We compare Danish, English and Japanese number names and show from
semiotic .... Table 1: Number names between 1 and 100 in Danish, English and ...
DANISH NUMBER NAMES AND NUMBER CONCEPTS Lisser Rye Ejersbo & Morten Misfeldt Danish School of Education, Aarhus University This paper raises some questions concerning the relation between Danish number names and digits in the canonical base 10 system. Our hypothesis is that in Danish, number names are more complicated than in other languages, and for this reason, Danish children have more difficulties learning and working with numbers. From this point of view, we investigate how the relation between numbers and number names differs from language to language and how the names influence the number concepts. We compare Danish, English and Japanese number names and show from semiotic and cognitive perspectives how the qualitative differences in how the first 100 numbers are named may give rise to linguistically determined differences in children’s concept of numbers and in the cognitive load of arithmetic processes. INTRODUCTION In this paper we investigate the Danish number names and their influence on how children may conceptualize numbers. We look at the interplay between the words we use to denote numbers and the way numbers are written in the canonical base 10 system (from here on base 10). In the Great Danish Encyclopedia, Danish number names are described as “Very old and reflecting a number concept that is primitive in relation to mathematical thinking” (Talord, 2009-2010/Our translation). Danish number names are very complex, deriving from old number systems using base 12 and base 20. In our investigation, we are especially interested in how the Danish number names can help or hinder the development of a practical concept of numbers and arithmetic competence. Denmark is one of the Scandinavian countries, and Danish is so similar to Swedish and Norwegian that the three languages are mutually intelligible - except for the number names. In both Swedish and Norwegian, the system for number names is similar to the English one. Comparative investigations between different linguistic communities provide examples of how a preschool child’s mother tongue influences his or her concept of numbers and understanding of place value (Miura et al., 1989; Miura et al., 1993; Miura et al., 1999). The focus of several investigations has been the connection between number names from 10-100 in different languages and the understanding of the place value in base 10. Denmark has never participated in such comparative studies. Our hypothesis is that the Danish names for the first 100 numbers are more complicated than those in other languages, and that for this reason Danish children have more difficulties learning and working with numbers. In this paper we test this hypothesis from three different types of perspectives: comparative perspectives
involving the analysis of the number names in Danish, English and Japanese, semiotic perspectives, and cognitive perspectives. COMPARING NUMBER NAMES The following table contains number names in Danish, English and Japanese. The table shows how some Danish names for numbers use 20 as the base reference, while the numbers written with digits follow base 10. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 30 32 40 43 50 54 60 65 70 76 80 87 90 98 100
Danish En To Tre Fire Fem Seks Syv Otte Ni Ti Elleve Tolv Tretten Fjorten Femten Seksten Sytten Atten Nitten Tyve Enogtyve Tredive Toogtredive Fyrre Treogfyrre Halvtreds Fireoghalvtreds Tres Femogtres Halvfjers Seksoghalvfjers Firs Syvog firs Halvfems Otteoghalvfems Hundrede
Explanation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 + 20 30 2 + 30 40 3 + 40 50 (2.5x20) 4 + 50 60 (3x20) 5 + 60 70 (3.5x20) 6 +70 80 (4x20) 7 + 80 90 (4.5x20) 8 + 90 100
English One Two Three Four Five six seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen Seventeen Eighteen Nineteen Twenty Twenty-one Thirty Thirty-two Forty Forty three Fifty Fifty-four Sixty Sixty-five Seventy Seventy-six Eighty Eighty-seven Ninety Ninety-eight Hundred
Explanation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 20 + 1 3x10 or 30 30 + 2 4x10 or 40 4x10 + 3 5x10 5x10 + 4 6x10 6x10 + 5 7x10 7x10 + 6 8x10 8x10 + 7 9x10 9x10 + 8 100
Japanese Ichi Ni San Shi Go Roku Sichi Hachi Kyu Juu Juu-ichi Juu-ni Juu-san Juu-shi Juu-go Juu-roku Juu-sichi Juu-hachi Juu-kyu Ni- Juu Ni- Juu-ichi San-Juu San- juu-ni Si- Juu Si- Juu-san Go-Juu Go- Juu-shi Roku- Juu Roku- Juu-go Sichi- Juu Sichi- Juu-roku Hachi- Juu Hachi- Juu-sichi Kyu- Juu Kyu- Juu-hachi Hyaku
Explanation 1 2 3 4 5 6 7 8 9 10 10 + 1 10 + 2 10 + 3 10 + 4 10 + 5 10 + 6 10 + 7 10 + 8 10 + 9 2x10 2x10 + 1 3x10 3x10 + 2 4x10 4x10 + 3 5x10 5x10 + 4 6x10 6x10 + 5 7x10 7x10 +6 8x10 8x10+7 9x10 9x10 + 8 100
Table 1: Number names between 1 and 100 in Danish, English and Japanese, combined with the underlying calculation ‘explained’ through these names.
In comparing Danish and English and Japanese, we look at two main issues: 1. The number of words that must be learned by rote in each language 2. The regularity of the spoken number system; that is, the degree to which the spoken number system corresponds to the written base 10 system
We can simply count the number of rote learning instances that are required to count to one hundred. The first 100 numbers of spoken Danish can be viewed as a system containing 28 basic signs, 1-9, 10-20, 30, 40, 50, 60, 70, 80, 90, 100, whereas the written base 10 contains 10 signs and a system. In that sense it takes 28 different words to count to 100 in Danish while in Japanese it only takes 11. In English, it is debatable whether there are 13, 20 or 28 different words to be learned by rote. If we accept that the three different forms of ten (ten, -teen and –ty) are the same, then there are only 13 words; if we accept only that –ty is equal to ten, then there are 20; and if we do not accept that –teen and –ty are the same as ten, then there are 28 words – as in Danish. In Danish, all the numbers from 13-19 end with a –ten, but the initial syllable does not sound like the numbers (from 3-9) that they refer to and we therefore count them as different words. The Danish decade number names from 20 to 90 are not directly connected to the numbers from 2-9 and must therefore be learned by rote without a system. Compared to English, the Danish number names are harder to systematize. In English, there is a system whereby all decades start with the number (e.g. twen-, thir-, four-, …) and end with a –ty. In Japanese there is regularity in the oral counting system exactly as in base 10 and in addition, the names for numbers between 0-9 are very short. Danish and English have the same regularity as base 10 starting with the number twenty (in Danish tyve): we start counting from 1-9 between each decade. In Japanese, this regularity starts already at 10, as in base 10. The English and Danish numbers between 10 and 20 follow another system. In both languages eleven (elleve in Danish) and twelve (tolv in Danish) do not follow the same system as the numbers between 13 and 19; in fact they are reminiscences of an old base 12 system. The numbers from 13 to 19 are combinations of an ordinal number with a –ten in Danish and a –teen in English. In both languages, the numbers between 13 and 19 are named in reverse order from the digits. This irregularity continues for the Danish numbers until 100. For instance, the Danish name for 83 is “three and ‘fours’” or in Danish "tre-og-firs" where ‘tre’ is Danish for ‘three’, ‘og’ is Danish for ‘and’, and ‘fire’ is Danish for ‘four’ (the inflection ‘firs’ means eighty); in numbers: 3 + 80. In English, this irregularity in the sequence of the syllables constituting the number words only exists in the numbers from 13-19. In summary, the Danish spoken number system is in reverse order from 10-100, the English spoken number system is reversed from 10-20 and regular from 20-100, while the Japanese system is regular from 10-100. COMPARATIVE PERSPECTIVES Several studies (Miura et al., 1989; Miura et al., 1993; Miura et al., 1999) comparing English-speaking American, Japanese, Chinese and Korean first graders’ (aged on average 6-7 years) cognitive representation of and understanding of place value confirmed that the Asian language speakers showed a preference for using base 10 representations to construct numbers, whereas English speakers showed a preference for using a collection of units. In the investigation, children were asked to construct
the numbers 11, 13, 28, 30 and 42 from sets of wooden blocks (ten blocks and unit blocks). The results showed that 91% of the American first graders used unit blocks to represent the numbers on their first try. In contrast, about 80% of the Asian children used ten blocks when representing the numbers on their first try. These differences in cognitive representation were mainly ascribed to language (Miura et al., 1993). The validity of this conclusion is challenged by the many other cultural and educational differences between Asian and Western children. Yet, children who share similar cultures and belong to similar school systems but have different mother tongues have also been investigated. For example, Dowker et al. (2008) compare English and Welsh students. These students have the same cultural conditions, but Welsh names for numbers are as regular as the Japanese. They conclude: No statistically significant differences were found between schools or age groups on the scaled score on either test (arithmetic and number skills). […] However, there were group differences in a specific area of arithmetical ability, notably, in ability to read and judge numbers pairs.
Because their investigation can more or less eliminate the cultural and educational differences we can conclude that the results indeed show an effect of linguistic differences: Welsh-speaking children find it easier than English-speaking children to read and compare two-digit numbers, suggesting that they are better at using the principles of place value.
This raises the question of why and how different languages influence number concepts and perhaps even the ability to learn simple arithmetic. SEMIOTIC PERSPECTIVES ON NUMBER NAMES The empirical studies show that some languages seem to support the development of concepts of numbers better than other languages, and our initial comparison of Danish, English and Japanese shows that the differences may be related to the degree to which the number names and the written numbers are ‘alike’. In order to view this case from a semiotic perspective we use the concept of iconicity (Stjernfelt, 2007), which describes the likeness of a sign to what it signifies, and the epistemological triangle (Steinbring, 2006), a model describing the relation between mathematical signs and mathematical concepts. Iconicity of the base 10 number system In the case of an iconic sign, the token relates to the object by similarity. Signs with strong iconicity typically “look like” their objects in some sense. For example, the roman number iii, which represents three, is typically considered iconic, because the three i's correspond to the cardinality of the number three. In order to describe how the base 10 numbers can be considered iconic we use Stjernfelt’s description of operational iconicity (Stjernfelt, 2007). The operational
criterion for iconicity denotes the way a sign or a system of signs allows us to experiment and learn about what the sign signifies by extracting information from the sign that was not deliberately included by the producer of the sign. Base 10 has some qualities that make it reasonable to consider written numbers as partially iconic signs. A base 10 number tells us how many of each power of 10 it contains. The system allows us to create increasingly larger numbers from the ten basic numbers; furthermore, the system allows us to easily decide which one of two given natural numbers is largest. Written base ten numbers fully reflect the system. Using the operational criterion of iconicity we can say that written numbers are iconic in the sense that it is easy to determine which one of two given (natural) numbers is largest; it is easy to create larger and larger numbers; and the written base 10 numbers support a range of arithmetic algorithms (addition, subtraction, multiplications etc.). There is in principle nothing to prevent spoken number names from resembling this kind of iconicity. As simple examples of iconic spoken numbers, the additive “bum bum bum” can represent “three” in the same way as the roman number iii, and saying “four times ten and 3” to represent 43 is iconic in the same way as writing “43” in base 10. As we have seen, this is the situation in Japanese and for some numbers in English, but not at all in Danish. Epistemology and the signification of numbers In order to understand how written numbers, spoken number names and concepts of numbers relate to each other we use the ‘epistemological triangle’, which connects conceptual entities to the signs that represent them and to mathematical objects in a reference context (Steinbring, 2006). Steinbring notes that there is in some cases exchangeability between the reference context and the sign/symbol, because the same sign can serve as a reference context for a mathematical concept (left side of the triangle) in some cases and as a representation of a mathematical concept (right side of the triangle) in other cases (Steinbring, 2006). Object, reference context
Sign/ symbol
concept
Figure 1: The epistemological triangle.
We can apply the epistemological triangle in order to understand how mother tongue influences number concepts, by viewing spoken words and written signs as reference
contexts for each other. Taking point of departure in the epistemological triangle, we can represent the situation as shown below: Base 10 system
≈ Written number
Spoken number
Number concept
Figure 2: The influence of written and spoken numbers on the number concept. The written numbers reflect the base 10 system completely.
The epistemological triangle shows that the relations between the written number and the spoken number in terms of how the written number signifies the spoken number and how the spoken number signifies the written number, can influence the number concepts that individuals develop. Since these relations, as we have argued, differ from language to language, we should expect a difference in the number concepts arising from the different number names. In English and especially in Danish the signifying relations between spoken and written numbers are more complicated than in Japanese. The specific effect on the number concepts for English and Danish speaking children cannot be inferred from this analysis, but to hypothesize that this leads to more complicated concepts of numbers and possibly even to problems in learning numbers seems reasonable. Using the concept of iconicity and the epistemological triangle for mathematical signs, we have seen that the written numbers resemble base 10, and using an operational criterion of iconicity, can be said to be iconic. We have also seen that different languages respect this iconicity in the spoken numbers to different degrees. What we have previously described as regularity between written base 10 numbers and spoken number names are cases in which the operational iconicity of the base 10 numbers are reflected in the spoken numbers as well. Furthermore, we have applied the epistemological triangle to a situation in which written and spoken signs for numbers are considered as reference contexts for each other. We infer that these two representations of numbers affect the number concept that pupils develop and hypothesize that this leads to more complicated number concepts for native speakers of languages in which the operational iconicity of the written numbers is less reflected in the spoken number names.
COGNITIVE PERSPECTIVES Learning to count and understand base 10 are cognitive challenges involving many small steps. We have chosen to focus on the following three aspects: 1. Oral counting 2. The cardinal principle of combining a name with a cardinal value 3. The combination of words for a number, its cardinal value and the digit sign Oral counting Developing familiarity with the symbolic number system begins with oral counting. Children start oral counting quite early, and it is not clear if they understand what they are doing when they count. Counting appears to be learned first as a linguistic routine through which the number names are perceived as ‘sign systems’ or cultural semiotic systems that enable the symbolic representation of knowledge (Goswami, 2008). We have seen that Japanese number names are brief and regular. English number names are somewhat longer and the regularity of the number system starts more or less at 20. In Danish, number names are generally even longer and the system has many irregular numbers. Studies of numbers and language (Sousa, 2008) show that a language with short number names loads the working memory (WM) less than a language with longer number names. There are no differences in the ability to count and numerate sets from 1 to 12 among Asian and American children, but from 13 to 100 Asian children are much better (Sousa, 2008). The development from being a novice to being an expert by gaining automaticity with numbers bigger than 12 seems to take longer if the language used has an irregular number system. Danish children have to learn many different and meaningless number words by rote, and we can now conclude that Danish number names load the children’s WM considerably when they are learning to count; in addition, there is clear evidence that syllables rated as more meaningful are easier to recall (Baddeley et al., 2009). A Danish investigation of children’s ability to count (N=140) at the age of 6 showed that more than 40% of the children stopped counting at a number ending with 9 (Lyngsted & Knudsen, 2007). We suspect that this is due to the irregular Danish system for naming decade numbers. The cardinal principle of combining a name with a cardinal value At 3-5 years of age, children understand more or less the five counting principles at least until the set of 10, even when they err in their counting. The five counting principles are (Gelman and Gallistel, 1978, here Siegler, 2003): 1. The one-one principle: Assign one and only one number word to each object. 2. The stable order principle: Always assign the numbers in the same order. 3. The cardinal principle: The last count indicates the number of objects in the set.
4. The order irrelevance principle: The order in which objects are counted is irrelevant. 5. The abstraction principle: The other principles apply to any set of objects. Children typically learn the names of numbers as a long list of words and demonstrate knowledge of the stable order principle by almost always saying the number words in a constant order and saying the last number with emphasis (ibid). The names are developed as sounds connected to the number of objects in the sets. The developmental shift to understanding the number name as a cardinal value requires a qualitative shift in children’s representation of numbers. The cardinal principle requires an understanding of the logic behind counting (Goswami, 2008) and the ability to judge the size of a set. It relies on a representation of quantitative information in which the coding of smaller quantities is different from the coding of larger quantities (ibid). Children are born with the capacity to cognize magnitudes (Halberta et al., 2008) and distinct numerical difference in small number values, called subitization (Dehane 1997). This means that when comparing two different sets, children are for the most part capable of pointing out which set is biggest, depending on the size of the sets and the differences between them; but counting and telling the number in words seem to be harder, especially with bigger numbers. Children’s conceptual understanding of numeration depends on their being able to make a connection between a number name and its cardinal value, which they learn to do by grouping and quantifying sets of objects (Thomas et al., 2002). To group a set means to divide the set into smaller equal groups. Miura et al. (1993) showed how American, French and Swedish children used units instead of ten sticks representing two-digit numbers, and we know from our own experiences that Danish children do the same; it seems that certain languages facilitate grouping in tens whereas others do not. The combination of words for a number, its cardinal value and the digit sign Learning how to connect the number word, its cardinal value and the digit sign is another challenge. As discussed, two different systems must be combined with different representations. Becoming an expert at combining these two systems means developing rapid access to an automatic use of written numbers and at the same time being able to multitask to solve other problems in parallel. If the two systems are iconic and support each other, the difficulties the child encounters in learning this skill will be minimized, as is the case for Japanese-speaking children. If the two systems are irregular and therefore conflict with each other, the child will have greater problems understanding and remembering the connection between the name, the cardinal value and the sign. Duval (2006) describes this situation as a conversion between registers, and observes that the conversions that seem to be easiest for students are the ones that are congruent, meaning that the representation in the starting register is transparent to the target register.
From our own observations of Danish first graders, we have seen that if the task is to say the name of a written number, say 63, children often repeat the ten, twenty, thirtyrhyme and use their fingers. The children stop when they get to their sixth finger and then they know the word. This may be seen as a kind of interfering process, which in cognitive terms means that two parallel processes are in conflict with each other. The semantic treatment demands too much attention, and therefore it is not possible to multitask and complete both processes at the same time (Baddeley, 2009). This interfering effect means that learning to combine spoken and written numbers takes Danish children much longer time to automate. The logic in the base 10 disappears in the Danish language, and therefore the combination of the names and the written digits has more or less to be learned by rote. CONCLUSION In this article we have investigated our hypothesis that Danish number names are more complicated than those of other languages, and therefore, that Danish children have more difficulties learning and working with numbers. We have shown through comparison of number names in different languages, combined with semiotic and cognitive arguments, how the differences in naming numbers may give rise to linguistically determined differences in how children learn number concepts as well as in the cognitive load of arithmetic processes. We have argued that the Danish number names create comparatively great cognitive load in relation to number comparison, counting and basic arithmetic. Furthermore we have shown how number concepts are influenced by the names of numbers, and that number concepts are especially simple when the iconicity in the written base 10 numbers is also present in the words used to signify numbers. In Danish there are two main reasons that this iconicity is not present in the number names: (1) the order of digits is reversed in the numbers between 11-99, and (2) the names of the number decades do not correspond to the number of tens they represent. We conclude that Danish children experience comparatively great difficulties learning numbers due to the system of naming numbers in Danish. REFERENCES Baddeley, A., Eysenck, M. W., Anderson, M. C. (2009): Memory. Psychology Press, Hove and NY Dehaene, S. (1997): The number sense. How the mind creates mathematics. Penguin Press, London Dowker, A., Bala, S., Lloyd, D. (2008): Linguistic influences on Mathematical Development: How Important Is the Transparency of the Counting System? In Philosophical Psychology, Vol. 21, No. 4, August 2008, 523-538
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