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

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.

Two different aspects of the development of number symbol representation have been outlined in this brief introduction: Production (more dominant in the work of Hughes, 1986) and recognition (central in the work of Bialystok, 1992). In the light of what has been presented, this study purports to extend our knowledge of children's production and recognition of number symbols. The study was conducted in Israel with Hebrew speaking children. In Israel, the Hindu-Arabic number system is used for representing numbers, and children are exposed to Hindu-Arabic numerals in their daily life (much like in United States, in Europe and in many other counties). Notably, Hebrew is written from right to left. This differentiation in directions between writing numbers (from left to right) and writing words (from right to left) might affect the development of the number writing. Hence, the research questions that guided our study are:

1. How do Israeli kindergarten children and first graders write number symbols?

2. What do Israeli kindergarten children and first graders consider as adequate, number symbols?

Methods

Participants

One hundred and fifty four Israeli children (48 kindergarten children and 106 first graders) participated in the study; half of them were boys and half were girls. All participants came from upper middle-class families with Hebrew as their mother tongue. The kindergarten children were aged between 5 and 6.4 years. They attended two nursery schools where they were exposed to the Hindu-Arabic number symbols but there was no formal instruction relating to these symbols. The first graders were aged between 6 and 7.2 years. They studied in five classes at two schools. During the first grade, these children formally begin to study the base 10 number system.

Tools and Procedure

A structured, individual interview was developed for this study: it consisted of two main sections, in accordance with two facets of representation knowledge: production and recognition. Each child was interviewed twice in a quiet room by the first author. The first interview dealt with the production of number symbols and the second with their recognition. Each interview lasted about 30 minutes.

Production: Writing numbers. Children were given four empty cards and asked to write one number on each card. The numbers were 5, 8, 13 and 20: two numbers below ten, and two above ten. These numbers were presented only orally in this fixed order.

Recognition: Identifying written number symbols. Five groups of symbols were presented to the children: number symbols (0, 5, 23, 1,000, 3,456, 44,444, 0.008, -75, 1/12); mixture of Hindu-Arabic numbers and other symbols (3[ALEPH]58, three numerals and a square between them, numeral (9) and rectangle); symbols of mathematical operations and of mathematical relations (+, =, >, x); letters and words (three written in Hebrew -[TEXT NOT REPRODUCIBLE IN ASCII]), a letter in Hebrew [TEXT NOT REPRODUCIBLE IN ASCII], and the letter w) and iconic and pictorial representations of numbers (five dots, four squares, three flowers, six stars, five houses, five rectangles). The interviewer presented each of the 25 cards to the child, one card at a time, in a mixed order (not according to their classification). She or he was asked to determine whether what appeared on the card was a correct way of writing a number. The interviewer explained the task to each child as follows:

At the end of the classification the child was encouraged to look at the two piles of cards and to make changes, if they so wished.

Results

Production: Writing Numbers

Almost all of the first graders produce adequate representations of the numbers (100% of the two numbers below 10 and 95% and 90% of the numbers 13 and 20, respectively). Almost all the preschoolers (93%) produce adequate representations of the number 8 and about half adequately wrote the number 5 and 13 (54% and 46% respectively). The number 20 was the most difficult one to produce for the preschoolers: 21% produce adequate presentations.

The level of difficulty in writing numbers could not be attributed to the numerical order of the numbers, as more children faced difficulties in writing 5 than in writing 8. The interviews revealed two types of difficulties: unacceptable ways of writing numbers and avoidance of writing the numbers.

Unacceptable ways of writing numbers. The difficulties in writing the numbers below 10 are characterized by the level of complexity of the graphic symbol. The preschool children represent the number 8 by writing two circles, one on top of the other (see Figure 1).

[FIGURE 1 OMITTED]

The drawing of 5 was more demanding as it requires coordinating two kinds of lines (straight and curved) and two directions of writing (vertical and horizontal or left and right) (see Figure 2).

[FIGURE 2 OMITTED]

The difficulties in writing the number above 10 are mostly related to the place value notation and to a mismatch between the directions of writing numbers and the directions of writing Hebrew words. Some preschool children (26%) wrote the number 13 in reverse order. About half of them wrote the number 13 from left to right in the order in which the number is pronounced in Hebrew (in Hebrew, like in English, when referring to the number 13, the 3 is said first and then the 1). The typical explanation that these children gave to their writing was: "I write it as I hear it, three and then ten ..." (see Figure 3).

[FIGURE 3 OMITTED]

The other half wrote the number 1 first, but they wrote the two digits of the number 13 from right to left. The resulting number was, therefore, 31. A typical verbal comment was, "Numbers are written in the same direction as words ... from right to left." These children over-generalized the rules regarding the direction of writing Hebrew to those of writing numbers (All the preschool children knew how to write some words in Hebrew, including their names and some other words). Similar writings and explanations were provided by the preschoolers to the unacceptable ways of writing 20 (see Figure 4).

Avoidance of writing the numbers. A substantial proportion of preschool children (about 40%) avoided writing some of the numbers (mainly those greater than 10). The explanations given by these children were of the type: "I know that I don't know how to write this number" and "I know it's with two numbers but I don't know how to write it." It seems that these children realized that the representational number system has its own principles and rules and that they were unfamiliar with these. This avoidance might imply an existence of an intermediate phase between the realization that numerical symbols represent numbers and the ability to produce such representations (between the third and the fourth stages that are described in Hughes, 1986). This phase of awareness of the existence of the rules for writing numbers coupled with a lack of knowledge of these rules perhaps accounts for the avoidance of writing.

[FIGURE 4 OMITTED]

Recognition: Identifying Written Number Symbols

Number symbols. The nine cards in this category included three with only numerals (5, 23, and 0) and six bearing numerals and other signs (3, 456; 44,444; 1,000, -75, 0.008 and 1/12). Almost all the children in both groups (at least 94%) immediately put the cards that included only numerals in the "number" box. The six cards that included numerals and a comma or a minus sign were regarded as numbers by about 85% of the children in the kindergarten group, whereas interestingly, a smaller number of the first graders accepted these representations (about 70%). The kindergarten children justified their acceptance of the cards that included numerals and other signs as numbers by saying, "There are numerals on the card." Those children in the first grade who chose not to regard these expressions as numbers typically explained that, "When you write numbers you don't add dots and lines." It may therefore be assumed that the preschoolers have not yet acquired the skill of differentiation between numerals and numbers. The first graders, on the other hand, are very much involved in the process of learning the number system and therefore insist that only numerals are to be used for writing numbers.

Mixture of Hindu-Arabic numerals and other symbols. More preschool children (about 60%) than first grade pupils (about 30%) accepted representations that included both numerals and other non-mathematical symbols (e.g., 3[ALEPH]58) as representing numbers. The children's explanations of their choices further support the assertion that the first graders tend to argue that numerals, and only numerals, should be used to write numbers. The preschool children, however, accepted representations that included numerals as well as other, mathematical or non-mathematical symbols, as numbers.

Symbols of mathematical operations and of mathematical relations. Most children in both groups (about 92%) knew that the signs +, x, =, and > were not numbers. Many of them noted that these signs are often used in mathematics, but that they are not numbers. Ben (a first grader), for instance, commented, "We write + when we write numbers, but it is not a number."

Letters and words. Practically all children knew that Hebrew letters and Hebrew words do not represent numbers. One preschool child provided no response, explaining that "I am not sure if these are numbers."

Iconic and pictorial representations of numbers. Most children in both age groups (about 75%) argued that iconic and pictorial representations are not numbers. Typical explanations given by those who did not accept iconic representations as numbers were: "These are drawings, not numbers." and "There is no number symbol so it's not a number." The explanation given by the children who did accept iconic representations as numbers related to counting: "You count and you know how many there are and then you know what the number is."

Summary and Discussion

This research expands the informal and formal knowledge base on preschool and first grade children's knowledge of the number system. The results suggest that the level of complexity of the graphic symbol and the order and direction of the number writing are two major factors that influence young children's production of numbers. Studies that were conducted with preschool children at various countries (England, Israel, and Spain) reported that at the age of four, children differentiate between letters and number symbols (e.g., Tolchinsky-Landsman, 1986; Tolchinsky-Landsman & Karmiloff-Smith, 1992). Our results are in accordance with these findings, yet our results also suggest that some Israeli preschool children and some children in the first grade do not differentiate between the directions of writing words in Hebrew and those of writing numbers. It is important to explore if this behaviour persists in higher grades or diminishes with age and/or schooling. Furthermore, attention should be paid to increasing young children's awareness of the appropriate directions of writing numbers. A related, important line of research, from an international perspective, is to explore the generality of this phenomenon, namely, to study if a similar behaviour is evident in the development of number production in other cultures that experience similar discrepancies between the direction of writing the prevalent language and the direction of writing the Hindu-Arabic number system (e.g., Arabic).

A phenomenon that was identified among preschool children in this study is that of avoidance of writing numbers. Hughes (1986) describes four stages of development of the symbolic representations of numbers. The avoidance of writing numbers might constitute an additional stage between the third stage and the fourth stage. At this stage children are aware both of the existence of rules for writing numbers that consists of more than one numeral, and of their own lack of familiarity with these rules. As a result of this awareness children might prefer not to attempt to write numbers. Issues related to this phenomenon, such as whether this is a general developmental phase, should be explored. Our findings may support the idea that children can represent numbers at or below their level of abstraction but not above this level (Kato, Kamii, Ozaki & Nagahhirg, 2002).

As regards recognition, our findings show that a substantial number of preschool children tend to identify expressions that contain Hindu-Arabic numerals as numbers, regardless of any other symbols that are present. In contrast, some first graders tended to identify expressions as numbers if they contained only Hindu-Arabic numerals and commas. It can therefore be maintained that when a concept has not yet been formally studied, there exists a tendency to define the elements in a given set of symbols (in this case, symbols that are regarded as numbers) in an over-inclusive way. However, at the first stages of formal instruction, there is a tendency to over-restrict the recognition of number symbols. This phenomenon was observed by other researchers in other contexts (e.g., in pretend play by Olson & Campbell, 1994). In the case of mathematics, and in the particular situation of determining if a given written expression is a number, it is important to study these processes more thoroughly, especially in light of the large body of research that has been carried out on students' frequent failure to identify numbers as such (see, for instance, Hart, 1981: on writing fractions, and Tirosh & Almog, 1989, on writing complex numbers). A related recommendation for instruction is to devote more attention to discussing the critical and non-critical properties of this symbolic system.

References

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Bialystok, E. (2000). Symbolic representation across domain in preschool children. Journal of Experimental Child Psychology, 76, 173-189.

Bialystok, E., & Cobb, J. (1996). Preschool children's understanding of the form and function of number notations. Canadian Journal of Behavioral Science, 28, 281-291.

Butterworth, B. (2000). What counts. New York: The Free Press.

Carruthers, E. & Worthington, M. (2005, September). Creativity and cognition: The art of children's mathematics. Paper presented at the meeting of the European Early Childhood Educational Research Association. Dublin, Ireland.

Dehaene, S. (1997). The number sense: How the mind creates mathematics. New York: Oxford University Press.

DeLoache, J. S., Miller, K. F., & Rosengren, K. S. (1997). The credible shrinking room: Very young children's performance with symbolic and on-symbolic relations. Psychological Science, 8, 308-313.

Dorfler, W. (2000). Means and meaning. In P. Coob, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 100-132). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Ginsberg, H. (1977). Children's arithmetic. New York: Van Nostrand

Hart, K. (1981). Children's understanding of mathematics. London: Murray.

Hughes, M. (1986). Children and number: Difficulties in learning and mathematics. Oxford: Basil Blackwell.

Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 53-74). Dordrecht, Netherlands: Kluwer Academic Publishers.

Kamii, C., Kirkland, L., & Lewis, B. A. (2001). Representations and abstraction in young children's numerical reasoning. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 24-33). Reston, VA: National Council of Teachers of Mathematics.

Kato, Y., Kamii, C., Ozaki, K., & Nagahiro, M. (2002). Young children's representations of groups of objects: The relationship between abstraction and representation Journal for Research in Mathematics Education, 33, 16-30.

Lesh, R., & Doerr, H. M. (2000). Symbolizing, communicating, and mathematizing: Key components of models and modeling. In P. Coob, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 361-384). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Mandler, J. M. (1992). The foundation of conceptual thought in infancy. Cognitive Development, 7, 273-285.

Munn, P. (1998). Symbolic function in preschoolers. In C. Donlan (Ed.), The development of mathematical skill (pp. 47-71). London: Taylor & Francis.

Nemirovsky, R., & Monk, S. (2000). "If you look at it the other way ...": an exploration into the nature of symbolizing. In P. Coob, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (pp. 177-224). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Olson, D. R., & Campbell, R. (1994). Representation and misrepresentation: On the beginning of symbolization in young children. In D. Tirosh (Ed.), Implicit and explicit knowledge: An educational approach. Norwood, New Jersey: Ablex.

Piaget, J. (1962). Play, dream and imitation in childhood. New York: Norton.

Thomas, G. V., Jolley, R. P., Robinson, E. J., & Champion, H. (1999). Realist errors in children's responses to pictures and words as representations. Journal of Experimental Child Psychology, 74, 1-20.

Tirosh, D., & Almog, N. (1989). Conceptual adjustment in progressing from real to complex numbers. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 18th International Conference for the Psychology of Mathematics Education (Vol. 3, pp 221-227). Paris: University de Paris.

Tolchinsky-Landsmann, L. (1986). The development of written language among preschoolers and first grade beginners. Unpublished doctoral dissertation. Tel Aviv University. Tel Aviv (In Hebrew).

Tolchinsky-Landsman, L., & Karmiloff-Smith, A. (1992). Children's understanding of notation as domain of knowledge versus referential-communicative tools. Cognitive Development, 7, 287-300.

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Wynn, K. (1992). Addition and subtraction by human infants. Nature, 358, 749-750.

Wynn, K. (2002). Do infants have numerical expectations or just perceptual preferences? Developmental Science, 2, 207-209.

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Nitza Mark-Zigdon

Levinsky College of Education

Dina Tirosh

School of Education, Tel-Aiv University

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.

Two different aspects of the development of number symbol representation have been outlined in this brief introduction: Production (more dominant in the work of Hughes, 1986) and recognition (central in the work of Bialystok, 1992). In the light of what has been presented, this study purports to extend our knowledge of children's production and recognition of number symbols. The study was conducted in Israel with Hebrew speaking children. In Israel, the Hindu-Arabic number system is used for representing numbers, and children are exposed to Hindu-Arabic numerals in their daily life (much like in United States, in Europe and in many other counties). Notably, Hebrew is written from right to left. This differentiation in directions between writing numbers (from left to right) and writing words (from right to left) might affect the development of the number writing. Hence, the research questions that guided our study are:

1. How do Israeli kindergarten children and first graders write number symbols?

2. What do Israeli kindergarten children and first graders consider as adequate, number symbols?

Methods

Participants

One hundred and fifty four Israeli children (48 kindergarten children and 106 first graders) participated in the study; half of them were boys and half were girls. All participants came from upper middle-class families with Hebrew as their mother tongue. The kindergarten children were aged between 5 and 6.4 years. They attended two nursery schools where they were exposed to the Hindu-Arabic number symbols but there was no formal instruction relating to these symbols. The first graders were aged between 6 and 7.2 years. They studied in five classes at two schools. During the first grade, these children formally begin to study the base 10 number system.

Tools and Procedure

A structured, individual interview was developed for this study: it consisted of two main sections, in accordance with two facets of representation knowledge: production and recognition. Each child was interviewed twice in a quiet room by the first author. The first interview dealt with the production of number symbols and the second with their recognition. Each interview lasted about 30 minutes.

Production: Writing numbers. Children were given four empty cards and asked to write one number on each card. The numbers were 5, 8, 13 and 20: two numbers below ten, and two above ten. These numbers were presented only orally in this fixed order.

Recognition: Identifying written number symbols. Five groups of symbols were presented to the children: number symbols (0, 5, 23, 1,000, 3,456, 44,444, 0.008, -75, 1/12); mixture of Hindu-Arabic numbers and other symbols (3[ALEPH]58, three numerals and a square between them, numeral (9) and rectangle); symbols of mathematical operations and of mathematical relations (+, =, >, x); letters and words (three written in Hebrew -[TEXT NOT REPRODUCIBLE IN ASCII]), a letter in Hebrew [TEXT NOT REPRODUCIBLE IN ASCII], and the letter w) and iconic and pictorial representations of numbers (five dots, four squares, three flowers, six stars, five houses, five rectangles). The interviewer presented each of the 25 cards to the child, one card at a time, in a mixed order (not according to their classification). She or he was asked to determine whether what appeared on the card was a correct way of writing a number. The interviewer explained the task to each child as follows:

I asked you, at the beginning, to write numbers on cards, like 8 and 13. I asked other children to do the same. I will show you cards that the others wrote. Look at each card and tell me if what is written on it is an acceptable (good, correct) way to write a number, If it is-- put it in the red box, if it is not, put it in the blue box. And while you do this, would you please explain why?

At the end of the classification the child was encouraged to look at the two piles of cards and to make changes, if they so wished.

Results

Production: Writing Numbers

Almost all of the first graders produce adequate representations of the numbers (100% of the two numbers below 10 and 95% and 90% of the numbers 13 and 20, respectively). Almost all the preschoolers (93%) produce adequate representations of the number 8 and about half adequately wrote the number 5 and 13 (54% and 46% respectively). The number 20 was the most difficult one to produce for the preschoolers: 21% produce adequate presentations.

The level of difficulty in writing numbers could not be attributed to the numerical order of the numbers, as more children faced difficulties in writing 5 than in writing 8. The interviews revealed two types of difficulties: unacceptable ways of writing numbers and avoidance of writing the numbers.

Unacceptable ways of writing numbers. The difficulties in writing the numbers below 10 are characterized by the level of complexity of the graphic symbol. The preschool children represent the number 8 by writing two circles, one on top of the other (see Figure 1).

[FIGURE 1 OMITTED]

The drawing of 5 was more demanding as it requires coordinating two kinds of lines (straight and curved) and two directions of writing (vertical and horizontal or left and right) (see Figure 2).

[FIGURE 2 OMITTED]

The difficulties in writing the number above 10 are mostly related to the place value notation and to a mismatch between the directions of writing numbers and the directions of writing Hebrew words. Some preschool children (26%) wrote the number 13 in reverse order. About half of them wrote the number 13 from left to right in the order in which the number is pronounced in Hebrew (in Hebrew, like in English, when referring to the number 13, the 3 is said first and then the 1). The typical explanation that these children gave to their writing was: "I write it as I hear it, three and then ten ..." (see Figure 3).

[FIGURE 3 OMITTED]

The other half wrote the number 1 first, but they wrote the two digits of the number 13 from right to left. The resulting number was, therefore, 31. A typical verbal comment was, "Numbers are written in the same direction as words ... from right to left." These children over-generalized the rules regarding the direction of writing Hebrew to those of writing numbers (All the preschool children knew how to write some words in Hebrew, including their names and some other words). Similar writings and explanations were provided by the preschoolers to the unacceptable ways of writing 20 (see Figure 4).

Avoidance of writing the numbers. A substantial proportion of preschool children (about 40%) avoided writing some of the numbers (mainly those greater than 10). The explanations given by these children were of the type: "I know that I don't know how to write this number" and "I know it's with two numbers but I don't know how to write it." It seems that these children realized that the representational number system has its own principles and rules and that they were unfamiliar with these. This avoidance might imply an existence of an intermediate phase between the realization that numerical symbols represent numbers and the ability to produce such representations (between the third and the fourth stages that are described in Hughes, 1986). This phase of awareness of the existence of the rules for writing numbers coupled with a lack of knowledge of these rules perhaps accounts for the avoidance of writing.

[FIGURE 4 OMITTED]

Recognition: Identifying Written Number Symbols

Number symbols. The nine cards in this category included three with only numerals (5, 23, and 0) and six bearing numerals and other signs (3, 456; 44,444; 1,000, -75, 0.008 and 1/12). Almost all the children in both groups (at least 94%) immediately put the cards that included only numerals in the "number" box. The six cards that included numerals and a comma or a minus sign were regarded as numbers by about 85% of the children in the kindergarten group, whereas interestingly, a smaller number of the first graders accepted these representations (about 70%). The kindergarten children justified their acceptance of the cards that included numerals and other signs as numbers by saying, "There are numerals on the card." Those children in the first grade who chose not to regard these expressions as numbers typically explained that, "When you write numbers you don't add dots and lines." It may therefore be assumed that the preschoolers have not yet acquired the skill of differentiation between numerals and numbers. The first graders, on the other hand, are very much involved in the process of learning the number system and therefore insist that only numerals are to be used for writing numbers.

Mixture of Hindu-Arabic numerals and other symbols. More preschool children (about 60%) than first grade pupils (about 30%) accepted representations that included both numerals and other non-mathematical symbols (e.g., 3[ALEPH]58) as representing numbers. The children's explanations of their choices further support the assertion that the first graders tend to argue that numerals, and only numerals, should be used to write numbers. The preschool children, however, accepted representations that included numerals as well as other, mathematical or non-mathematical symbols, as numbers.

Symbols of mathematical operations and of mathematical relations. Most children in both groups (about 92%) knew that the signs +, x, =, and > were not numbers. Many of them noted that these signs are often used in mathematics, but that they are not numbers. Ben (a first grader), for instance, commented, "We write + when we write numbers, but it is not a number."

Letters and words. Practically all children knew that Hebrew letters and Hebrew words do not represent numbers. One preschool child provided no response, explaining that "I am not sure if these are numbers."

Iconic and pictorial representations of numbers. Most children in both age groups (about 75%) argued that iconic and pictorial representations are not numbers. Typical explanations given by those who did not accept iconic representations as numbers were: "These are drawings, not numbers." and "There is no number symbol so it's not a number." The explanation given by the children who did accept iconic representations as numbers related to counting: "You count and you know how many there are and then you know what the number is."

Summary and Discussion

This research expands the informal and formal knowledge base on preschool and first grade children's knowledge of the number system. The results suggest that the level of complexity of the graphic symbol and the order and direction of the number writing are two major factors that influence young children's production of numbers. Studies that were conducted with preschool children at various countries (England, Israel, and Spain) reported that at the age of four, children differentiate between letters and number symbols (e.g., Tolchinsky-Landsman, 1986; Tolchinsky-Landsman & Karmiloff-Smith, 1992). Our results are in accordance with these findings, yet our results also suggest that some Israeli preschool children and some children in the first grade do not differentiate between the directions of writing words in Hebrew and those of writing numbers. It is important to explore if this behaviour persists in higher grades or diminishes with age and/or schooling. Furthermore, attention should be paid to increasing young children's awareness of the appropriate directions of writing numbers. A related, important line of research, from an international perspective, is to explore the generality of this phenomenon, namely, to study if a similar behaviour is evident in the development of number production in other cultures that experience similar discrepancies between the direction of writing the prevalent language and the direction of writing the Hindu-Arabic number system (e.g., Arabic).

A phenomenon that was identified among preschool children in this study is that of avoidance of writing numbers. Hughes (1986) describes four stages of development of the symbolic representations of numbers. The avoidance of writing numbers might constitute an additional stage between the third stage and the fourth stage. At this stage children are aware both of the existence of rules for writing numbers that consists of more than one numeral, and of their own lack of familiarity with these rules. As a result of this awareness children might prefer not to attempt to write numbers. Issues related to this phenomenon, such as whether this is a general developmental phase, should be explored. Our findings may support the idea that children can represent numbers at or below their level of abstraction but not above this level (Kato, Kamii, Ozaki & Nagahhirg, 2002).

As regards recognition, our findings show that a substantial number of preschool children tend to identify expressions that contain Hindu-Arabic numerals as numbers, regardless of any other symbols that are present. In contrast, some first graders tended to identify expressions as numbers if they contained only Hindu-Arabic numerals and commas. It can therefore be maintained that when a concept has not yet been formally studied, there exists a tendency to define the elements in a given set of symbols (in this case, symbols that are regarded as numbers) in an over-inclusive way. However, at the first stages of formal instruction, there is a tendency to over-restrict the recognition of number symbols. This phenomenon was observed by other researchers in other contexts (e.g., in pretend play by Olson & Campbell, 1994). In the case of mathematics, and in the particular situation of determining if a given written expression is a number, it is important to study these processes more thoroughly, especially in light of the large body of research that has been carried out on students' frequent failure to identify numbers as such (see, for instance, Hart, 1981: on writing fractions, and Tirosh & Almog, 1989, on writing complex numbers). A related recommendation for instruction is to devote more attention to discussing the critical and non-critical properties of this symbolic system.

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Nitza Mark-Zigdon

Levinsky College of Education

Dina Tirosh

School of Education, Tel-Aiv University

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Author: | Mark-Zigdon, Nitza; Tirosh, Dina |
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Publication: | Focus on Learning Problems in Mathematics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2008 |

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