This project aimed to find out more about ways that Year 4 children who had participated in the Early Numeracy Project (ENP) understood place value concepts and to develop a conceptual model to assist teaching and learning about place value.

Data were gathered firstly by a review of the literature about place value concepts. Two groups of Year 4 children were then observed working on place value activities: one that had not participated in the ENP and one that had. The first group was selected from National Educational Monitoring Project (NEMP) data gathered in 2001 and the activities used by the teacher / administrators with this group were then replicated with a small group of Year 4 children from my own school.

Two themes emerged from this study. The first was concerned with children’s ownership of learning and how this is an important aspect for teachers to consider when they are setting up learning environments and activities for children. The challenge for teachers is to develop realistic activities with which children can engage and to encourage children to explore flexible approaches to problem-solving. The second theme raised concerns about ability grouping children and the potentially limiting consequences that this method of classroom organisation might have on children’s learning experiences. The stages theory assumptions behind the ENP are challenged and an alternative pedagogical model is proposed.

“… providing teachers with a knowledge of how students acquire number concepts, an increased understanding of how they can assist students make progress and an effective means to assess students' levels of thinking in number.”

As briefly defined in Mathematics in the New Zealand Curriculum (1992), place value means,

“The value of the place a digit occupies, for example, in 57 the 5 occupies the tens place.” (p.214)

The concept of place value enables us to express an infinite range of numbers with only ten different digits. It is characterised by the following four mathematical properties:

Researchers have found that children at the Year 4 stage often have difficulty understanding place value concepts. Three suggested reasons for these confusions and gaps in knowledge are the English language (Cotter, 2000; Ross, 1995), children being taught place value too early (Kamii, 1986; Thompson, 2000 and Ross, 2002) and teachers being unsure about how best to teach place value (Young-Loveridge, 1998).

The English language does not provide consistent patterns with all of its numbers. The words that young children have to learn can make learning about numbers and place value more difficult than it might be for children who have other first languages, such as Japanese or te reo M_ori. In English, the concept of ten, for example, has three names: ten, -teen, and -ty. In te reo M_ori, the number concepts are integrated into the word for multi-digit numbers. For example, tekau ma rua (ten and two) is much more revealing of the part-whole nature of this number than is shown by the English word, twelve. Likewise, rua tekau ma rua (two tens and two) tells the young child more about the part-whole character of this number than is revealed by the word twenty two. According to Cotter (2000), counting to 100 in an Asian language requires knowledge of just eleven words, whereas an English-speaking child needs to know a total of twenty-eight words.

Kamii (1986), Thompson (2000) and Ross (2002) claim that when children are taught place value too early they may become confused by it if they are also engaged in counting activities. While their knowledge is situated within a counting-based model of number, children are not ready to move to working with collections-based, or part-whole, concepts. This focus on counting by ones is thought to interfere with the development of place value understanding. It can be contrasted with the approach of the Japanese school system where children are discouraged from using only one-by-one counting (Cotter, 2000) and are encouraged to see multi-digit numbers as part-whole concepts from an early stage.

Young-Loveridge (1998) argues that the best age to introduce place value is after the concept of ones has been established and the child has built up a network of relationships that will lead to the concept of ten. This approach is disputed by Cotter (2000) who sees children becominyg confused by an early unitary, or counting-by-ones focus. Clearly there is a range of opinion within the literature about how best to teach place value. Kamii (1986), Cobb and Wheatley (1988), Fuson et al. (1997), Beishuizen and Anghileri (1998) and Thompson (2000) support introducing place value gradually in the context of developing mental strategies for solving multi-digit addition and subtraction problems. These research findings support the approach taken by the ENP with its focus on building concepts and mental strategies. Within the literature there appears to be a variety of approaches to the teaching of place value, which can lead to confusion about best practice methods.

Effective teachers, according to Askew, Brown, Rhodes, Johnson and Wiliam (1997) have a clear “mental map” of how children develop concepts. Research has found, however, that many teachers are not as effective as they could be because they have only fragmented and vague mental models of children's development of place value knowledge (Jones et al., 1996). Higgins (2001) recommended, after studying the teaching and learning of place value in ten Wellington classrooms, that enhancing teachers' pedagogical content knowledge was the most effective way of improving learning outcomes about place value. Apart from teachers perhaps lacking confidence and competence in the teaching of mathematics, one of the reasons many teachers lack a clear mental model for the teaching of place value is that researchers have not agreed upon a best practice approach (Young-Loveridge, 1998).

Several models show what is thought to be the conceptual development of place value knowledge and strategies in children. Knowledge, in these models, shows a recognition and understanding of an increasing range of numbers and also a move from concrete to abstract concepts about number. Young-Loveridge (2001) notes that there are two broad concepts of number that are the basis of children's understandings when adding or subtracting multi-digit numbers. They are counting-based and collections-based models.

Counting-based models of number involve keeping one number intact when adding or subtracting. This model is reflected in Stage 4 (Advanced Counting) of the ENP where the child, adding two numbers together, can count on from one of the numbers and does not have to count both numbers individually. The Jump method for addition also reflects the counting-based model of number. This is shown in the Empty Number Line (in Diagram 1 below) that was developed by Beishuizen and Anghileri (quoted in Wright et al, 2002).It can be used as a visual prompt when adding or subtracting. Yackel (quoted in Young-Loveridge, 2001) argues that even when there is no obvious counting, solutions to problems such as these are still counting-based.

Collections-based models of number involve the partitioning of numbers (e.g. tens and ones) so that, when adding or subtracting, numbers of the same value can be combined separately. Resnick (quoted by Young-Loveridge, 2001) emphasised that developing an understanding of the part-whole properties of multi-digit numbers is perhaps the most important mathematical achievement of the early years at school.

Examples of collections-based models of number are described by Fuson et al (1997) and are also included in the part-whole stages of the ENP Number Framework. Other researchers (Ross, 1989, Cobb and Wheatley, 1988, Baroody, 1990 and Young-Loveridge, 1998) have developed similar models of place value understanding. They all follow a “stages theory” approach showing progression from unitary to part-whole, and from concrete to abstract concepts. As children become more sophisticated in their number knowledge they can recognise and problem-solve with an increasing range of numbers. Yackel (quoted in Young-Loveridge, 2001) argues that it is important for children to have access to both counting-based and collections-based models of number if they are to become flexible in their approach to problem-solving.