Validación de la propuesta

1.2.1.1 Innate numerical ability

In 1954, Tobias Dantzig wrote a book entitled ―Number, the language of science‖ which suggested that people were born with a faculty that the author referred to as ‗number sense‘ (Dantzig, 1954). The book was written whilst the Piagetian thought dominated, which was quite conservative with regard to young children‘s numerical abilities. It took nearly twenty years for Dantzig‘s insight to be confirmed.

In a pioneering experiment, Starky and Cooper (1980) showed that 4-6 month year olds were more likely to attend to visual arrays (dots) that had changed in numerosity using a ‗habituation-dishabituation‘ paradigm: they looked for longer at a novel stimulus (different number of objects). Clearly, with each change of numerosity, there were other factors that could vary such as the area or darkness of the array, therefore Starkey and Cooper tried to control for these by changing the arrangement of dots in each trial.

Nevertheless, Mix, Huttenlocher & Levine (2002) have emphasised the difficulty in ruling out other perceptual clues such as shape, size or density rather than quantity in children‘s judgements and suggested that children may be responding to continuous rather than discrete quantity. Wynn, Bloom, & Chiang (2002) have attempted to address this possibility by using a group of moving dots, controlling for factors such as area and contour, thereby showing that infants do indeed respond to numerosity.

Wynn‘s research has also demonstrated that infants are able to compute basic arithmetic consequences of adding and subtracting (e.g., 1+1, 2-1) again using the

‗habituation-dishabituation‘ paradigm. Although her assertions have been questioned (e.g., Cohen & Marks, 2002), the balance of evidence does suggest that infants are able to represent the numerosity of sets and carry out mental manipulations of these representations.

27 Infants do have an upper limit for their numerical concept: up to around 4 objects (Starkey & Cooper, 1980), which is most likely to reflect their ability to identify the numerosity of an array at a glance without counting. This perceptual ability is shared with adults and has been called subitising (Mandler & Shebo, 1982), which allows children and adults to enumerate small numbers (up to around 5) without having to count.

Research continues to develop our understanding of innate mechanisms that may provide the foundations of later ability, such as a possible internal number line that enables children and adults to approximate the addition and subtraction of larger numbers of perceptual objects (Gilmore, McCarthy, & Spelke, 2007). However, in order to succeed mathematically, it is necessary to understand how to manipulate and communicate mathematical symbols, starting with number words and progressing to more complex operations.

1.2.1.2 Pre-School Experience

Children‘s numerical ability is founded on their early experiences (Baroody, Eiland, &

Thompson, 2009). Jordan, Kaplan, Ramineni, & Locuniak (2009), for example, have demonstrated a strong and significant relationship between children‘s kindergarten (aged 5.5 years) number competence and their mathematical achievement three years later. The start of school does not however mark the beginning of children‘s numerical development, because children bring to school a range of skills and understanding gained from prior informal experiences (Canobi, 2007) such as the ability to add or subtract (Martin Hughes, 1981). Indeed, in a study of over 1,400 children in Australia, Clarke, Clarke, & Cheesman (1996) found that much of what had traditionally formed the maths curriculum for the first year of school was already understood by many children on arrival at school. When children enter school, they already therefore have some

28 knowledge of the number system and possibly some basic operations. However, their limited understanding of numbers will still make certain maths problems inaccessible. To identify these difficulties, it is important to examine the development of children‘s understanding of number words, and how this relates to their ability to apply this understanding to more complex problems.

1.2.2 Childr

When children first say the number words, they do so without understanding exactly what these words mean. Indeed, the words may just be part of an inseparable linguistic sequence, such as part of a nursery rhyme. Eventually, children not only learn the symbolic significance of these words, but also how they are related within a specific culturally determined decade system. Fuson (1992) identified specific stages to this development. These will be outlined before looking at one of the most difficult numerical concepts children have to learn – multidigit understanding.

Fuson identified five key levels of development: String, Unbreakable list, Breakable chain, Numerable chain and Bidirectional chain.

 String

Children initially learn the number words, possibly through songs or counting activities, and may be able to recite them, albeit not actually being able to distinguish individual number words within the linguistic ‗string‘.

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 Unbreakable list

In the next stage, children learn to identify the number words, allowing them to take part in counting activities which involve reciting these words in the correct order, using each word to correspond to each item counted. This one-to-one correspondence between object and count words is not however immediately clear; it is a skill that is developed.

Although children may become proficient in counting, this does not mean that they understand the significance of each count word. For this, children need to realise that the last word counted represents the whole set, i.e. ‗three‘ is not just the last word counted but represents three objects. The notion that number words refer to a set is referred to as the cardinal principle, leading researchers to talk about children‘s

‗understanding of cardinality‘.

In Fuson‘s Unbreakable stage, children make a key developmental step – they are able to enumerate quantities, through counting, or possibly subitising if the set is small, and understand that the number words can represent quantities. Children are consequently able to approach questions asking ‗how many?‘ However, their calculation strategies are limited, mainly because they do not yet understand that number words can be ‗broken‘. In other words, given two amounts to add, children will want to combine the amounts and ‗count-all‘, starting at the first object. Consequently, addition is still dependent on objects, or perceptual items as Fuson refers. In fact, young children can often find the actual question ‗how many?‘ difficult to understand unless there is a concrete referent (Hughes, 1986).

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 Breakable chain level

Children‘s first step away from their dependence on perceptual items in counting is integral to Fuson‘s Breakable chain level. This level refers to how children can ‗break‘ the sequence of numbers by using a number word to represent a quantity within an addition (or subtraction) sum. Instead of counting from one, children can begin ‗counting on‘

from the number word of the first addend (number to be added). The transition from count-all to count-on is considered to reflect a key conceptual step forward and various attempts have been made to evaluate interventions supporting this graduation. Secada, Fuson, & Hall (1983), for example, analysed this transition and identified three sub-skills:

a) counting up from an arbitrary point, b) shifting from the cardinal to the counting meaning of the first addend and c) beginning the count of the second addend with the next counting word. The authors demonstrated the success of measuring these three sub-skills on predicting counting-on behaviour and furthermore demonstrated the success of interventions supporting these skills. In order to assess children‘s ability to count-on, the authors examined children‘s strategies for adding two amounts when dots representing the first addend were visible and then hidden as illustrated in Figure 1.1, thereby emphasising how counting-on marks children‘s first steps away from depending on perceptual units to add amounts.

Figure 1.1: Materials used to assess children‟s ability to count-on (Secada et al., 1983)

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 Numerable chain level

In the Numerable chain level, both addends are described as embedded within the sum.

Consequently, when counting on the second addend, children are not dependent on perceptual items to know when to stop counting. Instead, they use the number word itself as a means to count to the result. For example given the sum 5 + 4, children count-on from 5 (6, 7, 8, 9), and stop counting when they know they have counted out the second addend. This example illustrates how children require a method of knowing how many they have counted-on in the absence of perceptual clues. Three methods have been proposed (Steffe, von Glaserfield, Richards, & Cobb, 1983) for how this can be managed:

a) by keeping track of the auditory pattern of the words counted-on; b) by using known finger patterns and matching each addend word to a finger extended as the word is said;

and c) by double counting (alternating between the amount counted-on and the total).

The skills needed for counting-on are relatively demanding for children, yet are highly significant in that they mark the point at which children become independent of external materials to carry out basic addition and subtraction problems. This progression from a reliance on physical objects has important theoretical implications for this thesis; and it is worth examining in greater detail how children are able to achieve this.

When counting-on, children have the dual task of keeping track of the amount counted-on and the total. This places considerable cognitive demands on children and could explain the almost universal strategy of children using fingers to help them without instruction to do so. Fingers help because they provide perceptual structures that children can enumerate without counting (Fuson, 1992). This is particularly the case for smaller numbers where, by raising fingers when counting-on, children can identify when to stop counting. However, even with fingers, counting-on is quite a demanding task,

32 especially if the amount to count-on is large. In order to manage larger numbers, children therefore need a means to simplify calculations. This can be achieved through more flexible strategies that can facilitate calculations requiring a greater understanding of numbers than is reflected in Fuson‘s Bidirectional chain level.

 Bidirectional chain

In Fuson‘s final level, the Bidirectional chain level, the whole number sequence becomes a series of embedded cardinal amounts, where each word is part of a series but is also a separable cardinal amount. Understanding how each number is composed of smaller cardinal amounts enables children to transform calculations to take advantage of the number facts that they have begun to learn. For example, the sum 7 + 8 can be decomposed to 7 + 7 + 1. As a result, knowledge of doubles allows children to transform the problem to 14 + 1 which places fewer demands on counting. It is also possible for children to use the decade structure to simplify calculations in a similar way.

For example the sum 8 + 9 might be broken down into 8 + 2 + 7. Children can consequently draw upon possible number knowledge of both the number bonds to ten, and their understanding of how ten plus units corresponds to teen numbers.

1.2.2.1 Base ten understanding

The developmental levels described above refer to understanding the structure of single digit numbers. A key challenge and great difficulty for children is in developing an understanding of multidigit numbers (Baroody, 1990; Fuson, 1990; Varelas & Becker, 1997) - how a number such as 16 or 47 is composed of two parts – tens and units.

33 Significantly, this symbolic system, which uses place value according to a base ten grouping, is a culturally defined system.

According to Fuson (1990), multidigit understanding is difficult because it requires children to understand not only how numbers can be partitioned according to the decade structure, but also how these values interrelate. Resnick (1983b) uses the term ‗Unique partitioning‘ to describe the more basic ability to partition multidigits into tens and units and ‗Multiple partitioning‘ to describe the ability to partition multidigits in non-standard ways that demonstrate the deeper understanding required for competence with multidigit calculations (e.g., a number such as 34 is not just composed of 3 tens and 4 units (unique partitioning) but can also be decomposed into 2 tens and 14 units). Such understanding is challenging; indeed, Resnick described the introduction to the decimal system as the most difficult (and important) instructional task in mathematics in the early years (1983b, p.126).

According to Nunes and Bryant (1996), the reason that children‘s understanding of the decimal structure does not develop until a later age is likely to be that they do not understand one or both of the two mathematical principles that underlie its structure.

These are a) that units can be of different sizes – for example, tens and units, and b) that any positive integer can be decomposed into two or more others that precede it in the ordinal list of numbers.

This understanding of how numbers can be decomposed into smaller numbers is reflected in Fuson‘s Bidirectional chain level and is also encompassed in other developmental models such as, for example, Saxton and Cakir (2006) who identify children‘s ability to partition single digit numbers as a significant predictor of base ten understanding or Jones et al. (1996) who describe the ability to partition single digit numbers in different ways as a key prerequisite for place value understanding.

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1.2.3 Summary

Children are born with certain perceptual mechanisms that allow them to make non-symbolic quantitative judgements. Although these may support later abilities, children need to learn the number words and, importantly, the structural relationship between them (e.g., how the number 7 can be broken down into 3 and 4). This understanding of how numbers can be partitioned into smaller numbers is referred to as additive composition (see following section) and allows children to decompose and recompose addition and subtraction problems, thereby providing them with more flexible and efficient calculation strategies. It is also possible that understanding additive composition provides a foundation for understanding how multidigits are composed.

According to Martins-Mourao & Cowan (1998), additive composition is thought to form a conceptual base for the development of children‘s elemental arithmetic and their understanding of the decade numeration system. This concept will therefore be examined in more detail.