own pace through direct teaching and through carefully sequenced steps. Teachers
emphasised the need to practise skills and children were encouraged to ‘try and try
again’. Learning was viewed as a mechanical result following rewards such as praise or
a smile, and children were considered to be passively storing information.
These principles led to the idea of sequenced and individualised subject material
broken down into discrete steps and, subsequently, to ‘programmed learning’.
However, there were a number of disadvantages to a behaviourist approach:- The experience of working with sequenced learning materials could be repetitive
and is unlikely to have motivated children. - There was no place for children to explore their personal thinking as they worked
through a series of pre-written cards or tasks.
Constructivism
Piaget’s scientific studies of individual children led to a view of the child as a ‘lone
scientist’. Piaget viewed children’s learning as biological development. Although
children were considered actively to construct their understanding of the world
through interaction with peers and their environment, the overall view was of chil-
dren working individually and at their own pace.
Piaget’s four ‘stages of development’ and experiments on conservation have been
very influential in the teaching of mathematics. Piaget’s hierarchical view of learn-
ing unintentionally influenced the content and use of mathematics schemes that
continue to be extensively used in schools and in some Early Years settings. In the
Early Years this led to a focus on sorting, sets, matching, one-to-one correspondence
and classification as precursors to work with numbers that has only recently been
questioned (Carruthers, 1997c; Thompson, 1997).
Piaget’s work also emphasised the idea of ‘readiness’: children were not believed to
be ‘ready’ to understand particular concepts until the appropriate developmental
stage had been reached. The child’s culture, the role of language and social interac-
tion were not emphasised in Piaget’s work.Social constructivism
Vygotsky believed that children actively construct their understanding through
solving problems in their own way. Children were believed to have a current level of
learning and a level that might be reached with the help of more knowledgeable
others (an adult, peer or older child): he termed this second level the ‘zone of prox-
imal development’. According to this theory, the child’s understanding is con-
structed through talk, social interaction and shared meaning.
The theory of social constructivism implies that teachers encourage children to
talk about their mathematical understanding. Social interaction is emphasised since
individual children construct their understanding through talk and interaction with
others. Sharing of ideas and meanings is negotiated with others. Learning that is22 Children’s Mathematics8657part 1b.qxd 04/07/2006 18:06 Page 22