aA model for teaching place value: the Early Numeracy Project (ENP)

Thomas and Ward also found that the advances that children made in place value strategies were greater at the lower stages of the Number Framework. It seemed that children moved through the counting-based levels more rapidly but that they took longer to become secure in their understandings of collections-based or part-whole concepts. The transition between early additive and advanced additive was, in fact, beyond the understanding of some children. This may mean that there might be larger gaps between the higher stages of part-whole understanding than there are between the early stages of counting. Children working at the Early Additive stage are operating with 2-digit numbers and then at the Advanced Additive stage are working with numbers that contain up to six digits. Clearly, coming to grips with these kinds of numbers will take time. Examples of the kind of problem-solving that children are doing at these stages are:

Example 1 (Early Additive): addition of two- or three-digit numbers. The groupings and strategies that could be used,

Counting strategies take up the first four stages of the Number Framework before children engage in part-whole thinking. As Cotter (2000) has suggested, this emphasis on counting by ones may interfere with the development of place value concepts. When a child uses a counting-based strategy to count a collection of objects by ones, he or she may lose the idea of the whole. By focusing on counting first it may be difficult to then integrate ideas about the part-whole nature of multi-digit numbers. Because of this, as well as because of the irregularities of the English language, children may remain stuck in thinking of multi-digit numbers as collections of ones rather than moving on to a concept of parts and wholes.

The children do not own the concepts that are contained in the Number Framework. The model that is embedded within it comes from the levels of conceptual development derived from the work of researchers, not the children themselves. As such, it is not a constructivist approach to learning. Fuson et al (1997) reported that after participating in classes where the learning of multi-digit concepts and procedures were treated as problem-solving activities, children did show significant gains in understanding about place value. The key element in the lessons that they observed was that the children were encouraged to come up with their own solutions, rather than relying on instructions on how to use procedures and rules.

Conscious thinking skills, which can be articulated by the child, are emphasised in many schools (Claxton, 1997). Claxton differentiates between conscious and unconscious thinking: in the former, reason and logic are used and articulated whereas in the latter, unconscious thoughts can lead to intuitive knowledge. He argues that intuition as a valid way of thinking has been neglected in schools. Slower types of thinking or contemplation, while not infallible, are better suited to understanding complex or ill-defined situations. As a result of repeated observations, the child gradually uncovers patterns that are embedded in, or distributed across, a wide variety of experiences. Perhaps the development of deep place value knowledge may be fostered by an intuitive, unconscious process as well as by an explicit conscious way of thinking. Encouraging children to explain their thinking, according to Claxton, may hinder concepts that are slowly developing in the unconscious mind.

The Number Framework sets out stages of learning about the number system. Within these stages, children are taught strategies to help them think about and solve number problems. Haskell (2001) claims that teaching strategies is insufficient because learning by this method is too often restricted to the context within which it was experienced. It makes little use of the prior learning of the child. He says that teaching that promotes transfer of learning to problems outside of the initial context involves looking at an idea or procedure many times in different ways, on different levels and contexts and in different examples. This supports the thinking of Claxton (1997) about the gradual building up of intuitive knowledge through the observation of patterns and examples. The ENP encourages teachers to make links for the children when they are discussing concepts. Haskell warns that although providing examples is important, it may increase procedural efficiency at the expense of a deep conceptual knowledge base. The examples may make the children more expert in “doing” problems without having to really understand the concepts that lie behind them. Children become proficient at problem-solving within the confines of the examples given but may not develop a broader understanding of the concepts as they present themselves in other settings. Greeno (1991) argues that mastery is more than the ability to convert a form of skill: it is characterised as global, transferable, flexible and owned by the learner.

The Diagnostic Interview is an assessment tool provided for teachers in the ENP. It enables them to group children within the stages of the Number Framework. Children are interviewed on a one-to-one basis with a bank of activities designed to show their level of number knowledge and skill. The measurement of knowledge using a stages approach is problematichowever, as it is difficult to measure the extent of knowledge that may lie between the stages. By defining a set of skills and knowledge in a teaching model and then assessing against them the teacher is at once limiting the possibility that the child can demonstrate a breadth of learning that is outside of the model. Children may have progressed laterally and this may not feature in the assessment.

The difficulty that children appear to have in moving from the Early Additive to the Advanced Additive level as reported by Thomas and Ward (2002) could perhaps be a consequence of the focus on learning strategies and getting to the next stage, rather than exploring concepts from a broader base. Perhaps the children need more time to construct their own ways of knowing before they can progress to the more expert levels of the Number Framework. Social constructivists, such as Greeno (1991) argue that learning is more than strategies: it is encouraged by exploring, relating and creating understandings.

Greeno (1991) argues that little has been proven about the way that learners progress from a level of learned competence to that of automatic expertise and suggests that lesson sequences other than those based on stages models may need to be considered. In order to understand more complex ideas Ertmer and Newby (quoted in Mergel) assert that a constructivist perspective is more effective. Clay (2001), commenting about reading, describes how children begin to use more than one strategy or resource simultaneously to problem-solve, using different types of information and showing alternative ways of using information. This model of learning is relevant to the development of place value concepts and describes an increasingly unconscious or automatic expertise. As a dynamic, exponential process, the idea of the sequential learning that is assumed by a stages approach becomes untenable.