During the course of this research project I have found no reason to dispute the understandings about place value that inform the Early Numeracy Project. Children develop their ideas about place value from early attempts to count objects through more advanced counting to an appreciation of the part-whole nature of multi-digit numbers. As their understanding develops they leave behind a reliance on using materials and become more abstract in their knowledge. Concepts that had been previously uncertain become accepted as givens and form the basis of an increasingly complex conceptual structure. When they encounter challenging problems that are unmanageable this structure allows children to fold back to earlier understandings that then enable the problem to be reconstructed in a more simple form. Increasing knowledge and skill allow for greater flexibility in problem solving and also eventually enable the children to create their own understandings. Whether these understandings develop in a linear, hierarchical manner however, is open to question.

The Number Framework of the ENP sets out stages of concept development about place value. As such, it supplies teachers with a useful model for understanding children’s place value concept development. An alternative pedagogical model for teachers could include a number of other areas, however, which may be useful to consider using in addition to the current model of teaching and learning. These are allowing for social rather than ability groupings within a class, encouraging ownership of learning and aiming for a balance between instruction about strategies and concepts and allowing place value to emerge from realistic activities with which the children engage.

Ability grouping is an outcome of a stages approach to learning which makes assumptions about the nature of knowledge. Knowledge is seen as developing in a relatively ordered fashion which does not seem to allow for children who may have quite advanced knowledge in one area and less in another. By placing them in groups (stages) at the start of a lesson sequence, the teacher may limit the kinds of experiences that a child has access to. A more flexible arrangement of groupings, alternatively, may allow the child to have a wider variety of experiences with which to consolidate his or her learning. As has been shown in this project, children’s levels of knowledge and skill are different and may go unrecognised by assessment tools such as the ENP Diagnostic Interview. Different levels of understanding are evident among children across a range of place value concepts. Intuitive modes of thinking have been discussed as a type of thinking that has, to an extent been left out of assessment practices. By relying on an assessment tool such as the Diagnostic Interview, teachers may inaccurately categorise ability and place a child in a group that limits the amount of learning that he or she is capable of. This research raises questions about the kinds of teaching strategies that teachers might use as alternatives to ability grouping.

The Number Framework identifies the knowledge and strategies that the children are to learn and shows teachers activities with which the children can practice. A focus of the lessons is the shared discussions that enable the children and teacher to establish meanings as a platform for communal understanding. The key aspect of the ENP in emphasising the importance of discussion and shared understandings is to be encouraged but the problem of ownership remains. The stages aspect of the ENP model assumes that children progress in a relatively linear manner and that teaching is best focused on defined skills and strategies. This approach may restrict the learning that the children are capable of.

The importance of encouraging a social learning culture in the classroom has already been examined and this is an aspect that seems to be a strength of the ENP programme. Teachers are encouraged to focus their discussions with children on understanding the previously established skills and strategies of The Number Framework and to make links between them, It is possible however that they may not go far enough in allowing children to create their own skills and strategies. In their model of recursive development, Pirie and Kieran (1994) suggest that the creative stage is the highest level of learning. It is implied that it is beyond the level of children operating at lower levels. A constructivist approach, however, would suggest that children can formulate owned concepts at any level, which may help them to develop a more secure understanding of place value.

Developing an understanding of place value that is based on the children’s own concepts is a key aspect of an emergent approach. It sits within a constructivist paradigm and acknowledges that children develop their concepts in a variety of ways. It allows for the child who may learn in a linear fashion and also for those who may follow less obvious paths to understanding. The strength of this approach is that the child’s understanding is likely to be better equipped to problem-solve in areas that are unfamiliar.

When I started this research project I had in mind the idea of constructing a model of learning to illustrate children’s understandings about place value at the Year 4 level. As the research progressed this idea evolved into that of developing a model to support teaching and learning. It recognises that learning is a complex process for both children and for teachers.

For children, the learning process encompasses constructing their own meanings from realistic experiences and discussing emerging meanings within the context of a classroom community. This also includes using a variety of approaches, being able to explain ideas in different ways and owning the concepts that are developed. It includes elements of an emergent model (as shown in the Candy Factory) and the ENP Number Framework.

For teachers, the learning process involves both the ability to conceptualise the children’s process (noted in the previous paragraph) and to develop strategies to support them in becoming inquiring learners. Rather than teaching skills and strategies in isolation they could be taught as part of an environment created by the teacher and children together. The challenge for teachers is to find suitable activities in consultation with children that are sufficiently realistic to encourage them to engage with them. The instructional sequence could then follow much the same path as identified in the Candy Factory - setting the scene, exploring, interpreting, creating, regrouping, recording and problem-solving with number. The Number Framework provides the teacher with support in understanding the aspects of place value that the children work through. As such it goes some way towards answering the concerns expressed by researchers in the introduction to this project about teachers lacking a mental map of how children progress in the development of place value concepts. The important insight for teachers, however, is to recognise that children may not progress through the identified knowledge and skills in a linear fashion but that they are likely to emerge as they are needed in problem-solving activities.

The model is focused on teaching and learning. Both teachers and children have their own understandings to develop and their own work to do. Teachers, through their own professional development, and children, within the context of their classroom activities, need to construct their own understandings. They need to work together to articulate their emerging confidence and knowledge and share their different ways of understanding and explaining the different aspects of place value. The conceptual model to support teaching and learning of place value at the Year 4 level, which has emerged from this study, is a pedagogical model. It recognises that the different ways of understanding that emerge from engagement with realistic classroom activities need to be conceptualised individually and collectively by both teachers and children.