Last year, I quoted seven-year-old Olumide’s description of how extensive classroom discussion “helps with getting my brain on paper” (“Magic Moments”, TES Teacher October 18). That approach to formative assessment, promoted on Ann Neesom’s courses, is as useful for maths as it is for writing. It enables teachers to rethink their use of assessment and find their children’s work transformed.

Jo Wratten, Year 6 teacher at Friars School in the London borough of Southwark, organised numeracy sessions in which each member of her class would be the teacher. She gave them a sheet with 15 completed addition sums, involving numbers of up to five digits. Jo told them that some answers were right and some wrong, and asked them to find which were which.

They were also asked to correct the sums as a teacher would, using a red pencil to write comments on each one. Jo suggested that they think how the feedback could be helpful to children like themselves. After summarising orally the kinds of mistakes they had found, the children worked in pairs, adding their written advice. They knew they needed to be aware of the relationship between understanding and learning.

Aldo’s approach was: “How can I explain in a way they could understand what to do next time, and not discourage them?”

Laura amplified this: “I want to make sure they got a message to help them remember and improve.” Comments focused on general understanding and specific errors. One good annotation was “show your working so I can see what you’ve been thinking.” The children applied this formative approach to what they had done themselves. Bimpe said: “You realise the teacher doesn’t always need to mark your work, if you’ve already done it.” Elizabeth added:

“It helped us to improve, by thinking about those kids who got things wrong. We reminded ourselves about good habits of mind.”

Margo Cassidy, a Year 2 teacher at St Joseph’s School in Camberwell, south London, wanted her children to reflect consciously on what they learned when doing calculations. She provided them with opportunities to talk about how they know whether their arithmetic was right or wrong.

The emphasis was on “reasoning about numbers” rather than simply working through examples. Margo assigned time in the afternoons to this discussion.

The children were told: “I’ll write a statement on the board and you’ll help me find if it’s true.” Margo wrote 43+29=74, and asked: “What would you do to find out if it’s a true statement?”

In response, the children initiated a series of further questions and answers. They talked about how they might use a number line and how they might avoid counting on nine, by jumping ten and counting back one. This series of questions homed in on the answer to the problem. At the same time, it explored the principles underlying the process of finding what a correct answer would be like.

Imran and Fidelia explained what “reasoning” meant in a focused and sensible way: “We investigate different ways of adding and taking away. We find out if things are true or false.” Akinkunmi added: “If the answer you find on the number line is the same as your sum, it’s usually true.”

Margo says it is important that children are not passive receptacles of teaching. Learning mathematics is an active process. “They’re in control of what they’re doing. Most of my assessment is done while they’re actually working. I can see their understanding taking shape, and hear the language they use to formulate what they have come to.”

Dilys Finlay is a Year 2 teacher at Charlotte Sharman School near Elephant and Castle, also in south London. She wanted her children to understand the relationship between clocks and numbers and made a new game for them to play. One card gave them a “start time” to set their clock at. They also had a set of four “time operations” which had to be performed on the clock face, beginning with the start time. These said things such as “25 minutes later” or “quarter of an hour earlier”. To begin with, sets of operations were either all “earlier” or all “later”, but subsequent sets mixed the two. The children had to work in pairs, discussing each move with each other to discover where the clock would be after the four operations were performed.

Dilys also asked: “Does it make a difference what order you do the cards in?” Natalie and Holly’s reply was: “We’ll have to go and see.” The children began to see that they were working with three parallel systems of representing time - verbal, numerical and visual - and that the parallels were consistent. Arran and Kaelyn rephrased Dilys’s question thus: “Will we still get the same finish time? Yes, we do!”

When asked why this was, the children developed a joint explanation: “We’ve all got the same cards to start with. We all start at the same place. It’s like doing sums with plus, even if you turn them around, you end up that it equals the same number.”

These children were collaborators with their teacher in an active process.

It was not assumed simply that teachers teach and learners learn in a simple act of transmission. Knowledge was seen as deriving from interaction between people whose minds are engaged and the children were helping construct this knowledge for themselves.

Working Inside the Black Box: assessment for learning in the classroom by Paul Black, Christine Harrison, Clare Lee, Bethan Marshall and Dylan Wiliam. Tel: King’s College, 020 7848 3189Ann Neesom’s website contains full accounts of the work described here:www.neesom-education.com Advice on assessment for learning by the QCA: www.qca.org.ukca5-14afl