“I love how we managed to get the #Edexcelmaths onto the news yet we couldn’t figure out if Karl made £24 or £10...”
So wrote 16-year-old Emily in June, as GCSE students who had just taken a non-calculator maths paper by the Edexcel exam board used the hashtag to air their disbelief at the challenging problems they had just faced. Unable to fathom how much money the fictional “Karl” gave to charity, establish whether “Mary” could afford some tiles or—most notoriously in the outraged chatter that followed—prove an equation based on “Hannah’s” taste in sweets, they vented their frustrations on the social media site. Their posts were picked up by reporters across the British press, and even discussed on the BBC’s Today programme the next morning.
Emily and her contemporaries’ plaintive selfies and indignant complaints (a petition calling for Edexcel to lower its grade boundaries reached nearly 40,000 signatures) were newsworthy because they tapped into a perennial debate about the mathematical ability of British youth. That Britain is bad at maths is now the consensus among the public, politicians and journalists. A nation with more Nobel laureates than anywhere other than the US, boasting a catalogue of iconic mathematicians from Charles Babbage to the codebreakers of Bletchley Park, has allowed innumeracy to become a defining national characteristic.
Successive governments have banged their heads against this problem for decades, and now the current one is having a go. It has put maths at the centre of its education reforms—it is keen, says Schools Minister Nick Gibb, ultimately to bring our standards in line with Shanghai, where, at 15, students are three years’ ahead. Students will return to school to begin work on a tougher GCSE curriculum—and by the time they hit sixth form they’ll have a matching A-Level to contend with. Some 16-19 year olds will start brand new “core maths” qualifications from September, and this year’s primary school trials of Shanghai teaching methods will be rolled out into secondaries. But are these the right solutions to Britain’s numbers problem? How bad is the situation, and will these latest improvements actually benefit us?
While some hysterical headlines overstep the mark, it isn’t unreasonable to gripe that we aren’t good enough with numbers. A 2014 analysis by the Organisation for Economic Cooperation and Development (OECD) found that the relatively high achieving children of British professionals performed worse on internationally standardised Pisa tests than the children of Singapore’s manual workers, who themselves trailed behind their peers from most other social classes in the south-east Asian region. A 2015 OECD study which looked at more than just Pisa test results—which some deem unreliable as a comparative tool—using a range of international assessments in maths and science put the United Kingdom 20th in the world. That is not too bad for Europe but is behind Germany, Ireland, Slovenia and a host of Asian nations. Our graduates have been ranked among the worst in the developed world for their maths skill, and those who study subjects other than maths at university often encounter little if any mathematics once they leave school.
We drop out early, too. In contrast to most other developed countries, where the majority of students study maths until 18, only one in five pupils in England, Wales and Northern Ireland do. The rest pull out at 16, just like Gibb himself (he wishes “somebody had said to me how important it is”), David Cameron, and (full disclosure) your writer. This deficiency has been highlighted by employers’ groups including the Confederation of British Industry and the Institute of Directors as contributing to a skills shortage in the UK: “Maths study for a retail apprentice and an A-Level student will be fundamentally different, but it is vital that both are undertaken,” the CBI said in 2012. As Education Secretary, Michael Gove (predecessor to the incumbent Nicky Morgan) said he wanted to reverse this, and hoped to have the “vast majority” of pupils studying maths to 18 by the end of the decade.
Campaign group National Numeracy says that the proportion of working adults with basic GCSE-level maths skills has dropped from 26 per cent in 2003 to 22 per cent in 2011. Four in five adults therefore have what they class as a low level of numeracy. A 2013 OECD report found that, in England, students leaving school with an array of impressive-sounding qualifications today had no greater mathematical ability than those who left school with much less to show for it in the 1960s and 70s. In 2011, 68 per cent of students got an A*-C grade at GCSE, but only 24 per cent of those got an equivalent grade in the more practical “skills for life” assessment.
Changing this isn’t about giving our politicians some new figures to wave around at international conferences. There are serious benefits to be had for individual children and for the country. National Numeracy estimates that our poor number skills cost the UK £20.2bn a year. The Royal Academy of Engineering says that the UK will need a million new science, engineering and technology professionals by 2020.
But perhaps more importantly, granting them a better know-ledge of maths is the best way to ensure children can take advantage of opportunities in new, high-skilled industries, and not be left behind by the pace of change. “The ability to understand and interpret data,” concludes the British Academy’s “Count Us In” report this year, “is an essential feature of life in the 21st century; vital for the economy, for our society and for us as individuals.”
In the eastern regions of North America, understanding mathematics is life or death for one species of insect. Broods of cicadas emerge from slumber into their forest habitats at confusing intervals: depending on the location, there might be 13 or 17 years of silence, punctuated for one summer by the chirping of thousands of bugs, whose unusual life cycle means they emerge only for six weeks before laying eggs and dying. The reason behind this is one of the mathematician Marcus du Sautoy’s favourite mathematical stories—a story he has told classes of children to excite them about what maths can do in the real world.
The cicadas’ timing is not incidental. Thirteen and 17 are both prime numbers, divisible only by themselves and one. Most predators, on the other hand, have a 2-10 year population cycle; they might be at their most numerous every two, four, six, eight or 10 years. A cicada which emerged every 12 years would, therefore, be far more likely to spring forth into the beak of a waiting bird. Students can attempt to work out a life cycle which will keep the cicadas safe, finally overcoming their dismay at continually being eaten by predators as they grasp through practice the principle which real cicadas know by instinct—and with it the notion of prime numbers.
Du Sautoy thinks such stories haven’t been told enough in conventional school curricula, rendering them boring. “I often compare it to learning a musical instrument,” he says. “If you just did scales and arpeggios you would give up your instrument very quickly… [and] you’d never play anybody some real music.”
We certainly have a cultural problem with maths. “The UK remains one of the few advanced nations where it is socially acceptable... to profess an inability to cope with mathematics,” Peter Williams, then Chancellor of Leicester University, concluded in a landmark report in 2008.
“There is a big myth holding millions and millions of people back across the UK—and that myth is that some people can do maths and some people can’t,” National Numeracy chief executive Mike Ellicock said last year. A very small proportion of the UK—5 per cent at most—suffers from dyscalculia, and has difficulty processing numbers because of brain abnormailities, but at least the rest of the country should be capable of doing basic maths.
Rob Eastaway, a mathematician who aims at popularising maths, says there’s a clear distinction between less and more able students. “The ones who fly through maths… they kind of get this notion of the abstract in maths. They discover patterns, they will see a beauty in a proof… they need no extra convincing.” But such students are comparatively rare: “You don’t have to go very far down before… there is a need to see what the relevance is to their lives.”
“if you think about life beyond school and beyond examinations, what do you do with mathematics? You use it for a particular purpose.”It’s an issue that the education community—and the government—is aware of, and there’s an ever-greater emphasis on problem solving in maths education using real-world scenarios. Eastaway has just staged a series of lectures for students working toward GCSEs called “What’s the Point?” One particularly popular session explained the concept of a “locus” (a path formed by a point moving in accordance with a rule) to show why rescuers searching for the vanished Malaysia Airlines flight MH370 were so uncertain of its whereabouts.
From this September, students will begin studying for a reformed GCSE qualification with a far greater emphasis on problem solving. “The new qualification actually demands that students think mathematically, rather than simply regurgitate routines that they’ve picked up on,” says Eddie Wilde, maths team manager at the exam board OCR, of their version of the new GCSE.
A greater focus on problem solving gets to the heart of what is most socially useful about learning maths. “If you don’t [incorporate problem solving] you don’t do the economy any service and you don’t do the students any service,” says Wilde, “if you think about life beyond school and beyond examinations, what do you do with mathematics? You use it for a particular purpose.” It could also address the perennial complaint from universities and employers that even when students leave school with maths qualifications, they struggle to apply their skills in unfamiliar situations. The social usefulness of mathematical problem solving is something which has long occupied the Cambridge University mathematician Timothy Gowers. “If you had a generation of people who were better trained in spotting bullshit,” he says—in other words, if you could educate them in thinking rationally in a manner which didn’t alienate non-mathematicians—“it would be harder for politicians to get away with the sort of rhetoric that they get away with now.” If children leave school taught only to do the sort of formulaic maths which comes up in many exams, their ability to think rationally about complex real-life problems will be diminished and the tone of public debate lowered. They’ll also be at a disadvantage in the workplace, where even for less obviously mathematical jobs—Gowers suggests a Post Office manager arranging Christmas cover—those who can use mathematical reasoning will be at an advantage.
In 2012, Gowers posted a blog outlining his thoughts on how Gove’s intentions to popularise the study of maths after 16 should be used as an opportunity. He wrote of his objections to a previous attempt to create a “Use of Mathematics” A-Level: “One might think that a course called ‘Use of Mathematics’ would teach you how to come up with mathematical models for real-life situations,” he argued, “but these questions did the opposite.”
Gowers’s complaint was that too many “real-world” maths questions would simply describe a situation, then ask a series of purely mathematical questions linked only spuriously to the context. Instead, he proposed questions which “start with the real world rather than starting with mathematics.” “My ideal question,” he explains to me, “would be a question with a yes/no answer that people are likely to have differing positions on and doesn’t mention mathematics at all, but in order to answer well you have to work out what mathematics might be useful.” Some examples on his blog include “What are the chances that at some point in the last five years somebody in the UK dreamt that a loved one had died, only for that loved one to die unexpectedly the very next day?” and “How much can we trust opinion polls?”
Gowers’s work was noticed by the Chief Executive of Mathematics in Education and Industry (MEI), a charity, and the DfE funded MEI to investigate how these ideas could influence a new course for students who don’t currently study maths after 16. MEI has now also produced two new qualifications with the exam board OCR; the Level 3 Certificates in quantitative reasoning and problem solving. Both will be taught this September as two of six new “core” qualifications launched by exam boards and the government and aimed at students who achieved at least a C in GCSE maths but don’t want to study the full A-Level.
The courses focus on maths you can use in everyday life and work. The idea has been welcomed by the British Academy, some universities and the Institute of Education’s Celia Hoyles, a former maths tsar. But the key measure of their success will be how many students actually take the courses. Universities and employers’ groups will also have a watchful eye on whether graduates of core maths courses are really better at adapting maths to a range of contexts. Students at more than 170 pilot schools and colleges which have been teaching core maths courses since last year will finish in 2016, which might give some early indication.
“All secretaries of state should know what seven eights are.”Eastaway thinks that we are moving in the right direction. But he warns that such an approach can be difficult for time-stretched teachers to take up. “You can solve problems quickly or you can take ages, and its very unstructuredness makes it really hard to fit that into a lesson,” he says. Lesson planning was cited by 38 per cent of teachers as contributing to unnecessary workload in a Department for Education consultation earlier this year; many might feel that they don’t have the time to devise innovative lesson structures. Hoyles would like teachers who will deliver the reformed maths curriculum to be given time to brush up on their skills and seek appropriate training.
“There’s really no substitute for just knowing that stuff and just knowing it instantly. All secretaries of state should know what seven eights are.” Andrew King, head of London primary school Chase Bridge, a former inspector and an author of children’s maths books, is broadly supportive of the rigour in the new primary maths curriculum. Introduced in 2014, it places a greater emphasis on students knowing certain fundamentals off by heart at a younger age. By the end of Year 2, they should be able to do simple additions using numbers up to 20 almost instantly, Year 4 should see them conquer the times tables up to 12, and they should be working easily with fractions, decimals and percentages by the end of primary.
This focus on repetition and recall is often maligned as evidence of this government’s Gradgrindian impulses. But having immediate access to formulas, algorithms and what King calls “number facts” can help pupils to deepen their understanding of maths, seeing patterns as they arise and thinking better on their feet.
The same thinking lies behind the government’s affection for Shanghai teaching methods. Earlier this year, 30 teachers from the region were flown in to schools across the UK to demonstrate the methods which make them the envy of western politicians—what Gibb describes as a “mastery model.”
“Every child in that class will be expected to reach a minimum, and a very high minimum, level of arithmetic,” says Gibb, who observed one of the lessons just before the election. The 35-minute lesson was spent teaching a class how to multiply double digit numbers ending in a zero. The contrast with traditional British teaching, he says, is that the teacher focused solely on working through one method, bit by bit, until every child understood it, “so that… they become fluent.”
It is tempting to see such methods as promoting blind rote learning. But the opposite is true, says Shahed Ahmed, head of Elmhurst Primary School in London, who accompanied Gibb’s predecessor Liz Truss on a visit to Shanghai. “We thought it would just be a lot of robotic types learning, chanting, and it’s not. It’s far from it,” says Ahmed. “They actually make sure there’s deeper understanding of the concepts in mathematics far, far more than we do.” Gibb also cites an OECD survey which puts Shanghai top internationally for financial literacy: “They don’t teach financial literacy… in Shanghai, but they’ve done very well in those financial literacy tests because they have a very sophisticated mathematical ability.”
Many experts have also voiced concerns that Britain and Shanghai are culturally too distinct for these teaching methods to be imported—for example, after-school tuition is much more common in Shanghai. Hoyles says there are important differences, but thinks that a compromise will emerge: “There’s no way our schools are going to be like Shanghai… They’re not,” she says, “[But] some of the people who’ve been over there have been wildly enthused by what they’ve seen… Hopefully [they’ll] keep the key mathematical ideas.”
Any reforms to maths education will suffer from something of a Catch 22: because our population isn’t great with numbers, it can be difficult to find new teachers—or train existing ones—to deliver a tougher curriculum. King points out that primary schools—where teachers are largely generalists—have limited access to maths specialists, adding that “if you’re teaching at the edge of your knowledge you don’t make a very effective teacher because it is unlikely that you will understand the deeper connections within maths.” He thinks there is currently “a little bit of a vacuum” in professional development. The Department for Education last year launched a network of “maths hubs” led by outstanding schools and colleges aimed at delivering training and spreading best practice, and hopes to bring 17,500 extra maths and physics teachers into the classroom. Of these, 15,000 would be non-specialist teachers who would be retrained.
The reforms to Britain’s maths teaching have the potential to be significant. That is not so much for their headline-grabbing toughness as for their focus on developing students’ reasoning skills and ability to apply maths. They will stand or fall on the ability of Britain’s teachers to teach a curriculum which demands greater subject knowledge and flexibility, and the willingness of government to help them do so. If both play their part well, the results could be spectacular. But it’s important that the country watches closely; we can’t tolerate many more years of national jokes along the lines of Hannah and her sweets.