Children\'s Mathematics

to understand. We saw new meanings in these marks that children make and we

knew that this was the key to children’s understanding of their mathematics. This

will help them make the connections into more abstract forms of mathematics. If

they are allowed this empowerment they will be active in their thinking of mathe-

matics. The teacher’s role in this assessment is crucial because this is what will

inform her knowledge of where children might be in their thinking and how she can

support and extend this.

Torrance (2001) identifies two kinds of assessment. In ‘convergent assessment’ the

major emphasis is to work out if the learner can do a task that has been previously

set, the features of which are detailed planning with no flexibility and that the ques-

tions and tasks are closed and restricted. This kind of assessment is teacher and cur-

riculum dominated and the child’s meaning is not taken into consideration. The

other type of assessment highlighted by Torrance was ‘divergent assessment’ which

focuses on the child’s thinking rather than the teacher’s agenda. It is about not a

testing mode but a way of finding out and uncovering what the children know, so

that the teacher can work from there. Torrance’s study made the distinction between

‘help’ questions and ‘testing’ questions. In testing questions the children gave the

answer that they thought the teacher wanted to hear. Some classrooms are domi-

nated by testing questions. This makes it difficult for children to reveal what they

know and work out their own meaning because they are too busy trying to work out

what the teacher means. Testing questions do not help the process of learning.

Testing questions look for right and wrong answers, they are intent on the product

not the process.

If we only look at children’s written mathematics in terms of right and wrong

answers then it will tell us nothing to support the child in her understanding or aid

the teacher in her teaching. If we believe that mathematics is not just a set of rules

to remember then we must also respond to children’s mathematical representations

in a more flexible way. Divergent assessment is a more suitable tool to analyse chil-

dren’s mathematical marks than convergent assessment. It is, therefore, this kind of

assessment that we are proposing to analyse children’s mathematical graphics and

written methods.

Divergent assessment (adapted from Torrance, 2001)

Assessment aims to discover what the learner knows, understands and can do. This

is characterised by:

Practical implications

1 Flexible or complex planning which incorporates alternatives.

2 Open forms of recording i.e. children’s own ways of representationing their

thinking.

3 An analysis of the learner and the curriculum from a learner-centred perspective.

4 Open questioning, open tasks and following children’s self-initiated enquiries.

5 Focus on aspects of the learners’ work which will yield insights into their current

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