In maths classes in Scotland, there is a tendency to give pupils the answer without allowing them the chance to figure things out for themselves, claims Chris Morrice, a maths teacher at Hyndland Secondary in Glasgow - a bit like shouting out the punchline halfway through a joke, but worse.
So when Mr Morrice wanted to teach S2 children about volume rather than giving them the algorithm v = l x b x h, he set them a homework task. They had to find a 25-litre fish tank online and note down its dimensions. In class the next day, working in groups, they were set the challenge of unravelling what they had to do with those three numbers - the length, breadth and height - in order to get the volume.
"Although it took them a long time to figure out the formula, it was more worthwhile and engaging for them and me," he says.
To improve Scottish children's performance in maths, this is the kind of lesson Education Scotland wants to see more of when its inspectors visit schools - pupils working in groups, collaborating with each other and problem solving.
There is still too much "procedural thinking" and not enough "exploratory talk" in maths lessons, says Graeme Logan, an assistant director at Education Scotland. Or too much "barren duplication" and "memorising of techniques" instead of problem solving, as Gill Stewart, director of qualifications development at the Scottish Qualifications Authority (SQA), puts it.
Children should be asking "why" more in maths, says Mr Logan, and be encouraged to debate, interact, interrogate and challenge each other during lessons.
This is an increasingly pressing issue because numeracy is one of the three areas, along with literacy and health and well-being, that all Scottish teachers are responsible for developing under Curriculum for Excellence.
But recently some educationalists have claimed that literacy and numeracy are not being placed on an equal footing. Numeracy is literacy's poor cousin, they argue.
In a recent letter to TESS, Professor Sally Brown, convener of the Royal Society of Edinburgh's education committee, expressed disappointment that regulatory body the General Teaching Council for Scotland was proposing new entry requirements for teacher training that included a Higher or equivalent in a language, but not in maths or a science. (Trainee teachers are already expected to have Higher English.)
"As they stand, the proposals send an alarming signal about the relative importance of science and maths in comparison with languages," Professor Brown wrote.
In the wake of this criticism, at a national numeracy conference earlier this month, Alasdair Allan, minister for learning, science and Scotland's languages, felt compelled to reaffirm the government's "strong commitment" to numeracy. There is the perception in some quarters, he said, that "numeracy is not given the same prominence as literacy in Scottish education, and in schools in particular".
But it was "a vital life skill", he said, without which young people would struggle with the increasing complexity and demands of modern life, from energy company tariffs and credit cards to the consequences of payday loans and online gambling. Indeed, the government was committed to making "significant progress" towards eradicating innumeracy by 2016, he said.
Perhaps the situation isn't as bad as some fear. According to Mr Logan, the teaching of maths and numeracy in Scottish schools is improving. Recent inspections and the Scottish Survey of Numeracy, published for the first time last year, found a wide variety of resources being used for maths in schools, including a broad range of ICT, he says. Textbooks were still being used in classrooms, but to supplement learning, not lead it, he adds.
There is, however, plenty of room for improvement, Mr Logan admits. As well as more dialogue and discussion in maths lessons, he calls on teachers to place the subject in "a real-life context" - for it to be used to plan trips and events, for instance.
"For children to be secure in their knowledge, we know that they have to be given the chance to apply their learning. We need to plan for that and create more opportunities. We're not seeing a lot of explicit examples of that at the moment."
Mr Logan also questions how clear teachers are about the standard of numeracy they should expect, especially in secondary schools.
The numeracy survey showed that while maths specialists and primary teachers were comfortable with teaching numeracy across the curriculum, one-third (30 per cent) of secondary teachers were not. It also showed that 32 per cent of S2 students were not performing at their level for maths, compared with 2 per cent at P4 and P7.
At that time, however, experts warned against concluding that numeracy learning was in very good shape in primary and that everything went badly wrong in the early secondary years. Learning in numeracy was cumulative, they stressed.
So how does this mixed report fair internationally? According to Dr Stewart, Scotland is "doing OK" in maths by comparison with similar economies - the Programme for International Student Assessment (PISA) survey in 2009 found that Scottish pupils' performance was about average. Children in Scotland performed as well as those in England, Slovenia and Sweden; not as well as those in Canada, Germany and New Zealand; and better than children in Italy, Spain and the US.
Participation rates in maths, meanwhile, are "relatively high", says Dr Stewart. In Scotland, 48 per cent of students study maths post-16, compared with 20 per cent of students in England, more than 90 per cent of students in Germany and Hong Kong and more than 65 per cent of those in New Zealand and the US, according to a report published by the Nuffield Foundation earlier this year.
Dr Stewart is optimistic that the new qualifications being introduced to support Curriculum for Excellence will increase participation further, including numeracy units, life-skill maths courses, Advanced Higher mathematics of mechanics, Advanced Higher statistics and the personal finance award.
Many of the new National qualifications, which most schools have already begun to teach, have been designed to improve numeracy skills in context - in social subjects, for instance, pupils will be required to show that they can analyse charts and data.
The numeracy survey carried out last year, which highlighted poor performance in S2, was not assessing a Curriculum for Excellence cohort, some commentators have pointed out. Next year's survey will give a clearer picture of where Scotland is and whether Curriculum for Excellence and its emphasis on improving numeracy is making a difference, they say.
But before that will come a survey by which many will set much greater store - the latest PISA international education survey from the Organisation for Economic Cooperation and Development. Due at the end of the year, it is likely to set the tone for the next few years.
32% - Proportion of S2 pupils not working at the appropriate stage in maths
Less than 2% - Proportion of P4 and P7 pupils not working at the appropriate stage in maths
More than 60% - Proportion of pupils in P4, P7 and S2 who said the activity they experienced most in maths was "listen to the teacher talk"
30% - Proportion of secondary teachers not confident that they understand how to teach numeracy across the curriculum
92% - Proportion of primary teachers confident to teach numeracy across the curriculum
48% - Proportion of students who study maths post-16 in Scotland
Sources: Towards universal participation in post-16 mathematics, Nuffield Foundation (2013); Scottish Survey of Numeracy, Scottish government (2012)
HOW GLASGOW SET OUT TO RAISE THE BAR IN LITERACY AND NUMERACY
Glasgow City Council has responsibility for 30 secondaries, 140 primaries, 47 special schools and 113 nurseries, serving a range of schools from the leafy suburbs to the inner city. In 2009, it set out to try to ensure a consistent level of maths teaching across the board.
"We had to tackle underachievement in literacy and numeracy," explains the authority's quality improvement officer for maths, Stephen Watters."We also had to grasp the idea of "responsibility of all" in Curriculum for Excellence (CfE)."
Under CfE, all teachers, regardless of subject specialism, are responsible for improving children's literacy, numeracy and health and well-being.
The authority set up a numeracy reference group with around 175 primary, secondary and early years practitioners split into three cross-sector groups - CfE early level and P1; CfE first level; and CfE second and third level. The groups were charged with working together to understand the experiences and outcomes and principles and practices.
"Getting the ethos of the groups right at the start was probably the hardest part," Mr Watters says.
People felt overwhelmed by documentation, he adds, so the teachers were given five days out of school over a six-month period, to reflect on new developments and materials being delivered.
Ultimately, the groups developed a skills continuum for each of the eight numeracy organisers, highlighting the main skills that teachers should be taking forward at the different levels.
"It was not about saying to teachers, 'This is what you must do.' It was about highlighting the skills that they should expect at each level to inform their planning," Mr Watters says.
The teams also developed descriptions of what a secure learner in maths looked like at the different CfE levels.
In February - roughly six months after the groups had been set up - they disseminated their learning across the local authority and delivered in-service training to colleagues.
As a result of this initiative, the city has raised understanding of the standards; professional capacity and autonomy; and built confidence. It has also developed links across the sectors, Mr Watters says.
In 2008, the proportion of children in Glasgow achieving at least SCQF level 3 in maths was 89 per cent; by last year that proportion had risen to 96 per cent.
Reference groups have now been set up to identify the precise characteristics of particular levels across all curricular areas.