Richard Dunne, Maths author and consultant, recently seen in the Channel 4 Dispatches programme Kids Don't Count
I am concerned that there is a massive problem with maths teaching in England. It has been made too complicated and places a ceiling on children's achievement.
Mathematics, with the wonderful clarity of its logical symbolic language, is intrinsically exciting. I want there to be greater emphasis on its being interesting in its own right. Of course, it must also be used for problem-solving and I appreciate that it sounds reasonable to teach maths in context. But I see time and time again how this confuses children.
It is this approach that I see as the cause of the problem in maths teaching. The emphasis on using real-life contexts makes maths difficult for children. Real-life was not designed for learning maths.
I want to see maths teaching based not on real-life but on a carefully designed learning system which mimics the essence of mathematics. It is the clarity of a learning system, accurately conveying the clarity of maths, that creates a love of maths for its own sake, and an understanding of maths that can be used to model real-life contexts.
I am sure that every teaching professional is committed to improving children's life-chances. Sadly, I see so much effort and money is being spent on trying to make better, more efficient, what is already being done. I am saddened by this tragic journey of doing more of the same, rather than doing something completely different. It is a journey of more regulation, more tracking, more anxiety and less maths.
The learning system approach deserves detailed study by teachers. I want to see it taught in initial training and studied as professional development, especially in post-graduate study. I want to see it examined for its emphasis on preventing misconceptions rather than mending them and for its capacity for making abstract ideas accessible.
Rob Eastaway, Co-author of Maths for Mums and Dads and past president of the Maths Association
First, in a lot of ways things have got a lot better. As someone who has been going into primary schools for 15 years, children's ability to do mental arithmetic has improved and it is much harder for a primary teacher to hide it if they are really weak at maths.
But the thing that always concerns me is that teachers might be competent in numeracy but have no love for maths, no hinterland. One solution to this is the Williams report proposing a maths specialist in every school. I would also love to see more people with maths degrees or A-level going into primary teaching.
One big fact I'm surprised by is the complete mismatch between methods that parents know and those taught in primary schools. Children are now being taught the grid method, chunking and Carroll diagrams. Parents may know methods for long multiplication and long division, but can't help their children because they don't do it that way.
Better communication between schools and parents would help parents understand how maths is being taught and would make a big difference to children.
There also does seem to be such a distraction caused by the Year 6 Sats tests - I hardly ever come across a school that has a good word to say for them. I once heard a teacher tell a colleague who did not have to do Sats that they were lucky because their Year 6s could have fun and do real maths.
Margaret Brown, Professor of mathematics education, King's College London
People are unsure about what they are supposed to be doing now. The national numeracy strategy was very prescriptive, but that has broken up now and schools are rethinking what they are doing.
There is a position now where the older teachers were well-trained in the strategy and new teachers have very little training. The strategies are coming to an end and consultants are beginning to leave already.
It is acknowledged that there were some things the strategy didn't do very well. Two things are most obvious.
First, helping low attaining children - the strategy moved too far and too fast. Now there is still some concern about how fast teachers should be moving and there is still a feeling that the curriculum moves on from one thing to another before children get hold of a subject.
The other thing was using and applying mathematics. At key stage 3 and key stage 4 there is a move to a problem-solving curriculum.
We need to get more genuine problem solving into the primary curriculum, and not just little word problems, such as: "You have five rows of chairs in four rows, how many chairs are there?"
This is not a problem, it is just deciding what operation to use. A problem is something like: "You are having a class party, how much food do you need?"
What is really important for maths teachers is to have a deep understanding of the maths you are teaching and a view of where it goes. But you don't need A-level maths. You don't have to know quadratic equations. But you do need a deep understanding of decimals and numbers.
Violet Avenell, Head of Wroughton Infants School, Swindon. As assistant head at her previous school, Moredon Primary in Swindon, her work on improving children's problem-solving skills won The TES award for outstanding numeracy initiative in 2009
I went to China in 2007 and noticed they had huge classes of children and no differentiation.
It looked at first as if children were sitting on their own, but then every lesson they would turn to each other in groups and talk about their learning.
I thought I'd try something similar with my less able Year 6 maths class. I took big sheets of colourful paper, which had problems on them. They were laminated so the children could write on and rub off their workings.
I asked them to work on the problems in groups and the effect was almost instantaneous. Not having to do it on their own they weren't under pressure and worried about appearing stupid. At the end of the year, 11 of the 24 students had level 5 and all had level 4.
It is good for teachers. You walk around and the pupils are talking and getting quite animated and you really pick up on misunderstandings. If you just sit and question them as a class, they won't put their hand up unless they know the answer, so it's harder to address those problems.
Jo Boaler, Professor of education at Sussex University and author of The Elephant in the Classroom
There are three main areas that are going wrong. One issue is that children in school have a much too narrow variety of maths.
It's about having methods demonstrated and reproducing them - giving children a very narrow view of maths that's part of the problem. It turns children off the subject and creates anxiety.
The second problem is there is too much too early. The curriculum has lots of methods and teachers respond with having to teach method after method. I don't blame teachers for what happens in schools, I blame the curriculum and tests. If you give teachers hundreds of methods and a child is tested on hundreds of methods, that is not helpful for children.
The third point is that maths is taught as an elitist subject. This is to do with the way students are grouped. You get ability groups in primary schools from age seven, you have two sets, sometimes three, which means half or two-thirds of the children are told they are no good at maths.
The irony is that all the research on the impact of setting shows it to be negative.
I understand it is easier for teachers - certainly a lot of parents think it is a good idea - but I know from research it holds back children terribly and it is very bad for their development.
When I followed 700 students in the US over four years, some in sets, some in mixed groups, aged between 13 and 18, those in mixed groups did significantly better.
The ones who advanced the most were the high achievers, who did significantly better than those in the top sets.
Talking about maths and often having to explain things to other children is very helpful in clarifying their own understanding.
Ann Dowker, Psychology lecturer at the University of Oxford, lead researcher on the Catch Up Numeracy intervention project and an adviser on the Government's Every Child Counts project
My view is that primary maths and maths in general is not in such dire straits that there has to be a significant improvement.
But there is one very important point, which is that there is still a rather significant tail of underachievers in maths.
Every Child Counts is addressing some of those issues.
I think that it is important to have interventions for children who are struggling in maths quite early on in their school career before they start seriously falling behind and particularly before they become anxious or even phobic about maths.
It is also important that interventions should take into account that mathematical ability is not a single entity.
There are a lot of components of mathematical performance and mathematical thinking that children and adults can perform very well in some of those areas while having difficulties in others.
I think mathematical competence is important in primary teachers but it is not a matter of even higher qualifications.
Some people have very high maths qualifications but do not necessarily communicate maths well to children and vice versa.
I also think there is a general problem with our culture, that maths at least until recently has not been valued to the same extent as literacy and children with mathematical difficulties did not get the same amount of interest as those with dyslexia until very recently.
Interviews by Helen Ward.