For many children, mathematics is seen as a ‘foreign language’; the symbols and expressions provide a formidable barrier to understanding of mathematical concepts.
Schleppegrell (2007) conducted a review of research by applied linguists and mathematics educators that highlighted the pedagogical challenges of mathematics. The review notes that since at least the mid-1980s researchers have been pointing to ways that language is implicated in the teaching of mathematics. A key influence has been the discussion by M.A.K. Halliday (1978) of the ‘mathematical register’. Halliday pointed
out that counting, measuring, and other ‘everyday’ ways of doing mathematics draw on ‘everyday’ language, but that the kinds of mathematics that students need to develop through schooling use language in new ways to serve new functions. (in Schleppegrell, 2007.) A summary of key linguistic features of the mathematics register is indicative of the different aspects of language involved.
Features of the classroom mathematics register
Multiple semiotic register
◗◗ mathematics symbolic notation
◗◗ oral language
◗◗ written language
◗◗ graphs and visual displays
Grammatical patterns
◗◗ technical vocabulary
◗◗ dense noun phrases
◗◗ being and having verbs
◗◗ conjunctions with technical meaning
◗◗ implicit logical relationships. (Schleppegrell, 2007)
Schleppegrell’s review identifies research that suggests the important role that teachers play in helping students to use language effectively, and the need for explicit teaching of language in mathematics. For example, O’Halloran (2000) recommends that teachers use oral language to unpack and explain the meanings in mathematics symbolism as a way of using the multi-semiotic nature of mathematics to help students draw on the different meaning making modes for understanding. Explicitly focusing students’ attention on the linguistic features can help students explore and clarify the technical meanings. Other research identified by Schleppegrell suggests that there are various strategies that teachers can use to support
students to move from the everyday language into the mathematics register
… by helping students recognize and use technical language rather than informal language when they are defining and explaining concepts; by working to develop connections between the everyday meanings of words and their mathematical meanings, especially for ambiguous terms, homonyms and similar-sounding words; and by explicitly evaluating students’ ability to use technical language appropriately. One way to evaluate this ability is by having students talk about mathematics as they solve problems, encouraging them to articulate patterns and generalizations. (Adams, 2003).
The ‘mathematical register’ is unique to mathematics, is highly formalized and includes symbols, pictures, words and numbers. (Kotsopoulos, 2007) A study that analysed transcripts from a Year 9 classroom demonstrated how the mathematical register can sound like a foreign language to students. The analysis showed that sixty words identified as belonging to the mathematical register were used more than 1500 times in 300 minutes of classroom transcription. In the following extract from a lesson on the order of operations, words belonging to the mathematical register are :
Teacher: This is our last topic in algebra, and it’s actually not going to be very different from the stuff you’ve
already done. … Do this topic, do the review exercises, and finish the morning with our review. Adding and subtracting polynomials. All right. Again, a lot of time I find people look at a question like that and they go home and say, “Look at all those terms, look at all those positives and negatives, look at all those exponents. I can’t do that.” … But all it would take is for them to take two seconds and look at it and realise, “Wait a minute, what’s the operation that I’m being asked to perform here? What’s the operation I’m being asked to perform? And how can I rely on prior knowledge?” Watch. What’s the operation here? [long pause; the teacher calls on a student whose hand is up]
Evan: Division. Brackets, basically, multiplication.
Teacher: I don’t think so.
Evan: Addition? What was that question again?
Teacher: You’ve got four choices. What is the operation here?
Evan: [shrugs]
Teacher: I don’t think so. To look at it, you’ve got a set of brackets, but the important part is what’s in between them. It’s positive, so you’re being asked to add this polynomial. What kind of polynomial is this? A trinomial … (Kotsopoulos, 2007)
This analysis led to the conclusion that to become proficient in mathematics, students need to participate in mathematical discussions and conversations in classrooms. This participation, in turn, will allow teachers to understand better whether students are making appropriate conceptual connections between words and their mathematical meanings. (Kotsopoulos, 2007). Zevenbergen (2001) discussed the forms of literacy associated with reading and interpreting mathematical texts, and identified several specific literacy demands:
◗◗ words used in a mathematics specific way, including terms such as tessellation, and words that exist in school mathematics and also in the world beyond school
◗◗ spatial terminology (for example, above, horizontal)
◗◗ the concise and precise expression in mathematics that can involve significant lexical density
◗◗ word problems that create complexity through the semantic structuring of the questions rather than the
mathematics.
One aspect of language that can cause confusion is the ambiguity of words that are different in meaning between the context beyond school and the mathematics classroom. Zevenbergen cites this example from a middle years classroom:
T: Can you calculate the volume of this box?
S: um .. [pause] .. no [has a puzzled look]
T: Do you know what volume is?
S: Yes, it is the button on the TV.
Words can have different meanings depending on the context in which they are used, as is evident in the case
above. Zevenbergen provides the following list, indicating the ambiguity in meaning between the context beyond school, and the context of the mathematics classroom:
angle average base below cardinal change common degree difference face figure improper leaves left make mean model natural odd parallel point power product proper rational real record right root sign similar square table times unit volume Adams, Thangata & King (2005) report on research highlighting the complexity of working with words used in mathematics that have multiple meanings. Mathematical language includes many words that sound the same as words with other meanings (or homophones), and many words that have the same spelling as everyday words, but have different meanings as mathematical terms. Table 1 provides examples of some of these words.
Mathematical words and homophonic partner
Mathematical term Homophonic partner
arc ark
chord cord
mode mowed
pi pie
plane plain
serial cereal
sine sign
sum some
These researchers suggest several key considerations in helping students deal with mathematical vocabulary, for example:
◗◗ it is essential that students have the opportunity to see, hear, say and write mathematical vocabulary in context
◗◗ students at all levels need opportunities to define mathematical terms in ways that make sense to them
◗◗ support students’ development of visual skills by encouraging them to use pictures and diagrams to help
understand mathematical language. (Adams, Thanagata & King, 2005.)
A further difficulty is found in the words used for the four main mathematical operations. Tout (1991) notes how analysis of words and phrases used for the operations of addition, subtraction, multiplication and division indicates how complicated it is for adults to solve problems and interpret real life situations. He offers the following examples:
Terms can overlap between different operations. For example, the phrase ‘how many’ is commonly used to
indicate division as in ‘how many fives in 25?’ But what about ‘how many are there between 5 and 25? Or ‘how many are five 25s?’ ‘How many’ can be used for any operation, but many students recognise it as division. Another complication is the multitude of different words used for the one operation. Taking subtraction as an example, the common words used would include: from, minus, take away, and subtract. But what about: difference between, less, reduce, remove, decrease, discount, take off, and various other phrases that call for the use of subtraction? (Tout, 1991) Complexity of working with words used in mathematics that have multiple meanings.
Mathematics is like a language, although technically it is not a natural or informal human language, but a formal, that is, artificially constructed language. Importantly, we use our natural everyday language to teach the formal language of mathematics. Sometimes we encounter problems when the technical words we use, as formal parts of mathematics, conflict with an everyday understanding or use of the same word, or related words. (Gough, 2007)
Mathematics uses many words in the English language that are already familiar to students in their everyday lives. Words such as ‘change’ have a specific mathematical meaning, but as they also have an everyday meaning, they are ambiguous in mathematics classrooms. Some examples are provided below. Students need to be taught new meanings for these already familiar words.
Mathematical words and their everyday usage
Mathematical term Everyday usage
angle point of view
concrete hard substance used in paving
figure shape of an object
odd strange
order place a request
property belonging to someone
rational sane
volume sound level
A research project (Landsell, 1999) tracked the progress of 5-year old children as they acquired new mathematical concepts in the classroom, and investigated their learning of new meanings of familiar words, or new words for the concepts. In the following transcript the teacher is introducing the word ‘change’ with the mathematical meaning of ‘money left over’.
T: You could buy it, couldn’t you? Go on, then, you buy it. That’s it … Right, you’ve taken the penny off, OK? And you’d have one penny left, wouldn’t you? One penny change. So that would be nice. …
R: Mmmmm.
T: What about the duck, that’s 5 p – Could you buy the duck?
R: Yes.
T: Yes, and what would happen if you did that?
R: I would have … I’d have two pennies change. The next day the teacher asked R whether her 10 p could buy an item worth 7 p:
R: Er, I could buy it because I’ve got 10 p change then I would have three pennies.
T: Right, you’ve got 10 p and you have three pennies change, that’s quite right, well done! The student demonstrated confusion with the terminology rather than the mathematical concept, and the teacher corrected her use of language and confirmed her calculation. A week later, R was more confident when talking about change. In this game, the teacher had 10 p to spend on items worth less than 10 p:
R: You could but the 8 p one.
T: Would I have any left?
R: Yes, you would have some change. And you could buy the 7 p one, the 5 p one, and the 1 p one.
The role of language in mathematics learning has been a matter of interest over many years. For example, Pimm (1987) explored some of the language issues that arise in attempting to teach and learn mathematics in a school setting. This wide-ranging exploration covered the implications involved in using the metaphor of mathematics as a language, as well as aspects of classroom communication. It also examined some common spoken interactions in mathematics classrooms. In 2000, in Making Sense of Word Problems, Verschaffel, Greer, and de Corte reported research on students’ unrealistic considerations when solving arithmetical word problems in school mathematics conditions. That the language and literacies of mathematics be explicitly taught by all teachers of mathematics in recognition that language can provide a formidable barrier to both the understanding of mathematics concepts and to providing students access to assessment items aimed at eliciting mathematical understandings.
Thus, given that many mathematical ideas and concepts are abstract or symbolic, children’s literature has a unique advantage in the mathematics classroom because these ideas and concepts can be presented within the context of a story, using pictures, and more informal, familiar language. … By integrating mathematics and literature, students gain experience with solving word problems couched in familiar stories and thus avoid struggling with unfamiliar vocabulary.
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