Friday, 29 December 2017

A Brief History of Mathematics Education in England


Introduction

The story of the history of mathematics education in England is also the story of a country moving from a largely laissez-faire position to a dictatorial one.  Since the 1858 Newcastle report, mathematics education has changed from a system of great diversity to a highly uniform system controlled from the centre.  It is also the story of a battle: a long and far reaching fight for dominance between a mathematics education that focusses on the procedural and one that focusses on the conceptual.

It seems almost inconceivable today to imagine Cambridge and Oxford universities taking no particular interest in mathematics, but until very recently this was the case.  Schools had existed in England for many hundreds of years, but they were almost exclusively provided by religious bodies and focused largely on readying their students to understand and spread their religion.  It was not until 1540, when Henry VIII dissolved the monasteries and friaries, that the way was paved for different types of schools.  Henry and his son, Edward VI, established new types of schools: Grammar Schools, moving education away from the Church and into the hands of the State and wealthy merchants, trade guilds and philanthropists, who all continued the movement, opening new Grammar Schools for centuries to come.

Yet these schools paid almost no attention to mathematics, instead focusing on a humanist education.  The study of mathematics in England remained one of pupilage, with a network of mathematical practitioners providing tutoring to those who could afford it.

The only two universities of the day, Cambridge and Oxford, placed no pressure on schools to teach mathematics, even during the period of superstar mathematics professors such as Newton.  The application of mathematics played an increasingly important role in farming, industry, society and government, but still the Grammar Schools ignored the subject.

It was only in the seaport towns in the seventeenth century that real change started to be seen.  Driven by their need to fill increasing demand for seaman who could navigate, many of these schools began to introduce mathematics beyond the basic arithmetic that had been taught to date.

But it was not until much later, at the beginning of the nineteenth century, that mathematics would become more widespread.  Many continental countries had been making advances in their education system and became the envy of influential merchants, professionals and the emerging industrialists.  Although considerable pressure was beginning to be put on politicians to act, the government retained its stance of not wanting to interfere with education, preferring to leave schooling to other, established organisations and associations.

In 1811, The Church of England used this position to establish the National Society, an umbrella for the schools in its care, much to the umbrage of the nonconformists, who later set up their own organisation, the British and Foreign School Society.

Pressure mounted for government to play a role, but the memory of the French Revolution was still strong and many in government worried that educating the working classes would lead to them becoming dissatisfied with their lot.  Despite these very vocal concerns, in 1833, government did make a new grant to both the National Society and the British and Foreign School Society to extend their work to the working classes.  There was much protest about this, particularly with regards to value for money – should the grant really be given?

These financial concerns led to the establishment of the school inspectorate in 1839, which would ensure that the grant was being well spent.

By this time, mathematics still had no significant place in schools and remained, at best, simple arithmetic.

Those working-class children who did attend school, stayed in schooling on average 1-2 years.

The 1830s also saw the establishment of the first teacher training colleges, moving the system away from one where the norm had been for pupils to progress to becoming teachers through experience alone.  In 1842, the first teacher training college specifically for women was opened.



The Newcastle Report

The first moves towards a mathematics education becoming commonplace in schools came in 1858 when the government established a public commission into elementary education.  Chaired by the Duke of Newcastle, the report gave the first informed overview of the state of mathematics teaching and learning.  At this time, 1824 public schools operated across England, of which only 69% taught any arithmetic, 0.8% Euclid, 0.8% algebra and 0.6% mechanics.  Interestingly, the picture was arguably worse in the private schools, with 33% teaching arithmetic, 1.2% Euclid, 1.4% algebra and 1.3% mechanics.  In the small proportion of schools where arithmetic was taught, it was completely inadequate in almost all cases.

Newcastle took a bold approach to his response.  Rather than suggesting reform of the system, he instead focused, just like the inspectorate, on value for money.  Newcastle recommended that the grants given to the public schools should take on a new format, with awards based on pupil attainment, what followed truly was a payment by results system.  This required the establishment of national standards, which could be examined, and led to the introduction of the first common curriculum for those schools wishing to receive grant funding.

In mathematics, the first curriculum included only arithmetic, with six standards covering only the four operations and exercises on money, weights and common measures.  In 1871, a new standard, Standard VI – Proportion and Vulgar and Decimal Fractions, was introduced.

In many cases, this was a retrograde step as schools chased the funding and removed other areas of mathematics to concentrate on training students to pass the exams.  The mathematics education for many had become solely concerned with the passing of tests, so much so that the respected school inspector Malcolm Arnold remarked that the system of jumping through hoops must be ‘trying to the intellectual life of the school’.

Even with such a basic and bland mathematics curriculum, by 1873, only 15% of pupils were passing the examinations.  It was widely accepted that the payment by results approach had helped to raise the attainment in the worst schools, but that the cost had been a race-to-the-middle, with excellent schools becoming less innovative and high attaining students being held back.  Teachers in the best schools were angered by the lack of autonomy they now felt and their inability to stretch bright children to true excellence.  After much debate, the system was abolished in 1897.



State Education

Meanwhile, following Malcolm Arnold’s presentation of data on illiteracy in the English army, calls for the State to begin providing its own schools grew.  Compared to the armies of France, with 27% illiteracy and Prussia, with just 2% illiteracy, the English army fell way behind with 57% of all personnel being functionally illiterate. The outcry led to the 1870 Education Act, which created school boards across the country tasked with building schools where denominational schools did not already exist.  Crucially, these new schools were also required to provide education up to the age of 13 and ten years after the act was passed, attendance became compulsory for all children.  At this time, most denominational schools still charged for attendance, but the move towards universal free education for all continued apace until, in 1918, all elementary education was free of charge.

Secondary education, for students wishing to continue studying after the elementary phase had settled into a system of three types by 1800.  The nine great public schools provided education to the upper class; the endowed grammar schools that emerged after Henry VIII were widespread and accessed by the middle classes and some of the working class; and a wider range of private schools for both boys and girls existed across the country for children who had not been able to get into grammar school.

The nine great public schools and the endowed grammar schools largely ignored mathematics, instead pursuing their tradition of a humanist curriculum.  However, Winchester and Eton took the lead and appointed their first mathematics masters in 1834.  But it was not until 1840 that these schools were finally given freedom to provide any curriculum that they wished.  It was by including a large element of Euclid on the curriculum that real change could come about.  Euclid was viewed as classical and therefore justifiable in a humanist curriculum.

As pressure continued to mount to provide a more mathematically literate workforce for the armed forces, upper-middle class parents became frustrated by the provision of the endowed grammar schools and, as a result, founded their own new schools.  These new proprietary schools provided a more modern education and the schools began to compete with the nine great public schools, leading to a commission in 1861, which led to wide ranging changes in the mathematics curriculum in these schools.

Given the importance and standing of the nine public schools, they very quickly established a reputation for excellence in the provision of a more modern mathematics education, with their masters authoring many textbooks and guidance on how to teach.  The impact on the other schools was enormous, with the grammar schools, proprietary schools and other private schools moving to emulate what they were achieving.

At this stage, secondary schools were largely self-organised and run.  Government played little role in determining what should be happening in these schools or how they should be arranged.  But in 1864, the Taunton Commission, led to the reorganisation of the system along three clearly defined lines.  Schools would be established for one of each of the three social classes.

The curriculum was also beginning to be much more clearly defined across the country.  In 1856, Cambridge and Oxford published new examinations for entry to their mathematics courses, which potential students would now have to pass, rather than simply entering a college based on a family recommendation.  These papers, copied by the newer universities too, laid the foundations for a national view of what should be included on the mathematics curriculum in schools.  Euclid, logarithms, algebra to simple equations, sequences, trigonometry, mensuration, and some mechanics, statics and dynamics now appeared in most schools.

In 1871, a group of progressive educators established the Association for the Improvement of Geometrical Teaching (AIGT), the world’s first subject association, as a body to challenge the emphasis on Euclid, which it felt was no longer the most relevant approach to teaching geometry.  The association published its own schemes for schools to follow, but had very little success.  The group continued to campaign and broadened its interests to other areas of mathematics and in 1894, it published its first Mathematics Gazette.  Three years later, the AIGT changed its name to The Mathematical Association (MA).

Another alternative available to schools was a curriculum designed by the Department of Science and Art (DSA), which was established following the 1851 Great Exhibition.  The DSA offered grants to schools to follow a more scientific and technical curriculum.  Many endowed schools, struggling financially, adopted the scheme.  By 1900, its impact was the inclusion on many schools’ mathematics curriculum of topics such as Cartesian and polar coordinates, scalars and vectors and differentiation.

In 1899, an auditor noticed that there had never actually been an Act of Parliament to allow for the funding of secondary schools and put a stop to the grants being paid, causing considerable chaos for some time and a long running legal battle, which the government lost and forced the passing of the 1902 Education Act.  This Act abolished the many, varied school boards and replaced them with a national structure of Local Education Authorities (LEAs), which could establish new secondary and technical schools, and brought about a long period of stability in education.

The level of education and training of teachers increased, as did their pay, and the view was that teachers were to be trusted to provide a high-quality education and given carte blanche to go about it how they saw fit.  The guidance for teachers even included the line:

“The only uniformity of practice in public elementary schools is that each teacher shall think for himself, and work out for himself such methods of teaching as may use his powers to advantage and be best suited to the particular needs and conditions of the school”

It would be almost a century until this expected diversity in practice would change.

The freedoms afforded to teachers were, however, largely quashed by the entrenched system of payment by results on the national standards, with teachers teaching to the test.

The 1902 Act created many free places at the grammar schools for pupils who could pass the scholarship examination at aged 11, so again teaching became focused on training students to pass these tests, with a heavy focus on arithmetic.

The establishment of state secondary education immediately raised the question of what form it should take. The answer was to base the new grammar school curriculum firmly on that of the old public and proprietary schools. The mathematics curriculum was still effectively determined by university entrance requirements and by the syllabuses of the various university examining boards. This meant that the differential calculus came to be taught in most grammar schools and high standards were expected of students.

The 1926 Hadow Report suggested that some form of post-primary education should be provided for all children and that three types of secondary schools were needed: grammar, ‘modern’ and technical (Trade) schools. Some local authorities attempted to put this plan into practice, but little was done to make such a scheme universal for the resources needed to put the proposals into practice were non-existent.

The 1938 Spens Report 1938 suggested ‘The content of school mathematics should be reduced’, its teaching suffered ‘from the tendency to stress secondary rather than primary aims’, it concentrated too much on ‘tricky problem solving’ rather than giving a ‘broad view’, the type and ‘rigour’ of the logic it presented had ‘not been properly adjusted to the natural growth of young minds’

The Norwood Committee of 1943 supported Hadow’s view of a tripartite system, but already signs were emerging that the system was not universally supported. One critic, S. J. Curtis, wrote, according to Norwood the Almighty had benevolently created three types of children in just those proportions which would gratify educational administrators and, moreover, which class a child belonged to was clearly to be observed by the age of 11.

The 1944 Act suggested the school leaving age should be raised to 15 and when possible to 16, but this took until 1972.

The Act also ensured that entry to state grammar schools was now to be solely on merit. Previously parents could purchase a state grammar school education for their children at a relatively small cost, provided the school would accept them. Now, middle-class parents of children who failed the ‘11+’ and who could not afford to send their children to a non-state school were often dissatisfied by the education provided by, and the status of, the new secondary modern schools. They were to play a part in replacing the tripartite system, in the 1970s, by local comprehensive schools: as was, more importantly, the growing belief that valid decisions concerning a child’s future could not be taken at the age of 11.

The percentages of children who attended a grammar school varied from 10% to 30% depending on their local authority, but in 1961, according to official figures, only 22.1% of pupils in England and Wales were in maintained (state) grammar schools. The technical schools, which were intended to supply pupils with a specialist form of practical education, had only 3.1% of pupils, and 10.4% were in independent or ‘direct grant’ schools (the latter occupying a middle position between state and independent schools: a position that ceased to exist when, later, such schools had to choose between becoming comprehensive or independent – the vast majority choosing to go private). This left the majority of pupils to attend the secondary modern schools created following the 1944 Education Act.




Curriculum Reform

In 1938, increasing concern about the practice of teaching to the test, rather than teaching for understanding, led the MA to establish a new committee, which would take a broad look at mathematics teaching. The Second World War disrupted the work and the committee did not restart until 1946.  By then, the psychological impact of the war, particularly on the upper classes, had led to the Education Act of 1944.  For the first time, a clear line between elementary and post elementary education was drawn.  A new system of primary (5-11) and secondary (post-11) education was established.

The MA member, Caleb Gattegno greatly influenced the association in dealing with mathematics and not just arithmetic.  He wrote:

Practice without the power of mathematical thinking leads nowhere; the power of mathematical thinking without practice is like knowing what to do but not having the skill of tools to do it; but the power of mathematical thinking supported by practice and rote learning will give the best opportunity for all children to enjoy and pursue mathematics as far as their individual abilities allow.

The 1950s saw a significant movement in mathematics education, which changed practices in primary schools and had great effect on the content of teacher training courses.  The MA claimed this was a result of the committee report, but in reality, more impact came from the many courses for teachers based on it.  These changes saw the introduction of new materials into schools, including Cuisenaire rods, Dienes’ Multibase Arithmetical Blocks (and later his logic blocks), amongst others.

Gattegno would go on to found the Association for Teaching Aids in Mathematics (ATAM 1952). Its journal Mathematics Teaching first appeared in 1955 and the name was changed to the Association of the Teachers Mathematics (ATM) in 1962

Far more important than the work of the subject associations was the Nuffield Primary Mathematics Project, established in 1964, and the publication in 1965 by the newly established Schools Council for the Curriculum and Examinations of Mathematics in Primary Schools, written by the renowned schools inspector, Edith Biggs.  So tireless was Biggs that it is estimated she personally trained more than 15% of all maths teachers in England.

Course design was still left to teachers, but in practice, teachers followed the courses presented by textbook authors who supplied series ‘based on’ the advice and guidance offered. Later a scheme based on individual learning was produced for pupils aged 7–13 by the School Mathematics Project, but these left the teacher with too little to contribute to lesson planning as well as providing students with too unvaried a diet.

Great improvements in the professional development of teachers came with the establishment by the Nuffield Project of Teachers’ Centres, which led to the LEA local centres. Unfortunately, these, along with many mathematics adviser posts, were to disappear in the late 1980s as LEA responsibilities and funds were cut.

The effects of these initiatives on the actual curriculum were varied.  Sets and multibase arithmetic came into many schools, but then gradually disappeared.  Data gathering and display came in and stayed.  More emphasis came to be placed on geometry and on number patterns in the hope that the latter would facilitate the later learning of algebra.  This naturally meant that less time was spent on the learning of arithmetic with the expected results and public reaction.  Often, and particularly after the National Curriculum was established following the 1988 Education Act, primary school children in England tended to be introduced to concepts far earlier than were children in other countries.

Throughout the 1960s dissatisfaction grew both with the curriculum that had not changed significantly for many years and the way that mathematics was being taught.  An MA report, published in 1959, was criticised by Cyril Hope, a leading figure in the ATM, for the backward-looking nature of the mathematical content, leading to the beginnings of frosty relationships between MA and ATM, which last to this day.

In 1962, the School Mathematics Project (SMP) (ages 11–18), Mathematics in Education and Industry (MEI) (ages 16–18) and the Midland Mathematics Experiment (MME) (ages 11–16) emerged.

MME differed from SMP and MEI in that, from its initiation, it directed its work to secondary modern schools in addition to grammar schools. It failed to make a lasting impact, not on mathematical grounds but because it lacked the money that SMP and MEI were able to attract and because the schools attached to it did not have the prestige and status of those connected with those two projects.

These reforms led to much new material such as co-ordinate geometry, probability and statistics entering the 11–16 curriculum.

These curricular innovations were made possible because of the freedom given to schools, or groups of schools, to create their own syllabus, provided that an examination board would agree to set examinations on it.  In the 1970s, it was estimated that about a third of secondary schools were still following a traditional-style syllabus, a third modern ones, and the remaining third hybrids.

Government did not like the muddled picture of mathematics curricula across the country and responded by creating bodies to over-see examinations.  This led to the drawing up of lists of ‘core’ items that had to be present in all types of curricula.  Such restrictions made innovation increasingly difficult.  Moreover, differences began to grow in what was taught, and how, to more able pupils in the better independent, fee-paying schools compared to those in state comprehensive ones.

Tirades about the ‘new’ mathematics were common, but it was the perceived lack of numeracy of young employee, highlighted in a speech by Prime Minister James Callaghan at Ruskin College in 1976, which was used to justify a significant change in government policy towards exerting tighter control over the curriculum.

However, the Labour government did not want to upset LEA allies and started by asking them to produce local guidelines.

In 1978, Callaghan commissioned Wilfred Cockcroft and a small team of respected experts, including Hilary Shuard and Elizabeth Williams, to carry out an enquiry.  Over the years of the enquiry, this team of progressive educators found an adult population fearful of maths and incapable of applying mathematics.

The Cockcroft Report was published in 1982.  This set out many ideas for improving the teaching of mathematics at all levels, but, like so many other reports of its type did little to solve any problems. More significant were the projects for low attainers that were established in its wake: the Low Attainers Mathematics Project (LAMP), Raising Achievement in Mathematics Project (RAMP) and the SMP Graduated Achievement Project.  Most LEAs appointed ‘Cockcroft Missionaries’ to disseminate the recommendations of the report and train teachers in specific pedagogical approaches.



The End of Diversity

In 1985, the government published a new white paper, ‘Better Schools’, which called for wide ranging reforms, including the introduction of a nationally set curriculum and new national examinations, GCSEs.  Ken Baker became Secretary of State for Education and Science in May 1986 and was tasked with moving the white paper recommendations through parliament to Royal Ascent into an Act.  The intention was clearly that all schools should have the same goals and that the freedoms and diversity that had existed for almost a century should cease.  After progressing through both Houses, the Education Act was passed in 1988, which established a National Curriculum in Mathematics for students aged 5–16 to be followed in all state-funded schools (but not necessarily in independent ones).

The hastily assembled curriculum was designed to fit into a controversial and untried scheme for testing students at various attainment levels at ages 7, 11, 14 and 16.  The years since then have seen continuous attempts to solve the problems created by the poorly designed curriculum, the testing proposals (used in incompatible ways reminiscent of ‘payment by results’, i.e. to assess pupils’ progress and to rank schools for accountability) and more general social changes.

Year groups were now labeled 1 – 11 and broken in to Key Stages.  The National Curriculum set out in detail what mathematics would be taught to tall pupils.  This content has remained fairly constant since, though the order has changed in 1991, 1995, 2000 and 2010 national curriculum

A Task Group on Assessment and Testing (TGAT) was established by Ken Baker to plot out a journey through levels of mathematics.  Initially, a child centred, progressive journey through 20 levels of mathematical ideas was proposed.  These levels were not linked to age groups or years, rather they attempted to describe the journey through mathematics as a progressive building of understanding and knowledge.

As the work progressed, Baker left the Department and was replaced by a new Secretary of State, Kenneth Clarke (note: John MacGregor served briefly and without consequence, between the two Kens).  Clarke did not like the child centred proposals and, after some wrangling, the TGAT was forced to produce a rationalised set of 10 National Curriculum levels, linked to the new Key Stages.  Level 2 described the mathematical content that should be secure by the average pupil at age 7, L3 by age 9, L4 by age 11, L5 by age 13 and L6 by age 16.

The DES had taken a further step towards central control by now requiring tests in each core
subject at each key stage with as far as possible a full coverage of the curriculum.  This was much closer to the original concept of Margaret Thatcher, which was of a national curriculum as a list of basic skills in literacy and numeracy and corresponding tests.

A national teacher boycott of all national assessment in 1993–94 had ensured that there was no longer any requirement for continuous assessment, and the tasks had become externally marked class tests.  Although this brought teachers gains in terms of workload, the fury that the boycott caused the Conservative Government would lead to even greater central control and a deep mistrust of teachers.

National assessment was finally introduced at Key Stage 2 in 1995.

League Tables of performance were introduced and published to show results in the externally marked tests.  It is likely, had the boycott not happened, teacher assessment would have continued and league tables would have never come about.

Teachers were still required to submit their judgement of pupil attainment, but government distrust led to these being largely ignored.  The reality, in many primary schools, became one of teachers not bothering to make the judgements and instead simply waiting for the external results to be published and copying them.

Year 2 and Year 6 quickly became about passing the tests, with the curriculum being driven by the assessments rather than a considered view of learning mathematics.

In 1995, a leaked set of TIMMS results showed primary number skills standards were worsening.  The blame was directed at teaching methods.  As a response, in 1996, mental arithmetic tests and non-calculator papers were added to all the end of key stage national tests.

Problems about pupil autonomy and progressive methods more generally were featured in research studies in Leicester, Inner London and Leeds; a report commissioned by the Secretary of State (known as the Three Wise Men report) (Alexander et al. 1992), brought these together, proposing more whole-class teaching in primary schools.

Gillian Shephard, the new Secretary of State, announced the launch of parallel National Numeracy and Literacy Projects involving schools in poorly performing LEAs.

The aim was to raise standards in basic skills by a prescribed programme for each year, reducing differentiation and including a high proportion of whole-class teaching.  The project launched in autumn 1996, with Anita Straker leading the mathematics strand, working through a large team of ‘numeracy’ consultants (many of whom had been Cockcroft Missionaries in the 80s).

Numeracy was redefined; where previously it had referred to the ability to apply number ideas and skills in employment and everyday life, it now was taken to mean mainly abstract number skills, both written and mental, together with solving routine artificial word problems. The National Numeracy Project relegated those parts of mathematics which dealt with anything other than pure number work, that is, measurement, space and shape, and data handling, introduced into most schools in the 1960s, to the margins, by producing, as well as new teaching methods for number, a framework specifying in detail a number curriculum which was to occupy most of the teaching time available.

Labour swept to power in 1997.  They had been following the NNP closely and continued to expand the project across the country, becoming the National Numeracy Strategy in 1999.  The Labour government went faster and further in centralising control over all aspects of mathematics education.  They set targets for the number of pupils who would reach the ‘age expectations’ in the national tests, in particular within five years 75 per cent should reach Level 4 of the national curriculum at the end of Key Stage 2 (age 11).

What had started as a set of levels devised in order to report each child’s attainment, with Level 4 defined as what could reasonably be attained by the broad average group of children at age 11, had now become the definition of a requirement that almost all children should reach.

Differentiated progress and differentiated teaching would no longer be tolerated as they were at odds with social justice and human rights; schools were now under pressure to meet externally set norms in national tests, whatever the nature of their intakes. If schools could not meet these norms, then they were not likely to be judged by Ofsted inspectors as delivering a satisfactory education, and would be threatened first with shame, having their names publicly listed as a ‘failing school’, and finally, if insufficient improvement was made, with closure.

The National Strategies contract was first delivered by CfBT, but passed to Capita in the early 2000s.  Attainment in primary maths appeared to rise and, in particular, the mathematics education being delivered in primary schools became highly standardised, with all pupils across England being taught precisely the same maths in much the same way.  This level of prescription increased enormously under the Capita contract, with many teachers effectively following a scripted lesson.

In 2004, Professor Adrian Smith published a report suggesting that the CPD of maths teachers needed to improve.  This report led the government to issue a tender for a new national centre, the NCETM.  The contract was won by Tribal Group.  My job was to operationalise the centre and ensure that all teachers in England had knowledge of and access to high quality CPD.

In 2010, the hugely costly Strategies contracts were abolished, with over 400 numeracy consultants made redundant.  The landscape of mathematics teacher CPD became fragmented and confused.  The NCETM contract continued, but on a much reduced basis.  This coincided with a reduced role for local authorities (which had been receiving significant funding from the National Strategies), leaving mathematics teachers without regional or national coordination.  The rise of Multi-Academy Trusts accelerated, with some of the larger trusts able to recruit former numeracy consultants to provide strategic leadership of mathematics, but the vast majority choosing instead to appoint amateurs to the roles.  This confusion allowed the government to, once again, increase the level of centralisation, with guidance on mathematics teaching being disseminated through their new Maths Hubs contract.

In a very short space of time, the National Numeracy Strategy content, which cost hundreds of millions of pounds to develop, was largely pushed out of schools to be replaced by inferior – yet DfE recommended – schemes and approaches.

From a progressive system which valued autonomy in teachers and pupils, which had remained largely unchallenged for 100 years, we have moved to a public education emphasis with very tight control from a small group of people in central government.


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