations in teachers subject knowledge and understanding of common misconceptions and how best to challenge and correct these with lower attaining pupils. Where teaching is at its strongest, teachers have a well developed understanding of the key mathematical concepts for a particular teaching group and appreciate how these can be linked together to enhance learning. They are then able to make appropriate decisions about relative emphasis and priority for the different aspects of the curriculum for each group. This ensures that their pupils make good progress over time, developing secure understanding of the mathematical ideas and confidence in their ability to succeed in mathematics. In many of the less effective lessons, the teaching moves on before pupils have understood the concept; the pressure to cover new content as quickly as possible results in shallow coverage and lack of depth in learning. Many teachers, especially non specialist teachers, therefore need to develop a deeper knowledge and understanding of key mathematical concepts and how to improve their pupils progression through these key ideas. Developing pupils skills in using and applying their mathematics Although using and applying mathematics has been a significant part of National Curriculum mathematics since its introduction, schools continue to find it difficult to equip all pupils with these skills. Guidance from the Key Stage 3 National Strategy has gone some way towards providing ideas and teaching strategies related to strengthening pupils reasoning skills, but this is only part of the issue. The capacity to reason, justify, explain and prove is central to being successful in mathematics. However, these qualities need to be explicitly developed and nurtured over time in just the same way as calculation skills or techniques for solving equations. Many teachers do not have a sufficiently secure understanding in the progression in these skills from one National Curriculum level to the next. In many cases, pupils are not given the opportunity to develop these skills over time and, by Key Stage 4, they are only addressed in relation to GCSE coursework assessment rather than as an integral part of all mathematics learning. Page 6
In schools where pupils progress in these skills is not good enough, whole class teaching often predominates at the expense of time for pupils to talk through and explore ideas and strategies or engage in practical work to reinforce or extend their learning. Often good mathematical problems or challenges are left until last rather than being part of the main learning. Many pupils do not therefore reach them and have little opportunity to solve problems or apply their mathematics in new contexts and develop the associated skills. In contrast, some schools have successfully incorporated these skills into their lessons for all classes. The key to their success lies in the quality of the schemes of work and planning of the mathematics curriculum. Features usually include: a clear commitment to developing pupils thinking, reasoning, problem solving and explanation skills schemes of work that incorporate these skills and encourage the development of pupils mathematical writing, particularly explaining reasoning in writing targeted professional development within the department to support these commitments a recognition that pupils will need time to think and reason in lessons an expectation that paired and group discussion and collaborative activities are needed in order to provide opportunities for thinking and reasoning explicit planning of using and applying mathematics into units of work in order to cover all related knowledge and skills explicit teaching of the skills needed to use and apply mathematics, and regular monitoring and assessment of pupils progress in these skills judicious use of opportunities in other subjects to develop and extend such skills. More schools need to develop high quality schemes of work and curriculum plans incorporating these skills, with teaching ideas and resources made explicit. Maximising the use and impact of assessment Last year s report drew attention to the discrepancy between assessment of learning and assessment for learning in mathematics. 3 This year s evidence suggests little progress. Despite making good use of Key Stage 2 assessment data to group pupils, and having comprehensive systems for assessing, tracking and recording pupils progress from one year to the next, several important weaknesses remain. Page 7
A major issue lies in the links between the teaching and what pupils already know and understand. In too many of the least effective lessons, inspectors observe a learning gap between the objectives of the lesson and the present levels of understanding of a significant proportion of the class. This is often a consequence of the teacher failing to establish pupils current levels of knowledge and understanding at the beginning of a topic and therefore failing to build on this appropriately. Though this often results in low expectations, on occasions, the objectives and expectations of the teacher are unhelpfully high. In both scenarios, pupils inevitably make insufficient progress in these lessons. Second, pupils are insufficiently involved in assessment of their own performance. As a consequence, they are unaware of their areas of strength and weakness and do not know what they need to do to improve and how they could go about this. A number of successful schools have addressed this problem by adopting techniques for self assessment and peer assessment, thus enabling pupils to develop a better appreciation of the standards required and what they need to do to improve their performance. Third, the quality of marking and feedback to pupils on their work is often inadequate. In the least effective practice, teachers do not probe understanding sufficiently to find the root of the misconception and are therefore unaware of what pupils need to do to address their misunderstandings. They subsequently provide bland comments on pupils work rather than providing a focus for improvement. At its best, marking and oral feedback concentrates on the key mathematical idea, provides targets for improvement that are achievable and presents these in a supportive way intended to maintain pupils self esteem. Fourth, too few schools make regular assessments of pupils attainment in attainment target 1 (AT1 using and applying mathematics) despite the requirement to make a teacher assessment judgement for mathematics based on pupils levels on each attainment target, and then aggregated, at the end of Key Stage 3. Inspection and research evidence suggests that using good strategies for formative assessment can lead to significant improvements in the performance of pupils. As the QCA publication Using assessment to raise achievement in mathematics at Key Stages 1, 2 and 3 makes clear, this is most effective when schools: 4 involve pupils in understanding the standards required and assessing their own performance in relation to these standards set learning targets for and with pupils to help them achieve these goals use effective questioning techniques to assess and further pupils learning use marking and feedback strategies to inform pupils about their progress and help them understand how to improve. Page 8
These features of effective assessment are similar to those identified by the recent research reported on the Assessment Reform Group website. 5 More mathematics departments need to develop effective strategies in each of these areas. Making effective use of information and communication technology As was the case in 2002/03, the use of information and communication technology (ICT) to support teaching and learning within mathematics remains weak. While there are examples of good practice, these need to be shared more widely to address the significant inconsistencies between schools as well as within mathematics departments. Access to ICT for mathematics classes remains a problem in too many schools and some departments do not have sufficient access to ICT to meet National Curriculum requirements or Key Stage 3 framework recommendations. The HMI report published in June 2004 provides examples of good practice in mathematics and ICT and suggestions for development that many schools need to consider. 6 References 1 Making mathematics count: report of the post-14 mathematics inquiry (www.dfes.gov.uk/mathsinquiry) the Department for Education and Skills response to Professor Adrian Smith s inquiry into post-14 mathematics education, DfES, 2004. 2 Ofsted subject report 2002/03, secondary mathematics (HMI 1978), Ofsted, 2004. 3 Ibid. 4 Using assessment to raise achievement in mathematics, Qualifications and Curriculum Authority, 2003. 5 Assessment Reform Group (www.assessment-reform-group.org.uk). 6 ICT in schools 2004: the impact of government initiatives: secondary mathematics (HMI 2185), Ofsted, 2004. Page 9