This section briefly describes what I mean by mathematical language. It also presents a discussion about the implications of mathematical language over the teaching and learning mathematics in multilingual classrooms. Furthermore, the section also discusses other issues such as the formal and informal mathematical language that needs to be considered in a mathematics classroom.
Pimm (1991) explains that there are many different relationships that can be highlighted between mathematics and language. Mathematics has its own register (Halliday, 1975; Pirie, 1998), rules, grammar, syntax, vocabulary, word order, synonyms, negations, conventions, abbreviations, sentence structure, and paragraph structure (Esty & Teppo, 1994, p. 1). Halliday (1975) specifies the notion of register as ‘a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings’. Lee & Fradd (1998) explain that appropriate use of key mathematical terminology is an indicator of the precision and sophistication of understanding. Therefore, part of learning mathematics is gaining control over the mathematics register so that one is able to talk like a mathematician (Pimm, 1991).
The mathematical language adds on to the complexity of teaching and learning mathematics in multilingual classroom in different ways. For example, mathematical language differs from the ordinary language in different ways. Mathematical language
has certain language features, for example, that cannot be matched with other languages. Halliday (1975) gives an example that “four from six leaves two” when interpreted is “6 – 4 = 2”. In addition, mathematical language includes everyday vocabulary that takes on a different meaning in mathematics; for example, words like set, point, table, and altogether (Halliday, 1975). Learners are expected to know and become familiar with this type of language, which they have to learn from the mathematics teachers in their classrooms.
Moreover, Morgan (1998) and Pimm (1991) explains that, while mathematics, when spoken, emerges in a natural language, when written, it makes varied use of a complex, rule-governed writing system mainly separate from that of the natural language into which it can be read. Such mathematical encoding includes symbol order, position, relative size and orientation (Pimm, 1991). Morgan (1998) calls this “writing system” as “mathematical academic writing” (p. 11). Which means that, teachers in a mathematics classroom have the duty of helping their learners to write mathematically that is, using symbols in a correct order. Furthermore, learners may attempt to read, write and understand the mathematical sentences in the same way that they read, write and understand standard narrative text. Learners may try to translate word by word between mathematical concepts and, in most cases, in a linear translation. One-to-one linear translations are not always appropriate since the way some mathematical concepts are expressed in words differs in its order from the way the concept is expressed in symbols. For example, the number a is five less than the number b, which the learner may mistakenly restate as a = 5 – b when it should be a = b – 5 (Jarrett, 1999).
Furthermore, mathematical concepts sometimes are made up of the relationship between two words, which are hard to understand and at the same time require the use of symbols in solving the problem. For example, phrases like ‘all numbers greater/less than X’. In the context of mathematics, “symbols can help to show structure, allow routine manipulations to become automatic and make reflection possible, by putting thoughts that one has with some stability, compactness and permanence, as objects which may be examined” (Pimm, 1991, p. 19). However, Pimm argues that the ‘concreteness’ of the symbols and the absence of obvious mathematical objects to act as
referents can lead many pupils to believe that the symbols are the mathematical objects. The technique of describing algorithms in terms of attributes of the symbols adds to the potential confusion.
Apart from the need for learners to be skilled in mathematical vocabulary and the mathematical writing system, learners will also be required to know the logical connectives (Dawe, 1983) in mathematical language. Mathematical language is mostly linked with connectors such as if... then, if and only if, because, and either... or which signal relationships between parts of a mathematical text. These words signal similarity or contradiction, cause and effect, reason and result, chronological or logical sequence (Jarrett, 1999). These words also serve to link propositions in reasoned argument (Dawe, 1983). Dawe states that, knowledge of logical connectives is so important, more especially for achievement on a mathematical test. Therefore, Dawe argues that, the development of the ability to use logical connectives for reasoning and argument is an important task for mathematics and science teachers. Thus mathematics teachers have an important task of helping their learners develop the ability to be able to use and interpret these logical connectors.
As can be seen from the mathematical language alone, teachers have an enormous task in trying to get their learners to learn mathematics, thereby accomplishing their education objectives in a mathematics classroom. These challenges take on added significance in the context where the language of learning and teaching is not the home language of both the learners and teachers. Most of the things mentioned in the preceding section are easily done if the LoLT is the home language of both the learners and the teachers. However, in most African classrooms and Malawi in particular, the LoLT is English which makes the teaching of mathematics even harder. There are many issues that emerge as they teach and that should be of concern. One of the issues is how mathematics teachers can make mathematics more comprehensible to their learners’ more especially those whose home language is different from the LoLT.
As pointed out earlier, it is not only the mathematical language that matters in a multilingual classroom. There is also an issue of learners moving from informal to
formal mathematical language that needs to be considered in a mathematics classroom as discussed in the following section.
Moving from informal to formal mathematical language
In a mathematics classroom, mathematics learning involves both informal and formal components (Setati & Adler, 2000). The language that people use to express mathematics in their everyday life is referred to as informal mathematical language and the standard use of terminology developed within a formal setting as formal mathematical language. Pimm (1987) explains that learners do not commonly explicitly hear or read much mathematics outside the classroom and so the mathematical language that they bring to a mathematics class is informal. In school settings, it is the formal mathematical language that is valued. Therefore, learners need to learn the distinction between the informal and formal way of talking mathematics. Learners learn formal mathematical language in a mathematics classroom through their mathematics teachers. In this case, it is the mathematics teacher in a mathematics classroom who acts as a model of how to speak mathematically for the learners. Hence one thing that a learner does in a mathematics classroom is to learn a range of accepted ways in which mathematics is to be communicated and discussed through their mathematics teacher. This presents mathematics teachers with a big task of taking the learner from an informal way of talking mathematics to a formal one. The challenge therefore, is for teachers to help learners to move from the use of informal to formal mathematical language in a mathematics classroom.
Pimm (1991) explains the two levels of which mathematics teachers may help their learners to move from informal mathematical language to formal mathematical language: to encourage learners to write down their informal mathematical language and then work on this language to formal mathematical language; and to work on the spoken informal language to a formal spoken language and then formal written language. However, Setati & Adler (2000) suggest that movement from informal to formal mathematical language in a multilingual classroom may go through three routes:
from spoken to written language; from main language to English; and from informal to formal mathematical language. This is shown in figure 3.1.
Figure 3.1: Alternative routes from informal spoken (in main language) to formal written (in English) mathematical language (Adapted from Setati & Adler, 2000, p. 250)
Setati & Adler (2000) continues to argue that one way is to encourage learners to write down their informal utterances in the main language, then write them in informal mathematical English and finally to work on making the written mathematical English more formal. In this case, the mathematics teacher works first on learners’ writing their informal mathematical thinking in both languages, and thereafter on formalizing and translating the written mathematics into the LoLT. Another possibility is to work first on translating the informal spoken mathematical language into spoken English and then to work on formalizing and writing the mathematics. Setati & Adler (2000) continue to argue that, while formal written mathematics in the learners’ main language(s) is possible, there is a variety of reasons why most multilingual teachers would not work on formalizing spoken and written mathematics in their main language because of (i) the mathematics register is not well developed in most of the African languages and (ii) due to the dominance of English this would generally be seen/interpreted as a waste of time. This reflects the huge work that mathematics teachers have in a multilingual
Informal spoken mathematics main language Formal spoken mathematics main language Informal spoken mathematics English LoLT Formal spoken mathematics English LoLT Informal written mathematics main language Formal written mathematics main language Informal written mathematics English LoLT Formal written mathematics English LoLT
classroom, helping learners to be able to use mathematical language in a language that is not their home or first language, and at the same time, they should be helped to move from informal to formal way of talking mathematics.
Moschkovich (1999) explains another way of how a teacher in a bilingual mathematics classroom in the USA supported the mathematical communication of his learners. The teacher supported the learners by revoicing, interpreting and rephrasing what learners were saying. For example, in the class that she was conducting her study, the teacher asked the learners to tell her something about a rectangle that is different from a triangle. One of the learners said that “the rectangle has a parallelogram and triangle does not have parallelogram” (p. 14). The teacher revoiced the learner’s statement as “this is not a parallelogram” (p. 14) meaning the triangle is not a parallelogram. In her paper, she indicates that revoicing kept the discussion mathematical. Thus, teachers in a multilingual classroom can revoice, interpret and rephrase the learners’ informal mathematical statements to formal mathematical language, thereby enabling learners to move from everyday language to a formal mathematical language in a classroom. The difficulty is, however, that sometimes a teacher and a learner may speak from different points of view (Moschkovich, 1999, p. 15). Furthermore, how do the teachers rephrase or revoice the learners, utterances in order to avoid embarrassing or exposing the learners or changing the meaning of the learners’ response so that the revoicing should not discourage the learners in trying to express themselves? What should the teacher focus on: what the learner says, how it is said, or both?
Another way of helping learners, as explained by Halai (2001), is that teachers may prepare tasks that are set using the language and everyday life experiences of the learners. The assumption is that using the learners’ language and everyday situations may facilitate learning and ease the need for translation. However, the use of everyday language in preparing tasks for the learners may lead to more difficulties and challenges, especially if the teacher ignores some of the unquestioned assumptions. Everyday language varies from learner to learner in a classroom because of reasons such as differences in age, stage of understanding, and exposure or background. Learners might come up with different meanings among themselves and also meanings
different from the teacher’s since familiarity with the LoLT is not the same for all the learners in the class. The challenge for a teacher is how to find “a balanced language and experiences” to fit all age groups with different language backgrounds in a mathematics classroom. All this requires a great deal of the teacher’s own “best” judgements. Another problem might be that everyday language for some learners, whether at school or at home, may not have prepared them for the kind of problems that they meet. For example some learners may want to be told and be directed rather than to do things on their own.
This literature shows that the teacher mediates between the learners and the mathematical language. However, a crucial part in this process of helping learners to move from informal to formal mathematical language is how much attention should be paid to the mathematics register and the LoLT so as to help learners move from the informal way of talking mathematics to a formal way? Khisty (1993) investigated how language was used by teachers to introduce new mathematical concepts to limited English proficient (LEP) and non-English proficient (NEP) learners. In her study, she noted that teachers gave little attention to the mathematics register. Khisty explains that very few mathematical words were actually spoken even though they would open the day's lesson with an obvious naming of the objectives of their lessons such as "adding like fractions" or "adding decimals". Khisty observed that, apart from these initial introductory statements and occasional corrections or affirmations of learner responses to problems, few mathematical words or sentences were said. She further explains that the teacher’s talk would contain few mathematical words or incomplete sentences or ambiguous phrases. She gives an example that the teachers often would read quantities as a series of single digits as in the case of: “Add one, three, seven, and eighty-two” instead of “Add one hundred and thirty-seven and eighty-two” (137 + 82). The question that arises from this insight is: how much attention should a teacher give to mathematical language in a mathematics classroom? Are the teachers themselves well aware of this situation? What about mathematics teacher educators?
This research implies that teaching and learning mathematics is difficult and means a lot of work for teachers to help their learners to gain relevant knowledge of mathematical
language which includes its register, procedures, terms and concepts. Learners also need to be helped to use the language to work effectively together, to share and negotiate meanings in their classrooms. It has been shown that, the issue of mathematical language has different implications in the teaching and learning of mathematics.
This literature provides insights and raises some questions regarding helping the mathematics teachers to see how they can help their learners move from informal mathematical language to formal mathematical language. First, what knowledge and skills do mathematics teachers need in order to be able to mediate between their learner's informal mathematical language and formal mathematical language? In the next section, I present a further discussion of how complex it is to teach mathematics in multilingual classrooms and some of the strategies that teachers use to overcome the challenges that they face in these classrooms.