Language and Cultural Issues in the Teaching and Learning of WNA

Abstract

Language and culture play a common, key role in conveying concepts in mathematics teaching and learning for mathematical thinking development. Linguistic transparency can foster the construction of mathematical meanings and support the understanding that occurs in learning discourse. A cross-cultural examination of languages should thus allow us to understand linguistic supports or limitations that may interfere with students’ learning and teachers’ teaching of mathematics. This chapter examines number naming and structure across languages and language issues related to whole number structure, arithmetic operations and key concepts of place value and equality from a linguistic perspective. It also specifically considers how the Chinese language has been linked with Chinese arithmetic in ancient and present times.

The first author was supported by Research Committee, University of Macau, Macao, China (MYRG2015-00203-FED). The opinions expressed in the article are those of the authors.

1 Introduction

1.1 Reflections on Language and Culture Before, During and After the Macao Conference

Language is an artefact used to communicate and think (see Chap. 9). Languages differ not only in pronunciation, vocabulary and grammar, but also the different ‘cultures of speaking’. Language plays a common, key role in conveying mathematics concepts for learning and teaching and the development of mathematical thinking. The features of language can help to make numerical concepts transparent and support the understanding that occurs in learning discourse. A cross-cultural examination of languages should thus allow us to understand the linguistic support and limitations that may foster/hinder students’ learning and teachers’ teaching of mathematics. This study examines number naming and structure across languages and language issues related to whole number structure, arithmetic operations and key concepts, and thus has important educational implications for whole number arithmetic.

As reported in Chap. 2, this study aims to foster awareness of the relevance of cultural diversity in the teaching and learning of whole number arithmetic and in related studies. As stated in ICMI Study 21 (Barwell et al. 2016, p. 17), ‘language and culture are closely and intimately related and cannot be separated’. Language and culture influence each other. Language is part of culture and plays an important role in it. A language not only contains a nation’s cultural background, but also reflects a national view of life and way of thinking. Hence, no discussion of language issues in whole number arithmetic can be separated from cultural background.

In this chapter, we address language and cultural issues based on different examples reported by the conference participants, which can be roughly divided as follows:

• The language of whole number arithmetic in Indo-European languages

• The colonial case in Africa

• The Chinese case

We also consider some of the educational implications.

A short outline of Chinese grammar for numbers is collected for the interested reader, who may in this way become acquainted with some background of the Chinese mathematics education. This special focus on the Chinese language and culture depends on:

• The ‘perfect’ match between everyday Chinese arithmetic and the mathematician’s arithmetic (Sun 2015).

• The very interesting organisation of Chinese curricula that has seemingly proceeded uninterrupted since classic times (see Chap. 5).

• The presence of some original strategies (e.g. variation problems) (Sun 2011, 2016).

• The performance of Chinese students and teachers (e.g. Geary et al. 1993; Ma 1999).

• The circumstance of us meeting together in China and seeing a Chinese first-grade lesson in person (see Chap. 11).

The presence of different cultural and linguistic traditions makes participants and readers aware of the choices that have been made throughout history regarding the teaching and learning of whole number arithmetic, creating an improved awareness of cultural diversity in mathematics. This diversity does not allow for the simple adoption of curricula developed by others, unless a careful process of cultural transposition (see Chap. 13) is started. An example at the end of the chapter (Sect. 3.4.5.2) illustrates this point.

1.2 Some Everyday Language Issues in Number Understanding

An assumption of the universality of whole number arithmetic has been predominant for both curriculum reformers and international evaluators. The so-called Hindu-Arabic system of numerals as number signs is considered the most effective computation tool and has consequently been adopted by countries around the world during the last century. However, in this chapter, we discuss how whole number arithmetic is not culture-free, but rather deeply rooted in local languages and cultures, and present the inherent difficulty of transposition from language and culture perspectives.

Information about how words are connected with whole number arithmetic may be found in many books (e.g. Menninger 1969; Zaslavsky 1973; Ifrah 1981; Lam and Ang 2004). We do not aim to summarise what can easily be found elsewhere. Our intention is to systematically collect some of the information and reflections shared between the conference participants who represented many cultural contexts and report on some of the features of their languages/cultures that have important educational implications.

Before approaching the topic of whole number arithmetic, we examine some of the studies conducted in the linguistic field, particularly in the field of pragmatics, where the contrast within the same culture between figurative meaning in everyday language and literal meaning in school arithmetic language is investigated. As the examples (Bazzanella, personal communication) come from very different languages and cultures, this phenomenon seems universal despite having different features.

The roots of this phenomenon may be found in ancient ages. In Poetics, Aristotle himself introduces the idea of the metaphor, which consists of giving a thing a name that belongs to something else. Among the different examples, one concerns numbers: “‘Indeed ten thousands noble things Odysseus did,’ for ten thousand, which is a species of many, is here used instead of the word ‘many’” (Levin 1982, p. 24).

In ancient China, ‘ten thousand’ (wàn, 万) was used in a figurative way, as in the proper name for ‘the Great Wall’ (万里长城, wàn lĭ chángchéng). This name literally means ‘ten thousand lĭ long wall’, where lĭ is an ancient unit of length used to highlight the immense length of the wall.¹

In the last decades, linguistic scholars have started to investigate the use of numbers (whole numbers) in everyday language. There are several issues involved in this usage, such as indeterminacy and approximation. Indeterminacy in language is commonly resorted to for a variety of reasons and takes several different forms (Krifka 2007; Bazzanella 2011). In some applications of numerals, exact numbers are used systematically to denote indefinite quantities. Such uses are linked to certain very specific numerals, either rather low ones (2, 3, 4, 5) or high but ‘round’ ones (100, 1000). They are hyperbolic in the sense that the number indicated cannot be true; moreover, several variants of one expression (100, 1000, 10,000, etc.) often coexist (Lavric 2010). Lavric (2010) collects several examples from European languages (English, French, German, Italian and Spanish) where the meaning of whole numbers is not the same as the numerals learnt in counting. Some expressions in some languages must be interpreted in an approximate way. For instance, the sentence ‘vuoi due spaghetti?’ (do you wish to have two spaghettis?) among Italian speakers means ‘do you wish to have some spaghetti?’ Hence, ‘two’ is not used in its cardinal meaning but means a general number of things. Round numbers (i.e. powers of ten, such as ten, a hundred, a thousand, ten thousand, a hundred thousand, a million) are also used in hyperbolic meaning: ‘I have told you a thousand times that you have to be prudent’. Fractions may be used to mean a very small number (‘half’ also sometimes means a part when the original is divided into two parts that may not be equal), with the numerator ‘one’, and a high and round numerator (e.g. ‘even a millionth of a second’) is used to minimise or a very close numerator and denominator used to maximise (e.g. ‘it is ninety-nine point nine percent certain’). Apart from European languages, in Mandarin Chinese, approximate numerical expressions are classified into two main types with different meanings: one with the discourse marker bā (吧) denoting the approximate quantity and the other without an explicit marker denoting the exact quantity (Ran 2010). These aspects are studied in linguistics for their effects on translations from one language to another, when literal translation is impossible. They are not usually considered in the literature on mathematics education, although they are important to the connection (continuity vs discontinuity) between everyday language and school language.

2 Place Value in Different School Languages and Cultures

2.1 Some Reported Difficulties in Understanding Place Value

How is number naming in daily language related to the structure of numbers in school mathematics? How do listeners in the mathematics classroom recognise numerical concepts? What are the cognitive bases of approximate uses? What are their effects on cognitive processes? We discuss these language issues related to the comprehension of place value in the following.

Place value is the most important concept in the so-called Hindu-Arabic system, as it has a long-term effect on the comprehension of number structure and calculations. It denotes the value of a digit depending on its place or position in the number. Each place has a value of ten times the place to its right. In Chinese literature, place value is emphasised as an understanding of numeration with different units (計數单位). Recording magnitude with different units in counting is called place value in English-speaking communities or positional notation in French-speaking communities. In place value, there are two inseparable principles (Houdement and Tempier 2015):

• The positional principle, where the position of each digit in a written number corresponds to a unit (e.g. hundreds stand in the third place)

• The decimal principle, where each unit is equal to ten units of the immediately lower order (e.g. one hundred = ten tens)

However, a range of studies shows that the teaching and learning of place value/numeration units is difficult. For example, Tempier (2013) finds low percentages of success of 104 French third graders (8- to 9-year-olds) in tasks involving relations between units: ‘1 hundred = … tens’ (48% success), ‘60 tens = … hundreds’ (31% success) and ‘in 764 ones there are …tens’ (39% success). Even in the fourth and fifth grades, no more than half of the students demonstrate an understanding that the ‘5’ in ‘25’ represents five of the objects and the ‘2’ the remaining 20 objects (Kamii 1986; Ross 1989).

Bartolini Bussi (2011) mentions a similar difficulty (see Sect. 9.3.2):

When 7-year-old students are asked to write numbers, a common mistake in transcoding from number words to Hindu-Arabic numerals shows up: some students write ‘10,013’ instead of ‘113’ as the zeroes on the right (100) are not overwritten by tens and units. (p. 94)

It should not be surprising that these students cannot grasp multi-digit addition and subtraction. Many curricula in the West list place value as positional knowledge only. For instance, Howe (2010) offers a critique of elementary curricula in the USA:

Place value…is treated as a vocabulary issue: ones place, tens place, hundreds place. It is described procedurally rather than conceptually.

Bass (see Chap. 19) uses the problem of counting a large collection to stimulate the development of grouping with multiple units, according to the concept of place value. Young-Loveridge and Bicknell (2015) advise supporting the comprehension of place value by providing meaningful multiplication and division at the same time. Place value is inherently multiplicative (Askew 2013; Bakker et al. 2014) and usually introduced as part of the addition and subtraction of multi-digit numbers before children have experienced meaningful multiplication and division. In Chap. 9, we report on artefacts designed and used to overcome some difficulties in the introduction of place value. Based on the studies that have been conducted, a language perspective on place value is rare in the mathematics education field. In this chapter, we wish to reconstruct some part of the history of place value while looking at it from the language perspective.

2.2 Transparency and Regularity of Number Languages: Some European Cases

In Europe, place value was introduced in the thirteenth century through the Arabic tradition and came into conflict with previous traditions (Menninger 1969; Lam and Ang 2004). This explains why the principle of place value continues to be a specific part of school curricula (Fuson and Briars 1990). Units of hundreds and thousands are always explicit, but units of ones and tens are always implicit and often missing in spoken languages. For example, units of ones and tens are not visible in ‘thirty-one’.

Examples in European languages show that many irregularities appear in their languages; they depend on the existence of more ancient representation (with non-ten bases) or on other linguistic properties where the combination of two words forces an abbreviation. Furthermore, the order of units may be different.²

In English, French and German, numbers have independent names up to 12, while in Italian the suffix dici appears with 11 and becomes a prefix with 17 (as in French). English and German are similar from 13 to 20 (with the suffix teen or zehn, meaning 10). But from 21 the order of reading units and tens in German is opposite to that in English until 99. In French there is the memory of base 20, e.g. 70 is soixante-dix; 80 is quatre-vingts. A similar yet more complex irregularity is present in Danish: the irregularity involves the number names between 10 and 20, the inversion of units and tens (as in German) and a memory of a base 20 system (see Chap. 5 and Ejersbo and Misfeltd 2015).

When expressing 76 + 83, for example, different languages hint at different words that make the column calculation more or less difficult.

• English: seventy-six plus eighty-three

• French: sixty-sixteen plus four-twentys-three

• Italian: seventy-six plus eighty-three

• Danish: three-and-a-half-twenty-six plus four-twenty-three

• Chinese: seven tens six plus eight tens three.

The transparency of the Chinese names is likely to foster students’ understanding of place value.

2.3 Post-colonial Cases: Africa and Latin America

Zaslavsky (1973) wrote her fundamental books on African mathematical tradition to contrast the scarce (if any) references to Africa in Menninger (1969). In a later study, Verran (2001) reports on the Yoruba approach to whole number arithmetic. At the Macao Conference, there were two scholars from Northern Francophone (Nadia Azrou) and South-Eastern Anglophone Africa (Veronica Sarungi) who reported on the story of whole number arithmetic in the schools in their postcolonial regions.

2.3.1 Algeria

Azrou (2015) reports the language situation in Algeria, where many different languages are spoken with different status: classical Arabic, Berber and French together with many different local dialects (see Chaps. 5 and 15). Besides the different number words, Azrou reports the different meanings of ‘digit’ vs ‘number’. Consider the following examples:

• A- 1,2,3,…,9

• B- 2781

• C- a series of digits to design a phone number, a car number and an address, e.g. the contact number for ICMI-49 30 20 37 24 30...

• In English, A are called digits, B numbers and C numbers.

• In Arabic, A are called رقم raqm (digit) or أرقام arqam (digits, the plural).

• In French, A are called chiffres, B nombres and C numéro.

• In Berber (Tamazight is one of the oldest languages of humanity), only one word (numro) is used for everything.

The relationship between the French dialect and Berber language (languages used in everyday life) presents a problem. The dialect and Berber language have kept one word (numro) to express everything. This is a problem for students, who confuse the mathematical concepts they learn at school (for both Arabic and French) with the street mathematics used in Berber in everyday life.

2.3.2 The Guatemalan Case

In Guatemala, the official language is Spanish, and the indigenous population comprises 41% of the total population. There are 25 linguistic communities grouped in 4 ‘pueblos’ (different groups of people), i.e. Ladino, Maya, Grifúna and Xinka, each with a unique identity, culture and language. Mayans comprise 81% of the indigenous population and have four linguistic communities. The formal recognition of the complex ethnic composition of this country was made in 1996 through the ‘Agreement of Peace’, which recognised people’s right to their cultural identities. As a consequence, the Ministry of Education set up a bilingual programme in which the teaching must be done bilingually, respecting the culture and values of the indigenous people. In 2005, there were 3800 bilingual schools, and many of their teachers could speak the indigenous language but could not write or read it. Because of the characteristics of this country, primary schools work with two number systems: vigesimal (base 20) for Mayan mathematics and decimal. Numbers are read and written according to the two systems and various languages. The Mayan numeration system uses three symbols: the dot to represent a unit (●), the bar to represent five units () and a third symbol to represent the zero, also called a shell or cocoa bean (). With the combination of these three symbols, the first 19 numbers are written using three rules. First, from one to four, points are combined. Second, five points form a bar. Third, bars are combined up to three.

2.3.3 Tanzania and Other East African Countries

Sarungi (personal communication) reports on the complex situation in the part of Africa colonised by the British Empire (East African countries). The diversity in learners’ first languages makes teaching mathematics in those languages difficult. For example, Tanzania has over 120 ethnic tribes with their own languages, although these belong to major language groups such as Bantu, Nilotic and Cushitic. At the same time, Kiswahili, which is a mixture of Bantu, Arabic and other languages such as Portuguese and English, has become the first language of tribes along the coast and islands of Zanzibar. In fact, Kiswahili is the national language of Tanzania and Kenya and is widely spoken in other East and Central African countries such as Burundi, Rwanda, Uganda and Democratic Republic of Congo.

In Tanzania, the language policy is to use Kiswahili as the medium of instruction in pre-primary and primary education (MOEVT 2014), even though Kiswahili is not the first language of many children, especially those in rural areas (Halai and Karuka 2013) and is learnt formally when entering school. In Uganda, the policy is to use ethnic or local languages in the first 3 years of primary school, although English is used in settings in which learners have diverse local languages (National Curriculum Development Centre n.d.). Research conducted in African contexts has pointed to the challenges of using a language that is not easily accessible to learners and even teachers in some cases, while the use of first languages in mathematics classrooms has been shown to foster more interactions between learners and teachers (Sepeng 2014).

Apart from the benefits of increased participation, the names of numbers in ethnic languages usually point to a base 10 structure (see Funghi 2016). Most African languages have a similar structure for numbers between 10 and 20, namely, ‘ten’ and ‘digit’, where digit stands for a number from one to nine inclusive. Moreover, the decades from 20 to 90 have a logical structure. The wording constitutes either ‘tens digit’ to signify how many tens are taken or ‘decade digit’, such as in Simbiti. Thus, a number like 34 in African ethnic language is literally formed as ‘tens three and four’ or mathematically ‘three tens and four’. Many children encounter the names of numbers to around 30 in a non-formal way by the time they start attending school. Thus, learning whole numbers in such ethnic languages could help learners to make sense of the structure of the numbers. However, there are challenges in taking advantage of these affordances. First, teachers may not be equipped to assist learners, due to unfamiliarity with the local language and its mathematical register (Chauma 2012). Moreover, the language policy may not be favourable to promoting the use of ethnic languages, as is the case for Tanzania, where Kiswahili is encouraged for purposes of national unity.

When widely spoken in a community, Kiswahili can bridge the gap between the multiplicities of languages, although its use for learning whole numbers can be a potential source of confusion for children, even if it is their first language. This is due to the origins of number names from Arabic and Bantu words, which results in an inconsistency in the naming of decades (for a comparison between Awahili and Arabic, see Funghi 2016).

For Bantu speakers, numbers from 1 to 20 present little problem, except for the names of 6, 7 and 9. However, non-Bantu speakers have to learn most of the names, although the structure from 11 to 20 is familiar. For most learners, there is an additional cognitive demand in learning the names of decades, which no longer adhere to the structure of Bantu or other ethnic languages but instead borrow words from Arabic. For example, there is very little link between the names for 30 and 3. In effect, children are required to learn new names for 20, 30, 40 and 50 in Kiswahili. It is only 60, 70 and 90 that have some link to 6, 7 and 9, respectively. Thus, asking children to write down a given two-digit number in words can result in confusion if, for instance, the child needs to remember the name for 30 (thelathini) and cannot infer it from its name, which is linked to its value (three tens). On a related point, the use of English in private pre-primary schools in Tanzania further complicates the matter, as the structure of numbers from 11 to 19 does not follow the known structure of Bantu and Kiswahili. Ultimately, even in contexts in which both learners and teachers speak the same language as the language of instruction, it is important to take into account the features of the common language that hinder or promote the learning of whole numbers in early years of schooling.

2.4 Towards Transparency: The Chinese Approach

Chinese young children perform better at facets of basic arithmetic, such as generating cardinal and ordinal number names (Miller et al. 2000), understanding the base 10 system and the concept of place value (Fuson and Kwon 1992), using decompositions as their primary backup strategy to solve simple addition problems (Geary et al. 1993) and calculation (Cai 1998). A comparative study (Geary et al. 1992) indicates that the addition calculating scores of Chinese students is three times that of American students. Specifically, Chinese students use more advanced strategies and exhibit faster retrieval speeds. American students use counting strategies (e.g. counting fingers or verbal counting) more frequently than their Chinese counterparts. Chinese students use retrieval strategies more frequently than their American counterparts (He 2015). However, most studies have provided various explanations for these findings, such as parents’ high expectations for education, the diligence of the students and the effectiveness of the teachers. Ni (2015) argues that elementary school curricula, textbooks, classroom instruction and the cultural values related to learning mathematics have contributed to the arithmetic proficiency of Chinese children and the establishment of arithmetic as a social-cultural system.

The 2013 PISA results in mathematics (from the test taken in 2012) showed that the highest performers were located in Asian countries, placing in the following order: (1) Shanghai (China), (2) Singapore, (3) Hong Kong (China), (4) Taiwan, (5) South Korea, (6) Macao (China) and (7) Japan. All of these countries have used languages that share the same ancient Chinese number tradition.

Some authors have studied the Chinese language and culture in mathematics education in the last few decades. For instance, ICMI Study 13 (Leung, Graf and Lopez-Real 2006) first focused on a comparison of East Asia and the West. It was followed by a trend of studies and volumes about Chinese tradition in mathematics education (Fan et al. 2004, 2015; Li and Huang 2013; Wang 2013). The specific issue of language has been addressed by many authors such as Galligan (2001) and Ng and Rao (2010), and other authors such as Fuson and Li (2009) and Xie and Carspecken (2007) have compared educational materials in China and the USA.

This phenomenon relates to a large number of teachers and students. In China, there are nearly 2.63 hundred million primary school students. Moreover, the ancient Chinese literature affected the development of mathematics in most East Asian countries (e.g. Japan, Korea, Vietnam) (Lam and Ang 2004) in terms of the convention of place value.

3 The Chinese Approach to Arithmetic

3.1 The Ancient History

The Chinese approach to numerals in primary schools shows consistency among the features of Chinese language, the names of numbers and the use of artefacts for representing numbers and computing (Chap. 5), which can be traced back to the tradition of teaching numbers in China in fourteenth century BCE (Guo 2010). The long tradition is reflected in a range of ancient Chinese arithmetic works, such as the official mathematical texts for imperial examinations in mathematics used a thousand years ago:

• The Suàn shù shū, Writings on Reckoning (算数书) (202–186 BCE)

• Zhoubi Suanjing (周髀算经) (100 BCE)

• The Nine Chapters on the Mathematical Art (九章算术) (100 BCE)

• The Sea Island Mathematical Manual (海岛算经) (about 225–295 CE)

• The Mathematical Classic of Sun Zi (孙子算经) (500 CE)

• The Mathematical Classic of Zhang Qiujian (张丘建算经) (500 CE)

• Computational Canon of the Five Administrative Sections (五曹算经) (1212 CE)

• Xia Houyang’s Computational Canons (夏侯阳算经) (1084 CE)

• Computational Prescriptions of the Five Classics (五经算术)

• Jigu Suanjing (缉古算经) (625 CE)

• Zuisu (缀术) (500 CE)

• Shushu jiyi (数术记遗) (about 200 CE)

In this section, we offer a short outline of Lam and Ang’s (2004) Fleeting Footsteps, a long history drawing on an important reference.

In the general history of numbers, the importance of the Chinese tradition is not always acknowledged. For instance, Ifrah (1981) claims that place value is an Indian invention. Dauben (2002) strictly criticises this error:

One Chinese source of which Ifrah is apparently unaware is the Sun Zi Suanjing 孙子算经 (The Mathematical Classic of Sun Zi), written around 400 CE. (p. 37)

This text has been available in an English translation since 1992 in Fleeting Footsteps, an edition prepared with extensive commentary by Lam and Ang, who later published a more extended edition (Lam and Ang 2004). This source not only gives a complete description of Chinese rod numerals, but also describes in detail ancient procedures for arithmetic operations. The most ambitious part of Lam and Ang’s study argues that the Hindu-Arabic number system had its origins in the rod numeral system of the Chinese. The most persuasive evidence Lam and Ang offer is the fact that the complicated, step-by-step procedures for carrying out multiplication and division are identical to the earliest but later methods of performing multiplication and division in the West using Hindu-Arabic numerals, as described in the Arabic texts of al-Khwārizmī, al-Uqlīdisī and Kūshyār ibn Labbān (see the extensive review in Lam and Ang 2004). Guo (2010) explains that the Chinese system was transmitted to India during the fifth to ninth centuries, to the Arabic empire in the tenth century and then to Europe in the thirteenth century through the Silk Road. In 1853, Alexander Wylie, Christian missionary to China, refuted the notion that ‘the Chinese numbers were written in words at length’ and stated that in ancient China calculation was carried out by means of counting rods and that ‘the written character is evidently a rude presentation of these’, showing both the arithmetic procedure and the decimal place value notation in their numeral system through the use of rods. Wylie believed that this arithmetic method invented by the ancient Chinese played a vital role in the advancement of all fields that required calculations. After being introduced to the rod numerals, he wrote:

Having thus obtained a simple but effective system of figures, we find the Chinese in actual use of a method of notation depending on the theory of local value [i.e. place value], several centuries before such theory was understood in Europe, and while yet the science of numbers had scarcely dawned among the Arabs. (p. 85)

In a review of the first edition of the Archives internationales d’histoire des sciences, Volkov (1996) writes that the book ‘may provoke a strong reaction from historians of European mathematics’. Nevertheless, Volkov emphasises one of the book’s great strengths:

The emphasis made by the authors on the great importance of studying Chinese methods of instrumental calculators as well as numerical and algorithmic aspects of Chinese mathematics, which otherwise cannot be understood properly. (p. 158)

Chemla (1998) suggests adopting a prudent attitude towards this controversy:

The nine chapters share with the earliest extant Indian mathematical writing (6^th c.) basic common knowledge, among which is the use of a place-value decimal numeration system. Such evidence allows no conclusion as to where this knowledge originated, a question which the state of the remaining sources may prevent us forever answering. Instead, it suggests that, from early on, communities practicing mathematics in both areas must have established substantial communication. (p. 793)

This historic origin could be helpful for understanding why the Chinese (Eastern individuals) have found it so easy to grasp this concept, why it is so late to develop in Europe, how number heritage has been shaped and how we can advise on the number practices or tools used to strengthen the comprehension of place value. In the following, we elaborate on the Chinese approach to arithmetic as representative of East Asia. We begin by considering some elements of the Chinese approach to numbers and computation and then discuss some of the educational implications.

Although ancient Chinese mathematicians did not develop a deductive approach, they made advances in inductive algorithm and algebra development (Guo 2010). The Zhoubi Suanjing (周髀算經), the oldest complete surviving mathematical text compiled between 100 BCE and 100 CE, contains a statement highlighting the analogy nature of Chinese tradition:

In relation to numbers, you are not as yet able to generalize categories. This shows there are things your knowledge does not extend to, and there are things that are beyond the capacity of your spirit. Now in the methods of the Way [that I teach], illuminating knowledge of categories [is shown] when words are simple but their application is wide-ranging. When you ask about one category and are thus able to comprehend a myriad matters, I call that understanding the Dao. … This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning …Whoever can draw inferences about other cases from one instance can generalize. (Quoted in Cullen 1996, pp. 175-176)

Since antiquity, the major focus of Chinese mathematics has been on numbers and computations as collections of prescriptions similar to modern algorithms. Mathematics is called shùxué 数学 (“shù” meaning ‘number’) in Chinese. Knotted cords and tallies (see Sect. 9.2.2) were mentioned in ancient Chinese literature (Martzloff 1997, p. 179), following multiplicative-additive rules. The Chinese used bamboo rods to count (see the information about counting rods in Sect. 9.2.2), and this activity fostered the creation of a systematic way to represent numbers. The first nine numerals formed by the rods are presented in Fig. 3.1.

Fig. 3.1

The Chinese Rod representation of the first nine numerals

According to Lam and Ang (2004), the number presentation principle was initially introduced as follows:

Numerals in tens, hundreds and thousands were placed side by side, with adjacent digits rotated, to tell each apart. For example, 1 was represented by a vertical rod, but 10 was represented by a horizontal one, 100 by a vertical one, 1000 by a horizontal one and so forth. Zero was represented by a blank space so the numerals 84,167 and 80,167 would be as shown [see Figure 3.2]. … Although most books on the early history of mathematics, especially the recent ones, have mentioned the Chinese rod numerals, they have failed to draw attention to a very important fact that the ancient Chinese had invented a positional NOTATION. Any number, however large, could be expressed through this place value notation which only required the knowledge of nine signs. I should add that in the current more sophisticated written form, a tenth sign, in the form of zero, is required (Lam and Ang 2004, p. 1)

Fig. 3.2

Rod representation of multi-digit numbers

The translation of the computation principle was initially introduced as follows:

In the common method of computation [with rods] (fán suàn zhĭ fǎ, 凡算之法), one must first know the positions (wèi, 位) [of the rod numerals]. The units are vertical and the tens horizontal, the hundreds stand and the thousands prostrate; thousands and tens look alike and so do ten thousands and hundred. (Lam and Ang 2004, p. 193)

A feature of Chinese mathematics is the ancient use of the counting rods (算筹, suàn chóu) on a table (counting board, jìshù bǎn, 计数板, Fig. 3.3). The counting board was used to make computations (arithmetic operations, extracting roots) and solve equations. The rules for using the counting board are carefully described by Lam and Ang (2004), who highlight the feature of introducing procedures in pairs: the procedure for subtraction is the inverse of that for addition, and the procedure for division is the inverse of that for multiplication. Chemla (1996) highlights that the position in the counting board is stable (see Sect. 3.3.4, which reports on the wording of the elements of arithmetic operations).

Fig. 3.3

An ancient drawing of a suàn pán (算 盘) with measurement units (https://commons.wikimedia.org/wiki/File:Ming_suanpan.JPG)

3.2 Chinese Language Foundation to Place Value

The concept of place value is dominantly used in counting rod or suàn pán (算 盘) and written numerals (Sun 2015). Moreover, place value can be traced to the use of base 10 and conversion rates for measurements, classifier grammar and the part-part-whole structure with radicals and characters in local language. This language origin can be helpful in understanding why Western students find it so difficult to grasp the concept of place value and why it has developed so late in Europe from a language perspective.

3.2.1 Base 10 and the Conversion Rate for Measurement

The Chinese system had a base 10 convention for representing quantities (Lam and Ang 2004; Martzloff 1997; Sun 2015). This was consistent with the conversion rate between the measurement units of length and volume, since the first emperor (qin shi huang 秦始皇) who unified the whole of China in third century BCE introduced a metric system for measurements.³ Except for weight units, units of length and volume had base 10 conversion rates. For example, the conversion rate of length units was expressed as follows (Lam and Ang 2004):

• 1 yin(引) = 10 zhang (丈) = 100 chi (尺) = 1000 cun (寸) = 10,000 fen (分) = 100,000 li (釐) = 1,000,000 hao (毫).

The conversion rate of weight units was:

• 1 liang (两) = 10 qian (钱) = 100 fen (分) = 1000 li (釐) = 10,000 hao (毫) = 100,000 si (丝).

The conversion rate of volume units was:

• 1 gong (斛) = 10 dou (斗); 1 dou (斗) = 10 sheng (升).

The ancient conversion rate of time units was 100 before the Western Zhou dynasty:

• 1 shi (时) = 100 ke (刻); 1 night (晝夜) = 5 geng (更).

The first money conversion rate was 10 in ancient China:

• 1 peng (朋) = 10 ke (貝).

Besides the measurement unit systems, the Chinese system had a base 10 convention for representing numerals using number characters and corresponding number units (Zou 2015). This can be ascribed to the Yellow Emperor in the sixth century book by Zhen Luan, Wujing suanshu (五經算術 Arithmetic in Five Classics) (Guo 2010). The first five number units, i.e. ge (個), shí (十), bǎi (百), qiān (千) and wàn (萬), always represent 1, 10, 10², 10³ and 10⁴, respectively. The other number units vary with different systems of number notation.

Shushu jiyi 《数术记遗》 written by XuYue (徐嶽) during the Eastern Han dynasty (50–200 CE) recorded the early number naming principle: the conversion rate of the down number (xiashu 下数), i.e. the standard number, was 10; the conversion rate of the middle number (zhongshu 中数), i.e. the large number, was 10,000; and the conversion rate of the up number (shangshu 上数), i.e. the largest number, was the square of the number unit.

Looking at decimal and fraction numbers, the following spoken numeration units were used to denote small orders of magnitude in Sunzi Suanjing (Lam and Ang 2004) in ancient China. The negative power of 10 was stressed in daily spoken numerals: 10⁻⁴絲 si, 10⁻³ 毫 hao, 10⁻² 釐 li and 10⁻¹分 fen.

3.2.2 Classifiers

All number units in the Chinese language are called classifiers (Liàngcí 量詞). In English, it is natural to use measurement words to describe the quantity of a continuous noun (i.e. to identify a specific unit to make the quantity countable). For example, in 1 m of cloth, 1 ml of water and 1 kg of meat, the measurement units of m, ml and kg are, respectively, required. However, it is natural not to use measurement words to describe the quantity of countable nouns (e.g. one apple, five ducks and three desks). There are hundreds of different classifiers, all of which reflect the objects to be counted. In Chinese, both uncountable and countable nouns need measurement words known as classifiers. Consider one ge (個) apple, five zhi (只) ducks and three zhang (張) desks, in which ge (“unit of fruit”), zhi (“unit of animal”) and zhang (“unit of object”) play the role of measurement word as units. This is a kind of Chinese grammar used to describe quantity (數量), which requires numbers and classifiers. The classifiers are called number units (Zou 2015), numeration units (Houdement and Tempier 2015), number ranks (Lam and Ang 2004) or number markers (Martzloff 1997).

The column units left of the suàn pán (算 盘) shown in Fig. 3.4 read from left to right as follows:

• Wan (萬 ten thousand), qian (千 thousand), bai (百 hundred), shi (十 ten), liang (兩 weight unit 1 liang = 1/16 jin), 錢 qian (10 qian = 1 liang), 分 fen (weight unit, 10 fen = 1 qian).

Fig. 3.4

The number 71,824, written by the mathematician Jia Xian during the Song dynasty (960–1279)

The column units right of the suàn pán shown in Fig. 3.4 read as follows from left to right:

• 一 ones, 石 shi (volume unit, 10 dou = 1 shi); 斗 dou, 升 sheng, 合 (1 斛 = 10 斗, 1 斗 = 10 升, 1 升 = 10 合).

The units are used as column units. Weight, volume and numeration units have the same position in a functioning calculation. This indicates that numeration units have the same role as that of measurement units in Chinese (Martzloff 1997).

Following Allan (1977), there are about 50 languages in the world with this feature, some in the Far East and some in other parts of the world. We discuss the case of the Chinese language, where numeral classifiers are systematically used, in detail as follows. Following Senft (2000), numeral classifiers are defined in the following way:

In counting inanimate as well as animate referents the numerals (obligatorily) concatenate with a certain morpheme, which is the so called ‘classifier’. This morpheme classifies and quantifies the respective nominal referent according to semantic criteria. (p. 15)

There are many classifiers in Chinese, as each type of counted object has a particular classifier associated with it. This is a weak rule, as it is often acceptable to use the generic classifier (gè, 个) in place of a more specific classifier. The generic classifier (gè, 个) is not translated into English, but may be considered as a kind of unit (a ‘one’). The generic classifier may be considered the prototype of units in place value representation.

Besides the generic classifier (gè, 个), other units of higher value have been introduced in Chinese to represent numbers: ten (shi 十), hundred (bǎi 百), thousand (qiān 千) and ten thousand (wàn 万). A very interesting example from ancient Chinese is given in Figure 1 in the Yongle Encyclopedia (1408).⁴ In the example shown in Fig. 3.4, the number 71,824 is represented to indicate the digit and number (or measurement) unit. In this case, the unit is 步 (bù, i.e. step, an ancient length unit).

In particular:

• The first line ‘七一八四二’ represents the number value ‘71,824’.

• The second line represents the units 萬 (wàn, ten thousand), 千 (qiān, thousand), 百 (bǎi, hundred), 十 (shí, ten) and 步 (bu, or ‘step’, an ancient length unit).

• The third line represents the number using the ancient rod numerals, hinting at the counting rods discussed previously. The number units have the same position as the measurement unit (bu).

Classifiers are used also in the recitation of numerals when counting objects, so that both oral and written numerals are kept consistent with each other.

In Fig. 3.5, ten-two hints at an addition procedure of 10 + 2, while two tens hints at a multiplication procedure of 2 × 10. Hence, the Hindu-Arabic number 24 is translated into the Chinese language as ‘two tens and four ones’ (二十四个).

Fig. 3.5

The oral counting in Chinese. One ones, two ones …; ten ones, one ten and one ones, one ten and two ones, …; two ten ones, two tens and one ones …

The legend shows a literal translation into English (numeral and classifier). In the translation, there is ambiguity between one (number) and one (classifier or unit), which in Chinese are written (一, 个) and said (yī and gè) in two different ways. The same happens for 10, which in English is both a number and a unit.

In other languages (e.g. Italian), the situation may be less ambiguous, as ‘uno’ and ‘dieci’ (numbers 1 and 10) are different from ‘unità’ and ‘decina’ (unit), but the use of terms like the latter in the reading of numbers is limited to school practice (decomposition of a given number in unit, ten, hundred and so on).

In Chinese, classifiers are also used in interrogative questions, e.g. 多少 (duōshǎo), which means ‘How much? How many?’, and the right classifier must follow. When this term is used in an arithmetic word problem, e.g. in additive problems, the same classifier is used for both the data and question. For example, if five zhi (只) ducks swim in a river, and then two zhi (只) ducks join them, how many zhi (只) ducks are there altogether? This example shows that zhi (只) must be used for both the data and question.

By identifying classifiers of quantity, concrete numbers with units of the same name (same classifiers) are defined like numbers (see Chap. 18 of this volume). A principle for arithmetic operations with like numbers is also constructed in everyday language (see Chap. 18 of this volume).

Principle of addition/subtraction: only like numbers can be directly added or subtracted. Two zhi (只) ducks can be added to three zhi (只) ducks. Two zhi (只) ducks cannot be added to three dozen da (打) or groups of ducks.

Principle of multiplication: only unlike numbers can be directly multiplied. For example, three groups of zhi (只) ducks swim at the river. Each group comprises four zhi (只) ducks. How many zhi (只) ducks are there in total? The answer is 4 zhi(只) ducks * 3 groups = 12 zhi (只) ducks.

Principle of division:

• With like numbers (measure division): for example, 12 zhi (只) ducks swim in the river. Each group comprises four zhi (只) ducks. In this case, 12 and 4 are like numbers. How many groups are there in total? The answer is 12 zhi (只) ducks/4 zhi (只) ducks = 3 groups.

• With unlike numbers (partitive division): for example, 12 zhi (只) ducks swim in the river. We plan to group them into three groups. Here, 12 and 3 are unlike numbers. How many zhi (只) ducks are in each group? The answer is 12 zhi (只) ducks/3 groups = 4 zhi (只) ducks per group.

Classifiers are one of the most important elements required in word problem-solving. Generally, Chinese curricula do not need a section to differentiate partitive division from measure division, as the grammar of classifiers is enough to introduce the distinction.

3.2.3 Radicals and the Part-Part-Whole Structure

Radicals (部首 bù shǒu ‘section headers’) constitute the basic writing unit. Most (80–90%) of Chinese characters are phonetic-semantic compounds, combining a semantic radical with a phonetic radical. Chinese words have a compound or part-part-whole structure. The compound can be seen in the structure of Chinese number words. For example, as shown previously, the Chinese refer to the number 12 as ‘ten-two’ rather than as a single word such as ‘twelve’.

The idea of a part-part-whole structure appears in a more general way in number computations. A number (a whole) may be conceived as the sum of two parts in different ways (see Fig. 3.6).

Fig. 3.6

The decomposition of 6 in many different ways as 5 + 1, 4 + 2 and so on (Mathematics Textbook Developer Group for Elementary School 2005, p. 42)

This idea may be connected with the use of artefacts (either counting boards jìshù bǎn, 计数板 with rod numerals 算筹; suàn chóu or suàn pán 算 盘). For instance, in the suàn pán, it is important to ‘make a ten’ by replacing two groups of five beads with one bead in the tens place if one has to calculate the following while also exploring the associative and commutative properties of addition:

• 15 + 7 = (10 + 5) + (5 + 2) = 10 + 5 + 5 + 2 = 10 + 10 + 2

The practice of composing/decomposing numbers is exploited to carry out very fast calculations (for a didactical example, see Chap. 11, Sect. 11.2).

3.3 Conceptual Naming of Fractions

The Nine Chapters on the Mathematical Art (九章算術; Jiǔzhāng Suànshù) was composed by several generations of scholars from the tenth to second century BCE, with its latest stage composed from the second century CE. According to Guo (2010), it gave the first fraction theory in the world. These are the procedures called he fen (addition 合分: Problems 7–9), jian fen (subtraction 減分: Problems 10–11), ke fen (comparison 課分: Problems 12–14), ping fen (arithmetic mean 平分: Problems 15–16), cheng fen (multiplication 乘分: Problems 19–25) and jing fen (division 經分⁵: Problems 17–18) (Sun and Sun 2012).

Martzloff (1997) observes, ‘In Chinese mathematics, by far the most common notion of fraction is that which comes from the notion of dividing a whole into an equal number of equal parts (sharing)’ (p. 192). He quotes examples such as 三分之二 (sān fēn zhī èr), meaning ‘two thirds’. The word 分 (fēn) suggests the idea of sharing, as etymologically its upper component bā (八) means ‘to share’, while its lower component represents a knife (刀, dāo). The order of reading (and writing) is denominator first and numerator second and may be literally translated as ‘of three parts, one’. Martzloff (1997) continues:

The denominator and the numerator are then respectively called fēn mǔ (分 母 the ‘mother’ of the sharing) and fēn zǐ (分 子 the ‘son’ of the sharing). The inventor of these expressions was thinking of a pregnant mother and her child, thus highlighting both the difference in size and the intimate link between the two terms. (p. 103)

According to Needham and Wang (1959) and Guo (2010), decimal fractions were called tiny numbers (微數 wēi shù), first developed and used by the Chinese in first century BCE by Liuhui (劉徽) (Guo 2010).

3.4 Arithmetic Operations

Here, we explain how addition and subtraction were introduced into Chinese tradition. The links between addition and subtraction were highlighted in the ancient textbooks. In 1274, Yang Hui observed, ‘Whenever there is addition there is subtraction’ (quoted in Siu 2004, p. 164).

This strict link is evident in the wording of operations. The strong regularity is evident in the following list:

• 加 – jiā – addition.

• 加数 – jiā shù – addend.

• 减 – jiǎn – subtraction.

• 减数 – jiǎn shù – subtrahend, literally ‘subtracting number’.

• 被减数 – bèi jiǎn shù – minuend, literally ‘subtracted numbers’.

• 乘法 – chéngfǎ – multiplication.

• 被乘数 – bèi chéng shù – literally ‘multiplied number’.

• 乘数 – chéng shù – literally ‘multiplying number’.

• 除法 – chúfǎ – division.

• 被除数 – bèi chú shù – dividend, literally ‘divided number’.

• 除数 – chúshù – divisor, literally ‘dividing number’.

  被 (bèi) is the most common word used in Chinese to create the passive verb form.

This regularity is meaningful, especially when compared with the wording in Western languages. Schwartzman (1994) points out that many English mathematics terms are borrowed from Greek and that Latin-derived terms bear no inherent meaning. For example, the English words ‘minuend’ and ‘subtrahend’, which come from Latin words and thus have little meaning today for English-speaking children in contrast with Chinese subtracted and subtracting numbers, directly embody the subtraction relationship without the exchange law. (The same is true in other Western languages.)

Addition and subtraction are carried out using counting rods (算筹 suàn chóu) (see Sect. 9.2.2) by simply grouping (組合 zǔhé making the bundle) or ungrouping (解組 jiě zǔ opening the bundle) the rods (see Figs. 3.7 and 3.8) (Mathematics Textbook Developer Group for Elementary School 2005).

Fig. 3.7

Addition in the Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005, vol. 2, p. 62.)

Fig. 3.8

Subtraction in the Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005, vol. 2, p. 68.)

When the abacus (suàn pán 算盘) is introduced, fingering is complex (Fig. 3.12) and wording may become different (Fig. 3.10):

• 进一 – jìn yī – forward (towards the unit of higher value, e.g. when 10 units becomes a ten

• 退一 – tuì yī – backward (towards the unit of lower value, e.g. when a ten becomes 10 units

Fig. 3.9

Representation of 123456789 in a Chinese suàn pán (Kwa 1922, p. 6)

Fig. 3.10

Wording on suàn pán: forward and backward

Fig. 3.11

Comparing numbers in the first grade: the prior content of the textbook (Mathematics Textbook Developer Group for Elementary School 2005, p. 5)

Fig. 3.12

Fingering in suàn pán: the correct method of moving the beads (Kwa 1922, p. 8)

The following images are taken from Kwa (1922), an old handbook of the Chinese abacus that was included as a gift for participants at the Macao Conference.

These features are interesting, as in both cases they emphasise the inverse relation between addition and subtraction, which are described by means of inverse verbs. Division is based on multiplication, as it is the inverse of multiplication and uses a scheme that is symmetric with respect to the multiplication performed in rod calculations (adapted from Martzloff 1997, p. 217) (Table 3.1).

Table 3.1 The symmetric scheme in Sunzi Suanjing

We analyse the differences from Western languages where this link is not highlighted as follows.

3.5 Mathematical Relational Thinking: Equality

A range of studies has advised emphasising not only numerical computation but also quantitative relationships (Ma 2015; Bass 2015; see also Chaps. 6 and 9 of this volume). The relational thinking of equality constitutes a central aspect of equations and algebra thinking (Cai and Knuth 2011). Equality is a key concept, but sometimes problems are presented in Western curricula. Li et al. (2008) show that Chinese curricula introduce the equal sign in a context of relationships and interpret the sign as ‘balance’, ‘sameness’ or ‘equivalence’. In the following, we review the history of the equal sign and address the approach to the relational view of the equal sign and equality.

3.5.1 The History of the Equal Sign ‘=’ in Europe

The equal sign (‘=’) was invented (and used in its relational meaning) in 1557 by Welsh mathematician Robert Recorde (in his work The Whetstone of Witte), who was fed up with writing ‘is equal to’ in his equations. He chose the two lines because ‘no two things can be more equal’ (Cajori 1928, p. 126).

The etymology of the word ‘equal’ is from the Latin words ‘aequalis’ (meaning ‘uniform’, ‘identical’ or ‘equal’) and ‘aequus’ (meaning ‘level’, ‘even’ or ‘just’).

The symbol ‘=’ was not immediately popular. The symbol ‘||’ was used by some, and ‘æ’ (or ‘œ’), from the Latin word ‘aequalis’ meaning ‘equal’, was widely used into the 1700s.

3.5.2 The History of the Equal Sign ‘=’ in China

It seems there was no ancient symbol for ‘=’ in Chinese, but the Chinese characters 等 děng (equality) for relational meaning and dé 得 (get the result) for procedural meaning were used broadly in ancient texts. Equality is related to the balance rule of yin-yang and the invariant principle of the I Ching. The basic procedures of substituting in the Chinese rod/suàn pán, substituting 5 by 5 ones, substituting two 5s by 10, substituting 10 by 1 ten, substituting 100 by 10 tens, substituting 1 thousand by 10 hundreds, etc., reflect the spirit of equality used in a broader, flexible way to some extent.

Such is the fundamental ancient Chinese mathematics spirit. ‘Simultaneous equations’ appears as one of the nine chapters of The Nine Chapters on the Mathematical Art (九章算术; Jiǔzhāng Suànshù) (Guo 2010). Spirit of equality is reflected in the ‘equalising’ and ‘homogenising’ theory (齐同原理), the first basic principle to deduce fractions, and ‘cutting and paste’ theory (割补原理), an explicit principle used when solving geometry problems involving area and volume in Liuhui’s commentary on The Nine Chapters (Guo 2010).

There are 256 instances of the character 得 (dé) and 11 instances of the character 等 (děng) in The Nine Chapters. The fifth problem in 方田 fangtian – rectangular fields – reads as follows:

The method for simplifying parts: What can be halved, halve them. As for what cannot be halved, separately set out the numbers for the denominator and numerator. Then alternately reduce them by subtraction. This is seeking for the equality. Simplify using this equal number. (Guo 2010, p. 99)

3.5.3 Chinese Approaches to the Relational Meaning of Equality

Ni (2015) reports that Chinese teachers are intolerant of errors where the relational (or conceptual) meaning of ‘=’ is replaced by a procedural (or operational) meaning, while US teachers consider such errors minor. She mentions Chinese textbooks in which one-to-one correspondence is used from the beginning to assist students in better understanding the equal, greater-than and less-than symbols to enhance the relational meaning of ‘=’ in contrast with ‘<’ and ‘>’ (Fig. 3.11).

This strategy is widespread in other countries (Alafaleq et al. 2015).

In general, the English expression ‘how many’ is translated into Chinese as ‘more or less’ (duōshǎo, 多少), which hints at the relational meaning and denotes a comparison of more than or less than (the imagined number). This expression is very common in arithmetic word problems. Like numerals, such expressions need a classifier (Sect. 3.3.2.2), highlighting the explicit connection between data and unknown values.

The variation approach to word problems presents another way to cope with the conceptual meaning of equality in China. Variation (变式, biàn shì) is a widely used approach that aims to discern the variance, invariance and sameness behind a group of problems and is regarded as the foundation of algebraic thinking and equations (Sun 2011, 2016). This approach is also closely related to the features of the Chinese language. Chinese is a tonal and logographic language, where each character has multiple meanings (一詞多義) and each word plays multiple roles in its context (一词多性). Teaching by variation is consistent with the needs of teaching the Chinese language. To learn to write Chinese and to increase their orthographic awareness, students must distinguish the similarities and differences of different characters that very often look similar to each other (Marton et al. 2010).

As variation problems enhance perceptions of variance and invariance or equality to solve word problems in Chinese curricula, they are regarded as one of the most important and explicit task design frameworks in China (Sun 2016). They refer to the ‘routine’ daily practice commonly accepted by Chinese teachers (Sun 2007, 2011; see also Cai and Nie 2007). Following Sun (2011), Bartolini Bussi et al. (2013, p. 550) describe a typical feature of these problems:

• One distinctive feature of word problems is to develop the ability to identify the invariant category of word problems (识类) it belongs to and discern different categories (归类), namely, discern the invariant elements from the variant elements between problems and recognize the ‘class’ every problem belong to. This pedagogy is generally called as biànshì (变式) in Chinese, where ‘biàn’ stands for ‘changing’ and ‘shì’ means ‘form’, can be translated loosely as ‘variation’ in English (Sun 2011). Some categories of biànshì are the following:

• OPMS (One Problem Multiple Solutions), where, for instance, the operation to solve the problem is carried out in different ways, with different grouping and ungrouping: 8 + 9 = (8 + 2) + 7; 8 + 9 = 7 + (1 + 9) and so on.

• OPMC (One Problem Multiple Changes, see the variation problem below in Italy (Bartolini Bussi et al. 2013)), where in the same situation some changes are introduced.

• MPOS (Multiple Problem One Solution), where the same operation can be used to solve different problems, as in summary exercises (Sun 2011).

Western curricula use various models (e.g. models of taking away and comparing) to introduce meanings of addition/subtraction, as well as strategies to solve word problems. On the contrary, in Chinese curricula, rather than approaching word problems separately, problem variation permits them to be introduced in a holistic way without the use of multiple models (Sun 2015). Cai and Nie (2007, p. 467) report on the frequency of teaching with variation in the Chinese classroom through a survey of 102 teachers (see Table 3.2).

Table 3.2 The frequency of teaching with variation in the Chinese classroom

An example of OPMS of addition with two digits is discussed in Chap. 11. Section 3.4.5.2 considers a transposition of additive variation problems to Italy.

4 Educational Implications

The above observations clearly point out some of the features of the Chinese arithmetic tradition:

• The inductive approach, where general principles of representing numbers and calculation are consistent with and derived from the specific case of the ones place (e.g. number operations in the tens/hundreds place are similar to number operations in the ones place).

• The tradition of calculation using specific cultural artefacts that also leave traces in the language.

• The variation tradition in word problems.

These features have important educational implications. Ma (1999) finds that the content knowledge of American and Chinese teachers is different. In particular, the strength of mathematics content knowledge is related to profound understanding of fundamental mathematics. According to Ma (1999):

The US teachers tended to focus on the particular algorithm associated with an operation, for example, the algorithm for subtraction with regrouping, the algorithm for multi-digit multiplication, and the algorithm for division by fractions. The Chinese teachers, on the other hand, were more interested in the operations themselves and their relationships. In particular, they were interested in faster and easier ways to do a given computation, how the meaning of the four operations are connected, and how the meaning and the relationships of the operations are represented across subsets of numbers – whole numbers, fractions, and decimals. When they teach subtraction with decomposing a higher value unit, Chinese teacher start from addition with composing a higher value unit. (p. 112)

Similar reflections may be applied to other Western curricula. Chinese curricula do not have a chapter on place value similar to American or European curricula; rather, place value appears in all chapters, along with reading and writing number activities as an overarching principle. Place value involves implicit core knowledge of the number unit in ancient literature (Zou 2015) and in Chinese curricula (Sun 2015), which is different from the calculation vocabulary or extended number procedures in chapters on calculation in the mandatory practices of American curricula (Howe 2011, 2015).

4.1 Place Value and Whole Number Operations

Chinese verbal counting is transparent and completely regular for place value representation. However, in the West, place value may be perceived as an artificial construct for written purposes, as communities do not use it in ordinary conversation; for Western students, it can be a learned concept, but not a native one. The abbreviation of ‘-teen’ numbers in English (13 to 19) and in other European languages cannot be easily decoded in terms of the place value of tens and ones, which hinders understanding of the ten-structured regroup aspects of a multi-digit calculation, i.e. addition with moving up a place/subtraction with moving back a place. This is consistent with the findings of Ho and Fuson (1998), who argue that the structure of the English language makes it more difficult to understand that ‘-teen’ numbers are composed of a ten and some ones. It also makes it more difficult to learn the advanced make-a-ten method of single-digit addition and subtraction that is taught to first graders in China and other East Asian countries (Fuson and Kwon 1992; Geary et al. 1993; Murata 2004; Murata and Fuson 2001, 2006). Actually, the positional and decimal principles mentioned in Chap. 5 (WG1) have been naturally embedded in Chinese numeration and everyday language since the third century BCE. Some scholars (e.g. Butterworth 1999) have interpreted this as a reason why Chinese students are at ease with place value for large numbers from the beginning. From ancient times until now, spoken Chinese whole numbers have been the same as written numbers, implying that the written numeral directly reflects its pronunciation and thus has not diverged from the spoken language. Place value is an unlearned activity, but it is an inherited concept like a mother language, where native speakers are often unaware of the complexities of their language. This may explain why all current Chinese curricula do not include the topic of place value (for a discussion, see Sect. 15.3).

4.2 Cardinal Numbers and Measure Numbers

From a conceptual perspective of numbers, Bass (see Chap. 19) points out that numbers and operations have two aspects: conceptual (what numbers are) and nominal (how we name and denote numbers). At least two possible pathways exist for the development of whole numbers: counting and measurement. Conceptually, numbers arise from a sense of quantity of some experiential species of objects: count (of a set or collection), distance, area, volume, time, rate, etc. To develop a conceptual understanding, Bass supports an approach to developing concepts of numbers using general notions of quantity and their measurement, in which the measurement ‘unit’ is key to knowing how much (or many) of the unit is needed to constitute the given quantity while measuring one quantity by another. Cardinal and measure numbers in Western languages appear very different from each other, as measure numbers require the choice of a unit. This is not the case in the Chinese language, where both are considered in the same way.

4.3 Fraction Names

The order of writing (and reading) a fraction in Western languages is ‘first numerator, then denominator’, and the denominator is usually named with ordinal (not cardinal) numbers, such as ‘two thirds’ (2/3) of three parts. (In Chinese naming, taking two parts indicates a part-whole relationship rather than ‘two thirds’.) This method of fraction naming generates some difficulties for learning the part-whole relationship. The names for fractions in Western languages are not so clear. Bartolini Bussi et al. (2014) and Pimm and Sinclair (2015) analyse this difficulty and make proposals for overcoming it.

4.4 Arithmetic Operations

Research studies have identified several difficulties that Western students have with algorithms. For instance, Fuson and Li (2009) point out that many students in the USA make the error of subtracting the smaller number in a column from the larger number even if the smaller number is on the top:

346

–157

211

This error may be reinforced by language confusion, as the names ‘minuend’ and ‘subtrahend’ do not emphasise the passive relationship between them (see above). However, this seems to be only a part of the story. Written algorithms for addition and subtraction were introduced in Europe by Leonardo Fibonacci in the thirteenth century. They hint at the actions performed on some kind of abacus (the Chinese suàn pán, the Japanese soroban, the Roman abacus or similar; see Menninger 1969). More recently, the spike abacus was introduced for teaching (see the figures in Chap. 9). In English and other Western languages, the operations in Figs. 3.7 and 3.8 are described using terms like ‘carrying’⁶ and ‘borrowing’.⁷ The same was not true when algorithms were introduced in ancient textbooks. In Liber abaci, the term ‘borrow’ is not used. Rather, a kind of compensation or invariance is suggested: to increase by 10 the units in the minuend and to increase by 10 the units of the subtrahend. In this process, the 10 to be added to the subtrahend must be ‘kept in hands’ (reservare in manibus, in Latin). This strategy was maintained in many method textbooks for primary schoolteachers in Italy until at least 1930.

Ross and Pratt-Cotter (2000, 2008) reconstruct the story of the word ‘borrowing’ in North America. They find the first occurrence in a textbook by Osborne in 1827, but observe that ‘the term borrow may be a misnomer since it suggests that something needs to be returned’ (p. 49). Fuson and Li (2009) criticise this word (which was used for more than one century), and Fuson uses the words ‘grouping’ (for addition), ‘ungrouping’ (for subtraction) and ‘regrouping’ (if necessary in both cases) in the Math Expression project.⁸

The situation is quite different in China. In teaching subtraction with regrouping, the majority of the Chinese teachers interviewed in Ma (1999) describe the so-called ‘borrowing’ step in the algorithm as ‘a process of decomposing a unit of higher value instead of saying “you borrow 1 ten from the tens place”’ (p. 8). One third-grade teacher explained why she thought the expression ‘decomposing a unit of higher value’ was conceptually accurate:

The term ‘borrowing’ can’t explain why you can take 10 to the ones place. But ‘decomposing’ can. When you say decomposing it implies that the digits in higher places are actually composed of those at lower places. They are exchangeable. The term ‘borrowing’ does not mean the composing-decomposing process at all. (p. 9)

The English terms ‘carrying’ and ‘borrowing’ are not related to each other. The French term ‘retenue’ is the same for both operations. ‘Retenue’ literally means ‘keep in mind’ (or ‘keep in hand’ in French) and hence hints at memory and not a concrete action. The origin may be traced back to the term used in medieval arithmetic ‘reservare in manibus’ (‘to keep in hands’). The use of the same term for different actions creates many difficulties for pupils (Soury-Lavergne, personal communication). This simple example shows that different cultures/languages may foster or hinder the understanding of meaning.

4.5 Mathematical Relational Thinking: Equality or Sameness

4.5.1 Some Reported Difficulties in the Understanding of Equality

Several studies have been carried out to examine the use of the equality symbol ‘=’ in mathematics education. Kieran (1981) studies the interpretation of the equality symbol in the early grades. In preschool, two intuitive meanings appear: the first (conceptual or relational meaning) concerns the relation between two sets with the same cardinality (hence an equivalence relation, according to the historical genesis), while the second concerns the set resulting from the union of two sets. The second is related to the interpretation of ‘+’ and ‘=’ in terms of actions to be performed (procedural or operational meaning). This latter view is reinforced through the use of pocket calculators and the transcription of the additions and results as they appear on the display. For instance, to add the following numbers in a notebook:

• 15 + 31 + 18

it is common to see the following:

• 15 + 31 = 46 + 18 = 64

This discussion was carried out in a third-grade classroom in Italy. Only some excerpts are reported. The teacher (Rosa Santarelli) posed the following problem:

• How many days for holidays last summer?

• Two pupils have solved the problem as follows:

• 30 – 10 = 20 + 31 = 51 + 31 = 82 + 15 = 97

• Do you think that this calculation is correct?

• STE: Yes, it is correct. They have thought about the months of holidays. Hence, this month has so many days, and they have put that month. In June we were at school for 10 days, hence 30 – 10. … [T]hen they have written the equal sign and then 20 and from that 20 they have started to count all the holidays. They have written +31, then 51, + 31 equals 82, + 15 (the days in September) equals 97. Then they have understood the result, they have written it. What they have done is right.

• Many pupils agree and reword the same process.

• TEACHER: But what does the sign ‘=‘mean in mathematics?

• GIO: Equal means that if you have 20 + 30 you put the equal sign and you get the result. The equal sign tells the result of an operation …

• CAR: If you wish to use this sign in an operation, you must put it at the end. If you make 5 + 5 = then you write 10.

• Other pupils reword the same statements.

• TEACHER: What does it mean ‘to be equal to’ in mathematics?

• ILA: It means that you get the result.

• SAM: Equal, in mathematics, is usually in the operations. It is used to get the result.

• …

• TEACHER: Is it correct to write ‘8 = 8’?

• GIO: No, it isn’t. You must write ‘+0’ or else one doesn’t understand. You need to put something.

• TEACHER: Hence, I make a mistake if I write ‘8 = 8’.

• GIO: Yes, you do. You should write ‘8 + 0 = 8’ or ‘8 – 0 = 8’. (Zan 2007, p. 79 ff., our translation)

This short excerpt confirms that the procedural meaning of the equality symbol is often dominant in primary schools, at the expense of the relational meaning. Ni (2015) argues that student errors such as considering the equal sign as an order to ‘do something’ for an answer probably contribute to the difficulty they experience later when learning algebra; students treat an algebraic equation as indicating not a mathematical relation, but an order to ‘do something’ to obtain an answer. This may have very bad consequences in secondary school, when algebraic expressions are in the foreground. It is not possible to interpret the following equation according to the conceptual meaning:

• x + 3 = 4

Teachers tacitly reinforce the procedural meaning when they do not take care of this issue.

4.5.2 Variation Problems in China and Italy

Bartolini Bussi et al. (2013) report an example of variation problems from the OPMC category, where all of the problems are collected in one 3 × 3 table (see also Sullivan et al. 2015, p. 88). In China, a collection of variation problems was given to second graders at the end of the school year as a kind of summary, with several different examples of problems presented during the school year. It was expected that the task would be solved in just one lesson due to the background knowledge of the students. Bartolini Bussi et al. used this task in some Italian schools, but a process of cultural transposition was needed (see Chap. 13 of this volume) (Table 3.3).

Table 3.3 An example of variation problems from the OPMC category Beijing Education Science Research Institute and Beijing Instruction Research Center for Basic Education (1996), vol. 4, p. 88

The most evident effect of this transposition was the time needed. It was not possible to solve the task in just one lesson. The task was the source of a longer process, where the students had to become familiar with this surprising way of considering several problems together and using schemes to find/represent the solution. During the process, the students started to focus on the relationships between operations rather than on the execution of operations and hence started reasoning algebraically. Some further experiments (Mellone and Ramploud 2015) are in progress now.

5 Concluding Remarks

The attention to differences in whole number approaches is increasing. It is worthwhile to mention at least the book by Owens (2015), with a chapter on visuospatial reasoning with numbers, and the book by Owens et al. (2017) on the history of number in Papua New Guinea and Oceania that details number systems other than base 10 systems.

The examples discussed in this chapter show that language plays a common, key role in conveying concepts in the teaching and learning of whole number arithmetic. A cross-cultural examination of languages should thus allow us to understand linguistic supports and limitations that may foster or hinder students’ learning and teachers’ teaching of mathematics.

The above discussion highlights that in many cases the Chinese way to develop whole number arithmetic seems to offer advantages for the construction of mathematical meanings: the attention to mathematical consistency and coherence seems larger than in the Western curricula. Yet the Chinese case shows that the difference is strongly related to linguistic and cultural features not shared by other cultural groups. This observation suggests that caution must be taken when trying to apply some of the Chinese methods in other countries, unless a careful process of cultural transposition is established.