The Mathematical Association

Response to the Post-14 Mathematics Inquiry

 

Terms of Reference

To make recommendations on changes to the curriculum, qualifications and pedagogy for those aged 14 and over in schools, colleges and higher education institutions to enable those students to acquire the mathematical knowledge and skills necessary to meet the requirements of employers and of further and higher education.

DfES Press Release: 25th November 2002

 

Summary of Key Points

The Mathematical Association strongly supports the aims of the Inquiry in seeking to raise standards of achievement in mathematics so that students are better prepared for employment and for further and higher education. However, we are concerned that changes to 'curriculum, qualifications and pedagogy' cannot on their own provide solutions to the very real problems facing mathematical education. Such changes can often be part of the problem and can distract attention from facing the underlying causes of the difficulties.

We have identified three key issues which are closely interrelated. These are summarised below and expanded in the main part of this response.

1. The recruitment and retention of mathematics teachers is a matter for very serious concern. Radical and sustained measures over many years are required following an analysis of the underlying causes of this very real crisis facing mathematics.

No matter what form proposals for reforming mathematical education take, effective implementation will only be possible if there is a substantial increase in the number of mathematics teachers with sufficient understanding of the subject and the enthusiasm and skill to teach it well. Without progress on this issue, real improvements in standards simply cannot be achieved.

2. The quality of teaching and learning of mathematics needs to be greatly improved. The content of the mathematics curriculum is broadly acceptable and does not require substantial change. The problem is not what is taught, but how it is taught, or, much more to the point, what is learnt.

A period of curricular stability is needed to allow real progress in improving the quality of teaching and learning by reducing the pressures on teachers so that they have time to think and plan and engage in sustained professional development. This would help to create the conditions which are necessary for success in easing recruitment and retention problems.

3. The current system of assessment and all the accompanying targets and league tables are having substantial ill effects on the teaching and learning of mathematics. A radical shift away from the current dominance of tests, examinations, targets and league tables is essential if standards in mathematics are to be improved.

The excessive emphasis on tests and examinations has the very serious effect of skewing all classroom activity towards the short term goal of maximising test results. Important aspects of mathematical learning that are hard to assess become optional in the eyes of both students and teachers. A substantial reduction in statutory testing is necessary if the quality of teaching and learning is to be improved and would contribute greatly to creating an environment in which mathematics teaching is a more congenial task.

The measure of our success is the extent to which mathematics lessons that stimulate interest and boost confidence become the common experience of all students, so that they are able to acquire knowledge and skills with understanding and can apply what they have learnt to a wide variety of challenging situations.

 

Introduction

. Mathematics equips pupils with a uniquely powerful set of tools to understand and change the world;

. Mathematics is important in everyday life;

. Mathematics is a creative discipline. It can stimulate moments of pleasure and wonder when a pupil solves a problem for the first time, discovers a more elegant solution to the problem, or suddenly sees hidden connections.

Adapted from the National Curriculum for Mathematics [1], p.14

The Mathematical Association agrees strongly with the statement about mathematics that appears in the National Curriculum, but we are concerned that the vision is not shared widely by the public in general, in part because it is not the view of the subject that students commonly acquire from school. The quality of mathematical education and the numbers choosing to study it post-16 are a cause for constant concern: both do need to be raised substantially, but we need to be clear what it is we are seeking to achieve. We are less concerned to change curriculum content than with the difficulty of ensuring that students learn mathematics meaningfully, so that they value it and can use the power it confers, not least to support the country's economic and scientific growth.

Mathematics teaching at all levels should communicate interest and enthusiasm and enable students to acquire knowledge, fluency, deep understanding and an ability to think independently. Students should regularly experience lessons which stimulate their interest and boost their confidence by enabling them to acquire knowledge and skills with understanding, alongside an ability to apply what they have learnt to a wide variety of challenging problems. The measure of our success is the extent to which this becomes the common experience of all students, and will be reflected by the extent to which students opt to study mathematics further.

The key to these issues is an abundance of well qualified and committed mathematics teachers who have a deep understanding of elementary mathematics and the personal and pedagogical skills to communicate effectively. The barriers to progress are closely linked to the factors which make it difficult to recruit and retain good mathematics teachers. These in turn are linked to general issues such as workload and pupil behaviour and more specific factors for mathematics related to curriculum, pedagogy and assessment. There are no simple or quick answers to the serious difficulties we face, but we are certain that progress will only be made if there is a very strong focus on deep underlying problems rather than on making superficial changes to structure and content. A pragmatic approach is needed which is clear about long term aims and realistic about ways of working successfully towards them.

 

Key Issue 1: Recruitment and retention of teachers.

The recruitment and retention of mathematics teachers is a matter for very serious concern. Radical and sustained measures over many years are required following an analysis of the underlying causes of this very real crisis facing mathematics.

1.1 The quality of mathematical education will not improve unless steps are taken to make teaching a profession that is congenial to people with the right blend of mathematical and communication skills. The current situation is that most schools are finding it very difficult to recruit and retain suitably qualified mathematics teachers and, as a consequence, many students are taught by non-specialists who lack deep understanding of elementary mathematics and how to teach it meaningfully.

1.2 The recruitment and retention of competent mathematics teachers is a very long standing problem and has been a matter of concern throughout the post-war period, with a rapid deterioration in the last 20 years and evidence to suggest that it will get much worse. There is a continuing failure to recruit sufficient people into initial teacher training in spite of sustained campaigns and greatly improved financial incentives. The effect of Curriculum 2000 on mathematics has added to the recruitment difficulties for mathematics and allied subjects in Higher Education and that will almost inevitably reduce the flow of well qualified people into teaching in the future. Furthermore, the age profile of the teaching profession is such that retirements in the coming years will be disproportionately high, alongside a continual loss to the profession of teachers of all ages.

1.3 Mathematics teachers will continue leaving the profession until there is a substantial improvement in their conditions of service. Whilst pay levels are a very significant factor and should be boosted substantially, other factors are often more important. The key issue is poor job satisfaction caused by workload, incessant change and pupil behaviour are key factors. Incessant change means that there is rarely time to establish what works well before the goal posts are moved. Many thoughtful and creative teachers, both young future curriculum leaders and those with a wealth of experience, become disillusioned by constant superficial changes and interference which devour time and undermine professional judgement. Recent reforms have tended to reduce the freedom to innovate and give excessive emphasis to tests and targets to the detriment of stimulating interest, promoting positive attitudes and encouraging effective learning. These factors apply to teachers of all subjects, but mathematics is in a particularly vulnerable position because of the cumulative effect of the shortage of good teachers over a long period and because, like English and Science, it is continually in the forefront of public attention and subject to constant criticism and change.

1.4 It is vitally important that the quality of entrants should not be compromised in the search for recruits to initial teacher training or in making appointments generally. Schools are commonly faced with an impossible situation where the choice is between an inadequate mathematics teacher or no teacher at all. Schools need advice and support as to how best to cope with this situation without increasing the burden on other hard pressed teachers.

1.5 Strenuous efforts must be made to improve the recruitment to Initial Teacher Training of graduates in mathematics and allied subjects, but even with such efforts the shortfall will remain for a long time, so it is important to explore alternative sources of recruitment. One such alternative is to attract back into the profession more of those who have left, including those who have moved to allied work within the education system. Another is to greatly increase the opportunities for people with degrees in non-mathematical subjects (including existing teachers of other subjects), who have an A level or equivalent in the subject, to train as mathematics teachers. A number of universities provide two year conversion courses for such people, but these have never been widely publicised at a national level and students are not properly funded for the first year of these courses. A relatively modest outlay on publicity and funding in this area could make a very significant difference.

1.6 The quality of many current mathematics teachers needs to be enhanced by providing a coherent and substantial programme of long term professional development. The large number of non-specialists who are currently teaching mathematics need sustained professional development to enhance their subject and pedagogical knowledge. Moreover, all mathematics teachers need frequent opportunities for professional development to update their knowledge, to share ideas with colleagues and to renew their enthusiasm. We fully endorse the recommendations of the ACME report on Continuing Professional Development [2], but would add that the major barrier, besides cost, to providing professional development for mathematics teachers is that of making the time available for them to attend courses of sufficient length to have real long term benefits.

 

Key Issue 2: Curriculum and Pedagogy

The quality of teaching and learning of mathematics needs to be greatly improved. The content of the mathematics curriculum is broadly acceptable and does not, as a priority, require substantial change. The major problem is not what is taught, but how it is taught, or, much more to the point, what is learnt.

2.1 In the longer term, there is scope for radical change to the curriculum and qualifications framework, but the priority for at least the next 10 years should be a period of relative stability to content and structures during which steps can be taken to deal with some of the deep underlying problems facing mathematics education. Some short term changes may be needed, but any such should be introduced in a considered way so that they do not add to teachers' workload and are not subject to subsequent major amendment. Any longer term suggestion that mathematics should become compulsory for all students beyond 16, although possibly attractive, should be weighed carefully as a priority against the need to improve substantially the quality of learning up to the age of 16. Quality of learning is much more central to the contribution mathematics can make to this country's future.

2.2 Good pedagogy is the key to raising standards and requires expert teachers together with a substantial shift in emphasis from rote learning to meaningful learning and from summative assessment to formative assessment (see the MA's Position Paper on Assessment in the Appendix).

2.3 Mathematics teaching should communicate interest and enthusiasm and enable students to acquire knowledge, fluency, deep understanding and an ability to think independently. Greater challenge is needed for all pupils, but particularly for able pupils, within the existing curriculum by providing 'enrichment for greater depth' rather than acceleration, which invariably encourages superficial learning (see 'Acceleration or Enrichment?' [3]). If students achieved their GCSE grades with a deeper understanding than at present, and were accustomed to being challenged to think for themselves and were expected to produce clear written mathematical arguments, then it is likely that the demands of AS and A level would be less daunting and a realistic course for many more students. Teachers need effective support in providing the breadth and depth appropriate to the most able (see also 3.6 below).

2.4 Where the KS3 National Strategy has been implemented in a non-prescriptive way it has done much to improve pedagogy, especially among non-specialist teachers, but the use of materials produced with the specific aim of boosting test performance needs to be reviewed. The effectiveness of the Strategy is unfortunately being undermined by the shortage of good teachers and the constant pressures, created by tests and league tables, to get what might seem to be good results, when these actually reflect inadequate levels of understanding amongst pupils. Improvements in pedagogy are needed at KS4 and beyond, but careful research is needed to determine the best form for any new initiatives so that expert teachers have the freedom to make their own decisions about how best to teach, whilst inexperienced and poorly qualified teachers are given suitable advice and support.

2.5 The current content of the National Curriculum is broadly acceptable, except that we are concerned about the emphasis that in practice is accorded to Data Handling, partly through GCSE coursework requirements and partly through the perception that it is simpler than other topics. It is disturbing to note that it is possible to obtain grade A on a GCSE paper without achieving much success on the algebra questions (40% of the marks are awarded for number and algebra and in recent years an A grade has been awarded for a marks of not much more than 50%). We have reservations about the type of task, the marking schemes and the way students are often prepared for coursework tasks and suggest that the role of coursework in mathematics should be carefully reviewed.

2.6 A two tier GCSE, where students enter for a pair of consecutive papers targeted at particular grades, is currently being piloted. This is a potentially welcome development, because it should lead to students who achieve a Grade C doing so through a common paper and to a greater proportion of students being entered for the tier which offers the higher grades.

2.7 The new mathematics Criteria for AS and A level are broadly acceptable, but provision for post-16 students not taking AS or A level mathematics is unsatisfactory. There are three categories of student where more appropriate provision is desirable:

. those who need mathematics as a backup to their study of other subjects at A level and beyond;

. those who need to retake GCSE to achieve at least a grade C, who often face a repetition of earlier failure and would benefit from different format and approaches whilst retaining the currency of the title GCSE (the material produced by LSDA on Algebra [4] is an excellent development here);

. those for whom GCSE grade C and FSMQs are not a realistic or appropriate target (the recent withdrawal of City and Guilds Numeracy has been a serious loss for this group).

2.8 AS and A level Further Mathematics (particularly the former in the short term) should be an available option for all students who would benefit. We would estimate that at least three times the current numbers being entered could benefit, but there is a difficulty in making conventional provision because numbers are low in schools with small sixth forms. Distance learning provision along the lines developed by MEI [5], and other possible models of delivery, should be encouraged. Funding should permit all students to benefit from such developments. Establishment of Further Mathematics modules as FSMUs would enhance participation. Greater encouragement, and recognition through funding, should be given to the Advanced Extension Award (AEA) and Sixth Term Examination Papers (STEP) so that the challenge they offer is available to all able students.

2.9 ICT can have a valuable role in improving the quality of learning in mathematics classrooms, but its successful use is totally dependent on expert teachers who have the necessary skills, as well as on suitable provision in all classrooms of high quality hardware, software and technical support. The use of improved ICT facilities cannot overcome the deficiencies caused by teacher shortage. The recent report for the TTA, produced in conjunction with The Mathematical Association by Adrian Oldknow [6] offers wide ranging advice and makes the potential clear.

 

Key Issue 3: Assessment: tests and targets.

The current system of assessment and all the accompanying targets and league tables are having substantial ill effects on the teaching and learning of mathematics. A radical shift away from the current dominance of tests, examinations, targets and league tables is essential if standards in mathematics are to be improved.

 

3.1 The purposes of a national assessment system should be to:

. certify the attainment of individual students;

. monitor national and school performance;

. improve the quality of teaching and learning.

We have serious doubts about the effectiveness of present assessment arrangements at all levels in achieving these purposes. Assessment takes two distinct forms - summative and formative - both of which are necessary, but it is crucially important that the right balance between them is achieved.

3.2 The frequency and consequent dominance of summative assessment in the form of tests and examinations has the very serious effect of skewing all classroom activity towards the short term goal of maximising test results. Important aspects of mathematical learning that are hard to assess become optional in the eyes of students and teachers. Emphasis on short and structured questions means that too many students fail to acquire the skill putting together clear mathematical arguments.

3.3 This dominance is exacerbated massively by targets set for LEAs, schools, teachers and pupils based on test results, including the impact of threshold assessments of teachers' performance. Monitoring national and school performance requires a very much lighter touch and it should be done much less publicly. Performance would be monitored much more effectively and acceptably if it was done by an independent body using sampling techniques, as used to be the case with the Assessment of Performance Unit (APU), and combined with more qualitative evidence obtained through a sensitive system of inspection whose priority was to offer immediate and specific advice for improvement.

3.4 National testing in some form at the end of key stages 2 and 3 (not key stage 1), together with GCSE, A level and other post-16 qualifications, can have a useful role, which might possibly be enhanced by making it non-statutory. However, additional national testing, mandatory or optional, beyond these 4 occasions in 13 years is excessive and increasingly counter-productive. It makes no educational sense for students to take 'high stakes' examinations in three consecutive years in the 14 to 19 phase (and this can become four consecutive years if students are accelerated to do GCSE in year 10). The expectation that students will take AS in four subjects at the end of year 12 is the source of the problems created by Curriculum 2000. The increasing demands to retake modules during an AS or A level course is creating an additional distraction and pressure: the regulations need to be reviewed and simplified.

3.5 The repercussions of Curriculum 2000 for mathematics have been very severe. The following table shows data based on the annual summaries of A level results which are now published by the Joint Council for General Qualifications. The second column gives the total number of A level subject entries each year. The third column shows the number of entries for A level mathematics and the fourth column shows those figures as a proportion of the total number of candidates. The last two columns show comparable data for English. After a significant decline in the early 1990s, which reflected the steep decline in the 18 year old population at that time, the proportions for mathematics had been fairly stable since 1993, although the number of students in full time education post-16 has increased substantially. The steep decline in 2002 for mathematics is in marked contrast to the situation with English.

 

Year	Entries	Maths	Maths %	English	English %
1989	661591	84744	12.8	68846	10.4
1990	684117	79747	11.7	74182	10.8
1991	699041	74972	10.7	79187	11.3
1992	731024	72384	9.9	86779	11.9
1993	734081	66340	9.0	89238	12.2
1994	732974	64919	8.9	88214	12.0
1995	725992	62188	8.6	86382	11.9
1996	739163	67442	9.1	86627	11.7
1997	777710	68880	8.9	95223	12.2
1998	794262	70554	8.9	94099	11.8
1999	783692	69945	8.9	90340	11.5
2000	771809	67036	8.7	86428	11.2
2001	748866	66247	8.8	76808	10.3
2002	701380	53940	7.7	72196	10.3

 

It will take a long time to repair the damage that has been done. QCA's revised criteria for AS and A level [7] give an opportunity to promote mathematics as more accessible than it has been, but the role of AS as a step on the way to A level, rather than as an alternative goal, needs careful review.

3.6 Ways should be found to ensure that students develop problem solving skills and deep understanding as an alternative to taking GCSE early. If formal assessment of this is thought to be necessary, one possibility is a form of 'double award' GCSE with an additional paper or papers like the Advanced Extension Award at A level, with more challenging questions and no additional content. Students in schools without the resources to run separate classes are not thereby disadvantaged. The continuity problems for the next stage created by acceleration would be avoided, if this were to be taken alongside normal GCSE papers.

3.7 The system as a whole, and mathematics in particular, needs far less emphasis on summative assessment - tests and associated targets - and far more emphasis on formative assessment where informal everyday classroom strategies are designed to improve the learning of individual students. Excessive formal testing distorts curricular aims and is demotivating and demoralising for both students and teachers.

 

 

Conclusion

The Mathematical Association is passionately concerned to raise the levels of achievement and commitment of all students by improving the quality of learning that takes place in every classroom. This can only be achieved if we have a committed workforce of expert teachers who have the time and resources to devote most of their energies to preparing for and working with students. The solutions to the very serious problems facing mathematical education involve creating a climate in which teachers can teach and students can learn effectively because they see mathematics as a worthwhile endeavour, rather than as something to be endured to get through the next test.

 

References

[1] DfEE/QCA (1999) Mathematics: The National Curriculum for England DfEE/QCA

[2] ACME (2002) Continuing Professional Development for Teachers of Mathematics The Royal Society and the Joint Mathematical Council

[3] UK Mathematics Foundation (2000) Acceleration or Enrichment? Serving the Needs of the top 10% in School Mathematics.

[4] Swan,M. and Green,M. (2002) Learning Mathematics through Discussion and Reflection: Algebra at GCSE (CD-ROM, video and booklet) Learning and Skills Development Agency (LSDA)

[5] MEI Enabling Access to Further Mathematics. Details will be found at www.mei-distance.com and a longer article describing the project is at ltsn.mathstore.ac.uk/newsletter/nov2002/pdf/furthermaths.pdf .

[6] Oldknow, A. et al ICT and Mathematics: a Guide to Teaching and Learning Mathematics 11 -19 To be found under Teachers Teaching with Technology at www.m-a.org.uk

[7] QCA (2002) GCE Advanced Subsidiary (AS) and Advanced (A) level Specifications: Subject Criteria for Mathematics To be found at .

 

24.02.03.

The Mathematical Association

259 London Road

Leicester

LE2 3BE

Telephone: O116 221 0013

Fax: 0116 212 2835

e-mail: office@m-a.org.uk

Website: www.m-a.org.uk

 

Appendix

The Mathematical Association

A Position Paper on Assessment

The purposes of a national assessment system should be to:

· certify the attainment of individual students;

· monitor national and school performance;

· improve the quality of teaching and learning.

We have serious doubts about the effectiveness of present assessment arrangements at all levels in achieving these purposes. Assessment takes two distinct forms - summative and formative - summarised in the table below. Both forms of assessment are necessary, but it is crucially important that the right balance is achieved between them.

Summative (assessment of learning) To give a summary of current performance, typically through numerical data from the results of tests and examinations. This provides a measure of the attainment of individual students and is one source of evidence in monitoring both school and national performance. Over-emphasis on tests, targets and comparisons with others encourages teaching that leads to superficial learning aimed at avoiding errors, rather than learning from them, and demoralises those who fail. Success with summative assessment is more likely when emphasis is given to enhancing thinking skills and the general classroom focus is on what students can do and how they can improve rather than on the marks, grades and levels that they have attained.

Formative (assessment for learning) To provide diagnostic evidence which teachers can use both to give feedback to students and to inform their future planning, and which students can use to help them learn more effectively by giving them ways to improve their knowledge, fluency and understanding. Such evidence is obtained in a wide variety of largely informal ways, involving oral and written classroom activities and tasks rather than exclusively through tests and examinations, although careful analysis of responses to these can be used formatively. Formative assessment can improve learning by focusing on misconceptions and the development of understanding and independent thinking.

Whilst there is some overlap between the two categories, they are essentially distinct. Current systems and many traditional practices, particularly in mathematics, give a disproportionate emphasis to summative assessment with little attention being given to the extensive possibilities of formative assessment for raising standards. Extensive work has been done by Paul Black and Dylan Wiliam of King's College, London, who have gathered evidence concerning the effectiveness of formative assessment. One of the most significant pieces of evidence they offer challenges the effectiveness of much current assessment practice both at national and classroom level. Evidence suggests that students take little notice of teachers' comments when they are accompanied by marks or grades. When marks are abandoned attitudes change and the focus is on how to improve learning rather than on how to get better marks.

Our concern about current assessment practices is centred on the unfortunate effects of the 'teaching to the test' which inevitably occurs when too much emphasis is given to marks or grades. Mathematics education suffers particularly because, with English and science, success in mathematics is rightly regarded as a critical component of education for the individual and for society at large. Current national testing policies and the accompanying 'high stakes' targets at all levels are skewing all classroom activity in the core subjects towards the short term goal of maximising test results. This is resulting in a serious narrowing of curricular aims because formal tests cannot assess all aspects of learning. Confining attention to what is readily testable has serious effects on the attitudes and morale of both teachers and students. The long term goals of mathematical education at all levels require the development of a knowledge, fluency and understanding which enables students to use and apply mathematical ideas with confidence and enjoyment.

Society does have to measure the performance of the education system and provide certification to indicate the attainment of individual students, but this should be done in ways which ensure that the long term goals of education are not compromised. To this end we would suggest the following:

monitoring national and school performance requires a very much lighter touch and should be done much less publicly;

national performance would be monitored much more effectively, and acceptably, if done by an independent body using sampling techniques, as used to be the case with the Assessment of Performance Unit (APU), together with using more qualitative evidence obtained through a sensitive system of inspection whose priority was to offer immediate and specific advice for improvement;

targets for schools, LEAs and individual teachers based on proportions of students achieving particular levels or grades, and the related league tables, should be abandoned: schools and teachers need constant encouraging advice on how to improve the quality of students' learning rather than exhortation and pressure to maximise test results;

the National Numeracy Strategy and the Mathematics Strand of the Key Stage 3 National Strategy are having many beneficial effects on achievement and attitudes which are in danger of being jeopardised by the pressure to give undue emphasis to preparation for tests;

the system requires a substantial change of emphasis from summative to formative assessment and from a focus on easily testable skills to embracing much wider and more long term goals;

a change in national culture is needed so that education is valued much more for its intrinsic benefits to the personal development of the individual as well as for the benefits an educated workforce confers on society.

National testing at the end of key stages 2 and 3, GCSE and A level together with other post-16 qualifications, have a useful and acceptable place, subject to the provisos above, but additional national testing, mandatory or optional, beyond these 4 occasions in 13 years is excessive and counter-productive. It is far too early to give formal tests to students in Key Stage 1 - as a consequence some students are inevitably labeled as 'failures' at the age of 7. At Key Stage 2, the principle of national tests is more acceptable, but steps must be taken to ensure that the overall balance of the curriculum is not distorted by excessive time spent on narrow preparation for tests in the core subjects and that pressures do not result in students being accelerated to achieve higher levels to the detriment of deeper understanding and breadth of knowledge. At Key Stage 3 the national tests in mathematics were initially widely accepted, and were possibly contributing to raising standards and monitoring effectiveness across the system, but the advent of targets has led to a much narrower concentration on maximising results.

GCSE is widely accepted as an acceptable form of summative assessment at the end of year 11. In mathematics, attainment of a grade C is an important basic qualification which should be retained and not compromised by provision of alternatives whose credibility is less likely to be widely accepted, although it may take a different 'mature' form at the post-16 stage. We strongly support moves to a two tier GCSE so that grade C will be achieved through a common route by all students. We do not support pressures to encourage early entry for GCSE in mathematics for the same reasons given above for Key Stage 2. We would prefer to see some sort of double or extended award for very able students to be taken at the same time as the basic GCSE examination.

A level does have some shortcomings but, like GCSE, it is widely recognized as a measure of achievement and as such should be retained for the foreseeable future. AS level is much more problematic in the light of the serious and ongoing shortcomings of Curriculum 2000, which have been disastrous for mathematics. Whilst we fully support the principle of broadening the post-16 curriculum, it makes no sense to have high profile national examinations in three consecutive years. It is clear that this is resulting in excessive pressures on students and excessive time devoted to narrow preparation for examinations with a consequent lack of enthusiasm for the potential opportunities to develop broader interests and deeper understanding.

To summarise, the system as a whole, and mathematics in particular, needs far less emphasis on summative assessment - tests and associated targets - and far more emphasis on formative assessment where informal everyday classroom strategies are designed to improve the learning of individual students. Excessive formal testing distorts curricular aims and is demotivating and demoralising for both students and teachers.

7.02.02.