Published August 2011.

What is a rich mathematical task?

Why would I want to use rich tasks in my maths lessons?

Where can I find rich mathematical tasks for primary children?

I wonder whether you have ever asked yourself any of the above questions. I am hearing from more and more primary teachers who would like to inject something 'extra' into their maths lessons. They each have an underlying reason or reasons to get in touch with NRICH:

- Some feel that they need a change of approach to reinvigorate their mathematics teaching generally;
- Some report that the children in their school do well, but find it difficult to apply their mathematical knowledge and skills to new situations;
- Some would like their pupils to enjoy mathematics more;
- Some are worried that they are not stretching the higher-attaining children;
- Some are concerned that the lower-attaining children are 'turned-off' maths, lack confidence and have almost given up.

Of course this is not an exhaustive list. What you might find surprising is that the professional development we offer at NRICH for all of the above scenarios has a common focus: rich mathematical tasks.

In this article, I will describe the start of a project, which began in the spring term of 2010. Pete Hall, the NCETM East of England Regional Coordinator, contacted me to tell me about a number of £1000 grants on offer to schools who wanted to develop their understanding and use of rich mathematical tasks. The application form was relatively straightforward to complete, requiring some detail about the theme (what it was and why it had been chosen); who would be involved and a commitment to contribute to the NCETM rich tasks community. Schools were required to give a breakdown of how the money would be spent and they promised to submit a short written report to NCETM on completion of the project.

Four schools in the east of England were successfully awarded a grant: Clover Hill Infants' School in Norwich, Harrold Lower School near Bedford, Lakenham Primary School also in Norwich and St Philip's Primary School in Cambridge. Each school decided to spend at least some of their money on professional development run by NRICH and I hope by describing what we have achieved so far, you may feel able to lead one or more staff meetings in your own school without necessarily paying for NRICH support!

At all four schools, I have led an initial workshop, varying in length from a staff meeting to a half day. In all cases, we have begun with having a go at an activity altogether. I feel it is important for everyone to engage in some mathematics - it reminds us what it is like to be a learner and it gives us a common experience (to some extent), which aids subsequent discussion. The problem that I have used in all four schools is Magic Vs. (Do have a go at it if you do not know it. The approach I took with the teachers is exactly the same as that suggested in the teachers' notes on the website.) We spent anything from about thirty to forty-five minutes actually working on the problem itself, with me taking the role of 'teacher', just as I would if I was with a group of children.

Having reached a suitable pausing point, we reflected on what we had done. What mathematical 'content' knowledge did we use as we tackled this problem? By this I mean the aspects of number, calculation, shape and space, data handling and/or measures I needed to know, or I came to know. In terms of the Magic Vs problem, the following list reflects the range of suggestions:

- Odd/even numbers
- Addition/subtraction
- Number bonds
- Consecutive numbers
- Multiplication/division (perhaps)
- Factors/multiples (perhaps)

Next, we asked ourselves what problem-solving strategies we found useful. Here are those that came up frequently:

- Using trial and improvement
- Noticing and explaining patterns
- Working systematically
- Making conjectures
- Tweaking/altering/varying
- Testing ideas
- Generalising
- Talking to each other

I often find it helpful to reflect on mathematical activities in this way, that is considering the 'content' and processes separately. In terms of the Magic Vs problem, it is interesting to note that the 'content' we used was fairly basic, possibly not going beyond that usually met in Key Stage 1. However, we used a vast range of strategies to solve the problem, some of which are rather sophisticated.

So, this led on to further discussion: what makes this Magic Vs problem so 'rich'? Suggestions included:

- It is easy to get started but also has the potential to be taken to high levels of mathematics (what NRICH terms 'low threshold, high ceiling')
- It has more than one answer
- It is 'open-ended', in the sense that although there are some answers, you can go on asking, and pursuing, your own questions
- The way to go about solving the problem is not immediately obvious
- It can be approached in many different ways
- It requires you to use a range of knowledge and skills
- It leads to generalisations
- It might deepen our understanding of odd/even numbers
- It is non-threatening (perhaps linked to the fact everyone can begin to have a go)

By specifically talking about these characteristics, the idea is not to suggest that every problem we use in the classroom should tick all these boxes. Instead, by raising awareness of a set of characteristics, we can understand how resources we already use might be tweaked to make them 'richer'.

This in turn leads to another important point. Although in each session, the participating teachers came up with reasons for Magic Vs being a rich mathematical task, these are not necessarily inherent in the problem itself. Would the teachers have thought it was rich if I had simply handed each one of them a piece of paper with the problem written on it and demanded they work in silence? Some may have reached similar conclusions, but I suspect some would not. So, the potential of a task to be rich is not enough in my opinion. There are two other elements (at least!).

If we want children to get better at solving mathematical problems, then we need to encourage them to think in a mathematical way and to have a range of strategies at their fingertips, which they can draw upon. Therefore, the questions and prompts we use, in conjunction with the tasks we provide, are crucial. In this first session with the teachers, I showed them the 'Primary Questions and Prompts for Mathematical Thinking' book, published by the ATM and give them a taster of its contents. I am a huge fan of this book. The authors define certain activities, which 'typify mathematical thinking', and suggest questions and prompts to encourage these. These suggestions are entirely context-free, in other words they could be used when children are working on any topic, from number to calculating to shape to measuring to data handling. So, the first element to consider in conjunction with using rich tasks is the way we question learners in the classroom.

The second element is what I term the classroom 'culture'. Rich tasks and good questioning will thrive in a classroom where children are encouraged to talk to each other, where they are happy to offer ideas without the fear of being wrong, where their opinions are welcomed. The culture of your classroom reflects your values so in all four schools, we discussed what we value in mathematics and how this affects the way we work in class.

Reflecting on all three of these inter-related aspects of mathematics teaching (rich tasks, questioning and classroom culture) is a lot to cram into a staff meeting, let alone half a day. And I threw in a quick tour of the NRICH website too! Along the way, we talked about the benefits of such an approach. Many children who are currently 'high-attaining' may feel uncomfortable when presented with such tasks. They may not be used to being challenged in mathematics. They may be used to knowing immediately what to do when faced with a problem. However, surely as teachers we have an obligation to equip our children with skills that will carry them in good stead as they get older? Encouraging an ethic of perseverance and the idea of relishing a challenge is part and parcel of mathematics teaching, although it is something that perhaps feels rather daunting as a teacher.

We arranged a date for me to return to each school so there was time for everyone to mull over the first session. All the participating teachers agreed to try out at least one rich task with their children in the intervening weeks. They will come to the second session prepared to talk about their experience: the things that went well and those that didn't go well; the surprises and the lessons learned. We hope then to find some ways forward for each school so that they can build on their achievements and continue to go from strength to strength.