The two groups had some similarities and some differences in the knowledge and skills that they showed about place value. Making a decision about whether a child had showed knowledge in a particular area was a challenge in some of the activities and I had to rely on my own knowledge and experience as a teacher as well as by referring to other sources. In some of the activities, I used the criteria provided by the NEMP (2001) report. For example, in the Motorway activity (see Appendix 3, p. 49), a reasonable estimate for 98 x 9 was considered, by NEMP, to be any number between 850 and 900. In some cases, during the interview, where the children's answers made it hard to tell what knowledge they were showing, I rephrased the question. This opportunity was only possible with the ENP children, however, not the NEMP group. In the case of Child E (NEMP activity: Population, see Appendix 3, p. 51) her initial answers suggested confusion between face and place values for the digits in 2495. Her last answer, however, revealed that she did in fact understand that the 2 was in the thousands place. It seemed that she had misunderstood the question rather than that she had lacked the knowledge necessary to answer it. Table 3.1 gives an overview of the broad similarities and differences that the children seemed to have in their knowledge of place value. It shows six areas of similarity and four areas of difference between the two groups.

In Items 4 and 6, although the outcome was the same, the methods used by the children differed. Item 4 is examined in more detail later in this chapter. Appendices 3 and 4 provide more reported data from the interviews and mention is made of a number of points that may be worthy of further study by other researchers.

I have chosen to report in this chapter on two themes that emerged as I analysed the data: ownership of learning and the potentially limiting effect of ability grouping on children's learning.

The theme of ownership of learning encompasses two main subthemes. They are concerned, firstly, with the problem for teachers of developing realistic activities and contexts for learning and, secondly, with encouraging flexible approaches to both teaching and learning.

Ownership of learning extends beyond the ownership of strategies and concept development to ownership of the activities and types of materials used as well. Haskell (2001) and Greeno (1991) suggest that mathematical problems should be situated in realistic contexts that are recognised by the child. This is to allow children to make use of their prior knowledge and to be better able to engage with and think of the problem as a whole. For example, when asked whether it was possible for there to be an equal number of girls and boys in a class of 26 children, Child C2 (NEMP activity: Girls and Boys, Appendix 3, pp. 46-47), explained that, in his opinion, there were normally more girls than boys in any given classroom. For this child the idea of equal numbers of boys and girls in a class was unrealistic and initially prevented him from engaging with the division operation that the teacher was trying to assess. Only when the question was rephrased did this child attempt to work out the answer. Setting up realistic activities without negotiating their possible meanings with the children presents a problem for the teacher.

Some of the activities used in the interviews showed how children can and do take ownership of their learning. The ways that the children used materials highlighted this. When asked to use place value blocks to make up a number and show a division operation (see NEMP activity: Girls & Boys, Appendix 3, pp. 46-47), the children demonstrated a variety of strategies. For example, Child C (in the ENP group) used her fingers to work out the problem and then made up the correct number of blocks afterwards. She seemed to have decided to use her own method as an intermediary step to solving the problem in the manner that I had requested. Child B and Child C4 worked out the answer mentally before using the blocks. These children seemed to have moved beyond the need to use concrete strategies in solving a question like this. It appeared that the question was unduly focused on getting them to use a concrete strategy and had lost sight of conceptual flexibility as an important skill worth encouraging. Teachers need to be aware of the limiting nature of questions such as these where the use of any type of materials or strategies could be mistaken for the end point rather than just one of many potential pathways of learning.

Claxton (1997) argues that, in order to develop a multi-pathway, flexible approach to problem-solving, children must be helped to develop a greater sense of ownership of knowledge and the knowledge-making process. There is a big difference between working with a rule that has been given by the teacher and one that is based on “owned” images and connections (Greeno, 1991). This was illustrated when the children were asked to mentally add 36 and 29 (NEMP activity: 36 + 29, see Appendix 3, p. 44). The ENP children all successfully used either the split or the jump method, while, of the three NEMP children who solved the problem, two of them used a mental algorithm. Despite using different strategies to add 36 + 29, there was not much difference in the end result between the two groups of children. The relative uniformity of the strategies used, together with the fact that none of the children in either group were able to suggest any alternative ways of solving this problem, suggest that the children may have been applying a strategy that had been taught rather than one that they had thought of themselves. This highlights a lack of breadth in the children's learning and raises the question of whether the children's learning had become too narrowly focused on learning set procedures rather than developing a more flexible approach to problem-solving. The findings from this activity made me curious to see how the children performed when working with larger numbers that might make these strategies less effective.

In the Motorway activity (Appendix 3, p.49) the children had to think of a way of estimating the product of 98 and 9. The familiarity that both groups had shown with the split method or mental algorithms did not appear to serve them well in this problem. Only one child (A4 from the NEMP group) managed to answer this question correctly with most of the children unable to demonstrate ways of using estimation as a strategy. One way of solving this problem involved rounding the 98 up to 100 and then multiplying 100 times 9, which I suspected was well within the children's capability as all of the ENP children were able to multiply 10 times 12 (see NEMP activity: Independent, Appendix 3, p. 48). Child D managed to work out 316 times 10 and yet could not solve the Motorway problem. I decided to test my theory that the children were capable of solving this problem but simply lacked the knowledge of rounding up to estimate. In the next interview I provided Child C (who was unsure of how to start) with a prompt to think of a number close to 98 that might make the problem easier for her and she was able to round 98 up to 100. When I asked her what 100 times 9 might be, she gave the correct answer. Without similar prompts the other children attempted to use an addition or multiplication algorithm. Although the algorithm method that they used had been successful with smaller numbers, the children were unable to work with this larger number and either gave up or arrived at an inaccurate estimate. This could confirm the concerns of researchers (Pirie & Kieran, 1994, Fuson et al, 1997 and Greeno, 1991) that too much emphasis on learning how to follow the procedures of number operations can narrow the focus and prevent children from thinking about alternative ways of solving a problem.

The problems that the children had suggest the need to encourage greater breadth of learning by creating realistic contexts for mathematical activity and for teachers to be open to the different ways that children choose to go about their learning. The focus in the ENP on getting children to think about numbers in different ways is helpful in this regard and needs to be further emphasised if children are to build a broader concept of number and place value. By building up a sense of their own ideas rather than just ones that have been taught to them, children may become more secure and better equipped to think of ways to solve problems of this type.

Three issues related to ability grouping are worth discussing in greater detail. In this section I show, firstly, that trying to fit children's learning to the curriculum is problematic for teachers. Secondly I discuss the challenge that teachers have in deciding when a child is ready to learn a concept and, thirdly, I question the assumption that there is a pre-set order for learning. In all of these examples, I am raising issues concerned with the validity and consequences of grouping children according to criteria which are based on a pre-set, stages model of learning.

A problem for teachers who are working within the stages paradigm is knowing how children's conceptual development fits in with the different stages described in the Mathematics curriculum or the Number Framework of the Early Numeracy Project. The ENP answers this need by supplying teachers with the Diagnostic Interview assessment tool. This enables teachers to sort children into ability groups according to the different stages of the Number Framework. Being placed in an ability group, however, may hinder the concept development of some children. I found an example of this in the 36 + 29 activity (see Appendix 3, p. 44). Child E had been placed by her teacher at the Advanced Counting level (Stage 4) of the Number Framework and appeared to have the weakest mathematical knowledge of the group yet her choice of method suggested otherwise. Despite this, she was able to perform subtraction operations (see NEMP activity: Speedo, Appendix 3, p. 52) as well or better than her peers who had been placed in a group that was working at the higher Early Additive level (Stage 5). In the case of Child E, her learning did not seem to fit with a linear, stages perspective.

Bowers and Nickerson (2000) suggest that forms of assessment, such as the ENP Diagnostic Interview, focus on the individual at the expense of shared learning. By isolating children from their more or less able peers, teachers could be limiting children's learning and discussion as well as threatening their self-esteem as mathematicians. Bowers (1999) described a shift to a view that mathematical activity is inherently social and cultural in nature. All children can learn from each other and although some learning is individual, Greeno (1991) suggests that social learning is the more effective way to learn.

Deciding when a child is ready to learn a concept is a problem that all teachers face. At the Advanced Additive stage of the Number Framework, children are described as being able to choose from a range of strategies to solve problems. As none of the children that I interviewed were working at this level in their classroom it was perhaps to be expected that none of them managed to provide more than one way of adding two 2-digit numbers together (see NEMP activity: 36 + 29, Appendix 3, p. 44). The lack of breadth in their problem-solving ability, although perhaps “acceptable” within a stages approach, may have hindered their potential to solve problems that were within their grasp. The implication of the stages approach is that children are not expected to be learning about, or showing understandings of, concepts that are “above their level”. Although this appears to have been the case with showing alternative strategies, the children showed on a number of occasions that they were operating at levels above their designated classroom groupings. An example of this, from the ENP group, was Child E who showed higher ability in some areas of knowledge and lower ability in others (see NEMP activities, Appendix 3: Speedo, p.52 and 36 + 29, p. 44). Her learning would seem to be at a variety of levels simultaneously and did not fit into a neat category or stage. The consequences of consigning her to a lower ability group could be boredom, reduced self-esteem and fewer learning opportunities.

Another example of children showing knowledge “above their level”, concerns four-digit numbers. In the ENP, children are not taught about groupings within 1000 until they reach the Advanced Additive stage. It would, therefore, be reasonable to expect that the ENP children, who were working at stages below this (Advanced Counting and Early Additive) would solve problems with smaller numbers more successfully than they could with larger numbers. Contrary to this however, Fuson et al (1997) suggest that teachers should focus on using larger (four-digit) numbers before smaller (two-digit) ones because of the irregularities of English language words. This allows children to construct multi-digit concepts with the more regular hundreds and thousands. An example of this would be practising multi-digit addition with 2300 + 6800 instead of 23 + 68. I was interested to explore this assertion further with the Speedo activity and find out whether the children were more successful when adding and taking away hundreds and thousands rather than two-digit numbers.

In this activity (NEMP activity: Speedo, see Appendix 3, p. 52), in the context of a car speedometer, the children were asked to add and subtract 1,10,100 and 1000 from a four-digit number. If the stages theory was correct for these children, they could be expected to be more successful with the smaller numbers and less so as the size of the numbers increased. The ENP group showed slightly more success overall and both groups were more successful when adding or subtracting 1000 from a four-digit number, as opposed to 10 or 100. The NEMP (2001) report also found that children were more successful in using ones and thousands than in using tens and hundreds. This activity suggested to me to be wary of using a lock-step, stages approach in my teaching. For some children, learning appears to follow different pathways that are not as predictable and ordered as a stages approach might have us believe.

The themes of ownership of learning and the limitations of ability grouping described so far highlight the importance of teachers encouraging children to develop a wide, multi-pathway approach to problem-solving that makes use of their own prior knowledge and understandings. In the next section of this chapter I discuss possible ways that teachers might address the problems of when to teach concepts and how to cater for children of all abilities. Firstly, I suggest how teachers might develop the learning culture of the whole class as a social group and, secondly, discuss an approach that seeks to address the problem of when to teach a concept and how to group children.