Our findings indicate that processing the syntax of language elicits the known substrate of linguistic competence, whereas algebraic operations recruit bilateral parietal brain regions previously implicated in the representation of magnitude. This double dissociation argues against the view that language provides the structure of thought across all cognitive domains. (p. 914)

Finally, some studies have pointed out that the sense of numbers is not only based on discrete approaches that rely on the one-one correspondence between external symbols and numerical representations, but also on approximate number of systems (e.g. the estimation of the numbers of two sets when subitising is not possible) that are based on the ratio between their cardinality and not on their difference (see Gallistel and Gelman (2000)). According to these studies, this continuous, analogic system emerged during our evolution and became encoded in our brains prior to the discrete approach.

These findings have introduced a fresh perspective on the issues of tradition and language and their roles as resources for arithmetic activities.

In particular, some major questions for the panel are:

#### **<IP_ADDRESS> Representing Numbers in Artefacts**

In the research on the semiotic representation of numbers, a specific stream of analysis concerns the calculation tools (typically, but not only, abaci) that incorporate both the specific representations of numbers and the corresponding practices for completing arithmetical operations (for a survey see Ifrah 2001). These tools are deeply intertwined with language and can be incorporated in the didactical designs used in primary school. Many teachers use the tools alongside modern technology to introduce concrete artefacts and their simulations in a virtual technological classroom environment. For example, Sinclair and Metzuyanim (2014) integrated such embodied and traditional representations using tablets based on the hypothesis that

the touch-screen devices enable an intuitive, embodied interface for conducting arithmetic. The devices are also suitable for young learners because they allow them to use their fingers and gestures to explore mathematics ideas and express mathematical understandings. Furthermore, Soury-Lavergne and Maschietto (2015) use an old Pascal machine to approach arithmetic in concrete and virtual ways in primary school. These and further examples are discussed in Chap. 9 of this volume.

These types of research pose the following interesting questions for the panel:

The panel consists of four scholars2 who are representative of the different cultural traditions of teaching numbers, namely, Nadia Azrou (mathematics teacher at the University of Yahia Fares in Medea, Algeria, and a PhD student in math education), Maria G. Bartolini Bussi (full professor in mathematics education at the University of Modena and Reggio Emilia, Italy), Sarah Inés González de Lora Sued (full professor in mathematics education at Pontificia Universidad Católica Madre y Maestra, República Dominicana) and Xu Hua Sun (assistant professor in education at the University of Macau, China). Man Keung Siu (honorary fellow, The University of Hong Kong) acted as the discussant.

### **15.2 Spoken and Written Arithmetic in Different Languages: The Case of Algeria**

Nadia Azrou

### *15.2.1 Post-colonial Countries: The Case of Algeria*

At the elementary level, numbers are learnt along with several technical concepts (e.g. the place value of digits, number line and decimal position system) that support learning or weaken it if not effectively acquired. Learning numbers and other basic arithmetic notions is also affected by culture and particularly by language. This is more visible in multicultural classes in schools that host migrants of different nationalities, but also in post-colonial countries such as Algeria, where history, cultural evolution and external and internal power influences have a direct influence on the school system.

2Sarah Inés González de Lora Sued was unable to take part in the panel for health reasons. However, she provided a text, which appears in this chapter.

How do we deal with such phenomena? The globalisation of school mathematics, which aims to unify the curricula in countries that have different cultural and linguistic backgrounds, and the assumption that learners have to submit to a country's dominant language, have been shown to have their limitations. From a different perspective, Usiskin (1992) claims that differences provide the best situation for curriculum development and implementation. Gorgorió and Planas (2001) point out that if language is the main carrier of a culture, then the 'language of the mathematics class' conveys the culture of the classroom as a social group doing mathematics, along with its norms and legitimate roles.

For teachers, this clearly represents a challenge, particularly for those who teach in a traditional, transmissible way. Most teachers presume that the 'normal' learning context is a monolingual classroom, that learners know the 'norms' of the school (which are usually shaped by the dominant culture) and that children already master the language of instruction. Given this situation, teachers should acknowledge the relevance of the issues related to cultural and linguistic diversity, understand how they influence the learning process and manage them to scaffold the children's learning in an effective way. In particular, teachers should be able to identify the possible difficulties that children experience when learning numbers in a language different from their mother tongue and to create opportunities to turn these difficulties into advantages. Moreover, I share the view of Gorgorió and Planas (2001) who believe that there is no classroom in which linguistic capital is equitably distributed. As a consequence, what may appear as an extremely 'different' setting not relevant for mainstream practice may be relevant for communication issues in all classrooms.

Further research is needed to clarify how mathematical language can be taught and to investigate the relationships among the 'language of the mathematics class', mathematical language and the process of construction of mathematical knowledge (Gorgorió and Planas, 2001). However, some elements and insights can already be provided to address the question of how teachers can concretely develop their task designs for basic notions of arithmetic by taking the linguistic characteristics and cultural roots of numbers into account. Some answers may be suggested by the analysis of the situation.

### *15.2.2 Number Naming, Place Value and Decimal Position System*

It is not unusual for a language to have irregularities in regard to number naming, and these irregularities are not the same in different languages. In Europe, for instance, every language possesses its own number naming system with its own set of irregularities. For example, similar to French, Spanish and Italian, the numbers between 13 and 19 in English have names that position the lowest place value digit first, in contradiction to the written form, which goes from left to right according to

the decreasing place value of digits and the other spoken numbers. Moreover, although in English the word 'ten' is nearly present in thirteen (13) to nineteen (19) as in Italian, this is not the case in French where 'ten' ('dix') does not appear in the numbers 12 to 16 (e.g. quatorze 14, seize 16), although 17 is pronounced 'dixsept'). In Arabic, the numbers from 11 to 99 are pronounced from the lowest place value digit to the highest and are in complete correspondence with the written form, which is presented from right to left. In Danish, seventy is named halvfjerds, which is a short for halvfjerd-sinds-tyve, meaning "fourth half times twenty", or "three score plus half of the fourth score" [3½ \* 20]. Moreover, as in German, there is no correspondence between the written and spoken forms for the numbers from 13 to 99, which are pronounced from the lowest place value digit to the highest. In French, the numbers between 81 and 99 are expressed as 'four-twenty' plus a number between 1 and 19. Traces of the contamination between different languages and the historical roots in old number systems can be detected in these irregularities and differences. However, irregularities, in the same oral language or when shifting from one language to another, may be a source of difficulty for children. Research suggests that in some Asian countries, Asian speaking children perform better with place value, counting and decimal system tasks due to their regular number naming systems (Miura et al. 1994). Nonetheless, irregularities and differences also provide students (under the guidance of the teacher) with opportunities to notice important characteristics of the decimal position system of writing of numbers, such as the position value of digits and reflect on them. For instance, with reference to the above examples, the teacher may exploit the differences between the irregular forms of spoken numbers within the same language (in the case of most European languages) and between how numbers are spoken in one language and another.

The case of Algeria is interesting. About 10 years ago, a political decision was made to write formulas and symbols from left to right with the Latin alphabet (in the past they were written from right to left with the Arabic alphabet) when teaching mathematics at all levels, while maintaining comments and names in classical Arabic (from right to left). This change has subsequently influenced how children conceive, understand and learn arithmetic. Thus, teachers should use this as an opportunity to allow children to realise that mathematics is not separate from culture and language and to understand that its evolution is also affected by the historical and political dynamics.

### *15.2.3 Mathematics Register*

As defined by Halliday & Hasan (1985), the mathematical register records how everyday language is used in new ways to serve the meanings of mathematical words, even though words such as 'double, less, more' may have different meanings in ordinary language from those in mathematics. These differences may have resulted in some children failing to solve problems caused by misunderstanding the text. For example, in Arabic, the verb used to express the multiplication operation is to 'beat' (thus scaring the children), that is, 'we multiply 2 by 4' is 'we beat 2 to 4'. In English, expressions such as 'twice as much as' or 'twice as less as' may sound ambiguous. Accordingly, children need to learn the language patterns associated with these words, how to construct concepts in mathematics and the implicit logical relationships, because when children construct mathematics concepts in everyday language, the relationships they come up with are often technically incorrect. Learning mathematics and the language of mathematics, that is, the mathematics register, is a challenge for all children. Teachers can facilitate the learning process by using the mathematical register effectively and working to build language in deliberate ways, moving from everyday to technical linguistic expressions of mathematical knowledge and using spoken language, reading and writing. If learners have difficulties in verbalising a mathematical process, the teacher can promote mathematical thinking by using their mother language to tackle mathematical problems (Adler 1997). Thus, it is highly recommended that teachers have some knowledge about their learners' languages and consider the norms and contexts in which words are used. Teachers should aim to develop the mathematics register in the languages in which the children are instructed. However, until this is done, teachers need to face and overcome the difficulties in translating mathematics concepts into students' home languages (Schleppegrell 2007).

### **15.3 From the Number Line to the Productive Dialogue Between Different Cultural Traditions: Italy and China**

Maria G. Bartolini Bussi

### *15.3.1 The Number Line*

The number line is a very popular teaching aid (Bartolini Bussi 2015; Electronic Supplementary Material: Bartolini Bussi 2017). Italian teachers can find specific references to the number line in the standards (MIUR 2012) for the mathematics curriculum. To begin with, the number line comprises whole numbers and it is then expanded to contain rational numbers. The following goals are stated at the end of the third grade (the first time the goals are explicitly listed):

The goals are summarised and reinforced at the end of primary school (fifth grade):

To represent the known numbers on the line and to use graduated scales in contexts that are meaningful for science and technique. (p. 62)

The last goal hints at a possible use of the number line as a modelling tool. A similar use is stated in the history curriculum, where, at the end of primary school (fifth grade), the following goal is stated:

To use the timeline to organize information, pieces of knowledge, periods and to detect sequences of events, concurrent events, duration of events. (p. 53)

The representation of numbers on a line is emphasised (at all levels) in the framework for the national assessment in mathematics (INVALSI 2012).

The general approach to the mathematics curriculum in Italy is stated in the programmes (MIUR 1985) for primary school:

The development of the concept of whole numbers must be roused exploiting the previous experience of students, like counting and recognizing numerical symbols in play and in family and social life. It is advisable to consider that the idea of whole number is complex and requires a multifaceted approach (order, cardinality, measuring, …); it is acquired at higher and higher levels of internalisation and abstraction during primary school and beyond.

This idea is widely shared and has been confirmed in other curriculum documents (e.g. MIUR-UMI 2001), which have strongly influenced the elaboration of the more recent standards (e.g. MIUR 2012).

In the number line, order and measuring are in the foreground. However, the other properties of whole numbers (e.g. cardinality, place value representation) are not supported by the number line and must be taught independently. This choice is consistent with a multifaceted approach in which different routes are explored in parallel to develop a complex concept of whole numbers.

In contrast, as argued by Sun (2015), the Chinese tradition of whole number arithmetic appears to place less emphasis on the number line and to foreground other properties of whole numbers (e.g. the part-part-whole and associative law) in constructing a consistent teaching path in which these properties are pursued in a systematic way, step by step and without deflection.

### *15.3.2 The Dialogue Between Cultures: Towards Cultural Transposition*

The panel discussion on the number line is a paradigmatic example of the process that occurs when scholars from different cultural backgrounds engage in a true dialogue. The point is not to determine the best, 'universal' choice but to understand how and why the mathematics curriculum was developed in one's own context. Jullien (1996) stated that, 'every thought, when coming towards the other, questions itself about its own unthought' (p. iii). In this sense, noticing the different approaches used in Italy and China serves as a prompt to start a cultural analysis of the content (Boero and Guala 2008). Cultural artefacts, when carefully analysed, reveal a lot of things about the culture that has produced them. To implement activities by using

cultural artefacts in a different culture, it is necessary to enter a process of *cultural transposition*, where:

the different cultural backgrounds generate possibilities of meaning and of mathematics education perspectives, that, in turn, organize the contexts and school mathematics practices in different ways. (Mellone and Ramploud 2015, p. 578)

### *15.3.3 Examples of Cultural Transposition*

The first example concerns the use of counting rods for the development of place value. Some years ago, we analysed some Chinese textbooks for the first grade and noticed the use of counting rods and bundles of rods to link numbers and quantities. The book for the first semester of the first grade comprised 120 pages (from September 1 to the end of January). After having presented the numbers from 1 to 10 together with addition, subtraction and word problems, the numbers between 10 and 20 were introduced. On p. 85, the following activity was presented (Fig. 15.1). This was the first activity in which numbers with two digits (from 11) were introduced.

The teacher says: 'First count ten little rods and bind them to get one bundle. How can you go on counting?' The boy answers: 'To combine together one ten and one, it is ten-one'. The right classifier is always used: 个(*gè*) for rods and 捆(*kǔn*) for bundles; 个(*gè*) again for one and for ten, which is the origin of place value (this volume, Chap. 3).

The process is supposed to be very fast, as if the student is able to produce the right name without any help from the teacher. This natural process is possible in China, because the way of recognising numbers in Chinese is part of everyday experience and completely transparent in relation to the place value (this volume, Chap. 3). No specific teaching processes are needed in school. In contrast, a specific teaching process is needed to design school practices in other languages/cultures. For instance, the names of numbers in Italian are irregular and not transparent, and hence it is not possible for a student to name a number with one bundle and one rod ('undici' in Italian, 'eleven' in English). It is necessary to plan two parallel processes before linking the rod and bundle representations to the names and the symbols in which the bundles are bound to construct the concept of ten as a higher order unit, and the Italian names for the numbers are learnt. Only later is it possible to link these processes to one another. Thus, more instruction time is needed than in the Chinese classroom.

An Italian teacher, Bruna Villa, produced another example of cultural transposition in the same content (see this chapter Sect. <IP_ADDRESS>; Electronic Supplementary Material: Arzarello 2017). She also introduced two parallel processes. At the beginning of the first grade, the teacher told a fairy tale of a small Chinese dragon who was visiting the classroom to teach the children how to say numbers. Hence, the students learnt to say the numbers in two ways:

**Fig. 15.1** The introduction of tens in the Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005)

### *The completely regular Chinese-like names: 11 is ten-one; 21 is two ten-one. The Italian irregular names: 11 is 'undici', that is, 'eleven'; 21 is 'ventuno', that is, 'twentyone'.*

The teacher evoked the images of two imaginary characters, the 'small Chinese dragon' (with narratives, drawings and even hats to wear when acting as a dragon) and the 'mom', representing the (Italian) adult voice from the students' everyday experience, for some months to avoid ambiguity and help the students to make sense of the experience. In this way, the teacher succeeded in teaching place value in an exciting but robust way while introducing some ideas about different cultural contexts.

Another example concerns the planned transposition to Italian classrooms of the word problem of cakes observed in the first grade of the Hou Kong School (this volume, Chap. 11). In this case, the task draws on Chinese practice to consider a number as a system of part-part-wholes in different ways (a variation of the 'one problem, multiple solutions' (OPMS) approach, Sun (2011)). Italian students are not accustomed to these kinds of tasks, hence the cultural transposition to Italian classes requires additional tasks to be introduced in parallel to the existing ones.

There are other examples of variation problems (e.g. Bartolini Bussi et al. 2013). After being examined in pilot studies, some curriculum material has been developed in Italy (see the Italian project *PerContare*, this volume, Chap. 7) in which the cultural analysis has been made explicit for teachers. This is a possible answer to the last question posed to the panel, namely, where do the 'traditional'activities of different cultural traditions come from? We were able to exploit the cultural transposition because the Italian standards leave some freedom for teachers to conduct pilot experiments. My questions are now: Is it possible to conduct pilot experiments in China that exploit new activities? And how can cultural differences be incorporated into the established approach, which is mainly based on textbooks with fixed content for each lesson?

### **15.4 The Role of Using the Cultural Roots of Numbers and Artefacts for Children Learning Whole Number Arithmetic in Latin America**

Sarah Inés González de Lora Sued

### *15.4.1 The Design of Learning Tasks for Arithmetic Based on the Linguistic and Cultural Roots of Numbers*

Most of the time, whole number arithmetic learning is not related to the reality in which children live, because the design of the learning tasks is divorced from the local cultural environment. Referring to the weak relationship between culture and mathematics in the classrooms, D'Ambrosio (2001) stated that:

when teachers do acknowledge a connection between mathematics and culture often they engage their students in multicultural activities merely as a curiosity. Such activities usually refer to a culture's past and to cultures that are very remote from that of the children in the class. (p. 308)

And he also pointed out:

As our students experience multicultural mathematical activities that reflect the knowledge and behaviors of people from diverse cultural environments, they not only may learn to value the mathematics but, just as important, may develop a greater respect for those who are different from themselves. (p. 308)

However, when mathematics, and whole number arithmetic in particular, is presented to students using examples that are a lively part of their culture (e.g. by introducing mathematical concepts and procedures through problematic situations drawn from their reality), the concepts become meaningful to them.

D'Ambrosio stresses that:

We can help students realize their full mathematical potential by acknowledging the importance of culture to the identity of the child and how culture affects how children think and learn. We must teach children to value diversity in the mathematics classroom and to understand both the influence that culture has on mathematics and how this influence results in different ways in which mathematics is used and communicated. We gain such an understanding through the study of Ethnomathematics. (p. 308)

Moreover, if whole number arithmetic is taught in isolation from the culture of segregated populations, such as the many indigenous populations of Latin America, the situation becomes more complex. In the 2013 report 'Intercultural citizenship: Contributions from the political participation of indigenous peoples in Latin America', the United Nations Development Programme (UNDP) indicated that there are approximately 50 million indigenous people in Latin America, about 10% of the total population. However, in countries such as Peru and Guatemala, indigenous people account for almost half of population, while in Bolivia they comprise over 60% of the total population. These indigenous peoples speak their own languages and many are marginalised because they do not speak Spanish. These cultures also have their own ways of conceptualising whole numbers. Many studies have been conducted on the mathematics of indigenous Latin American cultures. To teach the children of these populations, teachers need to be able to understand these mathematical approaches to whole number arithmetic. In the case of Guatemala, by law, all children must be taught using the Mayan number system *and* the base 10 system.

### *15.4.2 Numbers and Words*

Table 15.1 lists the names of the numbers from 1 to 20 in Mayan, Quechua Collao and Spanish. The fourth column lists the Latin words from where the Spanish names proceed, according to the Real Academy of the Spanish Language. It is interesting to note the patterns in the table. For example, from 1 to 10, the names within each language do not have any relationships among them. For Mayan numbers, this is also the case for numbers 11 and 12. However, from 13 to 19, the names are composite words of the names 3 to 9 and the name of 10 (lahun). In the case of Quechua Collao, from 11 to 19, the names are composite words with the name of 10, 'Chunka', the names of the numbers from 1 to 9 and the word 'niyuk'. In this case, the word for 10 comes first. In the case of Spanish names, from 11 to 15, the relationships between the names and the numbers are not clear. However, when you see the Latin roots of the words, as described in the dictionary of the Real Academy of Spanish Language, it seems the meaning is 'one and ten' for 11, 'two and ten' for 12, and so on. The dictionary of the Real Academy of Spanish Language does not include the roots of numbers 16 to 19 but their meaning is clearer and easier for children to understand (ten and six, ten and seven, and so on).

Moreover, in Quechua Callao, 30 is kimsa chunka, 40 tawa chunka, 50 pichqa chunka and so on, which can be interpreted as having the multiplicative meanings of three times ten, four times ten and so on. In Spanish, 30 is treinta and its Latin root is triginta, 40 is cuarenta and its Latin root is quadraginta, 50 is cincuenta and its Latin root is quinquaginta and so on with the same meaning. It seems that it would be easier for Quechua Collao children to learn the names of numbers from the patterns of the names.

**Table 15.1** Some pre-Columbian names of numbers

### *15.4.3 The Role of Semiotic Representation in Learning Whole Number Arithmetic*

Radford (2014) discussed the influence of representation and artefacts on knowing and learning, indicating that:

the problem of the epistemic and cognitive role of tools and signs continues to haunt us perhaps now more than ever by virtue of the unprecedented technological dimensions of contemporary life. And so here we are still trying to make sense of how we think and learn with and through signs and artefacts. We assume that interaction with reality plays a crucial role in learning. Meanings constructed from one's experience lead to a deeper understanding of theoretical constructions. From this perspective, human perception and action and, more generally, interaction with artefacts, are of crucial importance for learning and doing mathematics. (p. 406)

Arzarello et al. (2005) pointed out:

The embodied point of view of our 'acting is learning' stresses the importance of relating action and language to mental activity. Although such a claim is widely acknowledged from a theoretical point of view, our provocation consists in fostering its transposition in school practice. (p. 56)

In the Dominican Republic, for example, the use of concrete manipulatives, such as base-ten blocks and Cuisenaire rods, for learning whole numbers was only introduced in public schools around two decades ago (85% of schools in the country). This approach has been shown to motivate students and teachers to represent number concepts and operations and has produced higher levels of achievement when accompanied with teachers' professional development activities (González et al. 2015). In the case of Peruvian indigenous children, the use of the Yupana, an artefact used by the quipucamayos (accounting people in the Inca Empire), is more appropriate because it is grounded in their culture. There are several hypotheses with respect to how the Yupana was used by the Incas. Based on the hypothesis of Willliam Burns, in the 1980s Martha Villavicencio Ubillús (1990) developed a methodological sequence for the comprehensive learning of the decimal numeration system and the algorithms of the basic operations using the Yupana, which was first applied in the bilingual education experimental project of the Puno and is now used widely in bilingual schools in Perú. There are also technological applications for children to simulate arithmetic operations using the Yupana through computers, cellphones and tablets (Rojas-Gamarra and Stepanova 2015).

### **15.5 Chinese Arithmetic Tradition and Its Influence on the Current Curriculum3**

Xu Hua Sun

### *15.5.1 Chinese Arithmetic Tradition*

The ancient Chinese called mathematics arithmetic (算術 an art of computation), which possibly reflects China's long history and tradition of arithmetic. It is interesting to note that the Chinese traditional approach to whole numbers was mainly cardinal rather than ordinal. Ordinal numbers are formed by adding 第 dì ('sequence') before the number. The Chinese use cardinal numbers in certain situations in which English and other Western languages use ordinals. For example, whereas a person lives on the fourth floor in a building in English, in Chinese, it is said the person lives on four floor, not the fourth floor. In Chinese, the 20th of July is expressed as 20 July.

Figure 15.2 shows the world's earliest decimal multiplication table. Made from bamboo slips, the table dates from 305 BCE, during the Warring States period in China. Place value is the most overarching principle used in Chinese numerals and calculation tools (counting rods and the Chinese abacus), which could provide advantages as a regular system (foundation) for whole arithmetic/algebra development.

3This study was supported by Research Committee, University of Macau, Macao, China (MYRG2015–00203-FED). The opinions expressed in the article are those of the author.

**Fig. 15.2** The world's earliest decimal multiplication table

The Chinese term for mathematics is *shuxue* (數學) or *suanxue* (算學), which mean research on number or computation. Geometry was not included in the Chinese mathematics school curriculum until Euclid's *Elements* was introduced in the seventeenth century by Matteo Ricci (1552–1610) and Guangqi Xu (徐光啟)). The *Elements* mirrored the Chinese calculation tradition and was derived from the local world view. The ancient Chinese believed that the only way of knowing the world is through calculation, which is reflected in the *I Ching* (易經) in general. This is also expressed in the following quotation from the preface of *Sunzi Suanjing* (孫子算經):

Calculation is the whole of heaven and earth, the origins of all life, the beginning and end of all laws, father and mother of yin-yang, the beginning of all stars, the inner and outer of three lights, the standards of five elements, the beginning of four seasons, the origins of ten thousands matters, and the general principles of six arts. (Lam and Ang 2004, p. 29)

Shushu Jiyi (數術記遺) (the first Chinese book – 220 CE – about calculation and its tools) systematically recorded 14 calculation approaches and 13 calculation tools. Among them, only bead calculation and base two (related to the computer) calculation have been handed down to today. All mathematics books are based on the decimal place number system, in which place value is the most overarching principle used in Chinese numerals and calculation tools (this volume, Chap. 3).

### *15.5.2 The Chinese Arithmetic Curriculum*

Chinese mathematics education is famous for its stable basic education (Zhang 2006). The number line is rarely used in the Chinese curriculum, which may reflect some cultural trends in understanding number in China (Sun 2015).

The cardinal approach to whole numbers is embedded in the *suàn pán* (算盘, this volume, Chap. 9). The current Chinese curriculum standards mention *suàn pán* as a traditional tool for representing place value.4 Bead calculation is not required, although some books for students are available. However, the spike abacus (which is similar to the *suàn pán*) is widely used in the current curriculum as a heritage item (Fig. 15.3).

**Fig. 15.3** Spike abacus with rods in a Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005)

4 http://www.pep.com.cn/xxsx/jszx/xskcbj/201202/t20120224\_1103348.htm

**Fig. 15.4** Addition and subtraction with concept connection in a Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005, vol. 1)

It is worth noting that the Chinese curriculum aims to develop a system of numerative reasoning at an early stage. In addition to being closely related, implicit connections are made between the three concepts of addition, subtraction and number and the inverse relation between addition and subtraction (a − b = c equals a = b + c). The authors of Chinese textbooks link the concept of subtraction to that of addition even in the first calculation lesson (Sun 2011). Figure 15.4 shows the paradigmatic example of 1 + 2 = 3, 3 – 1 = 2 (Mathematics Textbook Developer Group for Elementary School 2005, pp. 20–21). The aim of the problem is to help learners understand the relationship between addition and subtraction and the meaning of 'equal'. The prototypical example from the Chinese textbook is an example of a *variation problem*. Further details of variation problems can be found in this volume (Chap. 11).

Chinese curriculum developers have connected the three core concepts of addition, subtraction and number in all of the chapters on addition and subtraction using the following explicit principles:

This approach not only promotes rote counting and memorising but also reasoning. In contrast, in many Western curricula, the ideas of number, addition and subtraction are presented in separate chapters, isolated from one another.

Drawing on 1 + 2 = 3; 3 – 1 = 2 (Fig. 15.4), Chinese curriculum designers also aim to elicit a similar property for tens and thousands (Fig. 15.5).

*1ten + 2tens = 3tens; 3tens – 1ten = 2tens. 1thousand + 2thousands = 3thousands; 3thousands – 1thousand = 2thousands.*

This kind of inductive generalisation is widely used in the Chinese curriculum (Sun 2016a, b) and represents a further example of the variation problem from units to tens to thousands (Mathematics Textbook Developer Group for Elementary School 2005).

**Fig. 15.5** From units to tens (the *left*) and thousands (the *right*) in the Chinese textbook (Mathematics Textbook Developer Group for Elementary School, 2005)

The Chinese curriculum for teaching number avoids counting as much as possible and is, therefore, different from the Western style of number teaching. The curriculum highlights the approach of composition/decomposition, which may have been inherited from ancient bead calculation. The approach is used seven times with 1–10 (decomposition of 4, 5, 6, 7, 8, 9, 10 in 7 lessons) in the Chinese curriculum (Sun 2013), which implicitly forms a core practice for learning number (Sun 2015). This approach aims to develop an implicit understanding of the associative and commutative laws, number properties and foundation of addition/subtraction operational flexibility. The partitions of ten are intensely studied, and numbers such as 12 are thought of from the beginning as one ten and two ones. The number names in Chinese are consistent with this method (for details see Chap. 3). The emphasis on partitioning and regrouping lends itself naturally to written algorithms.

The speed and accuracy of oral calculation are important requirements in the assessment of the Chinese curriculum standards. Great importance is traditionally attached to calculation because whole number is considered to be 'the first foundation of the whole subject' (Elementary Mathematics Department 2005, p. 1). This statement is consistent with the current Chinese curriculum standards, which state that:

Oral calculation is the basis for learning mathematics. It should have very important influence on students' basic written calculation capacity. Its training can help students develop capability of observation, capability of comprehensive thinking, capability of creativity, and capability of reaction.

This heavy emphasis on calculation skills is reflected in the higher requirements for speed and accuracy in oral (mental) and paper-and-pencil calculations. For example, addition and subtraction between 1 and 20 should be completed at 8–10 operations per minute and from 20–100 at 3–4 per minute, whereas multiplication within 1–10 should be conducted as a rate of 3–4 per minute and two-digit multiplication at 1–2 per minute, all with 90% accuracy.

Chinese mathematics evolved in relation to the problems of land measurement, commercial trade, architecture, government records and taxes. The systematic word problems in the current curriculum include simple and complex problems called variation problems. These problems aim to deepen the concept connections and operations and increase flexibility (举一反三) (e.g. Bartolini Bussi et al. 2013; Sun 2011, 2016a, b). Further discussion of the variation problem can be found in Chap. 11.

### *15.5.3 Concluding Remarks*

This brief presentation has highlighted the significant differences between the Chinese and Western curricula (in the USA and Europe) inherited from tradition. Further details on this topic can be found in this volume (Chap. 3). The aim of this contribution is to start a dialogue between cultures. Bartolini Bussi, Sun and Ramploud (2013) have argued that the goal of a true dialogue between scholars with different cultural backgrounds is not to determine the best 'universal' choice but to understand the development of one's own mathematics curriculum, which is helpful for questioning the unthought characteristics of the educational context.

### **15.6 Discussion**

Man Keung Siu

### *15.6.1 Introduction*

Most of the speakers on this panel have talked about the linguistic and cultural characteristics of the topic. As a mathematician, I will try to supplement the discussion by focusing more on the mathematical context. First, let me sketch a general framework of the teaching and learning of mathematics.

We begin with a world 'without mathematics'. This statement is to be taken with a grain of salt because mathematics is everywhere in our world and comes up frequently and unavoidably in our daily lives, perhaps even without our noticing it.  Later, we come to see the world after learning some elementary mathematics and forming the ideas of mathematical objects, notions, theories and techniques. Then we come to understand more 'formal' mathematical concepts as we refine our ideas of these mathematical objects, notions, theories and techniques. This is what Bill Barton labels NUC (near-universal conventional) mathematics (see also Sect. <IP_ADDRESS>). Finally, we try to apply the mathematics we have acquired to solve different kinds of problems. In some senses, NUC is culturally independent. However, when teaching mathematics, cultural transposition (as discussed in Sect. 15.3.2) is also a helpful concept.

Relatively few basic concepts are learnt in primary and secondary school, and these basic concepts come up time and again throughout a child's primary, secondary and even undergraduate education. Thus, I would like to extend the scope of whole number arithmetic and talk about other number systems. Here, my intervention is connected to the discussion of the Davydov approach in this volume (Chap. 19).

### *15.6.2 Numbers*

Let us begin with the notion of the number tree that often appears in school textbooks. Recalling our own learning experience, we can see that the concept of the number system is not presented in such a neatly packaged way in one stroke but is acquired in a vague and spiral fashion. Personally, I had been using the real numbers without much effort for many years in my secondary school days, but I did not know (and was not even aware that I did not know!) what real numbers were until I studied the subject as an undergraduate, and only gained a sufficient understanding when I came to teach mathematics as a young PhD student. The same process occurred in history as mankind went through the process of acquiring knowledge. Thus, it is likely that the number tree reveals itself in stages from kindergarten to primary school to secondary school to university as depicted below (Fig. 15.6).

The French mathematician Joseph-Louis Lagrange (1736–1813) referred to arithmetic and geometry as 'the wings of mathematics'. Another French mathematician, Henri Poincaré (1854–1912), commented on the construction of the real number system:

If arithmetic had remained free from all intermixture with geometry, it would never have known anything but the whole number. It was in order to adapt itself to the requirements of geometry that it discovered something else. (Poincaré 2003, p. 135)

**Fig. 15.6** Different number systems

### *15.6.3 The Chinese Translation of Euclid's Elements*

Here, I will try to address the issues raised by the other panellists about arithmetic and geometry. The panellists mentioned the discrete/continuous and algebraic/geometric characteristics of the number system and asked why the number line does not feature as prominently in the Chinese classroom as in the Western classroom. What M. Bartolini Bussi (Sect. 15.3) has in mind is the discrete number line, but allow me to extend the discussion to the real number line.

When the Ming official scholar Xu Guang-qi (徐光啟1562–1633) collaborated with the Italian Jesuit Matteo Ricci (利瑪竇1552–1610) in translating Euclid's *Elements* into Chinese at the beginning of the seventeenth century, the title of the book was set as *Jihe Yuanben* (幾何原本literally 'source of quantities'). '*Jihe'* is now the modern translation in Chinese of the term 'geometry'. Some people assume that this translation arose as a transliteration of the Greek word '*geometria'*. There are ample reasons to refute this assumption. Indeed, the Chinese translation of Book V of *Elements* makes it clear that '*jihe'* is used as a translation of a technical term for magnitude, while its connotation for Xu Guang-qi is the term 'how much or how many' that appears frequently in ancient Chinese mathematical classics (Siu 2011). Thus, from the beginning, Xu Guang-qi noticed the significance of magnitude in Western mathematics and the metric nature of Euclidean geometry, so much so that he selected this term and incorporated it as part of the title of the translated text.

### *15.6.4* **Jiuzhang Suanshu** *(*九章算術 *Nine Chapters on the Mathematical Art)*

In the ancient Chinese tradition, geometry and algebra, or shapes and numbers, were integrated. Let us look at Problem 12 in Chap. 4 of the Chinese mathematical classic *Jiuzhang Suanshu* (九章算術Nine Chapters on the Mathematical Art) compiled between the first century BCE and first century CE (Chemla & Guo, 2004). The problem states, 'Now given an area 55,225 [square] *bu*. Tell: what is the side of the square?' The text offers and explains an algorithm for extracting the square root. The following picture may explain the method more clearly (Fig. 15.7).

The text goes on to explain: 'If there is a remainder, [the number] is called unextractable, it should be defined as the side on which the square has the area of the *shi'*. It would be too much to claim that this indicates the awareness of an irrational number in this ancient epoch, but apparently it is the name of what is now called a surd. Thus, Chinese mathematicians in this ancient epoch knew about the estimate of the square root of an irrational number.

Let me cite another example in *Jiuzhang Suanshu* to show how algebra and geometry were integrated in the ancient Chinese mathematical tradition. Problem 20 in Chap. 9 states, 'Now given a square city of unknown side, with gates opening in the middle. 20 *bu.* from the north gate there is a tree, which is visible when one goes 14 *bu.* from the south gate and then 1775 *bu.* westward. Tell: what is the length of each side?' (Fig. 15.8). In modern-day mathematical language, we can solve this as a quadratic equation, *x*2+34*x*=71,000.

The method outlined in *Jiuzhang Suanshu* is an extension of the extraction of the square root, known as the extraction of the square root with an accompanying number (帶從開方法). Again, the following picture may explain the method more clearly (Fig. 15.9).

Even more interesting is the way the equation is set up in a geometric context (Fig. 15.10).

In offering these examples, my aim has been to let you see how algebra and geometry, shapes and numbers, come together in the ancient Chinese tradition. However, the number line does not seem to be a familiar representation in this tradition, perhaps as a result of its algorithmic nature.

**Fig. 15.7** Extraction of the square root

*15.6.5 Tongwen Suanzhi (*同文算指*)*

Primary mathematics education is important and difficult. It is difficult because no academic subject is easy; in particular, no one subject is easier than another. However, there is an additional reason, in my humble opinion, why it is difficult. Most people are affected by this difficulty, not just the primary school teachers and their pupils but most people, particularly those who are parents of primary school pupils. However, most people think that they know what primary mathematics

**Fig. 15.9** Extraction of the square root with accompanying number

education is and how it should be dealt with, because they were once primary school pupils. In other words, everybody considers themselves experts in this field! They do not think that they can learn from each other! As a semi-layman in this field, I am glad I have attended the ICMI Study 23. I have come to learn and, as the ancient Chinese classical text *Xueji* (學記) states, 'Teaching and learning help each other […] Teaching is the half of learning'. I am particularly glad to have joined the WG1 (Chap. 5) with culture as an emphasised component, and of course tradition, the subject of this panel.

My presentation in WG1 (Siu 2015a, b) concerns the book *Tongwen Suanzhi* (同 文算指), compiled under the collaboration of the Italian Jesuit missionary Matteo Ricci and the Ming scholar official Li Zhi-zao (李之藻1565–1630). This book first transmitted into China the art of *bisuan* (筆算 written calculation) (Siu 2015a, b). The term *tongwen*, meaning literally 'common cultures', indicates a deep appreciation of the common cultural roots of mathematics despite the different mathematical traditions.

In the preface of *Tongwen Suanzhi*, Xu Guang-qi wrote that:

The origin of numbers, could it not be at the beginning of human history? Starting with one, ending with ten, the ten fingers symbolise them and are bent to calculate them, [numbers] are of unsurpassed utility! Across the five directions and myriad countries, changes in cus-

**Fig. 15.10** Setting up a quadratic equation in a geometric context

toms are multitudinous. When it comes to calculating numbers, there are none that are not the same; that all possess ten fingers, there are none that are not the same.

In the preface to a reprinting of Matteo Ricci's *Tianzhu Shiyi* (天主實義*The True Meaning of the Lord of Heaven*), Li Zhi-zao wrote: 'Across the seas of the East and the West the mind and reasoning are the same [*tong*]. The difference lies only in the language and the writing'.

### **15.7 Conclusion**

The quotation from Li Zhi-zao that concludes the previous section deeply illustrates the rationale and problems of teaching/learning whole numbers in primary school. In fact, language and writing incorporate different cultural meanings according to which numbers are processed and conceived in different cultures. This is further reflected in the ways in which representations are used and interpreted in different cultural environments.

Many basic concepts of mathematics and whole numbers show the dramatic duality between mathematics as a universal language and its specific features of enculturation. The contributions from this panel widely illustrate this point.

This situation poses a great didactic challenge, which has the flavour of a paradox. On the one hand, the abstract universal concepts of mathematics are the goals of teaching and learning, but on the other hand, this goal can only be achieved by dealing with the concrete ways in which the concepts have been shaped by specific cultural tools, from oral and written words, to a variety of forms of representation (drawings, bodily expressions, etc.). This challenge constitutes the fascinating main feature of our work as mathematics educators and makes conducting research on this problem worthwhile.

Acquiring new knowledge on this issue is crucial because of the great social and economic changes the world is now facing. In recent years, economic globalisation, universal technological development and the related needs for manpower skills have provided strong historical motivations for introducing unified standards for mathematics in school. However, only a multicultural perspective allows us to consider the existence of different epistemological and cultural positions concerning mathematics and its cultural relevance and to realise the distance of the proposed curricular reforms from the mathematical cultures of different countries. It is important to base any teaching programme on its relationships with the cultures of the students and the personal contributions that they bring to the classroom. This will help avoid alienating the students from their cultural environment and allow them to engage in learning in a productive way.

The contributions of this panel pinpoint the crucial issues that need to be addressed to avoid the dangers of both the cultural refusal of innovation and of cultural alienation and of losing the cultural richness that exists in the different regions of the world.

### **References**

Barton, B. (2008). *The language of mathematics: Telling mathematical tales*. New York: Springer.

Barwell, R., Clarkson, P., Halai, A., Kazima, M., Moschkovich, J., Planas, N., Setati Phakeng, M.., Valero, P., & Villavicencio Ubillús, M. (Eds.). (2015). *Mathematics education and language diversity: The 21st ICMI study*. New York: Springer.

Bazzanella, C. (2011). *Numeri per parlare*. Bari: Laterza.

Bishop, A. J. (1991). *Mathematical enculturation: A cultural perspective on mathematics education*. Dordrecht: Kluwer Academic Publishers.

### **Cited papers from Sun, X., Kaur, B., & Novotna, J. (Eds.). (2015). Conference proceedings of the ICMI study 23: Primary mathematics study on whole numbers. Retrieved February 10, 2016, from www.umac.mo/fed/ICMI23/doc/Proceedings\_ICMI\_ STUDY\_23\_final.pdf**

Bartolini Bussi, M. G. (2015). The number line: A 'Western' teaching aid (pp. 298–306).

Sun, X. (2015). Chinese core tradition to whole number arithmetic (pp. 140–148).

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 16 Special Needs in Research and Instruction in Whole Number Arithmetic**

**Lieven Verschaffel Anna Baccaglini-Frank , Joanne Mulligan , Marja van den Heuvel-Panhuizen , Yan Ping Xin , and Brian Butterworth**

### **16.1 General Introduction**

Lieven Vershaffel

Many children have difficulties or problems with learning mathematics. While these difficulties or problems may occur at any stage in learners' mathematical development, by far the most attention of researchers and practitioners goes to the domain of early and elementary mathematics and, more specifically, to the domain of whole number arithmetic (WNA). Even though the issues of diagnosis of and instruction

A. Baccaglini-Frank Università di Pisa, Pisa, Italy

J. Mulligan Macquarie University, Sydney, NSW, Australia

M. van den Heuvel-Panhuizen Utrecht University, Utrecht, the Netherlands

Y.P. Xin Purdue University, West Lafayette, IN, USA

B. Butterworth (discussant) University College London, London, UK

**Electronic Supplementary Material** The online version of this chapter (doi:10.1007/978-3-319- 63555-2\_16) contains supplementary material.

L. Verschaffel (\*) Katholieke Universiteit Leuven, Leuven, Belgium e-mail<EMAIL_ADDRESS>
© The Author(s) 2018 375 M.G. Bartolini Bussi, X.H. Sun (eds.), *Building the Foundation: Whole Numbers in the Primary Grades*, New ICMI Study Series, https://doi.org/10.1007/978-3-319-63555-2\_16

for children with special mathematical learning needs are getting increasing research attention, research in this area is still lagging behind compared with other academic subjects such as reading. Hereafter, we list some major open questions for research and practice.

First, there is the terminological issue. Defining mathematical learning difficulties, problems or disabilities (hereafter abbreviated as MLD) is not an easy task (Berch and Mazzocco 2007). Despite the solid knowledge base that has been achieved in this field, more substantial progress in understanding and addressing MLD would be facilitated by establishing agreement on consistently used terminology and use of standardised criteria concerning the nature and seriousness of the disability. While certain definitions explicitly refer to a biologically based disorder, others emphasise the discrepancy between the child's mathematical achievement and his/her general intelligence as the main criterion, and others still focus on the response to intervention. But the field of MLD also lacks coherence and consensus about what constitutes 'mathematics' in MLD. Within MLD research, there is a history of predominance to focus on memorisation of arithmetic facts and automatisation of arithmetic procedures. A less (neuro)psychologically dominated and more interdisciplinary approach might bring a broader, more coherent and balanced perspective that takes into account both the views about mathematics learning as arithmetic and other equally important perspectives such as spatial and geometrical reasoning, mathematical relations and patterns and other forms of mathematical thinking with more potential towards abstraction and generalisation (Hord and Xin 2015; Mulligan 2011). Evidently, besides children with MLD, there are also other children requiring special mathematics educational support, but they are not diagnosed as MLD, such as children with intellectual disabilities; children with auditory, visual or motoric impairments; children with serious emotional and/or behavioural problems; or, finally, children with long-standing inappropriate instruction or environmental deprivation (De Smedt et al. 2013).

A second major concern of researchers in the field is to characterise the various cognitive mechanisms that are implicated in the development of MLD. Several cognitive explanations for the presence of MLD have been put forward. Most of the available research on MLD has dealt with domain-general cognitive factors, such as poor working memory and difficulties with the retrieval of phonological information of long-term memory. More recently (and against the background of findings from neuroimaging research), it has been proposed that MLD arises as a consequence of domain-specific impairments in number sense or the ability to represent and manipulate numerical magnitudes (Butterworth 2005; Landerl et al. 2004). For example, children with MLD have particular difficulties in comparing two numerical magnitudes and in putting numbers on a number line, both of which are thought to measure one's understanding of numerical magnitude. Although various cognitive candidates have been put forward to explain MLD, the existing body of data is still in its infancy. According to Karagiannakis et al. (2014), although the field has witnessed the development of many classifications, no single framework or model can be used for a comprehensive and fine interpretation of students' mathematical difficulties, not only for research purposes but also for informing mathematics educators. Starting from a multi-deficit neurocognitive approach and building on the

available literature, these authors have recently proposed a classification model for MLD describing four cognitive domains within which specific deficits may reside.

Third, initial accounts of MLD in the 1970s suggested that MLD was due to brain abnormalities. With the advent of modern neuroimaging techniques, researchers have begun to address this issue. There is converging evidence for the existence of a frontoparietal network that is active during number processing and arithmetic (Ansari 2008). Studies that examine this network in children with MLD are currently slowly but steadily emerging. These few studies consistently indicate that children with MLD have both structural and functional alterations in the above-mentioned frontoparietal network, particularly in the intraparietal sulcus, which is the brain circuitry that supports the processing of numerical magnitudes, and (pre)frontal cortex, which is assumed to have an auxiliary role in the maintenance of intermediate mental operations in working memory. Furthermore, it has been suggested that these brain abnormalities in children with MLD are probably of a genetic origin, yet the genetic basis of MLD remains largely unknown and no genes responsible for mathematics (dis)abilities have been identified. Studies in the field of medical genetics have revealed that some disorders of a known genetic origin, such as Turner syndrome and 22q11 deletion syndrome, show a consistent pattern of MLD. Furthermore, there is some early evidence of links to autism spectrum disorders and Asperger's.

The fourth and final issue relates to the question: what are appropriate educational interventions for children with MLD? Originally, general perceptuo-motor training was the dominant way of remediating learning disorders, but the effects of this type of training have been discounted. Interventions that target those specific components of mathematics with which a child with MLD has difficulty appear to be the most effective (Dowker 2008). Such intervention involves the assessment of a child's strengths and weaknesses in mathematics, and this profile is taken as an input to remediate specific components of mathematical skill. However, several major questions remain: what is the appropriate moment to diagnose MLD and to start specific interventions? Do MLD children profit more from individualised interventions organised out of the regular mathematics class or do they profit more from being integral part of the regular mathematics class? Do these children need a special kind of intervention or do they profit most from the same kind of instruction as children without MLD? More specifically, is conceptually based and constructivistoriented mathematics instruction also suitable for children with learning disabilities (Xin and Hord 2013; Xin et al. 2016)? Another issue is whether we do not have a blind spot when making assumptions about what children with MLD can do, rather than what they cannot do (Peltenburg et al. 2013). Finally, does the remedial instruction of children with MLD pay enough attention to other aspects of mathematics than whole number sense, such as to conceptual relationships that may develop from spatial reasoning? Clearly, it may not be productive to try to answer these major educational questions for all categories of children who have serious trouble with learning mathematics.

So, although the last decades have witnessed a serious growth in research into the diagnosis, remediation and prevention of MLD, much work remains to be done. Longitudinal research is needed to identify developmental precursors and to delineate developmental trajectories of MLD. The neural basis of these difficulties and their association with classroom performance certainly need to be further explored. Understanding the different characteristics of MLD at different levels – the behavioural, the cognitive and the neurobiological – will inform appropriate educational interventions. The design and evaluation of these remedial interventions needs to be a priority on the agenda for future research. These interventions may not only treat the difficulties but also prevent them. And, finally, there is a great need to look beyond diagnoses and interventions that are merely focused on counting and arithmetic to those also involve other aspects of mathematical thinking which hold more promise for abstraction and generalisation.

The goal of the ICMI Study 23 panel on special needs was to explore and discuss the above issues and challenges, with a strong emphasis on the last issue, namely, instructional goals and interventions for children with MLD. The panel consisted of four scholars with complementary specialisations in the domain of children with MLD and other special needs in the curricular domain of whole number arithmetic, namely, Anna Baccaglini-Frank ('La Sapienza' University, Rome, Italy), Joanne Mulligan (Macquarie University, Sydney, NSW, Australia), Marja van den Heuvel-Panhuizen (Utrecht University, Utrecht, The Netherlands) and Yan Ping Xin (Purdue University, West Lafayette, IN, USA), complemented by one of the keynote speakers of the ICMI Study 23 Conference, Prof. Brian Butterworth (University College, London, UK), a world-leading scholar in the domain of the (neuro)cognitive roots of dyscalculia and its treatment, who acted as discussant.

### **16.2 Does 'Dyscalculia' Depend on Initial Primary School Instruction?1**

Anna Baccaglini-Frank

In this contribution, I address the questions of whether (1) MLD children profit more from individualised interventions organised out of the regular mathematics class or from being an integral part of the regular mathematics class; (2) these children need a special kind of intervention or whether they profit most from the same kind of instruction as children without MLD; (3) the answers to the above questions are the same for all categories of children with MLD.

Let me start with the last question. Assuming that 'categories of children with MLD' is a well-defined construct (though I do not believe it yet is), in my opinion the answer is 'no'.

First of all, for the same child, the answers may vary at different stages of his/her life. For example, before any diagnosis is made (and some would argue, even after), many would probably claim that, at least initially (and perhaps always), the child should be in the 'regular' classroom and experience conceptually based,

1The study was possible thanks to the PerContare Project, coordinated by Fondazione ASPHI onlus, with the support of Compagnia di San Paolo and the operative support of Fondazione per la Scuola of Compagnia di San Paolo of Torino.

constructivist-oriented instruction (to use the same terms as in questions 2 and 3). But what if during a whole year, or even worse a whole school cycle (5–6 years), the child – for a variety of reasons – does not participate in the classroom discourse during the mathematics hours? This can be the case, for example, if the classroom culture is heavily based on written language and the child has not overcome difficulties related to the use of this medium, a frequent condition in cases of dyslexia.  Perhaps the child's environment would have been more aware of his/her difficulties with written language, if the child had spent time in a special education classroom, offering a context in which participation was fostered in a more appropriate way, leading to experiences of participation and success in mathematics.

However, after many years of (induced or voluntary) exclusion from mathematical discourse, throughout which the child – now adolescent – has never actually done mathematics, is it still appropriate to place him/her in a 'regular' classroom involving constructivist-oriented instruction that heavily builds on notions our student has never constructed? He/she will almost definitely fail mathematics for good.

On the other hand, it is possible that with an individualised remedial intervention that takes into account the student's (well-known) difficulties, he/she will rapidly regain confidence and start participating in a mathematical discourse that uses different means for acquiring and producing information and that can be appreciated by the teacher and by all the other participants in the mathematical discourse, even those within the 'regular' classroom. Throughout my experience in helping students learn mathematics in different settings, I have witnessed a number of cases similar to the prototypical one just described.

In the example, I mentioned difficulties in using written language; however, there can be many other cognitive conditions, such as a difficulty to remember procedures or facts, difficulties in encapsulating processes, difficulties in logical reasoning and many others that lead to experiencing failure and eventually to exclusion from the mathematical discourse produced in 'regular' classrooms. I believe it is fundamental (for the teacher, clinician or other educator) to identify these difficulties and 'work around' them, helping the student become aware of them while addressing and overcoming whichever ones are possible. Of course this is no trivial task and each student is quite different!

Returning to the last question, it also seems to be the case that, at a given point in time, different students can have different characteristics. For example, taking a cognitive perspective, it seems possible to regroup existing hypotheses on MLD into a fourfold model that can be used for describing students' mathematical (cognitive) learning profiles (Karagiannakis et al. 2014). Studies based on this assumption are showing that the profiles of students with similar (or identical) low scores on mathematical achievement tests (also those used for diagnosing MLD) are in fact different (Karagiannakis and Baccaglini-Frank 2014; Karagiannakis et al. 2018). In other words, the studies are suggesting that failure to overcome difficulties in mathematical learning, at a cognitive level, cannot always be associated with a single deficit in a particular domain of the model, nor can it be considered the consequence of *one* particular finalised combination (that is the same for all students) of deficits. This supports the claim that looking for a cognitive characterisation of all students with low achievement specifically in mathematics is not necessarily a fruitful direction of research.

What we can (and should) ask is, 'why do some children fail to overcome difficulties in mathematical learning that others do overcome?' Reasons may include students' cognitive characteristics, as a result of 'innate' inclinations that are shaped by immersion in society (as mentioned above), students' mathematical learning history, affective components of both the students and their teachers, teachers' choices about what mathematical content to present and the means they choose (or do not choose) to introduce it, the way MLD is viewed within school policies and teachers' perspectives, implicit or explicit assumptions on 'what' or 'how much' MLD students can learn, etc.

I believe that research conducted by mathematics educators should address how to minimise failure in mathematics due to children's individual specific learning characteristics, as early as possible – starting at least at the beginning of formal instruction (kindergarten or first grade in most countries). This is what we attempted to do in a 3-year project recently carried out in Italy (2011–2014). For this project, a team of mathematics educators and psychologists designed curricular material for mathematics, framed within the theories of semiotic mediation (Bartolini Bussi and Mariotti 2008) and embodied cognition (Gallese and Lakoff 2005), with the aim of providing all students (in first and second grade) with 'hands-on' (kinaesthetictactile) experiences that involve manipulation of physical artefacts to develop mathematical meanings (including procedures) from these and from consequent mathematical discussions.

For example, to help children learn what are known in English as the 'multiplication tables', children were introduced to the manipulation of rectangles cut out of squared paper (see Sect. 7.4.2 of Chap. 7). Children learned to cut and paste these rectangles together to figure out unknown products. The physical procedures were then carried out simply by drawing (in notebooks or on the blackboard), and eventually children started to use them with no further external support, as strategies of mental calculation (see the example in Sect. 7.4.2 of Chap. 7). A fundamental aspect of the mathematical activity stemming from activities such as the ones described is the sharing and discussing of strategies, during which all students were invited to (and did!) contribute.

In the episodes shown in part 4 of the video (Electronic Supplementary Material: Baccaglini-Frank 2017b), the teacher has asked the children to share strategies they used to figure out 8 × 6, showing their procedure on the blackboard. One student has decided to break the segment 8 into three parts (5, 2 and 1), which for him 'make it easy' because they are numbers he knows how to count by. He then counts up by 5s to obtain the first piece, mentally rotates the second piece and remembers that 6 + 6 = 12 and recognises the last piece as 1 × 6. So he finally adds 30 + 12 + 6. The student in general performs at an average-low level, but he was able to keep up with the class using the proposed activities and occasional extra practice at home.

Another student had decided to decompose 8 into 10 − 2 and describes her reasoning through 'ghost rectangles',2 a terminology that very quickly catches on in the classroom. These are rectangles that appear to make the calculation easier, but then they need to be taken away. She uses ghost rectangles to think of 8 as a part of 10, to reach the product 10 × 6 (= 60), and then subtract off 2 × 6 (= 12). In the final mental calculation (60 – 12), she makes a mistake: at first, she forgets to take a second 10 from 60 and ends up with 58 instead of 48. Then, prompted by the teacher, she quickly corrects the mistake. Both students seem to be very much at ease when implicitly using the distributive property (it was not presented formally).

An important finding of the project was that working with the experimental materials through first and second grade significantly reduced the number of children who could be classified as MLD by third grade (Baccaglini-Frank and Scorza 2013; Baccaglini-Frank and Bartolini Bussi 2015; Baccaglini-Frank 2015). Moreover, the children exposed to the PerContare teaching-learning experience developed a variety of strategies for addressing different mathematical situations. In particular, with respect to calculation, for these children, the acquisition of numerical facts occurred with greater accuracy, variety of strategies and eventually speed. The 'cost' was a 3-month lag in fact and automatisation compared with the higher performing children in the control classes.

Insisting on the finding that persistent use of particular curricular materials can significantly reduce the number of children who tested positively for dyscalculia in third grade, we find an apparent contradiction with the literature claiming that dyscalculia is an innate deficit. Indeed, our sample of students seems to show that testing positively for dyscalculia can depend on initial primary school instruction, an extremely 'cultural' experience. Of course, one can solve the dilemma in a number of ways, for example by attacking the effectiveness of diagnostic test batteries (at least those used in Italy) or the diagnostic criteria more in general, or by speaking more loosely of MLD without giving a clear definition, which indeed, unsurprisingly, has not yet been agreed upon across groups of research (e.g. Mazzocco and Räsänen 2013).

This brings me back to my earlier plea: as educators, we should continue studying why fewer students fail in mathematics when they participate in particular types of early mathematical experiences. Let us call these good practices. I believe that particular effort should be put in developing good practices and studying their effect with different samples of children. At this point, when a set of good practices has been identified, we can ask whether there are students who still fail in mathematics and set up studies to explore why this is the case, then possibly use such knowledge to further ameliorate the practices or, in parallel, develop ad hoc remedial interventions. My personal belief is that it is unlikely that many students now classified as MLD will benefit more from individualised interventions than from whole-class learning situations where they make use of good practices which afford multiple

2This may also be seen as a *pivot sign* according to the Theory of Semiotic Mediation (Bartolini Bussi and Mariotti 2008) and was exploited as such by the teacher.

means of participation in the mathematical discourse and which are considerate of the learning inclinations of all students. But of course this is yet an open question.

### **16.3 Are MLD Linked to a Lack of Underlying Awareness of Mathematical Patterns and Relationships that Are More Linked to Spatial Ability than Development of Number?**

#### Joanne Mulligan

To address some of the issues articulated in the introduction of this chapter, I will adopt an integrated perspective, in order to provide a more coherent view of the underlying cognitive bases of mathematical development and MLD based on an awareness of mathematical 'pattern and structure'. Rather than focusing on the domain of WNA, my research has focused on supporting the development of mathematical generalisation – through early identification of patterns and relationships and the development of interrelated spatial processes.

In recent years, mathematics education research has turned increasing attention to other research domains and interdisciplinary studies to explain and describe the wide variation in mathematical competence in the early years.

Studies of early mathematical competencies have largely emphasised children's numerical competencies, e.g. counting, subitising, representing number, numerical magnitudes and positioning on an empty number line (De Smedt et al. 2013; Fias and Fischer 2005). Another approach has focused on children's spontaneous focusing on number and quantitative relations (Hannula and Lehtinen 2005) found to be predictive of later achievement. Related studies highlight the critical role of perceptual subitising (McDonald 2015) and the spatial structuring of groups in arrays (Starkey and McCandliss 2014). Neurocognitive studies (Butterworth et al. 2011) also provide complementary evidence of the connection between the development of number and arithmetic and spatial processes. Number concepts depend on processes such as subitising (the rapid and accurate perception of small numerosities), comparison of numerical magnitudes, location on a number line, axis differentiation and symmetry (e.g. Dehaene 2009). Some interventions have incorporated some of these aspects for students with MLD and those performing below specified benchmarks, but with a focus on counting and arithmetic rather than underlying mathematical attributes.

The relative influence of the various components and how they interrelate in mathematical development, especially for students with MLD, remains unclear. Moreover the influence of one or more of these components on the individual's mathematical development may vary widely.

Recent developmental studies have indicated the positive impact of the early development of spatial skills on mathematical development (Verdine et al. 2014). Other studies in mathematics education highlight the sustained development of spatial reasoning skills from an early age – these are malleable and can be augmented

over time but can weaken if not supported (Davis 2015). Spatial ability has also been linked to development of patterning and early algebraic skills (Clements and Sarama 2011; Papic et al. 2011) and the relationship with other concepts such as number and measurement (Mulligan et al. 2013; Mulligan et al. 2015). This raises critical questions about the need for differentiated teaching, assessment and intervention programmes for learners with poor spatial skills, not exclusive to those identified with MLD.

The Australian Pattern and Structure Mathematics Awareness Project (see also Sect. 7.3.3 of Chap. 7) is a suite of related studies with 4–8-year-olds focused on the assessment of mathematical structures for children representing a wide range of abilities including children with MLD (Mulligan et al. 2013). These studies have taken into account the complexity of various components of mathematical competency by adopting a more integrated view: what are common salient features of early mathematical development? Does the ability to recognise patterns and structures reflect innate ability or can it be developed? Why do some children with MLD lack this ability? What is the role of spatial reasoning?

The project, spanning over a decade, involved the development and validation of an interview-based assessment instrument the *Pattern and Structure Assessment – Early Mathematics* (*PASA*) (Mulligan et al. 2015) and the evaluation of the *Pattern and Structure Mathematics Awareness Program* (*PASMAP*) (Xin et al. in press). On the basis of students' PASA responses drawn from a range of studies, five levels of structural development were identified and described: prestructural, emergent, partial, structural and advanced structural (see Mulligan et al. 2013). Students with low AMPS operated generally at the prestructural or emergent level: for example, they had difficulty subitising larger sets, recognising a unit of repeat in simple patterns or utilising the structural features of arrays. They were most likely to represent idiosyncratic or superficial features in their models, drawings and explanations.

Based on early studies on patterning, counting, the numeration system and multiplicative thinking, the research focused on identifying and describing common characteristics, later coined as the construct *Awareness of Mathematical Pattern and Structure* (*AMPS*)*.* AMPS has two interdependent components: one cognitive (knowledge of structure) and one meta-cognitive (a tendency to seek and analyse patterns). The AMPS construct involves the following structural components:

Remedial or intervention initiatives in early numeracy for students with MLD typically focus on number and arithmetic without paying attention to patterning and spatial processes. Yet increasing evidence from a number of disciplines points to other components contributing to numerical competence. Our studies have traced the early development of number and other mathematical concepts to the development of AMPS. The PASMAP intervention studies, which examine the development of spatial aspects of patterns and spatial structures across mathematics concepts, indicated that such features as differentiation of foreground/background, alignment (collinear or axis), unitising and equal grouping, transformation and recognition of shape and equal areas are critical to mathematical development. It was found that these aspects can be improved through intervention for some children with MLD (Mulligan et al. 2013).

The design of the PASMAP intervention takes account of the assessment (PASA) which measures the child's level of AMPS; however, the programme can be utilised in conjunction with other assessments and intervention strategies. The PASMAP intervention programme was designed and trialled with students with wide-ranging abilities including those with MLD. PASMAP focused on five structures described above and is flexible in its implementation because the teacher can target specific mathematical structures with which a child with MLD has most difficulty. The pedagogy is designed to move students towards identifying similarities and differences, with a view to representing and abstracting core structural elements. The use of visual memory to record spatial representations is emphasised.

I propose that development of the various components of mathematical competence described in the literature earlier must have interrelated influences on mathematical development, but there is a common underlying thread. I am not suggesting that components are simply amalgamated into the construct that we call AMPS. Our empirical evidence supports the promotion of structural features rather than emphasis on counting and arithmetic. Fine-grained analysis of children's development over time suggests a complex network of these PASMAP components: the common denominator is the ability to see patterns and structural features that are essentially or initially spatial in nature. Hence the importance of focusing on children's development of structures such as grouping and partitioning, unitising, subitising, colinearity and benchmarking numerical magnitudes.

Conceptual relationships in mathematics depend on AMPS: spatial structuring and recognising patterns may provide the inextricable link between spatial and number development. Our recent studies have linked a measure of AMPS to standardised measures of early numeracy (Mulligan et al. 2015). Further analysis utilising network analysis (Woolcott et al. 2015) provides visual links between AMPS structures as a map of connectivity. However, the role of spatial reasoning in the

development and use of AMPS is not fully understood; what our studies have described are domain-specific aspects of AMPS such as spatial structuring, partitioning and structuring linear, two-dimensional and three-dimensional space and relations between pattern and number.

Our future studies focus on evaluating the impact of various structures within the PASMAP intervention with children with MLD, moreover identifying critical differences for individuals in terms of AMPS over time. This may require a considered review of what constitutes critical components in early mathematical development and improved cross-disciplinary collaboration to inform research agendas and more effective pedagogical innovations.

### **16.4 It Is Time to Reveal What MLD Students Know, Rather than What They Do Not Know**

#### Marja van den Heuvel-Panhuizen

Good teaching starts with getting to know what students know. Although this applies to all students, it is particularly true for students who have mathematical learning difficulties (MLD). Unfortunately, the problem with these students is that they usually have low scores on mathematics tests, which may automatically lead to the conclusion that they are ignorant, that they are unable to solve demanding mathematical problems and that it cannot be expected that they can come up with their own solution methods. Unmasking these and other prejudiced ideas is of vital importance for MLD students, because it may open new opportunities for teaching them mathematics. However, the burning question is how we can reveal what MLD students *do* know. In this contribution, I will discuss some research findings that give rise to reconsidering the presumed limitations of MLD students.

My research activities in this field started at the beginning of the 1980s when I got acquainted with an approach to mathematics education that proposes to start from students' informal and context-related mathematical knowledge, to offer students models to eventually reach more general and formal levels of understanding, to go beyond the sole focus on whole number operations, but also includes other mathematical domains, to give students an active role in the learning process, to elicit reflection, to stimulate classroom interaction about different solution strategies and to aim not only at learning facts and skills, but also at gaining insight.

As a former special education teacher, I was surprised that special educationalists rejected this kind of teaching for students in special education. According to these educationalists, it would be better to teach MLD students only a fixed solution strategy; otherwise, they would get confused. Also, it would be better to teach MLD students bare number problems, because problems situated in contexts would make problems too complex for them. Furthermore, building on students' own informal solution methods would be an illusion, because MLD students can hardly come up with solutions by themselves (see more about these assumptions of special educationalists in van den Heuvel-Panhuizen 1986, 1996).

To challenge these, in my view, incorrect assumptions, in 1990, I set up a study (van den Heuvel-Panhuizen 1996) in two special education schools for mildly mentally retarded students. The age of the students was between 10.5 and 13 years. The students' mathematical levels were far behind their peers and lay between grades 2 and 4 of regular primary school. The topic I chose for this study was ratio, which is generally considered beyond the reach of students in schools for mildly mentally retarded students and which accordingly was not taught to the students who participated in the study. In order to provide evidence that this was an underestimation of their mathematical ability, I administered a test on ratio including 16 ratio problems all referring to contextual situations familiar to the students and not including formal notations of ratio. Instead, as Freudenthal suggested to me when I designed the test, I made use of the visual roots of ratio. The results revealed that the MLD students, without having had instruction on ratio, were quite able to solve the problems. The percentage of correct answers for the problems ranged from 13% to 64%. Both the teachers of these students and the experts (two special education school inspectors and two special educationists) who were asked to predict the students' scores in many cases underestimated them. Moreover, the scrap papers added to the test sheets showed clear traces of self-invented strategies and notations.

The often-heard claim that students who are weak at mathematics can be better taught only one fixed standard strategy for every operation (e.g. see Gelderblom 2008, p. 36: 'Letting students who are weak in mathematics discover strategies by themselves is fatal. Lead them by the hand, tell them which strategies they have to use'; translation into English by author) induced me and my PhD student Marjolijn Peltenburg in 2010 to set up a study in which we investigated how special education students solve subtraction problems up to 100. The standard strategy that MLD students are taught for this type of problems is the take-away strategy. On purpose, we also included in the subtraction test for this study problems that might elicit an adding-on strategy (e.g. bare number problems such as 62 − 58 and context problems with an adding-on context). What we found was that the MLD students, without being taught, made spontaneous use of the adding-on strategy. Moreover, they were rather flexible in what strategy they applied, and they were quite successful when applying the adding-on strategy (Peltenburg et al. 2012).

Besides offering MLD students assessment problems in which they could show their competence on topics that belong to the regular mathematics curriculum, we also did further research on a topic that is far beyond what is taught in special primary school education and even is lacking in regular primary school. In this research we investigated what happened when MLD students were asked to solve a number of combinatorics problems. Here we found that the MLD students in our study were equally successful in solving the combinatorics problems as their comparable peers in regular education, who were younger but at the same level of understanding number and operations. Moreover, on average the MLD students equally often applied a systematic strategy to find all possible combinations as the students in regular education, and in both school types, a significant increase in the use of systematic strategies could be observed (Peltenburg et al. 2013; Peltenburg 2012, Chap. 6).

**Fig. 16.1** Percentage problem in the DAE with optional auxiliary tools

Another avenue in our search for possibilities to make the hidden mathematical potential of MLD students visible is the use of a technology-enhanced assessment. For this we put a series of subtraction problems from a standardised test into an ICT environment and extended them with optional auxiliary tools. In one study, we used a digital interactive 100-board on which the students could represent the problems to be solved by dragging counters. In the other study, the optional auxiliary tool consisted of a digital interactive number line. Both studies showed that the proportions of correct answers were higher in the ICT-based test than on the standardised test (Peltenburg et al. 2010). This result appears rather obvious, but for teachers a test that not only tells them which students got which problems correct, but also which students made use of the auxiliary tools and how they used them, contains very valuable indications for further instruction. In fact, in this way the zone of proximal development of the students is opened. Moreover, we found that the MLD students were quite aware of whether they needed the help of the auxiliary tools. Students who made the most mistakes in the later administered standardised test more often chose to use an auxiliary tool in the earlier administered ICT-based test.

As a result of these positive experiences with optional auxiliary tools, this approach to assessment is now being further explored in the EU-funded FaSMEd project, which aims to research the use of technology in formative assessment classroom practices in ways that allow teachers to respond to the emerging needs of low-achieving students. The Dutch team of this project developed the Digital Assessment Environment (DAE) for mathematics education in the upper grades of primary school. Figure 16.1 shows an item on percentages with the optional auxiliary tools that can be chosen to solve this problem.

### **16.5 Conceptual Model-Based Problem-Solving: An Integration of Constructivist Mathematics Pedagogy and Explicit Strategy Instruction**

#### Yan Ping Xin

The question whether students with learning disabilities should be educated in the inclusive classroom or in a segregated instructional environment has always been a hot topic. Here, I use the term 'students with learning disabilities or difficulties in mathematics' (LDM) to include all students whose mathematics performance is ranked below the 35th percentile (Bryant et al. 2011), so not necessarily only students with a biologically based disorder. With this broad definition in mind, a more pertinent question would be: 'Do these children need a special kind of intervention or do they profit most from the same kind of instruction as children without LDM?' In particular, 'is conceptually-based and constructivist-oriented mathematics instruction also suitable for children with learning disabilities?' For the most part, it depends on (a) how we support these students with instructional strategies that address their needs and (b) how much support or scaffolding we provide for these students so that they are able to make sense of the mathematical concept or relations or, from the instructional point of view, whether we can make the mathematical discourse or reasoning process explicit to the students so they can grasp the concept or knowledge. To this end, regardless of the placement, it is more important to consider whether the instructional strategies we employ will provide the needed support or scaffolding that will allow these students to have meaningful access to mathematics.

As the outcome of a collaborative piece of work that integrates research-based practices from math education and special education, Xin, Tzur and Si (2008), with the project team, have developed an intelligent tutor, PGBM-COMPS © (Xin, Tzur and Si 2017), to support the learning of multiplicative problem solving for students with LDM. The intelligent tutor draws on three research-based frameworks: a constructivist view of learning from mathematics education, data (or statistical) learning from computer science, and conceptual model-based problem solving (COMPS) (Xin 2012), from special education, that generalises word problem underlying structures. Rooted in a constructivist perspective on learning, we focused on how a studentadaptive teaching approach (Steffe 1990), which tailors goals and activities for students' learning to their available conceptions, can foster advances in multiplicative reasoning. This approach is not based on a deficit view of students with learning disabilities; rather, it focuses on and begins from what they do know and uses task-based activities to foster transformation into advanced, more powerful ways of knowing.

The PGBM-COMPS tutor is made of two parts: (a) 'Please Go Bring Me…' (PGBM) turn-taking games designed to nurture a learner's *construction* of fundamental ideas in multiplicative reasoning (Tzur et al. 2013) and (b) COMPS (Xin 2012) that emphasises understanding and representation of word problem structures in mathematical model equations. In particular, the PGBM turn-taking games were designed to nurture a learner's *construction* of fundamental ideas such as 'number as a composite unit'. A basic version of the PGBM platform game involves sending a student to a box with Unifix Cubes to produce and bring back a tower made of a few cubes. After taking two-to-nine 'trips' for bringing same-sized towers, students are asked how many towers (i.e. composite units, CU) they brought, how many cubes each tower has (i.e. unit rate, UR) and how many cubes (1s) there are in all. The PGBM game was devised to promote learners' anticipated creation of and differentiation among 1s and CUs (Tzur et al. 2013). These two anticipations are crucial if the learner is to construct the mental operation of multiplicative double counting, which is fundamental to multiplicative reasoning. Multiplicative double counting integrates two counting sequences in a multiplicative problem situation (e.g. 'Please bring me a tower with 6 cubes in each…. If you brought me 5 such towers; how many cubes in all?'): one sequence that quantifies how many CUs (i.e. towers) were produced and one sequence that monitors the corresponding accumulation of 1s (i.e. total # of cubes) contained within those CUs (i.e. towers). Double counting is considered to be 'an advance over the more basic direct representation because it requires more abstract processing' (Kouba 1989, p. 152).

A variety of activities following a PGBM format were designed to promote students' construction of basic multiplicative concepts on the basis of continuous assessment of their existing knowledge and experiences. The learner will progress from a low to high level of tasks along the dimensions of (a) numerical numbers (e.g. 2, 5 or 10 - level I; 3 or 4 - level II and 6, 7, 8 or 9 - level III) involved in the problem and (b) cognitive demands of the task (i.e. operating with visible objects or invisible/covered objects with mental system).

On the other hand, COMPS generalises the understanding of multiplicative reasoning to the level of mathematical models. At this stage, students no longer rely on concrete models (such as cubes and towers) or drawing pictures or tally marks; the mathematical models directly drive the solution plan. The COMPS programme emphasises (a) the connection between the PGBM games (in the contexts of cubes and towers for instance) and the symbolic mathematical model equations, (b) students' representation of various multiplicative problem situations in the mathematical model equations and (c) development of the solution plan that is directly driven by the model equations. Figure 16.2 presents two sample screenshots from the PGBM-COMPS programme. The upper panel shows how the programme engages students in making the connection between the concrete modelling (cubes and towers) and the mathematical expression; the lower panel shows how the problem should be represented in the COMPS model to find a solution.

To evaluate the effect of the PGBM-COMPS © intelligent tutor, Xin et al. (2017) compared the effectiveness of the PGBM-COMPS programme with school teacher-delivered instruction (TDI) on enhancing the multiplicative reasoning and problem-solving skills of students with LDM. Results indicated that the improvement rate of the PGBM-COMPS group was much greater than that of the TDI group (effect size [ES] = 1.99 on researcher-developed multiplicative reasoning tasks; ES = 2.26 on a range of multiplication and division word problem-solving tasks involving large numbers). In addition, the group difference was shown on a commercial/published standardised test, the Stanford Achievement Test (SAT,

**Fig. 16.2** Sample screenshots of the PGBM-COMPS intelligent tutor system (Xin, Tzur and Si 2017)

Harcourt Assessment Inc. 2004): *Mathematics Problem Solving* subtest, favouring the COMPS group (ES = 1.23).

Given that the Common Core State Standards for Mathematics (National Council of Teachers of Mathematics 2012) demand much deeper content knowledge from teachers of mathematics, the preliminary findings of the above study are encouraging. The PGBM-COMPS intelligent tutor, which integrates the best practices from general mathematics education and special education, seems to yield better outcomes in enhancing participating students' multiplicative problem solving. Through the integration of heuristic instruction (that facilitates concept construction) and the explicit model-based problem-solving instruction, it seems that the PGBM-COMPS programmes have promoted generalised problem-solving skills of students with LDM.

From the foregoing, here comes my answer to the question "whether concetually based and constructivist-oriented mathematics instruction also suitable for children with learning disabilities? With appropriate scaffolding and support, students with LDM are able to engage in conceptually based and constructivist-oriented mathematics instruction. The promising outcome of the PGBM-COMPS intervention programme (Xin, Tzur and Si 2017) is just one example.

### **16.6 Discussion**

Brian Butterworth

If you want to get ahead, get a theory.

Verschaffel raised two fundamental issues in his introductory remarks to the panel. First, he asks what constitutes the 'mathematics' that MLD should address. Here I would like to start with a very simple approach. What constitutes 'a billable ICD-10-CM code that can be used to indicate a diagnosis for reimbursement purposes'? That is, what diagnosis will ensure that a child is entitled to special help for his or her mathematical difficulties? I take ICD 10 (The World Health Authorities list of diseases) because it is the clearest and most specific of the widely used classifications. In Sect. F81.2, the term used is a 'specific disorder of arithmetical skills'. This 'involves a specific impairment in arithmetical skills that is not solely explicable on the basis of general mental retardation or of inadequate schooling. The deficit concerns mastery of basic computational skills of addition, subtraction, multiplication, and division rather than of the more abstract mathematical skills involved in algebra, trigonometry, geometry, or calculus'. So, in this context, the answer to Verschaffel's question is simple: *arithmetic.* However, there are problems.

Notice that the ICD definition excludes an impairment in arithmetical skills that is solely explicable on the basis of general mental retardation.  Moreover, it excludes, in a later paragraph, 'arithmetical difficulties associated with a reading or spelling disorder'. Thus, the child cannot be both dyscalculic and dyslexic.

#### **Spatial Abilities**

ICD 10 does not mention spatial abilities, though it is known that, especially in the early years, there is a close link between them and arithmetical development (Rourke 1989). However, how this link operates is far from clear. Mulligan focuses on a specific set of spatial competences. In particular, she argues that a set of these underpins makes the conceptual relationships critical to arithmetical understanding. Specially designed interventions for weaknesses in this set of competences can make a big difference to the development of arithmetic.

#### **'Mental Retardation'**

Mental retardation does not prevent high levels of mathematical skill. We know from the study of savants, with very low measured IQ or with other indicators of limited cognitive ability, that they can be superb calculators (Butterworth 2006). We also know that IQ measures are poor predictors of mathematical competence, such that even individuals with very high measured IQ can be dyscalculic (Butterworth et al. 2011). In an original approach to this issue, van den Heuvel-Panhuizen reported studies she had carried out on the mathematical abilities in schools for children with special educational needs. Now these children will have low scores on standard tests and would be classified as MLD but would be excluded from a 'billable code' because of their measured IQ. Now it may well be that these children can be drilled to perform moderately well on arithmetical problems, but the question addressed is much more interesting: do they have the conceptual basis and cognitive ability to develop their own valid strategies for calculation?

van den Heuvel-Panhuizen has a clear answer to this question. 'What we found was that the MLD students, without being taught, made spontaneous use of the adding-on strategy. Moreover, they were rather flexible in what strategy they applied, and they were quite successful when applying the adding-on strategy.' They were also 'equally successful in solving the combinatorics problems'.

Verschaffel's second issue is what is the appropriate intervention for children with special needs, and this raises the ICD 10 exclusion criterion – 'inadequate schooling'. Now ICD 10 does not define this term, so it is not possible to determine whether the child is classified as MLD because of poor teaching. Baccaglini-Frank notes that this raises an important problem for the definition of MLD. She writes, 'Insisting on the finding that persistent use of particular curricular materials can significantly reduce the number of children who are positive to diagnostic tests for *dyscalculia* in third grade, we find an apparent contradiction with literature claiming that dyscalculia is an *innate deficit'*. I will return to this point below.

She also notes, quite properly, that 'if the classroom culture is heavily based on written language and the child has not overcome difficulties related to the use of this medium', then this could cause the child to fall behind in maths. The child, she says, would be better served in a 'special education classroom'. This would be another example of inadequate schooling.

Xin has developed an intelligent tutoring system designed to help all students with 'learning disabilities or difficulties in mathematics', by which she means students below the 35th percentile. For her, the nature of those difficulties, or their causes, appears not to be relevant. She argues that a conceptually based and constructivist-oriented mathematics instruction, developed for more typical students, is also suitable for children with learning disabilities. Her findings suggest that the intelligent tutoring system is more effective than teacher-delivered instruction.

#### **A Theory-based Approach**

At the root of the confusions about the criteria for MLD and about the appropriate intervention is the lack of theoretical perspective, and this is critical for understanding why a child fails to reach an expected level in maths.

Of course, what the expected level is will depend on social, economic and, importantly, political factors. One example is whether the educational authority – usually a government agency – recognises MLD as a 'billable' category. It may fail to do so out of ignorance, since mathematical competence is a proxy for intelligence or out of indolence, for example, if there is no parent pressure group to prod it into action. In the UK, and in many other countries, dyslexia is recognised precisely because there exist organisations that insist on its recognition. The authorities may not recognise MLD because it could entail a commitment to provide support for those assessed as MLD.

Without a theory, one is left with a criterion that could be set for economic or political reasons or could simply be arbitrary: 35th percentile, for example, or 1, 1.5, 2, 2.5 or 3 SDs below the population mean on a standardised test of arithmetic. None of these criteria tells you what the learner needs. The problem is compounded when one considers different populations. Consider an international comparison, for example, the PISA 2012 study. The proportion of children below level 2 in top ten countries was around 10%, but in the worst performing countries, it was between 60% and 75%. So what would count as MLD in Macao will be very different from what would count in Indonesia in terms of what the learner can and cannot do.

Here is the question that should be addressed: *why* is this child failing to understand what his or her classmates can understand? This is a theoretical question. For perhaps 5% of learners, the answer is that there is a deficit in very basic numerical concepts. That is, they will do poorly on tests that depend very little on the appropriateness of schooling or on social and economic status and even home background. These learners do poorly on tests of the enumeration of small sets of objects, typically displays of dots. They will be slower and less accurate than their peers, and this is a stable measure of individual difference and is a reliable predictor of the ease or difficulty of acquiring arithmetical competence (Reeve et al. 2012). Other very simple tests that rely little on education arrive at similar conclusions (Piazza et al. 2010). These are tests of a crucial component in the learner's 'starter kit' for acquiring basic numerical competence that I have called 'numerosity processing' and means the ability to estimate the number of objects in a set. Poor performance on tests of this ability points to a congenital *core deficit* in numerosity processing. Not only is performance on these tests independent of schooling, it is independent of intelligence, of working memory and of literacy (Landerl et al. 2004). We call this special need 'dyscalculia'.

The identification of a deficit in this core capacity has implications for intervention: more of the same, more slowly and with more repetition, does not work. As with dyslexia, specially designed interventions are needed, preferably using concrete materials, and adaptive digital games with virtual concrete materials, for much longer than would be needed with typically developing learners (Butterworth et al. 2011).

This approach also sheds light on the relationship between dyscalculia and other neurodevelopmental disorders. Dyslexia on this account cannot be a cause of dyscalculia, because it is due to a quite distinct core deficit, in most cases a phonological deficit (Butterworth and Kovas 2013; Landerl et al. 2009). This means that we must reject the ICD 10 exclusion criterion of reading disability and test for both core deficits.

Our approach also means that it is possible to have a core numerical deficit despite being highly intelligent, or indeed having low cognitive abilities. This is not to say that MLD may not have other causes, including inadequate schooling (an international problem), prematurity, poor diet and a difficult home environment (Benavides-Varela et al. 2016). In these cases, a different approach to intervention will be needed. In mathematical education, as in so many other things, one size does not fit all. Measure the customer first and then find the garment that fits best.

### **References**

### **Cited papers from Sun, X., Kaur, B., & Novotna, J. (Eds.). (2015). Conference proceedings of the ICMI study 23: Primary mathematics study on whole numbers. Retrieved February 10, 2016, from www.umac.mo/fed/ICMI23/doc/Proceedings\_ICMI\_ STUDY\_23\_final.pdf**

Baccaglini-Frank, A. (2015). Preventing low achievement in arithmetic through the didactical materials of the PerContare project (pp. 169–176).

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# **Chapter 17 Professional Development Models for Whole Number Arithmetic in Primary Mathematics Teacher Education: A Cross-Cultural Overview**

**Jarmila Novotná Maria G. Bartolini Bussi , Sybilla Beckmann , Maitree Inprasitha , Berinderjeet Kaur , Xu Hua Sun , Hamsa Venkat , and Mike Askew**

### **17.1 General Introduction1**

Jarmila Novotná

This chapter proceeds from the plenary panel on teacher education presented at the ICMI Study 23 conference in Macao (Novotná 2015). The main goal of the panel, as well as of this chapter, was to explore approaches to, and within, primary mathematics teacher education in different parts of the world and to discuss commonalities and differences in relation to broader cultural and curricular traditions.

J. Novotná (\*)

S. Beckmann University of Georgia, Athens, GA, USA

M. Inprasitha Khon Kaen University, Khon Kaen, Thailand

© The Author(s) 2018 399 M.G. Bartolini Bussi, X.H. Sun (eds.), *Building the Foundation: Whole Numbers in the Primary Grades*, New ICMI Study Series, https://doi.org/10.1007/978-3-319-63555-2\_17

1The paper was supported by the project Progres Q17 Příprava učitele a učitelská profese v kontextu vědy a výzkumu (Teacher preparation and teaching profession in the context of science and research).

Charles University, Prague, Czech Republic, and Laboratoire "Culture et Diffusion des Savoirs", Université de Bordeaux, Bordeaux, France e-mail<EMAIL_ADDRESS>
M.G. Bartolini Bussi Department of Education and Humanities, University of Modena and Reggio Emilia, Modena, Italy

The ICMI Study 23 was focused on the mathematical domain of WNA. There is broad agreement that deep understanding of school mathematics in general, and WNA in particular (for primary teachers), is critical. It follows from this that WNA provides a critical context for developing understandings and constructing arguments that adhere to the practices and norms of more advanced mathematics. One thrust of this chapter is to present and discuss examples from several parts of the world that approach this theme through a focus on primary mathematics teacher education on developing, discussing and applying mathematical models. But this thrust is set within the broader terms of different curricular approaches to WNA in different countries and regions and different cultures and structures regulating the ways in which primary mathematics teacher education (and primary teacher education more generally) is organised. In this broader curricular and cultural terrain, we take up two issues specifically. In the curricular approaches to WNA, we take up the issue of two ways of thinking about early number – in terms of length or location on a number line (where ordinality can be more emphasised, alongside cardinality) versus in terms of base ten structure (where place value structure and relationships can be more emphasised, alongside cardinality) – and use specific examples to point to number being presented in different ways in different countries. Secondly, we take up an issue that emerged from looking at the foci of different presentations at the ICMI Study 23 conference with antecedents in prior writing: of more individualist, decentralised and autonomous cultures and views of teacher learning versus more collective and centralised views of teacher learning. These latter differences have consequences for the models of teacher education that are possible in different contexts and for the ways in which teacher education is structured within the timetables of schooling.

In this chapter, we begin our discussion in the terrain of cultures associated with teacher education, before moving into curricular approaches to WNA. Key models for teacher education are then introduced and discussed to exemplify key contrasts. We then move into the detail of mathematical models that have been used within primary mathematics teacher education and use these examples to highlight attention to differences relating to length (i.e. models focusing on the ordinal aspects of number) and base ten structure (focusing on the cardinal aspects), as well as the

B. Kaur

National Institute of Education, Nanyang Technological University, Singapore, Singapore

X.H. Sun Faculty of Education, University of Macau, Macao, China

H. Venkat University of the Witwatersrand, Johannesburg, South Africa

M. Askew (discussant)

University of the Witwatersrand, Johannesburg, South Africa, and Monash University, Melbourne, Victoria, Australia

broader organisation of teacher education across these contexts. The key questions that we address in this chapter can therefore be summarised thus:

The panel consisted of six scholars representing different parts of the world, complemented by two discussants. All of them have rich experience with mathematics teacher education in their countries/regions. The panellists were (in alphabetical order) Maria G. Bartolini Bussi (University of Modena and Reggio Emilia, Italy), Sybilla Beckmann (University of Georgia, USA), Maitree Inprasitha (Khon Kaen University, Thailand), Berinderjeet Kaur (National Institute for Education, Singapore), Xu Hua Sun (University of Macau, China) and Hamsa Venkat (University of the Witwatersrand, South Africa). The discussants were Deborah Loewenberg Ball (University of Michigan, USA) and Mike Askew (University of the Witwatersrand, South Africa).

### **17.2 Cultural Views of Primary Teacher Learning and WNA**

Jarmila Novotná and Hamsa Venkat

Alexander (2009) proposes a view of pedagogy as profoundly cultural, while at the same time, exhibiting some continuities that appear to transcend place and time:

pedagogy does not begin and end in the classroom. It is comprehended only once one locates practice within the concentric circles of local and national, and of classroom, school, system and state, and only if one steers constantly back and forth between these, exploring the way that what teachers and students do in classrooms reflects the values of the wider society. (p. 924)

It follows from this that examining teacher development in the context of WNA also entails attention to broader curricular and cultural ideologies, in order to more completely understand their contents and approaches to supporting primary mathematics teaching.

Some broad differences in the emphasis of mathematics education research between the East and West have been noted in the papers presented in this ICMI study and in prior work. For example, Ma (1999) has noted the central role, and study by teachers, of carefully developed, standardised textbooks in Chinese primary mathematics education. This contrasts with a much more critical view of the standardisations inherent in textbook sequences, sometimes set within a broader culture that celebrates and emphasises the varied individualism of children's interests, understandings and learning pathways (Triandis and Trafimow 2001) and which was prevalent in much of the literature on the need for learner-centred education.

Moving into the terrain of WNA, Sun et al. (2013), comparing the ways in which addition and subtraction are presented in Chinese and Portuguese textbooks, noted the central role of textbooks in the Chinese professional education of teachers. They point to carefully varied sets of problems in which structural similarities of partwhole relations are at the fore and combinations of and with ten are collectively viewed as the key conceptual idea to be foregrounded in order for these ideas to become problem-solving tools for working with subsequent mathematical content. In base ten-oriented models of number, stick bundles or Dienes block-type manipulatives, triad diagrams representing a whole with its constituent parts and symbolic number sentences are commonly juxtaposed. Length- and location-based models using the number line predominate in early number tasks in the Portuguese textbooks in focus in this study, emphasising the direction and extent of motion associated with operations and quantities. There is less overt connection between addition and subtraction in the early examples shown in the Portuguese textbooks, with later work pointing to these as operations connected through the idea of inverse operations. Of interest is that while multiple methods are expected and promoted in the Chinese textbooks, all of these depend on ideas of decomposition, with specific selections often based around decimal structure. Beishuizen's (1993) emphasis on using number line models with compensatory steps as important higher-level efficiency moves, e.g. solving 65 – 38 with a jump backwards of 40 and a compensatory forward jump of 2, tends not to feature in the Chinese approach. The forward trajectories in the Chinese approach are seen as preparation for the focus on structure and relations needed for algebraic working, as well as for the decimal number structure. In contrast, the Realistic Mathematics Education approach emphasises the number line model as having strong associations backwards into counting and forwards into models and methods in the real number terrain (Anghileri 2006). These differences in goals and trajectories within the WNA context can be seen more broadly in the contrasts drawn between counting-based and structure-based approaches to early number (see for example Schmittau 2003).

Bartolini Bussi and Martignone (2013) have also pointed out the ways in which both mathematical models and structures had to be adapted in schooling in order to explore the 'transposition' of models beyond the ground of their cultural origins. The broader point we can make based on the ICMI study contributions is that some national traditions, more than others, foreground studies in which there is a broad focus on teaching and particular pedagogic tools and how such tools have both been developed and refined over the years. In these traditions, there are indications of greater homogeneity and 'taken as shared' views of both trajectories of mathematical content and approaches to support its learning. Backgrounded in these papers

from the ICMI Study 23 Proceedings was how teachers subsequently work with and make sense of such tools, with this occurrence perhaps reflecting the expectation of buy-in to the theories and models being promulgated, rendering a study of differences in take up less culturally normal.

The foci in these papers stand in stark contrast to those written by authors with a more typically 'Western' sensibility. For example, Askew (2015) presented a case study of a teacher in South Africa, focusing on how, in a lesson on place value, the teacher's actions, representations and discourse achieved coherence and connections. Similarly, Tempier (2013), cited in Chambris (2015), explores overlaps and contrasts in three teachers' handling of place values ideas in France, and Venenciano et al. (2015) present detailed excerpts of a teaching approach designed to support children's appropriation of ideas of unit and structural relations between units. While cardinality and place-value relations are at the fore of these studies, we draw attention to these studies here because of their common focus on to the ways in which mathematical ideas are taken up within teaching and learning.

This brief overview of culture and WNA models and approaches feeds into the range of cases that we present below dealing both with models of teacher education and mathematical models of WNA in teacher education. In some of these contexts, the professional development and mathematical models have broad cultural and historical support and are often accompanied by systemic sanction in the form of structural support for their take up in teacher development. In other countries, the cases reflect take up of approaches and models in more local initiatives, often accompanied by the need to build structural affordances for supporting take up. In each of the cases that follow, we provide an overview of the national/provincial context in relation to the extent of standardisation of curricula/textbooks and the structure of teacher education, with this data gathered from contributing authors, prior studies and the context forms that were collected with submissions for the ICMI 23 Study conference.

All themes dealt with during the ICMI Study 23 conference and in the volume are deeply linked with teacher education and development. This was both intentional and a natural property of the whole WNA domain. In the discussion document for the study conference, one of the basic questions was: how can each of the themes be effectively addressed in teacher education and professional development? In order to teach elementary mathematics effectively, there is a need for sound professional knowledge, both in mathematics and in pedagogy. In Theme 1 ('The why and what of whole number arithmetic'), the topic was explored from the perspective of teachers' mathematical content knowledge. Theme 2 ('Whole number thinking, learning and development') addressed cognitive aspects of WNA with attention paid to, among other aspects, integrating different perspectives into a more coherent view with consequences for teacher education and development. In Theme 3 ('Aspects that affect whole number learning'), teacher education played a central role and included examples of impact of the topic in teacher education and development (Canada and Thailand). In Theme 4 ('How to teach and assess whole number arithmetic'), teacher education and development were present explicitly or implicitly in all contributions, including examples of various pedagogic approaches, textbook organisation and used artefacts and a broader study explicitly acknowledging teachers' didactical or pedagogical knowledge and noting that neither teachers' extent of teaching experience nor their teacher education accounted for observed differences. Theme 5 ('Whole numbers and connections with other parts of mathematics') included the study of avenues through which whole number arithmetic learning might be supported in teacher education.

### **17.3 Primary Teacher Education Across the World**

### *17.3.1 US Experience2*

Sybilla Beckmann

#### **<IP_ADDRESS> Organisation of Primary Teacher Education in the USA**

General or local organisation: Primary teacher education is developed locally. Each state has its own guidelines or regulations for the preparation, certification and licensing of teachers. However, there are influential non-governmental accrediting bodies that operate nationally and issue standards for teacher education.

Teacher qualification: Primary teachers are usually generalists although there is growing interest in building specialist development.

Curriculum for primary mathematics: There is no national curriculum. Since 2010, the Common Core State Standards for Mathematics have been adopted by most states.

#### **<IP_ADDRESS> Key Questions About Teacher Knowledge in the USA**

A key concern in the mathematical education of primary and middle-grade teachers in the USA is how teachers can further their own disciplinary knowledge of mathematics while also studying deeply the mathematics that they will teach. According to a report issued jointly by national mathematical societies in the USA (Conference Board of the Mathematical Sciences 2012), teachers should know the mathematical topics they will teach and how these topics connect to others in earlier and later grades. Teachers should also know the ways of reasoning and constructing arguments in mathematics, how these ways of reasoning and argumentation apply at the elementary level and how to teach these ways reasoning and argumentation to students.

2Research was supported by the National Science Foundation under Grant No. DRL-1420307. The opinions expressed are those of the author and do not necessarily reflect the views of the NSF.

In the USA, the *Common Core State Standards for Mathematics* document (Common Core State Standards Initiative 2010), which has been adopted by most states, describes standards for mathematical practice for students in kindergarten through Grade 12. In particular, according to these standards, mathematically proficient students should be able to understand and use stated assumptions, definitions and previously established results in constructing arguments, and they should try to use clear definitions in discussion with others and in their own reasoning. Therefore even students in elementary school are expected to understand suitable definitions and use them in explaining and arguing for the validity of mathematical statements.