Early Childhood Numeracy: Building an Understanding of Part–Whole Relationships

Abstract

This paper looks at a key aspect of numeracy, quantification, the process for determining how many things are in a group. Things can be quantified by counting or by subitizing (knowing just by looking). Many mathematics educators see counting as the first step towards more advanced mathematical understanding. However, there is some evidence to suggest that, for some children, subitizing is well-established before counting. There seems to be reasonable agreement that children need to understand about the relationships between the parts and the whole (part—whole thinking). This paper looks at ways to support children's part—whole thinking in the early childhood years, and the use of egg cartons to create three-dimensional tens-frames and six-frames for this purpose. Collecting up treasures to put in the compartments of an egg carton capitalises on young children's ‘accumulation intent’, and helps them to appreciate the way that numbers are composed of other numbers.

Awareness of pattern and structure in early mathematical development

“…Many studies have implicitly or explicitly examined the role of pattern and structure in young children's understanding of number concepts and processes such as counting, subitising, partitioning, and numeration (Wright, 1994;Young-Loveridge, 2002). In their studies on numeration, Cobb, Gravemeijer, Yackel, McClain, and Whitenack (1997) described first graders' coordination of units of 10 and 1 in terms of the structure of collections.…”

“…Van Nes (2008) also found a strong link between developing number sense and spatial structuring in Kindergartners' finger patterns and subitising structures. Studies of partitioning and part-whole reasoning (Lamon, 1996;Young-Loveridge, 2002) indicate the importance of unitising and spatial structuring in developing fraction knowledge.…”

Structuring numbers 1 to 20: Developing facile addition and subtraction

“…In contrast to children who use counting by ones, in this solution none of the numbers need to be counted out to have meaning. This can be described as a part-whole construction of number (Resnick, 1983;Hunting, 2003;Young-Loveridge, 2002) --the ability to partition a whole number into number parts. Such part-whole thinking indicates a construction of the number sequence as a "bidirectional chain" (Fuson, 1992); or as an Explicitly Nested Number Sequence (Steffe & Cobb, 1988).…”

“…Development from counting strategies to facile non-counting strategies for addition and subtraction in the range 1 to 20 is regarded as an important accomplishment of early childhood mathematics (Resnick, 1983;Wright, 1994;Young-Loveridge, 2002). As well as facilitating calculation in the range 1 to 20, the non-counting strategies and part-whole thinking are required to calculate in higher decades (Heirdsfield, 2001;Treffers, 1991), and to understand multiplication and fractions (Olive, 2001;Resnick, 1983).…”

“…Drawing on the emergent modelling heuristic, we use instructional settings in which students can first structure context-bound models of combinations--such as identifying 5 red and 3 green dots as 8 dots--and then, reflect on their activity and generalise toward more formal reasoning about numbers--such as partitioning 8 into 5 + 3 to solve the written task 8 -3 without counting. We use the ten-frame setting for the range 1-10 (Bobis, 1996;Treffers, 1991;Young-Loveridge, 2002), and the arithmetic rack for the range 1-20 (Gravemeijer et al, 2000;Treffers, 1991) (see Instructional Settings below), settings which suggest a combination line-group structuring. Wright and colleagues (2006) have developed instructional procedures with these settings specifically for oneon-one interventions.…”