I wrote this report for my own CPD, but did you ever wonder with maths teaching in state schools is universally shit? Here’s the answers in my notes on the OFSTED (Office for Standards in Education, UK) report on Mathematics Teaching. This was published last year, May 2012:
Notes on the ‘Mathematics – Made To Measure’ report from OFSTED (May 2012)
8th March 2013
From reading this report, I have identified several key areas and reasons why mathematics teaching is systematically failing pupils within the state school education system, all of which impact onto teaching, learning and achievement within the Functional Skills mathematics programmes which form part of the apprenticeship pathway. These include:
· A lack of subject specialist teachers, especially for younger pupils and low achievers in secondary school
· No examination or how or why mathematical theory works and an over reliance on memorising techniques without ensuring understanding
· A lack of ‘full coverage’ within teaching mathematics, leading to topics being missed out from the curriculum
· Major problems with the way that both teachers and examinations mark and grade achievement in mathematics
· An over reliance on early entry into GCSE and the prioritising of C grade achievement over pursuit of A/A* grades
· The attitude that anyone who is incapable of achieving a C grade is not worth extra time and attention
· A lack of embedded teaching of mathematics within other areas of the curriculum
A lack of subject specialist teachers, especially for younger pupils and low achievers in secondary school
A common complaint from work based learners who have not achieved success in mathematics at school is that the quality of the teaching and attention they received at school was lacking. The report backs up this self assessment and offers reasons as to why this disparity of provision has arisen.
The ability to achieve in maths is not encouraged in those who develop at a slower rate. The report notes that the attainment of a child at 16 years of age ‘can largely be predicted by their attainment at age 11, and this can be tracked back to the knowledge and skills they have acquired by age 7’. It’s not unreasonable to expect that weaknesses in ability and understanding at a young age would be picked up and developed at a later stage during a child’s education, but this is not the case and too many children are experiencing a sub standard level of education, and therefore understanding and achievement in mathematics, as a result of the distribution of resources within state education, especially at secondary level.
The report notes some extremely that ‘Secondary pupils in lowest sets received the weakest teaching during inspections’ and 36% of the 2011 cohort achieved a D grade or lower in their GCSE examination. This is in spite of the frequent use of resits, early entry and double entry for GCSE maths examinations. Less experienced, temporary and non-specialist teachers were more likely to teach lower sets or younger pupils and one in seven lessons observed within these sets was judged to be inadequate. Nearly half of the schools observed had some inadequate teaching for mathematics. Teachers are struggling to improve their skills in teaching mathematics and very few schools are willing to provide curricular guidance for staff.
Those pupils who are preparing for external examinations, especially the high achievers, were more likely to receive better teaching from subject specialists, but the younger and less able pupils were more likely to be taught by non subject specialists, allowing schools to reserve those teachers which such skills for the higher KS4 sets, where the subjects being covered would be outside the expertise of a non specialist.
The weaknesses were not solely confined to subject knowledge, significant issues were identified within teacher confidence. Two lessons were observed where inaccuracies and alternative methods were identified by pupils and brought to the teacher’s attention only to be ignored and left un-tackled by the teachers in question. There were even examples of teachers using incorrect explanations for their classes which did not relate to the real life use of mathematics. Correct work was sometimes marked as incorrect or mistakes were unnoticed and work was marked as correct, and some pupils were given insufficient guidance by teachers as to how to utilise mathematical presentation to ensure that their working out was correctly completed. Teachers were not solely to blame, teaching assistants were identified as being lacking in their skills and confidence to tackle pupils’ understanding of mathematics and were not providing sufficient challenge, particularly in the lower key stages.
The report notes that there is ‘no quick fix’ for some of the issues identified which are leading to poor maths teaching, but short term planning is essential to support non specialists and these plans and approaches should align with what is used by others in the mathematics departments.
The outlook for this is not hopeful, as even the subject leaders at primary school did not teach noticeably better than the other teachers. The report even notes that ‘it is no longer the case that the subject leader is necessarily the most experienced and skilled teacher of mathematics in the school’ and some of those subject leaders ‘do not have sufficient depth of subject knowledge’ and they ‘lack confidence in and experience of mathematics across the primary age range’. If the teaching of mathematics is being guided by people who are not qualified, experienced or confident in delivering the subject, the hopes for children reaching secondary school on target for achieving their potential are slim at best.
The result of this combination makes it entirely possible that a child who does not received mathematics tuition from a subject specialist in primary school, and who therefore does not make adequate progression and is placed in a low set at secondary school, will not actually received any tuition from a subject specialist throughout their school experience. From the age of 4 to 16, it is entirely possible that a child’s experience of mathematical education will be defined by teachers who are not confident, sufficiently educated or experienced in delivering mathematics lessons, with no opportunity to rectify the damage that a poor experience at a young age may cause.
No examination or how or why mathematical theory works and an over reliance on memorising techniques without ensuring understanding.
The report notes that secondary school pupils, by and large, accept mathematics as ‘important but dull’ and that relatively few schools bothered to seek their pupils’ views on that makes mathematics learning successful. A key quote from a pupil surveyed by OFSTED notes that “you need to understand and not just do it. You think you know how to do it but you get to an exam and you can’t. You realise that nobody’s told you why it works and why you do what you do, so you can’t remember it.’ This simple fact is the root of many problems in learning mathematics, particularly within the largely kinaesthetic learning sector of work based learning. Understanding the reasons behind mathematical processes and being able to relate them to real life examples and needs are key to the success and progression of our learners. Many weaknesses were identified in maths teaching at state secondary schools, which indicated that the teachers did not understand the reasons behind mathematical processes either. The best practice identified was where ‘wrong answers were welcomed as an opportunity to explore how a misconception had arisen. Pupils did not fear making mistakes as they too recognised how unravelling an error helped their understanding’.
Many of our work based learners actively fear learning, especially when it comes to the traditionally academic subjects of maths and English. Sometimes their fear is expressed as disinterest, sometimes as reluctance, or a chronic lack of self confidence. Sometimes it even presents as anger. Tackling a fear of failure, combined with a thorough explanation of the ‘why’ factor of mathematics is the key to settling learners and opening their minds to the concept of learning.
The report notes that pupils regularly experienced lessons where the sense behind mathematics was not explained to them or developed collaboratively with them. These are the bad habits that we are required to unpick within work based learning in order to enable our learners to achieve at level 1 and at level 2. School lessons are not generally active. They are largely focused on listening to the teacher, copying down examples and questions being answered by the academic elite of the class, leaving the majority as silent spectators. These lessons are highly focussed upon visual and audio based learning. The vast majority of work based learners enrolled on apprenticeships will respond better to kinaesthetic activities, where they are encouraged to attempt examples for themselves, step by step. One to one teaching is an excellent facilitator for this type of teaching, as it allows for the learning to be broken down into steps with close contact from the tutor, allowing the learner to feel secure. Asking to learner to explain their reasons for the steps in their working out in another excellent way to embed their understanding. It forces the learner to admit when they are guessing and gives the tutor a clear indication of when a piece of learning has been fully embedded into their understanding.
The report notes that school teachers are particularly poor at allowing pupils ‘thinking time’: ‘Sometimes they were too quick to prompt or to answer their own questions…rather than leaving time for pupils to work out the first step for themselves.’ Leaving time for the learner to think about their response or approach to a question, whether verbal or audio, is crucial. Tutors must learn to be comfortable with silence, allowing thinking time and giving the learner chance to make mistakes and correct themselves. Highlighting areas where pupils are struggling, and encouraging pupils to identify those areas for themselves, are both crucial to furthering progression. Until a problem is identified, a solution cannot be approached. This, rather than the dictation of a mark scheme, is why the recording of working outs for maths is essential to assessing a learner’s progress. Sadly, new software which has been developed for homework purposes allows pupils to enter the answers to set tasks without displaying their working out, meaning that the teacher has no chance to assess their independent use of the methods for working out.
The report notes that this is still a weakness in schools, and that whilst teachers created opportunities for assessing pupil prior knowledge, they ‘varied in how well they interpreted the clues in pupils’ work and oral responses to pinpoint their difficulties precisely’. If a learner does not understand why they are making mistakes, or why their mistake does not fit with the process, they will never develop the independent ability to correct their own work. Identifying errors could, in the hands of a skilled teacher, lead to the identification of misconceptions, which once corrected could be used to enhance a pupils understanding.
However, it was noted that teachers were missing opportunities to link topics together effectively. Topics were tackled ‘without any connection being made’ and pupils did not appreciate the links between various topics. They were reduced to ‘memorising methods, because teachers emphasised emulating the worked example rather than why the methods worked.’ Topics must be linked together to enable learners to understand the breadth of mathematics as a subject. It was noted that tackling more than one part of a subject at once can cause confusion if these links are not made clear and explicit. The report cited an example of a lesson on area and perimeter where pupils were confused about what aspect of a diagram each measure. The report recommends that introducing the two concepts at different stages will ‘avoid such confusion’. Whilst time is of the essence when teaching work based learning, there is still a lot to be said for a well structured lesson where concepts are introduced, tackled and explained one at a time, and where learning on a single aim is consolidated before introducing a second, which is then linked back to the prior established learning. Whilst functional skills may require the learner to select from a range of skills, those skills still need to be tackled and familiarised individually, and then linked together once understanding is confirmed, before an appropriate selection can be made.
The report also noted that poor presentation could contribute to both a lack of understanding on the part of the pupil and missed opportunities on the part of the teacher to identify common errors and weaknesses in a pupil’s work and therefore their understanding. OFSTED noted that ‘no attention was given to how well work was set out, or whether correct methods and notation were used’. Notation is important, especially for measurement and finance, as this links the mathematical process to a real world concept. Enforcing the correct use of ‘£’ and ‘p’ within financial sums can enhance a pupil’s understanding of decimal places, and insistence upon the use of metric notation for measurement will enable a pupil to become more familiar with the metric system, especially if they did not learn the system whilst at school and have a primary familiarity with the Imperial systems of measurement. Bad habits in marking were also identified, where correct work appeared to have been marked incorrectly and vice versa, leading to confusion for pupils as to what they had accomplished in class and at home. Teachers comments were also more likely to refer to the quantity of work completed rather than the quality of the working or presentation.
A lack of ‘full coverage’ within teaching mathematics, leading to topics being missed out from the curriculum
Closely linked to the concerns about a lack of ‘joined up thinking’ in teaching maths is the absence of significant parts of the curriculum at school. The report noted that changes to the A level syllabus in the last ten years have ‘reduced the demand and breadth of content studied.’ However, there are times particularly at GCSE where the syllabus, regardless of its content, is not actually taught fully. This is in part due to flaws in the marking system for GCSE mathematics and the way in which the questions are allocated marks: ‘Part of the problem is that external assessment in mathematics…is generally based on a compensatory model: success with some questions in a test or examination compensates for poor performance on others, irrespective of the relative importance of the topic being assessed.’ The issues with the marking schemes and tracking that stem from this issue will be discussed more in the next section of this summary, but the main issue caused by this simple fact is a sense of complacency amongst maths teachers, even when teaching the brightest learners and most particularly when teaching the lower ability learners.
The report notes that ‘demanding algebra topics were too often a casualty of the limited time for completing GCSE’, especially in those schools who pushed learners to complete their exam by the end of year 10 instead of year 11. The ‘very strong emphasis’ on external assessment and performance measures by schools means that a rounded and thorough mathematical education is skipped over in favour of teaching to the test requirements, or rather teaching to the pass mark requirements. In schools which were deemed ‘less effective’, the full GCSE was not covered, and some topics were only tackled in a superficial way. The drawbacks of this approach have been noted by the pupils, as well as by OFSTED, as one was heard to comment about not studying any algebra during year 11 and another was heard commenting on the lack of depth in the GCSE study.
An over reliance on early entry into GCSE and the prioritising of C grade achievement over pursuit of A/A* grades
Pupils are frequently pushed to achieve C grades, and were even ‘accelerated’ to free standing maths qualifications after achieving B or C grades, rather than being encouraged to study for the additional year to achieve an A or A* grade.
Whilst this may only seem to affect a small proportion of learners, the implications for work based learning and the apprenticeship system are immense. With C and B grades expiring after five years, the early taking of these examinations at the expense of a learner’s ability to achieve an A grade is leading to a greater number needing to refresh their skills by taking the Functional Skills aspects on their apprenticeship. A learner who passes their maths GCSE with a C or even a B grade at the age of 15 will find their qualification defunct by the age of 20. This focus on external assessment and the exploitation of the compensatory marking system has lead to a culture of ‘home grown underachievement’ in secondary schools. The combination of these two facts will eventually lead to increased demand, and increased disaffection, for learners in the apprenticeship system.
Major problems with the way that both teachers and examinations mark and grade achievement in mathematics
The problems with compensatory marking are not limited to GCSE, and they have implications for Functional Skills too.
At level 1, three tasks are completed with 15 marks available for each one. The average pass mark is around 60%, which equates to approx 30 marks. It is therefore possible to completely fail one of the tasks, a whole third of the subject area and to still pass the paper. At Level 2 this is exacerbated further. Three tasks are completed with 20 marks available for each one. The pass mark can be as low as 50%, just 30 marks. It is possible to fail one task and only achieve half marks on a second and still achieve a pass at level 2.
Learners in mathematics become adept at focussing their efforts into their areas of strength and they are trained into this habit throughout the state school education system.
The problem with compensatory grading and specific topic achievement starts with the tracking system for pupils in primary and early secondary school with national curriculum levels. The most common way of tracking is for a teacher to assess a piece of marked work and assign it a curriculum level. However, the nature of mathematics content is ‘hierarchical’, and many individual topics are assigned discrete levels, and each level is made up of a range of topics. Success in one particular area does not mean that a pupil has achieved competency across a whole level, and it is not always possible to enhance understanding of a topic to a further level without acquiring a completely new set of learning: ‘for example, work on circumference and area of circles is generally considered to be a level 6 topic. Incorrect or misunderstood levels cannot be called ‘level 5’.’
From as early as Key Stage 1, the tracking and marking systems for mathematics are not compatible with the continuous progression of achievement favoured by standard methods of education. A sample piece of work does not give a true idea of a learner’s ability across the mathematics curriculum.
This has a knock on effect to our diagnostic grading system. Whilst a learner may achieve mid level 1, or even high level 1, the diagnostic test may still highlight key weaknesses in areas such as perimeter and area, mean and range or percentage calculation, all of which must be mastered before a learner can be guaranteed to pass level 1. The only way to assess the usage of those topics in combination, and to assess a learner’s ability to select an appropriate method to tackle an exam question, is to set them a mock exam. This largely makes the diagnostic tests redundant, as they do not indicate accurately whether a learner can independently assess a functional problem and select from a variety of methods to ensure that the task is completed effectively.
The attitude that anyone who is incapable of achieving a C grade is not worth extra time and attention
It is a common claim from learners on the apprenticeship system who did not achieve in maths and English at school: ‘I was in the bottom set at school and nobody cared about us.’ Whilst it does sound dismissive on the part of the learner, possibly simplistic and a little beyond belief, sadly the OFSTED report holds up their complaints in spectacular fashion.
The report notes in its introduction that intervention programmes of support need to be extended to ‘all pupils who were in need of support, not just those at key borderlines or about to take national assessments’. It is hard to believe that a school would prioritise the welfare of some pupils over others when it comes to the provision of support, but there is strong evidence to indicate that this is happening and that it is a widespread problem.
The report highlighted the tendency for subject leaders to teach the ‘higher attaining’ sets in Key Stage 4 (aged 14-16). However, ‘it was also the case that the lowest attaining pupils needed to make the most progress and therefore required the best teaching, but too often did not receive it. This is why in-school consistency in the quality of teaching is such a concern’. It is a bold statement and highlights a widespread problem, namely that: ‘pupils in the lowest sets typically learn less well, make less progress and attain low grades’ and this is, in the vast majority of cases, down to the quality of teaching and instruction they receive, which is more likely to be sub standard in the lower sets and across the low ability range.
Many schools were noted to be struggling to place subject specialists in all classes and priority was given to the key examination classes and high attaining students. Non specialists, temporary teachers and new teachers were more likely to lead the teaching of lower set classes. Timetable constraints sometimes meant that two teachers would share responsibility for a younger class. OFSTED specifically noted in their report that ‘in general, this does not aid coherent progression or good quality learning.’ A lack of consistency and conflict of methods would only serve to exacerbate the issues identified in earlier sections of this report, making it more likely that work will be mismarked, misunderstood and incorrectly assessed.
The vast majority of learners who enter the apprenticeship system in work based learning have not achieved level 2 at school, meaning that they are probably lacking a C grade GCSE in mathematics. If they were predicted to achieve less than a C grade in maths at school, their chances of receiving support to achieve those grades are not necessarily universal, as can be seen in the following diagram, taken from page 52 of the report:
The above diagram analyses a learner’s predicted performance at GCSE. Those in the central triangle are already on course to achieve Level 2 (5 A-C grades including maths and English). The intersects with two overlapping triangles are the pupils most likely to receive support, those who are on course in maths and/or English, but missing one of the other key criteria. Those in the single shaded area are next, if they are on course for one of the criteria, but missing the other two. Any success within these areas would lead to an increased league table position for the school and better chances of attracting further pupils to attend in future.
The diagram’s labels do not make it clear but there is a fourth category. Those outside of the Venn diagram. The pupils who are predicted to achieve less than 5 C grades in total, and who are not expected to achieve C grades in either maths or English. These are the learners who are most likely to be in the bottom sets across their education, and they are the least likely to receive support to raise their achievement levels: ‘Those placed outside the three circles …were least often the focus of support in the schools visited’ Perhaps most worryingly of all is the tendency to treat all pupils who are expected to achieve less than a C as being the same. The report noted that ‘too many pupils were gaining F grades when they had the potential to achieve D or E grades. These pupils were the least confident and self motivated’. Because it makes no difference to a school’s league table results whether a pupil achieves a D grade or an F grade, the focus on teaching quality is absolutely removed from those who have been deemed as unlikely to achieve a C grade. Standards slip further and no encouragement is given to pupils within this category to improve their achievement within the lower sets.
The benefits of setting are not universally beneficial in the first place. Some research points to the benefits being restricted to the more able pupils, with adverse effects on other pupil’s motivation and self confidence, particularly girls. For an industry with an intake which is heavily weighted towards women, this is startling news. The set a pupil is placed in ‘determines the mathematics he/she will encounter and potentially caps what he/she might attain’. With subject coverage already being in doubt due to the issues with the marking schemes and compensatory system, placing a pupil in a lower set and reducing the coverage of the curriculum even further reduces their chances of being able to catch up and progress to a higher set later in their school life. ‘Senior and subject leaders appeared to realise, perhaps too acceptingly, that ground would need to be made up in future if the pace of learning of younger and lower attaining pupils was affected by weaker teaching.’ However, it would appear that pupils are rarely given the chance to achieve this making up of ground. Being placed in a low set at an early age, perhaps due to incorrect or inaccurate grading or poor quality non specialist teaching in primary school, can condemn someone to a poor quality mathematical education, with no chance to extend achievement or improve attainment.
The lack of confidence, of both teachers and pupils, and poor attainment, by both teachers and pupils, endured by many apprenticeship learners has its roots deep within the school system and the complaints they voice about being placed in a low set and ignored, or struggling against a teacher who did not have a thorough understanding of the subject, should not be dismissed or ignored when they have been identified as common place and widespread problems, particularly in the secondary school system.
A lack of embedded teaching of mathematics within other areas of the curriculum
Whilst only touched upon in passing in this report, this idea has massive implications for the embedding of Functional Skills within the apprenticeship programmes. OFSTED noted that the implementation of Functional Skills had been ‘dogged with difficulties’ and many schools had ‘struggled to make adequate preparations.’ Much like many apprenticeship programmes, schools were treating it as a ‘bolt on activity rather than understanding that the development of these skills is part of behaving and reasoning mathematically, and therefore pervades good learning of mathematics’. The integration of mathematics into the wider curriculum is failing. With the abolition of GCSE coursework, pupils are not being given the opportunity to investigate tasks using a wide range of mathematical approaches and independent learning is not being adequately embedded. ‘A lack of emphasis on using and applying mathematics remained a weakness that is persistent’.
Unless mathematics is integrated into the wider curriculum, with investigative techniques being encouraged in subjects such as science, geography, design and technology, mathematics will continue to be seen as a standalone subject with little relevance to the real world. The use of mathematics must be integrated more fully into the apprenticeship system, with assessors being willing to encourage and suggest the use of mathematical processes within the teaching of modules, as an integrated process supported by a specialist teacher if necessary. Much like in schools, Functional Skills Mathematics is still seen as a ‘bolt on’ activity, rather than an important and embedded aspect of work based learning.
A link to the direct report is provided here:
The footnotes refer to page references within the original report.
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