Abstract
Background: Fluency in mathematical language is essential for learning mathematics. Teachers must understand and use their diverse mathematical knowledge, including language and communication difficulties inherent to mathematics instruction. According to recent South African research, Grade 1 teachers are not equipped to utilise learners’ linguistic skills for efficient learning of mathematics.
Objectives: This research investigates South African Grade 1 teachers’ mathematical language perceptions, experiences, and feelings. These Grade 1 teachers’ transcripts were analysed to discover their understanding of the language of mathematics.
Method: Exploratory, descriptive, and contextual research designs were used in conjunction with an adapted interactive qualitative analysis technique. Focus group interviews, individual interviews, and lesson observations, together with a purposive sampling technique, were used to gather the data from both public and private primary schools.
Results: The results showed that Grade 1 teachers view mathematics as a separate language with its own vocabulary and register. The findings highlighted the need to simplify the language of mathematics to enhance understanding.
Conclusion: This research concluded that language is essential to mathematics learning and that mathematics has its own register, which is acquired like any other additional language. To help isiXhosa learners understand mathematics in English, scaffolding strategies must be aligned with their linguistic demands.
Contribution: This article provides important recommendations for teachers who need to recognise the reality that English is the lingua franca and ensure isiXhosa home language-speaking learners receive the necessary support to acquire actual proficiency in the academic register of English for mathematical language learning.
Keywords: English Grade 1 classrooms; Grade 1 teachers; isiXhosa home language-speaking learners; mathematical language learning; scaffolding strategies; South African context; understanding of mathematical language.
Introduction
One of the primary reasons for South African learners’ poor performance in mathematics is that the significant role of language in education is overlooked in curriculum and teacher-training courses, resulting in a lack of language awareness. As a result, inadequate teaching strategies lead to language difficulties in curriculum areas such as mathematics (Robertson & Graven 2020:86). One of the crucial imperatives that was taken on by the South African government in 1994 was language planning. In this regard, as a way for the government to carry out this responsibility, the Department of Basic Education agreed to draft linguistic policies and identify strategies to improve the state of African languages (Department of Basic Education 2010; Yu & Dumisa 2015). In order to promote and develop African languages, the Pan-South African Language Board (PanSALB) was subsequently established. The PanSALB was founded as a statutorily mandated authority to oversee and enforce the implementation of all 11 official languages of South Africa. The PanSALB has organised lexicography components for each official language to ensure terminology uniformity and promote multilingualism (Mtsatse & Combrinck 2018:22). However, despite the importance of language planning on the reunification agenda, it was evident that much of the African language terminology had not been updated since the apartheid era and is still in use today (Mtsatse & Combrinck 2018:22). Moreover, according to Webb (2013), the condition of African languages has not changed significantly since 1996. Mtsatse and Combrinck (2018:22) claim that the government has produced much too little evidence that South Africa’s language policy has effectively promoted the ‘graphisation’ and ‘codification’ of African languages. Mtsatse and Combrinck assert that African languages have not been used meaningfully outside of formal schooling frameworks, such as in parliament, courts, academic institutions, schools, and the media in general. Due to the absence of official social systems in which African languages are utilised, it appears that Afrikaans and English maintain a higher linguistic status than African languages (Mtsatse & Combrinck 2018). Van Laren and Goba (2013) and Mtsatse and Combrinck claim that the standardisation of African languages is an ongoing difficulty because it is difficult to locate precise, widely accepted vocabulary for mathematics translated from English into an African language. It seems unlikely that academic registers in all African languages will be established in the near future, especially due to the influence and status of English as the lingua franca (Essien 2018; Robertson & Graven 2020). Consequently, concerns regarding equal educational opportunities to become a successful academic participant in mathematics and the necessary language skills required to maximise learners’ potential to really make sense of mathematics still persist (Engelbrecht et al. 2016; UNESCO 2016; UN 2018).
Significance of the study
Robertson and Graven (2020) claim that sustained and systematic investment is crucial in devising strategies to ensure that African language-speaking learners are provided the necessary support to achieve actual proficiency in the academic register of English for mathematical learning. Recent research, however, shows that there is a gap between what ‘good practice’ might look like in South African classrooms, building on what already happens in them, and the reality of these classrooms and how teachers perceive and deal with African language-speaking learners within their classrooms (Essien 2018; Mtsatse & Combrinck 2018; Robertson & Graven 2020).
This article is derived from a full doctoral thesis. However, as the aim of this research was to explore, describe, and understand what Grade 1 teachers’ understandings of mathematical language are in a South African context, the researchers only report on one of the sub-research questions, namely: ‘What are Grade 1 teachers’ understandings of mathematical language in the context of South Africa?’ In answering this research question, we add to a body of knowledge from which recommendations for Grade 1 teachers and their understanding of mathematical language were developed, allowing them to provide the necessary mathematical language support to isiXhosa home language-speaking learners receiving mathematics education in an English Grade 1 classroom.
The article begins with an introduction, followed by an explanation of the significance of the study, a literature review, and a theoretical framework. The research design and method are described, followed by the results and deliberation of the findings. The article concludes with a conclusion and recommendations.
Literature review
Understanding the relationship between language and mathematics
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:
‘A socially accepted association among ways of using language, other symbolic expressions, and artefacts of thinking, feeling, believing, valuing, and acting that can be used to identify oneself as a member of a socially meaningful group or social network, or signal [that one is playing] a socially meaningful role.’ (p. 131)
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).
Mathematical registers
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).
Everyday English versus mathematical English
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.
Teaching and learning mathematics in South African classrooms
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.
Theoretical framework
The interconnection between Vygotsky’s learning theory and the five strands of mathematical proficiency
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:
- conceptual understanding of the core knowledge of mathematics, learners, and pedagogical strategies required for teaching;
- procedural fluency in carrying out basic instructional procedures;
- strategic competence in planning effective instruction and solving problems that arise while teaching;
- adaptive reasoning in rationalising a decision; and
- a positive attitude (productive disposition) toward mathematics, teaching, learning, and improving mathematics practice.
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).
Research design and method
The researchers conducted an interpretive qualitative case study using Northcutt and McCoy’s (2004) interactive qualitative analysis (IQA) systems method to address the sub-research question: ‘What are Grade 1 teachers’ understandings of mathematical language in the context of South Africa?’ With the help of unstructured, open-ended focus group interviews, semi-structured individual interviews, and lesson observations, the researchers were able to identify a population, select a sample from that population, and then explore, describe, and understand teachers’ understandings of mathematical language in current mathematics practices in the context of South African English Grade 1 classrooms.
Setting
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.
Participants
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).
Data collection
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.
Data analysis
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 |
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