For many years only reading and writing of mathematical symbols were used during mathematics lessons, without any language skills required. This has changed in the last 20 years. The idea of communication in mathematics is expressed clearly in Principles and Standards for School Mathematics (National Council of Teachers of Mathematics (NCTM), 2000, p. 60): ‘They [learners] communicate to learn mathematics and they learn to communicate mathematically.’ If we re-examine the definition of language (Webster’s Universal College dictionary, 1997), it seems that mathematics language is a formalised set of conventions that are used for the specific purpose of problem-solving.

This study attempted to examine the interplay between mathematics language and natural language (in this study Hebrew, the formal language of studies at the college of education concerned). The research involved a group of pre-service mathematics teachers who were exposed to the new attitude in mathematical pedagogy, emphasising mathematical discourse and communication as well as views of mathematics as a language.

The pre-service teachers were asked to develop additional materials to help them whilst teaching mathematics in their classes.

The poem Look, we are two numbers (free translation and emphasis by D.P.) by Israeli poet Yehuda Amichai is an example of a literary composition which includes mathematical terms:

Look, we are two numbers,
Standing together and being added
or subtracted, because finally the sign
changes from time to time.
It was so difficult until we succeeded
to stand together, and we also knew
multiplications of happiness, fractions too,
as usually happens to numbers.
Even now, under us, the world is a fraction line –
don’t be alarmed, look how beyond
that line has now bloomed for us –
the common denominator. (Amichai, 1975, p. 36)

This is a love poem describing being a couple. Amichai made surprising use of the mathematics language and created original images for defining the relationship of a couple with its fluctuations ‘from time to time’. In describing the relationship he included metaphors, all taken from the world of the basic four operations of mathematics. The mathematical expressions mentioned in the poem embody opposites, reflecting contrasts in human and emotional relationships. He employed mathematical terms as ‘scaffolds’ for the everyday language. According to him, these terms are not merely mathematical, they also have a meaning in daily life. Applying words by moving and passing back and forth from the everyday to the mathematics language might enhance mathematics language and consequently students’ achievements in this field. Mathematics language should be fostered in various ways for that purpose.

Webster’s Universal College dictionary (1997) provides a number of definitions of the term ‘language’, two of which can be related to the definition of mathematics as a language. The first deals with communication using a system of arbitrary vocal sounds, written symbols, signs or gestures in conventional ways with conventional meanings. The second is any set or system of formalised symbols, signs, sounds or gestures used or conceived as a means of communicating. These definitions address the concept of language as a formalised or arbitrary means of communication comprising specific terminology such as signs, graphic representations, numbers and icons.

One should bear in mind that every language is based on unique symbols and syntax rules (Webster’s Universal College dictionary, 1997) which define the use of its symbols, which can serve to represent meanings in a specific area. If we relate to mathematics as a language built of symbols and syntax rules used for describing objects, actions and relations between sizes, it can be utilised both in writing and orally. This is done both by numerical representations and graphic or iconic representations for abstraction of reality, for use in axioms and in a linguistic-verbal representation. Hence, it becomes necessary to understand the mathematical word lexicon and terms in the teaching and learning process.

Mathematics encompasses unique linguistic forms. There is extensive use of key words which imply the four operations of mathematics (addition, subtraction, multiplication and division), and since the language of mathematics comprises symbols and signs, failing to distinguish between them on visual grounds may impede learning. For example, there can be confusion between 2 and 5, 6 and 9, the signs > and <, or between + and -. Moreover, there is a mix of various meanings of words used in everyday life on the one hand, and in mathematics lessons on the other (as in the poem of Amichai). All of this is difficult and confusing for learners (Cobb, Yackel & McClain, 2000).

Current literature supports the existence of four language components in mathematics learning: reading, writing, speaking and listening (Kazemi, 1998; NCTM, 2000; Siegel, Borasi, Sanridge & Smith, 1996; Usiskin, 1996). Despite the fact that mathematics is not perceived as a spoken language, the new standards for teaching mathematics indicate the need for verbalisation (‘mathematical discourse’) (Huinker & Laughlin, 1996).

Skemp (1972) argues that many difficulties and problems in mathematics education are due to the fact that many words have different meanings in different languages, and some words have different connotations in different countries. There are also some words which have double or even triple meaning in the same language. Thus, confusion and obstacles are to be expected. Skemp employed the French term faux amis to describe this, which means ‘false friends’. He gives the example of the word ‘biscuit’, which in the United States of America refers to a different item of pastry than in Britain. In the USA one should say ‘cookie’ in order to obtain the same item, ‘biscuit’, as in Britain.

Hence, during their lives learners acquire a variety of words, which they apply in order to express thoughts and ideas. Sometimes they use words which represent mathematical terms, whose ‘daily’ meaning is not in line with their mathematical meaning. All over the world, everyday words in English (such as field, group or set) are included in mathematics language − and this turns them into ‘false friends’. In Hebrew, for example, the word ‘cube’ has a different meaning in mathematics language and in daily life. When you tell a child in kindergarten to bring a cube from the cubes corner, this refers to one of the differently shaped game blocks with which they can build towers, bridges, et cetera. In mathematics language, however, this word refers unequivocally to a cube: a solid figure with six identical square faces, all right-angled.

According to Usiskin (1996) the reasons for experiencing problems in mathematics learning are related to aspects of its language. The fact that students are not exposed to the ‘language of mathematics’ at home or in their close environment makes teaching harder. It is important to start teaching mathematics at an early stage, so that it can become a mother tongue or second language for the learners. Moreover, mathematics is often taught out of context. This makes learning meaningless and inapplicable to the learners’ lives. As a result, the learners study a ‘dead’, useless language. In addition, teaching mathematics comprises the use of abstract concepts which are not always clear and meaningful to the learner. Usiskin argues that mathematics has much in common with other languages because of the following:

1. Mathematics does not only describe ideas, but also fosters the organisation of these ideas within the learner.
2. The number of symbols and signs in mathematics (e.g. ⊥, = , ≅) is similar to the number of letters in other languages
3. Mathematics has its own syntactic rules, with expressions such as ‘3+4’, and verbs (e.g. subtract).
4. Mathematics has a ‘private property’ of vocabulary like any other language, as well as its own unique features.
5. Mathematics lends and borrows words, like any other language. It makes use of the Latin alphabet in algebra, the Greek alphabet in geometry (e.g. ellipse, parabola), the word ‘radius’ from Latin, et cetera. It also lends words; for example, the word ‘triangle’ in mathematics refers to a two-dimensional shape whereas in everyday English it is also used to describe a romantic loop with three people involved.

The linguistic approach, which illustrates the difficulties in learning mathematics and the perception of mathematics as a language with all its aspects, has a strong relation to literacy activities (Patkin, Millet & Ezer, 2001).

The professional literature underscores that literacy activities can also be integrated into disciplines not related to the field of humanities. At the beginning of the 1980s Evans (1984) discussed the integration of writing activities in primary school mathematics lessons. She demonstrated that the attainments of students who participated in such a project were higher than those who did not. Moreover, Rose (1989) described a design to integrate literacy in mathematics at high schools in California, illustrating that writing in mathematics lessons allowed students to advance at their own pace whilst using their everyday language and personal experiences.

In recent years mathematics educators have encouraged practicing teachers and their students to develop, in addition to writing, a ‘mathematical discourse’ during mathematics lessons (Kazemi, 1998). They also advocate use of a reflective written record in order to promote mathematical thinking and learners’ ways of coping with mathematical material (Hart, Schultz, Najee-ullah & Nash, 1992; Norwood & Carter, 1992, 1994; Schiebelhut, 1994).

A study amongst different populations of mathematics teachers illustrates that integration of literacy in teaching may contribute to understanding of this subject and reduce fear of it (Ezer, Patkin & Millet, 1999). The question in a study which focused on integration of the fable The Lion and the Hunter by Lafontaine was: ‘What is the relation between Lafontaine’s quarter and the rational number quarter? Or what is the difference between mathematical justice and the jungle justice?’ In Lafontaine’s fable, four animals went hunting and agreed to divide the prey into four equal parts. According to mathematical justice, each animal was supposed to receive 1/4 of the prey. The lion, however, took everything because he made a ‘just’ division, according to the jungle laws. Thus, although the animals agreed about equal division, all of the parts went to the lion.

Findings indicated that the use of an intriguing fable that stimulates mathematical thinking led to an educational lesson which promoted a positive approach to mathematics teaching and use of mathematics language (Millet, Patkin & Ezer, 2002).

The present study explored a bank of words with multiple meanings and literacy activities by means of an intervention unit. The research questions addressed in this article are:

1. Which literacy activities can pre-service teachers create or use in order to be more aware of the problems with language and mathematics?
2. How do the reflections of the pre-service teachers provide insights into the importance of integrating literacy activities in the teaching of mathematics?

The first semester: The first session began with the poem of Amichai. The students were divided into small working groups of three or four people. In every subgroup one student had to read the poem aloud (in Hebrew). The second step in this working group was to write the poem in their own words, and then to mark all words in the poem which represented mathematics concepts and to explain the meaning of these words in the poem.

In the second session the students had to check the meaning of these words in a regular dictionary and in the dictionary of mathematics, and compare them to their explanations and definitions. In the following sessions each pre-service teacher had to choose a mathematical topic from the curriculum, and they were asked to read papers discussing the uses of everyday language in general and in mathematics in particular, for example communications in the language of mathematics (Kazemi, 1998; Rose, 1989; Schiebelhut, 1994; Usiskin, 1996). They were also requested to read studies dealing with the teaching of the chosen topic, its historical background, and related misconceptions and common mistakes.

The activity during the first semester consisted of consolidating the pre-service teachers in the two topics and exposing them to use and integration of literacy activities in teaching. They were taught to operate a programme which introduced literacy activities into mathematics lessons, namely leading mathematical conversations (verbalisation of mathematical relations), verbalisation of thinking processes in order to identify misconceptions and barriers, writing and reading students’ reflective written records, writing procedure assessment, performing inquiry assignments, developing strategies for understanding mathematical texts and interviewing students. At the end of each session all the pre-service teachers had to indicate all words with a different or double meaning in a list.

Semester break: The pre-service teachers were required to look for a short story, poem, folk tale or fable which included words in Hebrew with a double or triple meaning or one meaning in everyday life and another as a mathematics concept, and to write down the mathematics concepts in the chosen text. Then they had to choose two concepts and look for the theoretical background, including the component of the history of these concepts, the place of these concepts in the curriculum, and student mistakes and misconceptions relating to these concepts. They also had to design an intervention which comprised four to five lessons on the chosen topic using and integrating literacy activities.

Second semester: The pre-service teachers had to practice implementation of the programme in their own classes with children. The weekly sessions consisted of discussions, reflection and analysis of findings, that is executing the programme from theory to practice.

Throughout the course the pre-service teachers gathered an extensive collection of words which have one meaning in daily life and another in mathematics language. Table 1 shows examples from this ‘double-meaning words bank’ of the pre-service teachers, representing words with double meaning in each branch of mathematics: arithmetic, algebra, geometry, solid geometry, statistics, differential and integral calculus, et cetera. Their exact definitions were taken from Chambers dictionary (Magnusson, 1993) for everyday usage and the Penguin dictionary of mathematics (Nelson, 1998) for the mathematical definitions. It is important to mention that amongst these words are some which have a double meaning in mathematics and a double or triple meaning in ordinary language (e.g. ‘base’).

Another example which led to a very interesting activity was the word ‘division’ (e.g. when eight candy bars have to be divided between four children). In daily life the division must not necessarily be into four equal parts; there might even be an extreme situation where one child does not get any candy whereas another receives all eight bars. Conversely, in mathematics, if we divide eight candy bars between four children the tacit assumption is that the division is into four equal parts. Activity of this type is similar to that presented in the study of Millet, Patkin and Ezer (2002), which focused on a literacy activity in mathematics on the basis of Lafontaine’s fable, described earlier.

Another outcome was a bank of fables, poems and other types of literature which can be used in intervention units or sessions for the interplay of everyday language and mathematics language, such as fables by Aesop and Lafontaine, stories from the Bible, et cetera.

Analysis of the diaries revealed four key indices, namely pleasure, comprehension of the importance of knowledge about double meanings, a positive attitude towards the continuation of teaching in such a way, and a sense of self-confidence. The following excerpts from the diaries provide examples of each index:

I have undergone a change and I think that I am beginning to understand what is going on. The discussions and talks conducted during the entire academic year constituted part of my professionalisation process ... Until several days ago it was difficult for me to internalise and understand what was transpiring in my class. However, now when I am writing I actually think that one of the problems in my class stemmed from the fact that the students were not exposed to words with different meaning in different languages like the term ‘to reduce’. In Hebrew as in English, to reduce means making it smaller, to reduce weight etc. … in mathematics, to reduce means to write equal fractions in other way. [Pre-service teacher M]

I indicated that all the class students as well myself sensed a change following the talks about words with double meaning and the activities. [Pre-service teacher E]

… asking the students to bring examples of words with different meanings and explain them as part of the integration of literacy activities in lessons throughout the course, made me sense that the concept which was taught became clear and comprehensible. [Pre-service teacher Y]

To summarise, the findings indicated progress in terms of comprehension and ability to define words and terms correctly. Moreover, the pre-service teachers indicated a sense of pleasure resulting from progress achieved in the course as well as enhanced motivation to teach in this way.

An examination of the literature supports the existence of four language components in mathematics learning (Usiskin, 1996; Uso-Juan & Martinez-Flor, 2006). This study involved a group of pre-service mathematics teachers who were exposed to the new attitude in mathematical pedagogy, which emphasises mathematical discourse and communication and views mathematics as a language requiring use of oral and written components (Timor & Patkin, 2010).

The intervention unit exposed the pre-service teachers to numerous words with double or triple meanings which were ‘false friends’. When teaching a mathematical subject comprising words which have different meanings in everyday use and in mathematics, they discussed these words with their students. This process made them more attentive and sensitive to the possibility that double or triple meanings caused problems in the comprehension of their students.

Throughout the entire year whilst teaching in their classes they searched for words with different or confusing meanings, adding words from their own personal knowledge. The search for this word lexicon was performed by means of all the literacy activities described above. The students also directly addressed the students, asking them to search by themselves for short stories, poems, et cetera, to present them orally or in writing and to emphasise words with one or more meaning in everyday life and another in mathematics language. Integrating literacy activities into mathematics by raising awareness of the various meanings that words can have can, as reported in the diaries, eliminate mistakes (NCTM, 2000; Patkin & Gazit, 2011).

Speaking plays an important part in learning. It requires students to organise their thoughts and to focus them. Mathematical speech, which is a combination of everyday language and mathematics language, is manifested by reflective mathematical talks (Usiskin, 1996). As one of the pre-service teachers reported in his diary:

The talks about the different meanings helped me to be more accurate when making definitions and it influenced my way of teaching. It helped my students construct their mathematical comprehension and apply in practice their theoretical studies … [Pre-service teacher L]

Discussing questions like ‘Is it correct to refer to the wooden blocks from which kindergarten children build towers as “cubes”? ’ can demonstrate to the learners the risks embodied in failing to be meticulous in the application of words. There is no problem calling this section of the kindergarten the ‘cubes corner’ or the ‘solid figures corner’. One might say that we should not teach young children (up to age three years) the mathematical vocabulary because it is too difficult for them. However, just as young children learn to differentiate between different colours and the names of many animals, it is important to emphasise accuracy in mathematics terminology, otherwise they develop misconceptions and mistakes with these concepts.

We should take advantage of the opportunity to actively construct mathematical knowledge through mathematical talk, by means of which we identify difficulties, compare solutions, ask leading questions and present assumptions for discussion and thought (Siegler, 2006). Encouraging students to make use of verbalisation, writing reflective records, writing procedures and building stories which include mathematical problems creates conditions for the learning of mathematics whilst participating in social class processes.

During the course the pre-service teachers attested that the literacy activities in mathematics lessons not only enriched their personal knowledge but also extended their mathematical thinking. Using everyday language as scaffolds assisted them to teach the ‘second’ language – mathematics language. The course outcomes reinforced and enhanced the need for teaching of this kind. For example, the poem of Amichai (1975) ‘crossed’ the border and made use of mathematical terms, assuming that readers are familiar with these. The poem describes addition, subtraction and multiplication; it discusses fractions, two numbers, fraction line and common denominator. In the poet’s mind these terms are not merely mathematical terms but rather words with a double meaning.

To summarise, it is possible and recommended that learners be encouraged to move back and forth from one language − natural, everyday language − to a second language, mathematics language. Thus we can promote mathematical thinking and comprehension and, as a result, improve students’ attainments in this discipline. It is recommended that research of wider scope (including quantitative data) be conducted to validate the findings of this study.

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