Kindergarten and first graders' knowledge of the number symbols: production and recognition

Focus on Learning Problems in Mathematics, Wntr, 2008 by Nitza Mark-Zigdon, Dina Tirosh

The Hindu-Arabic number system represents amounts of objects by number symbols, without referring to other properties of these objects (e.g., color, size). That is to say that the number symbols, like many other symbols, are linked to the objects that they represent in an arbitrary but agreed upon manner with fixed representation rules (Vygotsky, 1978). Therefore, when the number system is used in a teaching-learning process, the child is required to perform a relatively complicated cognitive process. He has to refer to the meaning behind the symbols and to make the connections between the symbols and the quantities (Bialystok, 1992; DeLoache, Miller & Rosengren, 1997; Dorfler, 2000; Kaput, 1991; Lesh & Doerr, 2000, Thomas, Jolley, Robinson & Champion, 1999).

One question that is naturally raised regarding child's knowledge of the Hindu-Arabic number system is: What are the factors that determine the child's grasp of this symbolic system? One of the major factors that ought to be considered is the development of symbolic thinking. The development of symbolic thinking addresses the cognitive processes that take place in the structure of the mental representation during the change from the "unity level" to that of the "differentiation level" (Nemirovsky & Monk, 2000). In the early stages of the development of symbolic thinking, children are at the unity level. At this level, children believe that the symbolic representation reflects the nature of the object it represents. Thus, for example, children will write names of large objects with large letters (Thomas, Jolley, Robinson, & Champion, 1999). When differentiation occurs, the child separates between the object being represented and its symbolic representation. At this differentiation level the child understands that there is no connection between the size of the symbol and the size of the object that the symbol represents.

The ability of a child to employ symbols of numbers as symbols representing the mathematical meaning of the number is a result of a developmental process (Bialystok, 1992; Bialystok & Cobb, 1996; Worthington & Carruthers, 2003; Hughes, 1986, Munn, 1998; Worthington, 2003). However, newborns are cognitively equipped from the very outset to recognize objects and their quantities and almost immediately begin to accumulate knowledge about numbers (Butterworth, 2000; Dehaene, 1997; Wynn, 1992, 2002). Moreover, symbolization ability begins to develop in children from the earliest life stages (Kamii, Kirkland & Lewis, 2001; DeLoache, Miller & Rosengren, 1997; Mandler, 1992, Piaget 1962), and as a result, children acquire various types of knowledge about the written symbols of numbers (Bialystok, 1992; Carruthers & Worthington, 2005; Hughes, 1986). Tolshinsky-Landsmann (1986) found, for instance, that four year-olds differentiate between Hebrew letters and numerical symbols and that they consider one numerical symbol to be a number, whereas they do not think of one letter as a word (indeed, in Hebrew, one letter does not constitute a word). Tolshinsky-Landsmann and Karmiloff-Smith (1992) reported that children from England and from Spain at the age of about four distinguish between symbols that belong to the number system and those that do not.

As previously stated, the ability to attribute quantities to numerical symbols develops gradually. Two of the prominent researchers that have significantly contributed to our understanding of this developmental process are Bialystok (1992, 2000) who describes the different stages of the development of number symbolic thinking and Hughes (1986), who describes the development of the numerical symbolic representation.

Bialystok (1992) describes three hierarchical stages of number symbolic thinking. At the first stage children recite number sequences from their memory and employ the appropriate name for each number in the number sequence. At this stage children understand that counting is a way of describing quantities. At the second stage, children identify the written symbols of numbers and link them to the appropriate name of the number and the appropriate quantities. At this stage, the physical objects must be presented to the children together with the written symbols of numbers. At the third stage, children identify both the written and the spoken symbols of numbers. They are not dependent on a physical representation of the objects. The number symbols, by themselves, elicit the quantities that they represent.

The development of the conventional writing of the Hindu-Arabic number symbols is expounded in Hughes (1986) four-stage model. At the first stage, children represent numbers by means of idiosyncratic representations. These representations are neither linked to the symbol nor to the quantity of the objects that they represent. At the second stage, children illustrate numbers by means of pictographic representations. At this stage they employ the graphic expression appropriate to the quantity, shape, situation, color or direction of the objects. For example, children tend to draw five children to describe a situation related to five children. At the third stage, children employ iconic representations. Here they represent the number by means of a symbol system based on one-to-one correspondence between the number of shapes drawn and the given number of objects, such as lines or circles. At the fourth and final stage, the children represent the numbers by using the conventional, Hindu-Arabic number symbols.