Mulaudzi (2016) highlights the significance of language for effective mathematical teaching and learning. He emphasises that the language of learning and teaching (LoLT), through which mathematics is learned, creates the basis for mathematical teaching and learning to evolve in that language. Extensive research confirms that many language qualities influence cognitive development (Barwell 2016; Das 2020; Kotzé 2016; Robertson & Graven 2020; Sfard 2012). Despite the seeming complexity and inconsistency of the literature on the relationship between language and mathematics, there is a consensus that thinking occurs in some language or another (Robertson 2017). Vygotsky (1962) stressed that language and thought are inextricably linked and interdependent. Therefore, it can be argued that external talk brings a thought to life, whereas inner speech focuses energy on words to facilitate thought processing. In light of the latter premise, language influences the thought process. However, the act of a thought has its own structure, while the act of translating the thought into language is challenging. This is not an unconscious process because thought only develops and is expressed through language. Signs outside mediate thought, while word meanings internally mediate thought (Vygotsky 1962). Bruner (1975) highlights the significance of language as a medium for thought as well as its influence on cognitive development. Consequently, thought is tightly bound to language and must adapt to it (Robertson 2017).

The mathematics register is a distinct manner of using language and communicating information (Lee 2006; Pimm 1987). Thus, mathematical language is considered as a distinct ‘register’ inside an everyday language (e.g. English – ‘any language that has evolved spontaneously in humans via usage and repetition without conscious design or premeditation’) (Le Cordeur & Tshuma 2019:107). In addition to the specialised vocabulary used in mathematics, the mathematics register also includes the language used to describe or explain a specific sort of mathematical problem (Jourdain & Sharma 2016; Le Cordeur & Tshuma 2019). On the other hand, the mathematics register encompasses a great deal more than just lexicon and technical terms. It also includes phrases, words, and reasoning approaches in a specific context, all of which are expressed using everyday language (e.g. English) (Pimm 1987). The grammatical structure and lexicon of a specialised language (i.e. mathematics) allow for the communication of a wide range of concepts. Consequently, each language has its own mathematical register, which incorporates the various methods in which mathematical meaning is communicated in that language (Ní Ríordáin, Coben & Miller-Reilly 2015; Le Cordeur & Tshuma 2019).

As can be seen, the complicated ‘register’ of mathematics is comparable to that of a language, and, as such, it needs similar language-acquisition skills. In line with this perspective, Robertson and Graven (2019) argue that mathematics teachers must discover ways to incorporate second-language teaching and learning strategies into their pedagogical repertoires to increase inclusivity in mathematics classrooms. This adds a new layer to mathematics education and reinforces the view that mathematics content cannot be taught without language. Learning mathematics comprises the acquisition of the mathematics register automatically (Setati 2005). According to Meaney (2005), this enables learners to convey their mathematical reasoning with ease; ‘without this fluency, learners are limited in their ability to develop or rearrange their mathematical understandings’. Once they have mastered the mathematical register, learners will be able to listen, ask questions, and converse, as well as read, record, and engage in mathematics. Similarly, registers can be utilised in a variety of domains, as well as in any LoLT. Often, subtly, mathematics and ordinary language registers may be inhibited in an educational environment. Learners must therefore be able to recognise each register in order to decide which is being used at any given time, which is a difficult task for many English language learners (ELLs) in multilingual classrooms such as South Africa. Thus, it is essential to understand how different languages and registers function, as well as the fact that the usage of numerous languages and registers can aid a learner’s improvement in mathematical understanding (Ní Ríordáin et al. 2015).

Mathematics is a discourse and a type of communication, which is more than just language (Sfard 2012). Rymes (2016:5) describes discourse as ‘language in use’. Gee (1996) defines discourse as:

According to this explanation, discourses involve more than the use of specialised terminology and spoken or written language. Communities, viewpoints, values, and beliefs all contribute to discourse in this regard (Ní Ríordáin et al. 2015). Moschkovich (2012a:95) defines mathematics discourse practices as ‘social, cultural, and discursive because they come from communities and mark membership in different discourse communities’ to emphasise the notion that discourses are anchored in sociocultural practices. Furthermore, mathematical discourses are cognitive in character since they make use of thought, signs, tools, and meanings. Words, phrases, and texts have different meanings, uses, and intentions depending on the context. Mathematical discourses occur within the framework of practices that are connected to communities. Moschkovich (2012a) continues by stating that mathematical discourse practices are generated through actions, sense-making, focus of attention, and aims, and are anchored in social practices. As a result, multi-semiotic systems are included in mathematical discourse (e.g. speech, text, gestures, symbols, and visual images) (Arzarello et al. 2009; Robertson & Graven 2020).

The difficulty of functioning within English language registers is one of the most significant obstacles that ELLs experience while attempting to learn mathematics in an English-LoLT classroom (Robertson & Graven 2020). Halliday (1978) defines the mathematical register as a set of meanings suitable for a particular linguistic feature, along with the terminology and structures that represent these meanings. Thus, the register of mathematics is a sense of mathematical language-specific meanings. In mathematics classrooms, multiple registers are utilised. To succeed in mathematics, learners must not only be comfortable with and educated about their everyday English register, but also be fluent in several mathematical registers (Essien 2018; Moschkovich 2005; Schleppegrell 2011). For the understanding of mathematical registers and the ability to switch between them, strong language and metalinguistic skills are necessary. These skills are necessary for learners to communicate with their peers and cope with more complex mathematics (Mandy & Garbati 2014; Planas & Setati-Phakeng 2014; Robertson & Graven 2020).

As a new type of language that must be taught and mastered, mathematical registers provide a major challenge for ELLs (Setati 2008). This is a specific challenge in South Africa, where mathematics achievement is inadequate due to ELLs’ inadequate language skills, as they must master a specialised mathematical language to comprehend the mathematics curriculum’s content (Department of Basic Education 2011; Mulaudzi 2016; Sibanda & Graven 2018; Robertson & Graven 2020). The ELL must not only seek to learn in English while also learning to speak English but must also operate within the English mathematical register before mastering everyday English (Essien 2018; Mandy & Garbati 2014; Robertson & Graven 2020). Therefore, in order to fully understand mathematics in a classroom where English is the LoLT, ELLs may require more processing time compared to English mother-tongue speakers (Mulaudzi 2016; Setati 2008). Consequently, these learners may fail to keep up in mathematics because they spend too much time attempting to understand and switch between registers (Jourdain & Sharma 2016; Machaba 2018).

Furthermore, understanding mathematics-specific vocabulary is required for learners to be proficient in mathematics (Moschkovich 2005; Mulaudzi 2016; Robertson & Graven 2020). This mathematical vocabulary is not used in everyday English registers; hence, speakers of other languages have difficulty understanding it (Jourdain & Sharma 2016). Not only are the technical words and vocabulary of mathematics crucial for learners’ capacity to comprehend and interpret mathematics, but they also have a significant impact on their future mathematical development (Robertson & Graven 2020). Learning to operate in different registers poses significant difficulties for ELLs in this regard. This is particularly true when learning vocabulary because the meaning of words varies across registers (Mulaudzi 2016; Robertson 2017).

Regarding mathematical teaching and learning, learners already have a vocabulary. Although some LoLT terms may be known to ELLs, they may have different connotations in the context of mathematics (Morgan et al. 2014). Machaba (2018) contends that non-proficient learners of the LoLT will struggle to distinguish between the correct mathematical meaning of a common word and everyday English phrases. ‘Volume, multiplication, parallelogram, operation, even, and odd’ are examples of these terms (Ní Ríordáin et al. 2015:14). Therefore, it is crucial that both the learner and the teacher explore the numerous meanings and interpretations of mathematical terminology for them to comprehend the intended meanings and language patterns (Mulwa 2014). In addition to the terminology used in everyday English, mathematics includes several of its own unique terminologies (Le Cordeur & Tshuma 2019). Therefore, if learners wish to be successful in mathematics, they must learn these terms, their basic mathematical meanings, and how to use them in different situations.

The phenomenon of multilingualism can be observed in classrooms all around the world. However, only major regional or national languages are used in the majority of classrooms, often for pragmatic or political reasons (Gandara & Randall 2019). In the multilingual setting of South Africa, the presence of historically multilingual communities is seen in these schools (Barwell et al. 2016; Gandara & Randall 2019). In spite of this, the majority of South African learners study mathematics in English, as this is the dominant language that the learners’ parents ‘buy into’ as the medium of teaching (Robertson & Graven 2020). Consequently, there are many learners in South African schools who are not proficient in the LoLT, which is often English (Dale 2015; Robertson 2017; Robertson & Graven 2018).

Mulaudzi (2016) and Essien (2018) assert that English classrooms where mathematics is not taught in the learners’ home language (HL) are contexts in which there is a need for meaningful pedagogy on mathematics learning as well as the development of specialised skills that provide learners with a variety of opportunities to learn mathematics in such classrooms. In this context, Essien (2018) explicates Cummins’s (1979) claim that constructive cognitive development is achieved when a learner reaches a particular level of linguistic competence in a second language. For this reason, it can be assumed that language-deficient learners will be deprived of cognitive development. Furthermore, Essien found that learners in South African classrooms whose HL differs from the LoLT are unfamiliar with the linguistic structures encountered during mathematics instruction. Learners need to familiarise themselves with the structure of the mathematical language. This, however, necessitates that these learners understand both mathematical concepts and the language in which they are embedded. As a result, teachers are challenged to find a balance between English as the LoLT and the language of mathematics.

Both Vygotsky’s (1978) learning theory and Kilpatrick Swafford and Findell’s (2001) five-stranded model of mathematical proficiency, which place emphasis on the relationship between cognitive development and the general development of children’s mathematical proficiency, were used to interpret the empirical part of this research.

The mathematical proficiency model developed by Kilpatrick et al. (2001:5) consists of five ‘intertwining strands’ of mathematical proficiency, namely: ‘conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition’. Ramollo (2014:14) defines these strands as ‘various strands that incorporate knowledge about the interaction between the teacher, the learner, and the content and embrace the context for learning mathematics successfully’. Thus, each strand should address the needs of ELLs (i.e. isiXhosa HL-speaking learners) to become proficient in mathematics. Teaching for mathematical proficiency, on the other hand, necessitates the same interconnected strands from both teachers and learners. Thus, in order to teach for mathematical proficiency, a teacher must have the following skills:

The five strands of mathematical proficiency of Kilpatrick et al. (2001) and the learning theory of Vygotsky (1978) were combined to fit the context of South Africa since the majority of South African learners enter school speaking a language other than the LoLT, which presents a challenge for mathematics teaching and learning to most learners (Essien 2018; Robertson & Graven 2019).

A fundamental connection between Vygotsky’s (1978) theory of learning and Kilpatrick et al.’s (2001) five strands of mathematical proficiency is the development of cognitive competencies. The learning theory incorporated within Kilpatrick et al.’s five strands of mathematical proficiency is particularly applicable to this research since learners with inadequate LoLT fluency may regard language as a barrier to comprehending and becoming proficient in mathematics (Robertson & Graven 2020). According to Prediger et al. (2019), intellectual resources, such as language or other physical resources, serve an important role in teaching and learning for both communicative and conceptual reasons. Within each strand, Vygotsky’s theory of learning demonstrates ‘the relevance of language in learning and cognitive development’. In this regard, the teacher applies intellectual and physical resources (depending on the needs of the learner) as scaffolds (support strategies) to enhance learners’ mathematical understanding (Presmeg et al. 2016).

The researchers conducted this study in 2021 at public and independent primary schools in the Western Cape, Metro East Education District. The participating teachers ranged from newly qualified teachers to more experienced teachers teaching mathematics in English-LoLT Grade 1 classrooms. Although the LoLT of the classrooms is English, more than a quarter of the learners speak isiXhosa as their HL, of which 41% is the highest percentage of learners whose HL is not that of the classrooms’ LoLT. The teacher-learner ratio ranged between 1:29 and 1:37.

All Grade 1 teachers in the Western Cape constituted the study’s population. Using the technique of purposive sampling, the researchers selected four schools and 11 Grade 1 teachers (nine teachers representing three public primary schools and two teachers representing one independent primary school) as participants for this study (Creswell 2018; Cohen, Manion & Morrison 2018; Okeke & Van Wyk 2015). The participants were selected so as to offer the researchers a rich and comprehensive narrative of their perspectives, experiences, and feelings regarding their understanding of the language of mathematics in a South African context (Cohen et al. 2018; Mohajan 2018).

The researchers acquired their data using an adapted IQA data collection method. In this sense, original IQA research often collects data using two techniques, such as unstructured, open-ended focus group interviews and semi-structured individual interviews (Northcutt & McCoy 2004). Nonetheless, the researchers introduced a third data collection approach, namely field observations of mathematics lessons. Due to the COVID-19 pandemic’s restrictions on social distancing, we conducted two separate face-to-face, unstructured, open-ended focus group interviews instead of just one, with a total of 11 Grade 1 teacher participants (i.e. six Grade 1 teachers participated in Focus Group 1, and five Grade 1 teachers participated in Focus Group 2). These unstructured, open-ended focus group interviews informed the interview framework. The interview framework then guided the researchers’ data collection through the face-to-face mathematics lesson observations of 6 of the 11 individual participants (i.e. three Grade 1 teachers from Focus Group 1 and three Grade 1 teachers from Focus Group 2). After that, the interview framework helped guide the six semi-structured individual interviews, which took place online with the same six teachers whose mathematics lessons were observed.

The data analysis process consists of three distinct steps. Figure 1 depicts the data analysis process to show the relationship between the three processes.

The 11 participants of the unstructured, open-ended focus group interviews were involved in the first step of data analysis. During this interactive session, the participants of the focus group generated and recorded on index cards (inductive analysis) their perceptions, experiences, and feelings regarding the research statement (based on the sub-research question), namely: ‘Tell me what you think or feel or call to mind when I use the term “mathematical language”’. The brainstorming exercise was followed by a deductive analysis exercise in which the participants sorted and clustered the written cards into groups that represented the topics. The written ideas were accompanied by descriptive paragraphs that served as the framework for the interview.

The second step of the data analysis consisted of analysing the transcribed semi-structured individual interviews using John Stuart Mill’s analytic comparison technique as an analytical tool to detect patterns among themes (Neuman 2014). The third step entailed analysing the transcripts of the field observations of mathematics lessons again, using Mill’s analytic comparison technique (Neuman 2014) as an analytical tool to get a chronological understanding of what transpired in each participant’s mathematics lesson. The data analysis of the individual interview transcriptions was compared to the data analysis of the field notes from the mathematics lessons to create a synthesis. After gathering and analysing the data, the researchers performed a literature check, using recent and pertinent research as a backdrop for articulating and formulating the findings. The data was verified using Auberbach and Silverstein’s (2003) criterion of transferability and Maxwell’s five categories of validity in qualitative research (Maxwell 2016). The descriptive validity of the data utilised in this study was ensured using IQA interviewing techniques, data-gathering techniques, and member checking. Along with the descriptive validity, the interpretative validity includes the focus group interview members contributing their own data and analysis. Additionally, the verbal and non-verbal replies acquired from the participants to validate the information they provided during the focus group interviews were incorporated into the transcripts of the individual interviews and classroom observations. The researchers conducted a literature review to establish the theoretical validity after the focus group interview participants had identified the themes and sub-themes. Literature and direct quotations were used to support the themes and sub-themes. Consequently, conclusions were reached based on the findings of the data and the literature. Inferences were drawn from the data analysis, literature review, and theoretical framework to confirm the evaluative validity.

The researchers sought ethical approval from the Cape Peninsula University of Technology (CPUT) and the education authorities (Western Cape Education Department [WCED]) before beginning the investigation. The participants were given informed consent forms to sign and the researchers made it clear that their involvement was voluntary and that they could withdraw from the processes at any time. Participants remain anonymous and are simply identifiable by a number, for example Participant 1. Audio recordings and transcripts will be preserved in a secure safe for at least 5 years after the completion of this research.

By employing Mill’s analytic comparison technique (Neuman 2014), the analysis of the two focus groups’ themes revealed the first pattern: the researchers found that both themes had similar content or flash cards, but they had different names, namely ‘Mathematical language’ and ‘The language’. Thus, the researchers decided to group together the two themes as they had similar or complementary content (method of agreement) into one category, namely: ‘Mathematical language’. The theme, cards (thoughts), and descriptive paragraph indicated by the 11 teacher participants in both focus group interviews are shown in Table 1 and Table 2.

The semi-structured individual interviews were analysed to look for patterns. The researchers analysed the individual interview transcripts from the six participants individually to determine the following:

As previously mentioned, the category ‘Mathematical language’ was developed by grouping the themes ‘Mathematical language’ and ‘The language’ derived from the two focus group interviews (see Table 1 and Table 2). When the researchers deliberated on the category ‘Mathematical language’, the six participants, based on their personal perceptions, experiences, and feelings, expressed a variety of perspectives on their understanding of mathematical language. Participants’ responses were as follows:

Le Cordeur and Tshuma (2019) contend that mathematical language is perceived as a distinct register within an everyday language (e.g. English), and that learning mathematics entails the acquisition of the mathematics register itself (Setati 2005; Sibanda & Graven 2018). Therefore, the isiXhosa HL-speaking learner must learn both the LoLT, which is English, and the language of mathematics at the same time (Sibanda & Graven 2018).

Two of the participants agreed that mathematics has its own terminology, which is in line with the participants’ statements above. However, they stressed that mathematical terminology is difficult:

According to Jourdain and Sharma (2016) and Le Cordeur and Tshuma (2019), the specialised vocabulary employed in mathematics is one characteristic of the mathematics register. This unique terminology is used to describe or explain a specific type of mathematical problem. Ní Ríordáin et al. (2015) assert that learners from various cultural or linguistic backgrounds frequently struggle to comprehend mathematical problems due to a lack of familiarity with these terms.

In addition to the quotes provided by the participants thus far, they also acknowledged the significance of language to the understanding of mathematics. Mulaudzi (2016) argues that the LoLT in which mathematics is learned creates the foundation for the development of mathematical instruction and learning in that language. To be successful in mathematics, learners must not only be comfortable with and well informed of the everyday English register (i.e. LoLT), but also be fluent in numerous mathematical registers (Essien 2018; Moschkovich 2005; Schleppegrell 2011). Three participants emphasised the significance of comprehending the LoLT when introducing mathematical ideas:

This demonstrates that language (both the LoLT and the language of mathematics) plays a role in fostering mathematical comprehension (Barwell et al. 2016; Das 2020; Presmeg et al. 2016; Robertson & Graven 2020; Vygotsky 1962).

In addition, a participant’s response validated Le Cordeur and Tshuma’s (2019) assertion that the language of mathematics necessitates similar pedagogic procedures to those employed when learning any other language. Participant 9 stated the following:

Participant 9’s perspective above is consistent with Robertson and Graven’s (2019) argument that, in order to promote inclusivity in mathematics classrooms, mathematics teachers must find ways to incorporate second-language teaching and learning strategies.

In conclusion, the method of agreement reveals that all individual interviewees agreed that mathematics has its own language with its own vocabulary (i.e. mathematical terminology). In addition, they have agreed that language plays an essential role and must be simplified for mathematical understanding. The method of difference, on the other hand, showed that only one person thought that mathematical language could be learned in the same way as another language.

As one of the aims of this research study was to explore, describe, and understand what Grade 1 teachers’ understandings of mathematical language are in a South African context, the researchers could not obtain a full picture of the category ‘Mathematical language’ by just observing Grade 1 teachers teaching mathematics. Nevertheless, to a certain extent, the method of agreement revealed that the participants used visual representations for mathematical concepts, which form part of the language of mathematics. In this regard, all the teacher participants displayed mathematical information on a wall with all the mathematical terms together with pictures so that learners who speak isiXhosa could understand the mathematics vocabulary.

Moreover, the researchers found that Participants 1, 7, and 9 made the mathematics terms easier to understand by exposing learners to different words that meant the same thing. Participants 5 and 11 made the language easier when they introduced word sums, for example, by using simple, everyday English that was relevant to learners’ everyday lives. Participant 3 made the language easier to understand by giving learners the chance to revise terminology as a group before they did an individual activity about the same thing. The observations of these activities are displayed in Table 3.

Participant 9 asserted that mathematics can be taught and learned in the same way as any other language can be. In this instance, the method of difference revealed that all the participants were using various language strategies (e.g. semiotic systems), which are also used in second or third language acquisition. For example, all the teachers employed visual representations to help learners learn the language and concepts of mathematics.

Through analysing the transcripts of the classroom observations for the category ‘Mathematical language’ and comparing them to the transcripts of the individual interviews, the method of agreement showed that all six participants agreed that mathematics is made up of different terminology. They also agreed that language is a key part of understanding mathematics, which is why the participants tried to make mathematical terms easier to understand. During the analysis of the individual interview transcripts, the method of difference showed that only Participant 9 mentioned that strategies for teaching mathematics in a second language could be used. In line with what Participant 9 said above, it was noticed that all the other interview participants used second-language didactic strategies in their mathematics lessons without even realising it.

According to the data in Table 4, the category ‘Mathematical language’ comprises the theme ‘The language (of mathematics)’. All participants had a firm grasp of the concept that mathematics has its own register (mathematical terminology) and should be learned like any other language. In this regard, the participants asserted that the language of mathematics is a new language that both English-speaking and isiXhosa-speaking Grade 1 learners must learn. However, participants believed that isiXhosa learners lag behind English-HL learners because they must first comprehend the language in which mathematics is taught (i.e. English LoLT) before they can make sense of mathematics itself (Mulaudzi 2016; Sibanda & Graven 2018). In addition to the fact that mathematics has its own register, two participants stressed that mathematical terminology can be challenging for some learners because these terms are unknown to those from other cultural or linguistic backgrounds (Ní Ríordáin et al. 2015). As a result, the participants acknowledged the importance of language in mathematical understanding and emphasised the importance of understanding the English language when mathematical concepts are introduced (Barwell 2016; Bruner 1975; Das 2020; Moschkovich 2012a; Presmeg et al. 2016; Robertson & Graven 2020; Sfard 2008; Vygotsky 1962). Consequently, participants stressed the importance of simplifying (scaffolding) mathematical language for better understanding. They also emphasised the need for learners to hear both the English LoLT and the language of mathematics regularly in order to grasp and apply them when mathematical concepts are introduced (Essien 2018). Another participant was of the opinion that the language of mathematics necessitates the same pedagogical techniques utilised when learning any other language (Le Cordeur & Tshuma 2019; Robertson & Graven 2019).

According to Vygotsky (1931), language is associated with cognitive development and occurs in a sociocultural environment. In addition, based on Vygotsky’s theory of learning, Moschkovich (2012b) claims that the language of mathematics refers to communicative skills that are necessary and adequate for participation in mathematical discourse. In this context, Moschkovich (2012a) argues that the idea of ‘language’ must be expanded to include the interaction of multi-semiotic systems engaged in mathematical communication (i.e. natural language, mathematics symbol systems, and visual representations). Therefore, the processes of language development and conceptual mathematical understanding are interdependent and interconnected (Council of the Great City Schools [CGCS] 2016). To support isiXhosa HL-speaking learners in becoming proficient in English for mathematical learning, teachers must provide specific scaffolds (i.e. multi-semiotic systems) during mathematical discourse to match isiXhosa HL-speaking learners’ linguistic demands, thus fostering their acquisition of English mathematical proficiency (CGCS 2016; Walshaw 2017).