All the reports collected can be seen as attempting in some way to address concerns about mathematics, frequently referred to as ‘problems’. As one might expect, the particular concerns and priorities reflect the interests of the originator of the report. Overall the picture is far-reaching and complex. Harris provides an overview:

England is viewed as having a mathematics ‘problem’ at all age groups …. ‘the mathematics ‘problem’ in England is a multi-faceted one with many important stakeholders, all of whom can have different views of the problem’ (2012, p.5, 7).

There is a widespread view that in HE and the workplace, there are too few young people with the mathematical skills, knowledge and confidence at a sufficiently high level (ACME, 2011c, 2012a; British Academy, 2012). As the British Academy report states:

There is a skills deficit. In higher education, almost all disciplines require quantitative capacity, but students are often ill-equipped to cope with those demands. They then leave university with skills inadequate to the needs of the workplace – be it in business, public sector, or academia. Students are graduating with little confidence in using what skills they do have, having had little practice in applying them. Employers often lament the lack of quantitative skills in the workplace. (British Academy, 2012, p. 2)

Not only is the ‘skills deficit’ a problem for future growth, but it also seems that it is costing the country in real terms. Figures from KPMG, quoted in the National Numeracy report, National Numeracy for everyone, for life: Facts and figures’ (National Numeracy for everyone for life, 2012a) estimated that children’s lack of numeracy skills in primary schools costs the country £2.4bn every year and that annual GDP could be increased by 0.44% if pupils who currently fail to reach minimum OECD standards were brought up to that level.

Whereas the majority of the reports are mainly concerned with the mathematical activity of non-specialists, some also express concern over those young people who might pursue mathematically demanding academic study and enter mathematically demanding careers. As ACME suggests:

In short, the UK is not realising its potential in terms of growing and supporting able mathematicians. This is a huge waste that the UK can ill-afford, economically or socially, and it short changes individual young people: the problem requires urgent attention (2012a, p. 1).

Many agree that something must be done and that change is necessary. (ACME, 2012a; British Academy, 2012; Hodgen, Marks, & Pepper, 2013; National Numeracy for everyone for life, 2012a; NFER, 2013; Norris, 2012; Vorderman, Porkess, Budd, Dunne, & Rahman-hart, 2011). Their recommendations are outlined below in ‘Recommendations’. This section provides a synthesis of the details of the ‘problems’, or to be more precise, the symptoms. In other words, it outlines how we know there is a problem; within educational institutions at the level of schools, post-16, HE and also in society more generally, which includes the workplace.

## Within educational institutions

Whereas it is frequently argued that the attitudes of young people in school are a cause of the problems (see the section on causes), it is suggested here that negative attitudes towards mathematics also provide an indication that there is something wrong with mathematics education. First, young people become ‘disengaged’ (Harris, 2012; Norris, 2012; Vorderman et al., 2011) and fear mathematics (Vorderman et al., 2011). As Vorderman at al state:

It is not a case of a few students being lazy; … and it is not just that they fail GCSE but, much more seriously, that many of them end up immunised against mathematics, fearing and hating the subject. (2011, p. 53)

Further, it seems that young people lack confidence in their mathematical ability, identifying themselves as non-mathematicians, as is perhaps evidenced perhaps by an ‘I can’t do maths’ attitude (Harris, 2012; National Numeracy for everyone for life, 2012b). It is even suggested that those which good grades in mathematics lack confidence (ACME, 2012a)

More evidence of the problems at school level is provided by low levels of achievement in mathematics. Many reports quote low levels of achievement at GCSE (Harris, 2012; National Numeracy for everyone for life, 2012b; Vorderman et al., 2011).

Low levels of achievement at GCSE are perhaps more widely reported than at other school ages, but there are suggestions in some reports of concerns about pupils’ achievement and progress earlier in their school careers (Ofsted, 2011a; Vorderman et al., 2011). For example, the Vorderman report raises the concern that approximately a third of pupils fail to make progress in mathematics in the first year of secondary school.

In addition, it appears that the country is ‘significantly underachieving in terms of developing able mathematicians, and this situation is now critical’ (ACME, 2012a, p. 1). First, of those young people achieving a C grade at GCSE, relatively few choose to continue to study the subject (ACME, 2012b; Harris, 2012) (although there seems now to be a reversal in this trend according to media interpretations of the latest A level figures (15^{th} August 2013). This ‘encouraging trend’ is also identified in, for example, the ACME report ‘Increasing provision and participation in post-16 mathematics’ (ACME, 2012b)). Of those who choose to continue with mathematics as a A-level subject seem not to have the skills required by the subject:

The major criticism from those teaching AS and A level mathematics is that GCSE does not provide students with a good enough preparation, particularly in algebra. Consequently the transition is difficult for many students. Mathematics has one of the highest failure rates at AS level; many students under-perform or drop out at this stage. (Vorderman et al., 2011, p 54).

Jerrim and Choi (2013) also address the achievement levels of the highest achievers, suggesting that this group of students falls behind their counterparts in East Asia and Smithers’ detailed analysis of a number of tests strongly suggests that the percentage of students in England achieving at the highest levels is about half of the level of those reaching the same standards in the countries which do best in these tests.

A number of reports other than those by Jerrim and Choi and Smithers suggest, implicitly or explicitly, that England’s (and the UK’s in some cases) performance in international tests such as PISA and TIMMS provides evidence of the problems in mathematics education. (ACME, 2012a; National Numeracy for everyone for life, 2012a; Select Committee on Science and Technology, 2012; Vorderman et al., 2011). These reports cite England’s apparently low rankings within the countries tested. For example,

English policymakers have shown particular concern over England’s position of 28th, out of 65 countries, in the PISA 2009 mathematics assessment. (Jerrim & Choi, 2013, p. 4)

Further it is suggested that the country’s performance has deteriorated since 2000 (Vorderman et al., 2011) but at least one report (Smithers, 2013) urges caution in interpreting the results of these tests in this way.

As discussed elsewhere, post 16 participation in mathematics has, over the time period covered by the reports synthesised in this paper, risen high on the political agenda and a number of reports focus on this particular area of mathematics education.

It is generally recognised within the reports that the country has low levels of participation in post-compulsory mathematics and this is seen as a symptom of the problems with mathematics; for a variety of reasons young people are not choosing to continue studying mathematics. (ACME, 2011, 2012b; Harris, 2012; Hodgen & Marks, 2013; National Numeracy for everyone for life, 2012b; Nuffield, 2012; Select Committee on Science and Technology, 2012; Vorderman et al., 2011). The low levels of participation can also be seen as a cause of the problems as discussed elsewhere.

Some post 16 students who do not study mathematics engage in mathematics through other subject areas, such as the sciences, economics, computing and geography. It seems that these students may be held back within their A-level course by a lack of mathematical skills to fully engage with their courses (Nuffield, 2012; Score, 2012; Select Committee on Science and Technology, 2012).

A similar symptom is evident in higher education. A number of reports suggest that students enter undergraduate degree courses, under-prepared and ill-equipped to cope with the quantitative demands of their subjects. (ACME, 2011; British Academy, 2012; Nuffield, 2012; Select Committee on Science and Technology, 2012; Vorderman et al., 2011)(Parliamentary Office of Science and Technology, 2013). For example, as the Select Committee on Science and Technology stated in their report:

In 2006, the Royal Society argued that the gap between the mathematical skills of students when they entered HE and the mathematical skills needed for STEM first degrees was a problem which had become acute…. The evidence we received suggested that the problem remains. (2012, p. 15) (SC STEM p 15)

## Outside formal education

National Numeracy, quoting figures from the BIS Skills for Life Survey (2012), reported that ‘17 million adults in England (just under half the working-age population) are at ‘Entry Levels’ in numeracy – roughly equivalent to the standards expected in primary school.’ (2012b, p. 1). The same report provides evidence that this figure represents a decrease in numbers of adults with these skills over the time period 2003 to 2011.

Further symptoms, but related to so-called Level 2 (roughly equivalent to attainment at A*- C at GCSE), are presented in this and other reports. For example it quotes figures of 22% of working age population achieving this level and above and Ofsted claims that approximately half those who enter for these qualifications are successful (Ofsted, 2011b).

The figures in the reports are not always consistent, but the overall message is clear: a significant proportion of adults are lacking core numerical skills in terms of day to day living. As the All Party Parliamentary Group on Financial Education states: ‘around one in four economically active adults is functionally innumerate.’ (2011, p. 12)

These concerns relate to everyday adult living, but on the specific level of jobs and employment, the reports are much more vocal and within the workplace, symptoms of the problems in mathematics education are reported widely.

It is widely agreed that very many young people enter the workplace lacking the numerical skills demanded by modern jobs, with limited knowledge of mathematics and how it is used. (Ofsted, 2011b; Parliamentary Office of Science and Technology, 2013; Vorderman et al., 2011).

The message from employers is clear. It seems that they are unable to find enough employees with suitable qualifications (Harris, 2012; NFER, 2013; Norris, 2012; Parliamentary Office of Science and Technology, 2013; Vorderman et al., 2011). They complain that recruits at all levels lack essential skills, are not fluent in mathematics techniques, and make basic mistakes (ACME, 2011; British Academy, 2012; Hodgen & Marks, 2013; Vorderman et al., 2011) Various reports suggest that employees are unable to apply mathematical skills they know (ACME, 2011; Hodgen & Marks, 2013; Vorderman et al., 2011), that they are not able to use their mathematics in new situations (ACME, 2011; Score, 2012), are not able to communicate their understanding (even when it is present) (ACME, 2011c). Hodgen and Marks sum up the symptoms:

[T]his study finds many examples of people in the workplace using a ‘black-box’ approach to some mathematical techniques, where they lack the mathematical knowledge to understand fully the techniques they are using, to control the technology, and to understand and use the outputs. (2013, p. 2)

Whereas these concerns relate to the workplace more generally, there are also concerns specifically related to those who enter the workplace with a degree. First, employers again complain that they are not able to recruit sufficient numbers of graduates with skills in mathematics (STEM) (Clark-Wilson, Oldknow, & Sutherland, 2011; Select Committee on Science and Technology, 2012). Second, they complain that graduates lack the numerical skills and knowledge needed in the workplace, (British Academy, 2012; Parliamentary Office of Science and Technology, 2013; Select Committee on Science and Technology, 2012) and are not confident about using the skills they have (British Academy, 2012).

# References

ACME. (2011). *Mathematical Needs Mathematics in the workplace and in Higher Education*. London.

ACME. (2012a). *Raising the bar: developing able young mathematicians* (pp. 1–4). London.

ACME. (2012b). *Increasing provision and participation in post-16 mathematics*. London.

All Party Parliamentary Group on Financial Education. (2011). *Financial Education & the Curriculum*. London.

British Academy. (2012). *Society Counts: Quantitative Skills in the Social Sciences (A Position Paper)* (pp. 1–12). London.

Clark-Wilson, A., Oldknow, A., & Sutherland, R. (2011). *Digital technologies and mathematics education: A report from a working group of the Joint Mathematical Council of the United Kingdom*. London.

Department for Business Innovation and Skills. (2012). *The 2011 Skills for Life Survey : A Survey of Literacy , Numeracy and ICT Levels in England*. London.

Harris, J. (2012). *Rational Numbers*. London.

Hodgen, J., & Marks, R. (2013). *The Employment Equation: Why our young people need more maths for today’s jobs*. London.

Hodgen, J., Marks, R., & Pepper, D. (2013). *Towards universal participation in post-16 mathematics : lessons from high-performing countries*. London.

Jerrim, J., & Choi, A. (2013). *The mathematics skills of school children : How does England compare to the high performing East Asian jurisdictions?* (p. 39). London.

National Numeracy for everyone for life. (2012a). *The National Numeracy Challenge* (pp. 1–9). Lewes.

National Numeracy for everyone for life. (2012b). *National Numeracy for everyone, for life: Facts and figures*. Lewes.

NFER. (2013). *NFER Thinks: Improving young people’s engagement with science, technology, engineering and mathematics (STEM)*. Slough.

Norris, E. (2012). *Solving the maths problem: international perspectives on mathematics education*. London.

Nuffield. (2012). *Mathematics in A level assessments*. London.

Ofsted. (2011a). *Good practice in primary mathematics*. Manchester: Ofsted.

Ofsted. (2011b). *Tackling the challenge of low numeracy skills in young people and adults*. Manchester.

Parliamentary Office of Science and Technology. (2013). *STEM education for 14-19 year olds* (pp. 1–4). London.

Score. (2012). *Mathematics within A-level science 2010 examinations*. London.

Select Committee on Science and Technology. (2012). *Higher Education in Science, Technology, Engineering and Mathematics ( STEM ) subjects*. London.

Smithers, A. (2013). *Confusion in the ranks: how good are England’s schools?*. London.

Vorderman, C., Porkess, R., Budd, C., Dunne, R., & Rahman-hart, P. (2011). *A world-class mathematics education for all our young people*. London.

Interesting how similar the issues presented here are to those in science education – has establishment of a science/maths for all ethos through compulsory teaching and assessment up to age 16 actually itself put people off or is it the associated ‘teaching to the test’ culture or something else ….?

Well I wonder about the compulsory teaching and assessment – and there’s more to come with some kind of maths being compulsory to 18/19!

Calculators may not be used in my classroom unless I say so (Grades 10 to 12) and the improvement in mental agility and confidence with numbers has been remarkable. Many students are finding that they now stand a much better chance of finishing examination papers, too. I would ban calculators totally at primary level.

Hi Rob, thanks for commenting. Am I right in thinking that your view is that one of the ‘problems’ in maths education is that calculators are used too freely?

Hi Marie, and at too early an age. Multiplication tables, number bonds and ability to handle simple fraction calculations are a thing of the past. The fact that modern calculators give answers as improper fractions and in surd form also does not promote the ability to recognise common forms and understand simple calculations. Just today, I had several Grade 10 students asking me how I got 21/4 from 5+1/4 without using a calculator!

Rob, what do you think of the other ‘problems’ in maths education outlined above? For example, do you find that teachers tend to ‘teach to the test’?

Yes, teachers do often ‘teach to the test’, not only to achieve good looking results but also because very few have the knowledge and ability to extend the students. In South Africa there is a major problem with the lack of properly qualified teachers and this is not only in the so-called Township schools.

I think there are other reasons, however, that teachers ‘teach to the test’ – for example, the high status that is given to test results (so called “high-stakes testing regimes”).

About calculators: here’s a recentish email from brother who lectures maths at UKZN (in Maritzburg). I think mainly to Agric students.

After marking the Benchmark test I tried my students out in class.

What is 8 – 3 + 2?

There were about 90 present. About 15 abstained. Out of the remaining 75 or so, EXACTLY ONE (!!!) knew the right answer. This is because they have been taught BODMAS at school, so you do the addition first.

I’m fairly convinced a very high proportion of primary school teachers would also get it wrong. Since they (teachers & pupils) do everything on a calculator, nobody notices that something is wrong.

Now I asked them, what is 282-322? Everybody believes it’s 28-3+2, so according to them, it’s 23.

Everybody also agrees the order of multiplication doesn’t matter, so

282-322 = 28222-3= 28+2-3 = 27.

So now 23 = 27, that is 3 = 7.

This would be funny if it were not tragic.

Paddy

Chairman ECIPS (Eradicate calculators in primary school).

Teaching to test: I’m pretty sure that the high-achieving Asian countries that the UK is comparing itself with do this all the time. And I also think that it’d be almost impossible to eliminate. So perhaps the debate should be about how to work within that paradigm.

Perhaps changing the tests themselves would be more fruitful than changing the attitude towards them.

I completely agree. I think we need to accept that the tests dictate everything so if we want to change anything we need to change the tests.

I don’t feel that I do teach to the test at KS3. Problem solving tasks and projects are an integral part of our curriculum and we try to use a variety of lesson styles and approaches, as well as looking for ways to link concepts together and try out ideas that will not appear on exams. We do use worksheets as one approach and sometimes practise standard techniques in isolation. It’s not the case that I somehow limit what the class does based on how a topic may appear in a GCSE in a few years time.

John – Do you think your attitude has changed since the demise of the KS3 SATs?

No, not really. We still use SAT style questions in our internal summative testing. Heading towards May in year 9 there is a change in that we don’t practise using past papers and there is less use of mental tests. However, most of my KS3 teaching is much the same. Changes have been due to a greater interest in formative assessment and wider access to ideas via the web.

I actually like what you’ve acquired here, certainly like what you are saying and the way in which you say it.

The answer might be in the English spelling system which takes –at least– 3 or 4 years to master in order to read just common 7000 words! English has a notorious irregular (dare I say flawed) spelling system, frought with exceptions –in fact almost as many exceptions as words that follow the rules. There are twice as many spelling rules (or patterns) as in any other western languages! Of course, Asian students compensate by spending more hours in studying their complex language (as well), but also math, after school. In western countries, the opposite is happening. I have also noticed in N.-A. more and more text-based math books, which require students to be expert readers and considering the point I made earlier, the results are easy to extrapolate. Those 3 elements, alone, must account for the dismal state of acquisition of math. Of course, very few people are interested in tackling any of those issues. Those young people know NOTHING! 🙂

I would like to pick on your point Marie – changing the tests and assessments. Why do teachers teach the way they do? My feeling and experience from working with teachers suggests that teaching is usually informed by assessment frameworks upheld by institutions. If tests require students to recall facts, reproduce algorithms and use rules and procedures, that is what probably most teachers will do in their classes – to help their students pass the tests! As the case of Malta, this is further supported by the fact that students are set into ability groups and the syllabus is actually predominantly content based. Thus, what I see is a change in design – one that focuses primarily on studying assessment – types and models of implementation. My view is one that focuses on backward design – we actually need to start researching about, defining and training teachers to engage with different assessment methods before trying to implement changes to teaching.

Hi James. I couldn’t agree more. As our colleague Hugh Burkhardt says: ‘what you test is what you get’. I tend to think that changes in the curriculum are really only tinkering with things and won’t change anything much – but that a careful rethink of assessment might. However, this page really just reports on what has been identified in the reports as problematic – I try to keep my views out of it!