Numerous bright ideas have been suggested for making pupils better at maths. Recently, the English schools minister Nick Gibb called for a review of calculator use in primary schools, fearing they prevent children mastering basic arithmetic. He also argued that pupils need to spend more time on times tables, as "you can't expect children to cope with complicated quadratic equations if they don't know their times tables by heart".
However, neither solution appears in a list of recommendations to improve maths teaching, published recently in a major cross-European study.
The report, Mathematics Education in Europe: common challenges and national policies, was produced by Eurydice, the European Commission's education research network, which analysed a wide range of studies and results from 31 European Union countries.
It notes that there are many areas where all countries in Europe could improve to help them compete better with those at the top of the global rankings.
Some of the main conclusions are not directly relevant to the UK because schools or officials here are already doing them.
On the potential benefits of ICT, it argues that "research has not found conclusive evidence about any definite benefits" for maths teaching, but also cites a range of studies that have shown that computers can improve pupils' engagement with the subject.
Most EU countries seem to be making little use of computers. The only countries where significant proportions of teachers reported that 9- and 10-year-old pupils frequently used computers for practising mathematical skills were the Netherlands, England and Scotland.
However, most of the recommendations show areas where Britain could do better, too. The key conclusions for teachers are that they should use a greater variety of teaching methods, including looking at real-world problems, and that more needs to be done to make pupils engaged and motivated to learn maths.
"Research evidence suggests that effective mathematics instruction involves the use of a variety of teaching methods," the Eurydice report states.
"At the same time, there is general agreement that certain methods such as problem-based learning, investigation and contextualisation are particularly effective for raising achievement and improving students' attitudes toward mathematics."
The study it gives as evidence is a British one - Mathematics Matters, a one-year research project by the National Centre for Excellence in the Teaching of Mathematics (NCETM), published in 2008. It found that a variety of teaching methods were needed because maths involved many different types of learning. Four particularly valuable types of learning, highlighted by the centre - and by Eurydice - are:
fluency in recalling facts and performing skills;
conceptual understanding and interpretations for representations;
strategies for investigation and problem-solving;
appreciation of the power of mathematics in society.
The British study was led by Malcolm Swan, professor of mathematics education at the University of Nottingham. According to Professor Swan, fluency is perhaps best taught using traditional methods such as explanation, example and exercise. But the other skills require more varied approaches.
"Conceptual understanding is about knowing what things mean - being able to conjure up images and associations - so that 5.3 can be imagined as a spot on a number line, but also as a measurement. To get that kind of understanding, it's fairly well accepted that what is needed is some active classroom discussion," he says.
Professor Swan has produced a set of resources called Improving Learning in Maths to help create that kind of lesson. The unit on "understanding perimeter and area" begins with one of the greatest motivators known to humankind: "Show the group a large bar of chocolate (real if possible) ."
Pupils then work together to discover how different rectangles can have varying perimeters, even if the number of chocolate squares (the area) stays the same. "If you have a rich conceptual understanding, then if you forget a formula you can rebuild it yourself," says Professor Swan.
"I don't think you tend to forget things if you have come to understand them well."
Problem-solving differs from this in that the focus is on strategies and investigations rather than concepts.
An example lesson is to show pupils a sign on a vet's door that says: "In 18 months a cat can have 2,000 offspring - get your cat neutered."
"Is that sign reasonable?" asks Professor Swan. "We give students some figures, such as about how long a cat's pregnancy lasts, and ask them to go away and come up with an answer."
This time the lesson structure changes again. It is not just the answer that is important, but how students achieve it - and, importantly, how it can help them explore the power of mathematical methods.
Many teachers in Britain have long taught such enquiry and problem-based lessons.
"I'm inclined to think we're probably a bit ahead of many other European countries in using a variety of strategies," says Jane Imrie, deputy director of the NCETM. "There's been a big push here over the past 10 years or so to get teachers using a great variety of lessons and more interactive ways of teaching maths."
The Eurydice report makes it clear that employers want mathematical competence, not just basic numeracy. Competence refers to the ability to reason mathematically, to pose and solve mathematical questions and to apply mathematical thinking to real-life problems.
One of the biggest and most influential research projects in education in the past decade, Visible Learning by Professor John Hattie of Auckland University, found that "the use of real-world applications of mathematics has a very slightly negative impact".
One reason may be that supposed "real world" examples teachers use often add confusion - and are, in fact, unrealistic. Well-intentioned attempts to make maths problems less abstract grew more common in the 1970s and 1980s.
"It is great if maths classrooms can involve real data that students use and explore, such as shadows in the playground that they can measure when learning about shapes or trigonometry," says Jo Boaler of Stanford University. "But it is not good if they are given fake, pseudo- contexts."
In her book, The Elephant in the Classroom: helping children learn and love maths, Professor Boaler describes some of the "ridiculous" problems used in maths classrooms, where trains travel toward each other on the same tracks and people paint houses at identical speeds all day.
Making maths genuinely relevant is seen as vital for pupils' motivation. "At school, and also in wider society, mathematics is sometimes perceived as a difficult and abstract subject which involves learning a lot of processes and formulae," the Eurydice report states.
"Improving student motivation to learn mathematics is important for raising school attainment, for increasing the numbers of students choosing mathematics-related subjects beyond secondary education and for encouraging young people to pursue careers in fields requiring high levels of mathematical knowledge."
The report stresses the importance of using creative approaches to overcome any negative attitudes. One teacher in no doubt about the importance of motivating pupils is Matt Parker, who combines doing mathematical stand-up comedy with working as a "maths communicator" at Queen Mary, University of London.
"I have taught the bottom set Year 10 (S3)," he says. "The pupils who, every year, are still being forced to do the maths that they failed to do and didn't understand in Year 7 (P7). My strategy was to do lessons that didn't require previous knowledge - so we started with card tricks. The idea was to get their confidence back by showing them logic and patterns, then gradually move to tricks using basic number skills.
"We need mathematicians to drive the economy and it's a worthwhile investment to get those top kids and strengthen their skills, but it's also important to motivate everyone else, even those who may never do maths again."
The report says the majority of countries in Europe have changed their maths curricula over the past decade to a greater focus on competences and skills rather than content to be covered, and have increased the emphasis on cross-curricular links and problem-solving. It suggests the trend is positive, but says teachers have often failed to translate these ambitions to the classroom.
"There is an ongoing debate about whether the use of calculators improves or hinders student achievement in mathematics," it states, but does not suggest they should be prohibited for any age group. Most studies, including Visible Learning, "conclude that calculators might be useful, but only for specific activities", it adds.
England and Scotland are noted alongside Sweden and Norway as the countries with the fewest schools to prohibit calculators, but it is pupils in Denmark who use them the most.
"In general, even in those countries where calculators were widely allowed, teachers rarely reported using calculators often."
How much homework?
A strong link appears to exist in the UK between how much homework teachers give and how well pupils do at maths, the Eurydice report notes. However, the study suggests that schools in England are sensible to limit homework.
It refers to research with 9- and 10-year-olds showing that in England "the 18 per cent of students whose teachers reported giving relatively long homework assignments on a relatively frequent basis scored on average 552 points in mathematics; the 23 per cent in the medium category scored an average of 520 points, and the 59 per cent whose teachers gave little homework on average scored 499".
The report from Eurydice recommends the following:
Effective maths teaching should involve using a variety of methods;
Teachers should widen their repertoires of teaching methods;
Improving students' motivation can increase attainment and should not just be targeted at high-achieving students;
Teachers need greater support in how to provide feedback to students on assessments;
Achievement targets and monitoring are needed for low-achieving pupils;
The majority of European countries have revised their curricula to focus more on students' skills than on teaching content - but often that has not translated to the classroom;
Government policies on maths should be evidence-based.